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PHASE BEHAVIOR CURTIS H. WHITSON AND MICHAEL R. BRULÉ
MONOGRAPH VOLUME 20
SPE
HENRY L. DOHERTY SERIES
PHASE BEHAVIOR Curtis H. Whitson Professor of Petroleum Engineering U. Trondheim, NTH and Founder PERA a/s and
Michael R. Brulé President and Chief Executive Officer Technomation Systems Inc.
First Printing Henry L. Doherty Memorial Fund of AIME Society of Petroleum Engineers Inc. Richardson, Texas 2000 i
SPE Monograph Series The Monograph Series of the Society of Petroleum Engineers was established in 1965 by action of the SPE Board of Directors. The Series is intended to provide authoritative, uptodate treatment of the fundamental principles and state of the art in selected fields of technology. The Series is directed by the Society’s Monograph Committee. A committee member designated as Monograph Editor provides technical evaluation with the aid of the Review Committee. Below is a listing of those who have been most closely involved with the preparation of this monograph.
Monograph Review Committee Peter G. Christman, Shell Intl. E&P B.V., Monograph Editor David F. Bergman, Amoco Production Co. W. David Constant, Louisiana State U. A.S. Cullick, Landmark Graphics Corp. Gustave A. Mistrot III, Mistrot & Assocs. Teresa G. MongerMcClure, Marathon Oil Co. Franklin M. Orr Jr., Stanford U. Robert R. Wood, Shell Intl. E&P B.V. Aaron A. Zick, Zick Technologies
Monograph Committee (2000) Mary Jane Wilson, WZI, Chairperson Jesse Frederick, WZI Russell T. Johns, U. of Texas, Austin Medhat Kamal, Arco E&P Technology Mark Miller, U. of Texas, Austin Ken Newman, CTES L.S. Dan O’Meara Jr., U. of Oklahoma David Underdown, Chevron Production Technology Co.
Acknowledgments Many people contributed to the production of this monograph. It is first and foremost the product of the authors. I am sure that the effort was more significant than either author had anticipated, but they persevered and should be proud of the book they wrote. I want to thank R.R. Wood, who initiated the project, chose the authors, and formed a distinguished review committee. I succeeded Rob in 1990 and coordinated the efforts of A.A. Zick, G.A. Mistrot, T.G. MongerMcClure, D.F. Bergman, A.S. Cullick, and W.D. Constant, who reviewed every chapter from their own unique perspectives. F.M. Orr contributed significant reviews on selected chapters. It was a pleasure to work with such a talented group of engineers. I am confident that we kept the focus of the monograph on use by the working engineer. The book is meant to serve as a reference. As such, I hope it will be a valuable addition to the library of every petroleum engineer working in phase behavior. Peter G. Christman
Copyright 2000 by the Society of Petroleum Engineers Inc. Printed in the United States of America. All rights reserved. This book, or any part thereof, cannot be reproduced in any form without written consent of the publisher. ISBN 1555630871 ii
Dedication To Morris Muskat, a pioneer in the field of reservoir engineering, who made important contributions in the area of phase behavior.
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Acknowledgments We thank the SPE editorial staff, the Monograph Review Committee members, our professional colleagues, our students, and the petroleum industry at large for valuable assistance and input toward the completion of this monograph. In particular, we thank the two technical editors, Rob R. Wood and Peter G. Christman, and our staff editor, Flora Cohen. We have been strongly influenced by the pioneering phasebehavior research of Donald Katz, Muz Standing, and Ken Starling and the many others who have made invaluable contributions to the field. The scientific contributions of these engineers and their coworkers, together with contributions from the community of petroleum and chemical engineers, have laid the foundation for the material selected, synthesized, and presented in this monograph. We hope that all contributors have been correctly cited and given due credit for their contributions. We are confident that the material contained herein is valuable for dealing with engineering problems affected by phase behavior, both today and in the future. We use the technology presented in this monograph daily to solve problems for the industry and as the basis of our longterm research. Curtis H. Whitson Michael R. Brulé
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Table of Contents Chapter 1—Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 1.2 1.3 1.4 1.5
Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Historical Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scope and Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nomenclature and Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 1 2 2 2
Chapter 2—Volumetric and Phase Behavior of Oil and Gas Systems . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1 2.2 2.3 2.4 2.5
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 ReservoirFluid Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Phase Diagrams for Simple Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Retrograde Condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Classification of Oilfield Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Chapter 3—Gas and Oil Properties and Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.1 3.2 3.3 3.4 3.5 3.6
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Review of Properties, Nomenclature, and Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gas Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Oil Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IFT and Diffusion Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . KValue Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18 18 22 29 38 40
Chapter 4—EquationofState Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.1 4.2 4.3 4.4 4.5 4.6 4.7
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cubic EOS’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TwoPhase Flash Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phase Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SaturationPressure Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equilibrium in a Gravity Field: Compositional Gradients . . . . . . . . . . . . . . . . . . . . Matching an EOS to Measured Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47 47 52 55 62 63 65
Chapter 5—HeptanesPlus Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.1 5.2 5.3 5.4 5.5 5.6 5.7
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Molar Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . InspectionProperties Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CriticalProperties Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recommended C7) Characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Grouping and Averaging Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68 68 70 77 80 83 83
Chapter 6—Conventional PVT Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.1 6.2 6.3 6.4 6.5 6.6
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wellstream Compositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MultistageSeparator Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constant Composition Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Differential Liberation Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constant Volume Depletion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
88 88 91 93 95 97
Chapter 7—BlackOil PVT Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 7.1 7.2 7.3 7.4 7.5 7.6
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Traditional BlackOil Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modified BlackOil (MBO) Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications of MBO Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PartialDensity Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modifications for Gas Injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
109 109 110 116 118 119
Chapter 8—GasInjection Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 8.1 8.2 8.3 8.4 8.5
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miscibility and Related Phase Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LeanGas Injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EnrichedGas Miscible Drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CO2 Injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
121 122 128 131 135
Chapter 9—Water/Hydrocarbon Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 9.1 9.2 9.3 9.4
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Properties and Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EOS Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hydrates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
142 142 150 151
Appendix A—Property Tables and Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 Appendix B—Example Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 Appendix C—EquationofState Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Appendix D—Understanding Laboratory Oil PVT Reports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
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Chapter 1
Introduction 1.1 Purpose This monograph covers a wide range of topics related to phase behavior. Phase behavior is the behavior of vapor, liquid, and solids as a function of pressure, temperature, and composition. In this monograph, “vapor” is used interchangeably with “gas,” “liquid” refers to oil and water, and “solids” include hydrates, asphaltenes, and wax. We are concerned primarily with the volumetric behavior and composition of phases, including density and isothermal compressibility, and component distribution in each phase. For a mixture with a known composition, we need to determine the vapor/liquid equilibrium (VLE), including saturation conditions over a wide range of temperatures and pressures. Transport properties are also needed for flow calculations (e.g., viscosity in Darcy’s law and molecular diffusion coefficients in Fick’s law). Phase behavior has many applications in petroleum engineering. The reservoir engineer relies on pressure/volume/temperature (PVT) relations to calculate oil and gas reserves, production forecasts, and the efficiency of enhanced oil recovery (EOR) methods. Most reservoir calculations require PVT properties at reservoir temperature. Production engineers use phase behavior data for surface separator design and to calculate flow in pipe, where such calculations are made over a range of temperatures from surface to reservoir conditions. Petroleum engineering calculations generally are made at temperatures from 60 to 350°F and at pressures from about 15 to 15,000 psia. 1.2 Historical Review Gibbs1,2 and van der Waals3 stated the basic theory of phase behavior in the the late 1800’s and early 1900’s. They formulated the concepts and mathematical relations necessary to describe phase behavior. Katz and Rzasa4 published a comprehensive review of phase behavior literature from before 1860 to 1945. Muckleroy5 published a bibliography covering 1946 to 1960, and other bibliographies exist for work in phase behavior over the past 30 years.* Experimental data on reservoir fluids were scarce before the late 1930’s, when Katz et al.4,639 at the U. of Michigan, Sage and colleagues4073 at the California Inst. of Technology, and Eilerts et al.7478 at the U.S. Bureau of Mines (USBM) began significant research programs. For 10 years, during the 1950’s, a large amount of highquality experimental data was compiled on reservoir fluids. During the past 40 years, most phase behavior data have been measured by commercial service laboratories and major oil companies. *SPE Reprint Series No. 15 Phase Behavior gives a recent update of earlier bibliographies.
INTRODUCTION
These data have been used for engineering studies of primary depletion, waterflood evaluation, and gasinjection studies. Correlation of phase behavior data began in the 1940’s, with notable work by Standing and Katz,17,18 Bicher and Katz,25 Standing,79,80 Eilerts,78 Kennedy and colleagues,8185 and others. Although equations of state (EOS’s) had been available for more than 50 years (since van der Waals3 published the first cubic EOS in 1873) it was necessary to rely mostly on tables, figures, and chart correlations, such as nomograms. These correlations provided reliable property estimates for engineering calculations through the 1970’s. Subsequently, empirical equations representing these graphical correlations were developed and programmed for calculators and computer applications. With the introduction of electronic computers in the late 1940’s, application of complicated thermodynamic models became possible. In 1949, Muskat and McDowell86 published one of the earliest papers in the SPE/AIME Transactions on applications of this new generation of computers. These authors solved the twophase flash calculation with fixed K values for multistage separator design. Not until Redlich and Kwong87 introduced their classic cubic EOS in 1949 was it generally accepted that volumetric properties could be accurately predicted by use of theoretical models. Considerable advances were made in the 1950’s toward correlating volumetric properties of pure components with multiconstant EOS’s.88 By the early 1960’s, there was considerable activity in the application of sophisticated thermodynamic models to multicomponent VLE calculations, although most of this activity was in process engineering. In the 1960’s and 1970’s, Starling,89 Soave,90 and Peng and Robinson91 proposed several important modifications of existing EOS’s. Petroleum engineering EOS applications started seriously in the late 1970’s and early 1980’s, when EOSbased compositional reservoir simulators were introduced.92,93 At the same time, several methods were proposed for EOS fluid characterization of reservoir fluids, in particular for heptanes and heavier components.9496 Finally, in the 1980’s, supercomputers appeared and special solution techniques were developed for compositional simulators,93 thereby making possible fullfield, EOS compositional simulation. Today’s standard treatment of phase behavior in reservoir simulation is still based on formation volume factors (FVF’s) and surface gas/oil ratios (GOR’s). This will probably remain true for many years, in part because many problems can be solved adequately with a simple PVT formulation and in part because many petroleum engineers are not familiar with more complicated EOS models. This monograph treats both simple and complicated methods for estimating phase behavior. We suspect that the more complicated PVT 1
models will gradually become the standard, eventually replacing many of the simpler correlations.
with Chap. 6, Conventional PVT Experiments, and are included as a supplement to the discussion in that chapter.
1.3 Objectives This monograph provides the petroleum engineer with a tool to solve problems that require a description of phase behavior and specific PVT properties. These problems include calculating the FVF to determine original oil and gas in place and GOR’s, design of “optimal” surface separator conditions, and description of nearcritical phase behavior resulting from the injection of a gas that develops miscibility with a reservoir oil. Because of the dramatic evolution in computer technology, petroleum engineers can now study such phenomena as developed miscibility,97 compositional gradients,98 and nearcritical phase behavior99 with more sophisticated models. The quality of these models is sensitive to the EOS fluid characterizations. This monograph presents phase behavior concepts used in petroleum engineering and stateoftheart technology for more complex phase behavior models, such as cubic EOS’s. We hope the monograph will serve its purpose for many years to come.
1.5 Nomenclature and Units SPEapproved symbols are used throughout the monograph. Some of these symbols will be unfamiliar even to the seasoned SPE reader (as they are confusing even to the authors!). One of the most significant changes in nomenclature that we have introduced is the use of different subscripts for surface and reservoir phases. Traditionally, o, g, and w are used for oil, gas, and water at reservoir and at surface conditions, a practice that was difficult to follow in Chaps. 6 and 7. We have therefore introduced the subscripts o, g, and w for surface phases, retaining o, g, and w for reservoir phases. A better solution to this problem was not apparent, particularly because some quantities required subscripts for both reservoir and surface phases—e.g., the gravity of surface gas produced from reservoir oil (written g go in this monograph). To avoid confusion in the property correlations in Chap. 3, gas and oil specific gravities are still written g g and g o (instead of g g and g o) because specific gravity is always reported at standard conditions. We use customary oilfield units (psi, ft3 and bbl, °F and °R, and lbm). The oilfield unit for mass is pound, written “lbm” to avoid confusion with pounds force, written “lbf.” Pounds force is never used explicitly in this monograph. Conversion factors to SI units are included at the end of each chapter, and Appendix A provides a comprehensive discussion of units and unit conversion tables. Standard conditions are defined in this monograph as 60°F and 14.7 psia. We recognize that standard pressure varies geographically and the calculation of surface gas volumes in some areas must use the locally defined value for standard pressure. To accomplish this, some constants given in the monograph must be recalculated.
1.4 Scope and Organization The scope of this monograph is limited mostly to twophase, gas/oil phase behavior. Multiphase and vapor/solid phase behavior are discussed only briefly. Phase behavior related to chemical (surfactant and polymer) flooding is not covered because a detailed description would necessarily reduce coverage of problems more commonly encountered in petroleum engineering. We also think that this subject should be covered in a separate publication specifically within the context of chemical flooding technology. Chaps. 2 and 3 review the “nuts and bolts” of phase behavior principles, relevant PVT properties, and methods to solve most petroleum engineering problems. Useful correlations are presented for the most common PVT properties. Chap. 4 discusses cubic EOS’s, including the twophase flash, saturationpressure, and phasestability calculations and numerical methods used to solve these VLE calculations. The problem of “tuning” an EOS to match measured PVT data is also addressed. Chap. 5 describes the characterization of heavy components (“heptanes plus”) in reservoir fluids for EOS applications. Experimental and mathematical methods describing the heptanesplus material are presented, including splitting C7+ into petroleum fractions, estimating critical properties, and grouping an extended fluid characterization into a reduced number of pseudocomponents. Chap. 6 covers laboratory measurements of PVT properties and their application in engineering calculations. The standard PVT studies include constant composition (mass) expansion, differential liberation, constantvolume depletion, and the multistage separator test. Separator and bottomhole sampling methods for establishing wellstream composition are also discussed. Chap. 7 describes the blackoil PVT formulation and its extension to gas condensates, volatile oils, and gasinjection processes. The blackoil PVT formulation uses FVF’s and solution gas/oil ratios to relate phase and volumetric properties at reservoir conditions to surface volumes. Chap. 8 reviews the importance of phase behavior to gasinjection EOR processes. These processes include vaporizing, condensing, and the combined condensing/vaporizing miscibledrive mechanisms. CO2 immiscible and miscible drives and nitrogen injection are also reviewed. Chap. 9 covers the behavior of water/hydrocarbon phase and volumetric behavior, including mutual solubilities, water FVF and compressibility, and the treatment of hydrates. Appendix A gives tables of component properties, various other useful tables, and unit conversion factors. Appendix B includes more than 20 worked examples that range from simple calculations of ideal gas properties to detailed stepbystep EOS calculations for a ternary system. Appendix C gives two detailed EOS fluid characterizations, one for a gas condensate and another for a slightly volatile oil. Appendix D is a set of notes by M.B. Standing on understanding laboratoryoil PVT reports. These notes clearly belong 2
References 1. Gibbs, J.W.: The Collected Works of J. Willard Gibbs, Yale U. Press, New Haven, Connecticut (1948) 1. 2. Gibbs, J.W.: On the Equilibrium of Heterogeneous Substances, C. Works (ed.), Yale U. Press, New Haven, Connecticut (1928) Chap. 1. 3. van der Waals, J.D.: Continuity of the Gaseous and Liquid State of Matter (1873). 4. Katz, D.L. and Rzasa, M.J.: Biblography of Hydrocarbons Under Pressure 1860–1946, University Microfilms Inc. (1946). 5. Muckleroy, J.A.: Biblography on Hydrocarbons, 1946–1960, Gas Processors Assn. (1962). 6. Katz, D.L. and Hachmuth, K.K.: “Vaporization Equilibrium Constants in a CrudeOil Natural Gas System,” Ind. & Eng. Chem. (1937) 29, 1072. 7. Katz, D.L.: “Application of Vaporization Equilibrium Constants to Production Engineering Problems,” Trans., AIME (1938) 127, 159. 8. Katz, D.L., Vink, D.J., and David, R.A.: “Phase Diagram of a Mixture of Natural Gas and Natural Gasoline Near the Critical Conditions,” Trans., AIME (1939) 136, 106. 9. Katz, D.L. and Singleterry, C.C.: “Significance of the Critical Phenomena in Oil and Gas Production,” Trans., AIME (1939) 132, 103. 10. Katz, D.L. and Saltman, W.: “Surface Tension of Hydrocarbons,” Ind. & Eng. Chem. (January 1939) 31, 91. 11. Katz, D.L. and Kurata, F.: “Retrograde Condensation,” Ind. & Eng. Chem. (June 1940) 32, No. 6, 817. 12. Wilcox, W.I., Carson, D.B., and Katz, D.L.: “Natural Gas Hydrates,” Ind. & Eng. Chem. (1941) 33, No. 5, 662. 13. Katz, D.L.: “High Pressure Gas Measurement,” Refiner and Natural Gasoline Manufacturer (June 1942). 14. Carson, D.B. and Katz, D.L.: “Natural Gas Hydrates,” Trans., AIME (1942) 146, 150. 15. Kurata, F. and Katz, D.L.: “Critical Properties of Volatile Hydrocarbon Mixtures,” Trans., AIChE (1942) 38, 995. 16. Katz, D.L.: “Possibilities of Secondary Recovery for the Oklahoma City Wilcox Sand,” Trans., AIME (1942) 146, 28. 17. Standing, M.B. and Katz, D.L.: “Density of Natural Gases,” Trans., AIME (1942) 146, 140. 18. Standing, M.B. and Katz, D.L.: “Density of Crude Oils Saturated with Natural Gas,” Trans., AIME (1942) 146, 159. 19. Katz, D.L.: “Prediction of the Shrinkage of Crude Oils,” Drill. & Prod. Prac. (1942) 137. 20. Matthews, T.A., Roland, C.H., and Katz, D.L.: “High Pressure Gas Measurement,” Proc., Natural Gas Assn. of America (NGAA) (1942) 41. PHASE BEHAVIOR MONOGRAPH
21. Weinaug, C.F. and Katz, D.L.: “Surface Tension of MethanePropane Mixtures,” Ind. & Eng. Chem. (1943) 35, No. 2, 239. 22. Bicher, L.B. Jr. and Katz, D.L.: “Viscosities of the MethanePropane System,” Ind. & Eng. Chem. (1943) 35, 754. 23. Katz, D.L., Monroe, R.R., and Trainer, R.P.: “Surface Tension of Crude Oils Containing Dissolved Gases,” Trans., AIME (1943) 155, 624. 24. Standing, M.B. and Katz, D.L.: “Vapor/Liquid Equilibria of Natural Gas/Crude Oil Systems,” Trans., AIME (1944) 155, 232. 25. Bicher, L.B. Jr. and Katz, D.L.: “Viscosity of Natural Gases,” Trans., AIME (1944) 155, 246. 26. Katz, D.L., Brown, G.G., and Parks, A.S.: “NGAA Report on Sampling TwoPhase Gas Streams from High Pressure Condensate Wells,” Proc., NGAA (September 1945). 27. Katz, D.L. and Beu, K.L.: “Nature of Asphaltic Substances,” Ind. & Eng. Chem. (February 1945) 37, 195. 28. Katz, D.L.: “Prediction of Conditions for Hydrate Formation in Natural Gases,” Trans., AIME (1945) 160, 140. 29. Poettman, F.H. and Katz, D.L.: “CO2 in a Natural Gas Condensate System,” Ind. & Eng. Chem. (1946) 38, 530. 30. Brown, G.G. et al..: Natural Gasoline and the Volatile Hydrocarbons, NGAA, Tulsa, Oklahoma (1948) 24–32. 31. Kobayashi, R. and Katz, D.L.: “MethanenButaneWater System in Twoand ThreePhase Regions,” Ind. & Eng. Chem. (1948) 40, No. 5, 853. 32. Unruh, C.H. and Katz, D.L.: “Gas Hydrates of Carbon Dioxide/Methane Mixtures,” Trans., AIME (1949)186, 83. 33. Rzasa, M.J. and Katz, D.L.: “The Coexistence of Liquid and Vapor Phases at Pressures Above 10,000 psi,” Trans., AIME (1950) 189, 119. 34. Kobayashi, R. et al.: “Gas Hydrates Formation with Brine and Ethanol Solutions,” Proc., 30th Annual Convention of NGAA (1951). 35. Katz, D.L. and Williams, B.: “Reservoir Fluids and Their Behavior,” Amer. Soc. Petr. Geology Bulletin (February 1952) 36, No. 2, 342. 36. Katz, D.L.: “Possibility of Cycling Deep Depleted Oil Reservoirs After Compression to a Single Phase,” Trans., AIME (1952) 195, 175. 37. Kobayashi, R. and Katz, D.L.: “VaporLiquid Equilibria for Binary HydrocarbonWater Systems,” Ind. & Eng. Chem. (1953) 45, No. 2, 440. 38. Donnelly, H.C. and Katz, D.L.: “Phase Equilibria in the Carbon DioxideMethane System,” Ind. & Eng. Chem. (1954) 46, 511. 39. Katz, D.L. et al.: Handbook of Natural Gas Engineering, McGrawHill Book Co. Inc., New York City (1959). 40. Reamer, H.H., Sage, B.H., and Lacey, W.N.: “Volumetric Behavior of Hydrogen Sulfide,” Ind. & Eng. Chem. (1950) 42, 140. 41. Sage, B.H. and Olds, R.H.: “Volumetric Behavior of Oil and Gas from Several San Joaquin Valley Fields,” Trans., AIME (1947) 170, 156. 42. Olds, R.H., Sage, B.H., and Lacey, W.N.: “Partial Volumetric Behavior of the MethaneCarbon Dioxide System,” Fundamental Research on Occurrence and Recovery of Petroleum, API, Dallas (1943) 44. 43. Reamer, H.H. et al.: “Phase Equilibria in Hydrocarbon Systems—Volumetric Behavior of EthaneCarbon Dioxide System,” Ind. & Eng. Chem. (1945) 37, 688. 44. Sage, B.H. and Lacey, W.N.: “Partial Volumetric Behavior of the Lighter Paraffin Hydrocarbons in the Gas Phase,” Drill. & Prod. Prac. (1939) 641. 45. Sage, B.H. and Lacey, W.N.: “Thermodynamic Properties of the Light Paraffin Hydrocarbons and Nitrogen,” API Research Project 37, monograph, API, New York City (1950). 46. Sage, B.H., Hicks, B.L., and Lacey, W.N.: “Partial Volumetric Behavior of the Lighter Hydrocarbons in the Liquid Phase,” Drill. & Prod. Prac. (1938) 402. 47. Sage, B.H. and Lacey, W.N.: “Apparatus for Determination of Volumetric Behavior of Fluids,” Trans., AIME (1948) 174, 102. 48. Reamer, H.H., Sage, B.H., and Lacey, W.N.: “Volumetric and Phase Behavior of the MethanePropane Systems,” Ind. & Eng. Chem. (1950) 42, 534. 49. Sage, B.H., Lacey, W.N., and Schaafsma, J.G.: “Phase Equilibria in Hydrocarbon Systems: MethanePropane Systems,” Ind. & Eng. Chem. (1934) 26, 214. 50. Reamer, H.H., Sage, B.H., and Lacey, W.N.: “Volumetric and Phase Behavior of the Methanen ButaneDecane System,” Ind. & Eng. Chem. (1951) 43, 1436. 51. Sage, B.H. and Lacey, W.W.: Volumetric and Phase Behavior of Hydrocarbons, Gulf Publishing Co., Houston (1949). 52. Reamer, H.H., Sage, B.H., and Lacey, W.N.: “Phase Equilibria in Hydrocarbon Systems,” Ind. & Eng. Chem. (June 1951) 43, 1436. 53. Reamer, H.H., Sage, B.H., and Lacey, W.N.: “Phase Behavior in Hydrocarbon System,” Ind. & Eng. Chem. (1951) 43, 2515. 54. Olds, R.H. et al.: “Phase Equilibria in Hydrocarbon Systems. The ButaneCarbon Dioxide System,” Ind. & Eng. Chem. (1949) 41, 475. INTRODUCTION
55. Reamer, H.H. and Sage, B.H.: “Phase Equilibria in Hydrocarbon Systems—Volumetric and Phase Behavior of the nDecaneCO2 System,” J. Chem. Eng. Data (1963) 8, 508. 56. Reamer, H.H., Fiskin, J.M., and Sage, B.H.: “Phase Equilibria in Hydrocarbon Systems: Phase Behavior in the MethanenButaneDecane System at 160°F,” Ind. & Eng. Chem. (December 1949) 41, 2871. 57. Reamer, H.H., Sage, B.H., and Lacey, W.N.: “Phase Equilibria in Hydrocarbon Systems—Volumetric and Phase Behavior of the MethanenHeptane System,” Ind. & Eng. Chem. (1956) 1, 29. 58. Sage, B.H., Webster, D.C., and Lacey, W.N.: “Phase Equilibria in Hydrocarbon Systems,” Ind. & Eng. Chem. (1936) 28, 1045. 59. Reamer, H.H., Sage, B.H., and Lacey, W.N.: “Phase Equilibria in Hydrocarbon Systems—Volumetric and Phase Behavior of the MethaneCyclohexane System,” Ind. & Eng. Chem. (1958) 3, 240. 60. Sage, B.H. and Lacey, W.N.: “Effect of Pressure Upon Viscosity of Methane and Two Natural Gases,” Trans., AIME (1938) 127, 118. 61. Sage, B.H., Yale, W.D., and Lacey, W.N.: “Effect of Pressure on Viscosity of nButane and iButane,” Ind. & Eng. Chem. (1939) 31, 223. 62. Sage, B.H. and Lacey, W.N.: “Gravitational Concentration Gradients in Static Columns of Hydrocarbon Fluids,” Trans., AIME (1939) 132, 120. 63. Reamer, H.H., Sage, B.H., and Lacey, W.N.: “Volumetric and Phase Behavior of the MethanenButaneDecane System,” Ind. & Eng. Chem. (1947) 39, 77. 64. Reamer, H.H., Sage, B.H., and Lacey, W.N.: “Volumetric and Phase Behavior of the MethanenButaneDecane System,” Ind. & Eng. Chem. (1952) 44, 1671. 65. Sage, B.H., Lacey, W.N., and Schaafsma, J.G.: “Behavior of Hydrocarbon Mixtures Illustrated by a Simple Case,” API Bulletin (1932) 212, 119. 66. Sage, B.H.: Thermodynamics of Multicomponent Systems, Reinhold Publishing Co. (1965) 67. Sage, B.H. and Lacey, W.N.: Volumetric and Pha.se Behavior of Hydrocarbons, Stanford Press, Stanford, Connecticut (1939). 68. Sage, B.H. and Reamer, R.H.: “Volumetric Behavior of Oil and Gas From the Rio Bravo Field,” Trans., AIME (1941) 142, 179. 69. Olds, R.H., Sage, B.H., and Lacey, W.N.: “Phase Equilibria in Hydrocarbon Systems. Composition of the DewPoint Gas of the MethaneWater System,” Ind. & Eng. Chem. (1942) 34, No. 10, 1223. 70. Reamer, H.H. et al.: “Phase Equilibria in Hydrocarbon Systems. Composition of the DewPoint Gas in the EthaneWater System,” Ind. & Eng. Chem. (1943) 35, No. 7, 790. 71. Reamer, H.H. et al.: “Phase Equilibria in Hydrocarbon Systems. Compositions of the Coexisting Phases of nButaneWater System in the ThreePhase Region,” Ind. & Eng. Chem. (1944) 36, No. 4, 381. 72. Reamer, H.H., Sage, B.H., and Lacey, W.N.: “Phase Equilibria in Hydrocarbon Systems. nButaneWater System in the TwoPhase Region,” Ind. & Eng. Chem. (1952) 44, No. 3, 609. 73. Sage, B.H. and Lacey, W.N.: “Some Properties of the Lighter Hydrocarbons, Hydrogen Sulfide, and Carbon Dioxide,” API Research Project 37, monograph, API, New York City (1955). 74. Eilerts, C.K.: “The Reserve Fluid, Its Composition and Phase Behavior,” Oil & Gas J. (1 January 1947) 63. 75. Eilerts, C.K.: “Gas Condensate Reservoir Engineering, 1. The Reserve Fluid, Its Composition and Phase Behavior,” Oil & Gas J. (1 February 1947) 63. 76. Eilerts, C.K., Carlson, H.A., and Mullen, N.B.: “Effect of Added Nitrogen on Compressibility of Natural Gas,” World Oil (June 1948) 129. 77. Eilerts, C.K. et al.: “Phase Relations of a GasCondensate Fluid at Low Temperatures, Including the Critical State,” Pet. Eng. (February 1948) 19, 154. 78. Eilerts, C.K.: Phase Relations of Gas Condensate Fluids, Monograph 10, USBM, American Gas Assn., New York City (1957) I and II. 79. Standing, M.B.: “VaporLiquid Equilibria of Natural GasCrude Oil Systems,” PhD dissertation, U. of Michigan, Ann Arbor, MI (1941). 80. Standing, M.B.: “A PressureVolumeTemperature Correlation for Mixtures of California Oils and Gases,” Drill. & Prod. Prac. (1947) 275. 81. Alani, G.H. and Kennedy, H.T.: “Volumes of Liquid Hydrocarbons at High Temperatures and Pressures,” Trans., AIME (1960) 219, 288. 82. Kennedy, G.C.: “PressureVolumeTemperature Relations in CO2 at Elevated Temperatures and Pressures,” Amer. J. Sci. (April 1954) 252, 225. 83. Kennedy, H.T. and Bhagia, N.S.: “An EOS for Condensate Fluids,” JPT (September 1969) 379. 84. Little, J.E. and Kennedy, H.T.: “A Correlation of the Viscosity of Hydrocarbon Systems with Pressure, Temperature and Composition,” SPEJ (June 1968) 157; Trans., AIME, 243. 85. Nemeth, L.K. and Kennedy, H.T.: “A Correlation of Dewpoint Pressure With Fluid Composition and Temperature,” SPEJ (June 1967) 99; Trans., AIME (1967) 240. 3
86. Muskat, M. and McDowell, J.M.: “An Electrical Computer for Solving Phase Equilibrium Problems,” Trans., AIME (1949) 186, 291. 87. Redlich, O. and Kwong, J.N.S.: “On the Thermodynamics of Solutions, V: An Equation of State. Fugacities of Gaseous Solutions,” Chem. Rev. (1949) 44, 233. 88. Benedict, M., Webb, G.B., and Rubin, L.C.: “An Empirical Equation for Thermodynamic Properties of Light Hydrocarbons and Their Mixtures, I. Methane, Ethane, Propane, and nButane,” J. Chem. Phy. (1940) 8, 334. 89. Starling, K.E.: “A New Approach for Determining EquationofState Parameters Using Phase Equilibria Data,” SPEJ (December 1966) 363; Trans., AIME, 237. 90. Soave, G.: “Equilibrium Constants from a Modified RedlichKwong EOS,” Chem. Eng. Sci. (1972) 27, No. 6, 1197. 91. Peng, D.Y. and Robinson, D.B.: “A NewConstant EOS,” Ind. & Eng. Chem. Fund. (1976) 15, No. 1, 59. 92. Coats, K.H.: “An EOS Compositional Model,” SPEJ (October 1980) 363; Trans., AIME, 269. 93. Young, L.C. and Stephenson, R.E.: “A Generalized Compositional Approach for Reservoir Simulation,” SPEJ (October 1983) 727; Trans., AIME, 275. 94. Yarborough, L.: “Application of a Generalized Equation of State to Petroleum Reservoir Fluids,” Equations of State in Engineering and Re
4
search, K.C. Chao and R.L. Robinson Jr. (eds.), Advances in Chemistry Series, American Chemical Soc. (1978) 182, 386–439. 95. Whitson, C.H.: “Characterizing Hydrocarbon Plus Fractions,” SPEJ (August 1983) 683; Trans., AIME, 275. 96. Pedersen, K.S., Thomassen, P., and Fredenslund, A.: “Characterization of Gas Condensate Mixtures,” C7 Fraction Characterization, L.G. Chorn and G.A. Mansoori (eds.), Advances in Thermodynamics, Taylor & Francis, New York City (1989) 1. 97. Zick, A.A.: “A Combined Condensing/Vaporizing Mechanism in the Displacement of Oil by Enriched Gases,” paper SPE 15493 presented at the 1986 SPE Annual Technical Conference and Exhibition, New Orleans, 5–8 October. 98. Schulte, A.M.: “Compositional Variations Within a Hydrocarbon Column Due to Gravity,” paper SPE 9235 presented at the 1980 SPE Annual Technical Conference and Exhibition, Dallas, 21–24 September. 99. Coats, K.H.: “Simulation of Gas Condensate Reservoir Performance,” JPT (October 1985) 1870.
SI Metric Conversion Factors °F (°F*32)/1.8 +°C psi 6.894 757 E)00 +kPa
PHASE BEHAVIOR MONOGRAPH
Chapter 2
Volumetric and Phase Behavior of Oil and Gas Systems 2.1 Introduction Petroleum reservoir fluids are naturally occurring mixtures of natural gas and crude oil that exist in the reservoir at elevated temperatures and pressures. Reservoirfluid compositions typically include hundreds or thousands of hydrocarbons and a few nonhydrocarbons, such as nitrogen, CO2, and hydrogen sulfide. The physical properties of these mixtures depend primarily on composition and temperature and pressure conditions. Reservoir temperature can usually be assumed to be constant in a given reservoir or to be a weak function of depth. As oil and gas are produced, reservoir pressure decreases and the remaining hydrocarbon mixtures change in composition, volumetric properties, and phase behavior. Gas injection also may change reservoirfluid composition and properties. Katz and Williams1 give an excellent review of reservoir fluids and their general behavior under different operating conditions. The hydrocarbon phases and connate water sharing the pore volume (PV) in a reservoir are in thermodynamic equilibrium. Strictly speaking, hydrocarbons and water should be treated simultaneously in phasebehavior calculations. At typical reservoir conditions, the effect of connate water on hydrocarbon phase behavior can usually be neglected. Water can, however, affect the totalsystem phase behavior (for example, when hydrates form from naturalgas/water mixtures). This chapter covers only twophase, vapor/liquid phase behavior. Chap. 8 briefly covers three and fourphase systems (vapor/liquid/ liquid and vapor/liquid/liquid/solids) for lowtemperature CO2/oil and richgas/oil mixtures, and Chap. 9 gives the behavior of vapor and solids related to hydrates. Sec. 2.1 introduces the composition of petroleum reservoir fluids and emphasizes their chemical complexity. Because reservoir fluids are made up of many components, a detailed quantitative analysis is difficult to perform. Organic compounds found in reservoir fluids are expressed by a general formula that classifies even highmolecularweight compounds containing sulfur, nitrogen, and oxygen. This chapter also gives a historical review of the American Petroleum Inst. (API) supported projects that defined many of the compounds known today. Simple one and twocomponent phase behavior can be helpful in describing the effects of pressure, temperature, and composition on the reservoirfluid phase behavior. Sec. 2.2 presents pressure/temperature ( pT), pressure/volume ( pV), and pressure/composition ( px) phase diagrams of simple systems. The behavior of these idealized systems is qualitatively similar to the behavior of complex reservoir fluids, as Sec. 2.3 shows. VOLUMETRIC AND PHASE BEHAVIOR OF OIL AND GAS SYSTEMS
Retrograde condensation is perhaps the most unusual phase behavior that petroleum reservoir fluids exhibit.* Sec. 2.4 discusses the definition of retrograde condensation and the effect of retrograde condensation on the behavior of gascondensate reservoirs. Petroleum reservoir fluids can be divided into five general categories, in increasing order of chemical complexity: dry gas, wet gas, gas condensate, volatile oil, and black oil. However, the phase behaviors of gas condensates and volatile oils are considerably more complex than those of black oils. The component distribution in a reservoir fluid, not simply the number of components, determines how close a fluid is to a critical state. Complex phase behavior is typically associated with systems that are “near critical”: systems that usually contain 10 to 15 mol% of components that are heptanes and heavier (C7+). Since the early 1930’s, experimental data have been measured onfluids of each type listed above. Sec. 2.5 defines each fluid type by its pT diagram. Also, general characteristics of reservoir fluids are summarized in terms of composition and surface properties, such as GOR and stocktankoil gravity. 2.2 ReservoirĆFluid Composition The nature and composition of a reservoir fluid depends somewhat on the depositional environment of the formation from which the fluid is produced. Geologic maturation also influences reservoirfluid composition. Several theories offer explanations for the origin and formation of petroleum over geologic time; no single theory suffices to explain how oil and gas were formed in all reservoirs. One theory portrays reservoirs as giant hightemperature/highpressure reactors with catalytic rock surfaces that slowly convert deposited organic matter into oil and gas. Other theories hypothesize that oil and gas were formed from bacterial action on deposited organic matter. Other investigators maintain that oil and gas may be formed in the same geologic formation but that each fluid migrates to “traps” at different elevations because of fluiddensity differences and gravity forces. Crude oil and natural gas are composed of many chemical compounds with a wide range of molecular weights. Some estimates24 suggest that perhaps 3,000 organic compounds can exist in a single *Historically, retrograde condensation has been considered the most complex phasebehavior phenomenon observed by reservoir fluids. Perhaps equally intriguing are the phenomena of strong compositional gradients, the condensing/vaporizing miscible mechanism (Chap. 8), asphaltene precipitation, and lowtemperature, multiphase CO2 behavior.
1
TABLE 2.1—COMPOSITION AND PROPERTIES OF SEVERAL RESERVOIR FLUIDS Composition (mol%) Gas
NearCritical
Component
Dry Gas
Wet Gas
Condensate
Oil
Volatile Oil
Black Oil
CO2
0.10
1.41
2.37
1.30
0.93
0.02
N2
2.07
0.25
0.31
0.56
0.21
0.34
C1
86.12
92.46
73.19
69.44
58.77
34.62
C2
5.91
3.18
7.80
7.88
7.57
4.11
C3
3.58
1.01
3.55
4.26
4.09
1.01
iC4
1.72
nC4 iC5
0.50
0.28
0.71
0.89
0.91
0.76
0.24
1.45
2.14
2.09
0.49
0.13
0.64
0.90
0.77
0.43
nC5
0.08
0.68
1.13
1.15
0.21
C6(s)
0.14
1.09
1.46
1.75
1.61
C7 +
0.82
8.21
10.04
21.76
56.40
Properties MC g
7)
C 7)
K wC
7
130
184
219
228
274
0.763
0.816
0.839
0.858
0.920
12.00
11.95
11.98
11.83
11.47
1,490
300
38
24
GOR, scf/STB
∞
105,000
5,450
3,650
OGR, STB/MMscf
0
10
180
275
57
49
45
gAPI gg
0.61
0.70
0.71
0.70
0.63
psat, psia
3,430
6.560
7,015
5,420
2,810
0.0051
0.0039
2.78
1.73
1.16
9.61
26.7
30.7
38.2
51.4
Bsat, ft3/scf or bbl/STB ò sat, lbmńft
3
reservoir fluid. The lighter and simpler compounds are produced as natural gas after surface separation, whereas the heavier and more complex compounds form crude oil at stocktank conditions. Table 2.1 gives typical oilfield molar compositions for reservoir mixtures. The heavier components are usually lumped into a “plus” fraction instead of being identified individually. Chap. 5 discusses methods of quantifying and characterizing the components that make up the plus fraction—usually heptanesplus. Natural gas is composed mainly of lowmolecularweight alkanes (methane through butanes), CO2, hydrogen sulfide, nitrogen, and, in some cases, lesser quantities of helium, hydrogen, CO, and carbonyl sulfide.5 Most crude oils are composed mainly of hydrocarbons (hydrogen and carbon compounds). The broad spectrum of organic compounds found in petroleum during the formation of crude oil also includes sulfur, nitrogen, oxygen, and trace metals. Tars and asphalts are solid or semisolid mixtures that include bitumen, pitch, waxes, and resins. These highmolecularweight complex colloidal suspensions exhibit nonNewtonian rheology. The temperature and pressure gradients in a formation may cause reservoirfluid properties to vary as a function of depth. “Compositional grading” is the continual change of composition as a function of depth.68 In compositional grading, reservoir temperature may be near the critical temperature of reservoir fluid(s) at certain depths in the reservoir. Physically, the thermodynamic forces of individual components in a nearcritical mixture are of the same order of magnitude as gravity forces that tend to separate the lighter from the heavier components. The result can be a transition from an undersaturated gas condensate at the highest elevation to an undersaturated oil at the lowest elevation, with or without a visible phase transition from gas to oil (gas/oil contact). In petroleum refining, crude oil is often categorized according to its base and the hydrocarbon series (paraffin, naphthene, or aromatic) it contains in the highest concentration. Figs. 2.1 and 2.29 illustrate the types and relative amounts of hydrocarbon series that can be found in typical petroleumrefinery products. Nelson3 gives a full account of basic hydrocarbon chemistry and test methods that 2
have been used for many years to determine petroleum composition and inspection properties for refining purposes. The more common test methods include paraffin, naphthene, and aromatic; saturates, aromatics, resins, and asphaltenes; and Strieter (asphaltenes, resins, and oils) analyses; oil gravity in °API; Reid vapor pressure; trueboilingpoint distillation; flash, fire, cloud, and pour points; color; and Saybolt and Furol viscosities. Chap. 5 discusses some of these methods that are used in petroleum engineering. The empirical formula Cn H2n)h Sa Nb Oc can be used to classify nearly all compounds found in crude oil. The largest portion of crude oil is composed of hydrocarbons with carbon number, n, ranging from 1 to about 60, and h numbers ranging from h+)2 for lowmolecularweight paraffin hydrocarbons to h+*20 for highmolecularweight organic compounds (e.g., polycyclic aromatic hydrocarbons). Occasionally, sulfur, nitrogen, and oxygen substitutions occur in highmolecularweight organic compounds, with a, b, and c usually ranging from 1 to 3.2,10 Over the past 60 years, petroleum chemists have identified hundreds of the complex organic compounds found in petroleum. Beginning in 1927, Rossini and others11,12 conducted a lengthy investigation of the composition of petroleum [API Research Project 6 (API 6)] to develop and improve petroleumrefining processes. It took API 6 investigators almost 40 years to elucidate the composition of a single midcontinent crude oil from Well No. 6 in South Ponca City, Oklahoma. Because compounds with carbon numbers u12 could not be isolated from crude oils, during 1940–66, API Research Project 42 focused on synthesizing and characterizing model hydrocarbons with high molecular weights. These model compounds were used for identifying compounds that could not be isolated from crude oil. A crude oil compound with analytical responses that matched those of a synthesized model compound was inferred to have a similar chemical structure. Other API projects13 followed API 6, and increasingly more complex petroleum compounds were identified. API 48 focused on sulfur compounds, API 52 on nitrogen compounds, and API 56 on orPHASE BEHAVIOR
Fig. 2.1—Petroleum products identified according to carbon number.
ganometallic compounds. API 60 extended the work of API 6 to include petroleum heavy ends. In 1975, API stopped sponsoring basic research into the composition of petroleum. From 1975 to 1982, the petroleum engineering industry made additional advances in analytical techniques mainly because of the synfuels effort. The most sophisticated analytical techniques now in use include highly selective solvent extraction1416; simulated distillation; gel permeation, highperformance liquid,17 and supercritical chromatography18; and mass infrared, 13C nuclear magnetic resonance,19 and Fouriertransform infrared spectroscopy. The American Chemical Soc. Div. of Petroleum Chemistry provides a comprehensive review of this area of research every 2 to 3 years. Table 2.220 shows an example of a crudeoil distillate classified by h number (in the general formula Cn H2n)h Sa Nb Oc ) and probable structural type, which determines the range of possible n numbers. Within and across each hydrocarbon class, many isomers share h and n numbers. The alkane (paraffin) series (h+2) has completely saturated hydrocarbon chains that are chemically very stable. The alkene (olefin) and alkyne (acetylene) series (h+0 and h+*2) are composed of unsaturated, straightchain hydrocarbons. Because alkenes and alkynes are reactive, they are not usually found in naturally occurring petroleum deposits. The naphthene series (h+0), saturatedring or cyclic compounds, are found in nearly all crudes. The aromatic or “benzene” series (h+*6) are unsaturated cyclic compounds. Lowboilingpoint aromatics, which are also reactive, are found in relatively low concentrations in crude oil. Heavier crude oils are characterized by unsaturated polycyclic aromatic hydrocarbons with increasingly negative h numVOLUMETRIC AND PHASE BEHAVIOR OF OIL AND GAS SYSTEMS
bers. As molecular weight increases, these compounds assume varying degrees of fusedring saturation, with occasional hydrocarbon side chains. Sulfur, nitrogen, and oxygen can be substituted in the fused hydrocarbon rings to form heterocyclics or can occupy various positions on side chains.21 Metals, such as nickel and vanadium, can form organometallic compounds (porphyrins) in crude oil.2,10 Asphalts, bitumens, and tars are complex colloidal mixtures of carboids, carbenes, asphaltenes, and maltenes (resins and oils). Micellar structures of carboids, carbenes, and asphaltenes are formed by aromatic polycondensation reactions and are maintained in colloidal suspension by the maltenes. These fractions are separated according to their solubility or lack of solubility in certain lowmolecularweight solvents, such as propane, pentane, nhexane, and carbon disulfide. Fig. 2.316 shows a hypothetical chemical structure of an asphaltene. The bracket around the structure implies that the structure is repeated three times. Although asphalt mixtures are complex in composition and rheology, they follow certain molecularweight distributions that can be characterized as discussed in Chap. 5. Understanding the nature of asphaltenes is important in petroleum engineering because, even in low concentrations, asphaltenes can markedly affect reservoirfluid phase behavior.22 Because asphaltenes are polar and hydrogen bonding, they alter reservoir wettability by adsorbing onto the rock surface.23 This alteration of reservoir wettability may affect capillary pressure, relativepermeability relations, residual oil saturations, waterflood behavior, dispersion, and electrical properties. Figs. 2.2 and 2.3 vividly show that the composition of crude oil is considerably more complex than the Cn H2n)2 straightchain models commonly thought of as “oil.” This complexity 3
Fig. 2.2—Summary of hydrocarbons to be expected in crudeoil fractions (from Neumann et al.9).
should be borne in mind when modeling the phase behavior of complex reservoir fluids, particularly in gasinjection projects.23,24 2.3 Phase Diagrams for Simple Systems The dependence of volumetric and phase behavior on temperature, pressure, and composition is similar for simple (two and threecomponent) and complex (multicomponent) systems. Traditionally, the introduction to phase behavior of complex reservoir fluids starts with a description of the vaporpressure and volumetric behavior of single components. The introduction then proceeds to the behavior of two and threecomponent systems, and finally to the behavior of complex multicomponent systems. Part of the rationale for this procession lies in the Gibbs phase rule.25,26 The Gibbs phase rule states that the number of intensive variables (i.e., degrees of freedom), F, that must be specified to determine the thermodynamic state of equilibrium for a mixture containing n components distributed in P phases (gas, liquid, and/or solid), is F + n * P ) 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.1) Intensive (thermodynamic) variables, such as temperature, pressure, and density, do not depend on the amount of material in the system. Extensive variables, such as flow rate, total mass, or liquid volume, depend on the extent of the system. To attain equilibrium requires that no net interphase mass transfer can occur. Thus, the temperatures and pressures of the coexisting 4
phases must be the same and the chemical potentials of each component in each phase must be equal. A more stringent definition of phase equilibrium includes other forces in addition to chemical potential (e.g., gravity and capillarity). On the basis of Eq. 2.1, for a twophase, singlecomponent system, F+1 and only temperature or pressure needs to be specified to determine the thermodynamic state of the system. For a twophase, twocomponent system, F+2 and both temperature and pressure need to be specified to define the thermodynamic state of the mixture. Twophase binary systems allow one to focus on the effect of temperature and pressure on the composition and the relative amounts of each of the two phases, regardless of the composition of the overall mixture. The Gibbs phase rule implies that as the number of components increases to n in a twophase mixture, n*2 composition variables must be specified in addition to temperature and pressure. If more than two phases are present, then n*P variables must be specified in addition to temperature and pressure. Because reservoir fluids comprise many components, the number of variables that must be defined to determine the state of a reservoir fluid is conceptually unmanageable. Therefore, simple systems are often used to model the basic volumetric and phase behavior of crude oil mixtures. Note that the phase rule must be modified if other potential fields are considered. For example, if the force of gravity is considered, as PHASE BEHAVIOR
TABLE 2.2—DISTRIBUTION OF h SERIES FROM 698 TO 995°F DISTILLATE OF SWAN HILLS CRUDE OIL (Ref. 20) Mass h Series
Probable Type
*12
Naphthalenes
*14
Naphthenonaphthalenes and/or biphenyls
*16
Dinaphthenaphthalenes and/or
*18
Trinaphthenaphthalenes and/or
*20
Tetranaphthenaphthalenes and/or
*22
Pentanaphthenaphthalenes and/or
*24
Hexanaphthenaphthalenes and/or
naphthenobiphenyls dinaphthenobiphenyls trinaphthenobiphenyls tetranaphthenobiphenyls pentanaphthenobiphenyls *26
Heptanaphthenaphthalenes and/or
*28
Octanaphthenaphthalenes and/or
*4S
Tricyclic sulfides
*6S
Tetracyclic sulfides
*8S
Pentacyclic sulfides
*10S
Hexacyclic sulfides
hexanaphthenobiphenyls heptanaphthenobiphenyls
*8S
Thiaindanes/thiatetralins
*10S
Naphthenothiaindanes/thiatetralins
*12S
Dinaphthenothiaindanes/thiatetralins
*14S
Trinaphthenothiaindanes/thiatetralins
*10S
Benzothiophenes
*12S
Naphthenobenzothiophenes
is done when calculating compositional variation with depth, the phase rule is F+n*P)3.7 2.3.1 SingleComponent Systems. The pT curve shown in Fig. 2.4 is a portion of the vaporpressure curve for a typical hydrocarbon compound. Above and to the left of the curve, the hydrocarbon behaves as a liquid; below and to the right, the hydrocarbon behaves as a vapor. Saturated liquid and vapor coexist at every point along the vaporpressure curve. The curve ends at the critical temperature and critical pressure of the hydrocarbon (the “critical point”). Fig. 2.5 shows a 3D PVT diagram of a pure compound. The critical temperature of a single component defines the temperature above which any gas/liquid mixture cannot coexist, regardless of pressure. Similarly, the critical pressure defines the pressure above which liquid and vapor cannot coexist, regardless of temperature. Along the vaporpressure curve, two phases coexist in equilibrium. At the critical point, the vapor and liquid phases can no longer be distinguished, and their intensive properties are identical. For a multicomponent system, the definition of the critical point is also based on a temperature and pressure at which the vapor and liquid phases are indistinguishable. However, for a singlecomponent system, the twophase region terminates at the critical point. In a multicomponent system, the twophase region can extend beyond the system’s critical point (i.e., at temperatures greater than the critical temperature and pressures greater than the critical pressure). Fig. 2.627 illustrates the continuity of gas and liquid phases for pure components. In this figure, the darker shading corresponds to higher density. A sharp contrast in phase densities is readily apparent along the vaporpressure curve. As temperature increases along the vaporpressure curve, the discontinuity becomes harder to discern, until finally, at the critical point, the contrast in shading is hardly noticeable. Qualitatively, the behavior described by the shading in Fig. 2.6 is the same for multicomponent mixtures in the undersaturated region. VOLUMETRIC AND PHASE BEHAVIOR OF OIL AND GAS SYSTEMS
Fig. 2.3—Hypothetical structure of a petroleum asphaltene (after Speight and Moschopedis14).
pc
Fig. 2.4—pT diagram for a single component in the region of vapor/liquid behavior near the critical point ( pc +critical pressure and Tc +critical temperature).
Phase changes do not have to take place abruptly if certain temperature and pressure paths are followed. A process can start as a saturated liquid and end as a saturated vapor, with no abrupt change in phase. The path D–A–E–F–G–B–D in Fig. 2.4 is an example of a process that changes phases without crossing the vaporpressure curve. Pure components actually exist as a saturated “liquid” and “vapor” only along the vaporpressure curve. At other pressures and temperatures, the component only behaves “liquidlike” or “vaporlike,” depending on the location of the system temperature and pressure relative to the system’s critical point. Katz28 suggested calling a pure substance “singlephase fluid” at pressures greater than the critical pressure. Strictly speaking, the terms liquidlike and vaporlike should be used to describe undersaturated fluids. 5
Fig. 2.5—Threedimensional schematic of the PVT surface of a pure compound (source unknown).
ponent is a saturated liquid. Similarly, the saturation curve to the right of the critical point (Point B to Point C) defines the dewpoint curve, along which the component is a saturated vapor. For any temperature less than the critical temperature, successive decreases in volume will elevate the pressure of the vapor until the “dewpoint” (vapor pressure) is reached (Point B on Fig. 2.7). At these conditions, the component is a saturated vapor in equilibrium with an infinitesimal amount of saturated liquid. Further decreases in the volume at constant temperature will result in proportionate increases in the amount of saturated liquid condensed, but the pressure does not change (i.e., the system pressure remains equal to the vapor pressure). While more liquid is being formed, the total volume (at Point D) is being reduced. However, the densities and other intensive properties of the saturated vapor and saturated liquid remain constant as a consequence of the Gibbs phase rule. A simple mass balance further shows that the ratio of liquid to vapor equals the ratio of Curve B–D to Curve D–A. Further decreases in volume will condense more liquid until the bubblepoint is reached. At the bubblepoint, the system is 100% saturated liquid in equilibrium with an infinitesimal amount of saturated vapor. Further decreases in volume beyond the bubblepoint are accompanied by a large increase in pressure because the liquid is only slightly compressible. This is indicated by the nearly vertical isotherms on the left side of Fig. 2.7. In the undersaturated vapor region on the right side of the diagram, a large change in volume reduces pressure only slightly because the vapor is highly compressible.
Fig. 2.726 shows a pV diagram for ethane. The area enclosed by the saturation envelope represents the twophase region. The area to the left of the envelope is the liquidlike region, and the area to the right is the vaporlike region. Point C represents the critical point. The saturation curve to the left of the critical point (from Point A to Point C) defines the bubblepoint curve, along which the com
2.3.2 TwoComponent Systems. Twocomponent systems are slightly more complex than singlecomponent systems because both temperature and pressure affect phase behavior in the saturated region. Two important differences between single and twocomponent systems exist. The saturated pT projection is represented by a phase envelope rather than by a vaporpressure curve, and the criti
3,000
2,000
1,000
0
0
100
200
300
400
500
600
Temperature, °F Fig. 2.6—Continuity of vapor and liquid states for a single component along the vaporpressure curve and at supercritical conditions (after Katz and Kurata27). 6
PHASE BEHAVIOR
Specific volume, ft3/lbm Fig. 2.7—pV diagram for ethane at three temperatures (from Standing26).
cal temperature and critical pressure no longer define the extent of the twophase, vapor/liquid region. Fig. 2.829 compares the pT and pV behavior of pure compounds and mixtures. Fig. 2.926 is a pT projection of the ethane/nheptane system for a fixed composition. For a singlecomponent system, the dew and bubblepoint curves are one in the same; i.e., they coincide with the vaporpressure curve. In a binary (or other multicomponent) system, the dew and bubblepoint curves no longer coincide, and a phase envelope results instead of a vaporpressure curve. To the left of the phase envelope, the mixture behaves liquidlike, and to the right it behaves vaporlike. For binary or other multicomponent systems, the critical temperature and pressure are defined as the point where the dew and bubblepoint curves intersect. At this point, the equilibrium phases are physically indistinguishable. Also, in contrast to the singlecomponent system, two phases can exist at temperatures and pressures greater than the critical temperature and pressure. The highest temperature at which two phases can coexist in equilibrium is defined as the cricondentherm (Tangent b–b in Fig. 2.9). Similarly, the highest pressure at which two phases can coexist is defined as the cricondenbar (Tangent a–a). In the singlephase region, vapor and liquid are distinguished only by their densities and other physical properties. The region just beyond the critical point of a mixture has often been called the “supercritical” or “densefluid” region. Here, the fluid is considered to be neither gas nor liquid because the fluid properties are not strictly liquidlike or vaporlike. Kay30 measured the phase behavior of the binary ethane/nheptane system for several compositions, as Fig. 2.10 shows. On the left side of this figure, the curve terminating at Point C is the vapor pressure of pure ethane; the curve on the right, terminating at Point C7, is the vapor pressure of pure nheptane. Points C1 through C3 are the critical points of ethane/nheptane mixtures at different compositions. The dashed line represents the locus of critical points for the infinite number of possible ethane/nheptane mixtures. Each mixture composition has its own pT phase envelope. The three compositions shown, which are 90.22, 50.25, and 9.78 wt% ethane, represent a system that is mainly ethane, a system that is onehalf ethane and onehalf nheptane (by weight), and a system that is mainly nheptane, respectively. Several interesting features of binary and multicomponent systems can be studied from these three mixtures. As composition changes, the location of the critical point and the shape of the pT phase diagram also change. Note that the critical pressures of many (but not all) mixtures are higher than the critical pressures of the components composing the VOLUMETRIC AND PHASE BEHAVIOR OF OIL AND GAS SYSTEMS
mixture. With a mixture composed mainly of ethane, the critical point lies to the left of the cricondentherm. Such a system is analogous to a reservoir gascondensate system. As the percentage of ethane in the mixture increases further, the critical point of the system approaches that of pure ethane. The critical point for the mixture composed mostly of nheptane lies below the cricondenbar. This system is analogous to a reservoir blackoil system. As the percentage of nheptane increases, the critical point of the mixture approaches that of pure nheptane. With equal percentages of ethane and nheptane, the critical pressure is close to the cricondenbar of ethane and nheptane. As the concentration of each component becomes similar, the twophase region becomes larger. Other binaries provide additional insight into the effect of widely differing boiling points of the components making up the system. Fig. 2.1131 shows the vapor pressure of several hydrocarbons and the critical loci of their binary mixtures with methane. As the boiling points of the methane/hydrocarbon binary become more dissimilar, the twophase region becomes larger and the critical pressure increases. For binaries with components that have similar molecular structures, the loci of critical points are relatively flat. 2.3.3 Multicomponent Systems. Phase diagrams for naturally occurring reservoir fluids are similar to those for binary mixtures. Fig. 2.125 is the first pT phase diagram measured for a complex gascondensate system. This pT diagram is particularly useful because it exhibits oillike to gaslike behavior over a range of typical reservoir temperatures, from 80 to 240°F. Katz and coworkers32 used a glasswindowed cell to measure the distribution of gas and liquid phases throughout the twophase region and near the mixture’s critical point. Fig. 2.135 shows isotherms of volume percent vs. pressure that were measured to determine the twophase boundary and the volumepercent quality lines in the pT diagram in Fig. 2.12. 2.4 Retrograde Condensation Kurata and Katz33 give the most concise and relevant discussion of retrograde phenomena related to petroleum engineering. In 1892, Kuenen34 used the term “retrograde condensation” to describe the anomalous behavior of a mixture that forms a liquid by an isothermal decrease in pressure or by an isobaric increase in temperature. Conversely, “retrograde vaporization” can be used to describe the formation of vapor by an isothermal increase in pressure or by an isobaric decrease in temperature. Neither form of retrograde behavior occurs in singlecomponent systems. Fig. 2.14 is a constantcomposition pT projection of a multicomponent system. The diagram shows lines of constant liquid volume percent (quality). Although total composition is fixed, the respective compositions of saturated vapor and liquid phases change along the quality lines. The bubblepoint curve represents the locus of 100% liquid, and the dewpoint curve represents the locus of 0% liquid. The bubble and dewpoint curves meet at the mixture critical point. Lines of constant quality also converge at the mixture critical point. The regions of retrograde behavior are defined by the lines of constant quality that exhibit a maximum with respect to temperature or pressure. Fig. 2.14 shows that for retrograde phenomena to occur, the temperature must be between the critical temperature and the cricondentherm. Fig. 2.1535 illustrates the liquid volumetric behavior of a lean gascondensate system measured by Eilerts et al.3537 Fig. 2.12 shows the pT diagram of a reservoir mixture that would be considered a gas condensate if it had been discovered at a reservoir temperature of, for example, 200°F and an initial pressure of 2,700 psia. For these initial conditions, if reservoir pressure drops below 2,560 psia from depletion, the dewpoint will be passed and a liquid phase will develop in the reservoir. Liquid dropout will continue to increase until the pressure reaches 2,300 psia, when a maximum of 25 vol% liquid will have accumulated. According to Fig. 2.12, further pressure reduction will revaporize most of the condensed liquid. These comments assume that the overall composition of the reservoir mixture remains constant during depletion, a reasonable assumption in the context of this general discussion. In reality, howev7
Fig. 2.8—Qualitative pT and pV plots for pure fluids and mixtures; Vc +critical volume (after Edmister and Lee29).
er, the behavior of liquid dropout and revaporization differs from that suggested by constantcomposition analysis. The retrograde liquid saturation is usually less than the saturation needed to mobilize the liquid phase. Because the heavier components in the original mixture constitute most of the (immobile) condensate saturation, the overall molecular weight of the remaining reservoir fluid increases during depletion. The phase envelope for this heavier reservoir mixture is pushed down and to the right of the original phase diagram (Fig. 2.16); the critical point is shifted to the right toward a higher temperature. It is not unusual that a retrogradecondensate mixture under depletion will reach a condition where the overall composition would exhibit a bubblepoint pressure if the reservoir were repressured (i.e., the overall mixture critical temperature becomes greater than the reservoir temperature). This change in overall reservoir composition results in less revaporization at lower pressures. Fig. 2.17 shows the difference between constantcomposition and “depletion” liquiddropout curves. 8
2.5 Classification of Oilfield Systems One might assume that the name used to identify a reservoir fluid should not influence how the fluid is treated as long as its physical properties are correctly defined. In theory this is true, but in practice we are usually required to define petroleum reservoir fluids as either “oil” or “gas.” For example, regulatory bodies require the definition of reservoir fluid for well spacing and determining allowable production rates and fielddevelopment strategy (e.g., unitization). The classification of a reservoir fluid as dry gas, wet gas, gas condensate, volatile oil, or black oil is determined (1) by the location of the reservoir temperature with respect to the critical temperature and the cricondentherm and (2) by the location of the firststage separator temperature and pressure with respect to the phase diagram of the reservoir fluid. Fig. 2.18 illustrates how four types of depletion reservoirs for the same hydrocarbon system are defined by the location of the initial reservoir temperature and pressure. PHASE BEHAVIOR
Fig. 2.10—pT diagram for the C2/nC7 system at various concentrations of C2 (after Kay30).
Fig. 2.9—pT diagram for a C2/nC7 mixture with 96.83 mol% ethane (from Standing26).
Fig. 2.11—pT diagram for various hydrocarbon binaries illustrating the effects of molecularweight differences on criticalpoint loci (after Brown et al.31). VOLUMETRIC AND PHASE BEHAVIOR OF OIL AND GAS SYSTEMS
A reservoir fluid is classified as dry gas when the reservoir temperature is greater than the cricondentherm and surface/transport conditions are outside the twophase envelope; as wet gas when the reservoir temperature is greater than the cricondentherm but the surface conditions are in the twophase region; as gas condensate when the reservoir temperature is less than the cricondentherm and greater than the critical temperature; and as an oil (volatile or black oil) when the reservoir temperature is less than the mixture critical temperature. For a given reservoir temperature and pressure, Fig. 2.1938 shows the spectrum of reservoir fluids from wet gas to black oil expressed in terms of surface GOR’s and oil/gas ratios (OGR’s). A more quantitative classification is also given in Fig. 2.19 in terms of molar composition, by use of a ternary diagram. In the nearcritical region, gas condensates have a C7+ concentration less than [12.5 mol% and volatile oils fall between 12.5 to 17.5 mol% C7+. Retrograde gascondensate reservoirs26,39 typically exhibit GOR’s between 3,000 and 150,000 scf/STB (OGR’s from about 350 to 5 STB/MMscf) and liquid gravities between 40 and 60°API. The color of stocktank liquid is expected to lighten from volatileoil to gascondensate systems, although light volatile oils may be yellowish or waterwhite and some condensate liquids can be dark brown
Fig. 2.12—pT diagram for a gascondensate system (after Katz et al.5). 9
Fig. 2.14—Hypothetical pT diagram for a gas condensate showing the isothermal retrograde region.
Fig. 2.13—Volume isotherms for the gascondensate pT diagram in Fig. 2.12 (after Katz et al.5)
or even black. Color has not been a reliable means of differentiating clearly between gas condensates and volatile oils, but in general, dark colors indicate the presence of heavy hydrocarbons. In some cases, for condensate recovery from a surface process facility, the reservoir fluid is mistakenly interpreted to be a gas condensate. Strictly speaking, the definition of a gas condensate depends only on reservoir temperature. The definition of a reservoir fluid as wet or dry gas depends on conditions at the surface. This makes differentiation between dry and wet gas difficult because any gas can conceivably be cooled enough to condense a liquid phase. The classification of a fluid as an oil is unambiguous because the only requirement is that the reservoir temperature be less than the
BUBBLEPOINT
Fig. 2.15—Liquid volume (expressed as a liquid/gas ratio) behavior for a leangascondensate system (from Eilerts et al.35). 10
PHASE BEHAVIOR
Fig. 2.16—Change in phase envelope during the depletion of a gas condensate.
critical temperature. However, the distinction between a black oil and a volatile oil is more arbitrary. Generally speaking, a volatile oil is a mixture with a relatively high solution gas/oil ratio. Volatile oils exhibit large changes in properties when pressure is reduced only somewhat below the bubblepoint. In an extreme case, the oil volume may shrink from 100 to 50% with a reduction in pressure of only 100 psi below the bubblepoint. Blackoil properties, on the other hand, exhibit gradual changes, with nearly linear pressure dependence below the bubblepoint. Volatile oils typically yield stocktankoil gravities greater than 35°API, surface GOR’s between 1,000 and 3,000 scf/STB, and FVF’s (see Formation Volume Factors in Chap. 6) greater than [1.5 RB/STB. Solution gas released from a volatile oil contains significant quantities of stocktank liquid (condensate) when this gas is produced to the surface. Solution gas from black oils is usually considered “dry,” yielding insignificant stocktank liquids when produced to surface conditions.
For engineering calculations, the liquid content of released solution gas is perhaps the most important distinction between volatile oils and black oils. This difference is also the basis for the modification of standard blackoil PVT properties discussed in Chap. 7. A reasonable engineering distinction between black oils and volatile oils can be made on the basis of simple reservoir materialbalance calculations. If the total surface oil and gas recoveries calculated by a reservoir material balance with the standard blackoil PVT formulation are sufficiently close to the recoveries calculated by a compositional material balance, the oil can probably be considered a black oil (see Chap. 7). If calculated oil recoveries are significantly different, the reservoir mixture should be treated as a volatile oil by use of a compositional approach or the modified blackoil PVT properties outlined in Chap. 7. Several researchers40,41 have shown that a compositional material balance for depletion of volatileoil reservoirs may predict from two to four times the surface liquid reBubblepoint or dissolved gas reservoirs
Dewpoint or gas condensate reservoirs
SingleĆphase gas reservoirs
CVD lower because of loss of C7+ in early depletion stages
CCE has stronger revaporization at low pressures because of greater (initial) mass of gas remaining in cell
Fig. 2.17—Retrograde volumes for constantcomposition and constantvolume depletion experiments. VOLUMETRIC AND PHASE BEHAVIOR OF OIL AND GAS SYSTEMS
Fig. 2.18—pT diagram of a reservoir fluid illustrating different types of depletion reservoirs. 11
ple, the gas is probably saturated at initial reservoir conditions, and an equilibrium oil could exist at some lower elevation. Discovery of a saturated reservoir fluid will usually require further field delineation to substantiate the presence of a second equilibrium phase above or below the tested interval. This may entail running a repeatformationtester tool to determine the fluidpressure gradient as a function of depth, or a new well may be required updip or downdip to the discovery well. Representative samples of saturated fluids may be difficult to obtain during a production test.42 Standing26 discusses the situation of an undersaturated gas condensate sampled during a test where bottomhole flowing pressure drops below the dewpoint pressure. The produced fluid, which is not representative of the original reservoir fluid, may have a dewpoint equal to initial reservoir pressure. This situation would incorrectly imply that the reservoir is saturated at initial conditions and that an underlying oil rim may exist. References
OGR (STB/MMscf) GOR (scf/STB)
Fig. 2.19—Spectrum of reservoir fluids in order of increasing chemical complexity from wet gas to black oil (from Cronquist38).
covery predicted by conventional material balances that are based on the standard blackoil PVT formulation. Fluid samples obtained from a new field discovery may be instrumental in defining the existence of an overlying gas cap or an underlying oil rim. Referring to Fig. 2.20, if the initial reservoir pressure equals the measured bubblepoint pressure of a bottomhole or recombined sample, the oil is probably saturated at initial reservoir conditions. This implies that an equilibrium gas cap could exist at some higher elevation. Likewise, if the initial reservoir pressure is the same as the measured dewpoint pressure of a reservoir gas sam
}
Retrograde dewpoint = Resevoir pressure Bubblepoint
Fig. 2.20—pT phase diagram of a gascap fluid in equilibrium with an underlying saturated oil. 12
1. Katz, D.L. and Williams, B.: “Reservoir Fluids and Their Behavior,” Amer. Soc. Pet. Geologists Bulletin (February 1952) 36, No. 2, 342. 2. Smith, H.M. et al.: “Keys to the Mystery of Crude Oil,” Trans., API, Dallas (1959) 433. 3. Nelson, W.L.: Petroleum Refinery Engineering, fourth edition, McGrawHill Book Co. Inc., New York City (1958). 4. Nelson, W.L.: “Does Crude Boil at 1400°F?,” Oil & Gas J. (1968) 125. 5. Katz, D.L. et al.: Handbook of Natural Gas Engineering, McGrawHill Book Co. Inc., New York City (1959). 6. Muskat, M.: “Distribution of NonReacting Fluids in the Gravitational Field,” Physical Review (1930) 35, 1384. 7. Sage, B.H. and Lacey, W.N.: “Gravitational Concentration Gradients in Static Columns of Hydrocarbon Fluids,” Trans., AIME (1939) 132, 120. 8. Schulte, A.M.: “Compositional Variations Within a Hydrocarbon Column Due to Gravity,” paper SPE 9235 presented at the 1980 SPE Annual Technical Conference and Exhibition, Dallas, 21–24 September. 9. Neumann, H.J., PaczynskaLahme, B., and Severin, D.: Composition and Properties of Petroleum, Halsted Press, New York City (1981). 10. Thompson, C.J., Ward, C.C., and Ball, J.S.: “Characteristics of World’s Crude Oils and Results of API Research Project 60,” Report B7, Energy R&D Admin. (ERDA) (1976). 11. Rossini, F.D.: “The Chemical Constitution of the Gasoline Fraction of Petroleum—API Research Project 6,” API, Dallas (1935). 12. Rossini, F.D. and Mair, B.J.: “The Work of the API Research Project on the Composition of Petroleum,” Proc., Fifth World Pet. Cong. (1954) 223. 13. Miller, A.E.: “Review of American Petroleum Institute Research Projects on Composition and Properties of Petroleum,” Proc., Fourth World Pet. Cong. (1955) 27. 14. Speight, J.C. and Moschopedis, S.E.: “On the Molecular Nature of Petroleum Asphaltenes,” Trans., Advances in Chemistry, American Chemical Soc. (1981) 195, 1. 15. Speight, J.G., Long, R.B., and Trowbridge, T.D.: “Factors Influencing the Separation of Asphaltenes from Heavy Petroleum Feedstocks,” Fuel (1984) 63, 616. 16. Speight, J.G. and Pancirov, R.J.: “Structural Types in Petroleum Asphaltenes as Deduced from Pyrolysis/Gas Chromatography/Mass Spectrometry,” Liquid Fuels Technology (1984) 2, No. 3, 287. 17. Such, C., Brulé, B., and BalujaSantos, C.: “Characterization of a Road Asphalt by Chromatographic Techniques (GPC and HPLC),” J. Liquid Chrom. (1979) 2, No. 3, 437. 18. Fetzer, J.C. et al.: “Characterization of Carbonaceous Materials Using Extraction with Supercritical Pentane,” report, Contract No. DOE/ ER/0085429, U.S. DOE (1980). 19. Helm, R.V. and Petersen, J.C.: “Compositional Studies of an Asphalt and Its Molecular Distillation Fractions by Nuclear Magnetic Resonance and Infrared Spectrometry,” Analytical Chemistry (1968) 40, No. 7, 1100. 20. Dooley, J.E. et al.: “Analyzing Heavy Ends of Crude, Swan Hills,” Hydro. Proc. (April 1974) 53, 93. 21. Dooley, J.E. et al.: “Analyzing Heavy Ends of Crude, Comparisons,” Hydro. Proc. (Nov. 1974) 53, 187. 22. Katz, D.L. and Beu, K.L.: “Nature of Asphaltic Substances,” Ind. & Eng. Chem. (February 1945) 37, 195. 23. Monger, T.G. and Trujillo, D.E.: “Organic Deposition During CO2 and RichGas Flooding,” SPERE (February 1991) 17. 24. Bossler, R.B. and Crawford, P.B.: “MisciblePhase Floods May Precipitate Asphalt,” Oil & Gas J. (23 February 1959) 57, 137. PHASE BEHAVIOR
25. Gibbs, J.W.: The Collected Works of J. Willard Gibbs, Yale U. Press, New Haven, Connecticut (1948). 26. Standing, M.B.: Volumetric and Phase Behavior of Oil Field Hydrocarbon Systems, SPE, Richardson, Texas (1977). 27. Katz, D.L. and Kurata, F.: “Retrograde Condensation,” Ind. & Eng. Chem. (June 1940) 32, No. 6, 817. 28. Katz, D.L. and Singleterry, C.C.: “Significance of the Critical Phenomena in Oil and Gas Production,” Trans., AIME (1939) 132, 103. 29. Edmister, W.C. and Lee, B.I.: Applied Hydrocarbon Thermodynamics, second edition, Gulf Publishing Co., Houston (1984) I. 30. Kay, W.B.: “The EthaneHeptane System,” Ind. & Eng. Chem. (1938) 30, 459. 31. Brown, G.G. et al.: Natural Gasoline and the Volatile Hydrocarbons, NGAA, Tulsa, Oklahoma (1948) 24–32. 32. Katz, D.L., Vink, D.J., and David, R.A.: “Phase Diagram of a Mixture of Natural Gas and Natural Gasoline Near the Critical Conditions,” Trans., AIME (1939) 136, 165. 33. Kurata, F. and Katz, D.L.: “Critical Properties of Volatile Hydrocarbon Mixtures,” Trans., AIChE (1942) 38, 995. 34. Kuenen, J.P.: “On Retrograde Condensation and the Critical Phenomena of Two Substances,” Commun. Phys. Lab. U. Leiden (1892) 4, 7. 35. Eilerts, C.K.: Phase Relations of Gas Condensate Fluids, Monograph 10, USBM, American Gas Assn., New York City (1957) I and II. 36. Eilerts, C.K.: “Gas Condensate Reservoir Engineering, 1. The Reserve Fluid, Its Composition and Phase Behavior,” Oil & Gas J. (1 February 1947) 63.
VOLUMETRIC AND PHASE BEHAVIOR OF OIL AND GAS SYSTEMS
37. Eilerts, C.K. et al.: “Phase Relations of a GasCondensate Fluid at Low Tempertures, Including the Critical State,” Pet. Eng. (February 1948) 19, 154. 38. Cronquist, C.: “Dimensionless PVT Behavior of Gulf Coast Reservoir Oils,” JPT (May 1973) 538. 39. Moses, P.L.: “Engineering Applications of Phase Behavior of Crude Oil and Condensate Systems,” JPT (July 1986) 715. 40. Lohrenz, J., Clark, G.C., and Francis, R.J.: “A Compositional Material Balance for Combination Drive Reservoirs with Gas and Water Injection,” JPT (November 1963) 1233; Trans., AIME, 228. 41. Reudelhuber, F.O. and Hinds, R.F.: “Compositional MaterialBalance Method for Prediction of Recovery From Volatile Oil Depletion Drive Reservoirs,” JPT (1957) 19; Trans., AIME, 210. 42. Fevang, Ø. and Whitson, C.H.: “Accurate InSitu Compositions in Petroleum Reservoirs,” paper SPE 28829 presented at the 1994 European Petroleum Conference, London, 25–27 October.
SI Metric Conversion Factors °API 141.5/(131.5)°API) +g/cm3 bbl 1.589 873 E*01 +m3 ft3 2.831 685 E*02 +m3 °F (°F*32)/1.8 +°C gal 3.785 412 E*03 +m3 lbm 4.535 924 E*01 +kg psi 6.894 757 E)00 +kPa
13
Chapter 3
Gas and Oil Properties and Correlations 3.1 Introduction Chap. 3 covers the properties of oil and gas systems, their nomenclature and units, and correlations used for their prediction. Sec. 3.2 covers the fundamental engineering quantities used to describe phase behavior, including molecular quantities, critical and reduced properties, component fractions, mixing rules, volumetric properties, transport properties, and interfacial tension (IFT). Sec. 3.3 discusses the properties of gas mixtures, including correlations for Z factor, pseudocritical properties and wellstream gravity, gas viscosity, dewpoint pressure, and total formation volume factor (FVF). Sec. 3.4 covers oil properties, including correlations for bubblepoint pressure, compressibility, FVF, density, and viscosity. Sec. 3.5 gives correlations for IFT and diffusion coefficients. Sec. 3.6 reviews the estimation of K values for lowpressure applications, such as surface separator design, and convergencepressure methods used for reservoir calculations. 3.2 Review of Properties, Nomenclature, and Units 3.2.1 Molecular Quantities. All matter is composed of elements that cannot be decomposed by ordinary chemical reactions. Carbon (C), hydrogen (H), sulfur (S), nitrogen (N), and oxygen (O) are examples of the elements found in naturally occurring petroleum systems. The physical unit of the element is the atom. Two or more elements may combine to form a chemical compound. Carbon dioxide (CO2), methane (CH4), and hydrogen sulfide (H2S) are examples of compounds found in naturally occurring petroleum systems. When two atoms of the same element combine, they form diatomic compounds, such as nitrogen (N2) and oxygen (O2). The physical unit of the compound is the molecule. Mass is the basic quantity for measuring the amount of a substance. Because chemical compounds always combine in a definite proportion (i.e., as a simple ratio of whole numbers), the mass of the atoms of different elements can be conveniently compared by relating them with a standard. The current standard is carbon12, where the element carbon has been assigned a relative atomic mass of 12.011. The relative atomic mass of all other elements have been determined relative to the carbon12 standard. The smallest element is hydrogen, which has a relative atomic mass of 1.0079. The relative atomic mass of one element contains the same number of atoms as the relative atomic mass of any other element. This is true regardless of the units used to measure mass. According to the SI standard, the definition of the mole reads “the mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilograms of car18
bon12.” The SI symbol for mole is mol, which is numerically identical to the traditional g mol. The SPE SI standard1 uses kmol as the unit for a mole where kmol designates “an amount of substance which contains as many kilograms (groups of molecules) as there are atoms in 12.0 kg (incorrectly written as 0.012 kg in the original SPE publication) of carbon12 multiplied by the relative molecular mass of the substance involved.” A practical way to interpret kmol is “kg mol” where kmol is numerically equivalent to 1,000 g mol (i.e., 1,000 mol). Otherwise, the following conversions apply. 1 kmol
+ 1,000 mol + 1,000 g mol + 2.2046 lbm mol
1 lbm mol + 0.45359 kmol + 453.59 mol + 453.59 g mol 1 mol
+ 1 g mol + 0.001 kmol + 0.0022046 lbm mol
The term molecular weight has been replaced in the SI system by molar mass. Molar mass, M, is defined as the mass per mole (M+m/n) of a given substance where the unit mole must be consistent with the unit of mass. The numerical value of molecular weight is independent of the units used for mass and moles, as long as the units are consistent. For example, the molar mass of methane is 16.04, which for various units can be written M+ + + +
16.04 kg/kmol 16.04 lbm/lbm mol 16.04 g/g mol 16.04 g/mol
3.2.2 Critical and Reduced Properties. Most equations of state (EOS’s) do not use pressure and temperature explicitly to define the state of a system, but instead they generalize according to correspondingstates theory by use of two or more reduced properties, which are dimensionless.2 T r + TńT c , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.1a) PHASE BEHAVIOR
the following relation for volume fractions x vi, based on component densities at standard conditions ò i or specific gravities g i. x vi +
m ińò i
+
j
j
j+1
x i M ińg i
ȍ x M ńg j
j
j
x i M ińò i
ȍ x M ńò N
j
j
j
j+1
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.4)
N
j
+
N
j
j+1
+
n i M i ńò i
ȍ m ńò ȍ n M ńò N
j
j+1
Fig. 3.1—Reservoir densities as functions of pressure and temperature.
where the sum of x vi is unity. Having defined component fractions, we can introduce some common mixing rules for averaging the properties of mixtures. Kay’s5 mixing rule, the simplest and most widely used, is given by a molefraction average,
ȍz q . N
p r + pńp c , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.1b) V r + VńV c , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.1c) and ò r + òńò c, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.1d) where ò r + 1ńV r . Absolute units must be used when calculating reduced pressure and temperature. p c, T c, V c, and ò c are the true critical properties of a pure component, or some average for a mixture. In most petroleum engineering applications, the range of reduced pressure is from 0.02 to 30 for gases and 0.03 to 40 for oils; reduced temperature ranges from t1 to 2.5 for gases and from 0.4 to 1.1 for oils. Reduced density can vary from 0 at low pressures to about 3.5 at high pressures. Average mixture, or pseudocritical, properties are calculated from simple mixing rules or mixture specific gravity.3,4 Denoting a mixture pseudocritical property by q pc, the pseudoreduced property is defined q pr + qń q pc. Pseudocritical properties are not approximations of the true critical properties, but are chosen instead so that mixture properties will be estimated correctly with correspondingstates correlations. 3.2.3 Component Fractions and Mixing Rules. Petroleum reservoir mixtures contain hundreds of welldefined and “undefined” components. These components are quantified on the basis of mole, weight, and volume fractions. For a mixture having N components, i + 1, . . . , N, the overall mole fractions are given by zi +
ni
mi ń Mi
+
ȍ n ȍ m ńM N
N
j
j
j+1
,
. . . . . . . . . . . . . . . . . . . . . . . (3.2)
j
j+1
where n+moles, m+mass, M+molecular weight, and the sum of z i is 1.0. In general, oil composition is denoted by x i and gas composition by y i. Weight or mass fractions, wi , are given by wi +
mi
+
j+1
, . . . . . . . . . . . . . . . . . . . . . . . . (3.3)
N
j
j
i i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.5)
i+1
This mixing rule is usually adequate for molecular weight, pseudocritical temperature, and acentric factor.6 We can write a generalized linear mixing rule as
ȍf q N
i i
q+
i+1 N
ȍf
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.6) i
i+1
where f i is usually one of the following weighting factors: f i + z i, mole fraction (Kay’s rule); f i + w i , weight fraction; or f i + x vi, volume fraction. Depending on the quantity being averaged, other mixing rules and definitions of f i may be appropriate.7,8 For example, the mixing rules used for constants in an EOS (Chap. 4) can be chosen on the basis of statistical thermodynamics. 3.2.4 Volumetric Properties. Density, ò, is defined as the ratio of mass to volume, ò + mńV, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.7) expressed in such units as lbm/ft3, kg/m3, and g/cm3. Fig. 3.1 shows the magnitudes of density for reservoir mixtures. Molar density, ò M , gives the volume per mole: ò M + nńV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.8) Specific volume, v^, is defined as the ratio of volume to mass and is equal to the reciprocal of density. v^ + Vńm + 1ńò. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.9) Molar volume, v, defines the ratio of volume per mole, v + Vńn + Mńò + 1ńò M , . . . . . . . . . . . . . . . . . . . . . (3.10) and is typically used in cubic EOS’s. Molar density, ò M , is given by ò M + 1ńv + òńM, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.11)
n i Mi
ȍm ȍn M N
q+
j
j+1
where the sum of w i + 1.0. Although the composition of a mixture is usually expressed in terms of mole fraction, the measurement of composition is usually based on mass, which is converted to mole fraction with component molecular weights. For oil mixtures at standard conditions (14.7 psia and 60°F), the total volume can be approximated by the sum of the volumes of individual components, assuming idealsolution mixing. This results in GAS AND OIL PROPERTIES AND CORRELATIONS
and is used in the formulas of some EOS’s. According to the SI standard, relative density replaces specific gravity as the term used to define the ratio of the density of a mixture to the density of a reference material. The conditions of pressure and temperature must be specified for both materials, and the densities of both materials are generally measured at standard conditions (standard conditions are usually 14.7 psia and 60°F). g+
ò ǒ p sc, T scǓ , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.12a) ò ref ǒ p sc, T scǓ 19
Fig. 3.2—Reservoir compressibilities as functions of pressure. Fig. 3.3—Reservoir FVF’s as functions of pressure.
ǒò oǓ sc go + , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.12b) ǒò wǓ sc
and g g +
ǒò gǓ sc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.12c) ǒò airǓ sc
Air is used as the reference material for gases, and water is used as the reference material for liquids. Specific gravity is dimensionless, although it is customary and useful to specify the material used as a reference (air+1 or water+1). In older references, liquid specific gravities are sometimes followed by the temperatures of both the liquid and water, respectively; for example, g o + 0.823 60ń60 , where the temperature units here are understood to be in degrees Fahrenheit. The oil gravity, g API, in degrees API is used to classify crude oils on the basis of the following relation,
B+
V mixture ǒ p, T Ǔ . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.16) V product ǒ p sc, T scǓ
The units of B are bbl/STB for oil and water, and ft3/scf or bbl/Mscf for gas. The surface product phase may consist of all or only part of the original mixture. Primarily, four volume factors are used in petroleum engineering. They are oil FVF, B o; water FVF, B w; gas FVF, B g; and total FVF of a gas/oil system, B t, where
g API + 141.5 g * 131.5 . . . . . . . . . . . . . . . . . . . . . . . . . . (3.13a)
Bo +
Vo V + o , . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.17a) Vo (V o) sc
141.5 , . . . . . . . . . . . . . . . . . . . . . . . . (3.13b) g API ) 131.5
Bw +
Vw V + w , . . . . . . . . . . . . . . . . . . . . . . . . . . (3.17b) Vw (V w) sc
o
and g o +
where g o +oil specific gravity (water+1). Officially, the SPE does not recognize g API in its SI standard, but because oil gravity (in degrees API) is so widely used (and understood) and because it is found in many property correlations, its continued use is justified for qualitative description of stocktank oils. Isothermal compressibility, c, of a fixed mass of material is defined as
ǒ Ǔ
c + * 1 ēV V ēp
ǒ Ǔ
+ * 1^ ēv v ēp T ^
T
ǒ Ǔ,
+ * 1v ēv ēp
. . . . . (3.14)
T
where the units are psi*1 or kPa*1. In terms of density, ò, and FVF, B, isothermal compressibility is given by
ǒ Ǔ
1 ēò c+ò ēp
T
ǒ Ǔ,
+ 1 ēB B ēp
. . . . . . . . . . . . . . . . . . . . . (3.15)
T
where B is defined in the next section. Fig. 3.2 shows the variation in compressibility with pressure for typical reservoir mixtures. A discontinuity in oil compressibility occurs at the bubblepoint because gas comes out of solution. When two or more phases are present, a total compressibility is useful.8,9 3.2.5 BlackOil Pressure/Volume/Temperature (PVT) Properties. The FVF, B; solution gas/oil ratio, R s ; and solution oil/gas ratio, r s, are volumetric ratios used to simplify engineering calculations. Specifically, they allow for the introduction of surface volumes of gas, oil, and water into materialbalance equations. These are not standard engineering quantities, and they must be defined precisely. These properties constitute the blackoil or “beta” PVT formula used in petroleum engineering. Chap. 7 gives a detailed discussion of blackoil properties. 20
FVF, or simply volume factor, is used to convert a volume at elevated pressure and temperature to surface volume, and vice versa. More specifically, FVF is defined as the volume of a mixture at specified pressure and temperature divided by the volume of a product phase measured at standard conditions,
Bg +
Vg
ǒV gǓ
and B t +
+ sc
Vg , . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.17c) Vg
Vo ) Vg Vo ) Vg Vt + + ; . . . . . . . . . . . (3.17d) Vo (V o) sc (V o) sc
and the total FVF of a gas/water system is B tw +
Vg ) Vw Vt + . . . . . . . . . . . . . . . . . . . . . . . (3.17e) Vw (V w) sc
In Eq. 3.17, V o +oil volume at p and T ; V g +gas volume at p and T ; V w +water/brine volume at p and T ; V o +(V o) sc +stocktankoil volume at standard conditions; V w + (V w) sc +stocktankwater volume at standard conditions; and V g + ǒV gǓ sc+surfacegas volume at standard conditions. Because gas FVF is inversely proportional to pressure, a reciprocal gas volume factor, b g (equal to 1/ B g), is sometimes used, where the units of b g may be scf/ft3 or Mscf/bbl. Fig. 3.3 shows FVF’s of typical reservoir systems. Inverse oil FVF, b o (equal to 1/ B o) is also used in reservoir simulation. Wet gas and gascondensate reservoir fluids produce liquids at the surface, and for these gases the surface product (separator gas) consists of only part of the original reservoir gas mixture. Two gas FVF’s are used for these systems: the “dry” FVF, B gd, and the “wet” FVF, B gw (or just B g). B gd gives the ratio of reservoir gas volume to the actual surface separator gas. B gw gives the ratio of reservoir gas volume to a hypothetical “wet” surfacegas volume (the actual separatorgas volume plus the stocktank condensate converted to an equivalent surfacegas volume). Chap. 7 describes when B gd and B gw are used. The standard definition of B g + (p scńT sc)(ZTńp) (see Eq. 3.38) represents the wetgas FVF. PHASE BEHAVIOR
Fig. 3.4—Solution gas/oil ratios for brine, Rsw , and reservoir oils, Rs , and inverse solution oil/gas ratio for reservoir gases, 1/rs , as functions of pressure.
When a reservoir mixture produces both surface gas and oil, the GOR, R go, defines the ratio of standard gas volume to a reference oil volume (stocktank or separatoroil volume),
ǒV gǓ
V + g . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.18a) R go + Vo (V o) sc and R sp +
sc
ǒV gǓ
sc
(V o) sp
+
Vg (V o) sp
. . . . . . . . . . . . . . . . . . . . . . (3.18b)
in units of scf/STB and scf/bbl, respectively. The separator conditions should be reported when separator GOR is used. Solution gas/oil ratio, R s , is the volume of gas (at standard conditions) liberated from a singlephase oil at elevated pressure and temperature divided by the resulting stocktankoil volume, with units scf/STB. R s is constant at pressures greater than the bubblepoint and decreases as gas is liberated at pressures below the bubblepoint. The producing GOR, R p, defines the instantaneous ratio of the total surfacegas volume produced divided by the total stocktankoil volume. At pressures greater than bubblepoint, R p is constant and equal to R s at bubblepoint. At pressures less than the bubblepoint, R p may be equal to, less than, or greater than the R s of the flowing reservoir oil. Typically, R p will increase 10 to 20 times the initial R s because of increasing gas mobility and decreasing oil mobility during pressure depletion. The surface volume ratio for gas condensates is usually expressed as an oil/gas ratio (OGR), r og. r og +
(V o) sc
ǒV gǓ
sc
+
Vo + 1 . . . . . . . . . . . . . . . . . . . . . . . (3.19) Vg R go
The unit for r og is STB/scf or, more commonly, “barrels per million” (STB/MMscf). To avoid misinterpretation, it should be clearly specified whether the OGR includes natural gas liquids (NGL’s) in addition to stocktank condensate. In most petroleum engineering calculations, NGL’s are not included in the OGR. The ratio of surface oil to surface gas produced from a singlephase reservoir gas is referred to as the solution oil/gas ratio, r s. At pressures above the dewpoint, the producing OGR, r p is constant and equal to r s at the dewpoint. At pressures below the dewpoint, r p is typically equal to or just slightly greater than r s; the contribution of flowing reservoir oil to surfaceoil production is negligible in most gascondensate reservoirs. In the definitions of R p and r p, the total producing surfacegas volume equals the surface gas from the reservoir gas plus the solution gas from the reservoir oil; likewise, the total producing surface oil equals the stocktank oil from the reservoir oil plus the condensate from the reservoir gas. Fig. 3.4 shows the behavior of R p, R s , and 1ńr s as a function of pressure. GAS AND OIL PROPERTIES AND CORRELATIONS
Fig. 3.5—Reservoir viscosities as functions of pressure.
3.2.6 Viscosity. Two types of viscosity are used in engineering calculations: dynamic viscosity, m, and kinematic viscosity, n. The definition of m for Newtonian flow (which most petroleum mixtures follow) is m+
tg c , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.20) duńdy
where t+shear stress per unit area in the shear plane parallel to the direction of flow, du/dy+velocity gradient perpendicular to the plane of shear, and g c +a units conversion from mass to force. The two viscosities are related by density, where m+n ò. Most petroleum engineering applications use dynamic viscosity, which is the property reported in commercial laboratory studies. The unit of dynamic viscosity is centipoise (cp), or in SI units, mPa@s, where 1 cp+1 mPa@s. Kinematic viscosity is usually reported in centistoke (cSt), which is obtained by dividing m in cp by ò in g/cm3; the SI unit for n is mm2/s, which is numerically equivalent to centistoke. Fig. 3.5 shows oil, gas, and water viscosities for typical reservoir systems. 3.2.7 Diffusion Coefficients. In the absence of bulk flow, components in a singlephase mixture are transported according to gradients in concentration (i.e., chemical potential). Fick’s10 law for 1D molecular diffusion in a binary system is given by u i + * D i j ǒdC ińd xǓ , . . . . . . . . . . . . . . . . . . . . . . . . . . (3.21) where u i +molar velocity of Component i; D ij +binary diffusion coefficient; and C i +molar concentration of Component i + y iò M, where y i +mole fraction; and x+distance. Eq. 3.21 clearly shows that mass transfer by molecular diffusion can be significant for three reasons: (1) large diffusion coefficients, (2) large concentration differences, and (3) short distances. A combination of moderate diffusion coefficients, concentration gradients, and distance may also result in significant diffusive flow. Molecular diffusion is particularly important in naturally fractured reservoirs11,12 because of relatively short distances (e.g., small matrix block sizes). Lowpressure binary diffusion coefficients for gases, D oij , are independent of composition and can be calculated accurately from fundamental gas theory (Chapman and Enskog6), which are basically the same relations used to estimate lowpressure gas viscosity. No wellaccepted method is available to correct D oij for mixtures at high pressure, but two types of correspondingstates correlations have been proposed: D ij + D oij f(T r, p r) and D ij + D oij f(ò r). At low pressures, diffusion coefficients are several orders of magnitude smaller in liquids than in gases. At reservoir conditions, the difference between gas and liquid diffusion coefficients may be less than one order of magnitude. 3.2.8 IFT. Interfacial forces act between equilibrium gas, oil, and water phases coexisting in the pores of a reservoir rock. These forces 21
are generally quantified in terms of IFT, s; units of s are dynes/cm (or equivalently, mN/m). The magnitude of IFT varies from [50 dynes/cm for crudeoil/gas systems at standard conditions to t0.1 dyne/cm for highpressure gas/oil mixtures. Gas/oil capillary pressure, P c, is usually considered proportional to IFT according to the YoungLaplace equation P c + 2sńr, where r is an average pore radius.1315 Recovery mechanisms that are influenced by capillary pressure (e.g., gas injection in naturally fractured reservoirs) will necessarily be sensitive to IFT. 3.3 Gas Mixtures This section gives correlations for PVT properties of natural gases, including the following. 1. Review of gas volumetric properties. 2. Zfactor correlations. 3. Gas pseudocritical properties. 4. Wellstream gravity of wet gases and gas condensates. 5. Gas viscosity. 6. Dewpoint pressure. 7. Total volume factor. 3.3.1 Review of Gas Volumetric Properties. The properties of gas mixtures are well understood and have been accurately correlated for many years with graphical charts and EOS’s based on extensive experimental data.1619 The behavior of gases at low pressures was originally quantified on the basis of experimental work by Charles and Boyle, which resulted in the idealgas law,3 pV + nRT, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.22) where R is the universal gas constant given in Appendix A for various units (Table A2). In customary units, R + 10.73146
psia ft 3 , . . . . . . . . . . . . . . . . . . (3.23) ° R lbm mol
while for other units, R can be calculated from the relation
ǒ ǓǒT°R Ǔǒ Ǔǒmlbm Ǔ .
p R + 10.73146 unit psia
unit
V unit ft 3
unit
. . . . . . . . (3.24)
For example, the gas constant for SPEpreferred SI units is given by
ǒ
kPa 6.894757 psia
R + 10.73146
ǒ
3 0.02831685 m3 ft
Ǔ ǒ
Ǔ
ǒ1.8 °R Ǔ K
2.204623 lbm kg
Ǔ
kPa @ m 3 + 8.3143 . . . . . . . . . . . . . . . . . . . . . . . . . (3.25) K @ kmol The gas constant can also be expressed in terms of energy units (e.g., R+8.3143 J/mol@K); note that J+N@m+(N/m2) m3+Pa@m3. In this case, the conversion from one unit system to another is given by R + 8.3143
ǒEJ ǓǒTK Ǔǒmg Ǔ . unit
unit
unit
+
10.73146(60 ) 459.67) 14.7
+ 379.4 scfńlbm mol + 23.69 std m 3ńkmol .
. . . . . . . . . . . . . . . . . . . (3.27)
Second, the specific gravity of a gas directly reflects the gas molecular weight at standard conditions, gg +
ǒò gǓ Mg Mg sc + + ǒò airǓ M air 28.97 sc
and M g + 28.97 g g .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.28)
For gas mixtures at moderate to high pressure or at low temperature the idealgas law does not hold because the volume of the constituent molecules and their intermolecular forces strongly affect the volumetric behavior of the gas. Comparison of experimental data for real gases with the behavior predicted by the idealgas law shows significant deviations. The deviation from ideal behavior can be expressed as a factor, Z, defined as the ratio of the actual volume of one mole of a realgas mixture to the volume of one mole of an ideal gas, Z+
volume of 1 mole of real gas at p and T , volume of 1 mole of ideal gas at p and T . . . . . . . . . . . . . . . . . . . . (3.29)
where Z is a dimensionless quantity. Terms used for Z include deviation factor, compressibility factor, and Z factor. Z factor is used in this monograph, as will the SPE reserve symbol Z (instead of the recommended SPE symbol z) to avoid confusion with the symbol z used for feed composition. From Eqs. 3.22 and 3.29, we can write the realgas law including the Z factor as pV + nZRT, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.30) which is the standard equation for describing the volumetric behavior of reservoir gases. Another form of the realgas law written in terms of specific volume ( v^ + 1ńò) is pv^ + ZRTńM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.31) or, in terms of molar volume (v + Mńò), pv + ZRT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.32) Z factor, defined by Eq. 3.30, Z + pVńnRT, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.33) is used for both phases in EOS applications (see Chap. 4). In this monograph we use both Z and Z g for gases and Z o for oils; Z without a subscript always implies the Z factor of a “gaslike” phase. All volumetric properties of gases can be derived from the realgas law. Gas density is given by ò g + pM gńZRT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.34)
. . . . . . . . . . . . . . . . (3.26)
An ideal gas is a hypothetical mixture with molecules that are negligible in size and have no intermolecular forces. Real gases mimic the behavior of an ideal gas at low pressures and high temperatures because the mixture volume is much larger than the volume of the molecules making up the mixture. That is, the mean free path between molecules that are moving randomly within the total volume is very large and intermolecular forces are thus very small. Most gases at low pressure follow the idealgas law. Application of the idealgas law results in two useful engineering approximations. First, the standard molar volume representing the volume occupied by one mole of gas at standard conditions is independent of the gas composition. 22
ǒV Ǔ sc ǒv gǓ + v g + ng sc + RT p sc sc
or, in terms of gas specific gravity, by p gg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.35) ZRT For wetgas and gascondensate mixtures, wellstream gravity, g w, must be used instead of g g in Eq. 3.35.3 Gas density may range from 0.05 lbm/ft3 at standard conditions to 30 lbm/ft3 for highpressure gases. Gas molar volume, v g , is given by ò g + 28.97
v g + ZRTńp,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.36)
where typical values of v g at reservoir conditions range from 1 to 1.5 ft3/lbm mol compared with 379 ft3/lbm mol for gases at standard conditions. In Eqs. 3.30 through 3.36, R+universal gas constant. PHASE BEHAVIOR
Pseudoreduced Temperature
1 January 1941
Fig. 3.6—StandingKatz4 Zfactor chart.
Gas compressibility, c g , is given by
ǒ Ǔ
ēV g cg + * 1 V g ēp
ǒ Ǔ
+ 1p * 1 ēZ Z ēp
. . . . . . . . . . . . . . . . . . . . . . . . . . . (3.37) T
For sweet natural gas (i.e., not containing H2S) at pressures less than [1,000 psia, the second term in Eq. 3.37 is negligible and c g + 1ńp is a reasonable approximation. Gas volume factor, B g, is defined as the ratio of gas volume at specified p and T to the idealgas volume at standard conditions, Bg +
ǒTp Ǔ ZTp . sc
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.38)
sc
For customary units ( psc +14.7 psia and Tsc +520°R), this is B g + 0.02827 ZT p , . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.39) with temperature in °R and pressure in psia. This definition of B g assumes that the gas volume at p and T remains as a gas at standard conditions. For wet gases and gas condensates, the surface gas will not contain all the original gas mixture because liquid is produced GAS AND OIL PROPERTIES AND CORRELATIONS
after separation. For these mixtures, the traditional definition of B g may still be useful; however, we refer to this quantity as a hypothetical wetgas volume factor, B gw, which is calculated from Eq. 3.38. Because B g is inversely proportional to pressure, the inverse volume factor, b g + 1ńB g , is commonly used. For field units, p . . . . . . . . . . . . . . . . . . . . . . (3.40a) ZT p . . . . . . . . . . . . . . . (3.40b) and b g in Mscfńbbl + 0.1985 ZT b g in scfńft 3 + 35.37
If the reservoir gas yields condensate at the surface, the drygas volume factor, B gd, is sometimes used.20 B gd +
ǒTp ǓǒZTpǓǒF1 Ǔ, sc
sc
. . . . . . . . . . . . . . . . . . . . . . . (3.41)
gg
where F gg+ratio of moles of surface gas, n g , to moles of wellstream mixture (i.e., reservoir gas, n g); see Eqs. 7.10 and 7.11 of Chap. 7. 3.3.2 ZFactor Correlations. Standing and Katz4 present a generalized Zfactor chart (Fig. 3.6), which has become an industry standard for predicting the volumetric behavior of natural gases. Many empirical equations and EOS’s have been fit to the original StandingKatz chart. For example, Hall and Yarborough21,22 present an 23
accurate representation of the StandingKatz chart using a CarnahanStarling hardsphere EOS, Z + ap prńy, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.42) where a + 0.06125 t exp[* 1.2(1 * t) 2], where t + 1ńT pr. The reduceddensity parameter, y (the product of a van der Waals covolume and density), is obtained by solving f( y) + 0 + * ap pr )
y ) y2 ) y3 * y4 (1 * y) 3
* (14.76t * 9.76t 2 ) 4.58t 3)y 2 ) (90.7t–242.2t 2 ) 42.4t 3)y 2.18)2.82 t, with
* 1 ) 4y ) df(y) + dy (1 * y) 4 4y 2
4y 3
)
. . . . . . . . . (3.43)
y4
* (29.52t * 19.52t 2) 9.16t 3)y ) (2.18 ) 2.82t)(90.7t * 242.2t 2 ) 42.4t 3) y 1.18)2.82 t .
. . . . . . . . . . . . . . . . . . . . . . . . . (3.44)
The derivative ēZ/ēp used in the definition of c g is given by
ǒēZēpǓ
+ pa T
pc
ap ńy ƪ1y * df(y)ńdy ƫ . . . . . . . . . . . . . . . . . . . . (3.45) pr
2
An initial value of y+0.001 can be used with a NewtonRaphson procedure, where convergence should be obtained in 3 to 10 iterations for Ťf( y)Ť + 1 10 *8. On the basis of Takacs’23 comparison of eight correlations representing the StandingKatz4 chart, the Hall and Yarborough21 and the Dranchuk and AbouKassem24 equations give the most accurate representation for a broad range of temperatures and pressures. Both equations are valid for 1 x T r x 3 and 0.2 x p r x 25 to 30. For many petroleum engineering applications, the Brill and Beggs25 equation gives a satisfactory representation ("1 to 2%) of the original StandingKatz Zfactor chart for 1.2 t T r t 2. Also, this equation can be solved explicitly for Z. The main limitations are that reduced temperature must be u1.2 ([80°F) and t2.0 ([340°F) and reduced pressure should be t15 ([10,000 psia). The Standing and Katz Zfactor correlation may require special treatment for wet gas and gascondensate fluids containing significant amounts of heptanesplus material and for gas mixtures with significant amounts of nonhydrocarbons. An apparent discrepancy in the StandingKatz Zfactor chart for 1.05 t T r t 1.15 has been “smoothed” in the HallYarborough21 correlations. The Hall and Yarborough (or Dranchuk and AbouKassem24) equation is recommended for most natural gases. With today’s computing capabilities, choosing simple, lessreliable equations, such as the Brill and Beggs25 equation, is normally unnecessary. The LeeKesler,26,27 AGA8,28 and DDMIX29 correlations for Z factor were developed with multiconstant EOS’s to give accurate volumetric predictions for both pure components and mixtures. They require more computation but are very accurate. These equations are particularly useful in custodytransfer calculations. They also are required for gases containing water and concentrations of nonhydrocarbons that exceed the limits of the Wichert and Aziz method.30,31 3.3.3 Gas Pseudocritical Properties. Z factor, viscosity, and other gas properties have been correlated accurately with correspondingstates principles, where the property is correlated as a function of reduced pressure and temperature. Z + fǒ p r , T rǓ and m g ńm gsc + fǒ p r , T rǓ, . . . . . . . . . . . . . . . . . . . . . . . . . (3.46) 24
Fig. 3.7—Gas pseudocritical properties as functions of specific gravity.
where p r + pńp c and T r + TńT c. Such correspondingstates relations should be valid for most pure compounds when component critical properties p c and T c are used. The same relations can be used for gas mixtures if the mixture pseudocritical properties p pc and T pc are used. Pseudocritical properties of gases can be estimated with gas composition and mixing rules or from correlations based on gas specific gravity. Sutton7 suggests the following correlations for hydrocarbon gas mixtures. T pcHC + 169.2 ) 349.5g gHC * 74.0 g 2gHC . . . . . . . . . . . (3.47a) and p pcHC + 756.8 * 131g gHC * 3.6g 2gHC . . . . . . . . . . . (3.47b) He claims that Eqs. 3.47a and 3.47b are the most reliable correlations for calculating pseudocritical properties with the StandingKatz Zfactor chart. He even claims that this method is superior to the use of composition and mixing rules. Standing3 gives two sets of correlations: one for dry hydrocarbon gases ( g gHC t 0.75), T pcHC + 168 ) 325g gHC * 12.5g 2gHC . . . . . . . . . . . . . . (3.48a) and p pcHC + 667 ) 15.0 g gHC * 37.5g 2gHC , . . . . . . . . . . (3.48b) and one for wetgas mixtures ( g gHC y 0.75), T pcHC + 187 ) 330 g gHC * 71.5g 2gHC . . . . . . . . . . . . . . (3.49a) and p pcHC + 706 * 51.7g gHC * 11.1g 2gHC . . . . . . . . . . . (3.49b) The Standing correlations are used extensively in the industry; Fig. 3.7 compares them with the Sutton correlations. The Sutton and the Standing wetgas correlations for T pc give basically the same results, whereas the three p pc correlations are quite different at g g u 0.85. Kay’s5 mixing rule is typically used when gas composition is available.
ȍy M , N
M+
i
i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.50a)
i+1
PHASE BEHAVIOR
ƪǒ
and å + 120 y CO ) y H
ǒ
2
4 ) 15 y 0.5 H S * yH 2
2S
2S
Ǔ
Ǔ,
0.9
ǒ
* y CO ) y H 2
2S
Ǔ ƫ 1.6
. . . . . . . . . . . . . . . . . . . . . . . (3.52c)
* and p * are mixture pseudocriticals based on Kay’s mixwhere T pc pc ing rule. This method was developed from extensive data from natural gases containing nonhydrocarbons, with CO2 molar concentration ranging from 0 to 55% and H2S molar concentrations ranging from 0 to 74%. If only gas gravity and nonhydrocarbon content are known, the hydrocarbon specific gravity is first calculated from
ǒ
Ǔ
g g * y N M N ) y CO M CO )y H S M H S ńM air 2 2 2 2 2 2 . g gHC + 1 * y N * y CO * y H S 2
2
2
. . . . . . . . . . . . . . . . . . . . (3.53) Hydrocarbon pseudocriticals are then calculated from Eqs. 3.47a and 3.47b, and these values are adjusted for nonhydrocarbon content on the basis of Kay’s5 mixing rule.
ǒ
p *pc + 1 * y N * y CO * y H 2
2
2S
Ǔp
pcHC
) y N p cN ) y CO p c CO ) y H S p cH 2
Fig. 3.8—Heptanesplus (pseudo)critical properties recommended for reservoir gases (from Standing,33 after Matthews et al.32).
2
ȍy T N
i
ci , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.50b)
i+1
ȍy p N
and p pc +
i
ci ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.50c)
i+1
where the pseudocritical properties of the C7+ fraction can be estimated from the Matthews et al.32 correlations (Fig. 3.8),3 Tc C
7)
+ 608 ) 364 logǒ M C ) ǒ2, 450 log M C
and p c C
7)
7)
7)
* 71.2 Ǔ
* 3, 800Ǔ log g C
+ 1, 188 * 431 logǒ M C
ƪ
7)
7)
. . . . . . (3.51a)
* 61.1 Ǔ
) 2, 319 * 852 logǒ M C * 53.7 Ǔ 7)
ƫǒg
C 7)*
0.8 Ǔ.
. . . . . . . . . . . . . . . . . . . (3.51b) Kay’s mixing rule is usually adequate for lean natural gases that contain no nonhydrocarbons. Sutton suggests that pseudocriticals calculated with Kay’s mixing rule are adequate up to g g [ 0.85, but that errors in calculated Z factors increase linearly at higher specific gravities, reaching 10 to 15% for g g u 1.5. This bias may be a result of the C7+ criticalproperty correlations used by Sutton (not Eqs. 3.51a and 3.51b). When significant quantities of CO2 and H2S nonhydrocarbons are present, Wichert and Aziz33,31 suggest corrections to arrive at pseudocritical properties that will yield reliable Z factors from the StandingKatz chart. The Wichert and Aziz corrections are given by * * å , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.52a) T pc + T pc
p pc +
p *pcǒ Tpc* * å Ǔ * T pc ) yH
2S
ǒ1 * y Ǔå
, . . . . . . . . . . . . . . . . . (3.52b)
H 2S
GAS AND OIL PROPERTIES AND CORRELATIONS
2
2
2S
. . . . . . . . . . (3.54a)
and T *pc + (1 * y N * y CO * y H S)T pcHC 2
2
2
) y N T cN ) y CO T c CO ) y H S T cH S . 2
T pc +
2
2
2
2
2
2
. . . . (3.54b)
T c* and p *c are used in the WichertAziz equations with CO2 and H2S mole fractions to obtain mixture T pc and p pc. The Sutton7 correlations (Eqs. 3.47a and 3.47b) are recommended for hydrocarbon pseudocritical properties. If composition is available, Kay’s mixing rule should be used with the Matthews et al.32 pseudocriticals for C7+. Gases containing significant amounts of CO2 and H2S nonhydrocarbons should always be corrected with the WichertAziz equations. Finally, for gascondensate fluids the wellstream specific gravity, g w (discussed in the next section), should replace g g in the equations above. 3.3.4 Wellstream Specific Gravity. Gas mixtures that produce condensate at surface conditions may exist as a singlephase gas in the reservoir and production tubing. This can be verified by determining the dewpoint pressure at the prevailing temperature. If wellstream properties are desired at conditions where the mixture is singlephase, surfacegas and oil properties must be converted to a wellstream specific gravity, g w. This gravity should be used instead of g g to estimate pseudocritical properties. Wellstream gravity r p represents the average molecular weight of the produced mixture (relative to air) and is readily calculated from the producingoil (condensate)/gas ratio, r p; average surfacegas gravity g g ; surfacecondensate gravity, g o ; and surfacecondensate molecular weight M o . gw +
g g ) 4, 580 r p g o , . . . . . . . . . . . . . . . . . . . (3.55) 1 ) 133, 000 r p ǒ gńM Ǔ o
with r p in STB/scf. Average surfacegas gravity is given by N sp
ȍR gg +
pi g gi
i+1 N sp
ȍR
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.56) pi
i+1
where R pi +GOR of Separator Stage i. Standing33 presents Eq. 3.55 graphically in Fig. 3.9. When M o is not available, Standing gives the following correlation. 25
Solution Gas/Oil Ratio, scf/STB
Oil/Gas Ratio, STB/MMscf
Fig. 3.9—Wellstream gravity relative to surface average gas gravity as a function of solution oil/gas ratio and surface gravities.
M o + 240 * 2.22 g API . . . . . . . . . . . . . . . . . . . . . . . . . (3.57) This relation should not be extrapolated outside the range 45 t g API t 60. Eilerts34 gives a relation for ( gńM) o , ǒ gńMǓ + ǒ1.892 o * ǒ4.52
10 *3Ǔ ) ǒ7.35
10 *5Ǔg API
2 10 *8Ǔg API , . . . . . . . . . . . . . . . . . . . (3.58)
which should be reliable for most condensates. When condensate molecular weight is not available, the recommended correlation for M o is the Cragoe35 correlation, Mo +
6, 084 , g API * 5.9
. . . . . . . . . . . . . . . . . . . . . . . . . . (3.59)
which gives reasonable values for all surface condensates and stocktank oils. A typical problem that often arises in the engineering of gascondensate reservoirs is that all the data required to calculate wellstream gas volumes and wellstream specific gravity are not available and must be estimated.3638 In practice, we often report only the firststageseparator GOR (relative to stocktankoil volume) and gas specific gravity, R s1 and g g1, respectively; the stocktankoil gravity, g o ; and the primaryseparator conditions, p sp1 and T sp1. To calculate g w from Eq. 3.55 we need total producing OGR, r p, which equals the inverse of R s1 plus the additional gas that will be released from the firststage separator oil, R s), rp +
1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.60) ǒR s1 ) R s)Ǔ
R s) can be estimated from several correlations.37,39 Whitson38 proposes use of a bubblepoint pressure correlation (e.g., the Standing40 correlation), R s) + A 1g g) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.61a) and A 1 +
p ƪǒ18.2 ) 1.4 Ǔ10 ǒ sp1
0.0125g API*0.00091T sp1
Ǔ
ƫ
,
with p sp1 in psia, T sp1 in °F, and R s) in scf/STB. g g) is the gas gravity of the additional solution gas released from the separator oil. The Katz41 correlation (Fig. 3.10) can be used to estimate g g), where a bestfit representation of his graphical correlation is . . . . . . . . . . . . . . . . . . . . . . . . . . (3.62)
where A 2 + 0.25 ) 0.02g API and A 3 + * (3.57 26
Solving Eqs. 3.61 and 3.62 for R s) gives R s) +
A1 A2 ǒ1 * A 1 A 3Ǔ
10 *6)g API .
.
. . . . . . . . . . . . . . . . . . . . . . . . . . (3.63)
Average surface separator gas gravity, g g, is given by gg +
g g1 R s1 ) g g) R s) . R s1 ) R s)
. . . . . . . . . . . . . . . . . . . . . . (3.64)
Although the Katz correlation is only approximate, the impact of a few percent error in g g) is not of practical consequence to the calculation of g w because R s) is usually much less than R s1 . 3.3.5 Gas Viscosity. Viscosity of reservoir gases generally ranges from 0.01 to 0.03 cp at standard and reservoir conditions, reaching up to 0.1 cp for nearcritical gas condensates. Estimation of gas viscosities at elevated pressure and temperature is typically a twostep procedure: (1) calculating mixture lowpressure viscosity m gsc at p sc and T from ChapmanEnskog theory3,6 and (2) correcting this value for the effect of pressure and temperature with a correspondingstates or densegas correlation. These correlations relate the actual viscosity m g at p and T to lowpressure viscosity by use of the ratio m gńm gsc or difference ( m g * m gsc) as a function of pseudoreduced properties p pr and T pr or as a function of pseudoreduced density ò pr. Gas viscosities are rarely measured because most laboratories do not have the required equipment; thus, the prediction of gas viscosity is particularly important. Gas viscosity of reservoir systems is often estimated from the graphical correlation m gńm gsc + f(T r, p r) proposed by Carr et al.42 (Fig. 3.11). Dempsey43 gives a polynomial approximation of the Carr et al. correlation. With these correlations, gas viscosities can be estimated with an accuracy of about "3% for most applications. The Dempsey correlation is valid in the range 1.2 x T r x 3 and 1 x p r x 20. The LeeGonzalez gas viscosity correlation (used by most PVT laboratories when reporting gas viscosities) is given by44
1.205
. . . . . . . . . . . . . . . . . . . (3.61b)
g g) + A 2 ) A 3 R s) ,
Fig. 3.10—Correlation for separatoroil dissolved gas gravity as a function of stocktankoil gravity and separatoroil GOR (from Ref. 41).
mg + A1 where A 1 +
10 *4 expǒA 2 ò g 3Ǔ , A
. . . . . . . . . . . . . . . . . . (3.65a)
ǒ9.379 ) 0.01607M gǓT 1.5 209.2 ) 19.26M g ) T
,
A 2 + 3.448 ) ǒ986.4ńTǓ ) 0.01009M g , and A 3 + 2.447 * 0.2224A 2 , . . . . . . . . . . . . . . . . . . . . (3.65b) with m g in cp, ò g in g/cm3, and T in °R. McCain19 indicates the accuracy of this correlation is 2 to 4% for gg t1.0, with errors up to 20% for rich gas condensates with g g u 1.5. PHASE BEHAVIOR
Gas Gravity (air+1)
N2, mol%
CO2, mol%
H2S, mol%
g
o
Molecular Weight
Pseudoreduced Temperature, Tr Fig. 3.11—Carr et al.42 gasviscosity correlation.
Lucas45 proposes the following gas viscosity correlation, which is valid in the range 1 t T r t 40 and 0 t p r t 100 (Fig. 3.12)6: m gńm gsc + 1 )
A 2 p pr5 ) ǒ1 ) A 3 p pr4Ǔ *1
(1.245
where A 1 +
A 1 p 1.3088 pr
A
10 *3)
A
,
expǒ T pr
. . . . . . . (3.66a)
5.1726T *0.3286 pr
Ǔ 0.4489 expǒ3.0578T *37.7332 pr , T pr
A4 +
Ǔ 1.7368 expǒ2.2310T *7.6351 pr , T pr
Ǔ
,
where m gsc c + ƪ0.807T pr0.618 * 0.357 expǒ* 0.449T prǓ ) 0.340 expǒ* 4.058T prǓ ) 0.018ƫ ,
,
N
i
and A 5 + 0.9425 expǒ* 0.1853T pr0.4489Ǔ , . . . . . . . . . . . (3.66b)
GAS AND OIL PROPERTIES AND CORRELATIONS
ȍy Z and p pc + RT pc
A 2 + A 1ǒ1.6553T pr * 1.2723Ǔ , A3 +
ǒ Ǔ
1ń6
T pc c + 9, 490 M 3p 4pc
ci
i+1 N
ȍy v
,
. . . . . . . . . . . . . . . . . . . . . . . . (3.67)
i ci
i+1
with c in cp*1, T and T c in °R, and p c in psia. Special corrections should be applied to the Lucas correlation when polar compounds, such as H2S and water, are present in a gas mixture. The effect of H2S is always t1% and can be neglected, and appropriate corrections can be made to treat water if necessary. Given its wide range of applicability, the Lucas method is recommended for general use. When compositions are not available, correlations for pseudocritical properties in terms of specific gravity can be used instead. Standing2 gives equations for m gsc in terms of g g, temperature, and nonhydrocarbon content, m gsc + ǒ m gscǓ uncorrected ) Dm N ) Dm CO ) Dm H S , 2
2
2
. . . . . . . . . . . . . . . . . . . . (3.68a) 27
) A 7ǒ z C pr +p/pc ; T/Tc ; h in mp 4 c+0.176(Tc /M3/pc )1/6; Tc in K, pc in bar
ȳ C 7) Ǔ 3) A ȱ 8 ȧ ȧ ȲǒgC7) ) 0.0001 Ǔȴ M
7)
ȱ ) Aȧ Ȳǒ g
MC
7)
ȳ )A ȱ ȧ ȧǒg ) 0.0001 Ǔȴ Ȳ 2
MC
9
7)
3
10
C 7)
ȳ ȧ ) 0.0001 Ǔȴ
MC C 7)
7)
) A 11 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.69) where A1+*2.0623054, A2+6.6259728, A3+*4.4670559 10*3, A4+1.0448346 10*4, A5+3.2673714 10*2, A6+ *3.6453277 10*3, A7+7.4299951 10*5, A8+*1.1381195 10*1, A9+6.2476497 10*4, A10+*1.0716866 10*6, and A11+1.0746622 101. The range of properties used to develop this correlation includes dewpoints from 1,000 to 10,000 psia, temperatures from 40 to 320°F, and a wide range of reservoir compositions. The correlation usually can be expected to predict dewpoints with an accuracy of "10% for condensates that do not contain large amounts of nonhydrocarbons. This is acceptable in light of the fact that experimental dewpoint pressures are probably determined with an accuracy of only "5%. The correlation is generally used only for preliminary reservoir studies conducted before an experimental dewpoint is available. Organick and Golding50 and Kurata and Katz51 present graphical correlations for dewpoint pressure.
pr +0
pr +0
Fig. 3.12—Lucas45 correspondingstates generalized viscosity correlation (Ref. 6); h+dynamic viscosity and mp+micropoise+10*6 poise+10*4 cp.
3.3.7 Total FVF. Total FVF,3,17,46 B t, is defined as the volume of a twophase, gasoil mixture (or sometimes a singlephase mixture) at elevated pressure and temperature divided by the stocktankoil volume resulting when the twophase mixture is brought to surface conditions, Bt +
10 *3Ǔ ) ƪǒ1.709
where ǒ m gscǓ uncorrected + ǒ8.188
10 *6Ǔg gƫT * ǒ6.15
* ǒ2.062
Dm N + y N ƪǒ8.48 2
Dm CO + y CO ƪǒ9.08 2
10 *3Ǔ log g g , 10 *3Ǔƫ,
10 *3Ǔ log g g ) ǒ9.59
2
10 *3Ǔƫ ,
10 *3Ǔ log g g ) ǒ6.24
2
and Dm H2S + y H 2Sƪǒ8.49
10 *5Ǔ
10 Ǔƫ.
10 Ǔ log g g ) ǒ3.73 *3
*3
. . . . . . . . . . . . . . . . . . . (3.68b) Reid et al.6 review other gas viscosity correlations with accuracy similar to that of the Lucas correlation. 3.3.6 Dewpoint Pressure. The prediction of retrograde dewpoint pressure is not widely practiced. It is generally recognized that the complexity of retrograde phase behavior necessitates experimental determination of the dewpoint condition. Sage and Olds’46 data are perhaps the most extensive tabular correlation of dewpoint pressures. Eilerts et al.47,48 also present dewpoint pressures for several lightcondensate systems. Nemeth and Kennedy49 have proposed a dewpoint correlation based on composition and C7+ properties.
ƪ
ln p d+A 1 z C ) z CO ) z H S )z C )2(z C )z C ) ) z C 2
2
2
ƫ
) 0.4z C )0.2z N ) A 2 g C 1
) A 4T )ǒA 5z C 28
2
7)
MC
7)
3
6
ȱ ) Aȧ Ȳǒz
7)
4
ȳ ȧ ) 0.002 Ǔȴ zC
3
C 7)
5
Ǔ) A6ǒzC7)MC7)Ǔ2
1
Vo ) Vg Vo ) Vg + . . . . . . . . . . . . . . . . . . . . . . (3.70) Vo (V o) sc
B t is used for calculating the oil in place for gascondensate reservoirs, where V o + 0 in Eq. 3.70. Assuming 1 res bbl of hydrocarbon PV, the initial condensate in place is given by N + 1ńB t (in STB) and the initial “dry” separator gas in place is G + Nńr p , where r p +initial producing (solution) OGR. For gascondensate systems, Sage and Olds46 give a tabulated correlation for B t. R pT B t + p Z *,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.71)
where R p +producing GOR in scf/STB, B t is in bbl/STB, T is in °R, and p is in psia. Z* varies with pressure and temperature, where the tabulated correlation for Z* is well represented by p p 1.5 , . . . . . . . . (3.72) Z * + A 0 ) A 1p ) A 2p 1.5 ) A 3 ) A 4 T T * * 3 6 where A0+5.050 10 , A1+*2.740 10 , A2+3.331 10*8, A3+2.198 10*3, and A4+*2.675 10*5 with p in psia and T in °R. Although the Sage and Olds data only cover the range 600tpt3,000 psia and 100tTt250°F, Eq. 3.72 gives acceptable results up to 10,000 psia and 350°F (when gas volume is much larger than oil volume). When reservoir hydrocarbon volume consists only of gas, the following relations apply for total FVF. B t + B gd R p + B gw ǒR p ) C og Ǔ ,
. . . . . . . . . . . . . . . . . (3.73a)
C og + 133, 000 ǒg ońM oǓ , . . . . . . . . . . . . . . . . . . . . . . (3.73b) M o [ 6, 084ńǒ g API * 5.9Ǔ ,
. . . . . . . . . . . . . . . . . . . . . (3.73c)
and g API + 141.5ń(131.5 ) g o) ,
. . . . . . . . . . . . . . . . . (3.73d)
B gw +wetgas FVF in ft3/scf where B gd +dry gas FVF in (given by Eq. 3.38), C og +gas equivalent conversion factor in scf/ STB (see Chap. 7), and R p +producing GOR in scf/STB. ft3/scf,
PHASE BEHAVIOR
PR EOS Glasø Uncorrected Glasø Corrected
C7+ Watson Characterization Factor, KwC7+
C7+ Watson Characterization Factor, KwC7+
Fig. 3.13—Effect of paraffinicity, Kw , on bubblepoint pressure.
Standing3 gives a graphical correlation for B t using a correlation parameter A defined as 0.5
A + R p T 0.3 g ao , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.74) gg where a + 2.9 10 *0.00027 Rp. Standing’s correlation is valid for both oil and gascondensate systems and can be represented with log B t + * 5.262 *
47.4 , . . . . . . . . . (3.75a) * 12.22 ) log A *
matically.8 An accurate method is needed to correlate the bubblepoint pressure, temperature, and solution gas/oil ratio. Oil properties can be grouped into two categories: saturated and undersaturated properties. Saturated properties apply at pressures at or below the bubblepoint, and undersaturated properties apply at pressures greater than the bubblepoint. For oils with initial GOR’s less than [500 scf/STB, assuming linear variation of undersaturatedoil properties with pressure is usually acceptable.
. . (3.75b)
3.4.1 Bubblepoint Pressure. The correlation of bubblepoint pressure has received more attention than any other oilproperty correlation. Standing3,17,40 developed the first accurate bubblepoint correlation, which was based on California crude oils.
and A is given by Eq. 3.74. On the basis of data from North Sea oils, Glasø52 gives a correlation for B t using the Standing correlation parameter A (Eq. 3.74):
p b + 18.2ǒ A * 1.4 Ǔ, . . . . . . . . . . . . . . . . . . . . . . . . . . (3.78) where A + ǒR ńg Ǔ 0.83 10 ǒ0.00091T*0.0125g APIǓ, with R in scf/STB, T
ǒ
where log A * + log A * 10.1 *
log B t + ǒ8.0135
96.8 6.604 ) log p
Ǔ
10 *2Ǔ ) 0.47257 log A *
) 0.17351ǒlog A * Ǔ 2 , . . . . . . . . . . . . . . . . . . . . . (3.76) where A*+A p*1.1089. Either the Standing or the Glasø correlations for B t can be used with approximately the same accuracy. However, neither correlation is consistent with the limiting conditions B t + B o for V g + 0 . . . . . . . . . . . . . . . . . . . . . . . . . . (3.77a)
s
s
g
in °F, and p b in psia. Lasater53 used a somewhat different approach to correlate bubblepoint pressure, where mole fraction y g of solution gas in the reservoir oil is used as the main correlating parameter17: p b + A gT , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.79) g
with T in °R and p b in psia. The function A( y g ) is given graphically by Lasater, and his correlation can be accurately described by A + 0.83918
; y g x 0.6 . . . . . . . . (3.80a) 10 1.17664yg y 0.57246 g
and B t + B gd R p for V o + 0. . . . . . . . . . . . . . . . . . . . (3.77b)
and A + 0.83918
B t correlations evaluated at a bubblepoint usually will underpredict the actual B ob by [0.2.
133, 000 ǒ gńMǓ o where y g + 1 ) Rs
3.4 Oil Mixtures This section gives correlations for PVT properties of reservoir oils, including bubblepoint pressure and oil density, compressibility, FVF, and viscosity. With only a few exceptions, oil properties have been correlated in terms of surfaceoil and gas properties, including solution gas/oil ratio, oil gravity, average surfacegas gravity, and temperature. A few correlations are also given in terms of composition and component properties. Reservoir oils typically contain dissolved gas consisting mainly of methane and ethane, some intermediates (C3 through C6), and lesser quantities of nonhydrocarbons. The amount of dissolved gas has an important effect on oil properties. At the bubblepoint a discontinuity in the system volumetric behavior is caused by gas coming out of solution, with the system compressibility changing draGAS AND OIL PROPERTIES AND CORRELATIONS
10 1.08000yg y 0.31109 ; y g u 0.6, . . . . . (3.80b) g
ƪ
ƫ
*1
, . . . . . . . . . . . . (3.81)
with R s in scf/STB. In this correlation, the gas mole fraction is dependent mainly on solution gas/oil ratio, but also on the properties of the stocktank oil. The Cragoe35 correlation given by Eq. 3.59 is recommended for estimating M o when stocktankoil molecular weight is not known. Standing’s approach was used by Glasø52 for North Sea oils, resulting in the correlation log p b + 1.7669 ) 1.7447 log A * 0.30218(log A) 2 , . . . . . . . . . . . . . . . . . . . . (3.82)
ǒT Ǔ with p b in psia, T in °F and where A + ǒR sńg gǓ R s in scf/STB. Glasø’s corrections for nonhydrocarbon content and stocktankoil paraffinicity are not widely used, primarily be0.816
0.172
ńg 0.989 API
29
cause the necessary data are not available. Sutton and Farshad54 mention that the API correction for paraffinicity worsened bubblepoint predictions for gulf coast fluids. Fig. 3.13 gives an explanation for this observation. Fig. 3.13 shows the effect of paraffinicity (which is quantified by the Watson characterization factor, K w) on bubblepoint pressure; the figure is based on calculations with a tuned EOS for an Asian oil (solid circles). The oil composition is constant in the example calculation. The 12 C7+ fractions are each split into a paraffinic pseudocomponent and an aromatic pseudocomponent (i.e., 24 C7+ pseudocomponents). The paraffinic fraction was varied, and bubblepoint calculations were made. The variation in paraffinicity is expressed in terms of the overall C7+ Watson characterization factor. Also shown in the figure are the variation in solution gas/oil ratio and the oil specific gravity with K wC . 7) The actual reservoir oil has a K wC + 11.55, where the EOS 7) bubblepoint is close to the uncorrected Glasø bubblepoint prediction. When the correction for paraffinicity is applied, the correction gives a poorer bubblepoint prediction (even though the overall trend in bubblepoints is improved by the Glasø paraffinicity corrections). A quantitatively similar correction to the Glasø correction (but easier to use) is based on the estimate for Whitson’s55,56 Watson characterization factor, K w, and yields ǒ g oǓ
corrected+
ǒ g oǓ measuredǒ K wń11.9 Ǔ 1.1824.
. . . . . . . . . . . (3.83)
The corrected specific gravity correlation is used in the Glasø bubblepoint correlation instead of the measured specific gravity. An estimate of Kw for the stocktank oil must be available to use this correction. Vazquez and Beggs57 give the following correlations. For g API x 30,
pb
ȱ R ǒ +ȧ27.64ǒg Ǔ10 Ȳ
*11.172 g API
s gc
T)460
Ǔȳ ȧ ȴ
0.9143
,
In summary, significant differences in predicted bubblepoint pressures should not be expected for most reservoir oils with most of the previous correlations. The Lasater and Standing equations are recommended for general use and as a starting point for developing reservoirspecific correlations. Correlations developed for a specific region, such as Glasø’s correlation for the North Sea, should probably be used in that region and, in the case of Glasø’s correlation, may be extended to other regions by use of the paraffinicity correction. 3.4.2 Oil Density. Density of reservoir oil varies from 30 lbm/ft3 for light volatile oils to 60 lbm/ft3 for heavy crudes with little or no solution gas. Oil compressibility may range from 3 10*6 psi*1 for heavy crude oils to 50 10*6 psi*1 for light oils. The variation of oil compressibility with pressure is usually small, although for volatile oils the effect can be significant, particularly for materialbalance and reservoirsimulation calculations of highly undersaturated volatile oils. Several methods have been used successfully to correlate oil volumetric properties, including extensions of idealsolution mixing, EOS’s, correspondingstates correlations, and empirical correlations. Oil density based on blackoil properties is given by òo +
62.4g o ) 0.0136g g R s , . . . . . . . . . . . . . . . . . . . . (3.88) Bo
with ò o in lbm/ft3, B o in bbl/STB, and R s in scf/STB. Correlations can be used to estimate R s and B o from g o, g g, p, and T. StandingKatz Method. Standing3,17 and Standing and Katz58 give an accurate method for estimating oil densities that uses an extension of idealsolution mixing. ò o + ò po ) D ò p * D ò T , . . . . . . . . . . . . . . . . . . . . . . (3.89) where ò po is the pseudoliquid density at standard conditions and the terms Dò T and Dò p give corrections for temperature and pressure, respectively. Pseudoliquid density is calculated with idealsolution mixing and correlations for the apparent liquid densities of ethane
. . . . . . . . . . (3.84)
and, for g API u 30,
pb +
ƪ
ǒ Ǔ ǒ
R 56.06 g s 10 gc
T)460
ƫ
Ǔ
*10.393g API
0.8425
, . . . . . . . . . (3.85)
with p b in psia, T in °F and R s in scf/STB. These equations are based on a large number of data from commercial laboratories. Vazquez and Beggs correct for the effect of separator conditions using a modified gas specific gravity, g gc , which is correlated with firststageseparator pressure and temperature, and stocktankoil gravity.
ƪ
g gc + g g 1 ) ǒ0.5912
10 *4Ǔ g APIT sp log
p ǒ114.7 Ǔƫ, sp
. . . . . . . . . . . . . . . . . . . . (3.86) with T sp in °F and p sp in psia. Standing’s correlation can be used to develop field or reservoirspecific bubblepoint correlations. A linear relation is usually assumed between bubblepoint pressure and the Standing correlating coefficient. This is a standard approach used in the industry, and the Standing bubblepoint correlating parameter is well suited for developing fieldspecific correlations. Sometimes the solution gas/oil ratio is needed at a given pressure, and this is readily calculated by solving the bubblepoint correlation for R s . For the Standing correlation,
ƪ(0.055p )10 1.4)10
ƫ
0.0125g API
1.205
. . . . . . . . . (3.87)
System Density at 60°F and 14.7 psia, g/cm3
similar relations can be derived for the other bubblepoint correlations.
Fig. 3.14—Apparent liquid densities of methane and ethane (from Standing33).
Rs + gg
30
0.00091T
;
PHASE BEHAVIOR
g Fig. 3.15—Chart for calculating pseudoliquid density of reservoir oil (from Standing33).
and methane at standard conditions. Given oil composition x i, ò po is calculated from
ȍx M N
i
ò po +
i
i+1
ȍǒx M ńò Ǔ N
i
i
, . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.90)
i
i+1
where Standing and Katz show that apparent liquid densities ò i of C2 and C1 are functions of the densities ò 2) and ò po , respectively (Fig. 3.14). ò C + 15.3 ) 0.3167 ò C 2
2)
ò C + 0.312 ) 0.45 ò po , . . . . . . . . . . . . . . . . . . . . . . . (3.91) 1
C 7)
ȍxM i
where ò C
2)
+
i
i+C 2 C 7)
ȍ ǒx M ńò Ǔ i
i
,
. . . . . . . . . . . . . . . . . . . . (3.92)
i
i+C 2
GAS AND OIL PROPERTIES AND CORRELATIONS
with ò in lbm/ft3. Application of these correlations results in an apparent trialanderror calculation for ò po . Standing33 presents a graphical correlation (Fig. 3.15) based on these relations, where ò po is found from ò C3) and weight fractions of C2 and C1 (w C2 and w C1, respectively). Figs. 3.16 and 3.17 show the pressure and temperature corrections, D ò p and D ò T , graphically. D ò p is a function of ò po, and D ò T is a function of ( ò po ) D ò p ). Madrazo59 introduced modified curves for D ò p and D ò T that improve predictions at higher pressures and temperatures. Standing3 gives bestfit equations for his original graphical correlations of D ò p and D ò T (Eqs. 3.98 and 3.99), which are not recommended at temperatures u240°F; instead, Madrazo’s graphical correlation can be used. The correction factors can also be used to determine isothermal compressibility and oil FVF at undersaturated conditions. The treatment of nonhydrocarbons in the StandingKatz method has not received much attention, and the method is not recommended when concentrations of nonhydrocarbons exceed 10 mol%. Standing3 suggests that an apparent liquid density of 29.9 lbm/ft3 can be used for nitrogen but does not address how the nonhydrocarbons should be considered in the calculation procedure (i.e., as part of the C3+ material or following the calculation of ò C and ò C ). 2 1 Madrazo indicates that the volume contribution of nonhydrocar31
Density of System at 60°F and 14.7 psia, lbm/ft3 Fig. 3.16—Pressure correction to the pseudoliquid density at 14.7 psia and 60°F (from Ref. 59).
bons can be neglected completely if the total content is t6 mol%. Vogel and Yarborough60 suggest that the weight fraction of nitrogen should be added to the weight fraction of ethane. Using additive volumes and Eqs. 3.91 and 3.92, we can show that ò C and ò po can be calculated explicitly. Thus, the following is the 2) most direct procedure for calculating ò o from the StandingKatz method. 1. Calculate the mass of each component. m i + x i M i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.93) 2. Calculate V C C 7)
VC
3)
+
ȍ
i+C 3
3)
.
mi ò i , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.94)
where ò i are component densities at standard conditions (Appendix A). 3. Calculate ò C . 2)
* b ) Ǹb 2 * 4ac , . . . . . . . . . . . . . . . . . . . . (3.95) òC + 2) 2a 32
where a + 0.3167V C , b + m C * 0.3167 m C ) 15.3 V C , 3) 2 2) 3) and c + * 15.3m C . 2) 4. Calculate V C . 2)
VC
2)
+ VC
mC )ò 2 3) C 2
+ VC
3)
)
mC
2
15.3 ) 0.3167ò C
. . . . . . . . . . . . . (3.96) 2)
5. Calculate ò po. ò po +
*b ) Ǹb 2 * 4ac , . . . . . . . . . . . . . . . . . . . . . (3.97) 2a
where a + 0.45V C , b + m C * 0.45m C ) 0.312V C , and 2) 1 1) 2) c + * 0.312m C . 1) 6. Calculate the pressure effect on density. D ò p + 10 *3 ƪ0.167 ) ǒ16.181 * 10 *8 ƪ0.299 ) ǒ263
10 *0.0425òpoǓƫ p 10 *0.0603òpoǓƫ p 2. . . . . . (3.98) PHASE BEHAVIOR
Density of System at Pressure and 60°F, lbm/ft3 Fig. 3.17—Temperature correction to the pseudoliquid density at pressure and 60°F (from Ref. 59).
ò ga + 38.52
7. Calculate the temperature effect on density.
ƪ
D ò T + (T * 60) 0.0133 ) 152.4ǒò po ) D ò pǓ
Ǌ
* (T * 60) ǒ8.1 2
* ƪ0.0622
*2.45
ƫ
10 Ǔ *6
ǋ
10 *0.0764(òpo)D òp)ƫ .
. . . . . . . . . . . (3.99)
8. Calculate mixture density from Eq. 3.89. In the absence of oil composition, Katz41 suggests calculating the pseudoliquid density from stocktankoil gravity, g o, solution gas/ oil ratio, R s , and apparent liquid density of the surface gas, ò ga, taken from a graphical correlation (Fig. 3.18), ò po +
62.4g o ) 0.0136 R s g g
1 ) 0.0136ǒR s g gńò gaǓ
. . . . . . . . . . . . . . . . . . . (3.100)
Standing gives an equation for ò ga. GAS AND OIL PROPERTIES AND CORRELATIONS
10 *0.00326 g API
) (94.75 * 33.93 log g API) log g g , . . . . . . . . . . . (3.101) with ò ga in lbm/ft3 and Rs in scf/STB. AlaniKennedy 61 Method. The AlaniKennedy method for calculating oil density is a modification of the original van der Waals EOS, with constants a and b given as functions of temperature for normal paraffins C1 to C10 and isobutane (Table 3.1); two sets of coefficients are reported for methane (for temperatures from 70 to 300°F and from 301 to 460°F) and two sets for ethane (for temperatures from 100 to 249°F and from 250 to 460°F). Lohrenz et al.62 give AlaniKennedy temperaturedependent coefficients for nonhydrocarbons N2, CO2, and H2S (Table 3.1). The AlaniKennedy equations are summarized next. Eqs. 3.102b and 3.102c are in the original van der Waals EOS but are not used. p + RT * a2 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.102a) v*b v 33
where log a C
+ ǒ3.8405985
10 *3Ǔ M C
* ǒ9.5638281
MC 10 *4Ǔ g 7) ) 261.80818 T C
7)
7)
7)
) ǒ7.3104464
10
*6
Ǔ
M 2C 7)
) 10.753517 . . . . . . . . . . . . . . . . . . . . . (3.103a) and b C
7)
+ ǒ3.499274
10 *2Ǔ M C
) ǒ2.232395
10 *4ǓT * ǒ1.6322572
7)
* 7.2725403 g C
7)
MC 10 *2Ǔ g 7) C 7)
) 6.2256545, . . . . . . . . . . . . . . . . . . . . . . . (3.103b) with ò in lbm/ft3, v in ft3/lbm mol, T in °R, p in psia, and R+universal gas constant+10.73. Solution of the cubic equation for volume is presented in Chap. 4. Density is given by ò+M/v, where M is the mixture molecular weight and v is the molar volume given by the solution to the cubic equation. The AlaniKennedy method can also be used to estimate oil compressibilities. Rackett,63 Hankinson and Thomson,64 and Hankinson et al.65 give accurate correlations for purecomponent saturatedliquid densities, and although these correlations can be extended to mixtures, they have not been tested extensively for reservoir systems. Cullick et al.66 give a modified correspondingstates method for predicting density of reservoir fluids, The method has a better foundation and extrapolating capability than the methods discussed previously (particularly for systems with nonhydrocarbons); however, space does not allow presentation of the method in its entirety. Either the StandingKatz or AlaniKennedy method should estimate the densities of most reservoir oils with an accuracy of "2%. The AlaniKennedy method is suggested for systems at temperatures u250°F and for systems containing appreciable amounts of nonhydrocarbons (u5 mol%). Cubic EOS’s (e.g., PengRobinson or SoaveRedlichKwong) that use volume translation also estimate liquid densities with an accuracy of a few percent (e.g., the recommended characterization procedures in Chap. 5 or other proposed characterizations 67,68).
Fig. 3.18—Apparent pseudoliquid density of separator gas (from Standing,33 after Katz41).
R 2T 2 a i + 27 p ci , . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.102b) ci 64 RT b i + 1 p ci , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.102c) 8 ci
ȍx a N
a+
i
i
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.102d)
i+1
ȍx b , N
b+
i
i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.102e)
i+1
ai +
a 1i ) log a 2i; i 0 C 7) , T
. . . . . . . . . . . . . . . . . . (3.102f)
and b i + b 1iT ) b 2i ; i 0 C 7), . . . . . . . . . . . . . . . . . (3.102g)
TABLE 3.1—CONSTANTS FOR ALANIKENNEDY61 OIL DENSITY CORRELATION Component
a1
a2
N2
4,300
CO2
8,166
126.0
2.293
H2 S
13,200
0.0
b1
104
b2
4.49
0.3853
0.1818
0.3872
17.9
0.3945
C1 At 70 to 300°F At 300 to 460°F
9,160.6413 147.47333
61.893223 3,247.4533
*3.3162472 *14.072637
0.50874303 1.8326695
C2
34
At 100 to 250°F
46,709.573
At 250 to 460°F
17,495.343
*404.48844 34.163551
5.1520981
0.52239654
2.8201736
0.62309877
C3
20,247.757
190.24420
2.1586448
0.90832519
iC4
32,204.420
131.63171
3.3862284
1.1013834
nC4
33,016.212
146.15445
2.902157
1.1168144
iC5
37,046.234
299.62630
2.1954785
1.4364289
nC5
37,046.234
299.62630
2.1954785
1.4364289
nC6
52,093.006
254.56097
3.6961858
1.5929406
nC7
82,295.457
5.2577968
1.7299902
nC8
89,185.432
nC9
124,062.650
nC10
146,643.830
64.380112 149.39026
5.9897530
1.9310993
37.917238
6.7299934
2.1519973
26.524103
7.8561789
2.3329874 PHASE BEHAVIOR
3.4.3 UndersaturatedOil Compressibility. With measured data or an appropriate correlation for B o or ò o , Eq. 3.14 readily defines the isothermal compressibility of an oil at pressures greater than the bubblepoint. “Instantaneous” undersaturatedoil compressibility, defined by Eq. 3.15 with the pressure derivative evaluated at a specific pressure, is used in reservoir simulation and welltest interpretation. Another definition of oil compressibility may be used in materialbalance calculations (e.g., Craft and Hawkins69)—the “cumulative” or “average” compressibility defines the cumulative volumetric change of oil from the initial reservoir pressure to current reservoir pressure.
Compressibility at Bubblepoint +
pi
V oi c oǒ p Ǔ +
ŕ c ǒ p Ǔ dp o
Bubblepoint Plus 1,000 psia
p
V oi ǒ p i * pǓ
+*
ǒV1 ǓƪV p**Vpǒ pǓƫ. . . . . . . . . . . . . . . . . o
oi
i
oi
(3.104)
Bubblepoint Plus 2,000 psia
The cumulative compressibility is readily identified because it is multiplied by the cumulative reservoir pressure drop, p i * p R. Usually c o is assumed constant; however, this assumption may not be justified for highpressure volatile oils. Oil compressibility is used to calculate the variation in undersaturated density and FVF with pressure.
Fig. 3.19—Undersaturatedoilcompressibility correlation (from Standing33).
ò o + ò ob expƪc oǒ p * p bǓƫ [ ò ob ƪ1 * c oǒ p b * pǓƫ . . . . . . . . . . . . . . . . . . . . . (3.105a) and B o + B ob expƪc oǒ p b * pǓƫ [ B ob ƪ1 * c oǒ p * p bǓƫ ,
ǒV1 ǓƪV ǒpp*Ǔ *p V o
ob
b
ob
ƫ
. . . . . . . . . . . . . . . . (3.106)
Strictly speaking, the compressibility of an oil mixture is defined only for pressures greater than the bubblepoint pressure. If an oil is at its bubblepoint, the compressibility can be determined and defined only for a positive change in pressure. A reduction in pressure from the bubblepoint results in gas coming out of solution and, subsequently, a change in the mass of the original system for which compressibility is to be determined. Implicit in the definition of compressibility is that the system mass remains constant. Vazquez and Beggs57 propose the following correlation for instantaneous undersaturatedoil compressibility. c o + Ańp, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.107) where A+ 10 *5(5R sb ) 17.2T * 1, 180g gc ) 12.61g API* 1, 433), with c o in psi*1, R sb in scf/STB, T in °F, and p in psia. With this correlation for oil compressibility, undersaturatedoil FVF can be calculated analytically from B o + B ob( p bńp) A.
Constant A determined in this way is a useful correlating parameter, one that helps to identify erroneous undersaturated p V o data. Standing33 gives a graphical correlation for undersaturated c o (Fig. 3.19) that can be represented by
. . . . . . . . . . . . . . . . . (3.105b)
where consistent units must be used. These equations are derived from the definition of isothermal compressibility assuming that co is constant. When oil compressibility varies significantly with pressure, Eqs. 3.105a and 3.105b are not really valid. The approximations ò o [ ò ob [1 * c o( p b * p)] and B o [ B ob [1 * c o( p * p b)] are used in many applications, and to predict volumetric behavior correctly with these relations requires that co be defined by c o( p) + *
Bubblepoint Oil Density, lbm/ft3
. . . . . . . . . . . . . . . . . . . . . . . . . . . (3.108)
If measured pressure/volume data are available (see Sec. 6.4 in Chap. 6), these data can be used to determine A (e.g., by plotting V ońV ob vs. pńp b on loglog paper). Constant A can then be used to compute compressibilities from the simple relation c o + Ańp. GAS AND OIL PROPERTIES AND CORRELATIONS
c o + 10 *6 exp
ƪ
ƫ
ò ob ) 0.004347 ǒ p * p bǓ * 79.1 , (7.141 10 *4)ǒ p * p bǓ * 12.938 . . . . . . . . . . . . . . . . . . . (3.109)
psi*1,
ò ob in lbm/ft3, and p in psia. with c o in The AlaniKennedy EOS can also be solved analytically for oil compressibility, and Trube70 gives a correspondingstates method for determining oil compressibility with charts. Any of the correlations mentioned here should yield reasonable estimates of c o. However, we recommend that experimental data be used for volatile oils when c o is greater than about 20 10*6 psi*1. A simple polynomial fit of the relative volume data, V ro + V ońV ob , from a PVT report allows an accurate and explicit equation for undersaturatedoil compressibility. V ro + A 0 ) A 1 p ) A 2 p 2 . . . . . . . . . . . . . . . . . . . . . . . (3.110a)
ǒ Ǔ
ēV ro and c o + * 1 V ro ēp +
T
* ǒ A 1 ) 2A 2 p Ǔ A0 ) A1 p ) A2 p2
, . . . . . . . . . . . . . . . . . . . . (3.110b)
where A0, A1, and A2 are determined from experimental data. Alternatively, measured data can be fit by use of Eq. 3.108. 3.4.4 BubblepointOil FVF. Oil FVF ranges from 1 bbl/STB for oils containing little solution gas to about 2.5 bbl/STB for volatile oils. B ob increases more or less linearly with the amount of gas in solution, a fact which explains why B ob correlations are similar to bubblepoint pressure correlations. For example, Standing’s3,17,40 correlation for California crude oils is B ob + 0.9759 ) ǒ12 where A + R sǒg gńg oǓ
0.5
10 *5Ǔ A 1.2,
. . . . . . . . . . . . . . . (3.111)
) 1.25T. 35
Glasø’s52 correlation for North Sea crude oils is logǒ B ob * 1 Ǔ + * 6.585 ) 2.9133 log A * 0.2768ǒlog AǓ 2 , . . . . . . . . . . . . . . . . . . . . (3.112) where A + R s ǒg gńg oǓ 0.526) 0.968T. The Vazquez and Beggs57 correlation, based on data from commercial laboratories, is B ob + 1 ) ǒ4.677
10 *4ǓR s ) ǒ0.1751
ǒgg Ǔ * ǒ1.8106 API gc
10 *4Ǔ(T * 60)
ǒ Ǔ
g 10 Ǔ R s(T * 60) gAPI gc *8
4,000 2,000 1,000 800 600 400 200
100°F
100 80 60
120°F 140°F 160°F
40
. . . . . . . . . . . . . . . . . . (3.113a)
20 180°F
for g API x 30 and B ob + 1 ) ǒ4.67 * ǒ0.1337
10 ǓR s ) ǒ0.11 *4
ǒ Ǔ
g 10 Ǔ(T * 60) gAPI gc *4
ǒ Ǔ
g 10 *8ǓR s(T * 60) gAPI . . . . . . . . . (3.113b) gc
10 8 6
200°F 220°F
4 240°F 2 1 0.8 0.6
Sources of Data Beal (1946) Frick (1962)
0.4
for g API u 30, where the effect of separator conditions is included by use of a corrected gas gravity g gc (Eq. 3.86). The Standing and the VazquezBeggs correlations indicate that a plot of B o vs. R s should correlate almost linearly. This plot is useful for checking the consistency of reported PVT data from a differential liberation plot. Eq. 3.114,71 which performs considerably better for Middle Eastern oils, also suggests a linear relationship between B ob and R s. B ob+ 1.0 ) ǒ0.177342
10 *3Ǔ R s ) ǒ0.220163
R sǒ g gńg oǓ )ǒ4.292580 ) ǒ0.528707
10 *3Ǔ
10 *6Ǔ R s(T * 60)(1 * g o)
10 *3Ǔ(T * 60). . . . . . . . . . . . . . . . (3.114)
All three B ob correlations (Eqs. 3.113a, 3.113b, and 3.114) should give approximately the same accuracy. Sutton and Farshad’s54 comparative study of gulf coast oils indicates that Standing’s correlation is slightly better for B ob t 1.4 and that Glasø’s correlation is best for B ob u 1.4. 3.4.5 SaturatedOil Compressibility. Perrine8 introduces a definition for the compressibility of a saturated oil that includes the shrinkage effect of saturatedoil FVF, ēB ońēp, and the expansion effect of gas coming out of solution, B g(ēR sńēp),
ǒ Ǔ
ēB o co + * 1 B o ēp
) T
ǒ Ǔ.
1 B g ēR s 5.615 B o ēp
. . . . . . . . . (3.115)
T
c o is used in the definition of total system compressibility, c t . c t + c f ) c w S w ) c o S o ) c g S g , . . . . . . . . . . . . . . (3.116) where c f +rock compressibility. B g has units ft3/scf. R s is in scf/ STB, and B o in bbl/STB+saturatedoil FVF at the pressure of interest, at or below the original oil’s bubblepoint pressure (where both gas and oil are present). 3.4.6 Oil Viscosity. Typical oil viscosities range from 0.1 cp for nearcritical oils to u100 cp for heavy crudes. Temperature, stocktankoil density, and dissolved gas are the key parameters determining oil viscosity, where viscosity decreases with decreasing stocktankoil density (increasing oil gravity), increasing temperature, and increasing solution gas. Oil viscosity is one of the most difficult properties to estimate, and most methods offer an accuracy of only about 10 to 20%. Two approaches are used to estimate oil viscosity: empirical and correspondingstates correlations. The empirical methods correlate gassaturatedoil viscosity in terms of deadoil (residual or stocktankoil) viscosity and solution gas/oil ratio. Undersaturatedoil viscosity is related to bubblepoint viscosity and the ratio or differ36
0.2 0.1 0
10
20
30
40
50
60
Fig. 3.20—Beal deadoil (stocktankoil) viscosity correlation including data in Frick (from Standing33).
ence in actual and bubblepoint pressures. Correspondingstates methods use reduced density or reduced pressure and temperature as correlating parameters. 3.4.7 DeadOil (Residual or StockTankOil) Viscosity. Several correlations for deadoil viscosity are given in terms of oil gravity and temperature. Standing,3 for example, gives bestfit equations for the original Beal72 graphical correlation, m oD +
ǒ
7 0.32 ) 1.8 4.5310 g API
Ǔǒ
Ǔ
360 A , . . . . . . . . . (3.117) T ) 200
where A + 10 ƪ0.43)ǒ8.33ńg APIǓƫ . A somewhat modified version of the original correlation is given in Fig. 3.20 by Standing.33 Beggs18 and Beggs and Robinson73 give the following equation for the original Beal correlation, m oD + * 1 ) 10 ƪT
*1.163 expǒ6.9824*0.04658g
Ǔƫ .
API
. . . . . . (3.118)
Bergman* claims that the temperature dependence of the Beggs and Robinson correlation is not valid at lower temperatures (t70°F) and suggests the following correlation, based on viscosity data, for pure compounds and reservoir oils. ln ln( m oD ) 1) + A 0 ) A 1 ln(T ) 310), . . . . . . . . . (3.119) where A 0 + 22.33* 0.194gAPI ) 0.00033 g 2API and A 1 + * 3.20 ) 0.0185 g API . Glasø52 gives a relation (used in the paraffinicity correction of his bubblepoint pressure correlation) for oils with K w + 11.9. m oD + (3.141
10 10)T *3.444(log g API) [10.313(log T )*36.447]. . . . . . . . . . . . . . . . . . . . (3.120)
AlKhafaji et m oD +
al.74
give the correlation
10 4.9563*0.00488T , ǒ g API ) T ń 30 * 14.29 Ǔ 2.709
. . . . . . . . . . . (3.121)
with T in °F and m oD in cp for Eqs. 3.117 through 3.121. *Personal communication with D.F. Bergman, Amoco Research, Tulsa, Oklahoma (1992).
PHASE BEHAVIOR
Solution gas/oil ratio, scf/STB
Fig. 3.22—Liveoil (saturated) viscosity as a function of deadoil viscosity and solution gas/oil ratio (from Standing,33 after Beal72 correlation).
Fig. 3.21 shows deadoil viscosities calculated at 100°F for a range of paraffinicities expressed in terms of K w, together with the Bergman* and Glasø48 correlations. Fig. 3.21—Deadoil (stocktankoil) viscosities at 100°F for varying paraffinicity (from Ref. 33).
3.4.8 BubblepointOil Viscosity. The original approach by Chew and Connally76 for correlating saturatedoil viscosity in terms of deadoil viscosity and solution gas/oil ratio is still widely used.
Standing75 gives a relation for deadoil viscosity in terms of deadoil density, temperature, and the Watson characterization factor.
m ob + A 1 ǒm oDǓ A2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.123)
log( m oD ńò o) +
1 *2.17, A 3ƪK w*ǒ8.24ńg oǓƫ ) 1.639A 2*1.059 . . . . . . . . . . . . . . . . . . . (3.122a)
where A 1 + 1 ) 8.69 log T ) 460, 560 A 2 + 1 ) 0.554 log T ) 460 , 560
. . . . . . . . . . . . . (3.122b)
A 1 + 10.715(R s ) 100) *0.515 . . . . . . . . . . . . . . . . . . (3.124a)
. . . . . . . . . . . . . . . . . (3.122c)
and A 2 + 5.44(R s ) 150) *0.338 . . . . . . . . . . . . . . . . . . (3.124b)
ǒ2.87A 1 * 1Ǔg o A 3 + * 0.1285 , . . . . . . . . . . . . . . . (3.122d) 2.87A 1 * g o and ò o +
Fig. 3.22 shows the variation in m ob with m oD as a function of R s. The functional relations for A1 and A2 reported by various authors differ somewhat, but most are bestfit equations of Chew and Connally’s tabulated results. Beggs and Robinson. 73
go , 1 ) 0.000321(T * 60)10 0.00462gAPI
. . . . . . (3.122e)
with m in cp, T in °F, and ò in g/cm3 for Eqs. 3.117 through 3.122. Eqs. 3.122a through 3.122e represent a best fit of the nomograph for viscosity in terms of temperature, gravity, and characterization factor. Eq. 3.122e (at standard pressure and temperature) is a best fit of thermal expansion data for crude oils. Deadoil viscosity is one of the most unreliable properties to predict with correlations primarily because of the large effect that oil type (paraffinicity, aromaticity, and asphaltene content) has on viscosity. For example, the oil viscosity of a crude oil with K w + 12 may be 3 to 100 times the viscosity of a less paraffinic crude oil having the same gravity and K w + 11. For this reason the Standing correlation based on the Watson characterization factor is recommended when K w is known. Using an incorrectly estimated K w, however, may lead to a potentially large error in deadoil viscosity. GAS AND OIL PROPERTIES AND CORRELATIONS
Bergman. * ln A 1 + 4.768 * 0.8359 ln(R s ) 300) . . . . . . . . . . (3.125a) and A 2 + 0.555 )
133.5 . . . . . . . . . . . . . . . . . . (3.125b) R s ) 300
Standing. 3 A 1 + 10 *ǒ7.4 and A 2 +
10 *4ǓR s)ǒ2.2
10 *7ǓR 2s
. . . . . . . . . . . . . . . (3.126a)
0.68 0.25 0.062 ) ) . 10 ǒ8.62 10 *5ǓRs 10 ǒ1.1 10*3ǓRs 10 ǒ3.74 10*3ǓRs . . . . . . . . . . . . . . . . . . (3.126b)
Aziz et
al. 77
A 1 + 0.20 ) ǒ0.80 and A 2 + 0.43 ) ǒ0.57
10 –0.00081 RsǓ . . . . . . . . . . . . . . (3.127a) 10 –0.00072 RsǓ . . . . . . . . . . . (3.127b)
*Personal communication with D.F. Bergman, Amoco Research, Tulsa, Oklahoma (1992).
37
AlKhafaji et al.74 extend the ChewConnally76 correlation to higher GOR’s (up to 2,000 scf/STB). A 1 + 0.247)0.2824 A 0) 0.5657 A 20 * 0.4065 A 30 ) 0.0631 A 40 . . . . . . . . . . . . . . . . . . . (3.128a)
ƪǒ m * m oǓc T ) 10 *4ƫ
and A 2 + 0.894 ) 0.0546 A 0 ) 0.07667A 20 * 0.0736 A 30 ) 0.01008 A 40 ,
is therefore desired. Several correspondingstates viscosity correlations can be used for both oil and highpressure gas mixtures; the Lohrenz et al.62 correlation has become a standard in compositional reservoir simulation. Lohrenz et al. use the Jossi et al.82 correlation for densegas mixtures ( ò pru0.1),6
. . . . . . . . . . . . . . (3.128b)
where A 0 + log(R s) and R s + 0.1 yields A 1 + A 2 + 1. R s is given in scf/STB for Eqs. 3.124 through 3.128. Chew and Connally indicate that their correlation is based primarily on data with GOR’s of t1,000 scf/STB and that the scatter in A 1 at higher GOR’s is probably the result of insufficient data. Eqs. 3.128a and 3.128b are based on additional data at higher GOR’s. Eqs. 3.127a and 3.128b appear to be the most well behaved. An interesting observation by AbuKhamsin and AlMarhoun78 is that saturatedoil viscosity, m ob, correlates very well with saturatedoil density, ò ob . ln m ob + * 2.652294 ) 8.484462
ò 4ob ,
. . . . . . . . . . (3.129)
This behavior is expected from the Lohrenz et with ò ob in al.62 correlation discussed later. Although AbuKhamsin and AlMarhoun do not comment on the applicability of Eq. 3.129 to undersaturated oils, it would seem reasonable that their correlation should apply to both saturated and undersaturated oils. In fact, the correlation even appears to predict accurately deadoil viscosities, m oD, except at low temperatures for heavy crudes. Simon and Graue give graphical correlations for the viscosity of saturated CO2/oil systems (see Chap. 8).79
m o * m ob 0.56 + 0.024m 1.6 ob ) 0.038m ob . 0.001( p * p b) The Vazquez and
Beggs57
m o + m ob ǒ pńp bǓ A ,
. . . . . . . . (3.130)
) 0.0093324ò 4pr , . . . . . . . . . . . . . . . . . . . . . . (3.133a)
ǒ Ǔ
ò ò pr + ò
pc
where A + 2.6 p 1.187 expƪ* 11.513 * ǒ8.98 10 *5Ǔpƫ. A more recent correlation by AbdulMajeed et al.80 is
ƫ,
. . . . . . . . (3.132a)
where A + 1.9311 * 0.89941 ǒln R sǓ * 0.001194 g 2API ) 0.0092545 g API ǒln R sǓ.
. . . . . . . . . . . . . . . . . . (3.133b)
ò v , . . . . . . . . . . . . . . . . . . . . . . . . (3.133c) M pc
+
ȍ z m ǸM N
i
and m + o
i
i
i+1 N
ȍ z ǸM i
.
. . . . . . . . . . . . . . . . . . . . . . . (3.133d)
i
i+1
Pseudocritical properties T pc, p pc, and v pc are calculated with Kay’s mixing rule. Component viscosities, m i , can be calculated from the Lucas45 lowpressure correlation Eq. 3.67 or from the Stiel and Thodos83 correlation (as suggested by Lohrenz et al.62). m i c Ti + ǒ34
10 *5ǓT ri0.94 . . . . . . . . . . . . . . . . . . . . . . (3.134a)
for Tri x1.5, and m i c Ti + ǒ17.78
10 *5Ǔ(4.58T ri * 1.67) 5ń8 . . . . . . (3.134b)
for T ri u 1.5, where c Ti + 5.35ǒT ci M 3ińp 4ciǓ . Lohrenz et al.62 give a special relation for v c C of the C7+ fraction. 1ń6
7)
+ 21.573 ) 0.015122M C ) 0.070615M C
g , 7) C 7)
7)
* 27.656g C
7)
. . . . . . . . . . . . . . . . (3.135)
with m in cp, c in cp*1, ò in lbm/ft3, v in ft3/lbm mol, T in °R, p in psia, and M in lbm/lbm mol. The Lohrenz et al. method is very sensitive to mixture density and to the critical volumes of heavy components. Adjustment of the critical volumes of heavy (and sometimes light) components to match experimental oil viscosities is usually necessary.
. . . . . . . . . . . . . . . (3.132b)
Eq. 3.132 is based on the observation that a plot of log(m o * m ob) vs. log(p * p b) plots as a straight line with slope of [1.11. Because this observation appears to be fairly general, it can be used to check reported undersaturatedoil viscosities and to develop fieldspecific correlations. Sutton and Farshad54 and Khan et al.81 present results that indicate that the Standing equation gives good results and that the VazquezBeggs correlation tends to overpredict viscosities somewhat. AbdulMajeed et al.80 indicate that both the Standing and VazquezBeggs correlations overpredict viscosities of North African and Middle Eastern oils (253 data), and that their own correlation performed best for these data and for the data used by Vazquez and Beggs. 3.4.10 Compositional Correlation. In compositional reservoir simulation of misciblegasinjection processes and the depletion of nearcritical reservoir fluids, the oil and gas compositions may be very similar. A single viscosity relation consistent for both phases 38
,
7)
. . . . . . . . . . . . . . . . . . . . . . . . . . . (3.131)
logǒ p*p bǓ
1ń6
T pc where c T + 5.35 M 3p 4pc
v cC
correlation is
mo + m ob ) 10 ƪA* 5.2106 ) 1.11
+ 0.10230 ) 0.023364ò pr
) 0.058533ò 2pr * 0.040758ò 3pr
g/cm3.
3.4.9 UndersaturatedOil Viscosities. Beal72 gives the variation of undersaturatedoil viscosity with pressure graphically where it has been curve fit by Standing.2
1ń4
3.5 IFT and Diffusion Coefficients 3.5.1 IFT. Weinaug and Katz84 propose an extension of the Macleod85 relationship for multicomponent mixtures.
ȍ P ǒx Mò N
s 1ń4 go +
o
i
i+1
i
o
* yi
Ǔ
òg , . . . . . . . . . . . . . . . . . (3.136) Mg
with s in dynes/cm (mN/m) and ò in g/cm3. P i is the parachor of Component i, which can be calculated by group contributions, as shown in Table 3.2. For nalkanes, the parachors can be expressed by P i + 25.2 ) 2.86M i .
. . . . . . . . . . . . . . . . . . . . . . . . (3.137)
Several authors propose parachors for pure hydrocarbons that deviate from the groupcontribution values. For example, P C +77 is 1 often cited for methane instead of the groupcontribution value of P C +71. Likewise, P N +41 is often used for nitrogen instead of 2 1 the groupcontribution value of P N +35. Fig. 3.23 plots parachors 2 vs. molecular weight for pure components and petroleum fractions. PHASE BEHAVIOR
TABLE 3.2—PARACHORS FOR PURE COMPONENTS AND COMPOUND GROUPS
nparaffins Heptanes plus of Ref. 4 Gasolines Crude oil
Pure Component C1
71
C2
111
C3
151
C4 (also iC4)
191
C5 (also iC5)
231
C6
271
C7
311
C8
351
C9
391
C10
431
N2
35
CO2
49
H2 S
80
Group C
9.0
H
15.5
CH3
55.5
CH2 [in (CH2)n ]
40.0
N
17.5
O
20.0
S
49.1
Example: For methane, CH4. PC1=PC+4(PH)=9+4(15.5)=71.
Fig. 3.23—Hydrocarbon parachors.
Nokay86 gives a simple relation for parachors of pure hydrocarbons (paraffins, olefins, naphthenes, and aromatics) with a normal boiling point between 400 and 1,400°R and specific gravity t1, log P i + * 4.20895 ) 2.29319 log
ǒ Ǔ
T bi , . . . . . (3.138) g 0.5937 i
with T b in °R. Katz and Saltman87 and Katz et al.88 give parachor data for C7+ fractions measured by Standing and Katz,58,89 which are approximately correlated by P i + 35 ) 2.40M i . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.139) The API recommended procedure for estimating petroleum fraction IFT’s is based on an unpublished correlation.27 The graphical correlation can be expressed by 602(1 * T ri) 1.194 , si + K wi
ǒ Ǔ
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.141)
where ò sL ơ ò sv is assumed. The saturatedliquid density can be estimated, for example, with the Rackett63 equation. ò sLi +
P i + 11.4 ) 3.23 M i * 0.0022 M 2i .
M i p ci *ƪ1)ǒ1*TriǓ2ń7ƫ Z , RT ci Ri
. . . . . . . . . . . . . . . . . (3.142)
where Z Ri [ Z ci [ 0.291 * 0.08 w i
. . . . . . . . . . . . . . (3.143)
and R+universal gas constant. The parachors predicted from Eqs. 3.140 through 3.143 are practically constant for a given petroleum fraction (i.e., the temperature effect cancels out). GAS AND OIL PROPERTIES AND CORRELATIONS
. . . . . . . . . . . (3.144)
They also discuss the qualitative effect of asphaltenes on IFT and suggest that the parachor of asphaltic substances generally will not follow the relations of lighter C7+ fractions. Ramey91 gives a method for estimating gas/oil IFT with blackoil PVT properties. We extend the method here to include the effect of solution oil/gas ratio, r s. Considering stocktank oil and separator gas as the “components” ( o and g) making up reservoir oil and gas, respectively, the WeinaugKatz84 relation can be written
ƪ ǒMò Ǔ * y ǒMò Ǔƫ ) P ƪx ǒMò Ǔ * y ǒMò Ǔƫ,
s¼ go + P o x o
. . . . . . . . . . . . . . . . . . . . . . . (3.140)
where K w + T 1ń3 ńg, with T b in °R. The parachor can be estimated b with the Macleod relation, Mi P i + s 1ń4 ò sLi , i
Firoozabadi et al.90 give an equation that can be used to approximate the parachor of pure hydrocarbons from C1 through C6 and for C7+ fractions,
g
o
o
o
o
g
o
g
g
o
g
g
. . . . . . . . . . . . . . . . . . . (3.145a) where x o +
1 ) (7.52
1 , . . . . . . . . (3.145b) 10 *6)R sǒM ońg oǓ
x g + 1 * x o , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.145c) yo +
1 ) (7.52
1 , 10 *6)ǒ M ońg o Ǔr s
. . . . . . . . . . . (3.145d)
y g + 1 * y o , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.145e) òo +
62.4g o ) 0.0136g g R s , 62.4 B o
. . . . . . . . . . . . . . . . . . (3.145f)
ò g + 0.0932ǒ pM gńZT Ǔ , . . . . . . . . . . . . . . . . . . . . . . (3.145g) M o + x o M o ) x g M g , . . . . . . . . . . . . . . . . . . . . . . . . (3.145h) 39
M g + y o M o ) y g M g ,ĂĂ . . . . . . . . . . . . . . . . . . . . . . . . (3.145i)
)
M o + 6, 084ńg API * 5.9 , . . . . . . . . . . . . . . . . . . . . . (3.145j) M g + 28.97g g ,
. . . . . . . . . . . . . . . . . . (3.149b)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.145k)
P o + ǒ2.376 ) 0.0102g APIǓńM o ,
. . . . . . . . . . . . . . . (3.145l)
1.03587 1.76474 ) , expǒ1.52996T ijǓ expǒ3.89411T ijǓ
T ij +
T , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.149c) (åńk) ij
and P g + 25.2 ) 2.86M g , . . . . . . . . . . . . . . . . . . . . . (3.145m)
ǒ åńk Ǔ ij+ ƪǒ åńk Ǔ i ǒåńkǓ jƫ 1ń2 , . . . . . . . . . . . . . . . . . . . . (3.149d)
with ò in g/cm3, R s in scf/STB, B o in bbl/STB, T in °R, and p in psia and where x o and x g+mole fractions of the surfaceoil and gas components, respectively, in the oil phase, and y o and y g+mole fractions of the surfaceoil and gas components, respectively, in the gas phase. In the traditional blackoil approach r s + 0, simplifying the relations to those originally suggested by Ramey.91 Eq. 3.145 is useful in blackoil reservoir simulation and when compositional data are not available. The blackoil approach generally is not recommended for predicting gas/oil IFT’s unless the surfaceoil parachor has been fit to experimental IFT data (or to IFT’s calculated with compositions and densities from an EOS characterization by use of Eq. 3.136).
ǒ åńk Ǔ i + 65.3T ci Z 18ń5 ,
3.5.2 Diffusion Coefficients. Molecular diffusion in multicomponent mixtures is a complex problem. The standard engineering approach uses an effective diffusion coefficient for Component i in a mixture, D im, where D im can be calculated in one of two ways: (1) from binary diffusion coefficients and mixture composition or (2) from Component i properties and mixture viscosity. The first approach uses the Wilke92 formula to calculate D im. D im +
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.146)
N
j
ij
j+1 j0i
where y i +mixture mole fraction and D ij + D ji is the binary diffusivity at the pressure and temperature of the mixture. Sigmund93 correlates the effect of pressure and temperature on diffusion coefficients using a correspondingstates approach with reduced density. ò M D ij + 0.99589 ) 0.096016ò pr * 0.22035ò 2pr ò oM D oij ) 0.032874ò 3pr , . . . . . . . . . . . . . . . . . . . . . . . (3.147) where D ij +diffusion coefficient at pressure and temperature, ò pr+pseudoreduced density+ ò Mńò Mpc + ǒ òńM Ǔv pc , ò M +mixture molar density, ò oM D oij +lowpressure densitydiffusivity product, and v pc +pseudocritical molar volume calculated with Kay’s5 mixing rule. Note that the ratio ò M D ijńò oD oij is the same for all binary pairs in a mixture because ò pr is a function of only mixture density and composition. da Silva and Belery12 note that the Sigmund correlation does not work well for liquid systems and propose the following extrapolation for ò pru3. ò MD ij + 0.18839 exp(3 * ò pr) , . . . . . . . . . . . . . . . . (3.148) ò oM D oij which avoids negative D ij for oils at ò pru3.7 as estimated by the Sigmund correlation. Lowpressure binary gas diffusion coefficients,6 D oij , can be estimated from D oij + 0.001883
T 3ń2ƪǒ1ńM iǓ ) ǒ1ńM jǓƫ p os 2ijW ij
0.193 ) where W ij + 1.06036 T ij0.1561 expǒ0.47635T ijǓ 40
. . . . . . . . . . . . . . . . . . . . . . . . (3.149e)
s ij + 0.5ǒs i ) s jǓ , . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.149f) and s i + 0.1866
v 1ń3 ci Z 6ń5 ci
,
. . . . . . . . . . . . . . . . . . . . . . . . (3.149g)
with the diffusion coefficient, D oij , in cm2/s; molecular weight, M, in kg/kmol; temperature, T, in K, pressure; p, in bar; Lennard Jones 126 potential parameter, s, in Å; LennardJones 126 potential parameter, e/k, in K; and critical volume, vc , in m3/kmol and where Z c +critical compressibility factor and i and j+diffusing and concentrated species, respectively. To obtain the lowpressure densitydiffusivity product, we use the idealgas law, ò oM + p ońRT, to get D oij ò oM
+ ǒ2.2648
10
*5
Ǔ
T 1ń2ƪǒ1ńM iǓ ) ǒ1ńM jǓƫ
1ń2
s 2ij W ij
,
. . . . . . . . . . . . . . . . . . (3.150)
1 * yi
ȍ y ńD
ci
0.5
, . . . . . . . . (3.149a)
where ò and ò M have units g mol/cm3. The accuracy of the Sigmund correlation for liquids is not known, but the extension proposed by da Silva and Belery (Eq. 148) for large reduced densities does avoid negative diffusivities calculated by the Sigmund equation.94 Renner95 proposes a generalized correlation for effective diffusion coefficients of light hydrocarbons and CO2 in reservoir liquids that can be used as an alternative to the Sigmundtype correlation. *1.831 4.524 M *0.6898 ò 1.706 T , D im + 10 *9 m *0.4562 o Mi p i
. . . . . . . . . . . . . . . . . . . (3.151) with D in cm2/s and where m o +oil viscosity in cp, M i +molecular weight, ò Mi +molar density of Component i at p and T in g mol/cm3, p+pressure in psia, and T+temperature in K. This correlation is based on 141 experimental data with the following property ranges: 0.2t m ot134 cp; 16tM it44; 0.04t ò Mit7 kmol/m3; 14.7tp t2,560 psia; and 273tTt333 K, where i+CO2, C1, C2, and C3. Renner also gives a correlation for diffusivity of CO2 in water/ brine systems. D CO
2*w
+ ǒ6.392
6.911 10 3Ǔ m CO m w*0.1584, . . . . . . . . . (3.152) 2
with D in cm2/s and m in cp. 3.6 KĆValue Correlations This section covers the estimation of equilibrium K values by correlations and the calculation of twophase equilibrium when K values are known. The K value is defined as the ratio of equilibrium gas composition yi to the equilibrium liquid composition x i, K i 5 y ińx i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.153) K i is a function of pressure, temperature, and overall composition. K values can be estimated with empirical correlations or by satisfying the equalfugacity constraint with an EOS (see Chap. 4). Although the increasing use of EOS’s has tended to lessen interest in empirical Kvalue correlations, empirical methods are still useful for such engineering calculations as (1) multistage surface separation, (2) compositional reservoir material balance, and (3) checking the consistency of separatoroil and gas compositions. PHASE BEHAVIOR
Fig. 3.24—General behavior of a K value vs. pressure plot on loglog scale.
Several methods for correlating K values have appeared in the past 50 years. Most rely on two limiting conditions for describing the pressure dependence of K values. First, at low pressures, Raoult’s and Dalton’s laws3 can be used to show that K i [ p vi ǒ T Ǔńp, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.154) where p v +component vapor pressure at the system temperature. The limitations of this equation are that temperature must be less than the component critical temperature (because vapor pressure is not defined at supercritical temperatures) and that the component behaves as an ideal gas. Also, the equation implies that the K value is independent of overall composition. In fact, the pressure dependence of lowpressure K values is closely approximated by Eq. 3.154. The second observation is that, at high pressures, the K values of all components in a mixture tend to converge to unity at the same pressure. This pressure is called the convergence pressure96 and, for binaries, represents the actual mixture critical pressure. For multicomponent mixtures, the convergence pressure is a nonphysical condition unless the system temperature equals the mixture critical temperature.97,98 This is because a mixture becomes single phase at the bubblepoint or dewpoint pressure before reaching the convergence pressure. The loglog plot of K i vs. pressure in Fig. 3.24 shows how the idealgas and convergencepressure conditions define the Kvalue behavior at limiting conditions. For light components (where T u T ci ), K values decrease monotonically toward the convergence pressure. For heavier components (where T t T ci ), K values initially decrease as a function of pressure at low pressures, passing through unity when system pressure equals the vapor pressure of a particular component, reaching a minimum, and finally increasing toward unity at the convergence pressure. GAS AND OIL PROPERTIES AND CORRELATIONS
For reservoir fluids, the pressure where K values reach a minimum is usually u1,000 psia (Fig. 3.25), implying that K values are more or less independent of convergence pressure (i.e., composition) at pressures t1,000 psia. This observation has been used to develop general “lowpressure” Kvalue correlations for surfaceseparator calculations. 3.6.1 Hoffman et al. Method. Hoffman et al.99 propose a method for correlating K values that has received widespread application. Ki +
10
ǒA0 ) A1 Fi Ǔ p
or log K i p + A 0 ) A 1 F i , . . . . . . . . . . . . . . . . . . . . . . (3.155) where F i +
1ńT bi * 1ńT logǒ p cińp scǓ ; . . . . . . . . . . . . (3.156) 1ńT bi * 1ńT ci
T c +critical temperature; p c +pressure; T b +normal boiling point; p sc +pressure at standard conditions; and A 1 and A 0 +slope and intercept, respectively, of the plot log(K i p) vs. F i. Hoffman et al. show that measured K values for a reservoir gas condensate correlate well with the proposed equation. They found that trend of log(K i p) vs. F i is linear for components C1 through C6 at all pressures, while the function turns downward for heavier components at low pressures. Interestingly, the trend becomes more linear for all components at higher pressures. As Fig. 3.26 shows, Slope A 1 and Intercept A 0 vary with pressure. For low pressures, K i [ p vńp. With the Clapeyron vapor pressure relation,5 log(p v) + a * bńT results in A 0 + log(p sc) and A 1 + 1. These limiting values of A 0 and A 1 are close to the values found when A 0( p) and A 1( p) are extrapolated to p + p sc. Because 41
Fig. 3.25—K values at 120°F for binary and reservoirfluid systems with convergence pressures ranging from 800 to 10,000 psia (from Standing3).
K values tend toward unity as pressure approaches the convergence pressure, p K , it is necessary that A 0 + log(p K) and A 1 ³ 0. Several authors have noted that plots of log(K i p) vs. F i tend to converge at a common point. Brinkman and Sicking101 suggest that this “pivot” point represents the convergence pressure where K i + 1 and p + p K. The value of F i at the pivot point, F K, is easily shown to equal log(p Kńp sc). It is interesting to note that the wellknown Wilson102,103 equation, Ki +
Ǔ exp 5.37(1 ) w i)ǒ1 * T *1 ri , . . . . . . . . . . . . . . (3.157) p ri
is identical to the Hoffman et al.99 relation for A 0 + log(p sc) and A 1 + 1 when the Edmister104 correlation for acentric factor equation, 42
T bińT ci wi + 3 logǒ p cińp scǓ * 1 , . . . . . . . . . . . . (3.158) 7 1 * T bińT ci is used in the Wilson equation. Note that 5.37+(7/3) ln (10). Whitson and Torp100 suggest a generalized form of the Hoffman et al.99 equation in terms of convergence pressure and acentric factor.
ǒ Ǔ
p K i + pci K
A 1*1
Ǔƫ expƪ5.37 A 1 (1 ) w i)ǒ1 * T *1 ri , p ri . . . . . . . . . . . . . . . . . . . (3.159)
where A 1 +a function of pressure, with A 1 + 1 at p + p sc and A 1 + 0 at p + p K. The key characteristics of K values vs. pressure PHASE BEHAVIOR
log pK
TABLE 3.3—VALUES OF b AND Tb FOR USE IN STANDING LOWPRESSURE KVALUE CORRELATION
–
Component, i
bi (cycle°R)
Tbi °R
470
109
N2
Intercept A0
Slope A1
Pressure, psia
Fig. 3.26—Pressure dependence of slope, A1, and intercept, A0, in Hoffman et al. KpF relationship (Eq. 3.155) for a North Sea gas condesate NS1 (from Whitson and Torp100).
CO2
652
194
H2 S
1,136
331
C1
300
94
C2
1,145
303
C3
1,799
416
iC4
2,037
471
nC4
2,153
491
iC5
2,368
542
nC5
2,480
557
C6 (lumped)
2,738
610
nC6
2,780
616
nC7
3,068
669
nC8
3,335
718
nC9
3,590
763
nC10
3,828
805
For C7+ fractions, see Eqs. 3.161f through 3.161h
and temperature are correctly predicted by Eq. 3.159, where the following pressure dependence for A 1 is suggested. A 1 + 1 * (pńp K) A 2 , . . . . . . . . . . . . . . . . . . . . . . . . . . (3.160) where A 2 ranges from 0.5 to 0.8 and pressures p and p K are given in psig. Canfield105 also suggests a simple Kvalue correlation based on convergence pressure. 3.6.2 Standing LowPressure K Values. Standing106 uses the Hoffman et al.99 method to generate a lowpressure Kvalue equation for surfaceseparator calculations ( p sp t 1, 000 psia and T sp t 200°F). Standing fits A 1 and A 0 in Eq. 3.155 as a function of pressure using Kvalue data from an Oklahoma City crude oil. He treats the C 7) by correlating the behavior of K C as a function of 7) “effective” carbon number n C . The Standing equations are 7)
ǒA0 ) A1 Fi Ǔ
, . . . . . . . . . . . . . . . . . . . . . . . . (3.161a)
F i + b iǒ1ńT bi * 1ńTǓ,
. . . . . . . . . . . . . . . . . . . . . . . (3.161b)
K i + p1 10 sp
b i + logǒ p cińp scǓńǒ1ńT bi * 1ńT ciǓ , . . . . . . . . . . . . . . . (3.161c) A 0ǒ pǓ + 1.2 ) ǒ4.5
10
*4
Ǔp ) ǒ15
10
*8
Ǔp ,
10 *4Ǔp * ǒ3.5
10 *8Ǔp 2,
. . . . . . . . . . . . . . . . . . . (3.161e) nC
7)
+ 7.3 ) 0.0075T ) 0.0016p,
bC
7)
+ 1, 013 ) 324n C
and T bC
7)
7)
+ 301 ) 59.85n C
. . . . . . . . . . . . . (3.161f)
* 4.256n 2C 7)
7)
,
* 0.971n 2C
. . . . . . . (3.161g) ,
7)
. . . . . (3.161h)
with T in °R except when calculating n C (for n C , T is in °F) and 7) 7) p in psia. Standing suggests modified values of b i and T bi for nonhydrocarbons, methane, and ethane (Table 3.3). Glasø and Whitson107 show that these equations are accurate for separator flash calculations of crude oils with GOR’s ranging from 300 to 1,500 scf/STB and oil gravity ranging from 26 to 48°API. Experience shows, however, that significant errors in calculated GOR may result for lean gas condensates, probably because of inaccurate C 1 and GAS AND OIL PROPERTIES AND CORRELATIONS
3.6.3 GalimbertiCampbell Method. Galimberti and Campbell108,109 suggested another useful approach for correlating K values where log K i + A 0 ) A 1T ci2
. . . . . . . . . . . . . . . . . . . . . . . . . (3.162)
is shown to correlate K values for several simple mixtures containing hydrocarbons C 1 through C 10 at pressures up to 3,000 psia and temperatures from *60 to 300°F. Whitson developed a lowpressure Kvalue correlation, based on data from Roland,110 at pressures t1,000 psia and temperatures from 40 to 200°F, for separator calculations of gas condensates. A 0 + 4.276 * ǒ7.6
10 *4ǓT
) ƪ* 1.18 ) ǒ5.675
10 *4ǓTƫ log p , . . . . . . . . (3.163a)
Ǌ
A 1 + 10 *6 ǒ* 4.9563 ) 0.00955T Ǔ ) ƪǒ1.9094
2
. . . . . . . . . . . . . . . . . . (3.161d) A 1ǒ pǓ + 0.890 * ǒ1.7
C 7) K values. The Hoffman et al. method with Standing’s lowpressure correlations are particularly useful for checking the consistency of separatorgas and oil compositions.
* ǒ1.235
10 *5ǓT ) ǒ3.34
10 *3Ǔ
ǋ
10 *8ǓT 2ƫ p , . . . (3.163b)
T cC1 + 343 * 0.04p, . . . . . . . . . . . . . . . . . . . . . . . . . (3.163c) and T c C7) + 1, 052.5 * 0.5125T ) 0.00375T 2 , . . . . (3.163d) with p in psia, T in °F, and T c in °R. 3.6.4 Nonhydrocarbon K Values. Lohrenz et al.111 reported nonhydrocarbon K values as a function of pressure, temperature, and convergence pressure. ln K H
2S
ǒ
p + 1*p
K
Ǔ ƪ6.3992127 ) 1, 399.2204 T 0.8
* 0.76885112 ln p * *
ƫ
1, 112, 446.2 , T2
18.215052 ln p T
. . . . . . . . . . . . . . . . . . . . . (3.164a) 43
p 1, 184.2409 + ǒ1 * p Ǔ ǒ11.294748 * T 0.4
ln K N
2
K
Ǔ
* 0.90459907 ln p , . . . . . . . . . . . . . . . . . . . (3.164b)
ǒ
p ln K CO + 1 * p 2
0.6
K
ln p )
Ǔ ǒ7.0201913 * 152.7291 * 1.8896974 T
Ǔ
1, 719.2956 ln p 644, 740.69 ln p * , T T2 . . . . . . . . . . . . . . . . . . . (3.164c)
with p in psia and T in °R. For lowpressure Kvalue estimation, the first term in Eq. 3.164 simplifies to unity (assuming that 1 * pńp K [ 1) and the K values become functions of pressure and temperature only. However, these equations do not give the correct lowpressure value of ē(ln K i)ńē(ln p) + * 1 3.6.5 ConvergencePressure Estimation. For correlation purposes, convergence pressure is used as a variable to define the composition dependence of K values. Convergence pressure is a function of overall composition and temperature. Whitson and Michelsen112 show that convergence pressure is a thermodynamic phenomenon, with the characteristics of a true mixture critical point, that can be predicted with EOS’s. Rzasa et al.113 give an empirical correlation for convergence pressure as a function of temperature and the product (Mg) C . 7) Standing2 suggests that convergence pressure of reservoir fluids varies almost linearly with C 7) molecular weight. Convergence pressure can also be calculated with a trialanderror procedure suggested by Rowe.97,98,114 This procedure involves the use of several empirical correlations for estimating mixture critical pressure and temperature, pseudocomponent critical properties, and the K values of methane and octane. The Galimberti and Campbell108,109 Kvalue method is used to estimate K values of other components by interpolation and extrapolation of the C 1 and C 8 K values. This approach to convergence pressure is necessary if the K values are used for processes that approach critical conditions or where K values change significantly because of overall composition effects. The method cannot, of course, be more accurate than the correlations it uses and therefore is expected to yield only qualitatively correct results. For reservoir calculations where convergence pressure can be assumed constant (e.g., pressure depletion), a more direct approach to determining convergence pressure is suggested. With a Kvalue correlation of the form K i + K( p K, p, T ) as in Eq. 3.159, the convergence pressure can be estimated from a single experimental saturation pressure. For a bubblepoint and a dewpoint, Eqs. 3.165 and 3.166, respectively, must be satisfied. F ǒ p KǓ + 1 *
ȍ z K ǒp , p , TǓ + 0 N
i
i
K
b
. . . . . . . . . . . . . (3.165)
i+1
and Fǒ p KǓ + 1 *
ȍ N
zi
i+1 K i ǒ p K, p d , T Ǔ
+ 0, . . . . . . . . . . . (3.166)
where z i, p b, or p d and T are specified and p K is determined. The twophase flash calculation, with K values given, is discussed in Chap. 4 in the PhaseSplit Calculation section. References 1. The SI Metric System of Units and SPE Metric Standard, SPE, Richardson, Texas (June 1982). 2. van der Waals, J.D.: “Continuity of the Gaseous and Liquid State of Matter,” PhD dissertation, U. of Leiden (1873). 3. Standing, M.B.: Volumetric and Phase Behavior of Oil Field Hydrocarbon Systems, SPE, Richardson, Texas (1981). 4. Standing, M.B. and Katz, D.L.: “Density of Natural Gases,” Trans., AIME (1942) 146, 140. 44
5. Kay, W.B.: “Density of Hydrocarbon Gases and Vapors at High Temperature and Pressure,” Ind. Eng. Chem. (1936) No. 28, 1014. 6. Reid, R.C., Prausnitz, J.M., and Polling, B.E.: The Properties of Gases and Liquids, fourth edition, McGrawHill Book Co. Inc., New York City (1987) 388–485. 7. Sutton, R.P.: “Compressibility Factors for HighMolecular Weight Reservoir Gases,” paper SPE 14265 presented at the 1985 SPE Annual Technical Conference and Exhibition, Las Vegas, Nevada, 22–25 September. 8. Ramey, H.J. Jr.: “Rapid Methods for Estimating Reservoir Compressibilities,” JPT (April 1964) 447; Trans., AIME, 231. 9. Fetkovich, M.J., Reese, D.E., and Whitson, C.H.: “Application of a General Material Balance for HighPressure Gas Reservoirs,” SPE Journal (March 1998) 3. 10. Fick, A.: Am. Phys., Leipzig (1855) 170, 59. 11. Coats, K.H.: “Implicit Compositional Simulation of SinglePorosity and DualPorosity Reservoirs,” paper SPE 18427 presented at the 1989 SPE Symposium on Reservoir Simulation, Houston, 6–8 February. 12. da Silva, F.V. and Belery, P.: “Molecular Diffusion in Naturally Fractured Reservoirs—A Decisive Recovery Mechanism,” paper SPE 19672 presented at the 1987 SPE Annual Technical Conference and Exhibition, San Antonio, Texas, 8–11 October. 13. Amyx, J.W., Bass, D.M. Jr., and Whiting, R.L.: Petroleum Reservoir Engineering, McGrawHill Book Co. Inc., New York City (1960). 14. Christoffersen, K. and Whitson, C.H.: “Gas/Oil Capillary Pressure of Chalk at Elevated Pressures,” SPEFE (September 1995) 153. 15. Delclaud, J., Rochon, J., and Nectoux, A.: “Investigation of Gas/Oil Relative Permeabilities: HighPermeability Oil Reservoir Application,” paper SPE 16966 presented at the 1987 SPE Annual Technical Conference and Exhibition, Dallas, 27–30 September. 16. Katz, D.L. et al.: Handbook of Natural Gas Engineering, McGrawHill Book Co. Inc., New York City (1959) 69–93. 17. Standing, M.B.: OilSystem Correlations, P.P. Handbook (ed.), McGrawHill Book Co. Inc., New York City (1962). 18. Beggs, H.D.: “Oil System Correlations,” Petroleum Engineering Handbook, SPE, Richardson, TX (1987) Chap. 22. 19. McCain, W.D. Jr.: “ReservoirFluid Property Correlations—State of the Art,” SPERE (May 1991) 266. 20. Whitson, C.H. and Torp, S.B.: “Evaluating Constant Volume Depletion Data,” JPT (March 1983) 610; Trans., AIME, 275. 21. Hall, K.R. and Yarborough, L.: “A New EOS for Zfactor Calculations,” Oil & Gas J. (18 June 1973) 82. 22. Yarborough, L. and Hall, K.R.: “How to Solve EOS for Zfactors,” Oil & Gas J. (18 February 1974) 86. 23. Takacs, G.: “Comparisons Made for Computer Zfactor Calculations,” Oil & Gas J. (20 December 1976) 64. 24. Dranchuk, P.M. and AbouKassem, J.H.: “Calculation of ZFactors for Natural Gases Using Equations of State,” J. Cdn. Pet. Tech. (July–September 1975) 14, No. 3, 34. 25. Brill, J.P. and Beggs, H.D.: “TwoPhase Flow in Pipes,” paper presented at the U. Tulsa INTERCOMP Course, The Hague (1974). 26. Lee, B.I. and Kesler, M.G.: “A Generalized Thermodynamic Correlation Based on ThreeParameter Corresponding States,” AIChE J. (1975) 21, 510. 27. API Technical Data Book––Petroleum Refining, third edition, API, New York City (1977). 28. Starling, K.E., Mannan, M., and Savidge, J.L.: “Equation Predicts Supercompressibility for Wet, Sour Gases,” Oil & Gas J. (2 January 1989) 31. 29. Ely, J.F. and Hanley, H.J.M.: Ind. Eng. Chem. Fund. (1983) 22, 90. 30. Wichert, E. and Aziz, K.: “Compressibility Factor of Sour Natural Gases,” Cdn. J. Chem. Eng. (1971) 49, 267. 31. Wichert, E. and Aziz, K.: “Calculate Z’s for Sour Gases,” Hydro. Proc. (May 1972) 51, 119. 32. Matthews, T.A., Roland, C.H., and Katz, D.L.: “High Pressure Gas Measurement,” Petroleum Refiner (1942) 21, No. 6, 58. 33. Standing, M.B.: Petroleum Engineering Data Book, Norwegian Inst. of Technology, Trondheim, Norway (1974). 34. Eilerts, C.K.: “Gas Condensate Reservoir Engineering, 1. The Reserve Fluid, Its Composition and Phase Behavior,” Oil & Gas J. (1 February 1947). 35. Cragoe, C.S.: “Thermodynamic Properties of Petroleum Products,” U.S. Dept. of Commerce, Washington, DC (1929) 97. 36. Gold, D.K., McCain, W.D. Jr., and Jennings, J.W.: “An Improved Method for the Determination of the Reservoir Gas Gravity for Retrograde Gases,” paper SPE 17310 presented at the 1988 SPE Permian Basin Oil and Gas Recovery Conference, Midland, Texas, 10–11 March. PHASE BEHAVIOR
37. Gold, D.K., McCain, W.D. Jr., and Jennings, J.W.: “An Improved Method for the Determination of the Reservoir Gas Gravity for Retrograde Gases,” JPT (July 1989) 41, 747; Trans., AIME, 287. 38. Whitson, C.H.: “Discussion of An Improved Method for the Determination of the ReservoirGas SpecificGravity for Retrograde Gases,” JPT (November 1989) 1216. 39. Leshikar, A.G.: “How to Estimate Equivalent Gas Volume of Stock Tank Condensate,” World Oil (January 1961) 108. 40. Standing, M.B.: “A PressureVolumeTemperature Correlation for Mixtures of California Oils and Gases,” Drill. & Prod. Prac. (1947) 275. 41. Katz, D.L.: “Prediction of the Shrinkage of Crude Oils,” Drill. & Prod. Prac. (1942) 137. 42. Carr, N.L., Kobayashi, R., and Burrows, D.B.: “Viscosity of Hydrocarbon Gases Under Pressure,” Trans., AIME (1954) 201, 264. 43. Dempsey, J.R.: “Computer Routine Treats Gas Viscosity as a Variable,” Oil & Gas J. (16 August 1965) 141. 44. Lee, A.L., Gonzalez, M.H., and Eakin, B.E.: “The Viscosity of Natural Gases,” JPT (August 1966) 997; Trans., AIME, 237. 45. Lucas, K.: Chem. Ing. Tech. (1981) 53, 959. 46. Sage, B.H. and Olds, R.H.: “Volumetric Behavior of Oil and Gas from Several San Joaquin Valley Fields,” Trans., AIME (1947) 170, 156. 47. Eilerts, C.K.: Phase Relations of Gas Condensate Fluids, Monograph 10, U.S. Bureau of Mines, American Gas Assn., New York City (1957) 1 and 2. 48. Eilerts, C.K. et al.: “Phase Relations of a GasCondensate Fluid at Low Tempertures, Including the Critical State,” Pet. Eng. (February 1948) 19, 154. 49. Nemeth, L.K. and Kennedy, H.T.: “A Correlation of Dewpoint Pressure With Fluid Composition and Temperature,” SPEJ (June 1967) 99; Trans., AIME, 240. 50. Organick, E.I. and Golding, B.H.: “Prediction of Saturation Pressures for CondensateGas and VolatileOil Mixtures,” Trans., AIME (1952) 195, 135. 51. Kurata, F. and Katz, D.L.: “Critical Properties of Volatile Hydrocarbon Mixtures,” Trans., AIChE (1942) 38, 995. 52. Glasø, O.: “Generalized Pressure/Volume/Temperature Correlations,” JPT (November 1980) 785. 53. Lasater, J.A.: “Bubblepoint Pressure Correlation,” Trans., AIME (1958) 213, 379. 54. Sutton, R.P. and Farshad, F.F.: “Evaluation of Empirically Derived PVT Properties for Gulf of Mexico Crude Oils,” SPERE (February 1990) 79. 55. Whitson, C.H.: “Characterizing Hydrocarbon Plus Fractions,” paper SPE 12233 presented at the 1980 SPE European Offshore Petroleum Conference, London, 21–24 October. 56. Whitson, C.H.: “Characterizing Hydrocarbon Plus Fractions,” SPEJ (August 1983) 683; Trans., AIME, 275. 57. Vazquez, M. and Beggs, H.D.: “Correlations for Fluid Physical Property Prediction,” JPT (June 1980) 968. 58. Standing, M.B. and Katz, D.L.: “Density of Crude Oils Saturated With Natural Gas,” Trans., AIME (1942) 146, 159. 59. Madrazo, A.: “LiquidDensity Correlation of Hydrocarbon Systems,” Trans., AIME (1960) 219, 386. 60. Vogel, J.L. and Yarborough, L.: “The Effect of Nitrogen on the Phase Behavior and Physical Properties of Reservoir Fluids,” paper SPE 8815 presented at the 1980 SPE Annual Technical Conference and Exhibition, Tulsa, Oklahoma, 20–23 April. 61. Alani, G.H. and Kennedy, H.T.: “Volumes of Liquid Hydrocarbons at High Temperatures and Pressures,” Trans., AIME (1960) 219, 288. 62. Lohrenz, J., Bray, B.G., and Clark, C.R.: “Calculating Viscosities of Reservoir Fluids From Their Compositions,” JPT (October 1964) 1171; Trans., AIME, 231. 63. Rackett, H.G.: “EOS for Saturated Liquids,” J. Chem. Eng. Data (1970) 15, No. 4, 514. 64. Hankinson, R.W. and Thomson, G.H.: “A New Correlation for Saturated Densities of Liquids and Their Mixtures,” AIChE J. (1979) 25, No. 4, 653. 65. Hankinson, R.W. et al.: “Volume Correction Factors for Lubricating Oils,” Oil & Gas J. (28 September 1981) 297. 66. Cullick, A.S., Pebdani, F.N., and Griewank, A.K.: “Modified Corresponding States Method for Predicting Densities of Petroleum Reservoir Fluids,” paper presented at the 1988 AIChE Spring Natl. Meeting, New Orleans, 7–10 March. 67. Chien, M.C.H. and Monroy, M.R.: “Two New Density Correlations,” paper SPE 15676 presented at the 1976 SPE Annual Technical Conference and Exhibition, New Orleans, 5–8 October. 68. Ahmed, T.: Hydrocarbon Phase Behavior, first edition, Gulf Publishing Co., Houston (1989) 7. GAS AND OIL PROPERTIES AND CORRELATIONS
69. Craft, B.C. and Hawkins, M.: Applied Petroleum Reservoir Engineering, first edition, PrenticeHall, Englewood Cliffs, New Jersey (1959) 126–29. 70. Trube, A.S.: “Compressibility of Undersaturated Hydrocarbon Reservoir Fluids,” Trans., AIME (1957) 210, 241. 71. AlMarhoun, M.A.: “New Correlations for FVFs of Oil and Gas Mixtures,” PhD dissertation, King Fahd U. of Petroleum & Minerals (1990). 72. Beal, C.: “The Viscosity of Air, Water, Natural Gas, Crude Oil and Its Associated Gases at Oilfield Temperatures and Pressures,” Trans., AIME (1946) 165, 94. 73. Beggs, H.D. and Robinson, J.R.: “Estimating the Viscosity of Crude Oil Systems,” JPT (September 1975) 1140. 74. AlKhafaji, A.H., AbdulMajeed, G.H., and Hassoon, S.F.: “Viscosity Correlation for Dead, Live, and Undersaturated Crude Oils,” J. Pet. Res. (1987) 6, No. 2, 1. 75. Standing, M.B.: “UOP Characterization Factor,” TI program listing, available from C.H. Whitson, Norwegian Inst. of Science and Technology, NTNU, [email protected]. 76. Chew, J.N. and Connally, C.A.: “A Viscosity Correlation for GasSaturated Crude Oils,” Trans., AIME (1959) 216, 23. 77. Aziz, K., Govier, G.W., and Fogarasi, M.: “Pressure Drop in Wells Producing Oil and Gas,” J. Cdn. Pet. Tech. (July–September 1972) 38. 78. AbuKhamsin, S.A. and AlMarhoun, M.A.: “Development of a New Correlation for Bubblepoint Oil Viscosity,” Arabian J. Sci. & Eng. (April 1991) 16, No. 2A, 99. 79. Simon, R., Rosman, A., and Zana, E.: “PhaseBehavior Properties of CO2Reservoir Oil Systems,” SPEJ (February 1978) 20. 80. AbdulMajeed, G.H., Kattan, R.R., and Salman, N.H.: “New Correlation for Estimating the Viscosity of Undersaturated Crude Oils,” J. Cdn. Pet. Tech. (May–June 1990) 29, No. 3, 80. 81. Khan, S.A. et al.: “Viscosity Correlations for Saudi Arabian Crude Oils,” paper SPE 15720 presented at the 1987 SPE Middle East Oil Technical Conference and Exhibition, Manama, Bahrain, 8–10 March. 82. Jossi, J.A., Stiel, L.I., and Thodos, G.: “The Viscosity of Pure Substances in the Dense Gaseous and Liquid Phases,” AIChE J. (1962) 8, 59. 83. Stiel, L.I. and Thodos, G.: “The Viscosity of Nonpolar Gases at Normal Pressures,” AIChE J. (1961) 7, 611. 84. Weinaug, C.F. and Katz, D.L.: “Surface Tensions of MethanePropane Mixtures,” Ind. & Eng. Chem. (1943) 35, 239. 85. Macleod, D.B.: “On a Relation Between Surface Tension and Density,” Trans., Faraday Soc. (1923) 19, 38. 86. Nokay, R.: “Estimate Petrochemical Properties,” Chem. Eng. (23 February 1959) 147. 87. Katz, D.L. and Saltman, W.: “Surface Tension of Hydrocarbons,” Ind. & Eng. Chem. (January 1939) 31, 91. 88. Katz, D.L., Monroe, R.R., and Trainer, R.P.: “Surface Tension of Crude Oils Containing Dissolved Gases,” Pet. Tech. (September 1943). 89. Standing, M.B. and Katz, D.L.: “VaporLiquid Equilibria of Natural GasCrude Oil Systems,” Trans., AIME (1944) 155, 232. 90. Firoozabadi, A. et al.: “Surface Tension of Reservoir CrudeOil/Gas Systems Recognizing the Asphalt in the Heavy Fraction,” SPERE (February 1988) 265. 91. Ramey, H.J. Jr.: “Correlations of Surface and Interfacial Tensions of Reservoir Fluids,” paper SPE 4429 available from SPE, Richardson, Texas (1973). 92. Wilke, C.R.: “A Viscosity Equation for Gas Mixtures,” J. Chem. Phy. (1950) 18, 517. 93. Sigmund, P.M.: “Prediction of Molecular Diffusion at Reservoir Conditions. Part I—Measurement and Prediction of Binary Dense Gas Diffusion Coefficients,” J. Cdn. Pet. Tech. (April–June 1976) 48. 94. Christoffersen, K.: “HighPressure Experiments with Application to Naturally Fractured Chalk Reservoirs. 1. Constant Volume Diffusion. 2. GasOil Capillary Pressure,” Dr.Ing. dissertation, U. Trondheim, Trondheim, Norway (1992). 95. Renner, T.A.: “Measurement and Correlation of Diffusion Coefficients for CO2 and RichGas Applications,” SPERE (May 1988) 517; Trans., AIME, 285. 96. Hadden, S.T.: “Convergence Pressure in Hydrocarbon VaporLiquid Equilibra,” Chem. Eng. Prog. (1953) 49, No. 7, 53. 97. Rowe, A.M. Jr.: “Applications of a New Convergence Pressure Concept to the Enriched Gas Drive Process,” PhD dissertation, U. of Texas, Austin, Texas (1964). 98. Rowe, A.M. Jr.: “The Critical Composition Method—A New Convergence Pressure Method,” SPEJ (March 1967) 54; Trans., AIME, 240. 99. Hoffmann, A.E., Crump, J.S., and Hocott, C.R.: “Equilibrium Constants for a GasCondensate System,” Trans., AIME (1953) 198, 1. 45
100. Whitson, C.H. and Torp, S.B.: “Evaluating Constant Volume Depletion Data,” JPT (March 1983) ; Trans., AIME, 275. 101. Brinkman, F.H. and Sicking, J.N.: “Equilibrium Ratios for Reservoir Studies,” Trans., AIME (1960) 219, 313. 102. Wilson, G.M.: “Calculation of Enthalpy Data From a Modified RedlichKwong EOS,” Advances in Cryogenic Eng. (1966) 11, 392. 103. Wilson, G.M.: “A Modified RedlichKwong EOS, Application to General Physical Data Calculations,” paper 15c presented at the 1969 AIChE Natl. Meeting, Cleveland, Ohio. 104. Edmister, W.C.: “Applied Hydrocarbon Thermodynamics, Part 4: Compressibility Factors and Equations of State,” Pet. Ref. (April 1958) 37, 173. 105. Canfield, F.B.: “Estimate KValues with the Computer,” Hydro. Proc. (April 1971) 137. 106. Standing, M.B.: “A Set of Equations for Computing Equilibrium Ratios of a Crude Oil/Natural Gas System at Pressures Below 1,000 psia,” JPT (September 1979) 1193. 107. Glasø, O. and Whitson, C.H.: “The Accuracy of PVT Parameters Calculated From Computer Flash Separation at Pressures Less Than 1,000 psia,” JPT (August 1980) 1811. 108. Galimberti, M. and Campbell, J.M.: “Dependence of Equilibrium Vaporization Ratios (KValues) on Critical Temperature,” Proc., 48th NGPA Annual Convention (1969) 68. 109. Galimberti, M. and Campbell, J.M.: “New Method Helps Correlate K Values for Behavior of Paraffin Hydrocarbons,” Oil & Gas J. (November 1969) 64. 110. Roland, C.H.: “Vapor Liquid Equilibrium for Natural GasCrude Oil Mixtures,” Ind. & Eng. Chem. (1945) 37, 930. 111. Lohrenz, J., Clark, G.C., and Francis, R.J.: “A Compositional Material Balance for Combination Drive Reservoirs With Gas and Water Injection,” JPT (November 1963) 1233; Trans., AIME, 228. 112. Whitson, C.H. and Michelsen, M.L.: “The Negative Flash,” Fluid Phase Equilibria (1989) 53, 51.
46
113. Rzasa, M.J., Glass, E.D., and Opfell, J.B.: “Prediction of Critical Properties and Equilibrium Vaporization Constants for Complex Hydrocarbon Systems,” Chem. Eng. Prog. (1952) 2, 28. 114. Rowe, A.M. Jr.: “Internally Consistent Correlations for Predicting Phase Compositions for Use in Reservoir Composition Simulators,” paper SPE 7475 presented at the 1978 SPE Annual Technical Conference and Exhibition, Houston, 1–3 October.
SI Metric Conversion Factors Å 1.0* E*01 +nm °API 141.5/(131.5)°API) +g/cm3 bar 1.0* E)05 +Pa bbl 1.589 873 E*01 +m3 Btu/lbm mol 2.236 E)03 +J/mol cp 1.0* E*03 +Pa@s cSt 1.0* E*06 +m2/s dyne/cm 1.0* E)00 +mN/m ft 3.048* E*01 +m E*02 +m2 ft2 9.290 304* ft3 2.831 685 E*02 +m3 ft3/lbm mol 6.242 796 E*02 +m3/kmol °F (°F*32)/1.8 +°C °F (°F)459.67)/1.8 +K E)00 +cm2 in.2 6.451 6* lbm 4.535 924 E*01 +kg lbm mol 4.535 924 E*01 +kmol psi 6.894 757 E)00 +kPa E*01 +kPa*1 psi*1 1.450 377 °R 5/9 +K *Conversion factor is exact.
PHASE BEHAVIOR
Chapter 4
EquationĆofĆState Calculations 4.1 Introduction Cubic equations of state (EOS’s) are simple equations relating pressure, volume, and temperature (PVT). They accurately describe the volumetric and phase behavior of pure compounds and mixtures, requiring only critical properties and acentric factor of each component. The same equation is used to calculate the properties of all phases, thereby ensuring consistency in reservoir processes that approach critical conditions (e.g., misciblegas injection and depletion of volatileoil/gascondensate reservoirs). Problems involving multiphase behavior, such as lowtemperature CO2 flooding, can be treated with an EOS, and even water/hydrocarbonphase behavior can be predicted accurately with a cubic EOS. Volumetric behavior is calculated by solving a simple cubic equation, usually expressed in terms of Z+pv/RT, Z 3 ) A 2 Z 2 ) A 1 Z ) A 0 + 0, . . . . . . . . . . . . . . . . . . . . (4.1) where constants A0, A1, and A2 are functions of pressure, temperature, and phase composition. Phase equilibria are calculated with an EOS by satisfying the condition of chemical equilibrium. For a twophase system, the chemical potential of each component in the liquid phase mi (x) must equal the chemical potential of each component in the vapor phase mi ( y), mi ( x)+mi ( y). Chemical potential is usually expressed in terms of fugacity, fi , where mi +RT ln fi )li (T ) and li (T ) are constant terms that drop out in most problems.13 It is readily shown that the condition mi (x)+mi ( y) is satisfied by the equalfugacity constraint, fLi +fvi , where fugacity is given by R
f ln f i + ln i + 1 RT yi p
ŕ ǒēnēp * RTVǓ dV * ln Z.
. . . . . . (4.2)
i
V
Other thermodynamic properties, such as Helmholz energy, enthalpy, and entropy, can be readily defined in terms of the fugacity coefficient. Michelsen4 gives a particularly compact and useful discussion of the relation between thermodynamic properties aimed at making efficient EOS calculations. A component material balance is also required to solve vapor/liquid equilibrium problems: zi +Fv yi )(1*Fv )xi , where Fv +mole fraction of the vapor phase+nv /(nv )nL ). Integrating the component balance in the twophase flash calculation is discussed in Sec. 4.3.1. Solving phase equilibria with an EOS is a trialanderror procedure, requiring considerable computations. With today’s computers, however, the task is fast and reliable. The accuracy of EOS predictions has also improved considerably during the past 15 years, EQUATIONOFSTATE CALCULATIONS
during which emphasis has been on improved liquid volumetric predictions and treating the heptanesplus fraction (Chap. 5). This chapter provides the equations and algorithms necessary for calculating phase and volumetric behavior of reservoir fluids with a cubic EOS. Sec. 4.2 reviews the most important cubic equations, starting with van der Waals’5 EOS from 1873 and concluding with the method of volume translation, which has greatly improved the volumetric capabilities of cubic EOS’s. In Secs. 4.3 through 4.5, we present algorithms for solving vapor/ liquid equilibrium (VLE) problems, including the twophase flash, phasestabilitytest, and saturationpressure calculations. Reference is also made to methods for solving threephase and criticalpoint calculations. Sec. 4.6 deals specifically with compositional gradients with depth caused by gravity and thermal diffusion. Finally, Sec. 4.7 covers how to “tune” an EOS to match experimental PVT data (see also Appendix C). 4.2 Cubic EOS's Since the introduction of the van der Waals EOS, many cubic EOS’s have been proposed—e.g., the Redlich and Kwong6 EOS (RK EOS) in 1949, the Peng and Robinson7 EOS (PR EOS) in 1976, and the Martin8 EOS in 1979, to name only a few.915 Most of these equations retain the original van der Waals repulsive term RT/(v*b), modifying only the denominator in the attractive term. The RedlichKwong equation has been the most popular basis for developing new EOS’s. Another trend has been to propose generalized three, four, and fiveconstant cubic equations that can be simplified to the PR EOS, RK EOS, or other familiar forms. Kumar and Starling16,17 use the most general fiveconstant cubic EOS to fit volumetric and phase behavior of nonpolar compounds, although they do not apply the equation to mixtures. Most petroleum engineering applications rely on the PR EOS or a modification of the RK EOS. Several modified RedlichKwong equations have found acceptance, with Soave’s18 modification (SRK EOS) being the simplest and most widely used. Unfortunately the SRK EOS yields poor liquid densities. Zudkevitch and Joffe19 proposed a modified RK EOS, the ZJRK EOS, where both EOS constants are corrected by temperaturedependent functions, resulting in improved volumetric predictions. Yarborough11 proposed a generalized form of the ZJRK EOS for petroleum reservoir mixtures. The PR EOS is comparable with the SRK EOS in simplicity and form. Peng and Robinson7 report that their equation predicts liquid densities better than the SRK EOS, although PR EOS densities are usually inferior to those calculated by the ZJRK EOS. A distinct advantage of the PengRobinson and SoaveRedlichKwong equa1
van der Waals also stated the critical criteria that are used to define the two EOS constants a and b—namely, that the first and second derivatives of pressure with respect to volume equal zero at the critical point of a pure component.
ǒēpēvǓ
p c,T c,v c
+
ǒēēvpǓ 2
2 p ,T ,v c c c
+ 0. . . . . . . . . . . . . . . . . . . (4.5)
Martin and Hou21 show that this constraint is equivalent to the condition (Z*Zc )3+0 at the critical point. Fig. 4.1 shows the pv relation of a pure compound for TtTc , T+Tc , and TuTc , indicating the inflection point on the critical isotherm that represents the van der Waals critical criteria. Imposing Eq. 4.5 on Eq. 4.3 and specifying pc and Tc (as opposed to specifying two of the other critical properties), the constants a and b in the van der Waals equation are given by a + 27 64
R 2 T c2 pc
R Tc and b + 1 p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.6) 8 c The critical volume is given by vc +(3/8)( RTc /pc ), resulting in a constant critical compressibility factor. Zc +
pc vc + 3 + 0.375. 8 R Tc
. . . . . . . . . . . . . . . . . . . . . . . (4.7)
The van der Waals equation also can be written in terms of the Z factor (Z+pv/RT ). Z 3 * (B ) 1) Z 2 ) A Z * AB + 0 , Fig. 4.1—pV relation of a pure component at subcritical, critical, and supercritical temperatures.
tions, where a simple temperaturedependent correction is used for EOS constant A, is reproducibility. The ZJRK EOS’s rely on tables or complex functions to represent the highly nonlinear correction terms for EOS constants A and B. Peneloux et al.’s20 volumetranslation method modifies a twoconstant cubic equation by introducing a third EOS constant, c, without changing the equilibrium calculations of the original twoconstant equation. The volumetranslation constant c eliminates the inherent volumetric deficiency suffered by all twoconstant equations, and, for practical purposes, volume translation makes any twoconstant EOS as accurate as any threeconstant equation.1215 4.2.1 van der Waals5 Equation. van der Waals proposed the first cubic EOS in 1873. The van der Waals EOS gives a simple, qualitatively accurate relation between pressure, temperature, and molar volume. p + RT * a2 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.3) v*b v where a+“attraction” parameter, b+“repulsion” parameter, and R+universal gas constant. Comparing this equation with the ideal gas law, p+RT/v, we see that the van der Waals equation offers two important improvements. First, the prediction of liquid behavior is more accurate because volume approaches a limiting value, b, at high pressures, lim v ǒ p Ǔ + b , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.4)
p ³ R
where b is sometimes referred to as the “covolume” (effective molecular volume). The term RT/(v*b) dictates liquid behavior and physically represents the repulsive component of pressure on a molecular scale. The van der Waals equation also improves the description of nonideal gas behavior, where the term RT/(v*b) approximates ideal gas behavior ( p[RT/v) and the term a/v2 accounts for nonideal behavior. The a/v2 term reduces system pressure and traditionally is interpreted as the attractive component of pressure. 2
. . . . . . . . . . . . . . . (4.8)
p pr where A + a + 27 2 64 T r (RT) 2 and B + b
pr p + 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.9) RT 8 Tr
Abbott22 gives an interesting historical review of the van der Waals EOS, its strengths and weaknesses, and its analogy to other cubic EOS’s. 4.2.2 RedlichKwong6 Equations. The RK EOS is a p + RT * . . . . . . . . . . . . . . . . . . . . . . . . (4.10) v * b v (v ) b) or, in terms of Z factor, Z 3 * Z 2 ) ǒ A * B * B 2 Ǔ Z * AB + 0 and Z c + 1ń3 ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.11)
with EOS constants defined as R 2T 2 a + W oa p c a(T r), c
. . . . . . . . . . . . . . . . . . . . . . . . . . . (4.12a)
where W oa + 0.42748; RT b + W ob p c , c
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.12b)
+ 0.08664; p pr A+a + W oa 2 a(T r), . . . . . . . . . . . . . . . . . . . . (4.12c) (RT) 2 Tr
where
W ob
where a(T r) + T *0.5 ; r p pr and B + b + W ob . . . . . . . . . . . . . . . . . . . . . . . . . (4.12d) RT Tr The fugacity expression for pure components is
ǒ
Ǔ
f ln p + ln f + Z * 1 * ln(Z * B) * A ln 1 ) B . B Z . . . . . . . . . . . . . . . . . . . . (4.13) PHASE BEHAVIOR
TABLE 4.1—BIP’s FOR THE PR EOS AND SRK EOS PR EOS*
SRK EOS**
N2
CO2
H2 S
N2
CO2
H2 S
N2
—
—
—
—
—
—
CO2
0.000
—
—
0.000
—
—
0.120
—
H2 S
0.130
0.135
—
0.120†
C1
0.025
0.105
0.070
0.020
0.120
0.080
C2
0.010
0.130
0.085
0.060
0.150
0.070
C3
0.090
0.125
0.080
0.080
0.150
0.070
iC4
0.095
0.120
0.075
0.080
0.150
0.060
C4
0.095
0.115
0.075
0.080
0.150
0.060
iC5
0.100
0.115
0.070
0.080
0.150
0.060
C5
0.110
0.115
0.070
0.080
0.150
0.060
C6
0.110
0.115
0.055
0.080
0.150
0.050
C7 +
0.110
0.115
0.050‡
0.080
0.150
0.030‡
*Nonhydrocarbon
BIP’s from Nagy and
**Nonhydrocarbon
Shirkovskiy.24
Use for both the original PR EOS (Ref. 7) and modified PR EOS (Ref. 25).
BIP’s from Reid et al.3
†Not reported by Reid et al.3 ‡Should decrease gradually with increasing carbon number.
The cubic Zfactor equation can readily be solved with an analytical or a trialanderror approach.1,2 One or three real roots may exist, where the smallest root (assuming that it is greater than B) is typically chosen for liquids and the largest root is chosen for vapors. The middle root is always discarded as a nonphysical value. For mixtures, the choice of lower or upper root is not known a priori and the correct root is chosen as the one with the lowest normalized Gibbs energy, g *,23
ȍy N
g *y +
ln f i ǒ yǓ
i
i+1
ȍx N
and g *x +
ln f i ǒ x Ǔ , . . . . . . . . . . . . . . . . . . . . . . . . . (4.14)
i
i+1
where yi and xi +mole fractions of vapor and liquid, respectively, and fi +multicomponent fugacity given (for a vapor phase) by ln
B fi + ln f i + i (Z * 1) * ln(Z * B) B yi p )A B
ǒ
Bi 2 * B A
ȍy A N
j
j+1
Ǔ
ij
ǒ
Ǔ
ln 1 ) B . Z
. . . . . . . (4.15)
The traditional quadratic mixing rule is used for A, and a linear mixing rule is used for B. For a vapor phase with composition yi , these are given by
ȍȍ y y A N
A+
N
i j
ij ,
i+1 j+1
ȍy B , N
B+
i
i
i+1
and A ij + ǒ1 * k ijǓ ǸA i A j , . . . . . . . . . . . . . . . . . . . . . . . (4.16) where kij +binaryinteraction parameters (BIP’s), where kii +0 and kij +kji . Usually, kij +0 for most hydrocarbon/hydrocarbon (HC/ HC) pairs, except perhaps C1/C7) pairs. Nonhydrocarbon/HC kij are usually nonzero, where kij [0.1 to 0.15 for N2/HC and CO2/HC pairs (Table 4.1).3,24,25 EQUATIONOFSTATE CALCULATIONS
Many students of the RK EOS have been intrigued by its simplicity, accuracy, and the pleasure of deriving its thermodynamic properties. This has led to innumerable attempts to improve and extend the original RedlichKwong equation. Certainly hundreds, if not thousands, of technical papers and theses have been written about the RK EOS. With the advent of digital computers, this “craze” developed into what Abbott10 called the RedlichKwong decade (1967–77). Abbott claims that the remarkable success of the RK EOS results from its excellent prediction of the second virial coefficient (securing good performance at low densities) and reliable predictions at high densities in the supercritical region. This latter observation results from the compromise fit of densities in the nearcritical region; all components have a critical compressibility factor of Z c +1/3, where, in fact, Z c ranges from 0.29 for methane to t0.2 for heavy C7) fractions. The RedlichKwong value of Z c +1/3 is reasonable for lighter hydrocarbons but is unsatisfactory for heavier components. 4.2.3 SoaveRedlichKwong. Several attempts have been made to improve VLE predictions of the RK EOS by introducing a componentdependent correction term a for EOS constant A. Soave18 used vapor pressures to determine the functional relation for the correction factor used in Eq. 4.12, a + ƪ1 ) mǒ1 * T r 0.5 Ǔƫ
2
and m + 0.480 ) 1.574 w * 0.176 w 2. . . . . . . . . . . . . . (4.17) Acentric factor w is defined in Chap. 5, and values for pure components can be found in Appendix A. Table 4.1 gives nonhydrocarbon BIP’s for the SRK EOS as recommended by Reid et al.3; kij +0 is generally recommended for HC/HC pairs. The SoaveRedlichKwong equation is the most widely used RK EOS proposed to date even though it grossly overestimates liquid volumes (and underestimates liquid densities) of petroleum mixtures. The present use of the SRK EOS results from historical and practical reasons. It offers an excellent predictive tool for systems requiring accurate predictions of VLE and vapor properties. Volume translation (discussed in Sec. 4.2.6) is highly recommended, if not mandatory, when liquid densities are needed from the EOS. The Pedersen et al.26,27 C7) characterization method is recommended when the SRK EOS is used. 3
plicity and overall accuracy (particularly when used with volume translation). The ZJRK EOS is surprisingly accurate for both liquid and vapor property estimations, where its main disadvantage is the complexity of functions used to represent temperaturedependent corrections for the EOS constants A and B.
0.09 0.08
4.2.5 PengRobinson.7 In 1976, Peng and Robinson proposed a twoconstant equation that created great expectations for improved EOS predictions and improved liquiddensity predictions in particular. The PR EOS is given by
0.07 0.06 0.05
a . . . . . . . . . . . . . . (4.19) p + RT * v * b v(v ) b) ) b(v * b)
0.04 0.03
or, in terms of Z factor, 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1.0
Reduced Temperature
Z 3 * (1 * B)Z 2 ) ǒ A * 3B 2 * 2B ǓZ * ǒAB * B 2 * B 3Ǔ + 0 and Z c + 0.3074 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.20)
0.45
The EOS constants are given by
0.40
R 2T 2 a + W oa p c a, c
0.35 0.30
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.21a)
where W oa + 0.45724;
0.25
RT b + W ob p c , c
0.20
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.21b)
where W ob + 0.07780; 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1.0
Reduced Temperature Fig. 4.2—Temperature and componentdependent EOS terms W oaa(T r, w) and W oaa(T r, w) for the ZJRK EOS (from Yarborough11).
a + ƪ1 ) mǒ1 * ǸT rǓƫ ; . . . . . . . . . . . . . . . . . . . . . . (4.21c) 2
and m + 0.37464 ) 1.54226 w * 0.26992 w 2 .
. . . . . . (4.21d)
al.25
4.2.4 ZudkevitchJoffeRedlichKwong. Zudkevitch and Joffe19 proposed a novel procedure for improving the volumetric predictions of the RK EOS without sacrificing VLE capabilities of the original equation. They suggest that the EOS constants A and B should be corrected as functions of temperature to match saturated liquid densities and liquid fugacities. They show that vapor fugacities and fugacity ratios (K values) remain essentially unaffected and that their procedure does not greatly affect vapor densities. Shortly after the original modification appeared, Joffe et al.28 suggested that vapor pressures should be used instead of liquid fugacities. This is the approach used today in what is still referred to as the ZudkevitchJoffe modification, the ZJRK EOS. Haman et al.29 proposed the correction terms a and b for EOS constants A and B in equation form for pure paraffins. Yarborough11 proposed generalized a and b charts for petroleum reservoir fluids that include heavy petroleum fractions. R 2T 2 a + W oa p c T r*0.5 aǒ T r , w Ǔ c RT and b + W ob p c bǒ T r , w Ǔ . c
and Robinson and Peng30 proposed a In 1979, Robinson et modified expression for m that is recommended for heavier components (wu0.49). m + 0.3796 ) 1.485w * 0.1644w 2 ) 0.01667w 3 . . . . . . . . . . . . . . . . . . . . . (4.22) Fugacity expressions are given by f ln p +ln f + Z * 1 * ln(Z* B) *
and ln
A ln 2 Ǹ2 B
ƪ
Z ) ǒ1 ) Ǹ2ǓB
Z * ǒ1 * Ǹ2ǓB
ƫ
fi B + ln f i + i (Z * 1) * ln(Z * B) B yi p )
A 2 Ǹ2 B
ǒ
Bi 2 * B A
ȍyA N
j
j+1
Ǔƪ
ij
ln
Z ) ǒ1 ) Ǹ2ǓB
Z * ǒ1 * Ǹ2ǓB
ƫ
,
. . . . . . . . . . . . . . . . . . . . . . . (4.18)
. . . . . . . . . . . . . . . . . . . . (4.23)
Unfortunately, the temperaturedependent functions are complex because they are represented by higherorder polynomials or cubic splines (see Fig. 4.2). The behavior of these functions is highly nonlinear near Tr +1, and a discontinuity is introduced by setting the correction factors a+b+1 at Tr y1. A single set of a and b corrections is not used in the industry, making reproducing results from one version to another difficult. Preferably, a table of a and b correction factors should be provided when reporting a fluid characterization based on a ZJRK EOS. Two RedlichKwong modifications, the SRK EOS and ZJRK EOS, have found widespread application to petroleum reservoir fluids. The Soave equation is sometimes preferred because of its sim
where traditional mixing rules (Eq. 4.16) are used in the derivation of the multicomponent fugacity expression. The PR EOS does not calculate inferior VLE’s compared with the RK EOS equations, and the temperaturedependent correction term for EOS constant A is very similar to the Soave correction. The largest improvement offered by the PR EOS is a universal critical compressibility factor of 0.307, which is somewhat lower than the RedlichKwong value of onethird and closer to experimental values for heavier hydrocarbons. The difference between PR EOS and SRK EOS liquid volumetric predictions can be substantial, although, in many cases, the error in oil densities is unacceptable from both equations. Some evidence exists that the PR EOS underpredicts sat
4
PHASE BEHAVIOR
TABLE 4.2—JHAVERIYOUNGREN31 VOLUMETRANSLATION CORRELATION FOR C7) FRACTIONS WITH THE PR EOS s i + 1 * A 0ńM
A1 i
A0
A1
Paraffins
2.258
0.1823
Naphthenes
3.004
0.2324
Aromatics
2.516
0.2008
Hydrocarbon Family
TABLE 4.3—VOLUMETRANSLATION COEFFICIENTS (si +ci /bi ) FOR PURE COMPOUNDS FOR THE PR EOS AND SRK EOS Component
PR EOS
SRK EOS
N2
*0.1927
*0.0079
CO2
*0.0817
0.0833
Fig. 4.3—pV diagram of a pure component as calculated by a cubic EOS illustrating the van der Waals’s “loop” defining vapor pressure by the equalarea rule.
H2 S
*0.1288
0.0466
C1
*0.1595
0.0234
uration pressure of reservoir fluids compared with the SRK EOS, thereby requiring somewhat larger HC/HC (C1/C7)) BIP’s for the PR EOS. In review, the PengRobinson and SoaveRedlichKwong equations are the two most widely used cubic EOS’s. They provide the same accuracy for VLE predictions and satisfactory volumetric predictions for vapor and liquid phases when used with volume translation.
C2
*0.1134
0.0605
C3
*0.0863
0.0825
iC4
*0.0844
0.0830
nC4
*0.0675
0.0975
iC5
*0.0608
0.1022
nC5
*0.0390
0.1209
nC6
*0.0080
0.1467
nC7
0.0033
0.1554
nC8
0.0314
0.1794
nC9
0.0408
0.1868
nC10
0.0655
0.2080
4.2.6 Volume Translation. In 1979, Martin8 proposed a new concept in cubic EOS’s, volume translation. His application was to ease the comparison of his proposed generalized EOS with previously published equations. In an independent study, Peneloux et al.20 used volume translation to improve volumetric capabilities of the SRK EOS. Peneloux et al.’s key contribution was to show that the volume shift does not affect equilibrium calculations for pure components or mixtures and therefore does not affect the original VLE capabilities of the SRK EOS. Volume translation works equally well with any twoconstant EOS, as Jhaveri and Youngren31 show for the PengRobinson equation. Volume translation solves the main problem with twoconstant EOS’s, poor liquid volumetric predictions. A simple correction term is applied to the EOScalculated molar volume. v + v EOS * c,
ǒ f viǓ
modified+
ǒ
ȍx c N
i i
i+1
ȍy c , N
and v v + v vEOS *
i i
. . . . . . . . . . . . . . . . . . . . . . . (4.25)
i+1
and v EOS where v EOS v +EOScalculated liquid and vapor molar volL umes, respectively; xi and yi +liquid and vapor compositions, respectively; and ci +componentdependent volumeshift parameEQUATIONOFSTATE CALCULATIONS
and ǒ f LiǓ
Ǔ
ǒf viǓ original exp * c i p RT
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.24)
where v+corrected molar volume, vEOS+EOScalculated volume, and c+componentspecific constant. The shift in volume is actually equivalent to adding a third constant to the EOS but is special because equilibrium conditions are unaltered. This is readily seen for a pure component, where the van der Waals “loop” (Fig. 4.3) defines vapor pressure by making the areas above and below the p+pv line on a pv plot equal. Shifting the pv plot to the left or right along the volume axis does not change the equalarea (fugacity) balance, and it can be readily seen that vaporpressure predictions are unaltered by introducing the volumeshift term c. Peneloux et al.20 also show that multicomponent VLE is unaltered by introducing the correction term as a molefraction average. v L + v LEOS *
ters. When the volume shift is introduced to the EOS for mixtures, the resulting expressions for fugacity are
modified+
ǒ
(f Li) original exp * c i
Ǔ
p . RT
. . . . . . . . . . (4.26)
This implies that fugacity ratios are unaltered by the volume shift, ǒ f Li ń f viǓ modified+ ǒf Li ń f viǓ original .
. . . . . . . . . . . . . . . . . . (4.27)
Applications that require direct use of fugacity (e.g., compositionalgradient calculation and semisolid phase equilibrium) must include the volumetranslation coefficient in the fugacity expression. Also, the constant c can be temperature dependent but cannot include pressure or composition dependence without derivation of new fugacity expressions. Peneloux et al. propose that ci be determined for each component separately by matching the saturatedliquid density at Tr +0.7. ci can actually be determined by matching the EOS to any density value at a specified pressure and temperature. Jhaveri and Youngren31 write ci as a ratio, si +ci /bi , suggesting the following equation for C7) fractions, A
si + ci ń bi + 1 * A0 ń Mi 1 .
. . . . . . . . . . . . . . . . . . . (4.28)
Table 4.2 gives A0 and A1 values, and Table 4.3 gives si values for selected pure components that have been determined by matching 5
PR EOS
S C1 through C10 paraffins fit at T+0.7 — JhaveriYoungren for paraffins Methane Concentration, mol%
Fig. 4.4—Variation of volumetranslation parameter si +ci /bi vs. molecular weight.
the saturated liquid density at Tr +0.7. Fig. 4.4 shows the variation of si with M. Volume translation can be applied to any twoconstant cubic equation, thereby eliminating the volumetric deficiency suffered by all twoconstant equations. For practical purposes volume translation makes any twoconstant EOS as accurate as any threeconstant equation1215 (see Fig. 4.5). 4.3 TwoĆPhase Flash Calculation The isothermal twophase flash calculation is the workhorse of most EOS applications. The problem consists of defining the amounts and compositions of equilibrium phases, usually liquid and vapor, given the pressure, temperature, and overall composition. An inherent obstacle to solving this problem is not knowing whether two equilibrium phases form at the specified pressure and temperature. The mixture may exist as a single phase or may split into two or more phases. The algorithms presented in this section assume that a mathematical solution to the twophase problem exists: either a solution yielding equilibrium phase compositions or a “trivial” solution. Even when the results appear physically consistent, a rigorous check of the solution with the phasestability test (discussed in Sec. 4.4) may be required. Alternatively, defining the phase stability before a twophase flash calculation is made improves the reliability of the flash results but adds computations. Mathematically, the twophase flash calculation can be solved by either (1) satisfying the equalfugacity and materialbalance constraints with a successivesubstitution or NewtonRaphson algorithm32,33 or (2) minimizing the mixture Gibbs free energy function.34 The first approach is used almost exclusively because it is readily implemented with one of several iterative algorithms. Gibbs energy minimization has received less attention, and it is unclear whether it has any fundamental advantages over the simpler and more direct equalfugacity approach, at least for twophase problems. The usual constraint equations for solving the twophase flash problem are equal fugacities and a component/phase material balance. Assuming that all other forces are negligible (e.g., gravity), the criterion of thermodynamic equilibrium is that the chemical potential of Component i in Phase 1 equals the chemical potential of Component i in Phase 2; this is true for all Components i+ 1, . . . , N (and all phases). Fugacity, fi , is a useful expression for the chemical potential, mi , where mi +RT ln fi )li (T), and the equalchemicalpotential constraint can be written as f Li + f vi ,
i + 1, . . . , N.
. . . . . . . . . . . . . . . . . . . . . . . (4.29)
This constraint can be solved numerically by use of some measure of convergence, such as 6
Fig. 4.5—Comparison of measured and EOScalculated saturatedliquid densities of the binary system C1/C10 systems at 100°F; SW+SchmidtWenzel.14.
ȍǒff
Li
i+1
Ǔ t e, 2
N
vi
*1
. . . . . . . . . . . . . . . . . . . . . . . . . . . (4.30)
where e is a convergence tolerance (e.g., 1
10*13).
4.3.1 TwoPhase Split Calculation (RachfordRice35 Procedure). The component and phase materialbalance constraints state that n total moles of feed with Composition zi distribute into nv moles of vapor with Composition yi and nL moles of liquid with Composition xi without loss of matter or chemical alteration of the component species. The materialbalance constraints can be written as n + nv ) nL and n z i + n v y i ) n L x i , i + 1, . . . , N. . . . . . . . . . . . . . . . (4.31) Introducing the vapor mole fraction Fv +nv /(nL )nv ), Eq. 4.31 can be written as z i + F v y i ) (1 * F v) x i . . . . . . . . . . . . . . . . . . . . . . . . (4.32) Additionally, the mole fractions of equilibrium phases and the overall mixture must sum to unity.
ȍ y + ȍ x + ȍ z + 1. N
N
i
N
i
i+1
i
i+1
. . . . . . . . . . . . . . . . . . . . (4.33)
i+1
This constraint can be expressed as
ȍǒ y * x Ǔ + 0. N
i
i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.34)
i+1
Introducing the equilibrium ratio Ki , K i + y ińx i ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.35)
the number of unknowns can be reduced from 2 N)1 ( yi , xi , and Fv ) to N)1 (Ki and Fv ). By use of the component material balance (Eq. 4.31) and by replacing yi by Ki xi , Eq. 4.34 can be solved in terms of a single variable Fv . * 1) ȍǒy * x Ǔ + ȍ 1 )z (K + 0. F (K * 1) N
h(F v) +
N
i
i
i+1
i
i+1
i
v
. . . (4.36)
i
Eq. 4.36 is usually referred to as the RachfordRice35 equation. Fig. 4.6 shows the function h(Fv ) for a fivecomponent mixture. With feed composition and K values known, the only remaining unknown is Fv. h(Fv ) has asymptotes at Fv +1/(1*Ki ), where each K value gives an asymptote.36,37 Mathematically, it can be shown that the only physically meaningful solution of h(Fv )—i.e., where PHASE BEHAVIOR
Phase compositions are calculated from the materialbalance equations zi xi + F v (K i * 1) ) 1 zi Ki + Ki xi . F v (K i * 1) ) 1
and y i +
. . . . . . . . . . . . . . . . (4.41)
4.3.2 EOS TwoPhase Flash Algorithm. The flash calculation is initialized by estimating a set of K values; the Wilson39 equation is commonly used.
Ki +
Fig. 4.6—RachfordRice35 function h(FV ) for a fivecomponent mixture (from Ref. 37).
all Compositions xi and yi are positive—lies in the region FvmintFv tFvmax, where F v min +
1 1 * K max
and F v max +
1 . 1 * K min
. . . . . . . . . . . . . . . . . . . . . . . . . (4.37)
It can be shown that Fvmint0 and Fvmaxu1 if at least one K value is t1 and one K value is u1. This implies that the solution for h(Fv )+0 should always be limited to the region FvmintFv tFvmax. Because h(Fv ) is monotonic and the derivative hȀ(Fv )+dh/dFv can be expressed analytically, the NewtonRaphson algorithm is commonly used to solve for Fv . + F nv * F n)1 v
hǒF nvǓ hȀǒF nvǓ
hȀ(F v) + dh + * dF v
N
v
i+1
where dh + * d Fv
ȍ
Ǔƫ . . . . . . . . . . . . . . (4.42)
pr i
K values from this equation are not accurate at high pressures, which potentially cause the twophase flash to converge incorrectly to a trivial solution. Results from a phasestability test provide the most reliable Kvalue estimates for initializing the twophase flash but are relatively expensive to obtain. Reliable Kvalue estimates can be taken from a converged flash of the same mixture or a “related” mixture at a pressure and temperature not too far removed from the conditions of the present flash calculation. For example, in simulating a depletion experiment with an EOS, the K values at the saturation pressure can be used as initial estimates for the flash at the first depletion stage, the converged K values from this flash can be used for the flash at the second stage, and so on at lower pressures. With estimated K values, the RachfordRice35 equation is solved for Fv, with the search for Fv always bounded by Fvmin and Fvmax. F v min +
1 t0 1 * K max
and F v max +
1 u 1. . . . . . . . . . . . . . . . . . . . . . . (4.43) 1 * K min
Phase compositions are calculated from the materialbalance equations. Having calculated xi and yi , phase Z factors ZL and Zv and component fugacities fLi and fvi are calculated with the EOS. and Z v + F EOSǒ y, p, T Ǔ . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.44)
ȍ
z i (K i * 1)
N
i+1
zi + 0, v ) ci N
ǒ
Z L + F EOSǒ x, p, T Ǔ
2
ƪF v (Ki * 1) ) 1ƫ
2
,
. . . . (4.38)
where n+iteration counter. The first guess for Fv can be chosen arbitrarily as 0.5. In 1949, Muskat and McDowell38 proposed a solution to the twophase split calculation that is basically the same as the one proposed by Rachford and Rice35 but numerically more efficient. Introducing the quantity ci +1/(Ki *1), where ci +R for Ki +1, Muskat and McDowell proposed the following form of the function h(Fv ). h(F ) + ȍ F
ƪ
exp 5.37ǒ1 ) w i Ǔ 1 * T *1 ri
. . . . . . . . . . . . . . . . . . . . . . (4.39)
zi
ǒF v ) c i Ǔ
and f Li + F EOSǒ x, Z L, p, T Ǔ and f vi + F EOSǒ y, Z v, p, T Ǔ.
. . . . . . . . . . . . . . . . . . . . . . . (4.45)
The “normalized” Gibbs energy function, g *, of each phase is calculated from
ȍ x ln f N
g *L +
i
Li
i+1
ȍ y ln f N
and g *v +
i
vi ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . (4.46)
i+1
and the normalized mixture Gibbs energy is given by . 2
. . . . . . . . . . . . . . . . . (4.40)
g *mix + F v g *v ) (1 * F v)g *L .
. . . . . . . . . . . . . . . . . . . . (4.47)
If a Newton estimate from Eq. 4.38 with either the MuskatMcDowell or RachfordRice equations for h gives an estimate of Fv outside the range FvmintFv tFvmax, the Newton method should be replaced by interval bisection or modified regula falsi until convergence is achieved. Severe roundoff errors may cause any solution technique to fail when both K and z of one component are very small (e.g., KN +1 10*12 and zN +1 10*20).*
If multiple Zfactor roots are found for either phase, the root with the lowest Gibbs energy should be chosen.23 For example, if three liquid Zfactor roots were calculated ( Z L1, Z L2, and Z L3), the middle root, Z L2, is automatically discarded and the two Gibbs energy functions, g *L1 and g *L3 , are calculated; f L1i are calculated with Z L1, and f L3i are calculated with Z L3. If g *L3 t g *L1, Z L3 should be chosen; otherwise, choose Z L1 for g *L1 t g *L3 . Zick* suggests that this method of choosing the Zfactor root is not failsafe because, at early iterations in the flash calculation, the incor
*Personal communication with A.A. Zick, Zick Technologies Inc., Portland, Oregon (1991).
*Personal communication with A.A. Zick, Zick Technologies Inc., Portland, Oregon (1985).
i+1
EQUATIONOFSTATE CALCULATIONS
7
TABLE 4.4—SEQUENCE OF FLASH CALCULATIONS TO ENSURE CORRECT SOLUTION WITH MULTIPLE ROOTS Liquid ZL Root Chosen
Possible Order of Multiple Flash Calculations
Smallest
1
n
Largest n
3
n n
Smallest
Largest n
2
4
Vapor, Zv Root Chosen
n n n
rect root may have a lower Gibbs energy than the correct root. He proposes that the flash calculation be converged completely with a consistent choice of roots (e.g., the smallest root always chosen for the liquid phase and the largest root always chosen for the vapor phase). If multiple roots in either phase are detected during this flash calculation, a second, third, and potentially fourth flash calculation must be completed, as summarized in Table 4.4. The twophase solution with the lowest mixture Gibbs energy is chosen as the correct solution. With fugacities calculated for each phase, the fugacity constraint (Eq. 4.30) is checked. The recommended convergence tolerance is 1 10*13, although a less stringent value can be used in some applications. If convergence is not achieved, the K values can be modified with successive substitution. + K (n) K (n)1) i i
f Li(n) f (n) vi
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.48)
where the superscripts (n) and (n)1) indicate the iteration level. With new K values, the RachfordRice35 equation is solved again (with new values of Fvmin and Fvmax), phase compositions are calculated with the converged Fv value, phase Z factors and component fugacities are calculated from the EOS, and the fugacity constraint is rechecked. This iterative procedure is repeated until convergence is achieved. Three types of converged solutions can be obtained. 1. A physically acceptable solution is found with 0xFv x1, where Fv +0 corresponds to a bubblepoint condition, Fv +1 corresponds to a dewpoint condition, and 0tFv t1 indicates a twophase condition. 2. A physically unacceptable solution is found with Fv t0 or Fv u1,37 where the calculated equilibrium compositions satisfy the equalfugacity constraint and the mathematical materialbalance equation. This solution indicates that the mixture is thermodynamically stable as a single phase and will not split into two phases. For this solution, the calculated equilibrium compositions would coexist in thermodynamic equilibrium at the given pressure and temperature if they were mixed together in a physically meaningful proportion (creating, of course, a different mixture composition). 3. A socalled trivial solution is found where the calculated phase compositions are identical to the mixture composition and K values equal one (xi +yi +zi and Ki +1). The first solution is usually a “correct” solution. However, if a potential threephase solution exists, the twophase solution may represent only a local minimum in the mixture Gibbs energy surface and the mixture Gibbs energy may be reduced further by locating the threephase solution or another twophase solution. Michelsen32 suggests that this problem is best dealt with by use of phasestability analysis. Whitson and Michelsen37 refer to the second solution to the flash as a “negative” flash because one of the phase mole fractions is negative (and the other phase fraction is u1). Although this condition is physically unacceptable, the solution still has practical application. For example, phase properties and compositions are continuous across the phase boundaries. Also, a nontrivial negative flash solution indicates phase stability with the same certainty as the phasestability test, although the negative flash calculation requires better initial Kvalue estimates than does the phasestability test. A trivial solution to the flash calculation should always be checked with the phasestability test to verify that the mixture is in fact single phase. Trivial solutions arise for several reasons, the most 8
Fig. 4.7—pT phase envelope and envelopes indicating the limit of a nontrivial negative flash and a nontrival stability test for the binary C2/nC4 system (from Ref. 37).
common being poor initial Kvalue estimates (e.g., from the Wilson39 equation). A “valid” trivial solution occurs when twophase solutions do not exist. This occurs outside the pT envelope that Whitson and Michelsen define as the convergencepressure envelope, where Fv !"R in the negative flash (Fig. 4.7). Along the phase boundary and near a critical point, the NewtonRaphson flash technique tends to converge to a trivial solution more readily than do successivesubstitution methods. Finally, as Michelsen23 shows, the twophase flash never converges to a trivial solution with successive substitution under the following conditions. 1. The phasestability test indicates that the mixture is unstable. 2. The K values resulting from the stability test are used to initialize the flash calculation. 3. The mixture Gibbs energy g *1 mix at the first iteration is less than the mixture Gibbs energy g *z . The flash calculation initialized by a successful phasestability test is the safest solution method available, albeit more expensive than a direct twophase flash calculation. Successive substitution is the safest solution technique for the twophase flash problem, but it becomes slow when fugacity coefficients are strongly composition dependent. The method is particularly slow near phase boundaries and critical points, where many thousands of iterations may be required to reduce the convergence criterion to an acceptable value. Successive substitution can be accelerated with one of several methods as described in Refs. 33 and 40 through 43 among others. Michelsen32 recommends the general dominant eigenvalue method44 (GDEM); he shows that this method is particularly well suited for the twophase flash problem because two dominant eigenvalues are found near phase boundaries and the critical point. He recommends preceding each GDEM promotion (acceleration) with five successivesubstitution iterations, where the GDEM Kvalue correction is given by + ln K (n) ) ln K (n)1) i i
Du (n) * m 2 Du (n*1) i i 1 ) m 1 ) m 2 , . . . . . . . . . (4.49)
where Du i 5 ln ǒ f Lińf viǓ and m 1 + ǒb 02 b 12 * b 01 b 22Ǔńǒb 11 b 22 * b 12 b 12Ǔ , . . . . . . . (4.50a) m 2 + ǒb 01 b 12 * b 02 b 11Ǔńǒb 11 b 22 * b 12 b 12Ǔ ,
ȍ Du N
and b jk +
ǒ n*j Ǔ Du (n*k) . i i
. . . . . (4.50b)
. . . . . . . . . . . . . . . . . . . . (4.50c)
i+1
PHASE BEHAVIOR
m1 and m2 are coefficients reflecting the relative magnitudes of dominant eigenvalues l1 and l2. Michelsen suggests that promotions be rejected (or reduced) if the mixture Gibbs energy increases after a promotion. Zick* shows that the coefficients m1 and m2 calculated with Eqs. 4.50a and 4.50b can be seriously affected by roundoff error. He suggests that the substitution ejk 5(bjk *b12)/b12 eliminates the roundoff problem and that this transformation of variables results in promotion coefficients m1 and m2 that can be used even near a critical point. Also, the Michelsen32 suggestion to switch to a NewtonRaphson method after two GDEM iterations is unnecessary with the modified GDEM coefficients. For most practical reservoir applications, GDEM will converge in two to three promotions (11 to 16 iterations), with nearcritical problems requiring up to six promotions (31 iterations). In summary, the twophase flash calculation can be outlined with the following stepbystep procedure. 1. Estimate K values. 2. Calculate Kmin and Kmax. 3. Solve the RachfordRice phasesplit calculation (Eq. 4.36) for Fv, limited between Fvmin and Fvmax (Eq. 4.43). 4. Calculate phase compositions x and y (Eq. 4.41). 5. Calculate phase Z factors ZL and Zv from the EOS. 6. Calculate component fugacities fLi and fvi from the EOS. 7. Calculate phase Gibbs energy functions g *L and g *v (Eq. 4.46), determine the correct Zfactor roots of each phase (if multiple roots exist), and calculate the mixture Gibbs energy (Eq. 4.47). 8. Check the equalfugacity constraint (Eq. 4.30). 9. (a) If convergence is reached, stop. (b) If convergence is not reached, update the K values with the fugacity ratios (Eq. 4.48) or a GDEM promotion (Eq. 4.49); alternatively, use another acceleration technique or a NewtonRaphson Kvalue update. 10. Check for convergence at a trivial solution (Ki !1) with the condition
ȍǒln K Ǔ N
i
2
t 10 *4.
. . . . . . . . . . . . . . . . . . . . . . . . . . (4.51)
i+1
11. If a trivial solution is not detected, return to Step 2. Otherwise, confirm the trivial solution with a stability test. For reservoir simulation, a NewtonRaphson solution to the flash problem can be used because initial Kvalue estimates (from earlier timesteps and neighboring gridblocks) should be reliable, and the reduced computation time of a Newton method compared with an accelerated successivesubstitution method can be significant.45 Michelsen’s23 implementation of the NewtonRaphson method is considered a very efficient algorithm and is cited here directly from his original publication (with the exception of some Nomenclature changes). “The set of equations to be solved is e i(K) + ƪlnǒn vińn vǓ ) ln f viƫ * ƪlnǒn Lińn LǓ ) ln f Liƫ + 0, . . . . . . . . . . . . . . . . . . . . (4.52) where nvi and nLi +number of moles of Component i in the vapor and liquid phases, respectively. “The Jacobian matrix is given by J ij +
ēe i , ē ln K j
yielding J + B A *1 ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.56)
z n vn L with B ij + x yi d ij * 1 ) nv ) nL i i
ƪǒ Ǔ ǒ Ǔ ƫ ē ln f i ēn j
)
v
ē ln f i ēn j
L
. . . . . . . . . . . . . . . . . . . . (4.57) and A ij +
zi d * 1. . . . . . . . . . . . . . . . . . . . . . . . . . . (4.58) x i y i ij
“Since B is symmetric, we can use the decomposition B + LDL T, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.59) where L is unit lower triangular and D is diagonal with positive elements for a positive definite B. “Then, b + * AL *TD *1L *1e, . . . . . . . . . . . . . . . . . . . . . . . . (4.60) where the cost of the decomposition and the subsequent backsubstitution is only about half of that required for conventional solution of Eq. 4.54 by Gaussian elimination.” Application of the Michaelsen NewtonRaphson algorithm, as proposed here and without proper precautions, will lead to convergence problems near phase boundaries because both matrices become singular at phase boundaries and the solution will be severely affected by roundoff errors. 4.4 Phase Stability One of the most difficult aspects of making VLE calculations with an EOS is knowing whether a mixture will actually split into two (or more) phases at the specified pressure and temperature. Traditionally, this problem has been solved either by conducting a twophase flash or by making a saturationpressure calculation; both methods are expensive and not entirely reliable. In 1982, two papers32,46 showed how the Gibbs tangentplane criterion could be used to establish the thermodynamic stability of a phase [i.e., whether a given composition has a lower energy remaining as a single phase (stable) or whether the mixture Gibbs energy will decrease by splitting the mixture into two or more phases (unstable)]. Ref. 46 shows graphically how the Gibbs tangentplane criterion is used to establish phase stability of simple binary systems, and Ref. 32 gives an algorithm to establish phase stability numerically. This section on phase stability follows these references closely. Phase stability deals with the question of whether a mixture can attain a lower energy by splitting into two or more phases. The Gibbs energy for n moles of mixture Composition z i considered as a homogeneous phase is given by
ȍǒn m Ǔ N
Gz +
i z
i
i+1
ȍz m . N
+n
i
zi
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.61)
i+1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.53)
The mixture will split into two phases y and x if the mixture Gibbs energy, Gmix, is less than Gz , where Gmix is given by
and the correction b with bi +Dln Ki is found from Jb + * e. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.54)
ȍǒn m Ǔ N
G mix +
i
“The Jacobian matrix is calculated from
ȍǒn N
N ēe i ēn L k ēe i + , . . . . . . . . . . . . . . . . . (4.55) J ij + ēn L k ē ln K j ē ln K j k+1
ȍ
+
EQUATIONOFSTATE CALCULATIONS
) ǒn i m i Ǔ L ; m Li + m vi + m i
vi
) n LiǓm i
i+1
ȍ nƪF y ) (1 * F ) x ƫ m . N
+ *Personal communication with A.A. Zick, Zick Technologies Inc., Portland, Oregon (1985).
i v
i+1
v i
v
i
i
. . . . . . . . . . . . . . (4.62)
i+1
9
Mole Fraction Component 1 Fig. 4.9—px plot of a twocomponent mixture exhibiting various two and threephase equilibrium conditions (Ref. 46).
Fig. 4.8—Gibbs energy surface for a binary system.
The Gibbs tangentplane criterion considers the energy surface for a homogenous phase. In terms of overall mole fractions zi +ni /n with fugacities evaluated for z, the normalized Gibbs energy function, g * + GńRT, is given by
ȍz N
g *z +
i
ln f i (z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.63)
i+1
g *z is a normalized Gibbs energy for the mixture composition. For a binary mixture, the energy surface g * represents a curve that can be plotted vs. one of the mole fractions (Fig. 4.8). For a ternary system, the energy surface can be plotted in three dimensions ( g * vs. two of the mole fractions), but a graphical representation is not possible for systems with more than three components. Graphically, the condition of equilibrium for a binary system is established on a g * plot by drawing a straight line that is tangent to the TABLE 4.5—PHASES IN EACH PRESSURE INTERVAL Region
Phases Present
I
Only singlephase vapor, V
II
Singlephase liquid, L1 Twophase vapor/liquid, V/L1 Singlephase vapor, V
III
Singlephase liquid, L1 Threephase vapor/liquid/liquid, V/L1/L2 Singlephase vapor, V
IV
Singlephase liquid, L1 Liquid/liquid, L1/L2 Singlephase liquid, L2 Vapor/liquid, V/L2 Singlephase vapor, V
V
Singlephase liquid, L1 Liquid/liquid, L1/L2 Singlephase liquid, L2
10
curve at two (or more) compositions. A valid tangent plane cannot intersect the Gibbs energy surface anywhere except at the points of tangency. For example, the vapor/liquid tangent passes through the two points ǒx, g *LǓ and ǒy, g *v Ǔ in Fig. 4.8. The compositions through which the tangent passes are equilibrium phases that satisfy the equalfugacity condition. A physically acceptable twophase solution requires that the mixture composition lie between the two equilibrium compositions, xtzty. If z lies outside the compositions bounded by x and y (ztx or zuy), the materialbalance constraint is violated and the mixture is stable. Likewise, z+y and z+x indicate stable conditions for a mixture at its dewpoint and bubblepoint, respectively. When the overall composition z lies between the equilibrium compositions (xtzty), the mixture is unstable and will split into the two equilibrium phases with compositions y and x, having a mixture Gibbs energy given by g *mix + F vg *v ) (1 * F v)g *L. with g *mix t g *z . The value of g *mix is read directly from the tangent line at the mixture composition, and the vapor mole fraction F v is given by the distance from z to y, relative to the total distance between x and y ƪF v + (z * y)ń(x * y)ƫ. Baker et al.46 discuss the mathematical conditions associated with the Gibbs tangentplane criterion and illustrate the technique for a binary system that exhibits two and threephase behavior at various pressures and a fixed temperature. Fig. 4.9, a px diagram divided into five pressure intervals, is adapted from their example. Depending on the mixture composition, various combinations of the three potential phases [vapor (V), lower liquid (L1), and upper liquid (L2)] can form in each pressure interval. Table 4.5 shows the phases for each interval. Figs. 4.10A through 4.10G and 4.11A through 4.11F present Gibbs energy plots for Regions II, III, and IV together with the px diagram (Fig. 4.9). Fig. 4.10A shows the g * curve for a low pressure in Region II where only two “valleys” exist, and thereby only one tangent can be drawn. Equilibrium compositions are located at the two points where the tangent touches the g * curve, y and xL1, each of which is near the bottom of a valley. Figs. 4.10B through 4.10D show the g * curve for a higher pressure in Region II, where a middle valley develops between the two valleys exhibited in Fig. 4.10A. Only one valid tangent can be drawn, between the L1 and V valleys. This tangent is valid because it does not pass through the g * curve at compositions other than the points of tangency, xL1 and y. Two other tangents can be drawn, one yielding a liquid/liquid (L1/L2) solution between the left and middle valleys and the other yielding a liquid/vapor (L2/V) solution between the middle and right valleys. These two tangents are, however, invalid because they lie above the g * curve in violation of the tangentplane criterion. Such tangents represent false twophase solutions that satisfy the equalfugacity constraint but PHASE BEHAVIOR
Developing Second Liquid Phase “Valley”
Fig. 4.10A—Gibbs energy plot for the Baker et al.46 binary example, Region II.
yield only a local minimum in the mixture Gibbs energy. False twophase solutions are difficult to detect unless one has a priori knowledge of the actual equilibrium condition. Lowtemperatures and highCO2 concentrations are conditions associated with threephase behavior that may be susceptible to false twophase solutions. Fig. 4.10E shows the g * curve for the threephase pressure (Region III). A single line can be drawn that is tangent to three compositions ( y, xL1, and xL2). The threephase solution is physically valid for any composition lying between the lower liquid (xL1) and the vapor (y) compositions, with the relative amounts of each phase in a twophase mixture being determined by the overall composition. For ztxL1 and zuy, the mixture is stable and remains as a single phase. Fig. 4.10F shows the g * curve for a pressure in Region IV where the middle valley decreases relative to the left and right valleys. This creates a curve that has two valid tangents, one representing a L1/L2 solution and the other representing a LȀ2ńV solution. Valid twophase solutions are found for mixture compositions in either the L1/L2 interval, xL1tztxL2, or the LȀ2ńV interval xȀL2tzty. Mixture compositions outside these two intervals will remain as a stable single phase. The tangent that can be drawn between a lower liquid and vapor phase (dashed line) is not a valid twophase solution because the tangent lies above the g * curve in the middle region of compositions (Fig. 4.10G). However, this is a potential twophase solution that could readily be calculated and mistaken for a valid solution. In Figs. 4.10A through 4.10G, the tangentplane solutions that pass through compositions where g * is convex have been ignored. This follows from the observation that any mixture composition with the condition ǒē 2g *ńēz 2Ǔ t 0 is intrinsically unstable,37 and any search for a solution to the tangentplane criterion will move away from such “convex” solutions. Also, these tangents violate the tangentplane criterion because they lie above the energy surface (see Fig. 4.11A). Baker et al.’s46 graphical interpretation of stability analysis is particularly useful for describing the Gibbs tangentplane criterion but does not lend itself to being implemented as a numerical algorithm that can be used to calculate phase stability. Michelsen32 proposes an algorithm that determines whether a mixture will remain EQUATIONOFSTATE CALCULATIONS
Fig. 4.10B—Gibbs energy plot for the Baker et al.46 binary example: Region II, correct twophase V/L1 solution.
single phase or split into multiple phases. Michelsen’s algorithm is similar to a flash calculation but is faster and safer (accurate Kvalue estimates are not needed for the stability test). The Michelsen stability test is based on locating “secondphase” compositions that have tangent planes parallel to the tangent plane of the mixture composition. If any of the parallel tangent planes lie below the tangent plane of the mixture composition, the mixture is unstable and will split into at least two phases. If all compositions having parallel tangent planes lie above the mixture tangent plane or no composition has a parallel tangent plane, the mixture is stable as a single phase. In addition, if a composition (not equal to the mixture composition) lies on the same tangent plane as the mixture, the mixture is at a bubble or a dewpoint and the second phase is an incipient equilibrium phase. Figs. 4.11B through 4.11F graphically illustrate the Michelsen stabilitytest criteria for stable and unstable mixture compositions. The mathematical description of Michelsen’s stability test is not within the scope of this monograph, but his stabilitytest algorithm follows. Actually, two tests are usually required; one test assumes that the second phase is vaporlike, and the other assumes that the second phase is liquidlike. This corresponds to initializing the search for a second phase with two compositions where each search is conducted separately. The compositions used to initialize each search should represent “poor” guesses (i.e., very vaporlike and very liquidlike compositions) to expand the composition space being searched. One could conceivably use N stability tests (N+number of components), each starting with a pure component as the initial composition estimate, but this would be unnecessarily time consuming. Michelsen shows that locating a secondphase composition with a tangent plane parallel to the tangent plane of the mixture composition is equivalent to locating a composition y with component fugacities f yi equal to mixture component fugacities f zi times a constant, f zi + S + I, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.64) f yi 11
False V/L2 TwoPhase Equilibrium Condition
False L1/L TwoPhase Equilibrium Condition
Fig. 4.10C—Gibbs energy plot for the Baker et al.46 binary example: Region II, false twophase V/L2 solution.
Fig. 4.10D—Gibbs energy plot for the Baker et al.46 binary example: Region II, false twophase L1/L2 solution.
where I+constant. A successivesubstitution algorithm, summarized in the following procedure, can readily be used to solve the Michelsen stability test. Note that each test is conducted separately (e.g., converging the vaporlike search first, then converging the liquidlike search). 1. Calculate the mixture fugacities, f zi ; with multiple Zfactor roots, choose the root with the lowest g *. 2. Use the Wilson equation (Eq. 4.42) to estimate initial K values.
Ǔƫ expƪ5.37(1 ) w i)ǒ1 * T *1 ri . p ri
K 1i +
. . . . . . . . . . . . . (4.42)
3. Calculate secondphase mole numbers, Yi , using the mixture composition z i and the present Kvalue estimates. (Y i) v + z i (K i) v or (Y i) L + z i ń(K i) L .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.65)
4. Sum the mole numbers.
ȍǒY Ǔ N
Sv +
j v
j+1
ȍǒY Ǔ . N
or S L +
j L
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.66)
j+1
5. Normalize the secondphase mole numbers to get mole fractions, yi . (y i) v +
(Y i) v
ȍǒY Ǔ N
j v
j+1
12
+
(Y i) v Sv Fig. 4.10E—Gibbs energy plot for the Baker et al.46 binary example: Region III, correct threephase solution. PHASE BEHAVIOR
False VL1 TwoPhase Equilibrium Condition
Fig. 4.10F—Gibbs energy plot for the Baker et al.46 binary example: Region IV, two possible correct twophase solutions (L1/L2 or V/L2).
(Y i) L
or (y i) L+
ȍǒY Ǔ N
+
(Y i) L . SL
. . . . . . . . . . . . . . . . . . . . . (4.67)
j L
Fig. 4.10G—Gibbs energy plot for the Baker et al.46 binary example: Region IV, false twophase V/L1 solution.
Michelsen suggests that Step 9 of the successive substitution can be accelerated with the GDEM approach with one eigenvalue (only one eigenvalue approaches 1 near the critical point in a stability test). He recommends that four successivesubstitution iterations precede each promotion. The GDEM update is given by
j+1
l
6. Calculate the secondphase fugacities ( fyi )v or ( fyi )L from the EOS; with multiple Zfactor roots (for a given phase), choose the root with the lowest Gibbs energy g *. 7. Calculate the fugacityratio corrections for successivesubstitution update of the K values. (R i) v +
f zi
1
ǒf yiǓ L f zi
2
SL .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.68) 10*12).
t e. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.69)
9. If convergence is not obtained, update the K values. K (n)1) + K (n) R (n) . i i i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.70)
10. Check whether a trivial solution is being approached using the criterion N
i
2
ȍ ln R N
b 01 +
(n) ln i
R (n*1) , i
ȍ ln R
(n*1) i
ln R (n*1) , i
. . . . . . . . . . . . . . . . . (4.72)
i+1
i+1
ȍǒln K Ǔ
Ť
b 11 , 11 * b 01
N
N
i
Ťb
and b 11 +
8. Check whether convergence is achieved (e.g., et1
ȍ(R * 1)
l+
i+1
ǒf yiǓ v S v
or (R i) L +
ƪR (n) ƫ, + K (n) K (n)1) i i i
t1
10 *4.
. . . . . . . . . . . . . . . . . . . . . . (4.71)
i+1
11. If a trivial solution is not indicated, go to Step 3 for another iteration. EQUATIONOFSTATE CALCULATIONS
where the superscript (n) is the iteration counter. Table 4.6 summarizes the interpretation of the twopart stability test. The mixture (very likely) is stable if both tests yield Sx1, if both tests converge to a trivial solution, or if one test yields Sx1 and the other converges to a trivial solution. Theoretically, it is impossible to establish without a doubt that a mixture is stable until all compositions have been tested. However, both solutions indicating stability from the twopart Michelsen test usually ensures that a mixture is in fact single phase. On the other hand, only one test indicating Su1 is sufficient to determine that a mixture is definitely unstable. For an unstable solution, the resulting K values from the stability test can be used to initialize the twophase flash. Potentially both SL and Sv are u1, in which case the best initial K values for the flash are given by Ki +( yi )v /( yi )L +(Ki )v (Ki )L , requiring that both tests be completed (even though the first test positively indicates an unstable mixture). Fig. 4.12 shows Nghiem and Li’s47 EOS calculations identifying the phase boundary of a reservoir oil. Also shown is the envelope within the phase boundary (dashed line) where one of the stability 13
Region of Compositions Where Stability Test Converges Nontrivial
Fig. 4.11A—Gibbs energy plot for a hypothetical binary system showing a graphical interpretation of Michelsen’s32 phasestability test for region of compositions where stability test converges nontrivial.
Fig. 4.11B—Gibbs energy plot for a hypothetical binary system showing a graphical interpretation of Michelsen’s32 phasestability test for liquidlike feed, z, with one unstable condition, y, located.
tests converges to a trivial solution. The lower dashed line (starting from the critical point) shows where the liquidlike stability test converges to a trivial solution, and the upper dashed curve shows where the vaporlike stability test converges to a trivial solution. Inside the dashedcurve envelope, both the liquid and vaporlike stability tests converge to a nontrivial unstable solution (both SL and
Sv are u0). Fig. 4.13 illustrates the behavior of SL and Sv vs. pressure at a fixed temperature for this system. Michelsen’s phasestability test has many applications; the following summarizes the most important ones. 1. Determining whether a mixture composition is thermodynamically stable as a single phase. If the test indicates stability (assuming
Fig. 4.11C—Gibbs energy plot for a hypothetical binary system showing a graphical interpretation of Michelsen’s32 phasestability test for vaporlike feed, z, with one stable condition, y, located.
Fig. 4.11D—Gibbs energy plot for a hypothetical binary system showing a graphical interpretation of Michelsen’s32 phasestability test for liquidlike feed, z, with two unstable conditions, yL and yv, located.
14
PHASE BEHAVIOR
Fig. 4.11E—Gibbs energy plot for a hypothetical binary system showing a graphical interpretation of Michelsen’s32 phasestability test for vaporlike feed, z, with one unstable condition, yL , located.
that both liquid and vaporlike second phases have been tested), it is very likely that a twophase solution does not exist. 2. With at least one unstable solution, initializing the twophase flash calculation with K values determined from the unstable solution(s) of the stability test. This is particularly useful if K values from a converged flash at nearby conditions are not available. 3. Initializing and limiting the pressure range in a saturationpressure calculation (see Sec. 4.5). 4. Checking the stability of a converged twophase flash when threephase behavior is suspected (e.g., for lowtemperature and highCO2 systems). This requires, however, two modifications of the stability test: (a) choice of appropriate initial Kvalue estimates for the “third”phase search and (b) use of the converged twophase fugaci
Fig. 4.11F—Gibbs energy plot for a hypothetical binary system showing a graphical interpretation of Michelsen’s32 phasestability test for liquidlike feed, z, with one unstable condition, yv , located.
ties, feqi +fvi +fLi , instead of fzi in the new search (i.e., locate a third composition y so that feqi /fyi equals a constant S, with Sx1 indicating stability; Su1 would indicate an unstable condition for the twophase solution, thereby guaranteeing a multiphase solution).
TABLE 4.6—SUMMARY OF POSSIBLE PHASESTABILITYTEST RESULTS Second Phase VaporLike
ǒKiǓ Stable
Unstable
ǒyiǓ
+ z v i
v
LiquidLike
ǒKiǓ
l
+
zi
ǒyiǓ
l
Probable Number of Valleys on g*
TS
TS
1
SL x1
TS
2
TS
SL x1
2
Sv x1
SL x1
3
Sv u1
TS
2
TS
SL u1
2
Sv u1
SL u1
2
Sv u1
SL x1
3
Sv x1
SL u1
3
TS+trivial solution.
EQUATIONOFSTATE CALCULATIONS
Fig. 4.12—Phase and stabilitylimit envelopes for a reservoir oil; stability limit represents the condition when one of the stability tests first converges to a trivial solution (from Nghiem and Li47). 15
LIQUIDlIKE SECOND PHASE VAPORLIKE SECOND PHASE
Nghiem et al.48 use the condition of zero tangentplane distance, d TP +0, to solve for saturation pressure, psat, and incipientphase composition y. d TPǒ p sat , y Ǔ + ln
ǒȍ Ǔ N
Yi
+ 0.
. . . . . . . . . . . . . . . . . (4.77)
i+1
The recommended approach for determining saturation pressure is based on a slightly different approach proposed by Michelsen49; he uses the condition TS
TS
Qǒ p sat , y Ǔ + 1 *
ȍ z ƪf (z)ńf ǒ y Ǔƫ + 0 N
i
i
i
i+1
ȍ y ǒf N
+1*
i
Ǔ
zi ń f yi
i+1
ȍY , N
+1*
i
. . . . . . . . . . . . . . . . . . . . . . . (4.78)
i+1
where incipientphase mole fractions are defined by Fig. 4.13—Behavior of mole number sums from stability test, SL and Sv, vs. pressure for a fixed temperature; TS+trivial solution.
4.5 SaturationĆPressure Calculation For a mixture composition z at fixed temperature T, the saturationpressure calculation involves finding the pressure(s) where the mixture is in equilibrium with an infinitesimal amount of an incipient phase. In terms of a twophase flash, the saturation pressure defines a pressure where the vapor mole fraction, Fv, equals zero or one (Fv +0 at bubblepoint and Fv +1 at dewpoint). One way to locate the saturation pressure of a mixture would be to make a 1D search in pressure for Fv +0 or 1, where the twophase flash is converged at each pressure estimate during the search. Although this approach would be safe, it also would be very slow. Several alternative saturationpressure algorithms are available that are both efficient and reliable when used with stability analysis. The two conditions defining a saturation pressure are that the fugacities of all components are equal in both phases, f zi + f yi , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.73) and that the mole fractions of the incipient phase, y, equal unity,
ȍ y + 1. N
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.74)
i
Expressing the incipientphase mole fractions in terms of K values ( yi +zi Ki for a bubblepoint and yi +zi /Ki for a dewpoint), the traditional equations used to solve bubble and dewpoint calculations, respectively, are
ȍz K + 0 N
i
i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.75a)
i+1
ȍ z ńK + 0. N
and 1 *
i
i
. . . . . . . . . . . . . . . . . . . . . . . . . (4.75b)
i+1
In terms of stability analysis, the saturationpressure condition corresponds to finding a second phase with a tangent plane that is parallel to the mixture composition’s tangent plane, with zero distance between the two tangent planes. This is equivalent to the sum of incipientphase mole numbers equaling unity.
ȍ Y + 1. N
i
i+1
16
Yi
ȍY
.
N
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.79)
j
j+1
An efficient method to solve this equation uses a NewtonRaphson update for pressure and accelerated successive substitution (GDEM) for composition. The following procedure outlines this approach. 1. Guess a saturation type: bubble or dewpoint. An incorrect guess will not affect convergence, but the final K values may be “upside down.” 2. Guess a pressure p *. 3. Perform Michelsen’s stability test at p *. 4. (a) If the mixture is stable for the current value of p *, this pressure represents p * the upper bound of the search for a saturation pressure on the upper curve of the phase envelope. Return to Step 1 and try a lower pressure to look for an unstable condition. (b) With an unstable condition at p *, this pressure represents the lower bound in the search for a saturation pressure on the upper curve of the phase envelope. 5. Having found an unstable solution, use the K values from the stability test to calculate incipientphase mole numbers at bubbleand dewpoint with Eqs. 4.80a and 4.80b, respectively. Y i + z i K i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.80a) and Y i + z ińK i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.80b)
i+1
1*
yi 5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.76)
If two unstable solutions were found in the stability test, use the K values for the test with the largest mole number sum S. At this point, the initialization is complete and the iteration sequence begins. 6. Calculate the normalized incipientphase compositions. yi +
Yi
ȍY
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.81)
N
j
j+1
7. Calculate phase Z factors, Zz and Zy, and component fugacities, fzi and fyi , from the EOS at the present saturationpressure estimate. When multiple Zfactor roots are found for a given phase, the root giving the lowest Gibbs energy should be chosen. 8. Calculate fugacityratio corrections. f zi Ri + f yi
ǒȍ Ǔ N
Yj
*1
.
. . . . . . . . . . . . . . . . . . . . . . . . . (4.82)
j+1
PHASE BEHAVIOR
9. Update incipientphase mole numbers with the fugacityratio corrections,
ƫ l, Y i(n)1) + Y i(n) ƪR (n) i
. . . . . . . . . . . . . . . . . . . . . . . . . . (4.83)
where four iterations use successive substitution (l+1) followed by a GDEM promotion with l given by l+
Ťb
11
Ť
b 11 , * b 01
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.84)
ȍ ln R N
where b 01 +
(n) ln R (n*1) i i
4.6 Equilibrium in a Gravity Field: Compositional Gradients Gibbs53 was the first to give the formula for calculating compositional variation under the force of gravity for an isothermal system. The condition of equilibrium is satisfied by the constraint m i ǒ p ref , z ref , T Ǔ + m iǒ p, z, TǓ ) M i gǒh * h refǓ ,
i+1
ȍ ln R
Michelsen52 also has proposed a criticalpoint calculation algorithm that is, surprisingly, as fast or faster than a twophase flash calculation. The critical point is determined by a simple 2D search (in temperature and volume) with a function that requires only evaluation of the mixture fugacities.
N
and b 11 +
(n*1) ln R (n*1) . i i
i + 1, 2, . . . , N, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.89)
i+1
10. Calculate a new estimate of saturation pressure using a NewtonRaphson update. + p (n) p (n)1) sat sat *
where
Q (n)
ǒēQ Ǔ ēp
(n)
,
. . . . . . . . . . . . . . . . . . . . . . (4.85)
ȍ Y R ǒēfēp f1 * ēfēp f1 Ǔ N
ēQ + ēp
yi
i
i
i+1
zi
yi
. . . . . . . . . . . . . (4.86)
zi
is evaluated at Iteration (n). If searching for an upper saturation pressure, the new pressure estimate must be higher than p *. If the new estimate is lower than p *, go to Step 1 and use a new initialpressure estimate higher than the present p * value. 11. Check for convergence. Zick* suggests the following two criteria.
Ť
and
ȍY N
1*
i
i+1
ƪ
Ť
t 10 *13
ƫ
) ȍ lnln(R ǒY ńz Ǔ N
i
i
i+1
i
2
t 10 *8. . . . . . . . . . . . . . . . . . . . . (4.87)
In addition, check for a trivial solution using the criterion
ȍǒln Yz Ǔ N
i
i+1
i
2
t 10 *4 .
. . . . . . . . . . . . . . . . . . . . . . . . . . (4.88)
12. (a) If convergence is not achieved, return to Step 6. (b) If convergence is achieved, determine the saturation type by comparing the mole fraction of the heaviest component in the mixture with that in the incipient phase, where yN tzN indicates a bubblepoint with Ki +yi /zi and yN uzN indicates a dewpoint where Ki +zi /yi , or by comparing the density of the incipient phase with that of the feed. This algorithm can be modified to search for both lower and upper saturation pressures as well as saturation temperature at a specified pressure. Michelsen50 also gives an efficient procedure for calculating the entire phase envelope, including calculations through the critical point. More recently, he presented an approximate phaseenvelope algorithm51 that is up to 10 times faster than his original algorithm using a trialanderror solution directly for pressure and temperature (component fugacities do not need to be converged at each point on the phase envelope). Surprisingly, the results are extremely close to fully converged saturation conditions and provide excellent starting estimates for a rigorous saturationpoint calculation. He also shows that the approximate solution is always inside the phase envelope, thus representing an unstable thermodynamic condition. *Personal communication with A.A. Zick, Zick Technologies Inc., Portland, Oregon (1985).
EQUATIONOFSTATE CALCULATIONS
where mi +chemical potential of Component i, zref+homogeneous (singlephase) mixture at pressure pref at a reference depth href, and p+pressure and z+mixture composition at depth h. The entire system is at constant temperature (dT/dh+0). In 1930, Muskat54 provided exact solutions to Eq. 4.89 for a simplified EOS (ideal mixing). Numerical examples based on this oversimplified EOS led to the misleading conclusion that gravity has negligible effect on compositional variation in reservoir systems. In 1938, Sage and Lacey55 evaluated Eq. 4.89 using a more realistic EOS model. They provide examples showing significant variations of composition with depth for reservoir mixtures. Furthermore, they made the key observation that significant compositional variations should be expected in systems in the vicinity of a critical condition. From 1938 to 1980, the petroleum literature is apparently void of publications regarding calculation of compositional gradients. Several references during this period do, however, mention reservoirs exhibiting compositional variation. Schulte56 cites most of these references. He appears to be the first to solve Eq. 4.89 with a cubic EOS. His classic paper illustrates that significant compositional variation can result from gravity segregation in petroleum reservoirs. Schulte gives examples showing the effect of oil type (aromatic content) and interaction coefficients (used in the mixing rules of a cubic EOS) on compositional gradients. He also compares gradients calculated with the PengRobinson and SoaveRedlichKwong equations. In 1980, significant compositional gradients were reported in the Brent field, North Sea.5658 In the Brent formation of the Brent field, a significant composition gradient was observed, with the transition from gas to oil occurring at a saturated gas/oil contact (GOC). These papers also describe the unusual transition from gas to oil in the absence of a saturated GOC. This transition occurs at a depth where the reservoir fluid is a critical mixture, with a critical temperature equal to the reservoir temperature and a critical pressure less than the reservoir pressure. Apparently, the Statfjord formation in the Brent field is an example of a reservoir with such an “undersaturated GOC.” In 1983, Holt et al.59 presented a formulation of the compositionalgradient problem that includes thermal diffusion. Example calculations in this paper were, unfortunately, limited to binary systems. Numerous publications on the subject of compositional gradient were presented in 1984 and 1985.60,61 Most of these were field case histories; in fact, a special session of the 1985 SPE Annual Technical Conference and Exhibition was dedicated to this subject.6264 Hirschberg60 discusses the influence of asphaltenes on compositional grading. He uses a simplified twocomponent model with one component representing asphaltenes and the other representing the remaining deasphalted oil. He makes the observation that compositional grading in heavier oils ( go u0.85 or gAPIt35°API) can be strongly influenced by both the amount and the properties of asphaltenes, which implies that quantitatively accurate estimates of compositional grading resulting from asphaltenes are extremely difficult because of the strong dependence of calculated results on physical properties of the oil and asphaltene(s). Finally, Hirschberg discusses two mechanisms for the development of a tar mat. Riemens et al.61 present an interesting evaluation of the compositional grading in the Birba field, Oman. On the basis of isothermal gravity/chemical equilibrium (GCE) calculations and field measure17
ments of PVT data, they show that a significant compositional gradient exists. The authors also evaluate the possibility of injecting gas into the undersaturated oil zone where multicontact miscibility can develop. Montel and Gouel65 suggest an algorithm for solving the isothermal GCE problem. The procedure is only approximate because it calculates pressure with an incremental hydrostatic term instead of solving directly for pressure. They discuss the effect of fluid characterization on compositional grading and the effect of reservoir temperature and pressure. Finally, the authors suggest that including thermal diffusion may improve the reliability of calculated compositional gradients (although they do not include this effect in their study). Metcalfe et al.63 report measured variation of composition and physical properties of reservoir fluids in the Anschutz Ranch East field in the U.S. Overthrust Belt. These authors use an EOS to characterize the PVT behavior of the entire range of fluids sampled from the reservoir. However, instead of calculating the compositional variation using gravity/chemical equilibrium and the developed EOS characterization, they correlate compositional variation graphically on the basis of measured data. Creek and Schrader62 report compositional grading data for another Overthrust Belt reservoir, the East Painter field. Considerable data are presented together with comparison of measured compositional gradients and those calculated with the isothermal GCE model. They report difficulty in matching observed saturationpressure and GOR gradients. Finally, the authors indicate that most reservoirs along the Overthrust Belt have varying degrees of compositional grading. Belery and da Silva66 present a formulation describing the combined effects of gravity and thermal diffusion for a system of zero net mass flux. After assessing various approaches for treating thermal diffusion, they selected the Dougherty and Drickamer67 method. Belery and da Silva extend this formulation (originally valid only for binary systems) to multicomponent systems. They use a field example with EOS characterization and measured gradient data from the North Sea Ekofisk field to illustrate the gravity/thermal model. Because measured PVT gradients were very scattered (probably because of sampling problems), the comparison is not quantitatively accurate (with or without thermal diffusion). However, the calculations show qualitatively the effect of thermal diffusion and are the first such calculations reported for multicomponent systems. Wheaton68 discusses an isothermal GCE model that includes the influence of capillary pressure. The addition of capillary forces was apparently justified in an effort to assist in the initialization of reservoir simulators. Simulators use capillary pressure curves to initialize saturation and pressure distributions discretely in the vertical direction. Results of the calculated examples in Wheaton’s paper suggest that neglecting compositional variations in a gascondensate reservoir may result in large errors in the initial hydrocarbons in place. Obviously, these results are primarily a consequence of neglecting the compositional variation resulting from gravity/chemical equilibrium. Quantitatively similar results would have been obtained with or without the inclusion of capillary pressures. Finally, his observation that neglecting compositional gradients will lead to incorrect specification of initial oil and gas in place is equally applicable to gascondensate and oil reservoirs (i.e., practically any petroleum reservoir). In his discussion of Wheaton’s paper, Chaback69 makes the observation that nonisothermal effects can be on the same order of magnitude as gravity effects. More importantly, he notes that a nonisothermal system will never reach equilibrium (zero energy flux) even though a stationary (steadystate) condition of zero net mass flux is reached. Montel70 discusses compositional grading, including comments on treating thermal diffusion. He provides an equation for calculating the RayleighDarcy number that is used to indicate whether a fluid/rock system will experience convection (mechanical instability). Bedrikovetsky 71 gives an extensive discussion and formal mathematical treatment of compositional grading, including gravity, thermal, and capillary forces. The treatment yields complicated expressions, which, in a few cases, are solved for simple conditions (idealized EOS and binary systems). Many of the results are similar to those given by Muskat.54 No examples are given for multicomponent mixtures with a realistic thermodynamic model. 18
Recently, Faissat et al.72 gave a theoretical review of equilibrium formulations that include gravity and thermal diffusion. Belery and da Silva66 mention most of the formulations, but Faissat et al. formalize the thermaldiffusion term in a generic way. Unfortunately, calculations are not provided for comparing the different formulations. 4.6.1 Isothermal GCE. Eq. 4.89 gives the condition for isothermal GCE, which is sometimes written in differential form as dm i ) M i gdh + 0, i + 1, 2, . . . , N . This condition represents N equations. Together with the constraint that the sum of mole fractions z(h) must add to one,
ȍ z (h) + 1, N
i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.90)
i+1
it is possible to solve for composition z(h) and pressure p(h) at a specified depth h. Because chemical potential can be expressed as mi +RT ln fi )l(T ), Eq. 4.89 can be expressed in terms of fugacity. ln f i ǒ p ref , z ref , TǓ + ln f i ǒ p, z, T Ǔ ) 1 M i gǒh * h refǓ , RT i + 1, 2, . . . , N. . . . . . . . . . . . . . . . . . . . . . . . . . (4.91) For convenience, we define fi (h)+fi [ p(h),z(h),T ] and fi (href)+fi ( pref,zref,T ), yielding
ƪ
f i (h) + f iǒh refǓ exp *
ƫ
M i gǒh * h refǓ , RT
i + 1, 2, . . . , N. . . . . . . . . . . . . . . . . . . . . . . . . . (4.92) The volumetranslation method is widely used for correcting volumetric deficiencies of the original SoaveRedlichKwong and PengRobinson equations. The method involves calculating a linearly translated volume, vȀ, by adding a constant c to the molar volume, v, calculated from the original EOS, vȀ+v)c. Peneloux et al.20 show that the volume shift modifies the component fugacity as fi exp[ci ( p/RT)] (see Eqs. 4.26 and 4.96). This correction must be included in the fugacity expressions used for gradient calculations and also must be included in the pressure derivative of fugacity used in the recommended algorithm for solving the isothermal GCE problem. On the basis of the GibbsDuhem equation,53 combining the mechanicalequilibrium condition, dpńdh +* òg , with the GCE condition, Eq. 4.89, guarantees automatic satisfaction of the condition
ŕ ò(h)gdh . h
p(h) + pǒh refǓ )
. . . . . . . . . . . . . . . . . . (4.93)
h ref
Interestingly, the isothermal GCE equations are still valid and satisfy this condition when a saturated GOC is located between href and h [i.e., even when ò(h) is not a continuous function]. 4.6.2 Isothermal GCE Algorithm. Eqs. 4.89 and 4.90 represent equations similar to those used to calculate saturation pressure. Michelsen51 gives an efficient method for solving the saturationpressure calculation, which has been modified here to solve the GCE problem, Qǒ p, z Ǔ + 1 *
ȍ z ƪf ǒp N
~
i
i
Ǔ ń f i ǒ p, z Ǔƫ
ref, z ref
i+1
ȍY , N
+1*
i
. . . . . . . . . . . . . . . . . . . . . . . . . . (4.94)
i+1
where Y i + z i ƪ f i ǒ p ref, z refǓńf i ǒ p, z Ǔƫ . . . . . . . . . . . . . . . . . . (4.95) ~
ƪ
and f i ǒ p ref, z refǓ + f i ǒ p ref, z refǓ exp * ~
ƫ
M i gǒh * h refǓ . RT
. . . . . . . . . . . . . . . . . . . . (4.96) PHASE BEHAVIOR
An efficient algorithm for solving Eq. 4.94 uses a NewtonRaphson update for pressure and accelerated successive substitution this approach. (GDEM44) for composition. The following outlines ~ 1. Calculate fugacities of ~the reference feed f i ( y ref, z ref) and the gravitycorrected fugacity f i ( p ref, z ref) from Eqs. 4.26 and 4.96. This calculation needs to be made only once. Initial estimates of composition and pressure at h are simply values at the reference depth, z 1(h) + z ref and p 1(h) + p ref . 2. Calculate fugacities of the composition estimate z at the pressure estimate p. Calculate mole numbers from Eq. 4.95. Calculate fugacityratio corrections with f i ǒp ref , z refǓ ~
Ri +
f i ǒ p, zǓ
ǒȍ Ǔ N
*1
Yj
.
. . . . . . . . . . . . . . . . . . (4.97)
j+1
3. Update mole numbers using Eqs. 4.83 and 4.84. from Y (n)1) using 4. Calculate z (n)1) i i zi + Yi ń
ǒȍ Ǔ N
Yj .
. . . . . . . . . . . . . . . . . . . . . . . . . . (4.98)
j+1
5. Update the pressure estimate using a NewtonRaphson estimate. p (n)1) + p (n) *
where
ēQ + ēp
Q (n) ǒēQńēpǓ
ȍY R N
i
i+1
i
(n)
,
ǒēf ińēpǓ . f i ǒ p, z Ǔ
. . . . . . . . . . . . . . . . . . . . (4.99)
. . . . . . . . . . . . . . . . . . (4.100)
6. Check for convergence using Eq. 4.87. 7. Iterate until convergence is achieved. After finding the composition z(h) and pressure p(h) that satisfy Eqs. 4.89 and 4.90, a phasestability test32 must be made to establish whether the solution is valid. A valid solution is single phase (thermodynamically stable). An unstable solution indicates that the calculated z and p will split into two (or more) phases, thereby making the solution invalid. If the gradient solution is unstable, then the stabilitytest composition y should be used to reinitialize the gradient calculation. The starting pressure for the new gradient calculation can be pref or, preferably, the converged pressure from the gradient calculation that led to the unstable solution. Note that unstable gradient solutions usually occur only a short distance beyond a saturated GOC. Locating a potential GOC requires a trialanderror search. For a saturated GOC, three approaches might be considered: (1) stability tests, (2) negative flash calculations,37 or (3) saturationpressure calculations. The first and second methods should be the fastest, with the negative flash probably being faster because information from previous flash calculations can be used for initialization of subsequent flash calculations. Unfortunately, an algorithm based on either the stability test or negative flash results may suffer from the fact that only trivial solutions exist over a large part of the reservoir thickness. On the other hand, either method can be used efficiently to determine the saturated GOC once a nontrivial stability condition is found. If an undersaturated GOC exists (i.e., a transition from gas to oil through a critical mixture), only a search based on saturationpressure calculations can be used. The following algorithm is recommended for locating both saturated and undersaturated GOC’s. First, calculate the composition and pressure at the top (zT and pT ) and the bottom (z B and pB ) of the reservoir; then, calculate saturation pressures psT and psB . If the saturation types (bubblepoint/dewpoint) are the same at the top and bottom, then no GOC exists. Otherwise, a search for the GOC, hGOC, is made. A straightforward search algorithm would be interval halving based on the saturation type. At Iteration n, a solution with a dew+ h (n) for the point at depth h (n) would replace the top depth h (n)1) T next iteration, and a solution with a bubblepoint at a given depth + h (n) . The depth estimate would replace the bottom depth h (n)1) B ƫ. The ) h (n) for a given iteration is calculated from h (n) + 0.5ƪh (n) B T EQUATIONOFSTATE CALCULATIONS
number of iterations required to meet a tolerance dh would be 1.5 lnƪ(h T * h B)ńdhƫ . For example, only 13 gradient and saturationpressure calculations would be needed to achieve dh+0.33 ft for a total thickness (h T * h B)+1,640 ft. More efficient algorithms for locating the GOC can probably be developed, particularly if a nontrivial stability solution can be located. Alternatively, Michelsen’s52 criticalpoint algorithm or his new method for calculating accurate approximations for saturation pressure and temperature51 may provide a good starting point for developing an improved algorithm. Whitson and Belery73 give a detailed discussion of compositionalgradient calculations, including the application of isothermal and nonisothermal compositionalgradient algorithms to reservoir fluid systems ranging from a saturated lowGOR blackoil/drygas system to a nearcritical system. 4.7 Matching an EOS to Measured Data Most EOS characterizations (see Chap. 5) are not truly predictive74,75 because errors in saturation pressure are commonly "10%, those in densities are "5%, and compositions may be off by several mole percent for key components. Also, the EOS may predict a dewpoint incorrectly when the measured saturation condition is a bubblepoint, or vice versa. This lack of predictive capability by the EOS can be because of insufficient compositional data for the C7) fractions, inaccurate properties for the C7) fractions, inadequate BIP’s, or incorrect overall composition. The EOS characterization can be improved in a number of ways. First, however, the experimental data and fluid compositions should be checked for consistency (see Chap. 6). If the PVT data appear consistent and the fluid compositions are considered representative of the material that was analyzed in the PVT laboratory, modifying the parameters in the EOS to improve the fluid characterization will be necessary. Refs. 26 and 74 through 79 present methods for modifying the cubic EOS to fit experimental PVT data. Most of these methods modify the properties of fractions making up the C7) (Tc , pc , w, or direct multipliers on the EOS constants Wa and Wb ) and BIP’s kij between methane and C7) fractions. When an injection gas containing significant amounts of nonhydrocarbons is being studied, the kij between nonhydrocarbon and C7) fractions may also be modified. Some methods use nonlinear regression to modify the EOS parameters automatically.74,78,79 Others have tried simply to make manual adjustments to the EOS parameters through a trialanderror approach.75,77,80 The trend is now to automate the EOS modification procedure with nonlinear regression, including large amounts of measured PVT and compositional data.81 Coats and Smart74 recommend five standard EOS modifications: Wa and Wb of methane; Wa and Wb of the heaviest C7) fraction; and kij between methane and the heaviest C7) fraction. Additional parameters (nonhydrocarbon Wa and Wb and kij ) are used for systems with significant amounts of nonhydrocarbon components. Their approach differs from other methods in that they do not use volume translation. As a result, significant methane corrections had to be applied to EOS constants Wa and Wb . Using the Coats and Smart approach with the PR EOS typically results in multipliers of the EOS constants Wa and Wb ranging from 1.2 to 1.5 for methane and from 0.6 to 0.8 for the heaviest C7) fraction; kij of the methane/C7) heavy fraction varies from 0 to 0.3. The W corrections can be interpreted as modifications of the critical properties.75 With a somewhat untraditional regression approach, Coats and Smart minimize a sum of weighted absolute deviations using linear programming. They suggest weighting factors of 40 for saturation pressures, 10 for saturation densities, and 1 for most other data. Their results are impressive, showing excellent matches of nearcritical fluids, hydrocarbon and nonhydrocarbon gas injection in oils and retrograde condensate systems, and simple depletion data. With a twoconstant cubic EOS with volume translation, the modifications of EOS parameters (or critical properties) is typically only 5 to 10% compared with the "30 to 40% modifications required with the Coats and Smart approach without volume translation. This is explained by the initial predictions being much better with vol19
ume translation, thereby requiring fewer modifications to achieve the same quality fit of measured data. Interestingly, the same five standard regression parameters originally suggested by Coats and Smart can be used with an EOS that uses volume translation. However, the result is usually that methane corrections to Wa and Wb remain close to 1.0 and corrections to Wa and Wb for the heaviest C7) fraction range from 0.9 to 1.1. Therefore, it may be better to drop methane corrections to Wa and Wb and use instead one set of corrections to the Wa and Wb for the heaviest C7) fraction, and another set of corrections to the Wa and Wb for the nexttoheaviest C7) fraction. This approach is particularly helpful when matching liquiddropout curves with a “tail” (see Appendix C) or in multicontact vaporization experiments. Finally, an alternative to use of corrections to Wa and Wb directly would be to modify Tc and pc instead (modification of w is not recommended). Be aware, however, that the sensitivity of the minimization problem to Tc and pc is probably less than to Wa and Wb , thereby making the mathematical search for a minimum more difficult. Appendix C gives a thorough discussion of how nonlinear regression can be used to adjust EOS parameters systematically to fit measured PVT data. References 1. Edmister, W.C. and Lee, B.I.: Applied Hydrocarbon Thermodynamics, Gulf Publishing Co., Houston (1983). 2. Edmister, W.C. and Lee, B.I.: Applied Hydrocarbon Thermodynamics, second edition, Gulf Publishing Co., Houston (1984) I. 3. Reid, R.C., Prausnitz, J.M., and Polling, B.E.: The Properties of Gases and Liquids, fourth edition, McGrawHill Book Co. Inc., New York City (1987). 4. Michelsen, M.L.: “Partial Derivatives of Thermodynamic Properties from Equations of State,” report, Inst. for Kemiteknik, Denmark Technical U., Lyngby, Denmark (1981). 5. van der Waals, J.D.: Continuity of the Gaseous and Liquid State of Matter (1873). 6. Redlich, O. and Kwong, J.N.S.: “On the Thermodynamics of Solutions. V. An Equation of State. Fugacities of Gaseous Solutions,” Chem. Rev. (1949) 44, 233. 7. Peng, D.Y. and Robinson, D.B.: “A NewConstant Equation of State,” Ind. & Eng. Chem. (1976) 15, No. 1, 59. 8. Martin, J.J.: “Cubic Equations of State—Which?,” Ind. & Eng. Chem. (1979) 18, No. 2, 81. 9. Abbott, M.M.: “Cubic Equation of State,” AIChE J. (May 1973) 596. 10. Abbott, M.M.: “Cubic Equations of State: An Interpretive Review,” Equations of State in Engineering and Research, K.C. Chao and R.L. Robinson Jr. (eds.), Advances in Chemistry Series, Amer. Chemical Soc., Washington, DC (1978) 182, 47–97. 11. Yarborough, L.: “Application of a Generalized Equation of State to Petroleum Reservoir Fluids,” Equations of State in Engineering and Research, K.C. Chao and R.L. Robinson Jr. (eds.), Advances in Chemistry Series, Amer. Chemical Soc., Washington, DC (1978) 182, 386–439. 12. Usdin, E. and McAuliffe, J.C.: “A One Parameter Family of Equations of State,” Chem. Eng. Sci. (1976) 31, 1077. 13. Fuller, G.G.: “A Modified RedlichKwongSoave Equation of State Capable of Representing the Liquid State,” Ind. & Eng. Chem. (1976) 15, 254. 14. Schmidt, G. and Wenzel, H.: “A Modified van der Waals Type Equation of State,” Chem. Eng. Sci. (1980) 35, 1503. 15. Patel, V.C. and Teja, A.S.: “A New Cubic Equation of State for Fluids and Fluid Mixtures,” Chem. Eng. Sci. (1982) 37, No. 3, 463. 16. Kumar, K.H. and Starling, K.E.: “Comments on: “Cubic Equations of State—Which?,” Ind. & Eng. Chem. (1979) 19, 128. 17. Kumar, K.H. and Starling, K.E.: “The Most General DensityCubic Equation of State: Application to Pure Nonpolar Fluids,” Ind. & Eng. Chem. (1982) 21, 255. 18. Soave, G.: “Equilibrium Constants from a Modified RedlichKwong Equation of State,” Chem. Eng. Sci. (1972) 27, No. 6, 1197. 19. Zudkevitch, D. and Joffe, J.: “Correlation and Prediction of VaporLiquid Equilibrium with the RedlichKwong Equation of State,” AIChE J. (1970) 16, 112. 20. Peneloux, A., Rauzy, E., and Freze, R.: “A Consistent Correction for RedlichKwongSoave Volumes,” Fluid Phase Equilibria (1982) 8, 7. 21. Martin, J.J. and Hou, Y.C.: AIChE J. (1955) 1, 142. 20
22. Abbott, M.M.: “Thirteen Ways of Looking at the van der Waals Equation,” paper #31b presented at the 1988 AIChE Spring Natl. Meeting, New Orleans, 7 March. 23. Michelsen, M.L.: “The Isothermal Flash Problem. Part II. PhaseSplit Calculation,” Fluid Phase Equilibria (1982) 9, 21. 24. Nagy, Z. and Shirkovskiy, A.I: “Mathematical Simulation of Natural Gas Condensation Processes Using the PengRobinson Equation of State,” paper SPE 10982 presented at the 1982 SPE Annual Technical Conference and Exhibition, New Orleans, 26–29 September. 25. Robinson, D.B., Peng, D.Y., and Ng, H.Y.: “Capabilities of the PengRobinson Programs, Part 2: ThreePhase and Hydrate Calculations,” Hydrocarbon Proc. (1979) 58, 269. 26. Pedersen, K.S., Thomassen, P., and Fredenslund, A.: “Characterization of Gas Condensate Mixtures,” C7) Fraction Characterization, L.G. Chorn and G.A. Mansoori (eds.), Advances in Thermodynamics, Taylor & Francis, New York City (1989) 1. 27. Pedersen, K.S., Thomassen, P., and Fredenslund, A.: “SRKEOS Calculation for Crude Oils,” Fluid Phase Equilibria (1983) 14, 209. 28. Joffe, J., Schroeder, G.M., and Zudkevitch, D.: “VaporLiquid Equilibria with the RedlichKwong Equation of State,” AIChE J. (May 1970) 496. 29. Haman, S.E.M. et al.: “Generalized Temperature Dependent Parameters of the RedlichKwong Equation of State for VaporLiquid Equilibrium Calculations,” Ind. & Eng. Chem. Proc. Des. Dev. (1977) 16, No. 1. 30. Robinson, D.B. and Peng, D.Y.: “The Characterization of the Heptanes and Heavier Fractions,” Research Report 28, Gas Producers Assn., Tulsa, Oklahoma (1978). 31. Jhaveri, B.S. and Youngren, G.K.: “ThreeParameter Modification of the PengRobinson Equation of State To Improve Volumetric Predictions,” SPERE (August 1988) 1033. 32. Michelsen, M.L.: “The Isothermal Flash Problem. Part I. Stability,” Fluid Phase Equilibria (1982) 9, 1. 33. Nghiem, L.X. and Aziz, K.: “A Robust Iterative Method for Flash Calculations Using the SoaveRedlichKwong or the PengRobinson Equation of State,” paper SPE 8285 presented at the 1979 SPE Annual Technical Conference and Exhibition, Las Vegas, 23–26 September. 34. Trangenstein, J.A.: “Minimization of Gibbs Energy in Compositional Reservoir Simulation,” Chem. Eng. Sci. (1985) 12, 2847. 35. Rachford, H.H. and Rice, J.D.: “Procedure for Use of Electrical Digital Computers in Calculating Flash Vaporization Hydrocarbon Equilibrium,” JPT (October 1952) 19; Trans., AIME, 195. 36. Li, Y.K. and Nmhiem, L.X.: “The Development of a General Phase Envelope Construction Algorithm for Reservoir Fluid Studies,” paper SPE 11198 presented at the 1982 SPE Annual Technical Conference and Exhibition, New Orleans, 26–29 September. 37. Whitson, C.H. and Michelsen, M.L.: “The Negative Flash,” Fluid Phase Equilibria (1989) 53, 51. 38. Muskat, M. and McDowell, J.M.: “An Electrical Computer for Solving Phase Equilibrium Problems,” Trans., AIME (1949) 186, 291. 39. Wilson, G.M.: “A Modified RedlichKwong Equation of State, Application to General Physical Data Calculations,” paper 15c presented at the 1969 AIChE Natl. Meeting, Cleveland, Ohio. 40. Mehra, R.K., Heidemann, R.A., and Aziz, K.: “Computation of Multiphase Equilibrium for Compositional Simulators,” paper SPE 9232 presented at the 1980 SPE Annual Technical Conference and Exhibition, Dallas, 21–24 September. 41. Mehra, R.K., Heidemann, R.A., and Aziz, K.: “Computation of Multiphase Equilibrium for Compositional Simulation,” SPEJ (February 1982) 61. 42. Nghiem, L.X.: “A New Approach to QuasiNewton Method With Application to Compositional Modeling,” paper SPE 12242 presented at the 1983 SPE Symposium on Reservoir Simulation, San Francisco, 16–18 November. 43. Risnes, R., Dalen, V., and Jensen, J.I.: “Phase Equilibrium Calculations in the NearCritical Region,” Proc., European Symposium on EOR, Bournemouth, U.K. (1981). 44. Crowe, A.M. and Nishio, M.: “Convergence Promotion in the Simulation of Chemical Processesthe General Dominant Eigenvalue Method,” AIChE J. (1975) 21, 528. 45. Young, L.: “Equation of State Compositional Modeling on Vector Processors,” JPT (February 1991) 107. 46. Baker, L.E., Pierce, A.C., and Luks, K.D.: “Gibbs Energy Analysis of Phase Equilibria,” SPEJ (October 1982) 731; Trans., AIME, 273. 47. Nghiem, L.X. and Li, Y.K.: “Computation of Multiphase Equilibrium Phenomena With an Equation of State,” Fluid Phase Equilibria (1984) 17, 77. PHASE BEHAVIOR
48. Nghiem, L.X. and Li, Y.K.: “Application of Tangent Plane Criterion to Saturation Pressure and Temperature Computations,” Fluid Phase Equilibria (1984) 21, 39. 49. Michelsen, M.L.: “Saturation Point Calculations,” Fluid Phase Equilibria (1985) 23, 181. 50. Michelsen, M.L.: “Calculation of Phase Envelopes and Critical Points for Multicomponent Mixtures,” Fluid Phase Equilibria (1980) 4, 1. 51. Michelsen, M.L.: “A Simple Method for Calculation of Approximate Phase Boundaries,” Fluid Phase Equilibria (1994) 98, 1. 52. Michelsen, M.L.: “Calculation of Critical Points and Phase Boundaries in the Critical Region,” Fluid Phase Equilibria (1984) 16, 57. 53. Gibbs, J.W.: The Collected Works of J. Willard Gibbs, Yale U. Press, New Haven, Connecticut (1948) 1. 54. Muskat, M.: “Distribution of NonReacting Fluids in the Gravitational Field,” Physical Rev. (June 1930) 35, 1384. 55. Sage, B.H. and Lacey, W.N.: “Gravitational Concentration Gradients in Static Columns of Hydrocarbon Fluids,” Trans., AIME (1938) 132, 120. 56. Schulte, A.M.: “Compositional Variations Within a Hydrocarbon Column Due to Gravity,” paper SPE 9235 presented at the 1980 SPE Annual Technical Conference and Exhibition, Dallas, 21–24 September. 57. Bath, P.G.H., Fowler, W.N., and Russell, M.P.M.: “The Brent Field, A Reservoir Engineering Review,” paper EUR 164 presented at the 1980 SPE European Offshore Petroleum Conference and Exhibition, London, 21–24 October. 58. Bath, P.G.H., van der Burgh, J., and Ypma, J.G.J.: “Enhanced Oil Recovery in the North Sea,” Proc., 11th World Pet. Cong. (1983). 59. Holt, T., Lindeberg, E., and Ratkje, S.K.: “The Effect of Gravity and Temperature Gradients on Methane Distribution in Oil Reservoirs,” paper SPE 11761 available from SPE, Richardson, Texas (1983). 60. Hirschberg, A.: “Role of Asphaltenes in Compositional Grading of a Reservoir’s Fluid Column,” JPT (January 1988) 89. 61. Riemens, W.G., Schulte, A.M., and de Jong, L.N.J.: “Birba Field PVT Variations Along the Hydrocarbon Column and Confirmatory Field Tests,” JPT (January 1988) 83. 62. Creek, J.L. and Schrader, M.L.: “East Painter Reservoir: An Example of a Compositional Gradient From a Gravitational Field,” paper SPE 14411 presented at the 1985 SPE Annual Technical Conference and Exhibition, Las Vegas, 22–25 September. 63. Metcalfe, R.S., Vogel, J.L., and Morris, R.W.: “Compositional Gradient in the Anschutz Ranch East Field,” paper SPE 14412 presented at the 1985 SPE Annual Technical Conference and Exhibition, Las Vegas, 22–25 September. 64. Montel, F. and Gouel, P.L.: “A New Lumping Scheme of Analytical Data for Compositional Studies,” paper SPE 13119 presented at the 1984 SPE Annual Technical Conference and Exhibition, Houston, 16–19 September. 65. Montel, F. and Gouel, P.L.: “Prediction of Compositional Grading in a Reservoir Fluid Column,” paper SPE 14410 presented at the 1985 SPE Annual Technical Conference and Exhibition, Las Vegas, 22–25 September. 66. Belery, P. and da Silva, F.V.: “Gravity and Thermal Diffusion in Hydrocarbon Reservoirs,” paper presented at the 1990 Chalk Research Program, Copenhagen, 11–12 June.
EQUATIONOFSTATE CALCULATIONS
67. Dougherty, E.L. Jr. and Drickamer, H.G.: “Thermal Diffusion and Molecular Motion in Liquids,” J. Phys. Chem. (1955) 59, 443. 68. Wheaton, R.J.: “Treatment of Variation of Composition With Depth in GasCondensate Reservoirs,” SPERE (May 1991) 239. 69. Chaback, J.J.: “Discussion of Treatment of Variations of Composition With Depth in GasCondensate Reservoirs,” SPERE (February 1992) 157. 70. Montel, F.: “Phase Equilibria Needs for Petroleum Exploration and Production Industry,” Fluid Phase Equilibria (1993) 84, 343. 71. Bedrikovetsky, P.G.: Mathematical Theory of Oil and Gas Recovery, Petroleum Engineering & Development Studies, Cluwer Academic, Horthreht, Russia (1993) 4. 72. Faissat, B. et al.: “Fundamental Statements about Thermal Diffusion for a Multicomponent Mixture in a Porous Medium,” Fluid Phase Equilibria (1995) 100, 1. 73. Whitson, C.H. and Belery, P.: “Compositional Gradients in Petroleum Reservoirs,” paper SPE 28000 presented at the 1994 U. of Tulsa/SPE Centennial Petroleum Engineering Symposium, Tulsa, Oklahoma, 29–31 August. 74. Coats, K.H. and Smart, G.T.: “Application of a RegressionBased EOS PVT Program to Laboratory Data,” SPERE (May 1986) 277. 75. Whitson, C.H.: “Effect of C7) Properties on EquationofState Predictions,” SPEJ (December 1984) 685; Trans., AIME, 277. 76. Coats, K.H.: “Simulation of Gas Condensate Reservoir Performance,” JPT (October 1985) 1870. 77. Pedersen, K.S., Thomassen, P., and Fredenslund, A.: “On the Dangers of Tuning Equation of State Parameters,” paper SPE 14487 available from SPE, Richardson, Texas (1985). 78. Agarwal, R., Li, Y.K., and Nghiem, L.X.: “A Regression Technique With DynamicParameter Selection for Phase Behavior Matching,” SPERE (February 1990) 115. 79. Søreide, I.: “Improved Phase Behavior Predictions of Petroleum Reservoir Fluids From a Cubic Equation of State,” Dr.Ing. dissertion, Norwegian Inst. of Technology, Trondheim, Norway (1989). 80. Turek, E.A. et al.: “Phase Equilibria in CO2Multicomponent Hydrocarbon Systems: Experimental Data and an Improved Prediction Technique,” SPEJ (June 1984) 308. 81. Zick, A.A.: “A Combined Condensing/Vaporizing Mechanism in the Displacement of Oil by Enriched Gases,” paper SPE 15493 presented at the 1986 SPE Annual Technical Conference and Exhibition, New Orleans, 5–8 October.
SI Metric Conversion Factors °API 141.5/(131.5)°API) +g/cm3 bar 1.0* E)05 +Pa ft 3.048* E*01 +m °F (°F*32)/1.8 +°C °F (°F)459.67)/1.8 +K psi 6.894 757 E)00 +kPa *Conversion factor is exact.
21
Chapter 5
HeptanesĆPlus Characterization 5.1 Introduction Some phasebehavior applications require the use of an equation of state (EOS) to predict properties of petroleum reservoir fluids. The critical properties, acentric factor, molecular weight, and binaryinteraction parameters (BIP’s) of components in a mixture are required for EOS calculations. With existing chemicalseparation techniques, we usually cannot identify the many hundreds and thousands of components found in reservoir fluids. Even if accurate separation were possible, the critical properties and other EOS parameters of compounds heavier than approximately C20 would not be known accurately. Practically speaking, we resolve this problem by making an approximate characterization of the heavier compounds with experimental and mathematical methods. The characterization of heptanesplus (C7)) fractions can be grouped into three main tasks.1–3 1. Dividing the C7) fraction into a number of fractions with known molar compositions. 2. Defining the molecular weight, specific gravity, and boiling point of each C7) fraction. 3. Estimating the critical properties and acentric factor of each C7) fraction and the key BIP’s for the specific EOS being used. This chapter presents methods for performing these tasks and gives guidelines on when each method can be used. A unique characterization does not exist for a given reservoir fluid. For example, different component properties are required for different EOS’s; therefore, the engineer must determine the quality of a given characterization by testing the predictions of reservoirfluid behavior against measured pressure/volume/temperature (PVT) data. The amount of C7) typically found in reservoir fluids varies from u50 mol% for heavy oils to t1 mol% for light reservoir fluids.4 Average C7) properties also vary widely. For example, C7) molecular weight can vary from 110 to u300 and specific gravity from 0.7 to 1.0. Because the C7) fraction is a mixture of many hundreds of paraffinic, naphthenic, aromatic, and other organic compounds,5 the C7) fraction cannot be resolved into its individual components with any precision. We must therefore resort to approximate descriptions of the C7) fraction. Sec. 5.2 discusses experimental methods available for quantifying C7) into discrete fractions. Trueboilingpoint (TBP) distillation provides the necessary data for complete C7) characterization, including mass and molar quantities, and the key inspection data for each fraction (specific gravity, molecular weight, and boiling point). Gas chromatography (GC) is a lessexpensive, timesaving alternative to TBP distillation. However, GC analysis quantifies only the mass of C7) fractions; such properties as specific gravity and boiling point are not provided by GC analysis. HEPTANESPLUS CHARACTERIZATION
Typically, the practicing engineer is faced with how to characterize a C7) fraction when only z C7) the mole fraction, ; molecular weight, M C7); and specific gravity, g C7) , are known. Sec. 5.3 reviews methods for splitting C7) into an arbitrary number of subfractions. Most methods assume that mole fraction decreases exponentially as a function of molecular weight or carbon number. A more general model based on the gamma distribution has been successfully applied to many oil and gascondensate systems. Other splitting schemes can also be found in the literature; we summarize the available methods. Sec. 5.4 discusses how to estimate inspection properties g and Tb for C7) fractions determined by GC analysis or calculated from a mathematical split. Katz and Firoozabadi’s6 generalized single carbon number (SCN) properties are widely used. Other methods for estimating specific gravities of C7) subfractions are based on forcing the calculated g C7) to match the measured value. Many empirical correlations are available for estimating critical properties of pure compounds and C7) fractions. Critical properties can also be estimated by forcing the EOS to match the boiling point and specific gravity of each C7) fraction separately. In Sec. 5.5, we review the most commonly used methods for estimating critical properties. Finally, Sec. 5.6 discusses methods for reducing the number of components describing a reservoir mixture and, in particular, the C7) fraction. Splitting the C7) into pseudocomponents is particularly important for EOSbased compositional reservoir simulation. A large part of the computing time during a compositional reservoir simulation is used to solve the flash calculations; accordingly, minimizing the number of components without jeopardizing the quality of the fluid characterization is necessary. 5.2 Experimental Analyses The most reliable basis for C7) characterization is experimental data obtained from hightemperature distillation or GC. Many experimental procedures are available for performing these analyses; in the following discussion, we review the most commonly used methods. TBP distillation provides the key data for C7) characterization, including mass and molar quantities, specific gravity, molecular weight, and boiling point of each distillation cut. Other such inspection data as kinematic viscosity and refractive index also may be measured on distillation cuts. Simulated distillation by GC requires smaller samples and less time than TBP distillation.79 However, GC analysis measures only the mass of carbonnumber fractions. Simulated distillation results can be calibrated against TBP data, thus providing physical properties for the individual fractions. For many oils, simulated distillation 1
Cutoff (nparaffin) boiling point Midvolume (“normal”) boiling point
Dp
Dp
N2
N2
Fig. 5.2—TBP curve for a North Sea gascondensate sample illustrating the midvolumepoint method for calculating average boiling point (after Austad et al.7).
Fig. 5.1—Standard apparatus for conducting TBP analysis of crudeoil and condensate samples at atmospheric and subatmospheric pressures (after Ref. 11).
provides the necessary information for C7) characterization in far less the time and at far less cost than that required for a complete TBP analysis. We recommend, however, that at least one complete TBP analysis be measured for (1) oil reservoirs that may be candidates for gas injection and (2) most gascondensate reservoirs. 5.2.1 TBP Distillation. In TBP distillation, a stocktank liquid (oil or condensate) is separated into fractions or “cuts” by boilingpoint range. TBP distillation differs from the Hempel and American Soc. for Testing Materials (ASTM) D158 distillations10 because TBP analysis requires a high degree of separation, which is usually controlled by the number of theoretical trays in the apparatus and the reflux ratio. TBP fractions are often treated as components having unique boiling points, critical temperatures, critical pressures, and other properties identified for pure compounds. This treatment is obviously more valid for a cut with a narrow boilingpoint range. The ASTM D289211 procedure is a useful standard for TBP analysis of stocktank liquids. ASTM D2892 specifies the general procedure for TBP distillation, including equipment specifications (see Fig. 5.1), reflux ratio, sample size, and calculations necessary to arrive at a plot of cumulative volume percent vs. normal boiling point. Normal boiling point implies that boiling point is measured at normal or atmospheric pressure. In practice, to avoid thermal decomposition (cracking), distillation starts at atmospheric pressure and is changed to subatmospheric distillation after reaching a limiting temperature. Subatmospheric boilingpoint temperatures are converted to normal boilingpoint temperatures by use of a vaporpressure correlation that corrects for the amount of vacuum and the fraction’s chemical composition. The boilingpoint range for fractions is not specified in the ASTM standard. Katz and Firoozabadi6 recommend use of paraffin normal boiling points (plus 0.5°C) as boundaries, a practice that has been widely accepted. 2
Fig. 5.27 shows a plot of typical TBP data for a North Sea sample. Normal boiling point is plotted vs. cumulative volume percent. Table 5.1 gives the data, including measured specific gravities and molecular weights. Average boiling point is usually taken as the value found at the midvolume percent of a cut. For example, the third cut in Table 5.1 boils from 258.8 to 303.8°F, with an initial 27.49 vol% and a final 37.56 vol%. The midvolume percent is (27.49)37.56)/2+32.5 vol%; from Fig. 5.2, the boiling point at this volume is [282°F. For normalparaffin boilingpoint intervals, Katz and Firoozabadi’s6 average boiling points of SCN fractions can be used (see Table 5.2). The mass, m i, of each distillation cut is measured directly during a TBP analysis. The cut is quantified in moles n i with molecular weight, M i, and the measured mass m i, where n i + m ińM i. Volume of the fraction is calculated from the mass and the density, ò i (or specific gravity, g i), where V i + m ińò i . M i is measured by a cryoscopic method based on freezingpoint depression, and ò i is measured by a pycnometer or electronic densitometer. Table 5.1 gives cumulative weight, mole, and volume percents for the North Sea sample. Average C7) properties are given by
ȍm N
MC
7)
+
i
i+1 N
ȍn
i
i+1
ȍm N
and ò C
7)
+
i
i+1 N
ȍV
,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.1)
i
i+1
where ò C7) + g C7)ò w with ò w +pure water density at standard conditions. These calculated averages are compared with measured values of the C7) sample, and discrepancies are reported as “lost” material. Refs. 7 and 15 through 20 give procedures for calculating properties from TBP analyses. Also, the ASTM D289211 procedure gives details on experimental equipment and the procedure for conducting TBP analysis at atmospheric and subatmospheric conditions. Table 5.3 gives an example TBP analysis from a commercial laboratory. PHASE BEHAVIOR
TABLE 5.1—EXPERIMENTAL TBP RESULTS FOR A NORTH SEA CONDENSATE Upper Tbi (°F)
Average Tbi * (°F)
C7
208.4
194.0
90.2
0.7283
96
123.9
0.940
C8
258.8
235.4
214.6
0.7459
110
287.7
C9
303.8
282.2
225.3
0.7658
122
C10
347.0
325.4
199.3
0.7711
C11
381.2
363.2
128.8
C12
420.8
401.1
C13
455.0
C14
4.35
4.80
7.80
4.35
4.80
11.92
1.951
10.35
11.15
16.19
14.70
15.95
11.88
294.2
1.847
10.87
11.40
15.33
25.57
27.35
11.82
137
258.5
1.455
9.61
10.02
12.07
35.18
37.37
11.96
0.7830
151
164.5
0.853
6.21
6.37
7.08
41.40
43.74
11.97
136.8
0.7909
161
173.0
0.850
6.60
6.70
7.05
48.00
50.44
12.03
438.8
123.8
0.8047
181
153.8
0.684
5.97
5.96
5.68
53.97
56.41
11.99
492.8
474.8
120.5
0.8221
193
146.6
0.624
5.81
5.68
5.18
59.78
62.09
11.89
C15
523.4
509.0
101.6
0.8236
212
123.4
0.479
4.90
4.78
3.98
64.68
66.87
12.01
C16
550.4
537.8
74.1
0.8278
230
89.5
0.322
3.57
3.47
2.67
68.26
70.33
12.07
C17
579.2
564.8
76.8
0.8290
245
92.6
0.313
3.70
3.59
2.60
71.96
73.92
12.16
C18
604.4
591.8
58.2
0.8378
259
69.5
0.225
2.81
2.69
1.87
74.77
76.62
12.14
C19
629.6
617.0
50.2
0.8466
266
59.3
0.189
2.42
2.30
1.57
77.19
78.91
12.11
C20
653.0
642.2
45.3
0.8536
280
53.1
0.162
2.19
2.06
1.34
79.37
80.97
12.10
427.6
0.8708
370
491.1
1.156
20.63
19.03
9.59
100.00
100.00
2,580.5
12.049
100.00
100.00
100.00
Sum
Mi (g/mol)
gi **
2,073.1
Average
0.8034
Vi (cm3)
ni (mol)
wi (%)
SxVi %
xi %
C21)
mi (g)
Swi %
xVi %
Fraction
172
Kw
11.98
Reflux ratio+1 : 5; reflux cycle+18 seconds; distillation at atmospheric pressure+201.2 to 347°F; distillation at 100 mm Hg+347 to 471.2°F; and distillation at 10 mm Hg+471.2 to 653°F. Vi +mi /gi /0.9991; ni +mi /Mi ; wi +100 *Average taken at midvolume point. **Water+1.
mi /2073.1; xVi +100
Vi/2580.5; xi +100
ni /12.049; Swi +Swi ; SxVi +SxVi ; and Kw +(Tbi +460)1/3/gi .
Boiling points are not reported because normalparaffin boilingpoint intervals are used as a standard; accordingly, the average boiling points suggested by Katz and Firoozabadi6 (Table 5.2) can be used. 5.2.2 Chromatography. GC and, to a lesser extent, liquid chromatography are used to quantify the relative amount of compounds found in oil and gas systems. The most important application of chromatography to C7) characterization is simulated distillation by GC techniques. Fig. 5.3 shows an example gas chromatogram for the North Sea sample considered earlier. The dominant peaks are for normal paraffins, which are identified up to nC22. As a good approximation for a paraffinic sample, the GC response for carbon number Ci starts at the bottom response of nCi*1 and extends to the bottom response of nCi . The mass of carbon number Ci is calculated as the area under the curve from the baseline to the GC response in the nCi*1 to nCi interval (see the shaded area for fraction C9 in Fig. 5.3). As Fig. 5.47 shows schematically, the baseline should be determined before running the actual chromatogram. Because stocktank samples cannot be separated completely by standard GC analysis, an internal standard must be used to relate GC area to mass fraction. Normal hexane was used as an internal standard for the sample in Fig. 5.3. The internal standard’s response factor may need to be adjusted to achieve consistency between simulated and TBP distillation results. This factor will probably be constant for a given oil, and the factor should be determined on the basis of TBP analysis of at least one sample from a given field. Fig. 5.5 shows the simulated vs. TBP distillation curves for the Austad et al.7 sample. A 15% correction to the internalstandard response factor was used to match the two distillation curves. As an alternative to correcting the internal standard, Maddox and Erbar15 suggest that the reported chromatographic boiling points be adjusted by a correction factor that depends on the reported boiling HEPTANESPLUS CHARACTERIZATION
point and the “paraffinicity” of the composite sample. This correction factor varies from 1 to 1.15 and is slightly larger for aromatic than paraffinic samples. Several laboratories have calibrated GC analysis to provide simulateddistillation results up to C40. However, checking the accuracy of simulated distillation for SCN fractions greater than approximately C25 is difficult because C25 is usually the upper limit for reliable TBP distillation. The main disadvantage of simulated distillation is that inspection data are not determined directly for each fraction and must therefore either be correlated from TBP data or estimated from correlations (see Sec. 5.4). Sophisticated analytical methods, such as mass spectroscopy, may provide detailed information on the compounds separated by GC. For example, mass spectroscopy GC can establish the relative amounts of paraffins, naphthenes, and aromatics (PNA’s) for carbonnumber fractions distilled by TBP analysis. Detailed PNA information should provide more accurate estimation of the critical properties of petroleum fractions, but the analysis is relatively costly and timeconsuming from a practical point of view. Recent work has shown that PNA analysis3,1923 may improve C7) characterization for modeling phase behavior with EOS’s. Our experience, however, is that PNA data have limited usefulness for improving EOS fluid characterizations. 5.3 Molar Distribution Molar distribution is usually thought of as the relation between mole fraction and molecular weight. In fact, this concept is misleading because a unique relation does not exist between molecular weight and mole fraction unless the fractions are separated in a consistent manner. Consider for example a C7) sample distilled into 10 cuts separated by normalparaffin boiling points. If the same C7) sample is distilled with constant 10vol% cuts, the two sets of data will not 3
TABLE 5.2—SINGLE CARBON NUMBER PROPERTIES FOR HEPTANESPLUS (after Katz and Firoozabadi6) KatzFiroozabadi Generalized Properties LeeKesler12/KeslerLee13 Correlations
Tb Interval* Fraction Number
Defined Kw
Tc (°R)
pc (psia)
ą ąw
Riazi14
Defined
Vc (ft3/lbm mol)
Zc
Lower (°F)
Upper (°F)
Average Tb (°F) (°R)
ăăg*ă
M
6
97.7
156.7
147.0
606.7
0.690
84
12.27
914
476
0.271
5.6
0.273
7
156.7
210.0
197.4
657.1
0.727
96
11.96
976
457
0.310
6.2
0.272
8
210.0
259.0
242.1
701.7
0.749
107
11.86
1,027
428
0.349
6.9
0.269
9
259.0
304.3
288.0
747.6
0.768
121
11.82
1,077
397
0.392
7.7
0.266
10
304.3
346.3
330.4
790.1
0.782
134
11.82
1,120
367
0.437
8.6
0.262
11
346.3
385.5
369.0
828.6
0.793
147
11.84
1,158
341
0.479
9.4
0.257
12
385.5
422.2
406.9
866.6
0.804
161
11.86
1,195
318
0.523
10.2
0.253
13
422.2
456.6
441.0
900.6
0.815
175
11.85
1,228
301
0.561
10.9
0.249
14
456.6
489.0
475.5
935.2
0.826
190
11.84
1,261
284
0.601
11.7
0.245
15
489.0
520.0
510.8
970.5
0.836
206
11.84
1,294
268
0.644
12.5
0.241
16
520.0
548.6
541.4
1,001.1
0.843
222
11.87
1,321
253
0.684
13.3
0.236
17
548.6
577.4
572.0
1,031.7
0.851
237
11.87
1,349
240
0.723
14.0
0.232
18
577.4
602.6
595.4
1,055.1
0.856
251
11.89
1,369
230
0.754
14.6
0.229
19
602.6
627.8
617.0
1,076.7
0.861
263
11.90
1,388
221
0.784
15.2
0.226
20
627.8
651.2
640.4
1,100.1
0.866
275
11.92
1,408
212
0.816
15.9
0.222
21
651.2
674.6
663.8
1,123.5
0.871
291
11.94
1,428
203
0.849
16.5
0.219
22
674.6
692.6
685.4
1,145.1
0.876
305
11.94
1,447
195
0.879
17.1
0.215
23
692.6
717.8
707.0
1,166.7
0.881
318
11.95
1,466
188
0.909
17.7
0.212
24
717.8
737.6
726.8
1,186.5
0.885
331
11.96
1,482
182
0.936
18.3
0.209
25
737.6
755.6
746.6
1,206.3
0.888
345
11.99
1,498
175
0.965
18.9
0.206
26
755.6
775.4
766.4
1,226.1
0.892
359
12.00
1,515
168
0.992
19.5
0.203
27
775.4
793.4
786.2
1,245.9
0.896
374
12.01
1,531
163
1.019
20.1
0.199
28
793.4
809.6
804.2
1,263.9
0.899
388
12.03
1,545
157
1.044
20.7
0.196
29
809.6
825.8
820.4
1,280.1
0.902
402
12.04
1,559
152
1.065
21.3
0.194
30
825.8
842.0
834.8
1,294.5
0.905
416
12.04
1,571
149
1.084
21.7
0.191
31
842.0
858.2
851.0
1,310.7
0.909
430
12.04
1,584
145
1.104
22.2
0.189
32
858.2
874.4
865.4
1,325.1
0.912
444
12.04
1,596
141
1.122
22.7
0.187
33
874.4
888.8
879.8
1,339.5
0.915
458
12.05
1,608
138
1.141
23.1
0.185
34
888.8
901.4
892.4
1,352.1
0.917
472
12.06
1,618
135
1.157
23.5
0.183
35
901.4
915.8
906.8
1,366.5
0.920
486
12.06
1,630
131
1.175
24.0
0.180
36
919.4
1,379.1
0.922
500
12.07
1,640
128
1.192
24.5
0.178
37
932.0
1,391.7
0.925
514
12.07
1,650
126
1.207
24.9
0.176
38
946.4
1,406.1
0.927
528
12.09
1,661
122
1.226
25.4
0.174
39
959.0
1,418.7
0.929
542
12.10
1,671
119
1.242
25.8
0.172
40
971.6
1,431.3
0.931
556
12.10
1,681
116
1.258
26.3
0.170
41
982.4
1,442.1
0.933
570
12.11
1,690
114
1.272
26.7
0.168
42
993.2
1,452.9
0.934
584
12.13
1,697
112
1.287
27.1
0.166
43
1,004.0
1,463.7
0.936
598
12.13
1,706
109
1.300
27.5
0.164
44
1,016.6
1,476.3
0.938
612
12.14
1,716
107
1.316
27.9
0.162
45
1,027.4
1,487.1
0.940
626
12.14
1,724
105
1.328
28.3
0.160
*At 1 atmosphere. **Water+1.
produce the same plot of mole fraction vs. molecular weight. However, a plot of cumulative mole fraction,
ȍz
vs. cumulative average molecular weight,
ȍz M
i
Q zi +
j+1
ȍz
j
,ĂĂ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.2)
N
j+1
4
i
j
j
Q Mi +
j+1
ȍz
j
,
i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.3)
j
j+1
PHASE BEHAVIOR
TABLE 5.3—STANDARD TBP RESULTS FROM COMMERCIAL PVT LABORATORY Component
mol%
wt%
Density (g/cm3)
Gravity ągAPI
Molecular Weight
Heptanes Octanes Nonanes Decanes Undecanes Dodecanes Tridecanes Tetradecanes Pentadecanes plus
1.12 1.30 1.18 0.98 0.62 0.57 0.74 0.53 4.10
2.52 3.08 3.15 2.96 2.10 2.18 3.05 2.39 31.61
0.7258 0.7470 0.7654 0.7751 0.7808 0.7971 0.8105 0.8235 0.8736
63.2 57.7 53.1 50.9 49.5 45.8 42.9 40.1 30.3
96 101 114 129 144 163 177 192 330
*At 60°F. Note: Katz and Firoozabadi6 average boiling points (Table 5.2) can be used when normal paraffin boilingpoint intervals are used.
should produce a single curve. Strictly speaking, therefore, molar distribution is the relation between cumulative molar quantity and some expression for cumulative molecular weight. In this section, we review methods commonly used to describe molar distribution. Some methods use a consistent separation of fractions (e.g., by SCN) so the molar distribution can be expressed directly as a relationship between mole fraction and molecular weight of individual cuts. Most methods in this category assume that C7) mole fractions decrease exponentially. A more general approach uses the continuous threeparameter gamma probability function to describe molar distribution. 5.3.1 Exponential Distributions. The LohrenzBrayClark24 (LBC) viscosity correlation is one of the earliest attempts to use an exponentialtype distribution for splitting C7). The LBC method splits C7) into normal paraffins C7 though C40 with the relation z i + z C exp[A 1(i * 6) ) A 2(i * 6) 2], . . . . . . . . . . . . . (5.4) 6
where i+carbon number and z C6 +measured C6 mole fraction. Constants A1 and A2 are determined by trial and error so that
ȍz
7)
7)
+
i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.5)
MC
7)
+
i
i
. . . . . . . . . . . . . . . . . . . . . . . . (5.6)
i+7
are satisfied. Paraffin molecular weights (Mi +14i)2) are used in Eq. 5.6. A NewtonRaphson algorithm can be used to solve Eqs. 5.5 and 5.6. Note that the LBC model cannot be used when z C7) t z C6 and M C7) u M C40. The LBC form of the exponential distribution has not found widespread application. More commonly, a linear form of the exponential distribution is used to split the C7) fraction. Writing the exponential distribution in a general form for any Cn) fraction (n+7 being a special case), z i + z Cn exp A[(i * n)],
. . . . . . . . . . . . . . . . . . . . . . . . (5.7)
where i+carbon number, z Cn +mole fraction of Cn , and A+constant indicating the slope on a plot of ln z i vs. i. The constants z Cn and A can be determined explicitly. With the general expression M i + 14 i ) h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.8) for molecular weight of Ci and the assumption that the distribution is infinite, constants z Cn and A are given by
40
zC
ȍz M 40
and z C
z Cn +
i+7
14 M Cn) * 14(n * 1) * h
and A + lnǒ1 * z CnǓ
. . . . . . . . . . . . . . . . . . (5.9)
. . . . . . . . . . . . . . . . . . . . . . . . . . . (5.10)
R
so that
ȍz + 1 i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.11)
i+n
(a)
(b)
(c)
Fig. 5.3—Simulated distillation by GC of the North Sea gascondensate sample in Fig. 5.2 (after Austad et al.7). HEPTANESPLUS CHARACTERIZATION
Fig. 5.4—GC simulated distillation chromatograms (a) without any sample (used to determine the baseline), (b) for a crude oil, and (c) for a crude oil with internal standard (after MacAllister and DeRuiter9). 5
700 to 1,000°F Distillate
1,000 to 1,250°F Distillate
1,200°F Residue
Fig. 5.5—Comparison of TBP and GCsimulated distillation for a North Sea gascondensate sample (after Austad et al.7). R
and
ȍz M + M i
i
C n)
. . . . . . . . . . . . . . . . . . . . . . . . . . . (5.12)
i+n
are satisfied. Eqs. 5.9 and 5.10 imply that once a molecular weight relation is chosen (i.e., h is fixed), the distribution is uniquely defined by C7) molecular weight. Realistically, all reservoir fluids having a given C7) molecular weight will not have the same molar distribution, which is one reason why more complicated models have been proposed. 5.3.2 GammaDistribution Model. The threeparameter gamma distribution is a more general model for describing molar distribution. Whitson2,25,26 and Whitson et al.27 discuss the gamma distribution and its application to molar distribution. They give results for 44 oil and condensate C7) samples that were fit by the gamma distribution with data from complete TBP analyses. The absolute average deviation in estimated cut molecular weight was 2.5 amu (molecular weight units) for the 44 samples. The gamma probability density function is p(M) +
(M * h) a*1 exp Ǌ* ƪǒ M * h Ǔńbƫǋ b a G(a)
, . . . . . . . . (5.13)
where G+gamma function and b is given by b+
MC
7)
a
*h
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.14)
The three parameters in the gamma distribution are a, h, and M C7) The key parameter a defines the form of the distribution, and its value usually ranges from 0.5 to 2.5 for reservoir fluids; a+1 gives an exponential distribution. Application of the gamma distribution to heavy oils, bitumen, and petroleum residues indicates that the upper limit for a is 25 to 30, which statistically is approaching a lognormal distribution (see Fig. 5.628). The parameter h can be physically interpreted as the minimum molecular weight found in the C7) fraction. An approximate relation between a and h is 110 h[ . . . . . . . . . . . . . . . . . . . . . . . . . (5.15) 1 * ǒ1 ) 4 ńa 0.7Ǔ 6
Fig. 5.6—Gamma distributions for petroleum residue (after Brulé et al.28).
for reservoirfluid C7) fractions. Practically, h should be considered as a mathematical constant more than as a physical property, either calculated from Eq. 5.15 or determined by fitting measured TBP data. Fig. 5.7 shows the function p(M) for the Hoffman et al.29 oil and a North Sea oil. Parameters for these two oils were determined by fitting experimental TBP data. The Hoffman et al. oil has a relatively large a of 2.27, a relatively small h of 75.7, with M C7)+198; the North Sea oil is described by a+0.82, h+93.2, and M C7)+227. The continuous distribution p(M ) is applied to petroleum fractions by dividing the area under the p(M ) curve into sections (shown schematically in Fig. 5.8). By definition, total area under the p(M ) curve from h to R is unity. The area of a section is defined as normalized mole fraction z ińz C 7) for the range of molecular weights Mbi*1 to Mbi . If the area from h to molecularweight boundary Mb is defined as P0(Mb ), then the area of Section i is P0(Mbi )*P0(Mbi*1), also shown schematically in Fig. 5.8. Mole fraction zi can be written zi + zC
7)
ƪP ǒM Ǔ * P ǒM Ǔƫ . 0
bi
0
b i*1
. . . . . . . . . . . . . . . (5.16)
Average molecular weight in the same interval is given by Mi + h ) a b
P 1ǒM b iǓ * P 1ǒM b i*1Ǔ P 0ǒM b iǓ * P 0ǒM b i*1Ǔ
,
. . . . . . . . . . . (5.17)
where P 0 + Q S, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.18)
ǒ
Ǔ
1 , . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.19) and P 1 + Q S * a PHASE BEHAVIOR
The gamma distribution can be fit to experimental molardistribution data by use of a nonlinear leastsquares algorithm to determine a, h, and b. Experimental TBP data are required, including weight fraction and molecular weight for at least five C7) fractions (use of more than 10 fractions is recommended to ensure a unique fit of model parameters). The sumofsquares function can be defined as
a + 2.273 h + 75.7 M C + 198.4 7)
a + 0.817 h + 93.2 M C + 227
Fǒ a, h , b Ǔ +
7)
ȍ (D
N*1
Mi)
2
,
. . . . . . . . . . . . . . . . . . . . . . . (5.24)
i+1
where D Mi +
Fig. 5.7—Gamma density function for the Hoffman et al.29 oil (dashed line) and a North Sea volatile oil (solid line). After Whitson et al.27
where Q + e *y y a G(a), . . . . . . . . . . . . . . . . . . . . . . . . . (5.20) R
S+
ƪ
ȍ y Ȋ(a ) k) j
j
j+0
and y +
k+0
ƫ
,
. . . . . . . . . . . . . . . . . . . (5.21)
i
. . . . . . . . . . . . . . . . . . (5.25)
ȍw , i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.26)
j+1
vs. the cumulative dimensionless molecularweight variable,
8
i
.
Subscripts mod and exp+model and experimental, respectively. This sumofsquares function weights the lower molecular weights more than higher molecular weights, in accordance with the expected accuracy for measurement of molecular weight. Also, the sumofsquares function does not include the last molecular weight because this molecular weight may be inaccurate or backcalculated to match the measured average C7) molecular weight. If the last fraction is not included, the model average molecular weight, (M C7)) mod + h ) ab, can be compared with the experimental value as an independent check of the fit. A simple graphical procedure can be used to fit parameters a and h if experimental M C7) is fixed and used to define b. Fig. 5.10 shows a plot of cumulative weight fraction, Q wi +
Note that P0(Mb0+h)+P1(Mb0+h)+0. The summation in Eq. 5.21 should be performed until the last term is t1 10*8. The gamma function can be estimated by30
ȍA x
(M i) exp
i
*1
Mb * h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.22) b
Gǒ x ) 1Ǔ + 1 )
(M i) mod * (M i) exp
, . . . . . . . . . . . . . . . . . . . . . (5.23)
i+1
where A1+*0.577191652, A2+0.988205891, A3+*0.897056937, A4+0.918206857, A5+*0.756704078, A6+0.482199394, A7+ *0.193527818, and A8+0.035868343 for 0xxx1. The recurrence formula, G(x)1)+xG(x), is used for xu1 and xt1; furthermore, G(1)+1. The equations for calculating zi and Mi are summarized in a short FORTRAN program GAMSPL found in Appendix A. In this simple program, the boundary molecular weights are chosen arbitrarily at increments of 14 for the first 19 fractions, starting with h as the first lower boundary. The last fraction is calculated by setting the upper molecularweight boundary equal to 10,000. Table 5.4 gives three sample outputs from GAMSPL for a+0.5, 1, and 2 with h+90 and M C7)+200 held constant. Fig. 5.9 plots the results as log zi vs. Mi . The amount and molecular weight of the C26) fraction varies for each value of a.
Q *M i +
QM i * h . MC * h
. . . . . . . . . . . . . . . . . . . . . . . . . . . (5.27)
7)
Table 5.5 and the following outline describe the procedure for determining model parameters with Fig. 5.10 and TBP data. 1. Tabulate measured mole fractions zi and molecular weights Mi for each fraction. 2. Calculate experimental weight fractions, w i + (z i M i) B (z C 7)M C7)), if they are not reported. 3. Normalize weight fractions and calculate cumulative normalized weight fraction Q w i . 4. Calculate cumulative molecular weight Q M i from Eq. 5.3. 5. Assume several values of h (e.g., from 75 to 100) and calculate Q *M i for each value of the estimated h. 6. For each estimate of h, plot Q *M i vs. Q wi on a copy of Fig. 5.10 and choose the curve that fits one of the model curves best. Read the value of a from Fig. 5.10. 7. Calculate molecular weights and mole fractions of Fractions i using the bestfit curve in Fig. 5.10. Enter the curve at measured values of Q wi , read Q *M i , and calculate Mi from Mi + h ) ǒMC
7)
* hǓ
ƪǒ
Q wi * Q wi*1
Ǔ * ǒQ wi*1ńQ *M i*1Ǔƫ
Q wińQ *M i
.
. . . . . . . . . . . . . . . . . . . (5.28)
p(M)
h
A + z ińz C 7)
+ P 0ǒM biǓ * P 0ǒM bi*1Ǔ
h M bi A + P 0ǒM biǓ
h M bi*1 A + P 0ǒM bi*1Ǔ
Fig. 5.8—Schematic showing the graphical interpretation of areas under the gamma density function p(M) that are proportional to normalized mole fraction; A+area. HEPTANESPLUS CHARACTERIZATION
7
TABLE 5.4—RESULTS OF GAMSPL PROGRAM FOR THREE DATA SETS WITH DIFFERENT GAMMADISTRIBUTION PARAMETER a ăąąąăa+0.5
ăąąąăa+1.0
Mole
Molecular
Mole
Molecular
Mole
Molecular
Number
Fraction
Weight
Fraction
Weight
Fraction
Weight
1
0.2787233
94.588
0.1195065
96.852
0.0273900
99.132
2
0.1073842
110.525
0.1052247
110.852
0.0655834
111.490
3
0.0772607
124.690
0.0926497
124.852
0.0852269
125.172
4
0.0610991
138.758
0.0815774
138.852
0.0927292
139.038
5
0.0505020
152.796
0.0718284
152.852
0.0925552
152.963
6
0.0428377
166.819
0.0632444
166.852
0.0877762
166.916
7
0.0369618
180.836
0.0556863
180.852
0.0804707
180.883
8
0.0322804
194.848
0.0490314
194.852
0.0720157
194.859
9
0.0284480
208.857
0.0431719
208.852
0.0632969
208.841
10
0.0252470
222.864
0.0380125
222.852
0.0548597
222.826
11
0.0225321
236.870
0.0334698
236.852
0.0470180
236.814
12
0.0202013
250.875
0.0294699
250.852
0.0399302
250.805
13
0.0181808
264.879
0.0259481
264.852
0.0336535
264.797
14
0.0164152
278.883
0.0228471
278.852
0.0281813
278.790
15
0.0148619
292.886
0.0201167
292.852
0.0234690
292.784
16
0.0134879
306.888
0.0177127
306.852
0.0194514
306.778
17
0.0122665
320.890
0.0155959
320.852
0.0160543
320.774
18
0.0111762
334.892
0.0137321
334.852
0.0132017
334.770
19
0.0101996
348.894
0.0120910
348.852
0.0108204
348.766
0.1199341
539.651
0.0890834
466.000
0.0463166
420.424
20 Total
1.0000000
1.0000000
Average
200
For all three cases h + 90 and M C
+ 200.
7)
ǒ
Qw i Q *M i
*
Q w i*1 Q *M i*1
Ǔ
.
. . . . . . . . . . . . . . . . . . . (5.29)
+ 200
(
}
7)
h + 90 DM b + 14
0.8 (
V a + 2.0
MC
1.0
)
a + 0.5 a + 1.0
200
Fig. 5.11 shows a Q *M i * Q wi match for the Hoffman et al.29 oil with h+70, 72.5, 75, and 80 and indicates that a best fit is achieved for h+72.5 and a+2.5 (see Fig. 5.12). Although the gammadistribution model has the flexibility of treating reservoir fluids from light condensates to bitumen, most reservoir fluids can be characterized with an exponential molar distribution (a+1) without adversely affecting the quality of EOS pre)
For computer applications, Q wi and Q *M i can be calculated exactly from Eqs. 5.16 through 5.23 with little extra effort.
f
1.0000000 200
Mole fractions zi are given by z i + z C 7)
ăąąąăa+2.0
Fraction
0.6
0.4
0.2
0.0 0.0
0.2
0.4
0.6
0.8
1.0
Cumulative Normalized Mole Fraction, Qzi
Fig. 5.9—Three example molar distributions for an oil sample with M C 7)= 200 and h = 90, calculated with the GAMSPL program (Table A4) in Table 5.4. 8
Fig. 5.10—Cumulativedistribution type curve for fitting experimental TBP data to the gammadistribution model. Parameters a and h are determined with M C held constant. 7)
PHASE BEHAVIOR
TABLE 5.5—CALCULATION OF CUMULATIVE WEIGHT FRACTION AND CUMULATIVE MOLECULAR WEIGHT VARIABLE FOR HOFFMAN et al.29 OIL Q *Mi Component i
zi
ąSzi ą
Mi
zi Mi
ăSzi Mi ă
Qwi
QMi
h+70
h+72.5
h+75
h+80
h+85
7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
0.0263 0.0234 0.0235 0.0224 0.0241 0.0246 0.0266 0.0326 0.0363 0.0229 0.0171 0.0143 0.0130 0.0108 0.0087 0.0072 0.0058 0.0048 0.0039 0.0034 0.0028 0.0025 0.0023 0.0091
0.0263 0.0497 0.0732 0.0956 0.1197 0.1443 0.1709 0.2035 0.2398 0.2627 0.2799 0.2941 0.3072 0.3180 0.3267 0.3338 0.3396 0.3444 0.3483 0.3517 0.3545 0.3570 0.3593 0.3684
99 110 121 132 145 158 172 186 203 222 238 252 266 279 290 301 315 329 343 357 371 385 399 444
2.604 2.574 2.844 2.957 3.497 3.882 4.570 6.067 7.371 5.093 4.079 3.596 3.466 3.008 2.526 2.152 1.811 1.582 1.351 1.196 1.039 0.963 0.926 4.049
2.604 5.178 8.021 10.978 14.475 18.357 22.928 28.995 36.366 41.458 45.538 49.134 52.600 55.607 58.133 60.285 62.097 63.679 65.031 66.227 67.265 68.228 69.154 73.203
0.036 0.071 0.110 0.150 0.198 0.251 0.313 0.396 0.497 0.566 0.622 0.671 0.719 0.760 0.794 0.824 0.848 0.870 0.888 0.905 0.919 0.932 0.945 1.000
99.0 104.2 109.6 114.8 120.9 127.2 134.2 142.5 151.7 157.8 162.7 167.0 171.2 174.9 178.0 180.6 182.9 184.9 186.7 188.3 189.8 191.1 192.5 198.7
0.225 0.266 0.308 0.348 0.396 0.445 0.499 0.563 0.634 0.682 0.720 0.754 0.787 0.815 0.839 0.859 0.877 0.893 0.907 0.919 0.931 0.941 0.952 1.000
0.210 0.251 0.294 0.335 0.384 0.434 0.489 0.555 0.627 0.676 0.715 0.749 0.782 0.811 0.836 0.857 0.875 0.891 0.905 0.918 0.929 0.940 0.951 1.000
0.194 0.236 0.280 0.322 0.371 0.422 0.478 0.546 0.620 0.669 0.709 0.744 0.778 0.808 0.832 0.854 0.872 0.889 0.903 0.916 0.928 0.939 0.950 1.000
0.160 0.204 0.249 0.293 0.345 0.398 0.457 0.526 0.604 0.655 0.697 0.733 0.769 0.799 0.825 0.847 0.867 0.884 0.899 0.913 0.925 0.936 0.948 1.000
0.123 0.169 0.216 0.262 0.316 0.371 0.433 0.506 0.586 0.640 0.683 0.722 0.758 0.791 0.818 0.841 0.861 0.879 0.894 0.909 0.921 0.933 0.945 1.000
Total
0.3684
198.7
73.203
dictions. Whitson et al.27 proposed perhaps the most useful application of the gammadistribution model. With Gaussian quadrature, their method allows multiple reservoirfluid samples from a common reservoir to be treated simultaneously with a single fluid characterization. Each fluid sample can have different C7) properties when the split is made so that each split fraction has the same molecular weight (and other properties, such as g, Tb , Tc , pc , and w), while
1.0 Y h + 65
0.8
J h + 70 F h + 75 X h + 80
the mole fractions are different for each fluid sample. Example applications include the characterization of a gas cap and underlying reservoir oil and a reservoir with compositional gradient. The following outlines the procedure for applying Gaussian quadrature to the gammadistribution function. 1. Determine the number of C7) fractions, N, and obtain the quadrature values Xi and Wi from Table 5.6 (values are given for N+3 and N+5). 2. Specify h and a. When TBP data are not available to determine these parameters, recommended values are h+90 and a+1. 3. Specify the heaviest molecular weight of fraction N (recommended value is M N + 2.5M C7)). Calculate a modified b* term, b * + ǒ M N * h ǓńX N .
0.6
0.4
0.2
0.0 0.0
0.2
0.4
0.6
0.8
1.0
Cumulative Normalized Mole Fraction, Qzi
Fig. 5.11—Graphical fit of the Hoffman et al.29 oil molar distribution by use of the cumulativedistribution type curve. Bestfit model parameters are a = 2.5 and h = 72.5. HEPTANESPLUS CHARACTERIZATION
Fig. 5.12—Calculated normalized mole fraction vs. molecular weight of fractions for the Hoffman et al.29 oil based on the best fit in Fig. 5.11 with a = 2.5 and h = 72.5. 9
TABLE 5.6—GAUSSIAN QUADRATURE FUNCTION VARIABLES, X, AND WEIGHT FACTORS, W X
W
Three Quadrature Points (plus fractions) 1 2 3
7.110 930 099 29 10*1
0.415 774 556 783 2.294 280 360 279 6.289 945 082 937
2.785 177 335 69 10*1 1.038 925 650 16 10*2
Five Quadrature Points (plus fractions) 5.217 556 105 83 10*1 1 2 3 4 5
3.986 668 110 83 10*1
0.263 560 319 718 1.413 403 059 107 3.596 425 771 041 7.085 810 005 859 12.640 800 844 276
7.594 244 968 17 10*2 3.611 758 679 92 10*3 2.336 997 238 58 10*5
Quadrature function values and weight factors can be found for other quadrature numbers in mathematical handbooks.30
4. Calculate the parameter d. d + exp
ǒ
a b* *1 MC * h 7)
Ǔ
.
. . . . . . . . . . . . . . . . . . . (5.30)
5. Calculate the C7) mole fraction zi and Mi for each fraction. zi + zC
7)
ƪ W i f (X i)ƫ,
Mi + h ) b* Xi , and f(X) +
(X) a*1 ǒ1 ) ln dǓ a . G(a) dX
. . . . . . . . . . . . . . . . . . (5.31)
6. Check whether the calculated M C7) from Eq. 5.12 equals the measured value used in Step 4 to define d. Because Gaussian quadrature is only approximate, the calculated M C7) may be slightly in error. This can be corrected by (slightly) modifying the value of d, and repeating Steps 5 and 6 until a satisfactory match is achieved. When characterizing multiple samples simultaneously, the values of MN , h, and b* must be the same for all samples. Individual sample values of M C7) and a can, however, be different. The result of this characterization is one set of molecular weights for the C7) fractions, while each sample has different mole fractions zi (so that their average molecular weights M C7) are honored). Specific gravities for the C7) fractions can be calculated with one of the correlations given in Sec. 5.4 (e.g., Eq. 5.44), where the characterization factor (e.g., Fc ) must be the same for all mixtures. The specific gravities, g C7) , of each sample will not be exactly reproduced with this procedure (calculated with Eq. 5.37), but the average characterization factor can be chosen so that the differences are very small ( g"0.0005). Having defined Mi and gi for the C7) fractions, a complete fluid characterization can be determined with correlations in Sec. 5.5. 5.4 InspectionĆProperties Estimation 5.4.1 Generalized Properties. The molecular weight, specific gravity, and boiling point of C7) fractions must be estimated in the absence of experimental TBP data. This situation arises when simulated distillation is used or when no experimental analysis of C7) is available and a synthetic split must be made by use of a molardistribution model. For either situation, inspection data from TBP analysis of a sample from the same field would be the most reliable source of M, g, and Tb for each C7) fraction. The nextbest source would be measured TBP data from a field producing similar oil or condensate from the same geological formation. Generalized properties from a producing region, such as the North Sea, have been proposed.31 Katz and Firoozabadi6 suggest a generalized set of SCN properties for petroleum fractions C6 through C45. Table 5.2 gives an extended version of the KatzFiroozabadi property table. Molecular 10
weights can be used to convert weight fractions, wi , from simulated distillation to mole fractions, zi +
w i ńM i
ȍ w ńM
.
N
j
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.32)
j
j+7
However, the molecular weight of the heaviest fraction, C N, is not known. From a mass balance, M N is given by wN , . . . . . . . . . . . . (5.33) MN + N*1 ǒwC ńMC Ǔ * ǒwińMiǓ 7)
7)
ȍ i+7
where Mi for i+7,…, N*1 are taken from Table 5.2. Unfortunately, the calculated molecular weight M N is often unrealistic because of measurement errors in M C7) or in the chromatographic analysis and because generalized molecular weights are only approximate. Both w N and M C7) can be adjusted to give a “reasonable” M N, but caution is required to avoid nonphysical adjustments. The same problem is inherent with backcalculating M N with any set of generalized molecular weights used for SCN Fractions 7 to N*1 (e.g., paraffin values). During the remainder of this section, molecular weights and mole fractions are assumed to be known for C7) fractions, either from chromatographic analysis or from a synthetic split. The generalized properties for specific gravity and boiling point can be assigned to SCN fractions, but the heaviest specific gravity must be backcalculated to match the measured C7) specific gravity. The calculated gN also may be unrealistic, requiring some adjustment to generalized specific gravities. Finally, the boiling point of the heaviest fraction must be estimated. TbN can be estimated from a correlation relating boiling point to specific gravity and molecular weight. 5.4.2 Characterization Factors. Inspection properties M, g, and Tb reflect the chemical makeup of petroleum fractions. Some methods for estimating specific gravity and boiling point assume that a particular characterization factor is constant for all C7) fractions. These methods are only approximate but are widely used. Watson or Universal Oil Products (UOP) Characterization Factor. The Watson or UOP factor, Kw, is based on normal boiling point, Tb , in °R and specific gravity, g.32,33 T 1ń3 K w 5 gb .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.34)
Kw varies roughly from 8.5 to 13.5. For paraffinic compounds, Kw +12.5 to 13.5; for naphthenic compounds, Kw +11.0 to 12.5; and for aromatic compounds, Kw +8.5 to 11.0. Some overlap in Kw exists among these three families of hydrocarbons, and a combination of paraffins and aromatics will obviously “appear” naphthenic. However, the utility of this and other characterization factors is that they give a qualitative measure of the composition of a petroleum fraction. The Watson characterization factor has been found to be useful for approximate characterization and is widely used as a parameter for correlating petroleumfraction properties, such as molecular weight, viscosity, vapor pressure, and critical properties. An approximate relation2 for the Watson factor, based on molecular weight and specific gravity, is K w [ 4.5579 M 0.15178 g *0.84573 . . . . . . . . . . . . . . . . . . . (5.35) This relation is derived from the RiaziDaubert14 correlation for molecular weight and is generally valid for petroleum fractions with normal boiling points ranging from 560 to 1,310°R (C7 through C30). Experience has shown, however, that Eq. 5.35 is not very accurate for fractions heavier than C20. Kw calculated with M C7) and g C7) in Eq. 5.35 is often constant for a given field. Figs. 5.13A and 5.13B7 plot molecular weight vs. specific gravity for C7) fractions from two North Sea fields. Data for the gas condensate in Fig. 5.13A indicate an average K wC7)+11.99"0.01 for a range of molecular weights from 135 to 150. The volatile oil shown in Fig. 5.13B has an average K wC7)+11.90"0.01 for a range of molecular weights from 220 to PHASE BEHAVIOR
Molecular Weight, MC 7+
Fig. 5.13A—Specific gravity vs. molecular weight for C7) fractions for a North Sea GasCondensate Field 2 (after Austad et al.7).
255. The high degree of correlation for these two fields suggests accurate molecularweight measurements by the laboratory. In general, the spread in K wC7) values will exceed "0.01 when measurements are performed by a commercial laboratory. When the characterization factor for a field can be determined, Eq. 5.35 is useful for checking the consistency of C7) molecularweight and specificgravity measurements. Significant deviation in K wC7) , such as "0.03 for the North Sea fields above, indicates possible error in the measured data. Because molecular weight is more prone to error than determination of specific gravity, an anomalous K wC7) usually indicates an erroneous molecularweight measurement. For the gas condensate in Fig. 5.13A, a C7) sample with specific gravity of 0.775 would be expected to have a molecular weight of [141 (for K wC7)+ 11.99). If the measured value was 135, the Watson characterization factor would be 11.90, which is significantly lower than the field average of 11.99. In this case, the C7) molecular weight should be redetermined. Eq. 5.35 can also be used to calculate specific gravity of C7) fractions determined by simulated distillation or a synthetic split (i.e., when only mole fractions and molecular weights are known). Assuming a constant Kw for each fraction, specific gravity, gi , can be calculated from g i + 6.0108 M i0.17947 K w*1.18241 .
. . . . . . . . . . . . . . . . . (5.36)
Kw must be chosen so that experimentally measured C7) specific gravity, (g C7)) exp, is calculated correctly.
ǒgC7)Ǔ
exp
+
zC
7)
MC
7)
ȍǒz M ńg Ǔ N
i
i
.
. . . . . . . . . . . . . . . . . . . . . (5.37)
i
i+1
The Watson factor satisfying Eq. 5.37 is given by Kw +
ƪ
0.16637 g C zC
7)
MC
ȍz M
A 7) 0
7)
ƫ
*0.84573
,
. . . . . . . . . . . . . . . (5.38)
Molecular Weight, MC 7+
Fig. 5.13B—Specific gravity vs. molecular weight for C7) fractions for a North Sea VolatileOil Field 3B(after Austad et al.7).
Jacoby Correlation (Aromaticity Factor, Ja ) Present Correlation (Watson Factor, Kw )
Ja
Fig. 5.14—Specific gravity vs. molecular weight for constant values of the Jacoby aromaticity factor (solid lines) and the Watson characterization factor (dashed lines). After Whitson.25
Boiling points, Tbi , can be estimated from Eq. 5.36. 3
T bi + (K wg i) .
i
0.82053 . i
. . . . . . . . . . . . . . . . . . . . . . (5.39)
i+1
HEPTANESPLUS CHARACTERIZATION
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.40)
Unfortunately, Eqs. 5.36 through 5.40 overpredict g and Tb at molecular weights greater than [250 (an original limitation of the RiaziDaubert14 molecularweight correlation). Jacoby Aromaticity Factor. The Jacoby aromaticity factor, Ja , is an alternative characterization factor for describing the relative composition of petroleum fractions.34 Fig. 5.142 shows the original Jacoby relation between specific gravity and molecular weight for several values of Ja . The behavior of specific gravity as a function of molecular weight is similar for the Jacoby factor and the relation for a constant Kw. However, specific gravity calculated with the Jacoby method increases more rapidly at low molecular weights, flattening at high molecular weights (a more physically consistent behavior). A relation for the Jacoby factor is
N
where A 0 +
Kw
Ja +
g * 0.8468 ) ǒ15.8ńMǓ . . . . . . . . . . . . . . . . . . (5.41) 0.2456 * ǒ1.77ńMǓ 11
Cf typically has a value between 0.27 and 0.31 and is determined for a specific C7) sample by satisfying Eq. 5.37. 5.4.3 BoilingPoint Estimation. Boiling point can be estimated from molecular weight and specific gravity with one of several correlations. Søreide also developed a boilingpoint correlation based on 843 TBP fractions from 68 reservoir C7) samples, T b + 1928.3 * ǒ1.695 exp ƪ * ǒ4.922 ) ǒ3.462
Fig. 5.15—Specific gravity vs. carbon number for constant values of the Yarborough aromaticity factor (after Yarborough1).
or, in terms of specific gravity,
ǒ
Ǔ
g + 0.8468 * 15.8 ) J a 0.2456 * 1.77 . . . . . . . (5.42) M M The first two terms in Eq. 5.42 (i.e., when Ja +0) express the relation between specific gravity and molecular weight for normal paraffins. The Jacoby factor can also be used to estimate fraction specific gravities when mole fractions and molecular weights are available from simulated distillation or a synthetic split. The Jacoby factor satisfying measured C7) specific gravity (Eq. 5.37) must be calculated by trial and error. We have found that this relation is particularly accurate for gascondensate systems.27 Yarborough Aromaticity Factor. Yarborough1 modified the Jacoby aromaticity factor specifically for estimating specific gravities when mole fractions and molecular weights are known. Yarborough tries to improve the original Jacoby relation by reflecting the changing character of fractions up to C13 better and by representing the larger naphthenic content of heavier fractions better. Fig. 5.15 shows how the Yarborough aromaticity factor, Ya , is related to specific gravity and carbon number. A simple relation representing Ya is not available; however, Whitson26 has fit the seven aromaticity curves originally presented by Yarborough using the equation g i + expƪA 0 ) A 1 i *1 ) A 2 i ) A 3 ln(i)ƫ , . . . . . . . . . . (5.43) where i+carbon number. Table 5.7 gives the constants for Eq. 5.43. The aromaticity factor required to satisfy measured C7) specific gravity (Eq. 5.37) is determined by trial and error. Linear interpolation of specific gravity should be used to calculate specific gravity for a Ya value falling between two values of Ya in Table 5.7. Søreide 35 Correlations. Søreide developed an accurate specificgravity correlation based on the analysis of 843 TBP fractions from 68 reservoir C7) samples. g i + 0.2855 ) C f (M i * 66) 0.13 .
. . . . . . . . . . . . . . . (5.44)
10 5Ǔ M *0.03522 g 3.266 10 *3Ǔ M * 4.7685 g
10 *3Ǔ Mgƫ ,
. . . . . . . . . . . . . . . . . . . . . (5.45)
with Tb in °R. Table 5.8 gives estimated specific gravities determined with the methods just described for a C7) sample with the exponential split given in Table 5.4 (a+1, h+90, M C7)+200) and g C7)+0.832. The following equations also relate molecular weight to boiling point and specific gravity; any of these correlations can be solved for boiling point in terms of M and g. We recommend, however, the Søreide correlation for estimating Tb from M and g. Kesler and Lee. 12 M + ƪ* 12, 272.6 ) 9, 486.4g ) (4.6523 * 3.3287g)T bƫ ) Ǌ ǒ1 * 0.77084g * 0.02058g 2Ǔ
Ǔ ƪǒ1.3437 * 720.79T *1 b
ǋ 10 7ƫ T *1 b
) Ǌǒ1 * 0.80882g ) 0.02226g 2Ǔ
Ǔ ƪǒ1.8828 * 181.98T –1 b
ǋ. 10 12ƫ T *3 b
. . . . . . . . (5.46)
Riazi and Daubert. 14 M + (4.5673
10 *5)T b2.1962 g *1.0164 .
. . . . . . . . . . . . (5.47)
American Petroleum Inst. (API). 36 M + ǒ2.0438
10 2Ǔ T b0.118 g 1.88 expǒ0.00218T b * 3.07gǓ . . . . . . . . . . . . . . . . . . . . . (5.48)
Rao and
Bardon. 37
ln M + (1.27 ) 0.071K w) ln
ǒ22.311.8T Ǔ. ) 1.68K b
w
. . . . . . . . . . . . . . . . . . . . (5.49) Riazi and Daubert. 18 M + 581.96 T b0.97476 g 6.51274 expƪǒ5.43076 * 9.53384 g ) ǒ1.11056
10 *3ǓT b
10 *3ǓT bgƫ . . . . . . . . . . (5.50)
TABLE 5.7—COEFFICIENTS FOR YARBOROUGH AROMATICITY FACTOR CORRELATION1,26
12
Ya
A0
A1
0.0
*7.43855 10*2
*1.72341
0.1
*4.25800 10*1
0.2
A2
A2
1.38058 10*3
*3.34169 10*2
*7.00017 10*1
*3.30947 10*5
8.65465 10*2
*4.47553 10*1
*7.65111 10*1
1.77982 10*4
1.07746 10*1
0.3
*4.39105 10*1
*9.44068 10*1
4.93708 10*4
1.19267 10*1
0.4
*2.73719 10*1
*1.39960
3.80564 10*3
5.92005 10*2
0.6
*7.39412 10*3
*1.97063
5.87273 10*3
*1.67141 10*2
0.8
*3.17618 10*1
*7.78432 10*1
2.58616 10*3
1.08382 10*3 PHASE BEHAVIOR
TABLE 5.8—COMPARISON OF SPECIFIC GRAVITIES WITH CORRELATIONS BY USE OF DIFFERENT CHARACTERIZATION FACTORS gi for Different Correlations With Constant Characterization Factor Chosen To Match g C + 0.832 7)
Kw +12.080
Ja +0.2395
Ya +0.2794
Cf +0.2864
96.8
0.7177
0.7472
0.7051
0.7327
0.1052
110.8
0.7353
0.7684
0.7286
0.7550
0.0926
124.8
0.7511
0.7849
0.7486
0.7719
4
0.0816
138.8
0.7656
0.7981
0.7660
0.7856
5
0.0718
152.8
0.7789
0.8088
0.7813
0.7972
6
0.0632
166.8
0.7913
0.8178
0.7951
0.8072
7
0.0557
180.8
0.8028
0.8253
0.8075
0.8161
8
0.0490
194.8
0.8136
0.8318
0.8189
0.8241
9
0.0432
208.8
0.8238
0.8374
0.8294
0.8314
10
0.0380
222.8
0.8335
0.8423
0.8391
0.8380
Fraction
zi
1
0.1195
2 3
Mi
11
0.0335
236.8
0.8426
0.8466
0.8482
0.8442
12
0.0295
250.8
0.8514
0.8505
0.8567
0.8500
13
0.0259
264.8
0.8597
0.8539
0.8646
0.8554
14
0.0228
278.8
0.8677
0.8570
0.8722
0.8604
15
0.0201
292.8
0.8753
0.8598
0.8793
0.8652
16
0.0177
306.8
0.8827
0.8623
0.8861
0.8697
17
0.0156
320.8
0.8898
0.8646
0.8926
0.8740
18
0.0137
334.8
0.8966
0.8668
0.8988
0.8782
19
0.0121
348.8
0.9033
0.8687
0.9048
0.8821
20
0.0891
466.0
0.9514
0.8805
0.9468
0.9096
1.0000
200.0
0.8320
0.8320
0.8320
0.8320
Total
5.5 CriticalĆProperties Estimation
KeslerLee. 12
Thus far, we have discussed how to split the C7) fraction into pseudocomponents described by mole fraction, molecular weight, specific gravity, and boiling point. Now we must consider the problem of assigning critical properties to each pseudocomponent. Critical temperature, Tc ; critical pressure, pc ; and acentric factor, w, of each component in a mixture are required by most cubic EOS’s. Critical volume, vc , is used instead of critical pressure in the BenedictWebbRubin38 (BWR) EOS, and critical molar volume is used with the LBC viscosity correlation.24 Critical compressibility factor has been introduced as a parameter in three and fourconstant cubic EOS’s. Criticalproperty estimation of petroleum fractions has a long history beginning as early as the 1930’s; several reviews22,25,26,39,40 are available. We present the most commonly used correlations and a graphical comparison (Figs. 5.16 through 5.18) that is intended to highlight differences between the correlations. Finally, correlations based on perturbation expansion (a concept borrowed from statistical mechanics) are discussed separately. The units for the remaining equations in this section are Tb in °R, TbF in °F+Tb *459.67, Tc in °R, pc in psia, and vc in ft3/lbm mol. Oil gravity is denoted gAPI and is related to specific gravity by gAPI+141.5/g*131.5.
T c + 341.7 ) 811g ) (0.4244 ) 0.1174g)T b
5.5.1 Critical Temperature. Tc is perhaps the most reliably correlated critical property for petroleum fractions. The following criticaltemperature correlations can be used for petroleum fractions. Roess. 41 (modified by API36). T c + 645.83 ) 1.6667ƪgǒ T bF ) 100 Ǔƫ * ǒ0.7127
2
10 *3Ǔƪgǒ T bF ) 100 Ǔƫ .
HEPTANESPLUS CHARACTERIZATION
) (0.4669 * 3.2623g)
. . . . . . . . . . . . (5.52)
Cavett. 42 T c + 768.07121 ) 1.7133693T bF * ǒ0.10834003
2 10 *2ǓT bF
* ǒ0.89212579
10 *2Ǔ g APIT bF
) ǒ0.38890584
3 10 *6ǓT bF
) ǒ0.5309492 ) ǒ0.327116
2 10 *5Ǔ g APIT bF 2 . 10 *7Ǔ g 2APIT bF
. . . . . . . . . . . . . . . (5.53)
RiaziDaubert. 14 T c + 24.27871T b0.58848 g 0.3596 .
. . . . . . . . . . . . . . . . . . (5.54)
Nokay. 43 T c + 19.078 T b0.62164 g 0.2985 .
. . . . . . . . . . . . . . . . . . . . (5.55)
5.5.2 Critical Pressure. pc correlations are less reliable than Tc correlations. The following are pc correlations that can be used for petroleum fractions. KeslerLee. 12 ln p c + 8.3634 * 0.0566 g *
. . . . . . . . . . . (5.51)
10 5T *1 b .
ƪǒ
0.11857 0.24244 ) 2.2898 g ) g2
Ǔ
ƫ
10 *3 T b 13
Fig. 5.16—Comparison of criticaltemperature correlations for boiling points from 600 to 1,500°R assuming a constant Watson characterization factor of 12.
)
*
ƪǒ ƪǒ
0.47227 1.4685 ) 3.648 g ) g2 0.42019 ) 1.6977 g2
ƫ
Ǔ
10 *7 T 2b
ƫ
Ǔ
Fig. 5.17—Comparison of criticalpressure correlations for boiling points from 600 to 1,500°R assuming a constant Watson characterization factor of 12.
10 *10 T 3b .
. . . . . (5.56)
Cavett. 42 log p c + 2.8290406 ) ǒ0.94120109 * ǒ0.30474749 * ǒ0.2087611
10 *3ǓT bF
2 10 *5ǓT bF
10 *4Ǔ g APIT bF
) ǒ0.15184103
3 10 *8ǓT bF
) ǒ0.11047899
2 10 *7Ǔ g APIT bF
* ǒ0.48271599
10 *7Ǔ g 2APIT bF
) ǒ0.13949619
2 . . . . . . . . . . . . (5.57) 10 *9Ǔ g 2APIT bF
RiaziDaubert. 14 p c + ǒ3.12281
10 9ǓT *2.3125 g 2.3201 . . . . . . . . . . . . . . (5.58) b
5.5.3 Acentric Factor. Pitzer et
ǒǓ
p* w 5 * log pv * 1, c
al.44
LeeKesler. 13 (Tbr +Tb /Tc t0.8).
defined acentric factor as w+
– lnǒ p cń14.7Ǔ ) A 1 ) A 2 T *1 ) A 3 ln T br ) A 4 T br6 br
. . . . . . . . . . . . . . . . . . . . . . . . (5.59)
where p *v+vapor pressure at temperature T+0.7Tc (Tr +0.7). Practically, acentric factor gives a measure of the steepness of the vaporpressure curve from Tr +0.7 to Tr +1, where p *v /pc +0.1 for w+0 and p *v /pc +0.01 for w+1. Numerically, w[0.01 for methane, [0.25 for C5, and [0.5 for C8 (see Table A.1 for literature values of acentric factor for pure compounds). w increases to u1.0 for petroleum fractions heavier than approximately C25 (see Table 5.2). The KeslerLee12 acentric factor correlation (for Tb /Tc u0.8) is developed specifically for petroleum fractions, whereas the correlation for Tb /Tc t0.8 is based on an accurate vaporpressure correlation for pure compounds. The Edmister45 correlation is limited to pure hydrocarbons and should not be used for C7) fractions. The three correlations follow. 14
Fig. 5.18—Comparison of acentric factor correlations for boiling points from 600 to 1500°R assuming a constant Watson characterization factor of 12.
A 5 ) A 6 T *1 ) A 7 ln T br ) A 8 T br6 br
,
. . . . . . . . . . . . . . . . . . . . (5.60) where A1+*5.92714, A2+ 6.09648, A3+ 1.28862, A4+ *0.169347, A5+ 15.2518, A6+*15.6875, A7+*13.4721, and A8+ 0.43577. KeslerLee. 12 (Tbr +Tb /Tc u0.8). w + * 7.904 ) 0.1352K w * 0.007465K 2w ) 8.359T br ) (1.408 * 0.01063K w)T *1 br .
. . . . . . . (5.61)
Edmister. 45 logǒ p cń14.7Ǔ w+3 * 1. 7 ƪǒT cńT bǓ * 1ƫ
. . . . . . . . . . . . . . . . . . . . . (5.62) PHASE BEHAVIOR
5.5.4 Critical Volume. The HallYarborough46 criticalvolume correlation is given in terms of molecular weight and specific gravity, whereas the RiaziDaubert14 correlation uses normal boiling point and specific gravity. HallYarborough. 46 v c + 0.025 M 1.15 g *0.7935 .
. . . . . . . . . . . . . . . . . . . . . . (5.63)
RiaziDaubert. 14 v c + ǒ7.0434
10 *7Ǔ T b2.3829 g *1.683.
. . . . . . . . . . . . . (5.64)
Critical compressibility factor, Zc , is defined as Zc +
p cv c , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.65) RT c
where R+universal gas constant. Thus, Zc can be calculated directly from critical pressure, critical volume, and critical temperature. Reid et al.40 and Pitzer et al.44 give an approximate relation for Zc . Z c [ 0.291 * 0.08w.
Paraffin molecular weight, MP, is not explicitly a function of Tb , and Eqs. 5.67 through 5.73 must be solved iteratively; an initial guess is given by MP [
Tb . 10.44 * 0.0052T b
. . . . . . . . . . . . . . . . . . . . . (5.74)
Twu claims that the normalparaffin correlations are valid for C1 through C100, although the properties at higher carbon numbers are only approximate because experimental data for paraffins heavier than approximately C20 do not exist. The following relations are used to calculate petroleumfraction properties. Critical Temperature.
ǒ11 )* 2f2f Ǔ , 2
T c + T cP
T T
f T + Dg T
ƪ
* 0.362456 ) T b0.5
Ǔ ƫ
ǒ
0.0398285 * 0.948125 Dg T , T b0.5
. . . . . . . . . . . . . . . . . . . . . . . . . (5.66)
Eq. 5.66 is not particularly accurate (grossly overestimating Zc for heavier compounds) and is used only for approximate calculations.
and Dg T + exp[5(g P * g)] * 1.
. . . . . . . . . . . . . . . . . . (5.75)
Critical Volume.
ǒ11 )* 2f2f Ǔ , 2
5.5.5 Correlations Based on Perturbation Expansions. Correlations for critical temperature, critical pressure, critical volume, and molecular weight have been developed for petroleum fractions with a perturbationexpansion model with normal paraffins as the reference system. To calculate critical pressure, for example, critical temperature, critical volume, and specific gravity of a paraffin with the same boiling point as the petroleum fraction must be calculated first. Kesler et al.47 first used the perturbation expansion (with nalkanes as the reference fluid) to develop a suite of criticalproperty and acentricfactor correlations. Twu48 uses the same approach to develop a suite of criticalproperty correlations. We give his normalparaffin correlations first, then the correlations for petroleum fractions. Normal Paraffins (Alkanes).
ƪ
T cP + T b 0.533272 ) ǒ0.191017 ) ǒ0.779681
10
*3
(0.959468 10 2) ) ǒ0.01T bǓ 13
ƫ
10 *10ǓT b3
*1
,
. . . . . . . . . . . . . . . . . . (5.67)
2
. . . . . . . . . . . . . . . (5.68)
v cP + [ 1 * (0.419869 * 0.505839a * 1.56436a 3 ,
. . . . . . . . . . . . . . . . . . . . . . . . . (5.69)
g P + 0.843593 * 0.128624a * 3.36159a 3 * 13749.5a 12 ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.70)
and T b + exp(5.71419 ) 2.71579q * 0.28659q 2 * 39.8544q *1 * 0.122488q *2) * 24.7522q ) 35.3155q 2 , T where a + 1 * b T cP and q + ln M P .
0.466590 ) T b0 .5
ǒ
Ǔ ƫ
* 0.182421 ) 3.01721 Dg v , T b0.5
and Dg v + expƪ4ǒg 2P * g 2Ǔƫ * 1.
. . . . . . . . . . . . . . . . . . (5.76)
Critical Pressure.
ǒTT ǓǒVV Ǔǒ11 )* 2f2f Ǔ , 2
p c + p cP
)
c
cP
p
cP
c
p
ƪǒ
2.53262 * 46.1955 * 0.00127885T b T b0.5
Ǔ
Ǔ ƫ
ǒ
* 11.4277 ) 252.14 ) 0.00230535T b Dg p , T b0.5
and Dg p + exp[0.5(g P * g)] * 1. . . . . . . . . . . . . . . . . . (5.77) 1 ) 2f Ǔ, ln M + ln M ǒ 1 * 2f 2
M
) 36.1952a 2 ) 104.193a 4) ,
*8
ƪ
f v + Dg v
v
Molecular Weight.
p cP + (3.83354 ) 1.19629a 0.5 ) 34.8888a
* 9481.7a 14)]
v
f p + Dg p
ǓT b
10 *7ǓT b2 * ǒ0.284376
v c + v cP
. . . . . . . . . . . . . . . . . (5.71)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.72)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.73)
HEPTANESPLUS CHARACTERIZATION
P
ƪ ǒ
f M + Dg M x )
M
Ǔ ƫ
* 0.0175691 ) 0.193168 Dg M , T b0.5
x + 0.012342 * 0.328086 , T b0.5 and Dg M + exp[5(g P * g)].
. . . . . . . . . . . . . . . . . . . . . (5.78)
Figs. 5.16 through 5.18 compare the various criticalproperty correlations for a range of boiling points from 600 to 1,500°R. 5.5.6 Methods Based on an EOS. Fig. 5.1928 illustrates the important influence that critical properties have on EOScalculated properties of pure components. Vapor pressure is particularly sensitive to critical temperature. For example, the RiaziDaubert19 criticaltemperature correlation for toluene overpredicts the experimental value 15
a+1, h+90) with Gaussianquadrature or equalmass fractions or (2) the exponential distribution (Eq. 5.7). Specific gravities should be estimated with the Søreide35 correlation (Eq. 5.44), choosing Cf to match measured C7) specific gravity (Eq. 5.37). Boiling points should be estimated from the Søreide correlation (Eq. 5.45). For the PR EOS, we recommend the nonhydrocarbon BIP’s given in Chap. 4 and the modified ChuehPrausnitz54 equation for C1 through C7) pairs,
k ij
ȱ + Aȧ1 * Ȳ
ǒ
2v 1ń6 v 1ń6 ci cj v 1ń3 ) v 1ń3 ci cj
Ǔ ȳȴȧ B
,
. . . . . . . . . . . . . . . . (5.79)
with A+0.18 and B+6.
Tc underpredicted←
→Tc overpredicted
Deviation From Experimental Value, % Fig. 5.19—Effect of critical temperature on vaporpressure prediction of toluene with the PR EOS; AAD+absolute average deviation (after Brulé et al.28).
by only 1.7%. Even with this slight error in Tc , the average error in vapor pressures predicted by the PengRobinson49 (PR) EOS is 16%. The effect of critical properties and acentric factor on EOS calculations for reservoirfluid mixtures is summarized by Whitson.26 In principle, the EOS used for mixtures should also predict the behavior of individual components found in the mixture. For pure compounds, the vapor pressure is accurately predicted because all EOS’s force fit vaporpressure data. Some EOS’s are also fit to saturatedliquid densities at subcritical temperatures. The measured properties of petroleum fractions, boiling point, and specific gravity can also be fit by the EOS, as discussed later. For each petroleum fraction separately, two of the EOS parameters (Tc ; pc ; w; volumeshift factor, s; or multipliers of EOS constants A and B) can be chosen so that the EOS exactly reproduces experimental boiling point and specific gravity. Because only two inspection properties are available (Tb and g), only two of the EOS parameters can be determined. Whitson50 suggests fixing the value of w with an empirical correlation and adjusting Tc and pc to match normal boiling point and molar volume (M/g) at standard conditions. Critical properties satisfying these criteria are given for a wide range of petroleum fractions by the PR EOS and the SoaveRedlichKwong (SRK) EOS.22,23 A better (and recommended) approach for cubic EOS’s is to use the volumeshift factor s (see Chap. 4) to match specific gravity or a saturated liquid density and acentric factor to match normal boiling point. Other methods for forcing the EOS to match boiling point and specific gravity have also been devised. Brulé and Starling51 proposed a method that uses viscosity as an additional inspection property of the fraction for determining critical properties. This approach proved particularly successful when applied to the BWR EOS for residualoil supercritical extraction (ROSE).28 5.6 Recommended C7) Characterizations We recommend the following C7) characterization procedure for cubic EOS’s. 1. Use the Twu48 (or LeeKesler12) critical property correlation for Tc and pc . 2. Choose the acentric factor to match Tb ; alternatively, use the LeeKesler12/KeslerLee13 correlations. 3. Determine volumetranslation coefficients, si , to match specific gravities; alternatively, use Peneloux et al.’s52 correlation for the SRK EOS22,23 or Jhaveri and Youngren’s53 correlation for the PR EOS.49 When measured TBP data are not available, a mathematical split should be made with either (1) the gamma distribution (default 16
5.6.1 SRKRecommended Characterization. Alternatively, the Pedersen et al.55 characterization procedure can be used with the SRK EOS. 1. Split the plus fraction Cn) (preferably nu10) into SCN fractions up to C80 using Eqs. 5.7 through 5.11 and h+*4. 2. Calculate SCN densities ò i (gi + ò i /0.999) using the equation ò i+A0)A1 ln(i), where A0 and A1 are determined by satisfying the experimentalplus density, ò n), and measured (or assumed) density, ò n *1 ( ò6+0.690 can be used for C7)). 3. Calculate critical properties of all C7) fractions (distillation cuts from C7 to Cn*1 and split SCN fractions from Cn through C80) using the correlations T c + 163.12 ò ) 86.052 ln M ) 0.43475 M * 1877.4 , M , ln p c + * 0.13408 ) 2.5019 ò ) 208.46 * 3987.2 M M2 and m SRK + 0.48 ) 1.574 w * 0.176 w 2 + 0.7431 ) 0.0048122 M ) 0.0096707 ò * ǒ3.7184
10 *6ǓM 2. . . . . . . . . . . . . . . . . . . (5.80)
Note that the use of acentric factor is circumvented by directly calculating the term m used in the a correction term to EOS Constant A. 4. Group C7) into 3 to 12 fractions using equalweight fractions in each group; use weightaverage mixing rules. 5. Calculate volumetranslation parameters for C7) fractions to match specific gravities; pure component c values are taken from Peneloux et al.52 6. All hydrocarbon/hydrocarbon BIP’s are set to zero. SRK BIP’s given in Chap. 4 are used for nonhydrocarbon/hydrocarbon pairs. The two recommended C7) characterization procedures outlined previously for the PR EOS and SRK EOS are probably the best currently available (other EOS characterizations, such as the RedlichKwong EOS modified by Zudkevitch and Joffe,56 and some threeconstant characterizations should provide similar accuracy but are not significantly better). Practically, the two characterization procedures give the same results for almost all PVT properties (usually within 1 to 2%). With these EOScharacterization procedures, we can expect reasonable predictions of densities and Z factors ("1 to 5%), saturation pressures ("5 to 15%), gas/oil ratios and formation volume factors ("2 to 5%), and condensateliquid dropout ("5 to 10% for maximum dropout, with poorer prediction of taillike behavior just below the dewpoint). The recommended EOS methods are less reliable for prediction of minimum miscibility conditions, nearcritical saturation pressure and saturation type (bubblepoint or dewpoint), and both retrograde and nearcritical liquid volumes. Improved predictions can be obtained only by tuning EOS parameters to accurate PVT data covering a relatively wide range of pressures, temperatures, and compositions (see Sec. 4.7 and Appendix C). 5.7 Grouping and Averaging Properties The cost and computer resources required for compositional reservoir simulation increase substantially with the number of compoPHASE BEHAVIOR
nents used to describe the reservoir fluid. A compromise between accuracy and the number of components must be made according to the process being simulated (i.e., according to the expected effect that phase behavior will have on simulated results). For example, a detailed fluid description with 12 to 15 components may be needed to simulate developed miscibility in a slimtube experiment. With current computer technology, however, a fullfield simulation with fluids exhibiting nearcritical phase behavior is not feasible for a 15component mixture. The following are the main questions regarding component grouping. 1. How many components should be used? 2. How should the components be chosen from the original fluid description? 3. How should the properties of pseudocomponents be calculated?
the method is general and can be applied to any molardistribution model and for any number of C7) groups. In general, most authors have found that broader grouping of C7) as C7 through C10, C11 through C15, C16 through C20, and C21) is substantially better than splitting only the first few carbonnumber fractions (e.g., C7, C8, C9, and C10)). Gaussian quadrature is recommended for choosing the pseudocomponents in a C7) fraction; equalmass fractions or the Li et al.59 approach are valid alternatives. 5.7.2 Mixing Rules. Several methods have been proposed for calculating critical properties of pseudocomponents. The simplest and most common mixing rule is
ȍz q q + ȍz , i i
iŮI
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.83)
I
5.7.1 How Many and Which Components To Group. The number of components used to describe a reservoir fluid depends mainly on the process being simulated. However, the following rule of thumb reduces the number of components for most systems: group N2 with methane, CO2 with ethane, isobutane with nbutane, and isopentane with npentane. Nonhydrocarbon content should be less than a few percent in both the reservoir fluid and the injection gas if a nonhydrocarbon is to be grouped with a hydrocarbon. Five to eightcomponent fluid characterizations should be sufficient to simulate practically any reservoir process, including (1) reservoir depletion of volatileoil and gascondensate reservoirs, (2) gas cycling above and below the dewpoint of a gascondensate reservoir, (3) retrograde condensation near the wellbore of a producing well, and (4) immiscible and miscible gasinjection. Coats57 discusses a method for combining a modified blackoil formula with a simplified EOS representation of separator oil and gas streams. The “oil” and “gas” pseudocomponents in this model contain all the original fluid components in contrast to the typical method of grouping where each pseudocomponent is made up of only selected original components. Lee et al.58 suggest that C7) fractions can be grouped into two pseudocomponents according to a characterization factor determined by averaging the tangents of fraction properties M, g, and Ja plotted vs. boiling point. Whitson2 suggests that the C7) fraction can be grouped into NH pseudocomponents given by
i
iŮI
where qi +any property (Tc , pc , w, or M) and zi +original mole fraction for components (i+1,..., I) making up Pseudocomponent I. Average specific gravity should always be calculated with the assumption of ideal solution mixing.
ȍz M . g + ȍǒz M ńg Ǔ i
i
iŮI
. . . . . . . . . . . . . . . . . . . . . . . . . . . (5.84)
I
i
i
i
iŮI
Pedersen et al.55 and others suggest use of weight fraction instead of mole fraction. Wu and Batycky’s63 empirical mixingrule approach uses both the molar and weightaverage mixing rules and a proportioning factor, F, to calculate pcI , TcI , and wI . qI +
ȍf q ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.85)
i i
iŮI
where qI represents pcI , TcI , and wI and fi +average of the molar and weight fractions, f i + F q iz i ) (1 * F) q i w i and w i +
N H + 1 ) 3.3 log(N * 7), . . . . . . . . . . . . . . . . . . . . . (5.81)
zi Mi
ȍz M
,
N
j
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.86)
j
j+1
where N+carbon number of the heaviest fraction in the original fluid description. The groups are separated by molecular weights MI given by MI + MC
7
ǒM
N ń MC
7
Ǔ
1ńN H
,
. . . . . . . . . . . . . . . . . . . (5.82)
where I+1,..., NH . Molecular weights, Mi , from the original fluid description (i+7,..., N) falling within boundaries MI*1 to MI are included in Group I. This method should only be used when C7) fractions are originally separated on a carbonnumber basis and for N greater than [20. Li et al.59 suggest a method for grouping components of an original fluid description that uses K values from a flash at reservoir temperature and the “average” operating pressure. The original mixture is divided arbitrarily into “light” components (H2S, N2, CO2, and C1 through C6) and “heavy” components (C7)). Different criteria are used to determine the number of light and heavy pseudocomponents. Li et al. also suggest use of phase diagrams and compositional simulation to verify the grouped fluid description (a practice that we highly recommend). Still other pseudoization methods have been proposed60,61; Schlijper’s61 method also treats the problem of retrieving detailed compositional information from pseudoized (grouped) components. Behrens and Sandler62 suggest a grouping method for C7) fractions based on application of the Gaussianquadrature method to continuous thermodynamics. Although a simple exponential distribution is used with only two quadrature points (i.e., the C7) fractions are grouped into two pseudocomponents), Whitson et al.27 show that HEPTANESPLUS CHARACTERIZATION
with 0xFx1. A generalized mixing rule for BIP’s can be written k IJ +
ȍȍf f k
j ij ,
i
. . . . . . . . . . . . . . . . . . . . . . . . . (5.87)
iŮI jŮJ
where fi is also given by Eq. 5.86. On the basis of Chueh and Prausnitz’s54 arguments, LeeKesler13 proposed the mixing rules in Eqs. 5.88 through 5.92. v cI +
ƪ ȍȍ ǒ ƪ ȍȍ 1 8
T cI +
)
v 1ń3 cj
iŮI jŮJ
1 8v cI
2
i
,
. . . (5.88)
iŮI
ǒ
z i z j ǒT ci T cjǓ 1ń2 v 1ń3 ) v 1ń3 ci cj
iŮI jŮJ
ǒ Ǔ ǒȍ Ǔń ǒȍ Ǔ
B
ȍz
ƫǒ Ǔ
Ǔ ń ȍz 3
z i z j v 1ń3 ci
Ǔ
3
ƫ
2
i
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.89)
iŮI
wI +
zi wi
iŮI
zi ,
Z cI + 0.2905 * 0.085w I , and p cI +
Z cI R T cI v cI .
. . . . . . . . . . . . . . . . . . . (5.90)
iŮI
. . . . . . . . . . . . . . . . . . . . . . (5.91)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.92) 17
TABLE 5.9—EXAMPLE STEPWISEREGRESSION PROCEDURE FOR PSEUDOIZATION TO FEWER COMPONENTS FOR A GAS CONDENSATE FLUID UNDERGOING DEPLETION Original Component
Original
Number
Component
Step 1
Step 2
Step 3
Step 4
Step 5
1
N2
N2)C1*
N2)C1
N2)C1
N2)C1)CO2)C2*
N2)C1)CO2)C2
2
CO2
CO2)C2*
CO2)C2
CO2)C2
C3)iC4)nC4
C3)iC4)nC4
)iC5)nC5)C6*
)iC5)nC5)C6
3
C1
C3
C3
C3)iC4)nC4*
F1
F1
4
C2
iC4
iC4)nC4*
iC5)nC5)C6*
F2
F2)F3*
5
C3
nC4
iC5)nC5*
F1
F3
6
iC4
iC5
C6
F2
7
nC4
nC5
F1
F3
8
iC5
C6
F2 F3
9
nC5
F1
10
C6
F2 F3
11
F1
12
F2
13
F3 Regression Parameters
kij
1, 9, 10, and 11
1, 7, 8, and 9
1, 5, 6, and 7
1, 3, 4, and 5
1, 3, and 4
Wa
1
4
3
1
3
Wb
1
4
3
1
3
Wa
2
5
4
2
4
Wb
2
5
4
2
4
*Indicates the grouped pseudocomponents being regressed in a particular step.
Lee et al.58 and Whitson2 consider an alternative method for calculating C7) critical properties based on the specific gravities and boiling points of grouped pseudocomponents. Coats57 presents a method of pseudoization that basically eliminates the effect of mixing rules on pseudocomponent properties. The approach is simple and accurate. Coats requires the pseudoized characterization to reproduce exactly the volumetric behavior of the original reservoir fluid at undersaturated conditions. This is achieved by ensuring that the mixture EOS constants A and B are identical for the original and the pseudoized characterizations. First, pseudocritical properties ( pcI , TcI , and wI ) are estimated with any mixing rule (e.g., Kay’s64 mixing rule). Then W aI and W bI are determined to satisfy the following equations.
ƪȍ W aI +
iŮI
ȍ zi zj aiaj ǒ1 * kijǓ jŮJ
zi
iŮI
ǒR 2TcI2 ńpcIǓa I(TrI, w I)
ǒȍ Ǔń ǒȍ Ǔ zi bi
and W bI +
ƫńǒȍ Ǔ
2
iŮI
zi
iŮI
ǒRT cIńp cIǓb I(T rI, w I)
,
. . . . . . . . . . . . . . . . (5.93)
R 2T 2 where a i + W ai p ci a i (T ri, w i) ci RT and b i + W bi p ci b i(T ri, w i) . ci
. . . . . . . . . . . . . . . . . . . . . (5.94)
W ai and W bi may include previously determined corrections to the numerical constants W oa and W ob. This approach to determining pseudocomponent properties, together with Eq. 5.87 for k I J , is surprisingly accurate even for VLE calculations. Coats also gives an 18
analogous procedure for determining pseudocomponent vcI for the LBC24 viscosity correlation. Coats’ approach is preferred to all the other proposed methods. It ensures accurate volumetric calculations that are consistent with the original EOS characterization, and the method is easy to implement. 5.7.3 Stepwise Regression. A reducedcomponent characterization should strive to reproduce the original complete characterization that has been used to match measured PVT data. One approach to achieve this goal is stepwise regression, summarized in the following procedure. 1. Complete a comprehensive match of all existing PVT data with a characterization containing light and intermediate pure components and at least three to five C7) fractions. 2. Simulate a suite of depletion and multicontact gasinjection PVT experiments that cover the expected range of compositions in the particular application. 3. Use the simulated PVT data as “real” data for pseudoization based on regression. 4. Create two new pseudocomponents from the existing set of components. Use the pseudoization procedure of Coats to obtain WaI and WbI values, and use Eq. 5.87 for k I J . 5. Use regression to fine tune the W aI and W bI values estimated in Step 4; also regress on key BIP’s, such as (N2)C1)*C7), (CO2)C2)*C7), and other nonzero BIP’s involving pseudocomponents from Step 4. 6. Repeat Steps 4 and 5 until the quality of the characterization deteriorates beyond an acceptable fluid description. Table 5.9 shows an example fivestep pseudoization procedure. In summary, any grouping of a complete EOS characterization into a limited number of pseudocomponents should be checked to ensure that predicted phase behavior (e.g., multicontact gas injection data, saturation pressures, and densities) are reasonably close to the predictions for the original (complete) characterization. Stepwise regression is the best approach to determine the number and PHASE BEHAVIOR
properties of pseudocomponents that can accurately describe a reservoir fluid’s phase behavior. If stepwise regression is not possible, standard grouping of the light and intermediates (N2)C1, CO2)C2, iC4)nC4, and iC5)nC5) and Gaussian quadrature for C7) (or equalmass fractions) is recommended; a valid alternative is the Li et al.59 method. The Coats57 method (Eqs. 5.93 and 5.94) is always recommended for calculating pseudocomponent properties. References 1. Yarborough, L.: “Application of a Generalized Equation of State to Petroleum Reservoir Fluids,” Equations of State in Engineering and Research, K.C. Chao and R.L. Robinson Jr. (eds.), Advances in Chemistry Series, American Chemical Soc., Washington, DC (1978) 182, 386. 2. Whitson, C.H.: “Characterizing Hydrocarbon Plus Fractions,” SPEJ (August 1983) 683; Trans., AIME, 275. 3. Pedersen, K.S., Thomassen, P., and Fredenslund, A.: “SRKEOS Calculation for Crude Oils,” Fluid Phase Equilibria (1983) 14, 209. 4. Craft, B.C., Hawkins, M., and Terry, R.E.: Applied Petroleum Reservoir Engineering, second edition, PrenticeHall Inc., Englewood Cliffs, New Jersey (1991). 5. McCain, W.D. Jr.: The Properties of Petroleum Fluids, second edition, PennWell Publishing Co., Tulsa, Oklahoma (1990). 6. Katz, D.L. and Firoozabadi, A.: “Predicting Phase Behavior of Condensate/CrudeOil Systems Using Methane Interaction Coefficients,” JPT (November 1978) 1649; Trans., AIME, 265. 7. Austad, T. et al.: “Practical Aspects of Characterizing Petroleum Fluids,” paper presented at the 1983 North Sea Condensate Reservoirs and Their Development Conference, London, 24–25 May. 8. Chorn, L.G.: “Simulated Distillation of Petroleum Crude Oil by Gas Chromatography—Characterizing the HeptanesPlus Fraction,” J. Chrom. Sci. (January 1984) 17. 9. MacAllister, D.J. and DeRuiter, R.A.: “Further Development and Application of Simulated Distillation for Enhanced Oil Recovery,” paper SPE 14335 presented at the 1985 SPE Annual Technical Conference and Exhibition, Las Vegas, Nevada, 22–25 September. 10. Designation D158, Saybolt Distillation of Crude Petroleum, Annual Book of ASTM Standards, ASTM, Philadelphia, Pennsylvania (1984). 11. Designation D289284, Distillation of Crude Petroleum (15.Theoretical Plate Column), Annual Book of ASTM Standards, ASTM, Philadelphia, Pennsylvania (1984) 8210. 12. Kesler, M.G. and Lee, B.I.: “Improve Predictions of Enthalpy of Fractions,” Hydro. Proc. (March 1976) 55, 153. 13. Lee, B.I. and Kesler, M.G.: “A Generalized Thermodynamic Correlation Based on ThreeParameter Corresponding States,” AIChE J. (1975) 21, 510. 14. Riazi, M.R. and Daubert, T.E.: “Simplify Property Predictions,” Hydro. Proc. (March 1980) 115. 15. Maddox, R.N. and Erbar, J.H.: Gas Conditioning and Processing—Advanced Techniques and Applications, Campbell Petroleum Series, Norman, Oklahoma (1982) 3. 16. Organick, E.I. and Golding, B.H.: “Prediction of Saturation Pressures for CondensateGas and VolatileOil Mixtures,” Trans., AIME (1952) 195, 135. 17. Katz, D.L. et al.: Handbook of Natural Gas Engineering, McGrawHill Book Co. Inc., New York City (1959). 18. Riazi, M.R. and Daubert, T.E.: “Analytical Correlations Interconvert DistillationCurve Types,” Oil & Gas J. (August 1986) 50. 19. Riazi, M.R. and Daubert, T.E.: “Characterization Parameters for Petroleum Fractions,” Ind. Eng. Chem. Res. (1987) 26, 755. 20. Robinson, D.B. and Peng, D.Y.: “The Characterization of the Heptanes and Heavier Fractions,” Research Report 28, Gas Producers Assn., Tulsa, Oklahoma (1978). 21. Riazi, M.R. and Daubert, T.E.: “Prediction of the Composition of Petroleum Fractions,” Ind. Eng. Chem. Proc. Des. Dev. (1980) 19, 289. 22. Pedersen, K.S., Thomassen, P., and Fredenslund, A.: “Thermodynamics of Petroleum Mixtures Containing Heavy Hydrocarbons. 1. Phase Envelope Calculations by Use of the SoaveRedlichKwong Equation of State,” Ind. Eng. Chem. Proc. Des. Dev. (1984) 23, 163. 23. Pedersen, K.S., Thomassen, P., and Fredenslund, A.: “Thermodynamics of Petroleum Mixtures Containing Heavy Hydrocarbons. 2. Flash and PVT Calculations with the SRK Equation of State,” Ind. Eng. Chem. Proc. Des. Dev. (1984) 23, 566. 24. Lohrenz, J., Bray, B.G., and Clark, C.R.: “Calculating Viscosities of Reservoir Fluids From Their Compositions,” JPT (October 1964) 1171; Trans., AIME, 231. HEPTANESPLUS CHARACTERIZATION
25. Whitson, C.H.: “Effect of C7) Properties on EquationofState Predictions,” paper SPE 11200 presented at the 1982 SPE Annual Technical Conference and Exhibition, New Orleans, 26–29 September. 26. Whitson, C.H.: “Effect of C7) Properties on EquationofState Predictions,” SPEJ (December 1984) 685; Trans., AIME, 277. 27. Whitson, C.H., Andersen, T.F., and Søreide, I.: “C7) Characterization of Related Equilibrium Fluids Using the Gamma Distribution,” C7 ) Fraction Characterization, L.G. Chorn and G.A. Mansoori (eds.), Advances in Thermodynamics, Taylor & Francis, New York City (1989) 1, 35–56. 28. Brulé, M.R., Kumar, K.H., and Watansiri, S.: “Characterization Methods Improve PhaseBehavior Predictions,” Oil & Gas J. (11 February 1985) 87. 29. Hoffmann, A.E., Crump, J.S., and Hocott, C.R.: “Equilibrium Constants for a GasCondensate System,” Trans., AIME (1953) 198, 1. 30. Abramowitz, M. and Stegun, I.A.: Handbook of Mathematical Functions, Dover Publications Inc., New York City (1970) 923. 31. Haaland, S.: “Characterization of North Sea Crude Oils and Petroleum Fractions,” MS thesis, Norwegian Inst. of Technology, Trondheim, Norway (1981). 32. Watson, K.M., Nelson, E.F., and Murphy, G.B.: “Characterization of Petroleum Fractions,” Ind. Eng. Chem. (1935) 27, 1460. 33. Watson, K.M. and Nelson, E.F.: “Improved Methods for Approximating Critical and Thermal Properties of Petroleum,” Ind. Eng. Chem. (1933) 25, No. 8, 880. 34. Jacoby, R.H. and Rzasa, M.J.: “Equilibrium Vaporization Ratios for Nitrogen, Methane, Carbon Dioxide, Ethane, and Hydrogen Sulfide in Absorber Oil/Natural Gas and Crude Oil/Natural Gas Systems,” Trans., AIME (1952) 195, 99. 35. Søreide, I.: “Improved Phase Behavior Predictions of Petroleum Reservoir Fluids From a Cubic Equation of State,” Dr.Ing. dissertation, Norwegian Inst. of Technology, Trondheim, Norway (1989). 36. Technical Data Book—Petroleum Refining, third edition, API, New York City (1977). 37. Rao, V.K. and Bardon, M.F.: “Estimating the Molecular Weight of Petroleum Fractions,” Ind. Eng. Chem. Proc. Des. Dev. (1985) 24, 498. 38. Benedict, M., Webb, G.B., and Rubin, L.C.: “An Empirical Equation for Thermodynamic Properties of Light Hydrocarbons and Their Mixtures, I. Methane, Ethane, Propane, and nButane,” J. Chem. Phys. (1940) 8, 334. 39. Reid, R.C.: “Present, Past, and Future Property Estimation Techniques,” Chem. Eng. Prog. (1968) 64, No. 5, 1. 40. Reid, R.C., Prausnitz, J.M., and Polling, B.E.: The Properties of Gases and Liquids, fourth edition, McGrawHill Book Co. Inc., New York City (1987) 12–24. 41. Roess, L.C.: “Determination of Critical Temperature and Pressure of Petroleum Fractions,” J. Inst. Pet. Tech. (October 1936) 22, 1270. 42. Cavett, R.H.: “Physical Data for Distillation CalculationsVaporLiquid Equilibria,” Proc., 27th API Meeting, San Francisco (1962) 351. 43. Nokay, R.: “Estimate Petrochemical Properties,” Chem. Eng. (23 February 1959) 147. 44. Pitzer, K.S. et al.: “The Volumetric and Thermodynamic Properties of Fluids, II. Compressibility Factor, Vapor Pressure, and Entropy of Vaporization,” J. Amer. Chem. Soc. (1955) 77, No. 13, 3433. 45. Edmister, W.C.: “Applied Hydrocarbon Thermodynamics, Part 4: Compressibility Factors and Equations of State,” Pet. Ref. (April 1958) 37, 173. 46. Hall, K.R. and Yarborough, L.: “New, Simple Correlation for Predicting Critical Volume,” Chem. Eng. (November 1971) 76. 47. Kesler, M.G., Lee, B.I., and Sandler, S.I.: “A Third Parameter for Use in Generalized Thermodynamic Correlations,” Ind. Eng. Chem. Fund. (1979) 18, No. 1, 49. 48. Twu, C.H.: “An Internally Consistent Correlation for Predicting the Critical Properties and Molecular Weights of Petroleum and CoalTar Liquids,” Fluid Phase Equilibria (1984) No. 16, 137. 49. Peng, D.Y. and Robinson, D.B.: “A NewConstant Equation of State,” Ind. Eng. Chem. Fund. (1976) 15, No. 1, 59. 50. Whitson, C.H.: “Critical Properties Estimation From an Equation of State,” paper SPE 12634 presented at the 1984 SPE/DOE Symposium on Enhanced Oil Recovery, Tulsa, Oklahoma, 15–18 April. 51. Brulé, M.R. and Starling, K.E.: “Thermophysical Properties of Complex Systems: Applications of Multiproperty Analysis,” Ind. Eng. Chem. Proc. Des. Dev. (1984) 23, 833. 19
52. Peneloux, A., Rauzy, E., and Freze, R.: “A Consistent Correction for RedlichKwongSoave Volumes,” Fluid Phase Equilibria (1982) 8, 7. 53. Jhaveri, B.S. and Youngren, G.K.: “ThreeParameter Modification of the PengRobinson Equation of State To Improve Volumetric Predictions,” SPERE (August 1988) 1033; Trans., AIME, 285. 54. Chueh, P.L. and Prausnitz, J.M.: “Calculation of HighPressure Vapor– Liquid Equilibria,” Ind. Eng. Chem. (1968) 60, No. 13. 55. Pedersen, K.S., Thomassen, P., and Fredenslund, A.: “Characterization of Gas Condensate Mixtures,” C7) Fraction Characterization, L.G. Chorn and G.A. Mansoori (eds.), Advances in Thermodynamics, Taylor & Francis, New York City (1989) 1. 56. Zudkevitch, D. and Joffe, J: “Correlation and Prediction of VaporLiquid Equilibrium with the RedlichKwong Equation of State,” AIChE J. (1970) 16, 112. 57. Coats, K.H.: “Simulation of GasCondensateReservoir Performance,” JPT (October 1985) 1870. 58. Lee, S.T. et al.: “Experiments and Theoretical Simulation on the Fluid Properties Required for Simulation of Thermal Processes,” SPEJ (October 1982) 535. 59. Li, Y.K., Nghiem, L.X., and Siu, A.: “Phase Behavior Computation for Reservoir Fluid: Effects of Pseudo Component on Phase Diagrams and Simulations Results,” paper CIM 843519 presented at the 1984 Petroleum Soc. of CIM Annual Meeting, Calgary, 10–13 June.
20
60. Newley, T.M.J. and Merrill, R.C. Jr.: “Pseudocomponent Selection for Compositional Simulation,” SPERE (November 1991) 490; Trans., AIME, 291. 61. Schlijper, A.G.: “Simulation of Compositional Processes: The Use of Pseudocomponents in EquationofState Calculations,” SPERE (September 1986) 441; Trans., AIME, 282. 62. Behrens, R.A. and Sandler, S.I.: “The Use of Semicontinuous Description To Model the C7) Fraction in Equation of State Calculations,” paper SPE 14925 presented at the 1986 SPE/DOE Symposium on Enhanced Oil Recovery, Tulsa, Oklahoma, 23–23 April. 63. Wu, R.S. and Batycky, J.P.: “Pseudocomponent Characterization for Hydrocarbon Miscible Displacement,” paper SPE 15404 presented at the 1986 SPE Annual Technical Conference and Exhibition, New Orleans, 5–6 October. 64. Kay, W.B.: “The EthaneHeptane System,” Ind. & Eng. Chem. (1938) 30, 459.
SI Metric Conversion Factors ft3/lbm mol 6.242 796 °F (°F*32)/1.8 °F (°F)459.67)/1.8 psi 6.894 757 °R 5/9
E*02 +m3/kmol +°C +K E)00 +kPa +K
PHASE BEHAVIOR
Chapter 6
Conventional PVT Measurements 6.1 Introduction This chapter reviews the standard experiments performed by pressure/volume/temperature (PVT) laboratories on reservoir fluid samples: compositional analysis, multistage surface separation, constant composition expansion (CCE), differential liberation expansion (DLE), and constant volume depletion (CVD). We present data from actual laboratory reports and give methods for checking the consistency of reported data for each experiment. Chaps. 5 and 8 discuss special laboratory studies, including trueboilingpoint (TBP) distillation and multicontact gasinjection tests, respectively. Table 6.1 summarizes experiments typically performed on oils and gas condensates. From this table, we see that the DLE experiment is the only test never performed on gascondensate systems. We begin by discussing standard analyses performed on oil and gascondensate samples. 6.1.1 General Information Sheet. Most commercial laboratories report general information on a cover sheet of the laboratory report, including formation and well characteristics and sampling conditions. Tables 6.2 and 6.31,2 show this information, which may be important for correct application and interpretation of the fluid analyses. This is particularly true for wells where separator samples must be recombined to give a representative wellstream composition. Most of these data are supplied by the contractor of the fluid study and are recorded during sampling. Therefore, the representative for the company contracting the fluid study is responsible for the correctness and completeness of reported data. We strongly recommend that the following data always be reported in a general information sheet: (1) separator gas/oil ratio (GOR) in standard cubic feet/separator barrel, (2) separator conditions at sampling, (3) field shrinkage factor used ( + B osp), (4) flowing bottomhole pressure (FBHP) at sampling, (5) static reservoir pressure, (6) minimum FBHP before and during sampling, (7) time and date of sampling, (8) production rates during sampling, (9) dimensions of sample container, (10) total number and types of samples collected during the drillstem test, and (11) perforation intervals. 6.1.2 Oil PVT Analyses. Standard PVT analyses performed on reservoir oils usually include (1) bottomhole wellstream compositional analysis through C 7), (2) CCE, (3) DLE, and (4) multistageseparator tests. The CCE experiment determines the bubblepoint pressure and volumetric properties of the undersaturated oil. It also gives twophase volumetric behavior below the bubblepoint; however, these data are rarely used. The DLE experiment and separator test are used together to calculate traditional blackoil properties, B o and R s, for reservoirengineering calculations. Occasionally, 88
instead of a DLE study, a CVD experiment is run on a volatile oil. Also, the C 7) fraction may be separated into singlecarbonnumber cuts from C 7 through approximately C 20) by TBP analysis or simulated distillation (see Chap. 5). 6.1.3 GasCondensate PVT Analyses. The standard experimental program for a gascondensate fluid includes (1) recombined wellstream compositional analysis through C 7), (2) CCE, and (3) CVD. The CCE and CVD data are measured in a highpressure visual cell where the dewpoint pressure is determined visually. Total volume/ pressure and liquiddropout behavior is measured in the CCE experiment. Phase volumes defining retrograde behavior are measured in the CVD experiment together with Z factors and producedgas compositions through C 7). Optionally, a multistageseparator test can be performed as well as TBP analysis or simulated distillation of the C 7) into singlecarbonnumber cuts from C 7 to about C 20) (see Chap. 5). 6.2 Wellstream Compositions PVT studies usually are based on one or more samples taken during a production test. Bottomhole samples can be obtained by wireline with a highpressure container during either production testing or a shutin period. Alternatively, separator samples can be taken during a production test. Bottomhole sampling is the preferred method for most oil reservoirs, while recombined samples are traditionally used for gascondensate reservoirs.38 Taking both bottomhole and separator samples in oil wells is not uncommon. The advantage of separator samples is that they can be recombined in varying proportions to achieve a desired bubblepoint pressure (e.g., initial reservoir pressure); these larger samples are needed for special PVT tests (e.g., TBP and slim tube among others). 6.2.1 Bottomhole Sample. Table 6.4 shows the reported wellstream composition of a reservoir oil where C 7) specific gravity and molecular weight are also reported. In the example report, composition is given both as mole and weight percent although many laboratories report only molar composition. Experimentally, the composition of a bottomhole sample is determined by the following (Fig. 6.1). 1. Flashing the sample to atmospheric conditions. 2. Measuring the volumes of surface gas, V g , and surface oil, V o . 3. Determining the normalized weight fractions, w gi and w oi, of surface samples by gas chromatography. 4. Measuring surfaceoil molecular weight, M o , and specific gravity, g o . PHASE BEHAVIOR
TABLE 6.1—LABORATORY ANALYSES PERFORMED ON RESERVOIROIL AND GASCONDENSATE SYSTEMS
TABLE 6.2—EXAMPLE GENERAL INFORMATION SHEET FOR GOOD OIL CO. WELL 4 OIL SAMPLE
Laboratory Analysis
Oils
Gas Condensates
Bottomhole sample
D
d
Recombined composition
d
D
First well completed
C7+ TBP distillation
d
d
Original reservoir pressure at 8,692 ft, psig
C7+ simulated distillation
d
d
Original produced GOR, scf/bbl
Constantcomposition expansion
D
D
Production rate, B/D
Multistage surface separation
D
d
Differential liberation
D
Separator temperature, °F
N
Separator pressure, psig
CVD
d
D
Multicontact gas injection
d
d
D+standard, d+can be performed, and N+not performed.
z i + F g y i ) (1 * F g) x i ; . . . . . . . . . . . . . . . . . . . . . . . . (6.1) 1 , . . . . . . . . . . . . . . . . . . (6.2) 1 ) ƪ133, 300ǒ gńM Ǔ ońR sƫ
where R s +GOR V gńV o in scf/STB from the singlestage flash; yi +
ȍ ǒw
ǒ
ńM jǓ ) w g C
j0C 7)
xi +
7)
7)
w o ińM i
ǒ
ńM jǓ ) w o C
oj
j0C 7)
+
7)
ńM g C
Ǔ
; . . . . . . . . . (6.3)
wo C ǒ1ńM Ǔ * o
7)
7)
ȍ ǒw
ńM o C
7)
Ǔ
;
600 300 75 200
Oil gravity at 60°F, °API Datum
8,000 No Well Characteristics
Elevation, ft
610
Total depth, ft
8,943
Producing interval, ft
8,684 to 8,700
Tubing size, in.
27/8
Tubing depth, ft
8,600
PI at 300 B/D, BD/psi
1.1
Last reservoir pressure at 8,500 ft, psig
3,954* / /19
Reservoir temperature at 8,500 ft, °F
Shut in 72 hours Amerada
Normal production rate, B/D
300
GOR, scf/bbl
600
Separator pressure, psig
200
Separator temperature, °F
Ǔ
ojńM j
. . . . . . . . . . . . . . (6.5)
j0C 7)
Surface gas usually contains less than 1 mol% C 7) material consisting mainly of heptanes and octanes; M g C + 105 is usually a 7) good assumption. Surface oil contains less than 1 mol% of the light constituents C 1, C 2, and nonhydrocarbons. Lowtemperature distillation can be used to improve the accuracy of reported weight fractions for intermediate components in the surface oil ( C 3 through C 6); however, gas chromatography is more widely used. The most probable source of error in wellstream composition of a bottomhole sample is the surfaceoil molecular weight, M o , which appears in Eq. 6.2 for F g and Eq. 6.4 for x i . M o is usually accurate within "4 to 10%. In Chap. 5, we showed that the Watson characterization factor, K w, of surface oil (Eq. 5.35) should be constant (to within "0.03 of the determined value) for a given reservoir. Once an average has been established for a reservoir (usually requiring three separate measurements), potential errors in M o can be checked. A calculated K w that deviates from the fieldaverage K w by more than "0.03 may indicate an erroneous molecularweight measurement. Eqs. 6.1 through 6.4 show that all component compositions are affected by M o C , which is backcalculated from M o with Eq. 7) 6.5. Fortunately, the amount of lighter components (particularly C 1) in the surface oil are small, so the real effect on conversion from weight to mole fractions of the surface oil usually is not significant. 6.2.2 Recombined Samples. Tables 6.5 and 6.6 present the separatoroil and gas compositional analyses of a gascondensate fluid and recombined wellstream composition. The separatoroil composition is obtained by use of the same procedure as that used for bottomhole oil samples (Eqs. 6.1 through 6.5). This involves bringing the separator oil to standard conditions, measuring properties CONVENTIONAL PVT MEASUREMENTS
(m/d/y) 217*
Well status
. . . . . . . . . . (6.4)
(m/d/y) 4,100
Pressure gauge
ȍ ǒw
and M o C
/ /19
Date
wg i ń Mi gj
Cretaceous
Original gas cap
5. Converting w gi weight fractions to normalized mole fractions y i and x i . 6. Recombining mathematically to the wellstream composition, z i. Eqs. 6.1 through 6.5 give Steps 1 through 6 mathematically.
Fg +
Formation Characteristics Name
75
Base pressure, psia
14.65
Well making water, % water cut
0
Sampling Conditions Sample depth, ft
8,500
Well status
Shut in 72 hours
GOR Separator pressure, psig Separator temperature, °F Tubing pressure, psig
1,400
Casing pressure, psig Sampled by Sampler type
Wofford
*Pressure and temperature extrapolated to the midpoint of the producing interval+4,010 psig and 220°F.
and compositions of the resulting surface oil and gas, and recombining these compositions to give the separatoroil composition; Tables 6.5 and 6.6 report the results. Separator gas is introduced directly into a gas chromatograph, which yields weight fractions, w g . These weight fractions are converted to mole fractions, y i , by use of appropriate molecular weights; Tables 6.5 and 6.6 show the results. C 7) molecular weight is backcalculated with measured separatorgas specific gravity, g g .
Mg C
7)
+ w gC
7)
ǒ
1 * 28.97g g
Ǔ
*1
ȍ
wg i Mi
i0C 7)
. . . . . . . . (6.6)
89
TABLE 6.3—EXAMPLE GENERAL INFORMATION SHEET FOR GOOD OIL CO. WELL 7 GAS CONDENSATE Formation Characteristics Formation name
Pay sand
Date first well completed
/ /19
Original reservoir pressure at 10,148 ft, psig
(m/d/y) 5,713
Original producedgas/liquid ratio, scf/bbl Production rate, B/D Separator pressure, psig Separator temperature, °F Liquid gravity at 60°F, °API Datum, ft subsea
8,000 Well Characteristics
Elevation, ft KB
2,214
Total depth, ft
10,348
Producing interval, ft
10,124 to 10,176
Tubing size, in.
2
Tubing depth, ft
10,100
Openflow potential, MMscf/D Last reservoir pressure at 10,148 ft, psig
5,713
Date
/ /19
Reservoir temperature at 10,148 ft, °F
(m/d/y) 186
Status of well status
Shut in
Pressure gauge
Amerada Sampling Conditions
Flowing tubing pressure, psig
3,375
FBHP, psig
5,500
Primaryseparator pressure, psig
300
Primaryseparator temperature, °F
62
Secondaryseparator pressure, psig
20
Secondaryseparator temperature, °F
60
Field stocktankliquid gravity at 60°F, °API
58.5
Primaryseparatorgas production rate, Mscf/D
762.14
Pressure base, psia
14.696
Temperature base, °F
60
Compressibility factor, Fpv
1.043
Gas gravity (laboratory)
0.737
Gasgravity factor, Fg
0.902
Stocktankliquid production rate at 60°F, B/D
127.3
Primaryseparatorgas/stocktankliquid ratio In scf/bbl In bbl/MMscf
5,987 167.0
Sampled by
For the example PVT report (Tables 6.5 and 6.6), the separator gas/oil ratio, R sp, during sampling is reported as standard gas volume per separatoroil volume (4,428 scf/bbl). In this report, the units are incorrectly labeled scf/bbl at 60°F, where in fact the separatoroil volume is measured at separator pressure (300 psig) and temperature (62°F). The separatoroil formation volume factor (FVF), B osp, is 1.352 bbl/STB and represents the volume of separator oil required to yield 1 STB of oil (i.e., condensate). The equation used to calculate wellstream composition, z i, is z i + F gsp y i ) (1 * F gsp) x i ,
ǒ
F gsp + 1 )
2, 130ò osp M osp R sp
Ǔ
*1
, . . . . . . . . . . . . . . . . . . . . . (6.8)
ȍx M . . . . . . . . . . . . . . . . . . . . . . . . . . . . N
where M osp +
i
i
(6.9)
i+1
. . . . . . . . . . . . . . . . . . . . . (6.7)
where F gsp +mole fraction of wellstream mixture that becomes separator gas. In the laboratory report, F gsp is reported as “primary90
separator gas/wellstream ratio” (801.66 Mscf/MMscf), which is equivalent to mole per mole ( F gsp +0.80166 mol/mol). The reported value of F gsp can be checked with
ò osp in lbm/ft3 is calculated with a correlation (e.g., StandingKatz9) or with the relation (62.4g o ) 0.0136g g R s)ńB o , where R s and B o +separatoroil values in scf/STB and bbl/STB, respectively; PHASE BEHAVIOR
TABLE 6.4—WELLSTREAM (RESERVOIRFLUID) COMPOSITION FOR GOOD OIL CO. WELL 4 BOTTOMHOLE OIL SAMPLE Component H2 S CO2 N2 Methane Ethane Propane ibutane nbutane ipentane npentane Hexanes Heptanes plus Total
mol% Nil 0.91 0.16 36.47 9.67 6.95 1.44 3.93 1.44 1.41 4.33 33.29 100.00
wt% Nil 0.43 0.05 6.24 3.10 3.27 0.89 2.44 1.11 1.09 3.97 77.41 100.00
Density* (g/cm3)
°API*
Molecular Weight
DV g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.11) V osp
D R sp +
0.8515
34.5
218
and D R s +
DV g . Vo
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.12)
Total GOR is calculated by adding the stocktankoilbased GOR’s from each separator stage.
*At 60°F.
g o +stocktankoil density; and g g +gravity of gas released from the separator oil. Finally, the value of stocktankliquid/wellstream ratio in bbl/MMscf represents the separator barrels produced per 1 MMscf of wellstream. In terms of F gsp and separator properties, this value equals
ǒ
removed gas, n g ; and specific gravity of removed gas, g g. If requested, the gas samples can be analyzed chromatographically to give molar composition, y. The oil remaining after gas removal is brought to the conditions of the next separator stage. The gas is removed again and quantified by moles and specific gravity. Oil volume is noted, and the process is repeated until stocktank conditions are reached. Final oil volume, V o , and specific gravity, g o , are measured at 60°F. Table 6.7 gives results from four separator tests, each consisting of two stages of separation. The firststageseparator pressure is varied from 50 to 300 psig, and stocktank conditions are held constant at 0 psig and 75°F. GOR’s are reported as standard gas volume per separatoroil volume, R sp, and as standard gas volume per stocktankoil volume, R s, respectively.
Ǔ
bbl + 470(1*F gsp) M ospńò osp , . . . . . . . . . . . . . . (6.10) B osp MMscf where 470+(1 million scf/MMscf)/[(379 scf/lbm mol)(5.615 ft3/bbl)]. The separatoroil and gas compositions can be checked for consistency with the Hoffman et al.10 Kvalue method and Standing’s11 lowpressure Kvalue equations. 6.3 MultistageĆSeparator Test The multistageseparator test is performed on an oil sample primarily to provide a basis for converting differentialliberation data from a residualoil to a stocktankoil basis. Occasionally, several separator tests are run to help choose separator conditions that maximize stocktankoil production. Usually, two or three stages of separation are used, with the last stage at atmospheric pressure and nearambient temperature (60 to 80°F). The multistageseparator test can also be conducted for highliquidyield gascondensate fluids. Fig. 6.2 illustrates schematically how the separator test is performed. Initially, the reservoir sample is at saturation conditions and the volume is measured ( V ob or V gd ). The sample is then brought to the pressure and temperature of the firststage separator. All the gas is removed, and the oil volume at the separator stage, V osp, is noted together with the volume of removed gas, DV g ; number of moles of
N sp
Rs +
ȍǒD R Ǔ
s k
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.13)
k+1
Separatoroil FVF’s, B osp, are reported as the ratio of separatoroil volume to stocktankoil volume. B osp +
V osp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.14) Vo
Accordingly, the relation between separator gas/oil ratio and stocktank gas/oil ratio at a given stage is D Rs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.15) B osp
D R sp +
Because B osp u 1, it follows that R sp t R s. Bubblepointoil FVF, B ob , is the ratio of bubblepointoil volume to stocktankoil volume. B ob +
V ob . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.16) Vo
The average gas gravity, g g , is used in oil PVT correlations and to calculate reservoir densities on the basis of blackoil properties. The average gas gravity is calculated from N sp
ȍǒ g Ǔ ǒD R Ǔ g k
gg +
s k
k+1 N sp
ȍǒD R Ǔ
, . . . . . . . . . . . . . . . . . . . . . . . . . . (6.17)
s k
k+1
Fig. 6.1—Procedure for recombining singlestage separator samples to obtain wellstream composition of a bottomhole sample; BHS + bottomhole sampler, GC + gas chromatograph, FDP + freezingpoint depression, and DM + densitometer. CONVENTIONAL PVT MEASUREMENTS
91
TABLE 6.5—SEPARATOR AND RECOMBINED WELLSTREAM COMPOSITIONS FOR GOOD OIL CO. WELL 7 GAS CONDENSATE Separator Products Hydrocarbon Analysis Separator Liquid Component
(mol%)
CO2
Trace
N2
Separator Gas (mol%)
(gal/Mscf)
0.22
Wellstream (mol%)
(gal/Mscf)
0.18
Trace
0.16
0.13
Methane
7.78
75.31
61.92
Ethane
10.02
15.08
Propane
15.08
6.68
1.832
14.08 8.35
2.290
isobutane
2.77
0.52
0.170
0.97
0.317
nbutane
11.39
1.44
0.453
3.41
1.073
isopentane
3.52
0.18
0.066
0.84
0.306
npentane
6.50
0.24
0.087
1.48
0.535
Hexanes
8.61
0.11
0.045
1.79
0.734
34.33
0.06
0.028
6.85
3.904
100.00
100.00
2.681
100.00
9.159
Heptanes plus Total
HeptanesPlus Properties Oil gravity, °API
46.6
Specific gravity at 60/60°F
0.7946
Molecular weight
0.795
143
103
143
Parameters Calculated separator gas gravity (air+1.000)
0.735
Calculated gross heating value for separator gas at 14.696 psia and 60°F, BTU/ft3 dry gas
1,295
Primaryseparatorgas*/separatorliquid* ratio, scf/bbl at 60°F
4,428
Primaryseparatorgas/stocktankliquid ratio at 60°F, bbl at 60°F/bbl
1.352
Primaryseparatorgas/wellstream ratio, Mscf/MMscf
801.66
Stocktankliquid/wellstream ratio, bbl/MMscf
133.9
*Primary separator gas and liquid collected at 300 psig and 62°F.
TABLE 6.6—MATERIALBALANCE CALCULATIONS FOR GOOD OIL CO. WELL 7 GASCONDENSATE SAMPLE Liquid Composition at Specified Pressures (mol%) Component
At 3,500 psig
At 2,900 psig
At 2,100 psig
At 1,300 psig
At 605 psig
CO2
0.18
0.18
0.18
0.15
0.08
N2
0.13
0.08
0.06
0.03
0.01
C1
13.18
45.04
32.22
19.69
11.77
C2
8.12
14.05
13.99
12.32
7.44
C3
12.59
9.67
11.25
11.66
9.31
iC4
3.44
1.14
1.59
1.85
1.64
nC4
5.21
4.82
6.12
7.35
7.17
iC5
2.67
1.25
1.77
2.43
2.79
nC5
5.74
2.16
3.48
4.62
5.50
C6
8.47
3.11
4.55
6.40
8.37
C7+ Total M o, gńmol M oC 7), gńmol ò o, gńcm 3 92
40.27
18.51
24.79
33.50
45.91
100.00
100.00
100.00
100.00
100.00
96.6
54.1
64.3
78.2
95.6
168.8
160.1
152.1
149.9
150.3
0.3235
0.2642
0.1625
0.0892
0.0398 PHASE BEHAVIOR
where M i +molecular weight and ò i + component liquid density in lbm/ft3 at standard conditions (Table A1). The C 7) material in separator gases is usually treated as normal heptane. 6.4 Constant Composition Expansion
pst+14.7 psia Tst+60°F Fig. 6.2—Schematic of a multistageseparator test.
where ǒg gǓ k +separatorgas gravity at Stage k. This relation is based on the ideal gas law at standard conditions, where moles of gas are directly proportional with standard gas volume ( v g +379 scf/lbm mol). Table 6.8 gives the composition of the firststageseparator gas at 50 psig and 75°F. The gross heating value, H g , of this gas is calculated by Kay’s12 mixing rule and component heating values, H i, given in Table A1.
ȍy H . N
Hg +
i
i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.18)
i+1
Component liquid yields, L i , represent the liquid volumes of a component or group of components that can theoretically be processed from 1 Mscf of separator gas (gallons per million standard cubic feet). Li can be calculated from
ǒ Ǔ
M L i + 19.73y i ò i , . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.19) i
6.4.1 Oil Samples. For an oil sample, the CCE experiment is used to determine bubblepoint pressure, undersaturatedoil density, isothermal oil compressibility, and twophase volumetric behavior at pressures below the bubblepoint. Table 6.9 presents data from an example CCE experiment for a reservoir oil. Fig. 6.3 illustrates the procedure for the CCE experiment. A blind cell (i.e., a cell without a window) is filled with a known mass of reservoir fluid. Reservoir temperature is held constant during the experiment. The sample initially is brought to a condition somewhat above initial reservoir pressure, ensuring that the fluid is single phase. As the pressure is lowered, oil volume expands and is recorded. The fluid is agitated at each pressure by rotating the cell. This avoids the phenomenon of supersaturation, or metastable equilibrium, where a mixture remains as a single phase even though it should exist as two phases.1315 Sometimes supersaturation occurs 50 to 100 psi below actual bubblepoint pressure. By agitating the mixture at each new pressure, the condition of supersaturation is avoided, allowing more accurate determination of the bubblepoint. Just below the bubblepoint, the measured volume will increase more rapidly because gas evolves from the oil, yielding a higher system compressibility. The total volume, V t, is recorded after the twophase mixture is brought to equilibrium. Pressure is lowered in steps of 5 to 200 psi, where equilibrium is obtained at each pressure. When the lowest pressure is reached, total volume is three to five times larger than the original bubblepoint volume. The recorded cell volumes are plotted vs. pressure, and the resulting curve should be similar to one of the curves in Fig. 6.4.16 For a black oil (far from its critical temperature), the discontinuity in volume at the bubblepoint is sharp and the bubblepoint pressure and volume are easily read from the intersection of the pV trends in the single and twophase regions. Volatile oils do not exhibit the same clear discontinuity in volumetric behavior at the bubblepoint pressure. Instead, the pV curve is practically continuous in the region of the bubblepoint because the undersaturatedoil compressibility is similar to the effective twophase compressibility. This makes determining the bubblepoint of volatile oils in a blind cell difficult. Instead, a windowed cell
TABLE 6.7—SEPARATOR TESTS (RESERVOIRFLUID) OF GOOD OIL CO. WELL 4 OIL SAMPLE Separator Pressure (psia)
Separator Temperature (°F)
GORb (ft3/bbl)
GORc (ft3/bbl)
50 to 0
75
715
737
75
41
41
100 to 0
75
637
676
75
91
92
200 to 0
75
542
602
75
177
178
300 to 0
75
478
549
75
245
246
StockTank Gravity (°API)
40.5
40.7
40.4
40.1
FVFd (bbl/bbl)
1.481
1.474
1.483
1.495
aGauge. bIn cubic feet of gas at 60°F and 14.65 psi absolute per barrel of oil at indicated pressure and cIn cubic feet of gas at 60°F and 14.65 psi absolute per barrel of stocktank oil at 60°F. dIn barrels of saturated oil at 2,620 psi gauge and 220°F per barrel of stocktank oil at 60°F. eIn barrels of oil at indicated pressure and temperature per barrel of stocktank oil at 60°F.
CONVENTIONAL PVT MEASUREMENTS
Separator Volume Factore (bbl/bbl)
FlashedGas Specific Gravity
1.031
0.840
1.007
1.338
1.062
0.786
1.007
1.363
1.112
0.732
1.007
1.329
1.148
0.704
1.007
1.286
temperature.
93
TABLE 6.8—FIRSTSTAGE SEPARATORGAS COMPOSITION AND GROSS HEATING VALUE FOR GOOD OIL CO. WELL 4 OIL SAMPLE* Component
mol%
gal/Mscf
H2 S
Nil
CO2
1.62
N2
0.30
C1
67.00
C2
16.04
4.265
C3
8.95
2.449
iC4
1.29
0.420
nC4
2.91
0.912
iC5
0.53
0.193
nC5
0.41
0.155
C6
0.44
0.178
C7+
0.49
0.221
Total
100.00
TABLE 6.9—CCE DATA (RESERVOIRFLUID) FOR GOOD OIL CO. WELL 4 OIL SAMPLE Saturation (bubblepoint) pressure*, psig
0.02441
Thermal expansion of undersaturated oil at 5,000 psi+V at 220°F/V at 76°F
1.08790
Compressibility of saturated oil at reservoir temperature From 5,000 to 4,000 psi, vol/volpsi From 4,000 to 3,000 psi, vol/volpsi From 3,000 to 2,620 psi, vol/volpsi
Calculated gross heating value,
BTU/ft3
8.793 0.840 1,405
dry gas at 14.65 psia and 60°F *Collected at 50 psig and 75°F in the laboratory.
is used to observe visually the first bubble of gas and the liquid volumes below the bubblepoint. Reported data from commercial laboratories usually include bubblepoint pressure, p b ; bubblepoint density, ò ob, or specific volume, v ob(v + 1ńò); and isothermal compressibility of the undersaturated oil, co , at pressures above the bubblepoint (Table 6.9). The table also shows the oil’s thermal expansion, indicated by a ratio of undersaturatedoil volume at a specific pressure and reservoir temperature to the oil volume at the same pressure and a lower temperature. Total volumes are reported relative to the bubblepoint volume. V rt +
Vt . V ob
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.20)
Traditionally, isothermal compressibility data are reported for pressure intervals above the bubblepoint. In fact, the undersaturatedoil compressibility varies continuously with pressure, and, because V t + V o (V rt + V ro) for p u p b, oil compressibility can be expressed as
ǒ Ǔ
ēV rt c+ 1 V rt ēp
ǒ Ǔ;
ēV ro + 1 ēp V ro T
p u p b . . . . . . . . . . (6.21)
T
13.48 x 10 – 6 15.88 x 10 – 6 18.75 x 10 – 6
Pressure/Volume Relations*
Heating Value Calculated gas gravity (air+1.000)
2,620
Specific volume at saturation pressure*, ft3/lbm
Pressure (psig)
Relative volume (L)†
5,000
0.9639
4,500
0.9703
4,000
0.9771
3,500
0.9846
3,000
0.9929
2,900
0.9946
2,800
0.9964
2,700
0.9983
2,620**
1.0000
2,605
1.0022
2.574
2,591
1.0041
2.688
2,516
1.0154
2.673
2,401
1.0350
2.593
2,253
1.0645
2.510
2,090
1.1040
2.422
1,897
1.1633
2.316
1,698
1.2426
2.219
1,477
1.3618
2.118
1,292
1.5012
2.028
1,040
1.7802
1.920
830
2.1623
1.823
640
2.7513
1.727
472
3.7226
1.621
Y function‡
* ** 1
At 220°F. Saturation pressure. Relative volume+V/Vsat in barrels at indicated pressure per barrel at saturation pressure. ‡ Y function+( p *p)/(p sat abs)(V/Vsat*1).
The V rt function at undersaturated conditions may be fit with a secondĆdegree polynomial, resulting in an explicit relation for undersaturatedoil compressibility (see Chap. 3). Total volumes below the bubblepoint can be correlated by the Y function,16,17 defined as Y+
pb * p pb * p + , p(V rt * 1) pƪǒV tńV bǓ * 1ƫ
. . . . . . . . . . . . . . (6.22)
where p and p b are given in absolute pressure units. As Fig. 6.5 shows, Y vs. pressure should plot as a straight line and the linear trend can be used to smooth V rt data at pressures below the bubblepoint. Standing16 and Clark17 discuss other smoothing techniques and corrections that may be necessary when reservoir conditions and laboratory PVT conditions are not the same.
Fig. 6.3—Schematic of a CCE experiment for an oil and a gas condensate. 94
6.4.2 GasCondensate Samples. The CCE data for a gas condensate usually include total relative volume, V rt , defined as the volume of gas or of gasplusoil mixture divided by the dewpoint volume. Z facPHASE BEHAVIOR
at 290 psia
Fig. 6.4—Volume vs. pressure for an oil during a DLE test (after Standing16).
tors are reported at pressures greater than and equal to the dewpoint pressure. Table 6.10 gives these data for a gascondensate example. Reciprocal wetgas FVF, b gw, is reported at dewpoint and initial reservoir pressures, where these values represent the gas equivalent or wetgas volume at standard conditions produced from 1 bbl of reservoir gas volume. b gw + ǒ5.615
p T p 10 *3Ǔ p sc + 0.198 , sc ZT ZT
. . . . . . . . (6.23)
with b gw in Mscf/bbl, p in psia, and T in °R. Most CCE experiments are conducted in a visual cell for gas condensates, and relative oil (condensate) volumes, V ro, are reported at pressures below the dewpoint. V ro normally is defined as the oil volume divided by the total volume of gas and oil, although some reports define it as the oil volume divided by the dewpoint volume. 6.5 Differential Liberation Expansion The DLE experiment is designed to approximate the depletion process of an oil reservoir18 and thereby provide suitable PVT data to CONVENTIONAL PVT MEASUREMENTS
calculate reservoir performance.16,1921 Fig. 6.6 illustrates the laboratory procedure of a DLE experiment. Figs. 6.7A through 6.7C and Table 6.11 give DLE data for an oil sample. A blind cell is filled with an oil sample, which is brought to a single phase at reservoir temperature. Pressure is decreased until the fluid reaches its bubblepoint, where the oil volume, V ob , is recorded. Because the initial mass of the sample is known, bubblepoint density, ò ob, can be calculated. The pressure is decreased below the bubblepoint, and the cell is agitated until equilibrium is reached. All gas is removed at constant pressure. Then, the volume, DV g; moles, Dn g; and specific gravity, g g, of the removed gas are measured. The remaining oil volume, V o , is also recorded. This procedure is repeated 10 to 15 times at decreasing pressures and finally at atmospheric pressure. Residualoil volume, V or , and specific gravity, g or , are measured at 60°F. Other properties are calculated on the basis of measured data ( DV g , V o , Dn g , g g , V or , and g or), including differential solution gas/oil ratio, R sd ; differential oil FVF, B od ; oil density, ò o ; and gas Z factor, Z. For Stage k, these properties can be determined from 95
Bubblepoint Temperature °5F 80 163 185 205
Pressure psia 1,970 2,437 2,520 2,615
Volume cm3 82.30 86.88 87.92 89.05
Fig. 6.5—PVT relation and plot of Y function for an oil sample at pressures below the bubblepoint.
ȍ 379ǒDn Ǔ
calculations, volume factors, R s and B o , are used to relate reservoiroil volumes, V o, to produced surface volumes, V g and V o; i.e.,
k
g j
ǒR sdǓ + k ǒB odǓ k +
j+1
, . . . . . . . . . . . . . . . . . . . . . . . . (6.24)
V or ǒ V oǓ k V or
,
Rs + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.25)
ȍǒ28.97ń5.615ǓǒDn Ǔ ǒg Ǔ
Vg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.28) Vo
and B o +
k
V or(62.4g or) )
g j
ǒò oǓ + k
ǒ V oǓ 350g or )
k
ȍ 0.0764ǒDR k
+
g j
j+1
j+1
5.615ǒB odǓ k
Ǔ ǒg gǓ j
and (Z) k + ǒ1ńRTǓǒ pDV gńDn gǓ k ,
and B od + . . . . . . . . . . . . . . . . . . (6.27)
with V or and V o in bbl, R sd in scf/bbl, B od in bbl/bbl, DV g in ft3, p in psia, Dn g in lbm mol, ò o in lbm/ft3, and T in °R. Note that the subscript j+1 indicates the final DLE stage at atmospheric pressure and reservoir temperature. Reported oil densities are actually calculated by material balance, not measured directly. 6.5.1 Converting From Differential to StockTank Basis. Perhaps the most important step in the application of oil PVT data for reservoir calculations is conversion of the differential solution gas/oil ratio, R sd, and oil FVF, B od , to a stocktankoil basis.16,20 For engineering 96
Vg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.30) V or
, . . . . . . . . . . . . . . . . . . (6.26)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.29)
Differential properties R sd and B od reported in the DLE report are relative to residualoil volume (i.e., the oil volume at the end of the DLE experiment, corrected from reservoir to standard temperature). R sd +
sd j
Vo . Vo
Vo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.31) V or
The equations commonly used to convert differential volume factors to a stocktank basis are
ǒBB Ǔ
R s + R sb * ǒR sdb * R sdǓ and B o + B od
ǒBB Ǔ , ob
ob
. . . . . . . . . . . . . . . . . . (6.32)
odb
. . . . . . . . . . . . . . . . . . . . . . . . . . . (6.33)
odb
where B ob +bubblepointoil FVF, R sb +solution gas/oil ratio from a multistageseparator flash, and R sdb and B odb +differential volume factors at the bubblepoint pressure. The term ( B obńB odb), PHASE BEHAVIOR
TABLE 6.10—CCE DATA FOR GOOD OIL CO. WELL 7 GASCONDENSATE SAMPLE Pressure (psig)
Relative volume
Deviation Factor Z
6,000
0.8808
1.144
5,713*
0.8948
1.107**
5,300
0.9158
1.051
5,000
0.9317
1.009
4,800
0.9434
0.981
4,600
0.9559
0.953
4,400
0.9690
0.924
4,300
0.9758
0.909
4,200
0.9832
0.895
4,100
0.9914
0.881
4,000†
1.0000
0.867‡
3,905
1.0089
3,800
1.0194
3,710
1.0299
3,500
1.0559
3,300
1.0878
3,000
1.1496
2,705
1.2430
2,205
1.5246
1,605
2.1035
1,010
3.5665
Pressure/volume relations of reservoir fluid at 186°F. * Reservoir pressure. ** Gas FVF+1.591 Mscf/bbl. † Dewpoint pressure. ‡ Gas FVF+1.424 Mscf/bbl.
representing the volume ratio, V orńV o , is used to eliminate the residualoil volume, V or , from the Rsd and Bod data. Note that the conversion from differential to “flash” data depends on the separator conditions because B ob and R sb depend on separator conditions. Although, the conversions given by Eqs. 6.32 and 6.33 typically are used, they are only approximate. The preferred method, as originally suggested by Dodson et al.,22 requires that some equilibrium oil be taken at each stage of the DLE experiment and flashed through a multistage separator to give the volume ratios, R s and B o . This laboratory procedure is costly and timeconsuming and is seldom used. However, the method is readily incorporated into an equationofstate (EOS) based PVT program. 6.6 Constant Volume Depletion The CVD experiment is designed to provide volumetric and compositional data for gascondensate and volatileoil reservoirs producing by pressure depletion. Fig. 6.8 shows the stepwise procedure of a CVD experiment schematically, and Figs. 6.9A through 6.9D and Table 6.12 give CVD data for an example gascondensate fluid. The CVD experiment provides data that can be used directly by the reservoir engineer, including (1) a reservoir material balance that gives average reservoir pressure vs. recovery of total wellstream (wetgas recovery), sales gas, condensate, and natural gas liquids; (2) producedwellstream composition and surface products vs. reservoir pressure; and (3) average oil saturation in the reservoir (liquid dropout and revaporization) that occurs during pressure depletion. For many gascondensate reservoirs, the recoveries and oil saturation vs. pressure data from the CVD analysis closely approximate actual field performance for reservoirs producing by pressure depletion. When other recovery mechanisms, such as waterdrive and gas cycling, are considered, the basic data required for reservoir engineering are still taken mainly from a CVD report. This section provides a description of the data provided in a standard CONVENTIONAL PVT MEASUREMENTS
Fig. 6.6—Schematic of DLE experiment.
CVD analysis, ways to check the data for consistency,2325 and how to extract reservoirengineering quantities from the data.23,26 Initially, the dewpoint, p d , or bubblepoint pressure, p b , of the reservoir sample is established visually and the cell volume, V cell, at saturated conditions is recorded. The pressure is then reduced by 300 to 800 psi and usually by smaller amounts (50 to 250 psi) just below the saturation pressure of morevolatile systems. The cell is agitated until equilibrium is achieved, and volumes V o and V g are measured. At constant pressure, sufficient gas, DV g, is removed to return the cell volume to the original saturated volume. In the laboratory, the removed gas (wellstream) is brought to atmospheric conditions, where the amount of surface gas and condensate are measured. Surface compositions y g and x o of the produced surface volumes from the reservoir gas are measured, as are the volumes DV o and DV g , densities ò o and ò g and oil molecular weight M o . From these quantities, we can calculate the moles of gas removed, Dn g. D ng +
DV o ò o DV g ) . . . . . . . . . . . . . . . . . . . . . . . . . (6.34) Mo 379
These data are reported as cumulative wellstream produced, n p , relative to the initial moles n.
ǒnn Ǔ p
k
+ 1n
ȍ(Dn ) , k
g j
. . . . . . . . . . . . . . . . . . . . . . . . . (6.35)
j+1
where j+1 corresponds to saturation pressure and (Dn g) 1 + 0. The initial amount (in moles) of the saturated fluid is known when the cell is charged. The quantity n pńn is usually reported as cumulative wet gas produced in MMscf/MMscf, which is equivalent to mol/mol. Surface compositions y g and x o of the removed reservoir gas and properties of the removed gas are not reported directly in the laboratory report but are recombined to yield the equilibrium gas (wellstream) composition, y i , which also represents the equilibrium gas remaining in the cell. The C 7) molecular weight of the wellstream, M gC7), is backcalculated from measured specific gravity ( g w + g g ) and reservoirgas composition, y. C 7) specific gravity of the produced gas, g gC7) , is also reported, but this value is calculated from a correlation. Knowing the cumulative moles removed and its volume occupied as a singlephase gas at the removal pressure, we can calculate the equilibrium gas Z factor from Z+
pDV g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.36) D n g RT
A “twophase” Z factor is also reported that is calculated assuming that the gascondensate reservoir depletes according to the material balance for a dry gas and that the initial condition of the reservoir is at dewpoint pressure. 97
Fig. 6.7A—DLE data for an oil sample from Good Oil Co. Well 4; differential solution gas/oil ratio, Rsd .
ǒ Ǔǒ1 * GG Ǔ,
pd p + Zd Z2
pw
. . . . . . . . . . . . . . . . . . . . . . . . (6.37)
w
where G pw +cumulative wellstream (wet gas) produced and G w +initial wet gas in place. As defined in Eq. 6.37, the term G pwńG w equals n pńn reported in the CVD report. From Eq. 6.37, the only unknown at a given pressure is Z 2 , and the twophase Z factor is then given by Z2 +
p . . . . . . . . . . . . . . . . . . . . . . (6.38) ǒ p dńZ dǓƪ1 * ǒ n pńn Ǔƫ
Theoretical liquid yields, L i , are also reported for C 3) through C 5) groups in the produced wellstreams at each pressuredepletion stage. These values are calculated with
ǒ Ǔ
M L i + 19.73y i ò i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.39) i and by summing the yields of components in the particular “plus” group. For example, the liquid yield of C 5) material at CVD Stage k is given by
ǒL Ǔ C 5)
98
C 7) k
+
ȍ ǒL Ǔ
j k
j+i C 5
M + 19.73 ȍ ǒy Ǔ ǒ ò Ǔ . C 7)
j
j k
j+i C 5
j
. . . . . (6.40)
Table 6.13 gives various calculated cumulative recoveries based on the reservoir initially being at its dewpoint. The basis for the calculations is 1 MMscf of dewpoint wet gas in place, G w ; the corresponding initial moles in place at dewpoint pressure is given by G n + vw g +
1 10 6 scf + 2, 638 lbm mol. . . . . . . . . . (6.41) 379 scfńlbm mol
The first row of recoveries (wellstream) simply represents the cumulative moles produced, n pńn, expressed as wetgas volumes, G pw, in Mscf.
ǒ Ǔ
np G pw + nv g n
+ (2, 638 lbm mol)ǒ379 scfńlbm molǓ
ǒ1 + 1
ǒ Ǔ
np 10 3 MscfńscfǓ n 10 3
ǒnn Ǔ. p
. . . . . . . . . . . . . . . . . . . . . . . . . (6.42)
Recoveries in Rows 2 through 4 (Normal Temperature Separation, Total Plant Products in PrimarySeparator Gas, and Total Plant Products in SecondStageSeparator Gas) refer to production when the reservoir is produced through a threestage separator. Fig. 6.10 PHASE BEHAVIOR
Fig. 6.7B—DLE data for an oil sample from Good Oil Co. Well 4; differential oil FVF (relative volume), Bod .
illustrates the process schematically. The calculated recoveries are based on multistageseparator calculations that use lowpressure K values and a set of separator conditions chosen arbitrarily or specified when the PVT study is requested. 6.6.1 Recoveries: “Normal Temperature Separation.” Column 1: Initial in Place. In Column 1, Row 2a the stocktank oil in solution in the initial dewpoint fluid (N+135.7 STB) is calculated by flashing 1 MMscf of the original dewpoint fluid, G w , through a multistage separator. Rows 2b through 2d give the volumes of separator gas at each stage of a threestage flash of the initial dewpoint fluid: 757.87, 96.68, and 24.23 Mscf, respectively. The mole fraction of wellstream resulting as a surface gas F gg is given by G F gg + d + ǒ757.87 ) 96.68 ) 24.23 Mscfńlbm molǓ Gw
ǒ1
10 3 scfńMscfǓńǒ379 scfńlbm molǓ
+ 0.8788 lbm molńlbm mol,
. . . . . . . . . . . . . . . (6.43)
where G d +total separator “dry” gas and the corresponding mole fraction of stocktank oil is 0.1212 mol/mol. F gg is used to calculate drygas FVF (see Eq. 3.41). For the dewpoint pressure, this gives CONVENTIONAL PVT MEASUREMENTS
B gd + +
ǒ p scńT scǓǒ ZTńp Ǔ B gw + F gg F gg ǒ14.7ń520ǓǊ[0.867(186 ) 460)]ń4015ǋ 0.8788
+ 4.487
10 *3 ft 3ńscf. . . . . . . . . . . . . . . . . . . . (6.44)
The producing GOR of the dewpoint mixture for the specified separator conditions can be calculated as R p + G + ƪǒ757.87 ) 96.68 ) 24.23 Mscfńlbm molǓ N
ǒ1
10 3 scfńMscfǓƫń135.7 STBńlbm mol
+ 6, 476 scfńSTB .
. . . . . . . . . . . . . . . . . . . . . . . . (6.45)
The dewpoint solution oil/gas ratio, r sd, is simply the inverse of R p . r sd + r p + 1 Rp + 1.544
10 *4 STBńscf + 154.4 STBńMMscf. . . . . . . . . . . . . . . . (6.46)
Note that specific gravities of stocktank oil and separator gases are not reported for the separator calculations. 99
Fig. 6.7C—DLE data for an oil sample from Good Oil Co. Well 4; oil viscosity, mo .
Column 2 and Higher. On the basis of 1 MMscf of initial dewpoint fluid, Rows 2a through 2d give cumulative volumes of separator products at each depletion pressure ( N p, G p1, G p2, and G p3 ). The producing GOR of the wellstream produced during a depletion stage is given by
ǒR pǓ + k
ǒG p1 ) G p2 ) G p3Ǔ
k
* ǒG p1 ) G p2 ) G p3Ǔ
ǒN pǓ * ǒN pǓ k k*1
k*1
R p + Ǌ[(301.57 ) 20.75 ) 5.61) * (124.78 ) 12.09 ) 3.16)]
. . . . . . . . . . . . . . . . . . . . . . . . (6.48)
10 *5 STBńscf
+ 45.8 STBńscf . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.49) 100
* ǒG p1 ) G p2 ) G p3Ǔ
G w ƪǒ n pńn Ǔ k * ǒ n pńn Ǔ k*1ƫ
k*1
.
For p+2,100 psig, this gives F gg + [(301.57 ) 20.75 ) 5.61)*(124.78 ) 12.09 10 3Ǔńǒ1
10 6Ǔ(0.35096 * 0.15438)
+ 0.9558 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.51)
B gd +
ǒ14.7ń520Ǔƪ0.762(186 ) 460)ń2, 115ƫ 0.9558
+ 6.884
rs + rp + 1 Rp 1 + 4.58 21, 580 scfńSTB
k
The drygas FVF at 2,100 psig is
10 3Ǔǋń(24.0 * 15.4)
In terms of the solution oil/gas ratio,
+
ǒG p1 ) G p2 ) G p3Ǔ
) 3.16)]ǒ1
For 2,100 psig, this gives
+ 21, 850 scfńSTB.
ǒF ggǓ + k
. . . . . . . . . . . . . . . . . . (6.50) .
. . . . . . . . . . . . . . . . . . (6.47)
ǒ1
At a given pressure, the mole fraction of the removed CVD gas wellstream that becomes dry separator gas is given by
10 *3 ft 3ńscf . . . . . . . . . . . . . . . . . . . . (6.52)
In summary, the information provided in the rows labeled Normal Temperature Separation gives estimates of the condensate and salesgas recoveries assuming a multistage surface separation. For example, at an abandonment pressure of 605 psig, the condensate recovery is 35.1 STB of the 135.7 STB initially in place (in solution in the dewpoint mixture), or 26% condensate recovery. Drygas recovery is (685.02)37.79)10.40)+733.21 Mscf of the 878.78 PHASE BEHAVIOR
TABLE 6.11—DLE DATA FOR GOOD OIL CO. WELL 4 OIL SAMPLE
Pressure (psig)
Solution GOR (scf/bbl*)
Relative Oil Volume (RB/bbl*)
2,620 2,350 2,100 1.850 1,600 1,350 1,110 850 600 350 159 0
854 763 684 612 544 479 416 354 292 223 157 0
1.600 1.554 1.515 1.479 1.445 1.412 1.382 1.351 1.320 1.283 1.244 1.075 1.000**
Differential Vaporization Relative Oil Deviation Total Volume Density Factor Z (RB/bbl*) (g/cm3) 1.600 1.665 1.748 1.859 2.016 2.244 2.593 3.169 4.254 6.975 14.693
0.6562 0.6655 0.6731 0.6808 0.6889 0.6969 0.7044 0.7121 0.7198 0.7291 0.7382 0.7892
0.846 0.851 0.859 0.872 0.887 0.903 0.922 0.941 0.965 0.984
Gas FVF (RB/bbl*)
Incremental Gas Gravity
0.00685 0.00771 0.00882 0.01034 0.01245 0.01552 0.02042 0.02931 0.05065 0.10834
0.825 0.818 0.797 0.791 0.794 0.809 0.831 0.881 0.988 1.213 2.039
DLE Viscosity Data at 220°F Pressure (psig)
Oil Viscosity (cp)
5,000 4,500 4,000 3,500 3,000 2,800 2,620 2,350 2,100 1,850 1,600 1,350 1,100 850 600 350 159 0
0.450 0.434 0.418 0.401 0.385 0.379 0.373 0.396 0.417 0.442 0.469 0.502 0.542 0.592 0.654 0.783 0.855 1.286
Calculated Gas Viscosity (cp)
0.0191 0.0180 0.0169 0.0160 0.0151 0.0143 0.0135 0.0126 0.0121 0.0114 0.0093
Gravity of residual oil+35.1°API at 60°F. *Barrels **At
of residual oil.
60°F.
Mscf dry gas originally in place, or 83.4%. These recoveries can be compared with the reported wetgas (or molar) recovery of 76.787% at 605 psig. In addition to recoveries, the calculated results in this section can be used to calculate solution oil/gas ratio, r s, and drygas FVF, B gd , for modified blackoil applications. 6.6.2 Recovery: Plant Products. Rows 3 through 5 consider theoretical liquid recoveries for propane, butanes, and pentanesplus assuming 100% plant efficiency. Recoveries in Rows 3 and 4 are for the calculated separator gases from Stages 1 and 2 of the threestage surface separation. Recoveries in Row 5 are for the produced wellstreams from the CVD experiment and represent the absolute maximum liquid recoveries that can be expected if the reservoir is produced by pressure depletion. Fig. 6.10 illustrates the recovery calculations schematically. Liquid volumes (in gal/MMscf of initial dewpoint fluid) at CVD Stage k are calculated from
ǒ ǓƪȍǒDnn Ǔ ǒ y Ǔ ƫ,
M (L i) k + 19, 730 ò i i
k
g
i j
j+1
. . . . . . . . . (6.53)
Fig. 6.8—Schematic of CVD experiment.
j
CONVENTIONAL PVT MEASUREMENTS
101
Fig. 6.9A—CVD data for gascondensate sample from Good Oil Co. Well 7; liquiddropout curve, Vro .
where j + 1 represents the dewpoint, y i +compositions of wellstream entering the gas plant at various stages of depletion, M i +component molecular weights, and ò i + liquid component densities in lbm/ft3 at standard conditions (Table A1). Calculated liquid recoveries below the dewpoint use the moles of wellstream produced ( Dn gńn) and the compositions yi from the separator gas (Rows 3 and 4) or wellstream (Row 5) entering the plant. Column 1 (Initial in Place) gives the total recoveries assuming that the entire initial dewpoint fluid is taken to the surface and processed [i.e., k + 1 and (Dn gńn) 1 + 1 in Eq. 6.53]. Note that cumulative recovery of propanes from the firststage separator during depletion (1,276 gal) is larger than the liquid propane produced in the firststageseparator gas of the original dewpoint mixture (1,198 gal). This means that the stocktank oil from the separation of original dewpoint mixture contains more propane than the cumulative stocktankoil volumes produced by depletion and threestage separation. The results given in Rows 3 and 4 cannot be calculated from reported data because surface separator compositions from the threestage separation are not provided in the report. The results in Row 5 can be checked. As an example, consider the C 3 recoveries for the initialinplace fluid at 2,100 psig.
ǒL Ǔ C3
pd
+ 19, 730 ǒ44.09ń31.66Ǔƪ (1)(0.0837)ƫ + 2, 299 galńMMscf . . . . . . . . . . . . . . . . . . . . (6.54a)
102
ǒ Ǔ
and L C
3 2100
+ 19, 730 ǒ44.09ń31.66Ǔ [0.0825(0.05374) ) 0.0810(0.15438 * 0.05374) ) 0.0757(0.35096 * 0.15438)] + 754 galńMMscf. . . . . . . . . . . . . . . . . (6.54b)
For the C 5) recoveries at the dewpoint,
ǒL Ǔ C 5)
pd
+ 19, 730 [(72.15ń38.96) (0.0091) ) (72.15ń39.36) (0.0152) ) (86.17ń41.43) (0.0179) ) (143ń49.6) (0.0685)] + 5, 513 galńMMscf . . . . . . . . . . . . . . . . . . (6.55)
6.6.3 Correcting Recoveries for Initial Pressure Greater Than Dewpoint Pressure. All recoveries given in Table 6.13 assume that the reservoir pressure is initially at dewpoint. This assumption is made because initial reservoir pressure is not always known with certainty when PVT calculations are made. However, adjusting reported recoveries is straightforward when initial pressure is greater than dewpoint pressure. With Q Table as recoveries given in Columns 2 and higher in Table 6.13, Q d as hydrocarbons in place in Column PHASE BEHAVIOR
Dewpoint Pressure
Fig. 6.9B—CVD data for gascondensate sample from Good Oil Co. Well 7; equilibrium gas compositions, yi .
1 at dewpoint pressure, and Q as actual cumulative recoveries based on hydrocarbons in place at the initial pressure, Q + Qd
ƪ
ƫ
ǒ pńZǓ ǒ pńZǓ i * ; p y p d , . . . . . . . . . . . . (6.56) ǒ pńZǓ ǒ pńZǓ d
d
Q + Q Table ) DQ d ; p t p d , . . . . . . . . . . . . . . . . . . . (6.57) and DQ d + Q d
pńZ) ƪ((pńZ) * 1ƫ , i
. . . . . . . . . . . . . . . . . . . . (6.58)
d
where DQ d +additional recovery from initial to dewpoint pressure. For the example report, DQ d +
ƪ
ƫ
ǒ5, 728ń1.107Ǔ * 1 Qd ǒ4, 015ń0.867Ǔ
+ 0.1173 Q d ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . (6.59)
recalling that moles of material at dewpoint is 2,638 lbm mol, moles of material at initial pressure of 5,728 psig is n +2, 638(1 ) 0.1173) + 2, 947 lbm mol, and the basis of calculations is G w + 1.173 MMscf of wet gas in place at initial pressure of 5,728 psia. The cumulative wellstream produced at the dewpoint pressure of 4,000 psig is 0.1173(1, 000) + 117.3 Mscf. Recovery at 3,500 psig is 117.3 ) 53.74 + 171.0 Mscf. Likewise, wetgas recovery CONVENTIONAL PVT MEASUREMENTS
should be increased by 117.3 Mscf for all depletion pressures in the CVD table. For stocktankoil recovery, Q d + 135.7 STB, so DQ d + 15.9 STB. Stocktankoil recovery at 4,000 psig is 15.9 ) 0 + 15.9 STB; at 3,500 psig the recovery should be 15.9 ) 6.4 + 22.3 STB, and so on. On the basis of 1 MMscf wet gas at the dewpoint or 1.1173 MMscf at initial reservoir pressure, the laboratory hydrocarbon pore volume (HCPV), V pHClab, is the same. V pHClab + ǒG wB gwǓ d + ǒ1
10 6Ǔ
Ǌǒ Ǔƪ 14.7 520
0.867(186 ) 460) 4, 015
ƫǋ
+ 3, 943 ft 3 + ǒG w B gwǓ i + 1.1173
10 6
+ 3, 943 ft 3 .
Ǌǒ Ǔƪ 14.7 520
1.107(186 ) 460) 5728
ƫǋ
. . . . . . . . . . . . . . . . . . . . . . . . . . (6.60)
The actual HCPV of a reservoir is much larger than V pHClab, and the conversion to obtain recoveries for the actual HCPV is simply 103
Fig. 6.9C—CVD data for gascondensate sample from Good Oil Co. Well 7; equilibrium gas Z factor, Zg .
Q actual + Q lab
V pHCactual , V pHClab
. . . . . . . . . . . . . . . . . . . . . . . (6.61)
where Q lab +laboratory value given by Eqs. 6.55 and 6.57. As an example, suppose geological data indicate a HCPV of 625,000 bbl (82.45 acreft), or 3.509 106 ft3. Then, original wet gas in place is G w + 1.1173
6 10 6 3.509 10 3, 943
+ 994.3 MMscf . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.62) and condensate in solution at initial pressure is given by 6 N + 135.7(1.1173) 3.509 10 3, 943
+ 134, 900 STB .
. . . . . . . . . . . . . . . . . . . . . . . . . . (6.63)
6.6.4 LiquidDropout Curve. Table 6.11 and Figs. 6.9A through 6.9D show relative oil volumes, V ro, measured in the example CVD experiment. V ro is defined as the volume of oil, V o , at a given pressure divided by the original saturation volume, V s. This relative volume is an excellent measure of the average reservoiroil saturation (normalized) that will develop during depletion of a gascondensate 104
reservoir. Correcting for water saturation, S w , the reservoiroil saturation can be calculated from V ro with S o + (1 * S w)V ro .
. . . . . . . . . . . . . . . . . . . . . . . . . . . (6.64)
For most gas condensates, V ro shows a maximum near 2,000 to 2,500 psia. Cho et al.27 give a correlation for maximum liquid dropout as a function of temperature and C 7) mole percent in the dewpoint mixture. ǒ V roǓ
max
+ 93.404 ) 4.799 z C
7)
* 19.73 ln T , . . . . . . (6.65)
with z C7) in mole percent and T in °F. The correlation predicts (V ro) max +23.2% for the example condensate fluid compared with 24% measured experimentally (at 2,100 psig). Fig. 6.11 shows values of (V ro) max vs. T and z C7)from Eq. 6.65. Considerable attention usually is given to matching the liquiddropout curve when an EOS is used. Some gas condensates havewhat is referred to as a “tail,” where liquid drops out very slowly (sometimes for several thousand psi below the dewpoint) before finally increasing toward a maximum. Matching this behavior with an EOS can prove difficult, and the question is whether matching the tail is really necessary (see Appendix C). What really matters for reservoir calculations of a gascondensate fluid is how much original stocktank condensate is “lost” because of retrograde condensation in the reservoir. The shape and magniPHASE BEHAVIOR
Fig. 6.9D—CVD data for gascondensate sample from Good Oil Co. Well 7; wetgas material balance.
tude of liquid dropout reflects the change in producing oil/gas ratio, r p [ r s . A tail on a liquiddropout curve implies that the producing wellstream is becoming only slightly leaner (i.e., r s is decreasing only slightly). The cumulative condensate recovery is given by Gp
Np +
ŕ r dG , s
p
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.66)
0
where G p +cumulative dry gas produced. Cumulative condensate production is readily evaluated from a plot of r s vs. G p . One of the most important checks of an EOS characterization for any gas condensate, particularly one with a tail, is N p calculated from CVD data vs. N p calculated from the EOS characterization. It is alarming how much the surface condensate recovery can be underestimated if the tail is not matched properly. We do not recommend matching the dewpoint exactly with a liquiddropout curve that is severely overpredicted in the region where measured results indicate little dropout. If the EOS characterization cannot be modified to honor the tail of liquiddropout curve, it is preferable to underpredict the measured dewpoint pressure and match only the higher liquiddropout volumes. In summary, oil relative volume, V ro, is not important per se; however, the effect of liquid dropout on surface condensate production CONVENTIONAL PVT MEASUREMENTS
should be emphasized. In fact, the effect of shape and magnitude of liquid dropout on fluid flow in the reservoir is negligible, and any EOS match will probably have the same effect on fluid flow from the reservoir into the wellbore (i.e., inflow performance). 6.6.5 Consistency Check of CVD Data. Reudelhuber and Hinds24 give a detailed procedure for checking CVD data consistency that involves a materialbalance check on components and phases and yields oil compositions, density, molecular weight, and M C7). Together with reported data, these calculated properties allow K values to be calculated and checked for consistency with the Hoffman et al.10 method.11,28 Whitson and Torp’s23 materialbalance equations are summarized later. Similar equations can also be derived for a DLE experiment when equilibrium gas compositions and oil relative volumes are reported. Reported CVD data include temperature, T ; dewpoint pressure, p d , or bubblepoint pressure, p b ; dewpoint Z factor, Z d, or bubblepointoil density, ò ob . Additional data at each Depletion Stage k include oil relative volume, V ro; initial fraction of cumulative moles produced, n pńn; gas Z factor (not the twophase Z factor), Z; equilibrium gas composition, yi ; and equilibrium gas (wellstream) C 7) molecular weight, M g C7). The equilibrium gas density, ò g ; molecular weight, M g ; and wellstream gravity, g w + M gńM air , are readily calculated at each 105
TABLE 6.12—CVD DATA FOR GOOD OIL CO. WELL 7 GASCONDENSATE SAMPLE 2* Reservoir Pressure, psig 5,713**
4,000†
CO2
0.18
0.18
0.18
0.18
0.18
0.19
N2
0.13
0.13
0.13
0.14
0.15
0.15
0.14
C1
61.72
61.72
63.10
65.21
69.79
70.77
66.59
C2
14.10
14.10
14.27
14.10
14.12
14.63
16.06
C3
8.37
8.37
8.26
8.10
7.57
7.73
9.11
iC4
0.98
0.98
0.91
0.95
0.81
0.79
1.01
nC4
3.45
3.45
3.40
3.16
2.71
2.59
3.31
iC5
0.91
0.91
0.86
0.84
0.67
0.55
0.68
nC5
1.52
1.52
1.40
1.39
0.97
0.81
1.02
C7
1.79
1.79
1.60
1.52
1.03
0.73
0.80
C7+
6.85
6.85
5.90
4.41
2.00
1.06
1.07
100.00
100.00
100.00
100.00
100.00
100.00
100.00
Component, mol%
Total
3,500
2,900
2,100
1,300
605
0‡
0.21
Properties C7+ molecular weight
143
143
138
128
116
111
110
C7+ specific gravity
0.795
0.795
0.790
0.780
0.767
0.762
0.761
Equilibrium gas deviation factor, Z
1.107
0.867
0.799
0.748
0.762
0.819
0.902
Twophase deviation factor, Z
1.107
0.867
0.802
0.744
0.704
0.671
0.576
0.000
5.374
15.438
35.096
57.695
76.787
Wellstream produced, cumulative % of initial
93.515
From smooth compositions C3+, gal/Mscf
9.218
9.218
8.476
7.174
5.171
4.490
5.307
C4+, gal/Mscf
6.922
6.922
6.224
4.980
3.095
2.370
2.808
C5+, gal/Mscf
5.519
5.519
4.876
3.692
1.978
1.294
1.437
23.9
22.5
18.1
Retrograde Condensation During Gas Depletion Retrograde liquid volume,
0.0
3.3
19.4
12.6
% hydrocarbon pore space *Study conducted at 186°F. ** Original reservoir pressure. † Dewpoint pressure. ‡ 0psig residualliquid properties: 47.5°API oil gravity at 60°; 0.7897 specific gravity at 60/60°F; and molecular weight of 140.
Depletion Stage k [and at the dewpoint ( k + 1) for a gascondensate sample] from
ǒM gǓ + k
ȍ(y ) N
i k
Mi ,
. . . . . . . . . . . . . . . . . . . . . . . . . . (6.67)
i+1
ǒò gǓ + k
ǒ Ǔ
np (n t) k + 1 * n , k
p ǒM gǓ k (Z) k RT
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.68)
and ǒg gǓ k + ǒ g wǓ k +
ǒM gǓ
k
28.97
.
. . . . . . . . . . . . . . . . . . . . (6.69)
On a basis of 1 mol initial dewpoint fluid ( n + 1), the cell volume is
ǒn gǓ + k
ǒ p Ǔ ǒV gǓ k k
(Z) k RT
,
and (n o) k + (n t) k * ǒn gǓ k , . . . . . . . . . . . . . . . . . . . . . . . . (6.73) and moles and mass of the individual phases remaining in the cell at Stage k are given by
ȍǒDnn Ǔ ǒM Ǔ , k
(m t) k + M s *
g
g j
j +2
Z RT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.70) V cell + dp d for a gas condensate and M V cell + ò ob . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.71) ob for a volatile oil. Oil and gas volumes, respectively, at Stage k are (V o) k+ V cell (V ro) k and ǒV gǓ k + V cellƪ1 * (V ro) kƫ . . . . . . . . . . . . . . . . . . . . . (6.72) Moles and mass of the total material remaining in the cell at Stage k are given by 106
j
ǒm gǓ + ǒn gǓ ǒM gǓ , k k k and (m o) k + (m t) k * ǒm gǓ k .
. . . . . . . . . . . . . . . . . . . . . . (6.74)
In Eqs. 6.73 and 6.74,
ǒDnn Ǔ + ǒnn Ǔ * ǒnn Ǔ g
p
j
p
j
, . . . . . . . . . . . . . . . . . . . . (6.75)
j*1
M s +saturatedfluid molecular weight, and (n pńn) 1 + 0. Densities and molecular weights of the oil phase are calculated from PHASE BEHAVIOR
TABLE 6.13—CALCULATED RECOVERIES* FROM CVD REPORT FOR GOOD OIL CO. WELL 7 GASCONDENSATE SAMPLE Reservoir Pressure (psig) Initial in Place
4,000**
3,500
2,900
2,100
1,300
605
0
1,000
0
53.74
154.38
350.96
576.95
767.87
935.15
Stocktank liquid, bbl
135.7
0
6.4
15.4
24.0
29.7
35.1
Primaryseparator gas, Mscf
757.87
0
41.95
124.78
301.57
512.32
658.02
Secondstage gas, Mscf
96.68
0
4.74
12.09
20.75
27.95
37.79
Stocktank gas, Mscf
24.23
0
1.21
3.16
5.61
7.71
10.4
Propane, gal
1,198
0
67
204
513
910
1,276
Butanes, gal
410
0
23
72
190
346
491
Pentanes, gal
180
0
10
31
81
144
192
Propane, gal
669
0
33
86
149
205
286
Butanes, gal
308
0
15
41
76
108
159
Pentanes, gal
138
0
7
19
35
49
69
Propane, gal
2,296
0
121
342
750
1,229
1,706
Butanes, gal
1,403
0
73
202
422
665
927
Pentanes, gal
5,519
0
262
634
1,022
1,315
1,589
Wellstream, Mscf Normal temperature separation†
Total plant products in primary separator‡
Total plant products in secondstage separator‡
Total plant products in wellstream‡
* Cumulative
recovery per MMscf of original fluid calculated during depletion. **Dewpoint pressure. † Recovery basis: primary separation at 500 psia and 70°F, secondstage separation at 50 psia and 70°F, and stock tank at 14.7 psia and ‡ Recovery assumes 100% plant efficiency.
ǒò oǓ + k
(m o) k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.76) (V o) k
and (M o) k +
(m o) k (n o) k
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.77)
ƪ
(z i) k + 1 (z i) 1 * (n t) k
K values can be calculated from K i + y ińx i , and z i +overall composition of the mixture remaining in the cell at Stage k .
ȍǒDnn Ǔ ǒ y Ǔ k
g
ƫ
i j
j+2
j
. . . . . . . . . . . . (6.79)
C 7) molecular weight of the oil phase can be calculated from
and the oil composition is given by (n t) k(z i) k * ǒn gǓ k ǒ y iǓ k (x i) k + . . . . . . . . . . . . . . . . . . . . (6.78) (n t) k * ǒn gǓ k
70°F.
ǒM
o C 7)
Ǔ
(M o) k * k
+
ȍ (x )
i k
i0C 7)
ǒxC7)Ǔk
Mi . . . . . . . . . . . . . . . (6.80)
Table 6.6 summarizes these calculations for the sample gascondensate mixture.
(Separator Gas 1)
(Separator Gas 2)
Fig. 6.10—Schematic of method of calculating plant recoveries in a CVD report for a gas condensate. CONVENTIONAL PVT MEASUREMENTS
107
Nonphysical
Heptanes Plus, mol% Fig. 6.11—Calculated maximum retrograde oil relative volumes from the Cho et al.27 correlation.
The oil composition at the last depletion state (605 psig for the example condensate) can be measured, but it must be requested specifically. Also, the residualoil molecular weight, M or , and specific gravity, g or, remaining after depletion at atmospheric pressure are typically measured and reported as shown in Table 6.12. These values can be compared with calculated values by use of the materialbalance equations shown earlier. The materialbalance calculations are more accurate for rich gas condensates and volatile oils. In fact, obtaining reasonable materialbalance oil properties for lean gas condensates is difficult. Sometimes it is useful to modify the reported oil relative volumes (particularly those close to the dewpoint) to monitor the effect on calculated oil properties. An alternative materialbalance check that may be even more useful for determining data consistency (particularly for leaner gas condensates) involves starting with reported finalstage condensate composition, (x i) k+N, and adding back the removed gases, (y i) k , for each stage from k + N to k + 1. This results in the original gas composition, (z i) k+1 , which can be compared quantitatively with the laboratoryreported composition. References 1. “Core Laboratories Good Oil Company Oil Well No. 4 PVT Study,” Core Laboratories, Houston. 2. “Core Laboratories Good Oil Company Condensate Well No. 7 PVT Study,” Core Laboratories, Houston. 3. Flaitz, J.M. and Parks, A.S.: “Sampling GasCondensate Wells,” Trans., AIME (1942) 146, 13. 4. Katz, D.L., Brown, G.G., and Parks, A.S.: “NGAA Report on Sampling TwoPhase Gas Streams from High Pressure Condensate Wells,” (September 1945). 5. Reudelhuber, F.O.: “Sampling Procedures for Oil Reservoir Fluids,” JPT (December 1957) 15. 6. Clark, N.J.: “Sampling and Testing Oil Reservoir Samples,” JPT (Jan. 1962) 12. 7. Clark, N.J.: “Sampling and Testing Gas Reservoir Samples,” JPT (March 1962) 266. 8. Recommended Practice for Sampling Petroleum Reservoir Fluids, API, Dallas (1966) 44. 9. Standing, M.B. and Katz, D.L.: “Density of Natural Gases,” Trans., AIME, (1942) 146, 140. 10. Hoffmann, A.E., Crump, J.S., and Hocott, C.R.: “Equilibrium Constants for a GasCondensate System,” Trans., AIME (1953) 198, 1. 11. Standing, M.B.: “A Set of Equations for Computing Equilibrium Ratios of a Crude Oil/Natural Gas System at Pressures Below 1,000 psia,” JPT (September 1979) 1193. 12. Kay, W.B.: “The EthaneHeptane System,” Ind. & Eng. Chem. (1938) 30, 459. 13. Kennedy, H.T. and Olson, C.R.: “Bubble Formation in Supersaturated Hydrocarbon Mixtures,” Oil & Gas J. (October 1952) 271. 108
14. Silvey, F.C., Reamer, H.H., and Sage, B.H.: “Supersaturation in Hydrocarbon Systems: MethanenDecane,” Ind. Eng. Chem. (1958) 3, No. 2, 181. 15. Tindy, R. and Raynal, M.: “Are TestCell Saturation Pressures Accurate Enough?,” Oil & Gas J. (December 1966) 126. 16. Standing, M.B.: Volumetric and Phase Behavior of Oil Field Hydrocarbon Systems, eighth edition, SPE, Richardson, Texas (1977). 17. Clark, N.J.: “Adjusting Oil Sample Data for Reservoir Studies,” JPT (February 1962) 143. 18. Moses, P.L.: “Engineering Applications of Phase Behavior of CrudeOil and Condensate Systems,” JPT (July 1986) 715. 19. Amyx, J.W., Bass, D.M. Jr., and Whiting, R.L.: Petroleum Reservoir Engineering, McGrawHill Book Co. Inc., New York City (1960). 20. Craft, B.C. and Hawkins, M.: Applied Petroleum Reservoir Engineering, first edition, PrenticeHall Inc., Englewood Cliffs, New Jersey (1959). 21. Dake, L.P.: Fundamentals of Reservoir Engineering, Elsevier Scientific Publishing Co., Amsterdam (1978). 22. Dodson, C.R., Goodwill, D., and Mayer, E.H.: “Application of Laboratory PVT Data to Reservoir Engineering Problems,” Trans., AIME (1953) 198, 287. 23. Whitson, C.H. and Torp, S.B.: “Evaluating ConstantVolumeDepletion Data,” JPT (March 1983) 610; Trans., AIME, 275. 24. Drohm, J.K., Goldthorpe, W.H., and Trengove, R.: “Enhancing the Evaluation of PVT Data,” paper OSEA 88174 presented at the 1988 Offshore Southeast Asia Conference, Singapore, 2–5 February. 25. Drohm, J.K., Trengove, R., and Goldthorpe, W.H.: “On the Quality of Data From Standard GasCondensate PVT Experiments,” paper SPE 17768 presented at the 1988 Gas Technology Symposium, Dallas, 13–15 June. 26. Reudelhuber, F.O. and Hinds, R.F.: “Compositional Material Balance Method for Prediction of Recovery From VolatileOil DepletionDrive Reservoirs,” JPT (January 1957) 19; Trans., AIME, 210. 27. Cho, S.J., Civan, F., and Starling, K.E.: “A Correlation To Predict Maximum Condensation for Retrograde Condensation Fluids and Its Use in PressureDepletion Calculations,” paper SPE 14268 presented at the 1985 SPE Annual Technical Conference and Exhibition, Las Vegas, Nevada, 22–25 September. 28. Clark, N.J.: “Theoretical Aspects of Oil and Gas Equilibrium Calculations,” JPT (April 1962) 373.
SI Metric Conversion Factors °API 141.5/(131.5)°API) +g/cm3 bbl 1.589 873 E*01 +m3 Btu 1.055 056 E)00 +kJ cp 1.0* E*03 +Pa@s ft 3.048* E*01 +m E*02 +m3 ft3 2.831 685 °F (°F*32)/1.8 +°C gal 3.785 412 E*03 +m3 in. 2.54* E)00 +cm lbm mol 4.535 924 E*01 +kmol psi 6.894 757 E)00 +kPa *Conversion factor is exact.
PHASE BEHAVIOR
Chapter 7
BlackĆOil PVT Formulations 7.1 Introduction This chapter reviews blackoil pressure/volume/temperature (PVT) formulations, gives examples of their application, and provides guidelines for choosing the proper PVT formulation for a given reservoir. Sec. 7.2 reviews the traditional blackoil PVT formulation. The three basic PVT properties are introduced: solution gas/oil ratio, R s ; oil formation volume factor (FVF), B o; and gas FVF, B g. These properties define the PVT behavior of reservoiroil and gas mixtures by quantifying the volumetric behavior and the distribution of surfacegas and surfaceoil “components” as functions of pressure. Many reservoirs being discovered today are at great depths, with a higher percentage of these deep reservoirs containing gascondensate and volatileoil fluids. Treatment of these reservoirs requires modification of the standard PVT formulation, as Sec. 7.3 discusses. In particular, the additional property solution oil/gas ratio, r s, is introduced, together with a modified gas FVF. Sec. 7.4 covers the application of blackoil PVT properties to wellrate deliverability and materialbalance calculations. Sec. 7.5 discusses alternative blackoil PVT formulations, including the partialdensity approach. And finally, Sec. 7.6 briefly reviews some limited compositional formulations that are used in the simulation of gasinjection processes. 7.2 Traditional BlackĆOil Formulation It was already clear in the 1920’s that the engineering of oil reservoirs required knowledge of how much gas was dissolved in the oil at reservoir conditions and how much the oil would shrink when it was brought to the surface. It was also recognized that free gas at reservoir conditions would expand up to several hundred times when brought to surface conditions. Engineering quantities were needed to relate surface volumes to reservoir volumes and vice versa. Three properties evolved to serve this purpose: solution gas/oil ratio, R s ; oil FVF, B o; and gas FVF, B g. These properties are defined, respectively, by Rs +
volume of surface gas dissolved in reservoir oil , volume of stocktank oil from reservoir oil . . . . . . . . . . . . . . . . . . . . . (7.1a)
Bo +
volume of reservoir oil , volume of stocktank oil from reservoir oil . . . . . . . . . . . . . . . . . . . . (7.1b)
BLACKOIL PVT FORMULATIONS
and B g +
volume of reservoir gas . volume of surface gas from reservoir gas . . . . . . . . . . . . . . . . . . . . . (7.1c)
These three properties constitute the traditional blackoil PVT formulation, which has the following assumptions. 1. Reservoir oil consists of two surface “components,” stocktank oil and surface (total separator) gas. 2. Reservoir gas does not yield liquids when brought to the surface. 3. Surface gas released from the reservoir oil has the same properties as the reservoir gas. 4. Properties of stocktank oil and surface gas do not change during depletion of a reservoir. Fig. 7.11 illustrates schematically the relation between reservoir phases and surface components. This simplified PVT formulation is still the standard for most petroleum engineering applications. The traditional blackoil quantities, R s, B o, and B g, can be estimated with the correlations in Chap. 3 or can be calculated from differentialliberation and multistageseparator data (Chap. 6). The validity of the traditional blackoil PVT formulation depends primarily on the reservoiroil volatility. Any reservoir oil with less than [750 scf/STB initial solution gas/oil ratio can probably be treated with the traditional PVT formulation. Also, any oil reservoir that produces at higher than its bubblepoint pressure during most of the reservoir’s productive life can be treated with this formulation (e.g., strong waterdrive, gascapdrive, or waterflooded reservoirs). Volatile oils usually have an initial gas/oil ratio (GOR) greater than [1,000 scf/STB and an initial stocktankoil gravity u35°API. The following are the two main depletion characteristics of a volatileoil reservoir: (1) varying surface gravity of produced stocktank oil and (2) the percentage of produced stocktank oil coming from the flowing reservoir gas increases from zero initially to a significant percentage at depletion (potentially u90%). For most petroleum engineering calculations, the variation in stocktankoil gravity can be neglected. However, neglecting the surface oil that is produced from flowing reservoir gas may cause gross underestimation of the ultimate stocktankoil recovery. Fig. 7.2 shows the actual depletion characteristics of a volatileoil reservoir, where reservoir pressure decreases from 5,000 to 1,800 psia, produced surfaceoil gravity increases from 44 to 62°API, and producing GOR increases from 3,800 to 22,000 scf/STB. A good check of the traditional blackoil formulation is to compare reservoir materialbalance performance determined on the basis of standard blackoil PVT properties (e.g., a material bal1
Fig. 7.1—Schematic of traditional blackoil formulation relating reservoir phases to surface components.
ance2) with depletion characteristics calculated from a compositional material balance. The traditional blackoil formulation should not be used if the stocktankoil recoveries differ significantly (see Figs. 7.3 and 7.4). Fig. 7.5 is another plot that indicates the relative volatility of an oil. Differentialliberation relative oil volumes are plotted as shrinkage ( 1 * B odńB odb) vs. normalized pressure ( pńp b ), which indicates whether the shrinkage is rapid or slow. A curve that drops rapidly indicates a highly volatile oil. A “black” oil will tend to plot above the solid “unitslope” line shown in Fig. 7.5. 7.3 Modified BlackĆOil (MBO) Formulation Several modifications of the traditional blackoil formulation have been introduced to account for the surfaceliquid content in reservoir gases. Most formulations introduce an additional PVT property, the solution oil/gas ratio, r s, and a modified definition of the gas FVF. Fig. 7.6 shows schematically the relation between reservoir phases and surface components in the MBO formulation. Because this chapter gives a detailed description of the MBO PVT formulation, we have introduced a more concise nomenclature that distinguishes between reservoir and surface phases. Traditionally, we use the subscript o to represent both reservoir oil and stocktank oil and g to represent both reservoir gas and surface separator
Cumulative Surface Oil Produced, fraction
Fig. 7.3—Average reservoir pressure and producing GOR vs. cumulative oil for nearcritical oil Reservoir NS2; comparison of traditional and MBO formulations. 2
Fig. 7.2—Depletion characteristics of a volatileoil reservoir (adapted from Ref. 1).
Pressure, psia
Fig. 7.4—GOR’s vs. pressure for nearcritical Reservoir NS2 and volatileoil Reservoir NS3. PHASE BEHAVIOR
0 0.1
0.2 0.3 0.4
0.5 0.6 0.7
0.8
Fig. 7.6—Schematic showing relation between reservoir phases and surface phases (components) for MBO formulation.
0.9 1.0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Fig. 7.5—Oil shrinkage plot used to evaluate volatility of a reservoir oil (from Ref. 3).
gas. In this chapter, we use the following subscripts to distinguish between reservoir and surface phases: o+reservoiroil phase at p and T, g+reservoir gas phase at p and T, oo+stocktank oil from reservoir oil, go+surface gas from reservoir oil (“solution” gas), og+stocktank oil (condensate) from reservoir gas, gg+surface gas from reservoir gas, o+total stocktank oil, and g+total surface gas, where the overbar indicates a surfacephase (component). To avoid confusion, the standard term g w is used to represent the wellstream gravity of a reservoir gas (instead of g g). The four MBO PVT parameters, oil FVF, solution gas/oil ratio, drygas FVF, and solution oil/gas ratio are defined respectively as Bo +
Vo , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.2a) V oo
Rs +
V go , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.2b) V oo
B gd +
where Vo +reservoiroil volume, V oo +volume of stocktank oil produced from the reservoir oil, V go +volume of surface gas produced from the reservoir oil, V g +reservoir gas volume, V gg +volume of surface gas produced from the reservoir gas, and V og +stocktank oil (condensate) produced from the reservoir gas. Fig. 7.7 outlines one procedure for determining MBO properties. The equilibriumgas and oil phases from a depletion experiment [constant composition expansion, constant volume depletion (CVD), or differential liberation] are passed separately through a multistage separator. The MBO properties are calculated according to the definitions given in Eq. 7.2. Figs. 7.8 through 7.11 show MBO properties calculated with the WhitsonTorp4 method for the gas condensate, nearcritical oil, and volatile oils in Table 7.1. Refs. 5 through 11 provide alternative methods. 7.3.1 Surface Gravities. When a well produces both reservoir oil and gas, the composite surface gravities, g o and g g, will be an average of the surface gravities of the two reservoir phases, g oo and g go for the reservoir oil and g og and g gg for the reservoir gas. The average gas gravity is given by g g + F gg g gg ) ǒ1 * F ggǓ g go ,
. . . . . . . . . . . . . . . . . . . . (7.3)
Vg , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.2c) V gg
and r s +
V og , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.2d) V gg
Fig. 7.7—Schematic of the WhitsonTorp4 method for calculating MBO properties on the basis of depletion experiments and multistage separation. BLACKOIL PVT FORMULATIONS
Fig. 7.8—Solution GOR, Rs , vs. pressure for volatile reservoir Fluids NS1, NS2, and NS3 calculated with the WhitsonTorp4 method. 3
Fig. 7.9—Oil FVF, Bo , vs. pressure for volatile reservoir Fluids NS1, NS2, and NS3 calculated with the WhitsonTorp4 method.
Fig. 7.10—Solution OGR, rs , vs. pressure for volatile reservoir Fluids NS1 and NS3 calculated with the WhitsonTorp4 method.
where F gg +fraction of total surface gas produced from the reservoir gas.
Clearly, the assumption that g oo + g og + g o makes predicting the variation in overall stocktankoil gravity during depletion impossible. As Fig. 7.2 shows, this variation can be significant.
F gg +
V gg 1 * Rs ń Rp V gg + + . . . . . . . . . . . . (7.4) V gg ) V go Vg 1 * R sr s
The average stocktankoil gravity is given by g o + F oo g oo ) (1 * F oo ) g og , . . . . . . . . . . . . . . . . . . . (7.5) where F oo +fraction of total stocktank oil that comes from the reservoir oil. F oo +
1 * rs Rp V oo V oo + + , V oo ) V og Vo 1 * R sr s
. . . . . . . . . . . . (7.6)
with R p and R s in scf/STB and r s in STB/scf in Eqs. 7.4 and 7.6. Surface gravities g oo, g og, g go, and g gg are determined separately for the reservoiroil and reservoirgas phases from multistageseparator calculations. Because the compositions of reservoir oil and gas change during pressure depletion, the surface gravities also vary with pressure. The variation in g og and g go in Figs. 7.12 and 7.13 is typical of volatileoil and gascondensate mixtures. On the other hand, g oo and g gg usually do not vary significantly with pressure. Although the variation in surface gravities should be considered in engineering calculations, most MBO formulations assume that g oo + g og + g o + constant and g go + g gg + g g + constant.
. . . . . . . . . . . . . . . . . . . (7.7)
Fig. 7.11—Inverse drygas FVF, bgd (+1/Bgd ), vs. pressure for GasCondensate NS1 calculated with the WhitsonTorp4 method.
TABLE 7.1—SOLUTION OIL/GAS RATIO CALCULATED FROM FIELD STOCKTANKOIL GRAVITY COMPARED WITH EOSCALCULATED VALUES rs (STB/MMscf) Test Date
pR (psia)
Rp (scf/STB)
Rs ( scf/STB)
Bubblepoint
5,555
1,500
January 1979
4,455
June 1980
go
EOS g oo
EOS g og
1,500
0.8430
0.843
0.7595
2,215
1,006
0.8353
0.843
0.7467
62
61
3,685
3,840
768
0.8289
0.843
0.7401
43
44
November 1983
3,105
7,480
615
0.8189
0.843
0.7356
32
34
May 1987
2,683
9,480
514
0.8146
0.843
0.7325
28
29
From g o
EOS 100
Note: WhitsonTorp4 method used to calculate R s, g oo, g og, and r s in last column. g oo does not change appreciably with pressure and is therefore assumed constant.
4
PHASE BEHAVIOR
Fig. 7.12—Surfacegas gravities vs. pressure during depletion.
Fig. 7.13—Surfaceoil gravities vs. pressure during depletion.
Because the constantgravity assumption is widely used, it should be considered when determining the MBO properties R s, B o, B gd, and r s. For example, Coats8 gives a procedure for determining MBO properties of a gas condensate where the original mixture is first passed through a separator to determine the surface gravities; these gravities are assumed to be constant. A depletion experiment is then simulated with an equation of state (EOS), and the equilibrium gas from each depletion stage is passed through a separator to determine r s at the particular pressure. With constant surface gravities and r s as a function of pressure, B gd, B o, and R s, are determined so that reservoiroil and gas densities and the oil relative volumes from the depletion experiment are honored. Surfaceoil and gas gravities are used in reservoir simulators to convert B o, R s, B gd, and r s to reservoiroil and gas densities.
A drygas FVF, B gd (defined as the volume of reservoir gas divided by the volume of surface gas resulting after separation of the reservoir gas), is used for the MBO formulation.
òo + and ò g
62.4g oo ) 0.0136g go R s Bo
0.0764g gg ) 350g og r s + . . . . . . . . . . . . . . . . (7.8) B gd
Accurate phase densities can be important for reservoir processes where gravity affects the recovery mechanism (e.g., gravity drainage in naturally fractured reservoirs). Therefore, manual checking of MBO properties and surface gravities used as input for reservoir simulation is recommended to ensure that the reservoiroil and gas densities are calculated accurately. 7.3.2 Gas FVF. The traditional definition of gas FVF assumes that the number of moles of gas at the surface equals the number of moles of gas at reservoir conditions. This obviously is not correct if the reservoir gas yields condensate at the surface. The definition is still used, however, for liquidyielding reservoir gases and is called the “wet”gas FVF, B gw. The surface volume is a hypothetical wetgas volume consisting of the “dry”surfacegas volume and the surface condensate converted to an equivalent surfacegas volume. Vg . B gw + V gw
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.9)
With V g + n g ZRTńp and V gw + n g RT scńp sc, B gw is simply given by the traditional equation for gas FVF. B gw +
p sc ZT + 0.02827 ZT p , T sc p
. . . . . . . . . . . . . . . . . . (7.10)
where B gw is in ft3/scf, T is in °R, and p is in psia. BLACKOIL PVT FORMULATIONS
B gd +
Vg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.11) V gg
With V g + n g ZRTńp and V gg + n gg RT scńp sc, the drygas FVF can be written B gd +
ZT p sc ZT (1 ) C og r s) + 0.02827 p (1 ) C og r s) T sc p
+ B gw(1 ) C og r s),
. . . . . . . . . . . . . . . . . . . . . . . (7.12)
where r s is in STB/scf, B gd and B gw are in ft3/scf, T is in °R, and p is in psia. C og is a conversion from surfaceoil volume in STB to an “equivalent” surface gas in scf. C og + 379
ǒlbmscfmol Ǔ
+ 133, 000
ǒ
5.615
ft Ǔ ǒSTB 3
62.4
ǒ
Ǔ
g og lbm mol M og ft 3
Ǔ
g og scf . . . . . . . . . . . . . . . . . . . . . . . (7.13) M og STB
If condensate molecular weight, M og, is not measured, it can be estimated with the Cragoe12 correlation, Mo +
6, 084 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.14) g API * 5.9
The term (1 ) C og r s) *1 represents the mole fraction of reservoir gas that becomes dry surface gas after separation and usually ranges from 0.85 for rich gases to 1.0 for dry gases. Fig. 7.14 shows the behavior of the ratio as a function of pressure during depletion of a gas condensate and a volatile oil. 7.3.3 Solution Oil/Gas Ratio. The following simplified relation can be used to calculate r s in terms of reservoirgas specific gravity, g w. rs +
g w * g gg . 4, 600 g og * C og g w
. . . . . . . . . . . . . . . . . . . . . . (7.15)
gw is reported as a function of pressure in the differentialliberation experiment and is readily calculated from reported compositions in a CVD experiment. Assuming that g gg + g g and g og + g o , surface gravities usually are taken from a multistage separation of the original reservoir mixture and assumed constant throughout depletion. On the basis of field production data, r s can be calculated in terms of the actual measured stocktankoil gravity, g o. 5
Moles of reservoir oil and gas, respectively, in lbm mol are n o + n oo ) n go and n g + n og ) n gg , . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.21) V o F ooC oo , 379
where n oo +
V o F oo R s
n go +
379
n og +
,
V o ǒR p * R s F oo Ǔr sC og 379 V o ǒR p * R s F oo Ǔ
and n gg +
379
.
,
. . . . . . . . . . . . . . . . . . . . (7.22)
This yields no +
Fig. 7.14—Fraction of reservoir gas that becomes “dry” surface gas vs. pressure during depletion of a gas condensate and a volatile oil.
rs +
g o * g oo . . . . . . . . . . . . . . . (7.16) R sǒ g o * g ogǓ * R p ǒg oo * g ogǓ
Table 7.1 compares r s values from this relation (determined with field data from a volatileoil reservoir) with r s from EOS calculations. 7.3.4 Compositional Information. Engineering calculations based on blackoil properties actually contain more compositional information than is commonly used. Given the compositions of stocktank oil and separator gases, we can calculate oil and gas compositions (and K values) at reservoir conditions using blackoil PVT properties. Also, wellstream composition can be calculated from the producing GOR. To develop the compositional relations, we use a basis of V o stocktank barrels of total stocktank oil. Volume of reservoiroil and gas phases, respectively, is V o + 5.615 V o F oo B o and V g + V o B gd ǒR p * R s F ooǓ ,
. . . . . . . . . . . . . . . . . . . (7.17)
with V o and V g in ft3, R p and R s in scf/STB, B o in bbl/STB, and B gd in ft3/scf. F oo is the fraction of total stocktank oil that comes from the reservoir oil (Eq. 7.4). Mass of reservoiroil and gas phases, respectively, in lbm is m o + m oo ) m go and m g + m og ) m gg , . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.18) where m oo + 350 V o F oog oo ,
m og + 350V o ǒR p * R s F oo Ǔ g og , and m gg + 0.076V o ǒR p * R s F oo Ǔ g gg . . . . . . . . . . . . . . . (7.19)
379
,
. . . . . . . . . . . . (7.23)
with C oo and C og given by g oo M oo g og and C og + 133, 000 . M og C oo + 133, 000
. . . . . . . . . . . . . . . . . . . . . . . . (7.24)
On the basis of these relations, the mole fractions of surface components in the reservoir oil are xo +
n oo 1 + ǒ1 ) R s ń C oo Ǔ no n go
and x g +
no
+ 1 * x o, . . . . . . . . . . . . . . . . . . . . . . . . . (7.25)
and the mole fractions of surface components in the reservoir gas are yo +
n og ng
+
n gg
and y g +
ng
1 1 ) ǒr s C og Ǔ
*1
+ 1 * y o, . . . . . . . . . . . . . . . . . . . . . . . . . (7.26)
with K values K o + y ońx o and K g + y gńx g . Strictly speaking, Components o and g are not the same “species” and K values cannot be interpreted physically unless (1) the properties of surface oils from reservoir gas and oil are equal and constant and (2) the surface gases from reservoir gas and oil are equal and constant. The mole fraction of the wellstream that comes from the reservoir gas is F g + n gń(n g ) n o); therefore,
ƪ
F oo(C oo ) R s) Fg + 1 ) Ǔ (1 * F oo)ǒC og ) r *1 s
yi +
ƫ
*1
,
. . . . . . . . . . . (7.27)
y ggi ) ǒC og r sǓ x ogi 1 ) C og r s
and x i +
This yields m o + V o F oo ǒ350 g oo ) 0.076 R s g goǓ
6
V o ǒR p * R s F oo Ǔǒ1 ) r s C ogǓ
and n g +
with C oo, C og, and R s in scf/STB and r s in STB/scf. Compositions of reservoir oil, x i, and reservoir gas, y i, can be calculated from blackoil properties R s, r s, and surface properties by
m go + 0.076 V o F oo R sg go ,
and m g + V o ǒR p * R s F ooǓǒ350 g og r s ) 0.076 g gg Ǔ .
V o F ooǒ C oo ) R sǓ 379
. . . (7.20)
y goi ) ǒC oo ńR sǓ x ooi 1 ) C oo ńR s
, . . . . . . . . . . . . . . . . . . . . (7.28)
where y ggi +average composition of surface gases produced from the reservoir gas; x ogi +composition of surface oil produced from the reservoir gas; y goi +average composition of surface gases proPHASE BEHAVIOR
TABLE 7.2—EOSCALCULATED SEPARATORGAS AND –OIL COMPOSITIONS FROM THREESTAGE SEPARATION OF ORIGINAL DEWPOINT GAS AND EOSCALCULATED EQUILIBRIUM OIL Reservoir Gas
Reservoir Oil
Component
y sp1
y sp2
y sp3
y gg
x og
y go
x oo
CO2
0.026092
0.030059
0.036539
0.026388
0.000588
0.027475
0.000627
0.003265
6.12 10*7
10*7
N2
0.003552
0.002154
0.000362
0.003460
C1
0.827710
0.809814
0.389891
0.816791
0.002079
0.809754
0.002103
C2
0.083029
0.099069
0.209316
0.086288
0.006739
0.090387
0.006730
C3
0.033261
0.036388
0.183444
0.036976
0.022803
0.039307
0.022010
5.94
iC4
0.005535
0.005410
0.039898
0.006376
0.013315
0.006609
0.012297
nC4
0.010249
0.009582
0.077103
0.011882
0.036997
0.012158
0.033498
iC5
0.003145
0.002559
0.022571
0.003616
0.030334
0.003486
0.026016
nC5
0.002939
0.002287
0.020158
0.003355
0.036413
0.003183
0.030772
C6
0.002425
0.001577
0.012855
0.002673
0.081629
0.002398
0.066496
F1
0.001671
0.000953
0.007116
0.001798
0.135151
0.001612
0.111360
F2
0.000380
0.000141
0.000739
3.87 10*4
0.252945
0.000353
0.221341
F3
6.34 10*6
1.03 10*6
2.63 10*6
6.20 10*6
0.223155
6.30 10*6
0.230727
F4
7.62 10*9
3.40 10*10
2.76 10*10
7.37 10*9
0.120536
8.91 10*9
0.162246
10*15
10*16
4.10
F5
10*13
2.90
4.50
3.93
10*13
0.037312
5.90
10*13
0.073772
TABLE 7.3—RESERVOIR EQUILIBRIUM COMPOSITIONS CALCULATED FROM EOS AND FROM MBO PVT PROPERTIES WITH SURFACEGAS AND OIL COMPOSITIONS Dewpoint* y
Reservoir Pressure**
x
y
x
Component
Feed
EOS
EOS
MBO
EOS
MBO
CO2
0.0237
0.0237
0.0245
0.0251
0.0206
0.0189
N2
0.0031
0.0031
0.0034
0.0033
0.0018
0.0022
C1
0.7319
0.7319
0.7817
0.7774
0.5316
0.5517
C2
0.0780
0.0780
0.0791
0.0824
0.0737
0.0637
C3
0.0355
0.0355
0.0344
0.0363
0.0401
0.0338
iC4
0.0071
0.0071
0.0066
0.0067
0.0090
0.0084
nC4
0.0145
0.0145
0.0133
0.0131
0.0194
0.0190
iC5
0.0064
0.0064
0.0056
0.0049
0.0097
0.0107
nC5
0.0068
0.0068
0.0058
0.0050
0.0106
0.0120
C6
0.0109
0.0109
0.0088
0.0065
0.0194
0.0229 0.0367
F1
0.0157
0.0157
0.0115
0.0082
0.0325
F2
0.0267
0.0267
0.0158
0.0126
0.0704
0.0709
F3
0.0233
0.0233
0.0081
0.0108
0.0841
0.0737
F4
0.0126
0.0126
0.0015
0.0058
0.0573
0.0518
F5
0.0039
0.0039
0.0001
0.0018
0.0196
0.0236
C7+
0.0821
0.1302
0.0369
0.0393
0.2639
0.2567
*6,750 psia. **4,315 psia.
duced from the reservoir oil; x ooi +composition of surface oil produced from the reservoir oil; and C oo, C og, and R s are in scf/STB and r s is in STB/scf. Average surfacegas compositions y ggi and y goi are calculated separately with the relations
N sp
ȍǒy y ggi +
ggi
Ǔ ńǒ r sǓ j j
j+1 N sp
ȍǒ1ń r Ǔ
,
. . . . . . . . . . . . . . . . . . . . . . . . . . (7.29)
s j
j+1 N sp
ȍǒy y goi +
goi
Ǔ ǒ R sǓ j j
j+1 N sp
ȍǒ R Ǔ
s j
j+1
BLACKOIL PVT FORMULATIONS
where the subscript j indicates the separator stage. Stage GOR’s and OGR’s, (R s) j and (r s) j , respectively, are based on stocktank volumes. The four surface compositions (and gravities) are, in principle, functions of pressure. However, the average separatorgas compositions from reservoir oil and from reservoir gas may be similar, and 7
Fig. 7.15—Calculated compositions for reservoir gas based on MBO properties and surfacecomponent compositions; comparison with EOS compositions.
y ggi + y goi + constant is a reasonable assumption (as is g gg + g go + constant). These compositions are readily determined from a multistage flash of the original reservoir mixture (see Table 7.2). Table 7.3 and Figs. 7.15 through 7.17 show calculated reservoirphase compositions based on Eq. 7.26 for a gascondensate mixture. K values are also calculated (K i + y ińx i) and compared with EOS results for a simulated CVD experiment (Fig. 7.18). Wellstream composition, z i, can be calculated from reservoir phase compositions y i and x i. z i + y i F g ) x i ǒ1 * F gǓ ,
. . . . . . . . . . . . . . . . . . . . . . (7.30)
where F g is given by Eq. 7.25 in terms of producing GOR, R p (through the quantity F oo). Note that values of R s and r s used to calculate F g , y i , and x i must be evaluated at the same pressure. 7.4 Applications of MBO Formulation The MBO PVT approach has been limited mainly to reservoir simulation, although some applications have been reported in welltest
Fig. 7.16—Calculated methane mole fractions for reservoir oil and gas based on MBO properties and surfacecomponent compositions; comparison with EOS compositions.
analysis, inflow performance, and reservoir material balance. Multiphase flow in pipe is another obvious application. To aid in the use of MBO properties for volatile reservoir fluids, we present several engineering equations in terms of MBO properties. 7.4.1 Rate Equations (IPR)—Traditional BlackOil PVT. Inflowperformance relations (IPR’s) give the relation between total surface rates, q o and q g ; wellbore flowing pressure, p wf ; and average reservoir pressure, p R. For example, consider the radialflow equation for an undersaturated oil well.13 q o + q oo +
khǒp R * p wfǓ 141.2 m o B oƪlnǒr eńr wǓ * 0.75 ) sƫ
, . . . . . (7.31)
where q o is in STB/D, k +absolute permeability at irreducible water saturation, md; h +total reservoir thickness, ft; m o +oil viscosity, cp; B o +oil FVF, bbl/STB; r e +outer drainage radius, ft; r w +actual wellbore radius, ft; and s +total skin factor.
Modified BlackOil Properties EOS EOS EOS
Fig. 7.17—Calculated compositions for reservoir oil based on MBO properties and surfacecomponent compositions; comparison with EOS compositions. 8
Fig. 7.18—Calculated K values for reservoir oil and gas based on MBO properties and surface component compositions; comparison with EOS compositions. PHASE BEHAVIOR
The appropriate equations to calculate rates in the production system are14,15 q o + q oo ) q og +
kh 141.2 ƪlnǒr eńr wǓ * 0.75 ) sƫ
ŕ ǒmk B ) 5.615 mk Br Ǔdp pR
rg s
ro
o
o
gd
g
p wf
and q g + q go ) q gg +
kh 141.2 ƪlnǒr eńr wǓ * 0.75 ) sƫ
ŕ ǒkm BR ) 5.615 m kB Ǔdp , pR
ro
s
o
o
rg
g
. . . . . . . . . . . . (7.36)
gd
p wf
with q o in STB/D and q g in scf/D. The liquid and vapor rates in the tubing or reservoir are given by q o + q o F oo B o and q g + q o ǒR p * R s F oo Ǔ B gd ,
. . . . . . . . . . . . . . . . . . . (7.37)
where F oo +fraction of total surface oil coming from the flowing liquid (Eq. 7.6). F oo +
Fig. 7.19—Fraction of wellbore rate from reservoir oil, fraction of surface oil from reservoir oil, GOR, and pwf during depletion of a volatileoil reservoir.
For saturatedoil wells producing both reservoir oil and gas, the oilrate equation can be written terms of traditional blackoil PVT properties (r s + 0) as pR
q o + q oo +
kh
141.2 ƪlnǒr eńr wǓ * 0.75 ) sƫ
ŕ mk B dp. ro
o
o
1 * Rp rs q oo + . qo 1 * Rs rs
. . . . . . . . . . . . . . . . . . . . . . . (7.38)
PVT properties used to calculate q o and q g are evaluated at the pressure and temperature in the reservoir or the production tubing. Evaluation of the integrals in Eq. 7.36 is not straightforward. In fact, using only one of the two rate equations would be logical, depending on which phase was dominant. For a predominantly oil system, the oil rate in Eq. 7.36 should be used for q o and the gas rate could be calculated from the total producing GOR. Likewise, for a predominantly gas system, the gas rate in Eq. 7.36 should be used for q g and the oil rate can be calculated from the total producing GOR. Producing GOR would be available from materialbalance calculations. The volumetric fraction of reservoir fluids flowing as an oil phase at wellbore conditions is
p wf
. . . . . . . . . . . . . . . . . . . . (7.32)
qo Bo qo + + qo ) qg q o B o ) q g B gd
Total gas rate from a saturatedoil well is the product of the oil rate and total producing GOR. qg + qo Rp ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.33)
where q g is in scf/D and R p usually is available from materialbalance calculations. The rate of the oil phase flowing anywhere in the tubing or reservoir can be calculated as qo + qo Bo ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.34)
with q o in B/D and B o evaluated at a specific pressure and temperature. The flow rate of free gas at the same conditions is calculated from q g + q oǒR p * R sǓ
B gd , 5.615
. . . . . . . . . . . . . . . . . . . . . . (7.35)
with q g in B/D, q o in STB/D, R s and R p in scf/STB, and B gd in ft3/scf. R s and B gd are evaluated at the same pressure and temperature. 7.4.2 IPR—MBO PVT. Eqs. 7.32 and 7.33 are based on the traditional blackoil PVT formulation where reservoir gas is assumed to have no liquid content. For volatile reservoir fluids, the surface oil consists of surface oil from the flowing liquid and condensed from the flowing vapor. Likewise, the surfacegas rate consists of surface gas from the flowing vapor and released from the flowing liquid. BLACKOIL PVT FORMULATIONS
ƪ
1)
ǒR p * Rs F ooǓ Bgd 5.615 F ooB o
ƫ
*1
,
. . . . . . . . . . . . . . . . . . . . (7.39) where B o, R s, B gd, and r s are evaluated at the wellbore flowing pressure, p wf . For a volatileoil reservoir, the oil fraction will drop to less than 50% during depletion (see Fig. 7.19), marking the point when the gas phase becomes the dominant flowing phase. The relative amounts of reservoir oil and gas flowing at the wellbore should be considered in the interpretation of well tests and application of IPR’s. 7.4.3 Reservoir Material Balance—MBO PVT. Reservoir materialbalance relations for solutiongasdrive and drygas reservoirs are well known and widely used. Borthne16 presents a reservoir material balance based on MBO properties that can be used for black oils, volatile oils, and gas condensates. Modifications to the material balance that account for connate water with dissolved gas, water influx, and other such factors can be included readily. The basis of calculation is 1 bbl reservoir bulk volume. The conservationofmass equations for a singlecell material balance yields the following difference equations for reservoiroil and gas phases during a timestep Dt k + t k * t k*1 with a change in average pressure from ( p R) k*1 to ( p R) k . ǒ A oǓ k * ǒ A oǓ k*1) D N p + 0 and ǒ A gǓ k * ǒ A gǓ k*1 ) DG p + 0 ,
. . . . . . . . . . . . . . . . (7.40) 9
TABLE 7.4—MBO PROPERTIES FOR GAS CONDENSATE NS1 Pressure (psia)
Bo (bbl/STB)
Rs (scf/STB)
g oo
Bgd (ft3/scf)
rs (STB/MMscf)
g og
g gg
6,748.2
2.6490
3,005
0.7837
6,514.7
2.4693
2,662
0.7849
0.7155
0.004244
181.0
0.7689
0.7114
0.8958
0.7171
0.004205
158.2
0.7647
0.7110
6,014.7
2.2241
2,180
0.9051
0.7859
0.7208
0.004226
125.7
0.7575
0.7107
5,514.7
2.0495
0.9194
1,829
0.7859
0.7251
0.004333
102.4
0.7516
0.7106
0.9306
4,314.7 3,114.7
1.7427
1,211
0.7845
0.7397
0.004940
64.0
0.7399
0.7114
0.9516
1.5116
757
0.7832
0.7629
0.006371
39.3
0.7298
0.7139
0.9677
2,114.7
1.3525
456
0.7829
0.7927
0.009179
26.2
0.7224
0.7181
0.9772
1,214.7
1.2277
232
0.7843
0.8324
0.016214
21.2
0.7151
0.7268
0.9808
714.7
1.1651
124
0.7864
0.8663
0.028276
24.7
0.7088
0.7386
0.9771
g go
F gg
Water g+1.
where D N p and DG p +incremental quantities of total surface oil and total surface gas, respectively, produced during the timestep; Ao + f
ƪBS ) 5.615(1 *BS * S )r g ƫ o
w
o
gd
o
* s o
ƪ
e + ǒA gǓ k * ǒA gǓ k*1) DG p. . . . . . . . . . . . . . . . . . . . . (7.45)
ƫ
S o R s g *g 5.615(1 * S w * S o) and A g + f ; ) B gd Bo
. . . . . (7.41)
g og and g *o + g oo
g go and g *g + g . gg
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.42)
In Eqs. 7.40 through 7.42, D N p and A o are in STB/bbl, DG p and A g are in scf/bbl, R s is in scf/STB, B o is in bbl/STB, r s is in STB/scf, and B gd is in ft3/scf. Other quantities used in the materialbalance procedure are E o + 1 ) 5.615r s g *o E g + R s g *g ) 5.615 Rp + and
k rg m o B o k ro m g B gd
,
k rg m o B o , k ro m g B gd
ǒ S oǓ k +
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.43)
ǒ A oǓ k*1 * ǒDN pǓ k * ƪf(1 * S wi)r s g *oń B gdƫ
ƪfǒ1ńB o * r s g *o ń B gdǓƫ
k
.
^
ƪfǒR s g*gń Bo * 1ńBgdǓƫ
k
. . . . . (7.46)
k
4. Calculate (k rgńk ro) k from (S o) k . 5. Calculate (A o) k , (A g) k , (E o) k , and (E g) k . 6. Calculate DN po , incremental surface oil produced from reservoir oil, where D N po + D G pńE g and E g + 0.5[(E g) k ) (E g) k*1]. 7. Calculate D N p, incremental total surface oil produced, where D N po + D N pńE o and E o + 0.5[(E o) k ) (E o) k*1]. 8. Calculate the materialbalance error, . . . . . . . . . . . . . . . . . . . (7.47)
k
7.5 PartialĆDensity Formulation In 1965, Kniazeff and Naville7 presented the first approach to modeling gascondensate and volatileoil systems with a simplified compositional PVT formulation. They introduced four “partial densities” as PVT parameters in a radial, 1D numerical model to study the inflow performance of a Middle East gas–condensate field. The flow and conservation equations were written in terms of mass, where surface volumes were not considered directly. Partial densities, ò p , are defined as ò pij +
4. Calculate (k rgńk ro) k from (S o) k . 5. Calculate (A o) k , (A g) k , (E o) k , and (E g) k . 6. Calculate D N po , incremental surface oil produced from reservoir oil, where D N po + D N pńE o and E o + 0.5[(E o) k ) (E o) k*1].
m ij , Vj
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.48)
where m ij +surface mass of Component i in Phase j; V j +reservoir volume of Phase j; i+g and o+surface gas and oil, respectively; and j+g and o+reservoir gas and oil, respectively. The four partial densities, ò p , can be expressed as composite terms of MBO properties.
. . . . . . . . . . . . . . . . . . . . (7.44)
10
ǒA gǓ k*1 * ǒDG pǓ k * ƪfǒ1 * S wiǓ ń B gdƫ
9. If e is not sufficiently small, assume a new pressure ( p R) k and redo Steps 2 through 8.
with m o and m g in cp; R s, R p, and E g in scf/STB; r s in STB/scf; E o in STB/STB; B o in bbl/STB; and B gd in ft3/scf. PVT properties and porosity are (g *g ) k functions of pressure only. Application of these relations is outlined for an oil and a gascondensate reservoir. Oil Reservoir. 1. Specify (D N p) k , the total surface oil produced in STB/bbl of bulk volume. 2. Assume ( p R) k and calculate PVT properties and porosity: (B o) k , (R s) k , ( m o) k , (g *o ) k , (B gd) k , (r s) k , ( m g) k , (g *g ) k, and (f) k . 3. Calculate oil saturation ( S o) k from Eqs. 7.39 through 7.41. ǒ S oǓ k +
9. If e is not sufficiently small, assume a new pressure ( p R) k and redo Steps 2 through 8. GasCondensate Reservoir. 1. Specify (DG p) k, total surface gas produced in scf/bbl of bulk volume. 2. Assume ( p R) k and calculate PVT properties and porosity: (B o) k , (R s) k , ( m o) k , (g *o ) k, (B gd) k , (r s) k , ( m g) k , (g *g ) k, and (f) k . 3. Calculate oil saturation ( S o) k from Eqs. 7.39 through 7.41.
e + (A o) k * (A o) k*1 ) DN p .
DG p , DN p
k rg + fǒ S oǓ , k ro
7. Calculate DG p, incremental total surface gas produced, where DG p + D N po E g and E g + 0.5[(E g) k ) (E g) k*1]. 8. Calculate the materialbalance error,
òpgg+
0.0763 g gg , B gd
ò pog+
350 g og r s , B gd PHASE BEHAVIOR
Gas Injection Parameter, Gi, Mscf/bbl Oil in Cell
Fig. 7.20—Partial densities vs. pressure for GasCondensate NS1.
òpgo+
0.0136 g go R s , Bo
and ò poo+
62.4 g oo , Bo
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.49)
with ò p in lbm/ft3, B o in bbl/STB, R s in scf/STB, B gd in ft3/scf, and r s in STB/scf. Table 7.4 and Fig. 7.20 show the behavior of partial densities and their relation to MBO properties. From Eq. 7.47, we see that the variation in surface gravities with pressure is included directly in the definitions of the PVT properties. In fact, this is necessary to maintain an exact mass balance. Drohm and Goldthorpe9 and Drohm et al.10,11 indicate that a similar approach can be used for reservoir simulators on the basis of the MBO approach. They correct the MBO parameters with surface densities, which indicates that an exact mass balance can be maintained if the corrected properties ( B *o, R *s , B *gd, and r *s ) are used instead of the original parameters ( B o, R s, B gd, and r s ). B *o +
Bo , 62.4 g oo
R *s + R s
ǒgg Ǔ , go oo
B *gd +
B gd , 62.4 g gg
and r *s + r s
ǒgg Ǔ , og
Gas Injection Parameter, Gi, Mscf/bbl Oil in Cell
Fig. 7.21—Variation in blackoil PVT properties with gasinjection parameter Gi (adapted from Ref. 5).
The complexity of some formulations is disturbing because so many nonphysical quantities are used to correlate compositional effects. With the increasing speed of compositional simulators and the increase in available computing power, it is difficult to justify the effort to develop these highly empirical, pseudoPVT formulations for gasinjection projects where compositional effects are important. If a simplified formulation is used, it should be checked with a compositional formulation. Tang and Zick19 recently proposed and interesting and accurate pseudocompositional model with the computational speed of a blackoil model and the accuracy of an EOS model that is of particular interest in misciblegasinjection simulations. References
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.50)
gg
with densities in lbm/ft3, B o in bbl/STB, R s in scf/STB, B gd in ft3/scf, and r s in STB/scf. Reservoir models based on the DrohmGoldthorpe or the partialdensity approach still do not yield a consistent surfacevolume material balance unless surface gravities are considered pressure dependent. 7.6 Modifications for Gas Injection Cook et al.5 extend the MBO formulation for vaporizinggasinjection processes, where a gasinjection parameter, G i, is defined as the cumulative volume of injection gas entering a grid cell, divided by the gridcell volume. PVT properties B o, R s, B gd, and r s are correlated in tabular form vs. G i (see Fig. 7.21). Lo and Youngren,17 Whitson et al.,18 and others propose other extensions to the MBO formulation. BLACKOIL PVT FORMULATIONS
1. Woods, R.W.: “Case History of Reservoir Performance of a Highly Volatile Type Oil Reservoir,” JPT (October 1955) 156; Trans., AIME, 204. 2. Dake, L.P.: Fundamentals of Reservoir Engineering, Elsevier Scientific Publishing Co., Amsterdam (1978). 3. Cronquist, C.: “Dimensionless PVT Behavior of Gulf Coast Reservoir Oils,” JPT (May 1973) 538. 4. Whitson, C.H. and Torp, S.B.: “Evaluating Constant Volume Depletion Data,” JPT (March 1983) 610; Trans., AIME, 275. 5. Cook, R.E., Jacoby, R.H., and Ramesh, A.B.: “A BetaType Reservoir Simulator for Approximating Compositional Effects During Gas Injection,” SPEJ (October 1974) 471. 6. Spivak, A. and Dixon, T.N.: “Simulation of GasCondensate Reservoirs,” paper SPE 4271 presented at the 1973 SPE Annual Meeting, Houston, 10–12 January. 7. Kniazeff, V.J. and Naville, S.A.: “TwoPhase Flow of Volatile Hydrocarbons,” SPEJ (March 1965) 37; Trans., AIME, 234. 8. Coats, K.H.: “Simulation of GasCondensate Reservoir Performance,” JPT (October 1985) 1870. 11
9. Drohm, J.K. and Goldthorpe, W.H.: “Black Oil PVT Revisited—Use of Pseudocomponent Mass for an Exact Material Balance,” paper SPE 17081 available from SPE, Richardson, Texas (1987). 10. Drohm, J.K., Goldthorpe, W.H., and Trengove, R.: “Enhancing the Evaluation of PVT Data,” paper SPE 17685 presented at the 1988 SPE Offshore Southeast Asia Conference, Singapore, 2–5 February. 11. Drohm, J.K., Trengove, R., and Goldthorpe, W.H.: “On the Quality of Data From Standard GasCondensate PVT Experiments,” paper SPE 17768 presented at the 1988 SPE Gas Technology Symposium, Dallas, 13–15 June. 12. Cragoe, C.S.: “Thermodynamic Properties of Petroleum Products,” U.S. Dept. Commerce, Washington, DC (1929) 97. 13. Golan, M. and Whitson, C.H.: Well Performance, second edition, PrenticeHall Inc., Englewood Cliffs, New Jersey (1986). 14. Fetkovich, M.D. et al.: “Oil and Gas Relative Permeabilities Determined From Rate/Time Performance Data,” paper SPE 15431 presented at the 1986 SPE Annual Technical Conference and Exhibition, New Orleans, 5–8 October. 15. Boe, A., Skjaeveland, S., and Whitson, C.H.: “TwoPhase Pressure Test Analysis,” SPEFE (December 1989) 604; Trans., AIME, 287. 16. Borthne, G.: “Development of a Material Balance and Inflow Performance for Oil and GasCondensate Reservoirs,” MS thesis, U. Trondheim, Norwegian Inst. Technology, Trondheim, Norway (1986).
12
17. Lo, T.S. and Youngren, G.K.: “A New Approach to Limited Compositional Simulation: Direct Solution of the Phase Equilibrium Equations,” SPERE (November 1987) 703; Trans., AIME, 283. 18. Whitson, C.H., da Silva, F.V., and Søreide, I.: “Simplified Compositional Formulation for Modified BlackOil Simulators,” paper SPE 18315 presented at the 1988 SPE Annual Technical Conference and Exhibition, Houston, 2–5 October. 19. Tang, D.E. and Zick, A.A.: “A New Limited Compositional Reservoir Simulator,” paper SPE 25255 presented at the 1993 SPE Symposium on Reservoir Simulation, New Orleans, 28 February–3 March.
SI Metric Conversion Factors °API bbl ft3 °F lbm lbm mol psi
141.5/(131.5)°API) +g/cm3 1.589 873 E*01 +m3 2.831 685 E*02 +m3 (°F*32)/1.8 +°C 4.535 924 E*01 +kg 4.535 924 E*01 +kmol 6.894 757 E)00 +kPa
PHASE BEHAVIOR
Chapter 8
GasĆInjection Processes 8.1 Introduction For the past 50 years, gas injection has been used successfully in both oil and gascondensate reservoirs. Hydrocarbon recoveries have been increased over what could be obtained by primary drive mechanisms and waterflooding. It was recognized early that the phase and volumetric behavior of gas/oil systems during gas injection had an important effect on ultimate recovery efficiency. Recovery efficiency is defined as the product of areal and vertical sweep efficiencies and the microscopic displacement efficiency of the contacted reservoir volume. Fluid properties influence all three components of overall recovery efficiency. 1. Viscosities are found in the definition of mobility ratio, which affects areal and vertical sweep efficiency, including viscous fingering. 2. Phase densities define the degree of gravity segregation, which in turn affects vertical sweep efficiency by gravity bypassing (tonguing) in gravitydominated processes. 3. Interfacial tensions, viscosities, interphase mass transfer (i.e., vaporization and condensation), and miscibility affect the residual oil saturation (ROS) defining microscopic displacement efficiency. Gasinjection processes are designed to enhance the recovery of oil. The first application of gas injection was intended simply to maintain reservoir pressure at a level that would sustain existing production rates. Another purpose for pressure maintenance in gascondensate reservoirs was to avoid low liquids recovery resulting from retrograde condensation. Injection of lean gas consisting mainly of methane or nitrogen can, by vaporization, recover significant quantities of light and intermediate hydrocarbons (C5 through C12) from reservoir oil. Nitrogenrichgas injection can theoretically recover most of the hydrocarbons making up solution gas (C1 through C7). In gascondensate reservoirs, leangas injection can be miscible if reservoir pressure is above the dewpoint; otherwise, lean gas can revaporize liquids that drop out by retrograde condensation, which occurs when reservoir pressure drops below the dewpoint. In oil reservoirs, vaporization at sufficiently high pressure may develop an insitu gas that becomes sufficiently enriched in intermediate components to displace the reservoiroil miscibility; this process is called the vaporizinggas miscible drive.1 Miscibility also can be attained by injecting a gas that is enriched with liquefied petroleum gases (LPG’s)—mainly propane. Through phase equilibrium, the injected gas transfers the LPG’s to the reservoir oil, which is typically deficient in these intermediate components. Repeated contacts with enriched gas develops an oil that may become miscible with the injection gas; this process is traditionally called the enrichedgas, or condensinggas, miscible drive.1 GASINJECTION PROCESSES
Zick2 shows that another mechanism may develop from injection of enriched gas that results in misciblelike recoveries (u95%) without necessarily achieving a miscible condition. The combined condensing/vaporizing mechanism he describes is a process that exhibits a sharp nearmiscible “front.” A condensing mechanism occurs just ahead of the front, and a vaporizing mechanism trails the front. A practical consequence of this mechanism is that a lower enrichment level can be used for the injection gas than would be estimated from the traditional interpretation of the enrichedgas miscible drive process. Miscible displacement also can be achieved by a miscibleslug drive process, where a slug of propanerich mixture is injected and mixes miscibly with the reservoir oil on first contact. After a sufficient volume of slug has been injected [5 to 20% of reservoir pore volume (PV)], a dry gas is injected to drive the slug. The dry gas may be followed by continuous water injection or by a wateralternatinggas (WAG) injection sequence. Since the 1970’s, CO2 flooding has been considered one of the most promising gasinjection processes in the U.S.35 Major investments have been made to transport large quantities of CO2 in pipelines from CO2 reservoirs in Colorado and New Mexico to oil reservoirs in west Texas and Oklahoma and from Mississippi to Louisiana. CO2 flooding has been used successfully in a wide variety of oil reservoirs, with stocktankoil gravities ranging from 15 to 45°API, reservoir temperatures from 80 to 300°F, reservoir pressures from less than 1,000 to more than 4,000 psia, and in both sandstone and carbonate formations that vary in thickness from less than ten to more than several hundred feet. Recovery mechanisms involved with CO2 flooding include oil swelling, oilviscosity reduction, vaporization of intermediate to heavy hydrocarbons (C5 through C30), and development of multicontact miscibility. Other phase behavior exhibited by CO2/oil systems includes asphaltene deposition and threephase [vapor/liquid/liquid (VLL)] behavior in lowtemperature systems. Phase and volumetric behavior are important in both miscible and immiscible CO2 processes. All the gasinjection methods mentioned can be initiated as secondary or tertiary projects (i.e., following, in conjunction with, or as a replacement for a waterflood). The occurrence of large water saturations in tertiary and WAG processes does not appear to influence the role of phase and volumetric behavior on these EOR processes. However, CO2 solubility in water may affect oil recovery if the loss of CO2 to connate and injected water is significant. 1
Fig. 8.2—Phase behavior of the methane/butane/decane ternary, including criticalpressure curves for mixtures of fixed composition as functions of temperature (from Refs. 9 and 10). Fig. 8.1—Phase behavior of ethane/heptane system, including critical locus defining MMP conditions (from Ref. 8).
8.2 Miscibility and Related Phase Behavior Miscible gas displacement typically is characterized by high recoveries in slimtube displacement experiments. These recoveries are usually greater than 90% and somewhat less than the 100% theoretical recovery expected for “firstcontact”miscible displacement. The small ROS (2 to 10% of PV) is an immobile, highly viscous oil consisting mainly of heavy, nonvolatile hydrocarbons. Miscible gas displacement may also cause deposition of a solid asphaltene precipitate that can alter wettability and water injectivity.6,7 As a thermodynamic condition, miscibility is defined as the condition when two fluids are mixed in any proportion and the resulting mixture is a single phase. For example, gasoline and kerosene are miscible at room conditions, whereas stocktank oil and water are clearly immiscible.
rameter C + z C4ń(z C4 ) z C10), the locus of critical pressures indirectly defines the condition of miscibility as a function of temperature. For a specific temperature, Fig. 8.2 gives the composition dependence of critical pressure. Choosing, for example, 280°F and 2,000 psia, the composition corresponding to this critical condition is z C1 + 0.5 and C + 0.85 ( z C4 + 0.42, z C10 + 0.08). At 280°F and 3,000 psia, the critical composition is z C1 + 0.68 and C + 0.65 ( z C4 + 0.21, z C10 + 0.11). Knowing only the critical composition of a ternary system at a specific temperature and pressure does not directly determine whether two mixtures of the three components will be miscible. Graphically, a ternary composition diagram can be used to determine whether two mixtures are firstcontact miscible or whether the two mixtures can develop miscibility. Fig. 8.3 shows the ternary diagram for the meth
8.2.1 Binary Systems. For a binary system, the condition of miscibility is readily defined on a pressure/temperature ( pT) diagram. The dashed line in Fig. 8.1 represents the locus of critical points for all mixtures of ethane and heptane. The criticallocus curve for a binary system will always enclose the twophase region for all possible mixtures of the two components. Thus, for a binary mixture at a specific temperature, the pressure on the criticallocus curve represents the minimum pressure where miscibility can occur independently of overall composition. At all pressures greater than this minimum miscibility pressure (MMP), any mixture of the binary will form a single phase. Fig. 8.1 also shows that temperature increases MMP for a binary mixture at lower temperatures, but the effect reverses at higher temperatures, where MMP decreases with increasing the temperature. 8.2.2 Ternary Systems. The condition of miscibility for a ternary system can also be depicted on a pT diagram. Fig. 8.2 shows the curves defining critical pressure vs. temperature for the ternary system methane/butane/decane (C1/nC4/nC10). For a specific composition defined in terms of mole percent methane z C1 and the pa2
Fig. 8.3—Ternary composition diagram for C1/nC4/nC10 system at 280°F and 2,000 psia (data from Ref. 10). PHASE BEHAVIOR
Fig. 8.4A—Path of developed miscibility by the vaporizinggas miscible drive process for C1/nC4/nC10 system.
Fig. 8.4B—EOS calculated slimtube profiles for the vaporizinggas miscible drive process for C1/nC4/nC10 system (adapted from Ref. 2).
ane/butane/decane system at 280°F and 2,000 psia. The critical composition determined from Fig. 8.2 is shown as the critical point, C. Other compositional data for equilibrium systems at the same condition are plotted on the ternary diagram. These data define the phase envelope enclosing all compositions that will split into two phases when brought to this specific pressure and temperature. The upper
curve of the phase diagram defines the dewpoint curve, while the lower curve defines the bubblepoint curve. The dewpoint and bubblepoint curves join at the critical composition, C. A tieline is a straight line on a ternary diagram joining an equilibriumvapor composition with its equilibriumliquid composition
Fig. 8.5A—EOS calculated slimtube profiles for the enrichedgas miscible drive process for C1/nC4/nC10 system (adapted from Ref. 2).
Fig. 8.5B—Path of developed miscibility by the enrichedgas miscible drive process for C1/nC4/nC10 system.
GASINJECTION PROCESSES
3
(e.g., Line XY). Any system with an overall composition lying on this tieline will split into the same equilibriumliquid and vapor compositions defined by X and Y (e.g., overall compositions ZA and ZB ). Fig. 8.3 shows three tielines inside the phase envelope. Every point on the twophase envelope is connected to another point on the envelope by a tieline. A limiting tieline can be drawn through the critical composition (dashed line in Fig. 8.3). This critical tieline determines whether two mixtures in a ternary system can develop miscibility by a multiplecontact process. Strictly speaking, two fluids are firstcontact miscible if the line connecting the two compositions does not pass through the twophase envelope on a ternary diagram. In Fig. 8.3, the G1/G2, G2/O2, and O1/O2 mixtures are firstcontact miscible and the G1/O1, G1/O2, and G2/O1 mixtures are not. Some systems can develop miscibility by a multiplecontact process. The criterion for developed multicontact miscibility in a ternary system is that the two original mixtures lie on opposite sides of the critical tieline. The following paragraphs describe two methods of developing miscibility in a ternary system. G1 and O2 can develop miscibility by the vaporizinggas miscible drive process. Here, intermediate and heavy components (C4 and C10) are vaporized from the original oil, O2, into the lean gas, G1, making a richer gas that contacts O2 and develops an even richer gas that again contacts O2. Finally, the gas composition approaches Critical Composition C, which is miscible with O2. Fig. 8.4A shows the path of developed miscibility for this process on a ternary diagram. Fig. 8.4B shows simulated slimtube profiles of oil saturation, phase densities, and K values for the vaporizinggas miscible drive of the methane/butane/decane system determined with the PengRobinson11 EOS. G2 and O1 can develop miscibility by the traditional enrichedgas miscible drive process. Here the intermediate component (C4) in the original gas, G2, transfers to oil, O1. This enriched oil is made even richer by new contacts with G2 until the oil is modified so that its composition approaches Critical Composition C. This developed critical “oil” is miscible with G2. Fig. 8.5A shows the path of developed miscibility for this process on a ternary diagram. Fig. 8.5B shows simulated slimtube profiles of oil saturation, phase densities, and K values for enrichedgas miscible drive of the methane/ butane/decane system determined with the PengRobinson EOS. 8.2.3 Pseudoternary Diagrams for Multicomponent Systems. For a true threecomponent system, firstcontact and developed miscibility can be determined uniquely from a ternary diagram at a specific pressure and temperature. Pseudoternary diagrams are also used for multicomponent mixtures, where several components are grouped together and represented at each apex on the ternary diagram. This method is used despite the inherent limitation that multicomponent phase behavior cannot be represented uniquely with a ternary diagram. Strictly speaking, a ternary representation of a multicomponent system is valid only if the relative amounts of all components defining each pseudocomponent remain constant. This condition cannot be satisfied for oil systems, but the graphical representation is still used. Methane, N2, and CO2 are usually treated as the light pseudocomponent in a pseudoternary diagram, with ethane through hexanes treated as the intermediate pseudocomponent and heptanesplus as the heavy pseudocomponent. Sometimes CO2 is included with the intermediate components. The general characteristic of pseudoternary phase behavior described earlier (namely, that developed miscibility can be achieved if the injection gas and reservoir oil lie on opposite sides of the critical tieline) are applied directly to multicomponent systems. Unlike the pseudoternary phase envelope for a threecomponent system, the pseudoternary phase envelope for a multicomponent system is not unique. It must be developed from a sequence of multiple contacts, where the multicontact procedure starts with the original injection gas and the reservoir oil. Thereafter, the procedures for vaporizing and condensinggas drives are different. A forwardcontact procedure is used for the vaporizinggas drive, while a backwardcontact procedure is used for the enrichedgas drive. The forwardcontact procedure starts by mixing the injection gas with the reservoir oil to obtain a twophase mixture. The equilibriumgas and oil compositions provide two points and a tieline on 4
the pseudoternary diagram. The gas from the twophase mixture is then removed and put into contact with original reservoir oil to form a new twophase mixture, providing two more points and another tieline on the pseudoternary diagram. The process of removing equilibrium gas and mixing it with original reservoir oil is repeated until either (1) the enriched gas becomes miscible with the original reservoir oil or (2) the compositions of the equilibrium gas and equilibrium oil no longer change. If Condition 1 is achieved, the process is multicontact miscible and most of the phase envelope is established up to the critical point. If Condition 2 is achieved, the process is not multicontact miscible and only part of the phase diagram is established. When miscibility is not achieved, the reservoir oil is located on an extension of a tieline with the equilibrium mixtures and no further component exchange is achieved by mixing the equilibrium gas with the original reservoir oil. The pseudoternary diagram for the traditional enrichedgas drive process is developed by a backwardcontact procedure. This starts by mixing the injection gas with reservoir oil to obtain a twophase mixture. The equilibriumgas and oil compositions determine a point and a tieline on the pseudoternary diagram. The equilibrium oil is then put into contact with the original injection gas to form a new twophase mixture, yielding another point and tieline on the pseudoternary diagram. The process of mixing altered equilibrium oil with original injection gas is repeated until either Condition 1 or 2 (described in the preceding paragraph) is achieved. Interpretation of the miscibility condition is the same as that for the vaporizinggas drive process. Zick2 claims that the pseudoternary representation of enrichedgas injection may lead to erroneous interpretation of the actual recovery mechanism. He further claims that the traditional enrichedgas miscible drive (developed by the multicontact process just described) may rarely, if ever, occur in reservoir systems. His observations are covered in more detail in the Sec. 8.4. On the other hand, pseudoternary representation of the vaporizinggas miscible drive process probably gives a reasonable description of the actual displacement mechanism. Quaternary diagrams have also been used to describe multicontact displacement in multicomponent systems; however, the additional dimension added by the fourth component makes this graphical representation more difficult to understand. Also, the “uniqueness” (i.e., oversimplification) of a single critical tieline on a ternary diagram is no longer valid with a quaternary representation. In their discussion of N2 in a vaporizinggas miscible drive process (Fig. 8.6), Koch and Hutchinson12 give perhaps the most illustrative use of a quaternary diagram. 8.2.4 SlimTube Displacements. A single definition of multicontact miscibility has not been accepted for multicomponent systems. Most definitions relate to recovery performance curves from labora
Fig. 8.6—Illustration of phase relations for vaporizinggas miscible drive process with N2 as injection gas (from Ref. 12). PHASE BEHAVIOR
H2O From PositiveDisplacement Pump
CO2 Supply Cylinder SandPacked Coil
CO2
Test OIl Solvent
Backpressure Regulator
CapillaryTube Sight Glass
Well Test Meter 100cm3 Burette
Fig. 8.8A—Experimental slimtube for CO2 displacement of a west Texas 30°API oil showing effect of temperature on recoverypressure behavior (adapted from Ref. 13).
100
WellDesigned Slim Tubes: Miscible Recoveriesu95% 90
Fig. 8.7—Schematic of a slimtube displacement apparatus; sandpacked coil consists of 40ftlong, 1/4in.OD stainlesssteel tubing packed with 160/200mesh Ottawa sand (adapted from Ref. 13).
80
tory displacement tests. A slimtube apparatus is used in the displacement experiments. Most slim tubes consist of 0.25in.outerdiameter coiled tubing, from 25 to 75 ft in length, packed with uniform sand or beads and housed in a constanttemperature container. Fig. 8.7 is a schematic of a slim tube. Orr et al.14 summarize slimtube characteristics described in various miscible studies. Slimtube results are interpreted by plotting cumulative oil recovery vs. PV of gas injected. Two recoveries are usually reported, at breakthrough and after injection of 1.2 PV of gas. To determine the MMP, several slimtube experiments are conducted at varying displacement pressures. Recovery at 1.2 PV of gas injected is then plotted vs. displacement pressure. For immiscible displacements, where relative permeabilities and viscosities influence the recovery process, recovery increases with pressure. The recoverypressure curve starts to flatten when the displacement becomes near miscible. Depending on the type of displacement process, temperature, injection gas, and other factors, the transition from immiscible to miscible may be abrupt or gradual (Figs. 8.8A and 8.8B). Table 8.1 gives reservoiroil and gas compositions for Fig. 8.8B. Choice of the “break point” defining MMP is somewhat arbitrary. Some investigators use a specific recovery factor, such as 90% at 1.2 PV of gas injected, to define MMP. For CO2/oil systems, Holm and Josendal1619 use a definition of MMP that requires 80% recovery at breakthrough and 94% recovery at a producing gas/oil ratio (GOR) of 40,000 scf/bbl (occurring at approximately 1.1 to 1.3 PV injected gas). Color change and lack of multiphase production from the slimtube apparatus have also been used to define the MMP. Yellig and Metcalfe13 give a particularly good discussion of criteria for defining MMP on the basis of slimtube data for CO2 systems. Fig. 8.9 shows recovery vs. PV of gas injected for a CO2 miscible displacement; changing colors of the produced oil are noted on the curve (from dark to red to orange to yellow to clear). Fig. 8.10 shows the qualitative character of produced fluids from a series of slimtube tests at immiscible and miscible conditions for an enrichedgas displacement. The solid line indicates fluid density based on photoelectriccell output, and shading represents twophase production. It is generally accepted that slimtube displacements yield the most reliable information for defining true multicontact miscibility. Although the slimtubedetermined miscibility condition is affected almost exclusively by the phase behavior of the fluids being studied, this miscibility condition is not the same as “thermodynamic misci
50
GASINJECTION PROCESSES
Low Miscible Recoveries Indicate SlimTube Equipment Design Problems
70 60
40 21
23
25
27
29
31
33
35
37
39
41
43
Displacement Pressure, MPa
Fig. 8.8B—Experimental slimtube results for highpressure displacement of a reservoir oil showing effect of injected leangas composition on recoverypressure behavior (from Ref. 15).
TABLE 8.1—RESERVOIROIL AND INJECTIONGAS COMPOSITIONS FOR FIG. 8.8B Reservoir Component
Oil
Gas 1
Gas 2
Gas 3
H2 S
0.00
0.0
0.00
0.00
N2
0.06
1.2
0.35
0.31
CO2
2.71
0.0
0.00
0.00
C1
34.66
93.3
81.71
69.61
C2
6.96
3.0
9.03
12.18
C3
6.46
1.1
4.31
8.83
iC4
1.54
0.0
0.84
1.63
nC4
4.09
0.7
1.63
3.42
iC5
1.87
0.0
0.54
1.07
nC5
2.57
0.7
0.59
1.22
C6
3.58
0.55
0.00
C7
3.66
0.00
1.73
C8
3.46
0.43
C9
3.13
C10
2.61
C11+
22.64 5
Clear and Yellow Yellow Orange Red Dark
Fig. 8.9—Experimental slimtube recovery vs. PV injected CO2 curve indicating change in color (as viewed in sight glass) of produced fluid after breakthrough (adapted from Ref. 13).
bility.” Slimtube experiments are relatively fast and simple to conduct, they do not require expensive equipment, and the experimental procedure can be automated readily with standard dataacquisition tools. The risingbubble apparatus has also been suggested as a method to arrive at an indication of true miscibility2124; however, we are skeptical of this claim for the condensing/vaporizing drive mechanism. Zhou and Orr25 appear to share this skepticism. 8.2.5 MultipleContact Pressure/Volume/Temperature (PVT) Experiments. Although the slimtube displacement experiment is the preferred method for determining the MMP of an injection gas, it does not provide controlled measurements of the system phase and volumetric behavior. Various PVT experiments can be used to supplement slimtube measurements for miscible displacement projects. Also, PVT experiments provide the only means of obtaining important data, such as viscosities, densities, compositions, and K values. Multicontact PVT data are particularly useful for tuning an equation of state (EOS) or any other PVT model that may be used in reservoir simulation. PVT experiments designed for gasinjection processes involve multiple contacts of injection or equilibrium gas with original reservoir oil or previously contacted equilibrium oil. The swelling test (Fig. 8.11) is the most common multicontact PVT experiment. In this experiment, injection gas is mixed with original reservoir oil in varying proportions, with each mixture quantified in terms of a molar percentage of injection gas (e.g., 20 mol% CO2 indicates that 0.2 moles of CO2 has been mixed with 0.8 moles of reservoir oil). The saturation pressure and phase volumes at more than and less than the saturation pressure are measured for each mixture. The data are presented in a pressure/composition ( px) diagram, as in Figs. 8.12 and 8.13. Pressure/volume plots are also used, usually as crossplots to determine quality lines on a px diagram. Occasionally, compositions of equilibriumoil and gas phases are determined for some mixtures in a swelling test (usually those at pressures close to the expected operating pressure of the injection project and those close to the critical point on the px diagram). The forward and backwardcontact PVT experiments (Fig. 8.14) also provide useful phase and volumetric data for gasinjection studies. The forwardcontact experiment follows the procedure described earlier for the vaporizinggas miscible displacement process. That is, the equilibrium gas from each contact is removed and mixed with more of the original reservoir oil. The amount of gas mixed with original oil at each contact may vary, but the amount should not affect the results significantly. The developed gas should eventually reach miscibility with the original reservoir oil if the experiment is conducted at a pressure greater than the MMP. Otherwise, the forwardcontact 6
Fig. 8.10—Produced wellstream character indicated by solid line representing density from photoelectriccell output; shading indicates twophase production (from Ref. 20).
experiment gives information about how efficiently the developed gas vaporizes the original oil without achieving miscibility. The backwardcontact experiment follows the procedure described for the enrichedgas miscible drive process. Here the equilibrium oil resulting from a given contact is mixed with more of the original injection gas. According to the traditional interpretation of the enrichedgas miscible drive process, miscibility should develop between the original injection gas and the altered reservoir oil. Benham et al.27 present backwardcontact PVT results that indicate miscibility can be achieved by this process (Fig. 8.15). On the other hand, Zick2 presents backwardcontact PVT experiments and EOS simulations that convincingly show that miscibility is not achieved by this process even at pressures considerably higher than the MMP determined by slimtube experiments (Figs. 8.16 and 8.17). The backwardcontact experiment also can be used to investigate revaporization of retrograde condensate by lean injection gas. Figs. 8.18 and 8.19 show swelling and backwardcontact experimental data reported by Vogel and Yarborough.28 Here, N2 was used in the backwardcontact experiment to revaporize retrograde liquid that formed when a leangas condensate was brought into contact with 50% N2 in a swelling test. Vogel and Yarborough also give experimental results for the effect of N2 on a reservoir oil. Nitrogen was mixed with the original reservoir oil in varying proportions (0, 0.144, 0.5, and 1.5 PV of N2 per PV of original reservoir oil). For a given N2/oil mixture, the system was brought to equilibrium at a specified pressure. The equilibrium gas was completely removed and discarded. The equilibrium oil was analyzed chromatographically, and a differential liberation experiment was conducted on the oil to determine solution gas/oil ratio and oil volumetric properties. Figs. 8.20 through 8.24 present some of the results from these experiments. PHASE BEHAVIOR
Fig. 8.11—Schematic of swelling test.
8.2.6 Calculation Algorithms. Several methods have been proposed for calculating MMP by multicontact calculations with an EOS or Kvalue model.27,29,30 These methods typically involve either a forward or backwardcontact mixing procedure, with the intention of simulating either a vaporizing or condensinggas drive process, respectively. Metcalfe et al.31 proposed a more rigorous calculation approach based on Cook et al.’s32,33 multicell vaporization model. With this approach, fluid mixing along a series of connected “cells” is used to simulate the development of miscible conditions with time (Fig. 8.25). Initially, all cells are filled with reservoirfluid composition. Then, a volume of injection gas (approximately 20% of a cell volume) is mixed with the contents of the first cell and brought to equilibrium. Part of the resulting equilibrium gas and oil is mixed with
the contents of Cell 2 and brought to equilibrium, part of the resulting equilibrium gas and oil from Cell 2 is mixed with Cell 3, and so on. Finally, production is recorded from the last cell; typically, approximately 50 cells are used. This series of calculations constitutes one “batch” or “timestep.” The calculations are repeated with a new volume of injected gas, and the compositional changes in one or more cells are monitored with time. Metcalfe et al. plot the results on a ternary diagram to apply the critical tieline approach for establishing the condition of miscibility. Injectiongas composition can change at each timestep, there
Bubblepoint curve
, vol% liquid
BUBBLEPOINTS DEWPOINTS
N2 in Painter Reservoir, mol%
Fig. 8.12—Experimental px diagram for mixtures of lean natural gas with a Block 31 Devonian reservoir oil (adapted from Ref. 1).
GASINJECTION PROCESSES
Fig. 8.13—Experimental px diagram for mixtures of N2 with a Painter reservoir oil (adapted from Ref. 26).
7
INJECTION GAS CRUDE INJECTION GAS
CRUDE
CRUDE
INJECTION GAS
INJECTION GAS
CRUDE
Fig. 8.14—Schematic of forward and backwardcontact experiments.
by allowing the study of miscibleslug displacement and the effect of driving an enriched gas with a cheaper lean gas. Metcalfe et al. propose three methods for determining which phases are passed from one cell to the next and the amount of each phase passed. The first method, originally proposed by Cook et al.,32,33 passes all equilibrium gas from cell to cell, simulating the vaporizinggas (forwardcontact) process. The second method passes only enough gas and oil to the next cell to ensure that the remaining mixture in the current cell fills the cell volume. The third passes equilibrium gas and oil according to the mobility ratio (krg /kro ) (mo /mg ), where the relative permeability ratio krg /kro is entered as a function of saturation. For any miscible displacement process, the second and third methods should give the same conditions of miscibility in the limit of small injection volumes and a large number of cells. The first method is valid only for a vaporizinggas drive mechanism. Short of simulating a slimtube displacement, the multicell calculation approach is probably the best generalized scheme for determining the conditions required to develop miscibility. It should give the same conditions of developed miscibility as slimtube results if the multicell calculations are interpreted correctly. Calculation methods based strictly on forward or backwardcontact procedures are not recommended.27,29,30 Johns et al.34,35 present analytical simulation results based on the method of characteristics for three and fourcomponent systems that verify the mechanisms of the condensing/vaporizing drive mechanism originally described by Zick.2 Practically, their approach is limited to fourcomponent systems and is more difficult to program than a onedimensional (1D) slimtube or batchtype experiment. Wang and Orr36 recently proposed a generalized algorithm for computing MMP on the basis of complex tieline analysis
Fig. 8.16—Experimental slimtube recoveries at 1.2 PV injected gas vs. injection pressure for Reservoir Oil A displaced by Solvent A (adapted from Ref. 2).
8
Intermediates, mol%
Fig. 8.15—Experimental backwardcontact PVT data for enrichedgas/reservoiroil system (adapted from Ref. 27).
founded in the theory fo the method of characterisitics; the CN mechanism can be the method of developed miscibility. 8.3 LeanĆGas Injection Leangas injection with methane and N2rich gases has been used for reservoir management during primary production, as an alternative to waterflooding for secondary recovery, and in gravitystable tertiary projects. Successful projects include (1) pressure maintenance in oil reservoirs to maintain productivity, sometimes by developing an artificial gas cap; (2) gravitystable displacement in dipping, highpermeability oil reservoirs; (3) reservoirvoidage replacement to maintain the oil/water contact in a strongwaterdrive reservoir; (4) recovery of upstructure “attic” oil and gas in strongwaterdrive reservoirs; (5) highpressure multicontact miscibility in oil reservoirs; and (6) partial and full pressure maintenance in gascondensate reservoirs. Some of the more important justifications for leangas injection include gas availability, better injectivity in lowpermeability reservoirs, conservation or environmental constraints, and superior oil recoveries compared with alternative EOR methods. Not all applications of leangas injection require special treatment of phase
Fig. 8.17—EOScalculated multiple backwardcontact PVT experiment for Oil A and Solvent A at 900 psi higher than MMP pressure indicated from experimental and simulated slimtube results (adapted from Ref. 2).
PHASE BEHAVIOR
0% (gas only, no added N2) 10% cumulative added N2 30% cumulative added N2 50% cumulative added N2
Liquid, vol%
0% (gas only, no added N2) 10% cumulative added N2 30% cumulative added N2 50% cumulative added N2
Liquid, vol%
Fig. 8.18—Effect of N2 on the phase behavior of two gascondensate reservoir fluids: equilibrium flash volumetric expansion tests run at 381K and no material removed from pV cell (from Ref. 28).
and volumetric behavior. However, highpressure injection in oil reservoirs and gas cycling in partially depleted condensate reservoirs do require detailed knowledge of how the injected gas behaves with the reservoir fluids. 8.3.1 VaporizingGas Miscible Drive. Several highpressure, leangas miscible projects have been reported for light oils with stocktankoil gravities greater than 35°API and with operational flooding pressures greater than approximately 3,500 psia.1 Lean gases contain mostly methane or N2, with methanerich gases also containing smaller quantities of ethane and C3+ components. Nitrogenrich gases include flue gas, consisting of approximately 88% N2 and 12% CO2, and pure N2 generated from cryogenic air separation. Lean gases tend to vaporize intermediate hydrocarbons in the C5 to C12 range, depending on the pressure and injectiongas composition. Nitrogen also tends to “trade places” with the solution gas in an oil, thereby improving natural gas recovery. At sufficiently high
pressure, lean gas can develop an insitu gas that is sufficiently rich in intermediate components (C2 through C4) to develop miscibility with the reservoir oil. Another condition for developed miscibility by the vaporizinggas drive process is that the reservoir should not have an initial freegas saturation. That is, in a vaporizinggas drive mechanism the gas saturation must always be zero ahead of the miscible displacement front. Pressure is usually the primary design parameter in a vaporizinggas miscible drive project. Other considerations include slug size, WAG ratio, and producer/injector pattern. Methanerich injection gases tend to develop miscibility at slightly lower pressures than N2rich gases, depending mainly on the methane content in the reservoir oil. Also, the price differential between N2 and lean natural gas can be significant, so the methane/N2 ratio may be a valid design parameter in some leangas miscible projects. Experimental Calculated
Cumulative PV N2 Contacted 0.0 PV N2
0.144 PV N2
0.50 PV N2 1.50 PV N2
N2 Injective, cumulative PV Fig. 8.19—Revaporization of retrograde condensate by multiple contacts with N2 (adapted from Ref. 28).
GASINJECTION PROCESSES
Fig. 8.20—Experimental oil formation volume factors (FVF’s) for mixtures of N2 with a reservoir oil (adapted from Ref. 28).
9
Experimental Experimental
Calculated
Calculated Cumulative PV N2 Contacted 0.0 PV N2 0.144 PV N2 0.50 PV N2
Cumulative PV N2 Contacted 1.50 PV N2
0.50 PV N2
1.50 PV N2 0.144 PV N2
0.0 PV N2
Fig. 8.21—Experimental solution GOR data for mixtures of N2 with reservoir oil (adapted from Ref. 28).
Stalkup1 reports only one MMP correlation for the vaporizinggas drive miscible process. This correlation gives MMP as a function of reservoiroil bubblepoint pressure; reduced temperature of the reservoir oil; and mass fraction of three groups in the reservoir oil: (1) C2 through C6 plus CO2 and H2S, (2) C1 plus N2, and (3) C7+. 8.3.2 Gas Cycling. Gas cycling in condensate reservoirs has been used for the past 50 years to minimize lossses in liquid recovery. When reservoir pressure drops below the dewpoint pressure in a gascondensate reservoir, liquids condense and remain primarily as an immobile phase. The produced wellstream becomes leaner (as reflected by a decreasing condensate yield), and overall condensate recovery may be as low as 15 to 20%. Further depletion at pressures less than approximately 2,000 psia may revaporize some of the “lost” retrograde condensate, but this additional recovery is usually not significant. To maximize liquid recovery, reservoir pressure should be kept higher than the dewpoint pressure to avoid retrograde condensation. Typically, this is achieved by reinjecting produced gas that has been separated and processed for condensate and natural gas liquids
Fig. 8.22—Experimental oil density data for mixtures of N2 with reservoir oil (adapted from Ref. 28).
(NGL’s). Because the produced gas is not sufficient to replace the reservoir voidage caused by production, makeup gas must be obtained to achieve full pressure maintenance. If the reservoir is initially undersaturated (i.e., the initial pressure is greater than the dewpoint pressure), reinjecting only the produced gas is acceptable until reservoir pressure approaches the dewpoint. The economics of delaying gas sales to increase condensate recovery may be prohibitive. Alternatives to delayed gas sales include reinjection of only part of the produced gas, purchasing cheaper makeup gas for reinjection, and replacing producedgas reinjection
Experimental Data Before Cycling With N2 After Cycling With N2
Experimental Calculated
20 PV
Cumulative PV N2 Contacted 1.50 PV N2 0.50 PV N2 0.144 PV N2 0.0 PV N2
Fig. 8.23—Experimental oil viscosity data for mixtures of N2 with reservoir oil (adapted from Ref. 28). 10
Fig. 8.24—Change in heptanesplus distribution for oil that has been in contact with 20 PV of N2: effect of cycling, simulated true boiling point analysis by temperatureprogrammed gas chromatography (adapted from Ref. 28). PHASE BEHAVIOR
(a)
Gas to Next Cell
Gas
Gas Oil
Oil
Oil
Oil
Original Cell Condition
(b)
Final Cell Condition Gas (and Oil) to Next Cell
Gas (and Oil)
Gas Oil
Oil
Oil
Oil
Original Cell Condition
Final Cell Condition Gas (and Oil) to Next Cell
(c)
Gas (and Oil)
Gas Oil
Gas Oil Original Cell Condition Cell 1
Cell 2
Gas (and Oil) Gas (and Oil) Injection Gas Original Original Oil Oil Batch 1 Injection Gas Batch 2 Injection Gas Batch N
Gas (and Oil) Oil
Gas (and Oil) Oil
Fig. 8.26—Effect of N2 and lean natural gas on the dewpoint pressure of a gascondensate reservoir fluid (from Ref. 37).
Cell NN
ters used to optimize recovery and other factors affecting an enrichedgas drive project.
Gas (and Oil) Original Oil
Gas (and Oil)
Oil
Oil
Cumulative Gas Injected, scf/RB
Gas Oil Final Cell Condition
Oil
Gas (and Oil) Oil
Gas (and Oil)
Gas (and Oil) Oil
Fig. 8.25—Schematic of multicell calculation method: (a) stagnant oil, (b) moving excess oil, and (c) oil and gas moved by phase mobilities (adapted from Ref. 31).
8.4.1 Traditional Mechanism. Some difference of opinion exists concerning the actual displacement mechanism responsible for high recoveries reported in slimtube experiments with enriched gases. The traditional enrichedgas displacement mechanism is based on an interpretation of a pseudoternary diagram, where miscibility is developed by repeated contacts of the injection gas with the oil found at the point of injection. The following is Benham et al.’s27 description (based on their Fig. 3) of this traditional interpretation of the enrichedgas miscible displacement process (Fig. 8.27). “Assuming that a phase diagram of this general shape is an appropriate representation, the mechanism for obtaining miscibility may be illustrated by reference to Fig. 3. This figure has been pre
with injection of flue gas or N2. Generating large quantities of N2 cryogenically on location has been demonstrated in several large gascycling projects.1 Nitrogen has also been used as makeup gas to ensure full pressure maintenance. In the early 1980’s, studies27,32 showed that N2 caused substantial condensation of liquids when mixed with a gascondensate mixture (Figs. 8.18 and 8.26). This behavior caused concern that N2 might worsen the problem of retrograde condensation if used to maintain pressure in condensate reservoirs. Subsequent displacement and multicontact tests showed that practically all the liquid condensed by initial contact with N2 was revaporized by later contacts with the N2 (Fig. 8.19).28,3739 Slimtube recoveries with N2 displacing a gas condensate showed behavior similar to that of methanerich gas displacements, with both gases yielding practically 100% total hydrocarbon recovery. 8.4 EnrichedĆGas Miscible Drive Miscible displacement projects with enriched injection gas are reported in the literature for reservoir oils with stocktankoil gravities ranging from 30 to 45°API.1 Typical flooding pressures range from 1,500 to 4,000 psia. Enriched gases usually contain methane, ethane, and varying quantities of LPG components C3 through C4. CO2 also may be found in the injection gas without significantly affecting the miscibility condition. Reservoir displacement pressure and the degree of LPG enrichment are the two main design parameGASINJECTION PROCESSES
Fig. 8.27—Pseudoternary representation of the traditional enrichedgasmiscible drive process (adapted from Ref. 27). 11
Fig. 8.28—Benham et al.26 chart for determining maximum methane content in an injection gas for miscibility to develop according to traditional enrichedgas miscible oil withM C +240 at 3,000 5) psia (adapted from Ref. 27).
pared to demonstrate the mechanism involved in obtaining miscibility between reservoir fluid represented by Point R and an enriched gas represented by Point RG. The reservoir fluid is in the twophase region and has a liquid phase of composition (m) and a vapor phase of composition (a). As gas is first injected, it will tend to move both liquid and vapor until eventually the gas velocity is greater than the liquid velocity. The first mixing will be between liquid (m) and rich gas (RG). The overall composition of this mixture could be Point a. This mixture separates into two phases represented by Points n and b. As more rich gas is injected, it displaces the gas (b) and mixes with the liquid (n). These may mix to an overall composition (b), which separates into liquid (o) and vapor (c). Injection of more rich gas will result in displacement of the vapor (c) and mixing of the liquid (o) with the injection fluid (RG) to form the mix (g). This continues until injection fluid (RG) mixes with the liquid (t), at which time a miscible displacement begins. Injection fluid (RG) miscibly displaces the liquid (t), which miscibly displaces the liquid (s), which miscibly displaces r, etc. The gases will also be miscibly displaced by the rich gas; therefore, a completely miscible displacement has been achieved. The liquids will gradually build up in saturation with displacement until a completely singlephase miscible displacement is achieved. “It may be shown that the leanest mixture that will give a miscible displacement is represented by a point on the extension of the limiting tieline (AB), which passes through the critical point (C).” Benham et al. use this interpretation of the displacement mechanism to develop a series of working curves for estimating the degree of enrichment required to attain MMP for a given reservoir oil. Their graphical correlations require (1) average molecular weight of the reservoiroil C5+ mixture, (2) average molecular weight of the C2+ components that will be used to enrich the injection gas, and (3) reservoir temperature. With these three data, the appropriate charts are entered to obtain the allowable methane concentration in the injection gas. Each chart represents an MMP; charts are provided for MMP’s of 1,500, 2,000, 2,500, and 3,000 psia (Fig. 8.28). A plot of LPG enrichment vs. MMP can then be made for design calculations. Zick2 reports an MMP of 3,100 psia at 185°F for his Reservoir Fluid A with an injection gas consisting of 39 mol% methane (20% methane mixed with 80% Solvent A containing 23.5 mol% C1). Fig. 8.29 plots slimtube recovery at 1.2 PV gas injected vs. dilution of the solvent with pure methane. With M C2)+40 for the reported solvent and M C5)+260 for the reservoir oil, the Benham et al. charts give a maximum methane content for the injection gas somewhat greater than 50 mol%. That is, the Benham et al. charts indicate that MMP can be achieved at 3,000 psia with the solvent diluted 35% with methane. Fig. 8.29 indicates that the experimental slimtube recovery is only 65% for this injectiongas composition. The Kuo16 MMP correlation predicts a similar overestimation of methane dilution. 8.4.2 Combined Condensing/Vaporizing Mechanism. Zick proposes an alternative mechanism to explain the misciblelike recoveries that can be achieved by displacing a reservoir oil with enriched gas. The 12
Fig. 8.29—Experimental slimtube recoveries at 3,100 psig as functions of solvent dilution with methane for Reservoir Oil A (depleted to 3,000 psig) and Solvent A at 185°F (adapted from Ref. 2).
mechanism is a combination of (1) a leading front that enriches original oil with light intermediates found in the original injection gas and middle intermediates (C5 through C30) that have been vaporized from the reservoir oil behind the front and (2) a trailing front of injection gas that vaporizes middleintermediate components. A sharp transition zone separates condensing and vaporizing fronts. This transition zone is near miscible, or perhaps miscible in the absence of dispersion, recovering practically all the reservoir oil with only a small ROS.
Fig. 8.30—EOScalculated slimtube profiles for condensing/ vaporizinggas drive of Reservoir Oil A by Solvent A (adapted from Ref. 2). PHASE BEHAVIOR

FourPhase
Injection Gas, mol% Fig. 8.31—Multiphase behavior for mixture of 81.72 mol% (67.99 vol%) enriched driving gas (32% C1, 37% C2, and 30% C3) and a reservoir oil at pt2,000 psia and 105°F (adapted from Ref. 40).
Fig. 8.30 shows the profile of oil saturation, phase densities, and K values for an enrichedgas displacement calculated by an EOS slimtube simulator. Five regions are readily identified in this figure. On the basis of the proposed condensing/vaporizing mechanism, these five regions can be summarized as follows. 1. Original oil zone. 2. A leading twophase front with net condensation of intermediate components. The gas contains lightintermediate components found in the original injection gas and middleintermediate components that have been vaporized from the reservoir oil. 3. A sharp transition zone with nearmiscible behavior. The front side of the transition zone (toward Zone B) shows dramatic condensation of intermediate and heavy components. The back side of the transition zone (toward Zone D) shows highly efficient vaporization of intermediate and heavy components. Only a small ROS is left behind the transition zone. 4. A trailing front of enriched gas, which vaporizes middleintermediate components found in the remaining residual oil. 5. A stripped ROS, in equilibrium with the injection gas, remains behind. Little if any mass transfer occurs here. The residual oil consists of a heavy, nonvolatile material and the components making up the injection gas. The net mass transfer of components between the gas and oil phases is reflected by the slope of the K values plotted vs. distance. Net condensation from the gas phase into the oil phase occurs where the slope dKi /dx is negative for middleintermediate and heavy components (Zone B). Net vaporization from the oil phase into the gas phase occurs where the slope dKi /dx is positive for the middleintermediate and heavy components (Zone D). Zick2 gives a fairly detailed summary of the condensing/vaporizing mechanism. With experimental and simulation results, he shows that the traditional enrichedgas miscible drive mechanism cannot explain misciblelike recoveries for three different reservoiroil/enrichedgas systems. His arguments basically hinge on the observation that the oil that should first become miscible with an enriched gas (i.e., the oil nearest the point of injection) does not become miscible in multicontact PVT experiments or in simulations of slimtube displacements. He writes, “When the enriched gas comes into contact with the oil, the light intermediates condense from the gas into the oil, making the oil lighter. The equilibrium gas is more mobile than the oil, so it moves on ahead and is replaced by fresh injection gas, from which more light intermediates condense, making the oil even lightGASINJECTION PROCESSES
Injection Gas, mol% Fig. 8.32—px diagram showing multiphase behavior for an enriched gas (32% C1, 37% C2, and 30% C3) mixed with a reservoir oil at 105°F (adapted from Ref. 40).
er. If this kept occurring until the oil was light enough to be miscible with the injection gas, it would constitute the condensinggas drive mechanism. However, this is unlikely to occur with a real reservoir oil. As the light intermediates are condensing from the injection gas into the oil, the middle intermediates are being stripped from the oil into the gas. Since the injection gas contains none of these middle intermediates, they cannot be replenished in the oil. After a few contacts between the oil and the injection gas, the oil becomes essentially saturated in the light intermediates, but it continues to lose middle intermediates, which are stripped out and carried on ahead by the mobile gas phase. The light intermediates of the injection gas cannot substitute for the middle intermediates the oil is losing. So after the first few contacts make the oil lighter by net condensation of [light] intermediates, subsequent contacts make the oil heavier by net vaporization of [middle] intermediates. Once this begins to occur, the oil no longer has a chance of becoming miscible with the gas. Ultimately, all the middle intermediates are removed and the residual oil will be very heavy, containing only the heaviest, nonvolatile fraction and the components present in the injection gas.” Zick goes on to explain how high recoveries can be obtained with enrichedgas displacement without necessarily achieving true miscibility. Regardless of whether true miscibility develops, he insists that the miscibility (or near miscibility) that does occur is not developed according to the traditional enrichedgas drive mechanism (i.e., between the injection gas and the oil at the point of injection). Instead, he proposes the combined condensing/vaporizing mechanism. He claims that reaching miscibility by the traditional enrichedgas process requires higher displacement pressures (or higher enrichment levels) than the MMP (or minimum miscibility enrichment) determined by slimtube experiments (Figs. 8.17 and 8.20). A characteristic of the combined condensing/vaporizing mechanism is that a freegas saturation always exists ahead of the front and that some ROS is found behind the front. Novosad and Costain21 and Novosad et al.22 describe a displacement mechanism for enrichedgas drive that differs from both the 13
TABLE 8.2—CO2 PHYSICAL PROPERTIES M (g+1.52)
44.01
Tc , °F
88
pc , psia
1,070
ò c, gmńcm (lbmńft ) 3
3
0.469 (29.2)
Zc
0.274
w (Pitzer acentric factor)
0.239
Tb ,°F, “dry ice” at 1 atm
*110
CO2 equivalent 1 ton, Mscf 1 lbm, scf
17.2 8.6
condensinggas and the combined condensing/vaporizinggas drive mechanisms. On the basis of their interpretation, they propose a simple risingbubble apparatus to determine the enrichment level required to develop miscibility for a given oil. This experimental technique implies, however, a type of vaporizinggas drive mechanism that would not seem to apply for most enrichedgas displacements. Even so, the experimental results they provide seem to give reasonable conditions of developed miscibility for the highly undersaturated oils used in their studies. 8.4.3 Multiphase Behavior. Enrichedgas injection at low temperatures may yield complex multiphase VLL/solid (VLLS) behavior. Shelton and Yarborough40 present a thorough study of multiphase behavior for a reservoir oil in contact with a rich gas consisting of 32% methane, 37% ethane, and 30% propane at 105°F. Figs. 8.31 and 8.32 show some of their study results. The multiphase VLL behavior and asphaltene/wax precipitation are strikingly similar to those of CO2/oil systems at the same temperature.40,41 Although experimental evidence is lacking, multiphase behavior probably can be anticipated when the system temperature is not
Fig. 8.34—CO2 Z factor (from Refs. 42 and 43).
14
Fig. 8.33—CO2 density (from Refs. 42 and 43).
Fig. 8.35—CO2 viscosity (from Refs. 42 and 43).
PHASE BEHAVIOR
TABLE 8.3—CO2 DENSITY* (from Ref. 42) Pressure (bar) Temperature (°F)
25
50
75
100
150
200
250
300
68
0.0527
0.1423
0.8100
0.8550
0.9010
0.9335
0.9600
0.9832
86
0.0499
0.1251
0.6550
0.7820
0.8500
0.8887
0.9190
0.9460
104
0.0476
0.1135
0.2305
0.6380
0.7850
0.8415
0.8771
0.9077
122
0.0456
0.1052
0.1932
0.3901
0.7050
0.7855
0.8347
0.8687
140
0.0437
0.0984
0.1726
0.2868
0.6040
0.7240
0.7889
0.8292
158
0.0421
0.0930
0.1584
0.2478
0.5040
0.6605
0.7379
0.7882
176
0.0406
0.0883
0.1469
0.2215
0.4300
0.5935
0.6872
0.7466
194
0.0391
0.0845
0.1381
0.2019
0.3730
0.5325
0.6359
0.7040
212
0.0378
0.0810
0.1305
0.1877
0.3330
0.4815
0.5880
0.6630
230
0.0366
0.0778
0.1239
0.1765
0.3040
0.4378
0.5443
0.6230
248
0.0354
0.0749
0.1187
0.1673
0.2800
0.4015
0.5053
0.5855
266
0.0344
0.0722
0.1141
0.1595
0.2620
0.3718
0.4718
0.5517
284
0.0334
0.0697
0.1094
0.1525
0.2465
0.3470
0.4419
0.5200
302
0.0325
0.0674
0.1054
0.1461
0.2337
0.3267
0.4151
0.4925
320
0.0316
0.0653
0.1018
0.1403
0.2229
0.3089
0.3918
0.4680
*In gm/cm3.
more than approximately 50°F higher than the critical temperature of the injection gas. For example, the pseudocritical temperature of the enriched gas used by Shelton and Yarborough was 60°F and significant multiphase behavior was observed at 105°F. CO2 systems exhibit multiphase behavior up to approximately 130°F, about 40°F higher than the critical temperature of CO2. Accordingly, an injection gas rich in NGL’s probably experiences multiphase behavior
Fig. 8.36—CO2 phase diagram (from Refs. 42 and 43). GASINJECTION PROCESSES
and asphaltene precipitation at higher reservoir temperatures than a less enriched gas does. 8.5 CO2 Injection 8.5.1 CO2 Physical Properties. CO2 is a stable, nontoxic compound found in a gaseous state at standard conditions. For petroleum applications, CO2 exists either as a gas or as a liquidlike su
Fig. 8.37—CO2 FVF (from Refs. 42 and 43). 15
StockTank Oil Molecular Weight StockTank Oil Specific Gravity
1.06 1.04 1.02 1.00 0.98 0.96 0.94 0.92 0.90 0.88 0.86 0.84 0.82 0.80 0.78
Fig. 8.39—Correlation for swelling of a dead stocktank oil when saturated with CO2 (adapted from Ref. 44).
cant. Corrosion in CO2 floods, particularly in WAG projects, requires special attention.
0
0.1 XCO2
0.2
0.3
0.4
0.5
0.6
in Oil With UOP K=11.7
Fig. 8.38—Correlation for solubility of CO2 in dead stocktank oils (adapted from Ref. 44).
percritical fluid. Table 8.2 gives the key physical properties of CO2. Figs. 8.33 through 8.35, respectively, show density, Z factor, and viscosity of CO2 as functions of pressure and temperature. Table 8.3 gives tabular data for the density of pure CO2. Fig. 8.36 shows the phase diagram of CO2 with an extrapolation of the critical isochor. The critical isochor defines supercritical conditions where phase density equals the critical density of 0.47 g/ cm3. Later, we show that CO2 density at reservoir conditions is the main parameter that determines MMP of CO2 with reservoir oils. In fact, the critical isochor drawn in Fig. 8.36 gives a close approximation of the YelligMetcalfe13 correlation for CO2 MMP. Fig. 8.37 gives the reservoir barrels occupied by 1 Mscf of CO2 as a function of pressure and temperature. For most CO2 projects, approximately 2 Mscf of CO2 is required to fill 1 res bbl PV. Typically, approximately 5 to 10 Mscf of CO2 is the “gross utilization” required to recover an additional 1 bbl of stocktank oil by the CO2miscible flooding process; gross utilization is driven strongly by economics and may differ from these typical values. As much as half of the injected CO2 may remain in the reservoir at the end of a CO2 flood. CO2, when mixed with water, forms carbonic acid. This acidic byproduct may affect injectivity in carbonate reservoirs, but the corrosive effect on steel tubulars and surface equipment may be signifi16
8.5.2 Immiscible CO2/Oil Behavior. CO2 flooding has been applied successfully in viscous, heavyoil reservoirs. Oil swelling and oilviscosity reduction are the two primary mechanisms in immiscible CO2 displacement. Lowpressure reservoirs and reservoirs with stocktankoil gravities less than approximately 30°API are typical candidates for immiscible CO2 displacement. Gravitystable displacement with CO2 also may be an efficient immiscible process. Simon and Graue44 give generalized graphical correlations for solubility (Fig.8.38), swelling (Fig. 8.39), and viscosity reduction for “dead” stocktank oils saturated with CO2 (Fig. 8.40). Reported accuracies for the solubility and swelling correlations are 2 and 0.5%, respectively, and 12% deviation is reported for the viscosity correlation. Fig. 8.38 shows that CO2 solubility in crude oils increases with decreasing temperature. Solution gas/oil ratio in scf/STB can be calculated from CO2 mole fraction, x CO2 , in a CO2/oil mixture from R s + 133, 000
x CO2 go . M o 1 * x CO
. . . . . . . . . . . . . . . . . . . (8.1)
2
At temperatures less than approximately 200°F, the correction to solubility based on the universal oil products (Watson) characterization factor is less than 2% for most reservoir oils (11.4 t K w t 12.4). Fig. 8.39 shows the swelling factor, expressed as the ratio of CO2saturated stocktankoil volume divided by original stocktankoil volume. Swelling increases with increasing CO2 solubility and with decreasing stocktankoil molar volume (M ońg o). Oilviscosity reduction (Fig. 8.40) is substantial for all APIgravity stocktank oils at pressures up to approximately 750 psia; the effect diminishes at higher pressures because of reduced CO2 solubility. Highviscosity oils are affected the most by CO2 solubility; oil viscosity may be reduced by as much as two orders of magnitude. Practically, Simon and Graue’s correlations are valid only for heavier oils ( g API t 25 and m o t 5 cp) without solution gas and at temperatures greater than approximately 120°F. Fig. 8.39 shows the effect of solution gas on oil swelling. The SimonGraue correlations cannot be used to calculate solubility and swelling in reservoir oils containing solution gas and also do not predict the draPHASE BEHAVIOR
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1 Fig. 8.41—Effect of solution gas on swelling of a reservoir oil by CO2 (adapted from Refs. 16 through 19). 0
Fig. 8.40—Effect of CO2 on oil viscosity (adapted from Ref. 44).
matic change in solubility and swelling behavior exhibited by some oils at lower temperatures. 8.5.3 Miscible CO2/Oil Behavior. Fig. 8.41 shows the swelling behavior of a stocktank oil reported by Holm,5,45 Holm and Josendal,1619 and Holm and O’Brien.46 The experiment starts with a constantvolume visual cell initially filled approximately onethird with stocktank oil. CO2 is added in increments, and the cell is rocked for each mixture until equilibrium is reached. The final pressure and oil volume are noted, and the oil volume, relative to the initial oil volume, is plotted vs. pressure. When a certain pressure is reached, the oil phase, which was being swollen by increasing amounts of dis
Fig. 8.42—Volumetric behavior of Cabin Creek stocktank oil as CO2 is added to a constantvolume visual cell (adapted from Refs. 16 through 19). GASINJECTION PROCESSES
solved CO2, suddenly decreases in volume. Significant extraction of intermediate and heavy components (C5 through C30) from the oil phase into the upper CO2rich phase causes this dramatic change in oil volumetric behavior. At sufficiently high pressures, the CO2rich phase may even become heavier than the oil (hydrocarbonrich liquid) phase, resulting in phase inversion (Fig. 8.42). Holm and Josendal note that the observed discontinuity in swelling behavior is caused by a change in the behavior of CO2rich phase from vaporlike to liquidlike that is almost coincident with the pressure required to develop miscibility in slimtube measurements. Qualitatively, the change in behavior of the CO2rich phase is analogous to the volumetric change that occurs for a pure component when pressure passes through the vapor pressure. That is, the CO2rich phase behaves like a vapor at pressures below the “vapor pressure” and like a liquid at higher pressures. Once the CO2rich phase attains liquidlike behavior, it extracts intermediate and heavy components from the oil, as would be expected with a liquid solvent. The temperature required for the CO2rich phase to exhibit sharp, discontinuous volumetric behavior depends on the oil but is usually
Fig. 8.43—Volumetric behavior of MeadStrawn stocktank oil as CO2 is added to a constantvolume visual cell at different temperatures (adapted from Refs. 16 through 19). 17
Fig. 8.44—Volumetric behavior and slimtube results for the MeadStrawn and Fansworth stocktank oils at 135°F (adapted from Refs. 16 through 19).
less than 150°F. Fig. 8.43 shows swelling/extraction experiments for the 41°API MeadStrawn crude oil at different temperatures. Holm and Josendal note that miscibility develops at a pressure only slightly higher than the pressure where the character of swelling changes (i.e., where significant hydrocarbon extraction starts). They also point out that the sharpness of change in the swelling behavior is more pronounced at lower temperatures and is coincident with the sharpness in change from immiscible to miscible displacement indicated on a recoverypressure curve from slimtube experiments (Fig. 8.44). Holm and Josendal correlate miscibility development with the density of pure CO2. They indicate that light oils can develop miscibility at conditions where CO2 has a density as low as 0.4 g/cm3 (critical CO2 density is 0.47 g/cm3) and that most oils will develop miscibility at conditions where CO2 density ranges from 0.5 to 0.7 g/cm3 (Fig. 8.45). Their correlation for MMP shows that the CO2 density required to develop miscibility depends primarily on the amount of gasoline and gas/oil components (C5 through C30) found in the stocktank oil. Their correlation uses weight percent (w C5 through w C30)ńw C5)as a correlating parameter, with typical values ranging from 70 to 80% requiring CO2 densities ranging from 0.65 to 0.55 g/cm3 to develop
Fig. 8.45—Density of CO2 required to develop miscibility for various oils at temperatures from 90°F to 190°F (adapted from Refs. 16 through 19).
miscibility. Fig. 8.46 shows the HolmJosendal correlation for MMP. Stalkup1 covers other correlations for CO2 MMP. The distribution of components in the C5 through C30 cut of an oil also affects the MMP, but Holm and Josendal do not include this effect directly in their correlation. They do show, however, that the fraction of gasolines (C5 through C12) in the C5 through C30 cut has a measurable effect. Typically, gasolines make up 40 to 50 wt% of the C5 through C30 cut. Higher gasoline content will decrease the MMP, and lower gasoline content will increase the MMP. The type of hydrocarbons (paraffinic vs. aromatic) making up the C5 through C30 material in a crude oil has negligible effect on MMP. Aromatic oils appear to have slightly lower MMP’s than paraffinic oils, all other conditions being the same. Nitrogen and light C1 through C4 hydrocarbons in the reservoir oil generally have a negligible effect on CO2 MMP if the MMP is less than the reservoiroil bubblepoint pressure. The light components in the reservoir oil are extracted ahead of the miscible front in a CO2 process (Fig. 8.47). The bank of light components does not affect the extraction process or developed miscibility. Yellig and Metcalfe13 point out that the MMP of a reservoir oil equals the bubblepoint of that oil if the bubblepoint pressure is greater than the MMP determined for a lowGOR sample of the same stocktank oil. However, this is true only when considering the traditional vaporizinggas drive mechanism. With the condensing/vaporizing mechanism, the MMP can be lower than the bubblepoint pressure. Methane, N2 , and C2 through C4 hydrocarbons mixed with the CO2 injection gas affect MMP significantly. Methane and N2 tend 135°F and 1,800 psi—near miscible
0.33 PV
0.17 PV
2,500 psi—multicontact miscible
0.15 PV
0.15 PV
2,500 psi—first contact miscible
C5C30 Content of Oil, (C5*C30)/C5+, wt% 0.24 PV
Fig. 8.46—CO2 MMP correlation equals the pressure corresponding to the CO2 density from the chart at reservoir temperature; MMP may be less than the oil bubblepoint for a C/V miscible mechanism.
18
Fig. 8.47—Schematic of distribution of components in CO2 displacement at miscible and nearmiscible conditions based on slimtube simulation results (adapted from Refs. 16 through 19).
PHASE BEHAVIOR
Fig. 8.48—Experimental recoveries from slimtube displacements for a Wasson stocktank crude oil displaced by a CO2 slug pushed by N2 at 1,250 psig and 107°F with no gas in solution and 100ft coil (adapted from Ref. 47).
to increase MMP, while NGL’s tend to decrease MMP. However, a sufficiently large PV of injected CO2 can be followed by N2 or lean gas without affecting MMP (Fig. 8.48). 8.5.4 Multiphase Behavior. CO2/oil systems exhibit multiphase VLLS behavior similar to that described earlier for enrichedgas/oil systems.48 Threephase VLL behavior is limited to reservoir temperatures less than approximately 130°F, pressures from 1,000 to 1,500 psia (somewhat less than the MMP), and CO2 concentrations greater than approximately 50 mol%. At other conditions, vapor/ liquid or liquid/liquid behavior is expected, with the upper phase containing mainly CO2 and the lower phase containing mostly hydrocarbons and some dissolved CO2. Asphaltene precipitation can occur over a relatively large range of pressures and CO2 concentrations (Fig. 8.49), usually including the VLL region. The three phases in a CO2/oil VLL system include a CO2rich vapor (the upper phase), a CO2rich liquid (the middle phase) containing some hydrocarbons, and a hydrocarbonrich liquid (the lower phase) containing C5+ with some dissolved natural gas and CO2. Consider a CO2/oil mixture in the threephase region in Fig. 8.50. Moving up in pressure through the lower twophase region, the hydrocarbonrich liquid is in equilibrium with a CO2rich vapor phase. Near 1,000 psia (the dewpoint of the CO2rich vapor phase), a CO2rich liquid phase appears. As pressure increases through the threephase region, the volume of the CO2rich liquid phase increases, mostly at the expense of the CO2rich vapor phase. At few hundred psi higher than the onset of threephase behavior (the bubblepoint of the CO2rich liquid phase), the CO2rich vapor phase disappears. The onset of threephase behavior in a CO2/oil system is related to the upper CO2rich phase behaving like a component at its vapor pressure (see the earlier discussion on miscibility). Because the CO2rich phase is actually a mixture, the transition from vaporlike to liquidlike behavior occurs over a narrow range of pressures compared with the abrupt change experienced at the vapor pressure of a pure component. The volume of hydrocarbonrich liquid increases because of swelling in the lowpressure, vapor/liquid region and through the threephase region. When the CO2rich phase completes its transition from vaporlike to liquidlike behavior at the top of the threephase region, the oil phase stops swelling and starts shrinking as a result of strong extraction of C5 through C30 components by the CO2rich liquid phase. Practically, the effect of threephase behavior on the CO2 displacement process is small and can be ignored when modeling field performance. The threephase region usually is located in geological layers that have experienced CO2 breakthrough, some distance GASINJECTION PROCESSES
CO2 in Mixture, mol%
Fig. 8.49—Experimental px diagram for west Texas reservoir oil and up to 95% CO2 injection gas showing large region of asphaltene precipitation (from Ref. 4).
away from the producing wells, where reservoir pressure is between 1,000 and 1,500 psia. The threephase region may, however, cause serious problems for compositional simulators based on a twophase vapor/liquid equilibrium (VLE) algorithm.50 The problem
LOWER LIQUID PHASE, vol%
CO2 in Mixture, mol% Fig. 8.50—Experimental px diagram for Wasson crude oil and CO2 injection gas at 105°F (from Ref. 49). 19
8.5.5 CO2/Water Behavior. Chap. 9 covers methods for estimating CO2/water behavior. The two primary design considerations in a CO2injection project related to CO2/water phase behavior are the treatment of corrosion resulting from the formation of carbonic acid when CO2 mixes with water and the loss of injected CO2 resulting from the saturation of connate and injected water with CO2. Fig. 8.51 shows CO2 solubility in water and brines at 100°F. In fact, CO2 solubility in water is not very sensitive to temperature at temperatures greater than 100°F. Also, solubility increases only slightly at pressures greater than approximately 3,000 psia. Salinity has a significant effect on CO2 solubility, reducing the solubility in brine by approximately 30% for every 100,000 ppm of total dissolved solids. Water density and viscosity change only slightly when saturated with CO2. Enick and Klara53 reported on the effect of CO2 solubility in brine on compositional simulation of CO2 flooding. References
Fig. 8.51—Solubility of CO2 in pure water and NaCl brines at 100°F (adapted from Ref. 4; data from Ref. 52).
arises because, thermodynamically, the flash algorithm is searching for an equilibrium condition with only two phases. If the mixture being flashed actually exhibits threephase behavior according to the thermodynamic model being used, the VLE algorithm must choose one of several valid twophase solutions. Any of these twophase solutions satisfies the equilibrium constraints, but the twophase solutions merely represent local minimums in the Gibbs free energy, while the threephase solution represents a global minimum (see Chap. 4). Numerical instabilities arise when a gridblock oscillates between one twophase solution and another. In CO2 flooding, asphaltene precipitation could be a more serious multiphase problem than threephase VLL behavior. First, asphaltene precipitation occurs over a wider range of pressures and CO2 compositions, potentially causing reduced injectivity and productivity. Several authors40,48 have provided laboratory measurements showing that asphaltene precipitation occurs over a wide range of conditions. Unfortunately, few investigators have documented the quantitative effect of asphaltenes on reservoir performance. Christman and Gorell6 give results that indicate that reduced injectivities experienced in many tertiary CO2 projects can be modeled without accounting for reduced permeability and altered wettability caused by asphaltene precipitation. Still, serious operational problems associated with asphaltenes have been reported in field operations. Monger and Trujillo41 report on a comprehensive study of the deposition of organic solids during CO2 and richgas flooding. Few thermodynamic models have been suggested for predicting asphaltene precipitation. Kawanaka et al.51 propose a technique for predicting organic deposition of asphaltene, wax, and other solidlike materials that may precipitate from reservoir oils. The model uses a continuous distribution for the solid phase, and the authors provide results that give reasonable predictions for misciblesolvent processes. Finally, they give a comprehensive review of literature on asphaltene precipitation, measurements, and thermodynamic models for prediction of VLS phase behavior. 20
1. Stalkup, F.I. Jr.: Miscible Displacement, Monograph Series, SPE, Richardson, Texas (1984) 8. 2. Zick, A.A.: “A Combined Condensing/Vaporizing Mechanism in the Displacement of Oil by Enriched Gases,” paper SPE 15493 presented at the 1986 SPE Annual Technical Conference and Exhibition, New Orleans, 5–8 October. 3. Klins, M.A.: CO2 Flooding, Basic Mechanisms, and Project Design, Intl. Human Resources Development Corp., Boston (1984). 4. Goodrich, J.H.: “ Target Reservoirs for CO2 Miscible Flooding,” Report DOE/MC/0834117, U.S. DOE, Washington, DC (1980). 5. Holm, L.W.: “Status of CO2 and Hydrocarbon Miscible Oil Recovery Methods,” JPT (January 1976) 76. 6. Christman, P.G. and Gorell, S.B.: “Comparison of LaboratoryObserved and FieldObserved CO2 Tertiary Injectivity,” JPT (February 1990) 226; Trans., AIME, 289. 7. Harvey, M.T., Shelton, J.L., and Kelm, C.H.: “Field Injectivity Experiences With Miscible Recovery Projects Using Alternate RichGas and Water Injection,” JPT (September 1977) 1051. 8. Kay, W.B.: “ The EthaneHeptane System,” Ind. Eng. Chem. (1938) 30, 459. 9. Katz, D.L. et al.: Handbook of Natural Gas Engineering, McGrawHill Book Co. Inc., New York City (1959). 10. Sage, B.H., Lacey, W.N., and Schaafsma, J.G.: “Behavior of Hydrocarbon Mixtures Illustrated by a Simple Case,” API Bulletin (1932) 212, 119. 11. Peng, D.Y. and Robinson, D.B.: “A NewConstant EquationofState,” Ind. Eng. Chem. Fund. (1976) 15, No. 1, 59. 12. Koch, H.A. Jr. and Hutchinson, C.A. Jr.: “Miscible Displacements of Reservoir Oil Using Flue Gas,” Trans., AIME (1958) 213, 7. 13. Yellig, W.F. and Metcalfe, R.S.: “Determination and Prediction of CO2 Minimum Miscibility Pressures,” JPT (January 1980) 160; Trans., AIME, 269. 14. Orr, F.M. Jr. et al.: “Laboratory Experiments To Evaluate Field Prospects for CO2 Flooding,” JPT (April 1982) 888. 15. Auxiette, G. and Chaperon, I.: “Linear Gas Drives in HighPressure Oil Reservoirs Compositional Simulation and Experimental Analysis,” paper SPE 10271 presented at the 1981 SPE Annual Technical Conference and Exhibition, San Antonio, Texas, 4–7 October. 16. Holm, L.W. and Josendal, V.A.: “Mechanisms of Oil Displacements of CO2,” JPT (December 1974) 1427; Trans., AIME, 257. 17. Holm, L.W. and Josendal, V.A.: “Effect of Oil Composition on MiscibleType Displacement by CO2,” paper SPE 8814 presented at the 1980 SPE/DOE Symposium on Enhanced Oil Recovery, Tulsa, Oklahoma, 20–23 April. 18. Holm, L.W. and Josendal, V.A.: “Discussion of Determination and Prediction of CO2 Minimum Miscibility Pressures,” JPT (May 1980) 870. 19. Holm, L.W. and Josendal, V.A.: “Effect of Oil Composition on MiscibleType Displacement by Carbon Dioxide,” SPEJ (February 1982) 87. 20. Kuo, S.S.: “Prediction of Miscibility for the EnrichedGas Drive Process,” paper SPE 14152 presented at the 1985 SPE Annual Technical Conference and Exhibition, Las Vegas, Nevada, 22–25 September. 21. Novosad, Z. and Costain, T.G.: “New Interpretation of Recovery Mechanisms in Enriched Gas Drives,” J. Cdn. Pet. Tech. (March–April 1988) 21, No. 2, 54. 22. Novosad, Z., Sibbald, L.R., and Costain, T.G.: “Design of Miscible Solvents for a Rich Gas Drive—Comparison of Slim Tube and Rising Bubble Tests,” J. Cdn. Pet. Tech. (January–February 1990) 29, No. 1, 37. PHASE BEHAVIOR
23. Poettmann, F.H., Christiansen, R.L., and Mihcakan, I.M.: “Discussion of Methodology for the Specification of Solvent Blends for Miscible EnrichedGas Drives,” SPERE (February 1992) 154. 24. Sibbald, L.R., Novosad, Z., and Costain, T.G.: “Authors’ Reply to Discussion of Methodology for the Specification of Solvent Blends for Miscible EnrichedGas Drives,” SPERE (February 1992) 156. 25. Zhou, D. and Orr, F.M. Jr.: “Analysis of RisingBubble Experiments To Determine Minimum Miscibility Pressures,” SPE Journal (March 1998) 19. 26. Peterson, A.V.: “Optimal Recovery Experiments With N2 and CO2,” Pet. Eng. Intl. (November 1978) 40. 27. Benham, A.L., Dowden, W.E., and Kunzman, W.J.: “Miscible Fluid Displacement—Prediction of Miscibility,” Trans., AIME (1960) 219, 229. 28. Vogel, J.L. and Yarborough, L.: “ The Effect of Nitrogen on the Phase Behavior and Physical Properties of Reservoir Fluids,” paper SPE 8815 presented at the 1980 SPE Annual Technical Conference and Exhibition, Tulsa, Oklahoma, 20–23 April. 29. Jensen, F. and Michelsen, M.L.: “Calculation of First Contact and Multiple Contact Miscibility Pressures,” In Situ (1990) 14, 1. 30. Luks, K.D., Turek, E.A., and Baker, L.E.: “Calculation of Minimum Miscibility Pressure,” SPERE (November 1987) 501; Trans., AIME, 283. 31. Metcalfe, R.S., Fussell, D.D., and Shelton, J.L.: “A Multicell Equilibrium Separation Model for the Study of MultipleContact Miscibility in RichGas Drives,” SPEJ (June 1973) 147; Trans., AIME, 255. 32. Cook, A.B. et al.: “Effects of Pressure, Temperature, and Type of Oil on Vaporization of Oil During Gas Cycling,” Report RI 7278, U.S. Bureau of Mines, Washington, DC (1969). 33. Cook, A.B., Walter, C.J., and Spencer, G.C.: “Realistic K Values of C7+ Hydrocarbons for Calculating Oil Vaporization During Gas Cycling at High Pressure,” JPT (July 1969) 901; Trans., AIME, 246. 34. Johns, R.T., Orr, F.M. Jr., and Dindoruk, B.: “Analytical Theory of Combined Condensing/Vaporizing Gas Drives,” paper SPE 24112 presented at the 1992 SPE/DOE Symposium on Enhanced Oil Recovery, Tulsa, Oklahoma, 22–24 April. 35. Johns, R.T., Fayers, J.F., and Orr, F.M. Jr.: “Effect of Gas Enrichment and Dispersion on Nearly Miscible Displacement in Condensing/Vaporizing Drives,” paper SPE 24938 presented at the 1992 SPE Annual Technical Conference and Exhibition, Washington, DC, 4–7 October. 36. Wang, Y. and Orr, F.M. Jr.: “Analytical Calculation of Minimum Miscibility Pressure,” Fluid Phase Equilibria (1997) 139, 101. 37. Moses, P.L. and Wilson, K.: “Phase Equilibrium Considerations in Using Nitrogen for Improved Recovery From Retrograde Condensate Reservoirs,” JPT (February 1981) 256; Trans., AIME, 271. 38. Donohoe, C.W. and Buchanan, R.D. Jr.: “Economic Evaluation of Cycling GasCondensate Reservoirs With Nitrogen,” JPT (February 1981) 263; Trans., AIME, 271. 39. Renner, T.A. et al.: “Displacement of a RichGas Condensate by Nitrogen: Laboratory Corefloods and Numerical Simulations,” SPERE (February 1989) 52; Trans., AIME, 287.
GASINJECTION PROCESSES
40. Shelton, J.L. and Yarborough, L.: “MultiplePhase Behavior in Porous Media During CO2 or RichGas Flooding,” JPT (September 1977) 1171. 41. Monger, T.G. and Trujillo, D.E.: “Organic Deposition During CO2 and RichGas Flooding,” SPERE (February 1991) 17; Trans., AIME, 291. 42. Kennedy, G.C.: “PressureVolumeTemperature Relations in CO2 at Elevated Temperatures and Pressures,” Amer. J. Sci. (April 1954) 252, 225. 43. Kennedy, J.T. and Thodos, G.: “ The Transport Properties of CO2,” AIChE J. (1961) 7, 625. 44. Simon, R. and Graue, D.J.: “Generalized Correlations for Predicting Solubility, Swelling, and Viscosity Behavior of CO2/Crude Oil Systems,” JPT (January 1965) 102; Trans., AIME, 234. 45. Holm, L.W.: “CO2 Requirements in CO2 Slug and Carbonated Water Oil Recovery Processes,” Prod. Monthly (September 1963). 46. Holm, L.W. and O’Brien, L.J.: “CO2 Test at the MeadStrawn Field,” JPT (April 1971) 431. 47. O’Leary et al.: NitrogenDriven CO2 Slugs Reduced Costs,” Pet. Eng. Intl. (May 1979) 130. 48. Orr, F.M. Jr., Yu, A.D., and Lein, C.L.: “Phase Behavior of CO2 and Crude Oil in LowTemperature Reservoirs,” SPEJ (August 1981) 480. 49. Gardner, J.W., Orr, F.M. Jr., and Patel, P.D.: “ The Effect of Phase Behavior on CO2Flood Displacement Efficiency,” JPT (November 1981) 2067. 50. Perschke, D.R., Pope, G.A., and Sepehrnoori, K.: “Phase Identification During Compositional Simulation,” paper SPE 19442 available from SPE, Richardson, Texas (1989). 51. Kawanaka, S., Park, S.J., and Mansoori, G.A.: “Organic Deposition From Reservoir Fluids: A Thermodynamic Predictive Technique,” SPERE (May 1991)185. 52. McRee, B.C.: “How It Works, Where It Works,” Pet. Eng. Intl. (November 1977) 52. 53. Enick, R.M. and Klara, S.M.: “Effects of CO2 Solubility in Brine on the Compositional Simulation of CO2 Floods,” SPERE (May 1992) 253.
SI Metric Conversion Factors atm 1.013 250 E)05 +Pa °API 141.5/(131.5)°API) +g/cm3 bar 1.0* E)05 +Pa bbl 1.589 873 E*01 +m3 cp 1.0* E*03 +Pa@s ft 3.048* E*01 +m E*02 +m3 ft3 2.831 685 °F (°F*32)/1.8 +°C in. 2.54* E)00 +cm lbm 4.535 924 E*01 +kg psi 6.894 757 E)00 +kPa ton 9.071 847 E*01 +Mg *Conversion factor is exact.
21
Chapter 9
Water/Hydrocarbon Systems 9.1 Introduction The connate or “original” water found in petroleum reservoirs usually contains both dissolved salts (consisting mainly of NaCl) and solution gas (consisting mainly of methane and ethane). Initial water saturation can range from 5 to 50% of the pore volume (PV) in the netpay intervals of a reservoir (where production is primarily oil and gas). Higher water saturations are found in the aquifer and where water has swept oil or gas during a waterflood. From a reservoirdepletion point of view, the amount of water connected with a reservoir is as important as the properties of the water, particularly in materialbalance calculations where water expansion (compressibility times water volume) may contribute significantly to pressure support.1,2 From a production point of view, water mobility is important, requiring determination of water saturations, water viscosity, and formation volume factor (FVF). For surfaceprocessing calculations, water composition, water content in the produced wellstream, and conditions where water and hydrocarbons coexist must be defined. The three most important aspects of phase behavior involving water/hydrocarbon systems are mutual solubilities of gas and water, volumetric behavior of reservoir brines, and hydrate formation and treatment. Sec. 9.2 presents pressure/volume/temperature (PVT) correlations for water/hydrocarbon systems. Standard PVT properties—solution gas/water ratio, Rsw ; isothermal water compressibility, cw ; water FVF, Bw ; water viscosity, mw ; and water content in gas, rsw —are correlated in terms of pressure, temperature, and salinity by use of graphical charts and empirical equations. Correlations for water/hydrocarbon interfacial tension (IFT), s wh, are also presented. At very high temperatures and pressures, some correlations and the existing waterproperty data base are not adequate. Equations of state (EOS’s) have been used with reasonable success in predicting mutual solubilities and phase properties of hydrocarbon/water systems up to 400°F and greater than 10,000 psia,38 as discussed in Sec. 9.3. The effect of salinity on gas/water phase behavior has also been treated to some extent by the EOS methods.9 Sec. 9.4 covers the physical structure of hydrates and how to calculate conditions under which hydrates form. Hydrate formation can have a significant effect on production and surfacefacilities equipment and even on deep drilling. Water/hydrocarbon phase diagrams give the conditions of initial hydrate formation. These diagrams are particularly useful for designing a production system to avoid hydrate formation. The formation of hydrates can also be estimated with vapor/solid equilibrium ratios. WATER/HYDROCARBON SYSTEMS
9.2 Properties and Correlations Like all reservoir fluids, formationwater properties depend on composition, temperature, and pressure. Reservoir water is seldom pure and usually contains dissolved gases and salts. Total dissolved solids (TDS), usually consisting mainly of NaCl, ranges from 10,000 to [300,000 ppm; seawater salinity is [ 30,000 ppm. Water is limited as to how much salt it can keep in solution. The limiting concentration for NaCl brine is10 C *sw + 262, 180 ) 72T ) 1.06T 2 ,
. . . . . . . . . . . . . . . . (9.1)
with T in °C and C w in ppm. If reservoir temperature is known but a water sample cannot be obtained, this relation gives the limiting salinity of the reservoir brine. Salinity of a brine usually is less than 80% of the value given by Eq. 9.1. Otherwise, the best estimate of brine salinity can be taken from a neighboring reservoir in the same geological formation. Scale buildup in tubing and surface equipment is caused by the precipitation of salts in produced brine,11 usually calcium carbonate, calcium sulfate (e.g., gypsum), barium or strontium sulfates, and iron compounds. Temperature, pressure, total salinity, and salt composition are the primary variables determining the severity of scaling. Note that Eq. 9.1 should not be used to detect conditions that result in scale buildup. Dissolved gas in water is usually less than 30 scf/STB (approximately 0.4 mol%) at normal reservoir conditions. The effect of salt and gas content on water properties can be important, and the following discussion gives methods to estimate fluid properties in terms of temperature, pressure, dissolved gas, and salinity. Methods for estimating PVT properties of formation water usually are based on initial estimates of the purewater properties at reservoir temperature and pressure that are then corrected for salinity and dissolved gas. 9.2.1 Salinity. The cations dissolved in formation waters usually include Na+, K+, Ca++, and Mg++, and the anions include Cl *, SO** 4 , . Most formation waters contain primarily NaCl. Susand HCO ** 3 pended salts, entrained solids, and corrosioncausing bacteria may also be present in reservoir waters, but these constituents usually do not affect formationwater PVT properties. The geochemistry of formation waters can be useful in detecting foreignwater encroachment and in determining its source. Table 9.1 gives example compositions of reservoir brines. Salinity defines the concentration of salts in a saline solution (brine) and may be specified as one of several quantities: weight fraction, w s; mole fraction, x s; molality, c sw; molarity, c sv; parts per million by weight, C sw; and parts per million by volume, C sv. Table 1
TABLE 9.1—EXAMPLE COMPOSITIONS OF FORMATION BRINES DodsonStanding13 Component
Seawater (ppm)
Brine A (ppm)
Brine B (ppm)
Arun Field (mg/L)
Gulf Coast Frioa (mg/L)
Kansas Wilcoxb (mg/L)
Kansas Wilcoxa (mg/L)
Sodium (Na)
10,560
3,160
12,100
5,212
40,600
10,800
142,500
Calcium (Ca) Magnesium (Mg)
400
58
520
80
5,100
790
14,400
1,270
40
380
5
1,000
5,560
68,500
Sulfate (SO4)
2,650
0
5
262
110
80
300
Chloride (Cl)
18,980
4,680
20,000
7,090
69,100
10,870
142,600
140
696
980
1,536
990
20
530
0
0
130
0
0
0
3
65
0
0
0
0
80
350
515
0
0
0
0
0
0
Bicarbonate (HCO3) Iodide (I) Bromide (Br) Others Total
34,580
8,630
34,110
14,190
116,900
28,200
369,180
Specific gravity
1.0243c
1.006d,e
1.024d,e
1.014d
1.086d,e
1.015d
1.140d
aMaximum saltcontaining composition reported for field/formation. bMinimum saltcontaining composition reported for field/formation. cAt 20°F. dAt 60°F. eEstimated with Eq. 9.3.
TABLE 9.2—DEFINITIONS OF SALT CONCENTRATIONS Symbol
Unit
Definition
Weight fraction
ws
g/g
m sńǒm s ) m owǓ
Mole fraction
xs
g mol/g mol
n sńǒn s ) n owǓ
Molality
csw
g mol/kg
10 3n sńm ow
Molarity
csv
g mol/L
10 3n sńV w
Quantity
ppm, weight basis
Csw
mg/kg
10 m sńǒm s ) m owǓ
ppm, volume basis
Csv
mg/L
10 6n sńV w
6
o m s + mass salt, m o w + mass pure water, n s + moles salt, n w + moles pure water, and V w + volume brine mixture.
9.2 formally defines these quantities; in the table, m s +mass of salt in grams, m ow+mass of pure water in grams, n s +moles of salt in gram moles, n ow+moles of pure water in gram moles, and V w+volume of the brine mixture in cubic centimeters. Some common conversions for the various concentrations are C sv + ò wC sw ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.2a)
C C sw + ò sv + 10 6 w s , w c sw +
17.1 , 10 6 C *1 sw * 1
and C sw +
. . . . . . . . . . . . . . . . . . . . . . . . . (9.2b) . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.2c)
10 6 , 17.1 c *1 sw ) 1
. . . . . . . . . . . . . . . . . . . . . . . (9.2d)
where the Eqs. 9.2c and 9.2d apply for NaCl brines. If brine density, ò w, at standard conditions (14.7 psia and 60°F) is not reported, it can be estimated from the RoweChou12 density correlation for NaCl. ò wǒ p sc, T scǓ + ǒ1.0009 * 0.7114w s )
0.26055 ws2
Ǔ
*1
,
with ò w in g/cm3 and w s in weight fraction TDS. For many engineering applications, ò w+1 g/cm3 is assumed and the mass of salt is considered negligible compared with the mass of pure water, resulting in the approximate relations c sv [ c sw + c s ,
2
10 *6ǓC s ,
ln x i + ln f i * ln H i *
. . . . . . . . . . . . . . . . . . . . . . . (9.4)
v~ i ǒ p * p vwǓ , RT
. . . . . . . . . . . . . (9.5)
where x i+solubility of gas Component i in water, f i +partial fugacity, H i+Henry’s constant, and v~ i +modified molar volume. H i and v~ i are nonlinear functions of temperature. Cramer18 uses a similar approach to correlate gas solubilities for methane/water and methane/NaClbrine systems over a wide range of pressures, temperatures, and salinities. At reservoir conditions, the solubility of methane in water and the effect of salinity are the most important variables affecting water properties. The following empirical equation gives a reasonable fit of the Culberson and McKetta14,19 solubility data for methane in pure water at conditions 100tTt350°F and 0tpt10,000 psia,
ƪȍǒȍ Ǔ ƫ 3
x C + 10 *3
3
A i jT j p i ,
i+0
. . . . . . . . . . . . . . . (9.6)
j+0
where A00+0.299, A01+*1.273 10*3, A02+0.000, A03+0.000, A10+2.283 10*3, A11+*1.870 10*5, A12+7.494 10*8, A13+*7.881 10*11, A20+*2.850 10*7, A21+2.720 10*9, A22+*1.123 10*11, A23+1.361 10*14, A30+1.181 10*11, A31+*1.082 10*13, A32+4.275 10*16, and A33+*4.846 10*19, with T in °F and p in psia. Gas solubility expressed as a solution gas/water ratio, R sw at standard conditions is R sw + 7, 370
C sw [ C sv + C s ,
10*6 applies for NaCl brines.
9.2.2 Gas Solubilities in Water/Brine. The solubility of natural gases in water is rather complicated to estimate from empirical correlations. However, the effect of gas solubility usually is minor except at high temperatures. At temperatures less than approximately 300°F and pressures less than 5,000 psia, solubility usually is less than 0.4 mol%, or approximately 30 scf/STB. According to Dodson and Standing’s13 results, this amount of dissolved gas causes an increase of approximately 25% in water compressibility (e.g., from 3.8 10*6 to 4.8 10*6 psi*1). Experimental gas solubilities for C1 through C4 hydrocarbons, nonhydrocarbons, natural gas, and a few binaries and ternaries are available in the literature. Figs. 9.1 through 9.3 present some of these data. Kobayashi and Katz17 give a method for estimating gas solubilities in pure water based on Henry’s law for dilute solutions.
1
. . . . . . . . . . . . . . . . . . . . . (9.3)
and c s [ ǒ17.1
where the constant 17.1
xg [ 7, 370 x g , . . . . . . . . . . . . . . . . . (9.7) 1 * xg
with R sw in scf/STB. PHASE BEHAVIOR
ks +
lim
cs ³ 0
ƪ
c *1 log s
ƫ
ǒf R Ǔ i w , ǒf R Ǔo i w
. . . . . . . . . . . . . . . . . (9.8)
where k s+Setchenow constant, c s+salt concentration, and (f R i )w o and (f R i ) w+fugacity coefficients of Component i at infinite dilution in the salt solution and in pure water, respectively. Both molality and molarity have been used in the literature for defining Setchenow constants; however, molality, c sw, is now considered to be the preferred concentration. The unit for the Setchenow constant is M*1 (i.e., kg/g mol), where M+molarity. The ratio of infinitedilution fugacity coefficients is traditionally assumed to give an accurate estimate of the ratio of solubilities, yielding the relation xg R sw ǒ *k c [ x o + 10 s s [ 10 * 17.1 R osw g
Fig. 9.1—Gassolubility data for methane in pure water (adapted from Ref. 14).
10 *6Ǔk s C s
,
. . . . . . . . . (9.9)
where R osw +solubility of gas in pure water and R sw +solubility of gas in brine. For k su0, the gas solubility is less in brines than in pure water, a fact that has led to the term “saltingout coefficient” for k s. The Setchenow constant is more or less independent of pressure but is a strong function of temperature. Cramer18 gives a detailed treatment of Setchenow (and Henry’s) constants for the C1/NaCl system using data at temperatures up to 570°F and pressures up to 2,000 psia. He proposes the temperature dependence of k s shown in Fig. 9.4. This figure also shows values of k s reported elsewhere for the C1/NaCl system, illustrating the relatively large uncertainty in saltingout coefficients, even for such a welldefined system. Søreide and Whitson9 give a bestfit relation for the Cramer correlation.
Amirijafari and Campbell20 give experimental component solubilities and an empirical method for calculating the total gas solubility of the C1/C2/C3 ternary mixture. However, for most applications gas solubility can be estimated by assuming that the gas consists only of methane. A standard twophase flash calculation with a cubic EOS gives a surprisingly accurate prediction of gas solubilities, as discussed in Sec. 9.3. This approach is the recommended procedure for estimating gas solubilities of hydrocarbon/water/brine mixtures at high pressures and temperatures. 9.2.3 Salinity Correction for Solubilities. Refs. 9 and 21 give the Setchenow (sometimes written Secenov) relation for correcting hydrocarbon solubility in pure water for salt content.
Pressure, psia
Fig. 9.2—Gassolubility data for natural gas in pure water (adapted from Ref. 13). WATER/HYDROCARBON SYSTEMS
Pressure, psia
1,000
Fig. 9.3—Gassolubility data for CO2 in pure water (adapted from Refs. 15 and 16). 3
f f f Methane V V V Ethane Propane nbutane
Fig. 9.4—Temperature dependence of the Setchenow (saltingout) coefficient for light hydrocarbons (Ref. 9).
(k s) C
1*NaCl
+ 0.1813 * ǒ7.692 ) ǒ2.6614
10 *4ǓT
10 *6ǓT 2 * ǒ2.612
10 *9ǓT 3,
. . . . . . . . . . . . . . . . . . . . (9.10) M*1
and T in °F. Using relations suggested by Pawliwith k s in kowski and Prausnitz21 relating k s of methane to k s of other hydrocarbons, Søreide and Whitson9 propose the following relation for Hydrocarbon i. k si + (k s) C
1*NaCl
) 0.000445ǒ T bi * 111.6 Ǔ ,
. . . . . . . (9.11)
with k s in M*1 and the normal boiling point, T bi, in K. Fig. 9.4 shows the temperature dependence of k s for light hydrocarbons (C2 through C4) based on Eqs. 9.10 and 9.11. Clever and Holland22 give saltingout correlations for C1/NaCl and CO2/NaCl systems. The correlation for CO2/NaCl is (k s) CO
2*NaCl
+ 0.257555 * ǒ0.157492
* ǒ0.253024
10 *3ǓT
10 *5ǓT 2 ) ǒ0.438362
10 *8ǓT 3 ,
. . . . . . . . . . . . . . . . . . (9.12) M*1.
The temperature range for Eq. 9.12 is with T in K and k s in 40tTt660°F. The Setchenow coefficient varies somewhat with pressure for the CO2/NaCl system, thereby making Eq. 9.12 less accurate than hydrocarbon/NaCl correlations. Fig. 9.5 illustrates the effect of salts other than NaCl on lowpressure solubilities by use of lines of equal gas solubility vs. molality of the salt, where NaCl is the reference salt. 9.2.4 Equilibrium Conditions in Oil/Gas/Water Systems. All phases (oil, gas, and water) in a reservoir are initially in thermodynamic equilibrium. This implies that the water phase contains finite quantities of all hydrocarbon and nonhydrocarbon components found in the hydrocarbon phases and that the hydrocarbon phases contain a finite quantity of water. The amount of lighter compounds (C1, C2, N2, CO2, and H2S) in the water phase can be significant and depends mainly on the amount of each component in the hydrocarbon phase(s). The amount of C3+ hydrocarbons found in water is usually small and can be neglected. The K value representing the ratio of the mole fraction of Component i in the hydrocarbon phase to the mole fraction of Component i in the water phase ( K i + z i,HCńx i,aq) is approximately constant at a given pressure and temperature, independent of overall hydrocar4
Fig. 9.5—Lines of equal gas solubility for various salts with NaCl as a reference (adapted from Ref. 23).
bon composition and whether the hydrocarbon is single phase or two phase. For example, the amount of methane dissolved in water for a methanerich natural gas will be higher than the amount of methane dissolved in water for an oil (above its bubblepoint). Furthermore, the amount of methane dissolved in water for a gas/oil system with overall methane content of 40 mol% will probably be about the same as for a singlephase oil with 40 mol% methane. An oil that is undersaturated (with respect to gas) is still in equilibrium with the water phase. When pressure is lowered, a new equilibrium state is reached between the undersaturated oil and water. The result is that some of the methane will move from the water to the oil (without free gas forming); i.e., the solution gas/water ratio decreases. At some lower pressure, the oil will reach its bubblepoint and further reduction in pressure will yield two sources of free gas: gas coming out of solution from the oil and gas coming out of solution from the water. Therefore, for an undersaturatedoil reservoir, the solution gas/ water ratio of reservoir brine will decrease continuously from the initial reservoir pressure to the reservoiroil bubblepoint pressure and even further at lower pressures. Correspondingly, the reservoiroil solution gas/oil ratio will increase (albeit slightly) from initial to bubblepoint pressure and then decrease below the bubblepoint. An EOS must be used to quantify the changing solution gas/water and solution gas/oil ratios in this situation. Fig. 9.6 shows calculations with an EOS that illustrate the relative gas solubility in a reservoir oil and a reservoir gas. The oil and gas compositions are in equilibrium at approximately 3,500 psia. At higher pressures, the gas solubility in water is higher in the gas/water system than in the oil/water system. At less than 3,500 psia, three phases will exist in either system and the twophase flash calculation gives only approximate solubilities on the basis of treating the hydrocarbon as a single phase. 9.2.5 Water/Brine FVF and Compressibility. The FVF of reservoir water, Bw, depends on pressure, temperature, salinity, and dissolved gas. Fig. 9.7 gives Dodson and Standing’s13 results for pure water with and without solution gas. Contrary to saturatedoil volumetric behavior, the liquid volume of a gassaturated water increases with decreasing pressure. That is, the expansion caused by isothermal compressibility is larger than the shrinkage caused by gas coming out of solution. The pressure dependence of Bw that Dodson and Standing give for gassaturated water/brine applies to all gas and oil reservoirs that have appreciable solution gas. Even if the oil is undersaturated, as discussed earlier, the solution gas/water ratio decreases continuousPHASE BEHAVIOR
T+258°F Gas/Water
Oil/Water
Gas/Oil Bubblepoint, 3,500 psia
Fig. 9.6—Gas dissolved in water for reservoiroil/water and reservoirgas/water systems, EOS twophase calculations.
ly from the initial pressure to the oil bubblepoint pressure and further thereafter. This precludes the pressure dependence of water FVF shown in Fig. 9.8, where a discontinuity occurs at some bubblepoint condition. The only way a reservoir brine could have this behavior is if the hydrocarbons that originally saturated the brine had migrated away completely and the reservoir pressure subsequently increased with further burial (creating an undersaturated condition for the brine with respect to hydrocarbon components). The FVF of brine at atmospheric pressure, reservoir temperature, and without dissolved gas, B ow, is ò wǒ p sc, T scǓ v oǒ p sc, TǓ B ow + o + w . v wǒ p sc, T scǓ ò wǒ p sc, TǓ
. . . . . . . . . . . . . . . . (9.13)
Long and Chierici24,25 give experimental data and correlations for the density of pure water and NaClbrine solutions, although the proposed correlations extrapolate poorly at temperatures greater than approximately 250°F. Kutasov26 gives several accurate correlations for FVF’s of pure water, but the equation for Bw results in a constant isothermal compressibility that is independent of pressure. Rowe and Chou12 give the following correlation for water and NaClbrine specific volume at zero pressure (also applicable at atmospheric pressure). v woǒ p sc, TǓ +
Fig. 9.7—FVF of pure water with and without natural gas (adapted from Ref. 13).
which, when integrated, gives B *wǒ p, TǓ +* ln o B wǒ p sc, TǓ
p
ŕ c ǒ p, TǓ dp. * w
. . . . . . . . . . . . . . . . (9.16)
0
With the compressibility data reported by Rowe and Chou covering the conditions 70tTt350°F, 150tpt4,500 psia, and 0t w st0.3, a general correlation for the compressibility of a brine (without solution gas), c *w, is c *wǒ p, T Ǔ + ǒ A 0 ) A 1 p Ǔ
*1
,
1 + A 0 ) A 1w s ) A 2w 2s , ò woǒ p sc, TǓ
where A 0 + 5.91635 * 0.01035794T ) ǒ0.9270048
10 *5ǓT 2
* 1, 127.522T *1 ) 100, 674.1T *2 , A 1 + * 2.5166 ) 0.0111766T * ǒ0.170552
10 *4ǓT 2 ,
and A 2 + 2.84851 * 0.0154305T ) ǒ0.223982
10 *4ǓT 2 ,
. . . . . . . . . . . . . . . . . . . . (9.14) v wo
in cm3/g, T in K, and w s in weight fraction of NaCl. The efwith fect of pressure on FVF can be calculated by use of the definition of water compressibility,
ǒ Ǔ
ēB w c *w + * 1 B w ēp
C s ,T ,
. . . . . . . . . . . . . . . . . . . . . . . (9.15)
WATER/HYDROCARBON SYSTEMS
Fig. 9.8—Effect of gas solubility on water FVF at saturated and undersaturated conditions, EOS twophase calculations. 5
Fig. 9.10—Effect of CO2 solubility (in terms of saturation pressure) on water viscosity.
9.4.6 Water/Brine Viscosity. Fig. 9.9 presents the viscosities of pure water and NaCl brines as functions of temperature and salinity. The following equations (except for the pressure correction A0) are presented by Kestin et al.,29 who report an accuracy of "0.5% in the range 70tTt300°F, 0tpt5,000 psia, and 0t C swt300,000 ppm (0t c swt5 M). m w + ǒ1 ) A 0 pǓm *w ,
Fig. 9.9—Water/NaClbrine viscosity as a function of temperature and salinity.
where A 0 + 10 6ƪ0.314 ) 0.58w s ) ǒ1.9 *ǒ1.45
A1 +
ǒ
A 1) 1 p A0
Ǔ
ǒ1ńA1Ǔ ,
. . . . . . . . . . (9.18)
. . . . . . . (9.19)
with R sw in scf/STB. This relation fits the DodsonStanding data at 150, 200, and 250°F but overpredicts the effect of dissolved gas at 100°F. Dodson and Standing also give a correction for the effect of dissolved gas on water/brine compressibility. c wǒ p, T, R swǓ + c *wǒ p, TǓ ǒ1 ) 0.00877 R swǓ ,
. . . . . . . . (9.20)
with R sw in scf/STB. This relation is valid only for undersaturatedoil/water systems at higher than oil bubblepoint pressure. For gas/ water systems and saturatedoil/water systems, the total compressibility effect is given by the Perrine formula,28
6
T, R sw
A2 +
ȍa
mo log m ow + w20
1.5Ǔ , B wǒ p, T, R swǓ + B *wǒ p, TǓǒ1 ) 0.0001 R sw
ǒ Ǔ
i 1i c sw
,
i 2 ic sw
,
i+1
where A0 and A1 are given by Eq. 9.17. Eq. 9.18 results in water and brine densities that are within 0.5% of values given by Rogers and Pitzer’s27 highly accurate correlation for 60tTt400°F, 0tpt15,000 psia, and 0t C st300,000 ppm. For the same range of conditions, Eq. 9.17 calculates isothermal compressibilities within approximately 5% of Rogers and Pitzer’s values. With Dodson and Standing’s13 data for pure water saturated with a natural gas, an approximate correction for dissolved gas on water/ brine FVF at saturated conditions is
ēB w c tw + * 1 B w ēp
ȍa
[0.8 ) 0.01(T * 90) exp(* 0.25c sw)],
3
. . . . . . . . . . . . . . . . . (9.17)
w s in weight fraction of NaCl. with Solving Eq. 9.16 for the FVF of a brine without solution gas, B *w, gives +
A 0 + 10
10 *6ǓT 2ƫ
B owǒ p sc, TǓ
w20
*3
3
10 *4ǓT
in psi*1, p in psia, T in °F, and
B *wǒ p, TǓ
w
i+1
and A 1 + 8 ) 50w s * 0.125w sT, c *w
mo m *w log m o + A 1 ) A 2 log m ow ,
ǒ Ǔ.
B g ēR sw ) 1 5.615 B w ēp
T
4
i+1
ǒ20 * T Ǔ i , 3i 96 ) T
and m ow20 + 1.002 cp,
. . . . . . . . . . . . . . . . . . . . . . . . . . (9.22)
where a11+3.324 10*2, a12+3.624 10*3, a13+*1.879 10*4, a21+*3.96 10*2, a22+1.02 10*2, a23+*7.02 10*4, a31+ 1.2378, a32+*1.303 10*3, a33+3.060 10*6, a34+2.550 10*8, with m in cp, T in °C, and p in MPa. Kestin et al.’s pressure correction A0 contains 13 constants and does not extrapolate well at high temperatures. The pressure correction for A0 in Eq. 9.22 is more wellbehaved, with only small deviations from the original Kestin et al. correlation at low temperatures. The effect of dissolved gas on water viscosity has not been reported. Intuitively, one might suspect that water viscosity decreases with increasing gas solubility, although Collins30 suggests that dissolved gas may increase brine viscosity. As Fig. 9.10 shows, systems saturated with CO2 show an increase in viscosity with increasing gas solubility. 9.2.7 Solubility of Water in Natural Gas. Fig. 9.11 shows the solubility of pure water in methane. McKetta and Wehe31 give two chart inserts for correcting purewater solubilities for salinity and gas gravity (based mainly on Dodson and Standing’s13 values). A bestfit equation for these charts is y w + y ow A g A s, ln y ow +
. . . . (9.21)
ȍa
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.23a)
0.05227p ) 142.3 ln p * 9, 625 T ) 460
* 1.117 ln p ) 16.44,
. . . . . . . . . . . . . . . . . (9.23b) PHASE BEHAVIOR
Dewpoint of Natural Gas J.J. McKetta and A.H. Wehe, U. of Texas (1958)
Fig. 9.11—Water solubility in natural gases, including gascomposition and salinity effects (adapted from Ref. 31).
Ag + 1 )
(1.55
g g * 0.55 10 4)g g T *1.446 * (1.83
10 4) T *1.288
,
. . . . . . . . . . . . . . . . . . (9.23c) A s + 1 * ǒ2.222 and A s + 1 * ǒ3.92
10 *6ǓC s ,
. . . . . . . . . . . . . . . . . (9.23d)
10 *9 ǓC 1.44 s ,
. . . . . . . . . . . . . . . (9.23e)
with T in °F, p in psia, and C s in ppm or mg/L. Eq. 9.23 yields an absolute average deviation of 2.5% for y ow, with a maximum error WATER/HYDROCARBON SYSTEMS
less than 10% for 100tTt460°F and 200tpt10,000 psia. Eq. 9.23d is from the DodsonStanding correlation and is not recommended. Eq. 9.23e is from the Katz et al.32 correlation and is recommended. Mole fraction of water in gas, y w, can be converted to a water/gas ratio, r sw , with r sw + 135
yw [ 135y w , 1 * yw
. . . . . . . . . . . . . . . . . . . (9.24)
where r sw is in STB/MMscf. Replacing the constant 135 with 47,300 yields r sw in lbm/MMscf. 7
ow
Fig. 9.12—Water/brine/oil IFT data correlated with the McLeod parameter (adapted from Ref. 33).
Temperature, °F
The correction term for salinity that Dodson and Standing13 proposed is based on limited results for one lowsalinity brine. The Katz et al.32 salinity correction is based on lowering of vapor pressure for brine solutions at 100°C, where the assumption is made that p vw(100°C, C s) yw y ow [ p ovw(100°C) ,
. . . . . . . . . . . . . . . . . . . . . . . . . (9.25)
where p vw+brine vapor pressure and p w+purewater vapor pressure, both measured at 100°C. Very little data are available to confirm these two salinity corrections. However, EOS calculations indicate that the Katz et al. correlation is probably valid up to M[3; at higher molalities, the EOScalculated ratio y wńy is less than that predicted by the Katz et al. correlation (see Sec. 9.3). Finally, water dissolved in reservoir gas and oil mixtures will not contain salts (i.e., it is fresh water), a fact that can help in identifying where produced water comes from. 9.2.8 Water/Brine/Hydrocarbon IFT. The IFT of water/hydrocarbon systems, Ds wh , varies from approximately 72 dynes/cm for water/ brine/gas systems at atmospheric conditions to 20 to 30 dynes/cm for water/brine/stocktankoil systems at atmospheric conditions. The variation in s wh is nearly linear with the density difference between water and the hydrocarbon phase Ds wh (i.e., Ds wo or Ds wg ), where s wh+72 dynes/cm at Ds wh + Ds wg +1. This can be expressed in equation form as s wh + s o ) (72 * s o)Dò wh , . . . . . . . . . . . . . . . . . . . (9.26) where s o+intercept at Dò wh +0. Ramey33 proposes a correlation for s wh based on the Macleod parameter s ¼ńDò. This parameter was plotted vs. Dò (Fig. 9.12) with data for brines with stocktank oil, saturated and undersaturated reservoir oils, and natural gases. Eq. 9.26, where s o+15, represents Ramey’s graphical correlation surprisingly well. A nearexact fit of his correlation is s wg + 20 ) 36 Dò wh .
Fig. 9.13—Water/brine/oil IFT data correlation (adapted from Ref. 34).
ported by various authors show considerable scatter, and it seems that any correlation will give only approximate IFT values for such systems until consistent data become available. Mutualsolubility effects of gas dissolved in water and water dissolved in gas may affect IFT’s, perhaps explaining some of the difference in methane/ brine and methane/water IFT’s in Fig. 9.14. Otherwise, the seemingly erratic behavior of some water/brine/oil IFT data may be explained by aromatic compounds and asphaltenes. Also, crudeoil samples exposed to atmospheric conditions for long periods of time may experience oxidation that can affect IFT measurements.
. . . . . . . . . . . . . . . . . . . . . . . (9.27)
Ramey’s data that do not lie on his general correlation are accurately represented by Eq. 9.26, with s o ranging from 5 to 30. Fig. 9.13 shows a graphical correlation for s wg given by Standing34 for water/ brine/methane systems (apparently based on Hocott’s35 naturalgas/brine data). Firoozabadi and Ramey36 consider the IFT of water and hydrocarbons using data for distilled water and pure hydrocarbons. They arrive at a graphical relation similar to Ramey’s33 original correlation, with the addition of reduced temperature as a correlating parameter. Unfortunately, their correlation does not predict water/ brine/oil IFT’s with more accuracy than the original Ramey correlation (or Eq. 9.27). As Fig. 9.14 shows, water/gas IFT’s re8
Salt Concentration, ppm
Fig. 9.14—Methane/water and methane/brine IFT’s. PHASE BEHAVIOR
which can be used for 0.44t T rt0.72 (60tTt400°F). Alternatively, the SøreideWhitson9 relation for a H O can be used with the 2 PengRobinson 38 EOS (PR EOS).
ƪ
1.1Ǔ ǒ a 0.5 H O + 1 ) 0.453 1 * T r H O 1 * 0.0103c sw 2
ǒ
2
) 0.0034 T r*3 *1 H O
Fig. 9.15—Purewater and NaClbrine vaporpressure curves.
9.3 EOS Predictions Mutual solubilities and volumetric properties of water/hydrocarbon systems can be predicted with reasonable accuracy with one of several modifications to existing cubic EOS’s. Other types of EOS’s also have been applied to these systems but do not show a clearly superior predictive capability. Although cubic EOS’s are not widely used for reservoir water/hydrocarbon systems, this approach eventually is expected to replace the empirical correlations currently being used. To improve vaporpressure predictions of water (and solubilities of water in the nonaqueous phase), Peng and Robinson37 proposed a modified correction term, a (applied to EOS Constant a), for water. aH
2O
+ ƪ1.008568 ) 0.8215ǒ1 *
0.5 T rw
Ǔƫ , 2
. . . . . . . . (9.28)
2
Ǔƫ .
. . . . . . . . . . . . . . . . (9.29)
Eq. 9.29 predicts purewater vapor pressures within 0.2% of steamtable values for 0.44t T rwt1 (i.e., Tu60°F) and can be used to predict vapor pressures of NaCl solutions with the same accuracy. Fig. 9.15 shows vapor pressures of pure water and NaClbrine solutions reported by Haas.10 With a correction for salinity in the a term, the predicted water solubilities in nonaqueous phases are expected to improve. The most important modification of existing cubic EOS’s for water/hydrocarbon systems is the introduction of alternative mixing rules for EOS Constant A, where different binaryinteraction parameters (BIP’s), k ij, are used for the aqueous and nonaqueous (hydrocarbon) phases. Peng and Robinson37 propose a simple EOS modification for hydrocarbon/water systems; namely, they define two sets of k ij: k ij,HC for the hydrocarbon phase(s) and k ij,aq for the aqueous phase. EOS Constant A is therefore calculated differently for the hydrocarbon and aqueous phases,
ȍȍy N
A HC +
N
i,HC y j,HC
A i A j ǒ1 * k i j,HCǓ
i+1 j+1
ȍȍx N
and A aq +
N
i,aq x j,aq
A i A j ǒ1 * k i j,aqǓ ,
. . . . . . . . . (9.30)
i+1 j+1
TABLE 9.3—RECOMMENDED BIP’s FOR THE PR EOS TO PREDICT SOLUBILITIES IN WATER/HYDROCARBON SYSTEMS* Aqueous Phase k ij, aq + ǒ1 ) a 0c swǓA 0 ) ǒ1 ) a 1c swǓA 1T ri ) ǒ1 ) a 2c swǓA 2T 2ri ,
Hydrocarbons
where a 0 + 0.017407, a 1 + 0.033516, a 2 + 0.011478 A 0 + 1.112 * 1.7369w *0.1 , A 1 + 1.1001 ) 0.83w i i A2+*0.15742*1.0988wi , i+hydrocarbons, and j+water/brine. 0.75Ǔ Ǔ ǒ k ij, aq + * 1.70235ǒ1 ) 0.025587c 0.75 sw ) 0.44338 1 ) 0.08126c sw T ri ,
N2
where i+N2 and j+water/brine.
Ǔ k ij, aq + * 0.31092ǒ1 ) 0.15587c 0.75 sw
CO2
Ǔ ) 0.2358ǒ1 ) 0.17837c 0.98 sw T ri * 21.2566 exp(* 6.7222T r * c sw), where i+CO2 and j+water/brine. k ij, aq + * 0.20441 ) 0.23426T ri , where i + H 2S and j + waterńbrine.
H2 S
Nonaqueous Phase i
kij ,HC, where j+water
C1
0.4850
C2
0.4920
C3
0.5525
C4
0.5091
C5)
0.5000
N2
0.4778
CO2
0.1896
H2 S
0.19031*0.05965Tri
Acentric factors w used in developing hydrocarbon/water BIP’s are C1+0.0108, C2+0.0998, C3+0.1517, and C4+0.1931. *Modified PengRobinson a term for water/brine, Eq. 9.29.
WATER/HYDROCARBON SYSTEMS
9
Brine Salinity, Csw fff 0 V V V 0.86 1.71 2.57 +++ 3.42 KKKK 5.13
D D D Katz et al.32 Correlation V V V Calculated at 100°F Calculated at 250°F
Fig. 9.16—Predicted gasphase water solubilities for methane/ NaClbrine mixtures at 250°F determined with the general aw term (Eq. 9.31).
respectively, where y i,HC +hydrocarbon composition (gas or oil) and x i,aq+waterphase composition. Using two sets of k ij has been applied successfully to correlate mutual solubilities of hydrocarbon/ water and nonhydrocarbon/water binary systems. Table 9.3 gives recommended k ij relations for aqueous and nonaqueous phases for the PR EOS, where these interaction coefficients must be used with the general a H O relation (Eq. 9.29). The CO2/water/brine correla2 tion gives the best results at pressures less than approximately 5,000 psia because data in this region have been given more weight in development of the correlation. Considerable data on solubilities of hydrocarbon and nonhydrocarbon gases in brine solutions were used in making the salinity corrections for aqueousphase k ij. Similar data were not available for solubilities of water in the nonaqueous phase for mixtures containing brines. Until more data become available, it will be necessary to assume that the effect of salinity is adequately treated by the modified a H O term (Eq. 9.30). 2 Fig. 9.16 shows predicted water solubilities for methane/NaClbrine mixtures with varying salt concentration. The predicted reduction in water solubility for mixtures containing brine, relative to solubility for mixtures containing pure water, is more or less independent of pressure and temperature. Fig. 9.17 correlates the ratio y wńy ow calculated by the modified PR EOS (with a H O from Eq. 2 9.30) vs. salinity. The effect of salinity is clearly less than that predicted by the DodsonStanding13 correlation (Eq. 9.23d), whereas the Katz et al.32 correlation (Eq. 9.23e) appears to be consistent with the EOS calculations up to M[3. Simultaneous application of aqueous and nonaqueousphase interaction coefficients requires modification of the standard EOS implementation (which uses a single set of k ij). Figs. 9.18 through 9.22 show the accuracy of this approach for mutualsolubility predictions of binaries and naturalgas/water/brine mixtures, suggesting that the required modification is probably warranted. A standard implementation of the PR EOS can still be used with the BIP’s in Table 9.3. If only gas solubility in the water phase is needed, accurate gas solubilities can be predicted with the aqueousphase k ij,aq for both phases; however, calculated hydrocarbonphase composition will not be accurate. Likewise, if only water solubility in the hydrocarbon phase is needed, the hydrocarbonphase k ij,HC can be used for both phases, but calculated aqueousphase compositions will not be accurate in this case. Fig. 9.23 compares experimental solubilities for the methane/water system with results predicted by the modified PR EOS (with two sets of k ij) and by the original PR EOS with a single set of k ij. 10
Fig. 9.17—Effect of salinity on gasphase water solubility for methane/NaClbrine mixtures determined with the general aw term (Eq. 9.31).
Composition and densitydependent mixing rules have also been proposed for modifying cubic EOS’s for water/hydrocarbon systems. Panagiotopoulos and Reid’s39 linear compositiondependent mixing rule has received considerable interest. Unfortunately, as Kistenmacher and Michelsen40 point out, it violates several fundamental thermodynamic conditions. Enick et al.8 propose temperaturedependent correction terms for both EOS Constants A and B of water, together with a linear compositiondependent mixing rule for Constant A. With this approach, they successfully describe multiphase equilibria for a multicomponent water/oil/CO2 system. Several noncubic EOS’s35,41,42 have been proposed for water/ hydrocarbon systems, including conventional activitycoefficient models that are limited to relatively low pressures and more general electrolyte EOS models. However, these models do not appear to be better than the simpler modifications of cubic EOS’s. 9.4 Hydrates Gas hydrates are solutions of gases in crystalline solids called clathrates. Gas molecules occupy the void spaces (cages) in the watercrystal lattice. Hydrates can form at temperatures considerably higher than the freezing point of pure water. For example, in highpressure wells (more than 15,000 psia), hydrates have been observed at temperatures much higher than 100°F. Hydrates resemble wet snow and, like ice, will float on water. In the oil field, hydrates look like a grayish snow cone. When hydrate “snow” is tossed on the ground, the hydrocarbons escaping can be heard easily, giving the impression that the hydrocarbons were physically trapped in the snow. The distinctive crackling sound is in fact caused by escaping naturalgas molecules rupturing the crystal lattice of the hydrate molecules. Hydrates were discovered in 1810 by Davy and were investigated only as curiosities of physical chemistry for many years thereafter.43 In 1888, Villard became the first to determine the existence of hydrates with typical components of natural gas, such as methane, ethane, and propane.43 However, the real push to measure hydrate phase behavior did not begin until the 1930’s when Hammerschmidt44 pointed out that hydrates were the culprits that were choking wellhead and production equipment in gas fields. He also suggested ways to inhibit their formation. Although hydrate inhibition has been practiced for more than 50 years, the severe conditions encountered in arctic and deep drilling have sparked a new wave of interest in measurement of hydrate formation and inhibition at these conditions. Although the kinetics and fluid mechanics of hydrate formation and dissociation are not covered here, they are nonetheless important in deepwater drilling operations. Because vast deposits of natuPHASE BEHAVIOR
Mole Fraction Water in Vapor Phase, yw
Fig. 9.18—EOS predictions of mutual solubilities for methane/ water system determined with different sets of BIP’s for aqueous and nonaqueous phases.
ralgas hydrates exist in the Arctic, a great deal of Russian research has been conducted on both the kinetics and thermodynamics of hydrate formation and dissociation.43 Recovery of natural gas entrapped in these vast hydrate deposits in permafrost regions (by hydrate dissociation) has also been studied recently.45 The three most widely used calculation methods for predicting hydrate formation are (1) the vapor/solid Kvalue method of Katz and his coworkers4651 and equations fitting the developed Kvalue charts; (2) methods of Campbell and his coworkers5254; and (3) combined methods based on statistical thermodynamics (van der Waals and Platteeuw55) for the hydrate phase and EOS’s for the fluid phases. These methods are discussed later. 9.4.1 Crystallography of Hydrates. In the presence of a freewater phase, hydrates will form below a certain temperature often referred to as the “hydrate temperature.” Hydrate crystals generally grow only in the presence of a freeliquidwater phase at typical oilfield conditions. Hydrates can also form in the presence of a densevaporwater phase at temperatures sufficiently low to ensure hydrogen bonding. The general conditions under which hydrates form include gas at or below its water dewpoint (which can yield the freewater phase necessary for hydrate formation in the system) and conditions at moderately low temperature or high pressure. With respect to components WATER/HYDROCARBON SYSTEMS
Mole Fraction Natural Gas, xng Fig. 9.19—EOS predictions of mutual solubilities for naturalgas/water system determined with different sets of BIP’s for aqueous and nonaqueous phases.
normally found in natural gas, hydrate formation has been observed and measured only for the light constituents found in natural gas: C1 through C4 alkanes (including iC4), N2, CO2, and H2S. Fig. 9.24 shows a schematic of the naturalgas hydratecrystal lattice. Two common types of hydratecrystal structures have been proposed from interpretation of results of von Stackelberg and Müller’s56 Xray diffraction studies of hydrates. Structure I is usually a bodycentered lattice, and Structure II has a diamond lattice. Structures I and II have different sized cages (i.e., void spaces). In Structure I hydrates, methane can fill the smaller cages, while the larger cages can be filled only by larger hydrocarbon molecules, such as ethane. The cages in Structure II hydrates are larger, allowing entrapment of propane and ibutane in addition to methane and ethane. Fig. 9.25 summarizes the components and corresponding size ranges that fit into Structure I and II cavities. Light components, such as methane, ethane, and CO2, form Structure I hydrates; nitrogen and the heavier alkanes, such as propane, nbutane, ibutane, and neopentane, form Structure II hydrates. Enough cages must be filled with hydrocarbon molecules to stabilize the crystal lattice. Because all the cages do not have to be full, 11
Mole Fraction Natural Gas, xng Mole Fraction CO2 in Aqueous Phase, xCO 2
Fig. 9.20—EOS prediction of gas solubility for CO2/water/brine systems at 302°F determined with different sets of BIP’s for aqueous and nonaqueous phases; symbols+experimental and lines+calculated.
the molecular weight of a clathrate hydrate is not fixed. The “vacancy” of the hydratecrystal lattice depends on which “guest” naturalgas molecules happen to be available to occupy the void locations between the interstices of the host water molecules and on the conditions under which the crystal lattice is formed. Thus, the presence of methane and ethane leads only to the formation of Structure I hydrates and the presence of methane, ethane, and propane leads to the formation of a mixture of Structure I and II hydrates. The general trends of hydrate formation can be qualitatively predicted for a particular naturalgas component. The two important factors in formation of the two different structures of hydrates are size and solubility of the naturalgas molecules. The rate of clathration is partially dependent on solubility because the more soluble a gaseous component is in water, the higher the probability that it will be “caught” in a cage as the hydrate crystal is being formed. The size of the guest molecule not only determines the structure type but also the rate of formation. For example, comparing the rate of clathration of methane with that of ethane, a higher pressure is required to form pure methane hydrates than pure ethane hydrates, even though methane is considerably more soluble in water than ethane. The reason is that methane is a smaller molecule that is more difficult to entrap as the cage of the crystal lattice closes. Furthermore, hydrates form more readily from naturalgas mixtures than from pure components because the range of molecular sizes in natural gas has a higher probability of filling enough cavities to stabilize the hydratecrystal lattice. Researchers only recently proved that butanes are hydrate formers. McLeod and Campbell53 and others showed that the butanes are hydrate formers when methane is present to occupy the smaller cavities in Structure II hydrates. They found that, like ethane and propane and in contrast to npentane, the butanes lower the hydrateforming pressure. Hydrates with nbutane are very unstable, and, at pressures higher than 10,000 psia, nbutane behavior reverts to that of a nonhydrate former. Alkanes with a higher carbon number than nbutane are not believed to form hydrates. 9.4.2 Phase Diagrams for Hydrates. At cryogenic temperatures and subatmospheric pressures, phase diagrams show a multitude of hydrate forms. We cover only the simpler phase diagrams that represent the most common conditions encountered in subsurface engineering and in surface facilities. The temperature and pressure conditions for hydrate formation in surface gasprocessing facilities are generally much lower than those considered in production and reservoir engineering. The 12
Mole Fraction Natural Gas, xng
Fig. 9.21—EOS prediction of gas solubility for naturalgas/brine system determined with different sets of BIP’s for aqueous and nonaqueous phases.
conditions of initial hydrate formation are often given by simple pT phase diagrams for water/hydrocarbon systems. In 1885, Roozeboom defined a lower hydrate quadruple point, Q 1 ( IńL wńHńV), and an upper quadruple point, Q 2 ( L wńHńVńL HC), as on Fig. 9.26.43 His nomenclature for the phases is I+pure ice, L w+liquid water, L HC+liquid hydrocarbon, V+vapor, and H+hydrate. The quadruple point defines the condition at which four phases are in equilibrium. Because the Gibbs phase rule leads to zero degrees of freedom for this system, the values of these quadruple points (Table 9.4) for the eight naturalgas hydrate formers are unique and invariant and provide a quantitative basis for classification of hydrate formers. Each quadruple point is at the intersection of four threephase lines. The lower quadruple point, Q 1, represents the transition of L w to I. As temperature decreases to Point Q 1, hydrates cease forming from vapor and liquid water and are forming from vapor and ice. The upper quadruple point, Q 2, is the approximate intersection of Line L wńHńV with the vapor pressure of the hydrate former and represents the upper temperature limit for hydrate formation for that component. Some of the lighter naturalgas components, such as methane and nitrogen, do not have an upper quadruple point, so no upper temperature limit exists for hydrate formation. This is the reason that hydrates can still form at high temperatures (up to 120°F) in the surface facilities of highpressure wells. PHASE BEHAVIOR
Experimental kij Different for Each Phase (modified aw), kij Same for Each Phase (modified aw), kij Same for Each Phase (original aw) kij+0.485 (nonaqueous phase) kij+*0.260 (nonaqueous phase)
Mole Fraction Nitrogen in Aqueous Phase, xN 2
Fig. 9.22—EOS prediction of gas solubility for N2/NaClbrine system at 217°F determined with different sets of BIP’s for aqueous and nonaqueous phases; symbols+experimental and lines+PR EOS predicted.
Fig. 9.27 shows the main area of hydrate formation in petroleumengineering applications. Line FEG represents the naturalgasmixture dewpoint curve. The dewpoint line is analogous to the vaporpressure curves of the individual components in Fig. 9.26. Point E is the maximum hydrateforming temperature (analogous to the quadruple points, Q 2, of the individual components in Fig. 9.26). The hydrate curve is Line BE. At the intersection of the dewpoint and hydrate curves, the hydrate curve for many naturalgas systems becomes nearly vertical and establishes the maximum hydrateforming temperature. For a naturalgas system with very high concentrations of methane, such as encountered in the deep naturalgas plays in the Anadarko basin, the maximum hydrateforming temperature may be essentially nonexistent (observe that no Q 2 exists for the methane curve in Fig. 9.26). The general approach to hydrate prediction in most engineering applications is to determine Hydrate Line BE and the position of Dewpoint Line FEG on Line BE. Sec. 9.4.3 discusses calculation methods. Fig. 9.28 shows Deaton and Frost’s58 data for hydrateformation conditions for methane/propane mixtures. These data show how hydrateformation conditions for natural gas are strongly dependent on the propane concentration. The general effect of increasing propane concentration is to lower the hydrateforming pressure and to increase the hydrateforming temperature. Katz et al.32 and Wilcox et al.49 developed Fig. 9.29 to determine hydrateforming conditions for natural gas at different specific gravities. Because Fig. 9.29 is based on gas gravity, it is particularly useful as a quick guide to estimate the hydrate temperature for a natural gas. Fig. 9.29 should not be used if CO2 or H2S is present at a combined concentration y1 mol%. At pressures less than 12,000 psia, the JouleThompson expansion of a natural gas, for example, across a separator choke, reduces the temperature of the gas. Katz et al.32 present charts (Figs. 9.30 through 9.32) that show the maximum permissible expansion of natural gases before hydrate formation occurs. 9.4.3 Calculation Method of Katz and Coworkers.32,47,49,51 By applying the analogy of vapor/liquid equilibrium K values to a solid solution, Carson and Katz47 and Wilcox et al.49 developed the concept of a vapor/solid K value for predicting the temperature and pressure conditions under which hydrates form or dissociate. K i (v*s)
y + x i , i (s)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.31)
WATER/HYDROCARBON SYSTEMS
Experimental kij Different for Each Phase (modified aw), kij Same for Each Phase (modified aw), kij Same for Each Phase (original aw) kij+0.485 (nonaqueous phase) kij+*0.260 (nonaqueous phase)
Fig. 9.23—Predicted mutual solubilities of methane/water system at 100°F determined with the modified PR EOS with one and two sets of kij ; all kij from Table 9.3.
where K i(v*s)+vapor/solid equilibrium value of Component i, y i+gas composition, and x i(s)+mole fraction of Component i in the solid on a waterfree basis. Calculation of hydrateformation temperature is analogous to calculation of a dewpoint temperature (discussed in Chap. 3). A gas in the presence of a freewater phase will form a hydrate if
ȍK y N
i
y 1.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.32)
i (v*s)
i+1
Conversely, hydratedissociation temperature can be treated like a bubblepoint calculation. A hydrate will dissociate if
ȍx K N
i
i (v*s)
y 1.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.33)
i+1
Because K i(v*s) is based on the mole fraction of a guest naturalgas component in the solidphase hydrate mixture on a waterfree basis, the concept of K i(v*s) is only an approximation of the original 13
Hydrate Former 3Å
Cavities Occupied
No Hydrates
A Kr N2 O2
4Å
52/3 H2O
512 + 51264 SII
53/4 H2O
512 + 51262 SI
CH4 Xe; H2S
5Å
CO2
C2H6
6Å
(CH2)3O
Fig. 9.24—Schematic of hydratecrystal lattice; circles represent water molecules, lines represent hydrogen bonds (from Ref. 52).
definition of vapor/liquid equilibrium ratios, K i. For example, the concept of the vapor/solid K value cannot be used to calculate hydratephase splits or equilibriumphase compositions. The vapor/ solid K value can be used only to predict the temperature or pressure where hydrates form or dissociate. However, on the basis of component K i(v*s) values, where the naturalgas components will concentrate can be determined qualitatively. If K i(v*s) for a naturalgas component is greater than unity (nitrogen is a typical example), the component will tend to concentrate in the gaseous phase rather than in the hydrate phase. If K i(v*s) is less than unity (for example, propane), the component will tend to concentrate in the hydrate. Katz and his coworkers provide K i(v*s) nomograms for several naturalgas components as functions of temperature and pressure. Sloan57 developed the following polynomialfit equation of the KatzCarson charts, which can be used to estimate K i(v*s). ln K i(v*s) + A 0 ) A 1 T ) A 2 p ) A 3 T *1 ) A 4 ńp ) A 5 pT ) A 6 T 2 ) A 7 p 2 ) A 8 ǒ pńTǓ ) A 9 ln ǒ pńT Ǔ ) A 10 p *2 ) A 11 ǒ Tńp Ǔ ) A 12 ǒ T 2ńp Ǔ ) A 13 ǒpńT 2Ǔ ) A 14 ǒTńp 3Ǔ ) A 15 T 3 ) A 16 ǒp 3ńT 2Ǔ ) A 17 T 4 . . . . . . . . . . . . . . . . . . . (9.34) Table 9.5 gives the values of Constants A0 through A17. K i(v*s) for nonhydrate formers are assumed to be infinity in the calculation. The original work assumed that nitrogen and butanes were not hydrate formers, which was subsequently shown to be incorrect. However, fairly reliable estimates can be obtained by assuming that the K i(v*s) for nitrogen and the butanes are also infinity as long as the pressure is less than approximately 1,000 psia. This method becomes less reliable for pressures higher than 1,000 psia. 9.4.4 Calculation Methods of Campbell and Coworkers.5254 To address the pressure limitations of the K i(v*s) method of Katz and his coworkers as well as the hydratetemperaturedepression effects of molecules too large to fit into the cavities of the hydrate crystal, Campbell and his coworkers5254 developed additional empirical procedures. In general, these methods can be used for quick estimates of hydrateformation temperatures when pressures exceed the 1,000psia limitation of the K i(v*s) method. The TrekellCampbell54 14
51262 SI
72/3 H2O
cC3H6
C3H8 isoC4H10
7Å
51264 SII
17 H2O
nC4H10
No SI or SII Hydrates
8Å Fig. 9.25—Summary of naturalgas components fitting into Structure I and II (SI and SII, respectively) cavities (from Ref. 57).
method covers pressures from 1,000 to 6,000 psia, and the McLeodCampbell53 method covers pressures from 6,000 to 10,000 psia. The TrekellCampbell method calculates additive effects of gas molecules on the hydrateforming temperature of methane. They give eight nomograms, six of which give positive displacements as functions of pressure for C3, nC4, and iC4 and two that give negative corrections (depression) for nonhydrate formers, such as C5+. This method is strictly empirical and must be used with caution, but it is useful as a quick estimate at pressures up to 6,000 psia. McLeod and Campbell developed another method to predict hydrateformation temperatures at very high pressures encountered in deepgaswell drilling. They prepared a very simple correlation based on a modified Clapeyron equation to describe the energy of phase transition at pressures 6,000 to 10,000 psia. T + 3.89 ǸC
ȍy C , N
and C +
i
i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.35)
i+1
where T is in °R, y i+gas molar composition, and Table 9.6 gives the hydrateformer constants C i for C1 through C4 hydrocarbons. The hydrateprediction methods of Campbell and his coworkers are mostly empirical but do provide a reliable answer when computer programs for the more theoretical models described in the next secPHASE BEHAVIOR
van der Waals and Platteeuw’s statisticalmechanical solidsolution theory of clathrates. These authors developed an adsorption model based on statistical mechanics to derive a relation for the chemical potential of water in the hydrate phase. Their method is based on an equation that relates the chemical potential of water in the hydrate structure in much the same way that chemical potential of a component is related to the activity of a component in a mixture. m i + m oi ) RT ln a i ,
. . . . . . . . . . . . . . . . . . . . . . . . . . (9.36)
where m i+chemical potential of pure Component i (see Chap. 4) and a i+activity of Component i in the mixture. van der Waals and Platteeuw propose the following Langmuir adsorptionisotherm analogy that accounts for the microscopic hydrate structure. m wH + m wMT ) RT
ȍn i
ci ln
ǒ
1*
ȍy j
Ǔ
ji
,
. . . . . . . . (9.37)
where m wH+chemical potential of water in the filled hydrate, m wMT +chemical potential of water in the empty hydrate, n ci +number of Type i cavities per water molecule in hydratecrystal lattice, and y ji+fraction (probability) of Type j molecule occupying Type i cavity. The Langmuir adsorption theory is applicable because “clathration” and “declathration” are analogous to adsorption and desorption, respectively. The probability term, y ji, depends on the interaction between the guest gas molecule and its “cage” (the “site” by analogy with the original Langmuir theory). The term y ji also depends on the fugacities of the components in the gas phase, which can be calculated with an EOS. Parrish and Prausnitz59 were the first to extend the van der Waals and Platteeuw statisticalmechanics model to multicomponent systems. They used the Kihara potential to calculate the Langmuir constants. John et al.60 and Schroeter et al.61 also used the Kihara potential to calculate the Langmuir constants. Erickson and Sloan62 developed a calculation procedure using the van der Waals and Platteeuw model. A computer program (CSMHYD) is included with Ref. 62, and a complete description of the calculation algorithm and computerprogram flow chart are also provided. Ref. 63 provides a description and algorithm for a similar approach. These methods are fairly difficult to program from the literature; therefore, Ref. 62 with the program diskettes is recommended. Other researchers have developed calculation methods based on the van der Waals and Platteeuw in combination with an EOS. Ng et al.64 and Robinson and Mehta65 made predictions of hydrateformation conditions using the PR EOS and developed a computerbased method that is available through the Gas Processors Assn. Schroeter et al.61 used the Benedict et al. EOS66 to model the fluid phase in hydrate calculations with sourgas (including H2S) systems. Munck et al.67 used the SoaveRedlichKwong EOS with the van der Waals and Platteeuw adsorption model to calculate fugacities of liquid and gaseous phases in equilibrium with hydrates. They used the Michelsen68 stability algorithms (see Chap. 4) to develop a computer program that predicts hydrateformation conditions without prior knowledge of the phases. To account for the effects of nonelectrolyte inhibitors, Munck et al.67 used the UNIQUAC activitycoefficient model. They obtained good agreement for hydrates in equilibrium with North Sea reservoir fluids.
MT
Fig. 9.26—Hydrateformation conditions for naturalgas hydrate formers (from Ref. 57).
tion are not available. They can also be used as a check of the more sophisticated estimation methods (also described in the next section). 9.4.5 van der Waals and Platteeuw55 Model. Most modern computerbased methods of predicting hydrate formation are based on WATER/HYDROCARBON SYSTEMS
9.4.6 Water Content of Vapor in Equilibrium With Hydrates. The concentration of water in the vapor phase in equilibrium with hydrate is usually very small, on the order of 0.001 mol% or less. Phase diagrams and nomograms for determining the water content of vapor in equilibrium with hydrates are complicated by metastable equilibrium in the gas/ice region and are cumbersome to use for the many possible combinations of compositions. Song and Kobayashi69 present a mathematical approach for determining the water content of gases in the vapor/hydrate region. They studied methanerich and CO2rich systems, which are especially important in EOR operations (Fig. 9.33). Sloan55 proposes a slight improvement to the Kobayashi et al.50 method and provides the necessary equations, along with an extensive table of coefficients and an example of how to use the method. 15
TABLE 9.4—QUADRUPLE POINTS FOR NATURALGAS HYDRATE FORMERS (from Ref. 57) Lower Quadruple, Q1
Upper Quadruple, Q2
NaturalGas
Temperature
Pressure
Temperature
Component
(°F)
(psi)
(°F)
Pressure
C1
29.9
371.7
C2
31.9
76.9
58.4
491.7
C3
31.9
24.9
42.2
80.6
iC4
31.9
16.4
35.3
24.2
CO2
31.9
182.2
49.7
N2
29.8
2,079.5
H2 S
31.4
13.5
(psi) No Q2
652.5 No Q2
85.2
324.7
nC4 does not form hydrate by itself; it requires the presence of a “help gas.”
Fig. 9.27—Characteristics of hydrateforming naturalgas mixture at typical production conditions (from Ref. 52).
Algorithms for predicting hydrocarbon concentration in vapor in equilibrium with hydrate are also discussed. Methods for predicting hydrateformation conditions have improved, and the prediction of hydrateformation conditions with the van der Waals and Platteeuw49 method can be used reliably. In extreme operating conditions, such as those encountered in deep drilling, calculation methods for predicting hydrate formation may not be reliable. In these situations, laboratory measurements are recommended. 9.4.7 Hydrate Inhibition. Hammerschmidt 70 presented a relation for predicting the depression of the hydrateforming temperature of natural gases in contact with dilute aqueous solutions of antifreezes, such as methanol and glycols (e.g., ethylene glycol). Hammerschmidt’s equation originates from the relationship for determining the colligative properties (in this case, freezing or hydrateforming point) of an ideal solution. DT [ 2, 335 16
w , 100 M * w M
. . . . . . . . . . . . . . . . . . . (9.38)
Fig. 9.28—Hydrateformation conditions for methane/propane/ water mixtures (from Ref. 57).
with DT in °F, M+molecular weight of the antifreeze agent (e.g., M+32 for methanol), and w+weight percent of the antifreeze agent in solution. Fig. 9.34 shows how the hydrateformation temperature is depressed with the addition of methanol to water and a typical naturalgas component. Use of the Hammerschmidt equation should be restricted to sweet natural gases with antifreeze concentrations of less than 0.20 mol%. Campbell51 suggests that for glycols, the factor 2,335 should be replaced by 4,000. For concentrated methanol solutions, like those used to free a pluggedup tubing string in a highpressure well, Nielsen and Bucklin71 propose the following modification of the Hammerschmidt equation. DT + * 129.6 lnǒ1 * x MeOHǓ ,
. . . . . . . . . . . . . . . . . . (9.39) PHASE BEHAVIOR
Initial Temperature, °F
Fig. 9.29—Temperature and pressure conditions of hydrate formation for natural gases (from Ref. 32).
where DT+depression of the hydrateforming temperature in °F and x MeOH+mole fraction of methanol inhibitor. References 1. Craft, B.C. and Hawkins, M.: Applied Petroleum Reservoir Engineering, first edition, PrenticeHall Inc., Englewood Cliffs, New Jersey (1959). 2. Fetkovich, M.J., Reese, D.E., and Whitson, C.H.: “Application of a General Material Balance for HighPressure Gas Reservoirs,” SPE Journal (March 1998) 3. 3. Li, Y.K. and Nghiem, L.X.: “Phase Equilibria of Oil, Gas, and Water/ Brine Mixtures From a Cubic Equation of State and Henry’s Law,” Cdn. J. Chem. Eng. (June 1986) 64, 486. 4. Carroll, J.J. and Mather, A.E.: “Equilibrium in the System WaterHydrogen Sulfide: Modelling the Phase Behavior with an Equation of State,” Cdn. J. Chem. Eng. (1989) 67. 5. Michel, S., Hooper, H.H., and Prausnitz, J.M.: “Mutual Solubilities of Water and Hydrocarbons From an Equation of State. Need for an Unconventional Mixing Rule,” Fluid Phase Equilibria (1989) 45. 6. Firoozabadi, A. et al.: “EOS Predictions of Compressibility and Phase Behavior in Systems Containing Water, Hydrocarbons, and CO2,” SPERE (May 1988) 673. WATER/HYDROCARBON SYSTEMS
Fig. 9.30—Maximum permissible expansion of 0.6gravity natural gas without hydrate formation (from Ref. 32). 7. Nutakki, R. et al.: “Calculation of Multiphase Equilibria for WaterHydrocarbon Systems at High Temperature,” paper SPE 17390 presented at the 1988 SPE/DOE Enhanced Oil Recovery Symposium, Tulsa, Oklahoma, 17–20 April. 8. Enick, R.M., Holder, G.D., and Mohamed, R.: “FourPhase Flash Equilibrium Calculations Using the PengRobinson Equation of State and A Mixing Rule for Asymmetric Systems,” SPERE (November 1987) 687. 9. Søreide, I. and Whitson, C.H.: “PengRobinson Predictions for Hydrocarbons, CO2, N2 and H2S With Pure Water and NaClBrines,” Fluid Phase Equilibria (1992). 10. Haas, J.L. Jr.: “Physical Properties of the Coexisting Phases and Thermochemical Properties of the H2O Component in Boiling NaCl Solutions,” Geological Survey Bulletin (1976) 1421A and B. 11. Patton, C.C.: Oil Field Water Systems, Campbell Petroleum Series, Norman, Oklahoma (1981). 17
Initial Temperature, °F
Initial Temperature, °F
Fig. 9.31—Maximum permissible expansion of 0.7gravity natural gas without hydrate formation (from Ref. 32). 12. Rowe, A.M. Jr. and Chou, J.C.S.: “PressureVolumeTemperatureConcentration Relation of Aqueous NaCl Solutions,” J. Chem. Eng. Data (1970) 15, 61. 13. Dodson, C.R. and Standing, M.B.: “Pressure, Volume, Temperature and Solubility Relations for Natural GasWater Mixtures,” Drill. & Prod. Prac. (1944) 173. 14. Culberson, O.L. and McKetta, J.J. Jr.: “Phase Equilibria in Hydrocarbon/ Water Systems. III The Solubility of Methane in Water at Pressures to 10,000 psi,” Trans., AIME (1951) 192, 223. 18
Fig. 9.32—Maximum permissible expansion of 0.8gravity natural gas without hydrate formation (from Ref. 32). 15. Wiebe, R. and Gaddy, V.L.: “The Solubility of Carbon Dioxide in Water at Various Temperatures from 12 to 40°C and at Pressures to 500 Atmospheres,” J. Amer. Chem. Soc. (1940) 62, 815. 16. Wiebe, R. and Gaddy, V.L.: “Vapor Phase Composition of Carbon DioxideWater Mixtures at Various Temperatures and at Pressures to 700 Atmospheres,” J. Amer. Chem. Soc. (1941) 63, 475. 17. Kobayashi, R. and Katz, D.L.: “VaporLiquid Equilibria for Binary HydrocarbonWater Systems,” Ind. Eng. Chem. (1953) 45, No. 2, 440. 18. Cramer, S.D.: “Solubility of Methane in Brines From 0 to 300°C,” Ind. Eng. Chem. Proc. Des. Dev. (1984) 23, No. 3, 533. PHASE BEHAVIOR
TABLE 9.5—VALUES OF COEFFICIENTS A0 THROUGH A17 IN EQ. 9.34 Coefficients Component
A0
A1
A2
A3
A4
A5
CH4
1.63636
0.0
0.0
31.6621
*49.3534
5.31 x 10*6
C2 H6
6.41934
0.0
0.0
*290.283
2,629.10
0.0
C3 H8
*7.8499
0.0
0.0
47.056
0.0
*1.17 x 10*6
iC4H10
*2.17137
0.0
0.0
0.0
0.0
0.0
nC4H10
*37.211
0.86564
0.0
732.20
0.0
0.0
N2
1.78857
0.0
*0.001356
*6.187
0.0
0.0
CO2
9.0242
0.0
0.0
*207.033
0.0
4.66 x 10*5
H2 S
*4.7071
0.06192
0.0
82.627
0.0
*7.39 x 10*6
A6
A7
A8
A9
A10
A11
CH4
0.0
0.0
0.128525
*0.78338
0.0
0.0
C2 H6
0.0
9.0 x 10*8
0.129759
*1.19703
*8.46 x 104
*71.0352
C3 H8
7.145 x 10*4
0.0
0.0
0.12348
1.669 x 104
0.0
iC4H10
1.251 x 10*3
1.0 x 10*8
0.166097
*2.75945
0.0
0.0
nC4H10
0.0
9.37 x 10*6
*1.07657
0.0
0.0
*66.221
N2
0.0
2.5 x 10*7
0.0
0.0
0.0
0.0
CO2
*6.992x 10*3
2.89 x 10*6
*6.223 x 10*3
0.0
0.0
0.0
H2 S
0.0
0.0
0.240869
*0.64405
0.0
0.0
A12
A13
A14
A15
A16
A17
CH4
0.0
*5.3569
0.0
*2.3 x 10*7
*2.0x 10*8
0.0
C2 H6
0.596404
*4.7437
7.82 x 104
0.0
0.0
0.0
C3 H8
0.23319
0.0
*4.48 x 104
5.5 x 10*6
0.0
0.0
iC4H10
0.0
0.0
*8.84 x 102
0.0
*5.7 x 10*7
*1.0 x 10*8
0.0
9.17 x 105
0.0
4.98x 10*6
*1.26 x 10*6
0.0
5.87 x 105
0.0
1.0 x 10*8
1.1x 10*7
2.55 x 10*6
0.0
0.0
0.0
nC4H10
0.0
N2
0.0
CO2
0.27098
0.0
0.0
8.82 x 10*5
H2 S
0.0
*12.704
0.0
*1.3x 10*6
TABLE 9.6—COEFFICIENTS FOR EQ. 9.35 AS FUNCTIONS OF PRESSURE HydrateFormer C Values Pressure (psia)
C1
C2
C3
iC4
nC4
6,000
18,933
20,806
28,382
30,696
17,340
7,000
19,096
20,848
28,709
30,913
17,358
8,000
19,246
20,932
28,764
30,935
17,491
9,000
19,367
21,094
29,182
31,109
17,868
10,000
19,489
21,105
29,200
30,935
17,868
19. Culberson, O.L. and McKetta, J.J. Jr.: “Phase Equilibria in Hydrocarbon/ Water Systems. IV Vapor Liquid Equilibrium Constants in the Methane/ Water and Ethane/Water Systems,” Trans., AIME (1951) 192, 297. 20. Amirijafari, B. and Campbell, J.M.: “Solubility of Gaseous Hydrocarbons Mixtures in Water,” SPEJ (February 1972) 21; Trans., AIME, 253. 21. Pawlikowski, E.M. and Prausnitz, J.M.: “Estimation of Setchenow Constants for Nonpolar Gases in Common Salts at Moderate Temperatures,” Ind. Eng. Chem. Fund. (1983). 22. Clever, H.L. and Holland, C.J.: “Solubility of Argon Gas in Aqueous Alkali Halide Solutions,” J. Chem. Eng. Data (July 1968) 13, No. 3, 411. 23. Markham, A.E. and Kobe, K.A.: “The Solubility of Carbon Dioxide and Nitrous Oxide in Aqueous Salt Solutions,” J. Amer. Chem. Soc. (1941) 63, 449. 24. Long, G. and Chierici, G.L.: “Compressibilité et Masse Specifique des Eaux de Gisement dans les Conditions des Gisements. Application à Quelques Problemes de ‘Reservoir Engineering’,” Proc., Fifth World Petroleum Congress (1959) 187. 25. Long, G. and Chierici, G.: “Salt Content Changes Compressibility of Reservoir Brines,” Pet. Eng. (July 1961) B25. WATER/HYDROCARBON SYSTEMS
26. Kutasov, I.M.: “Correlation simplifies obtaining downhole brine density,” Oil & Gas J. (5 August 1991) 48. 27. Rogers, P.S.Z. and Pitzer, K.S.: “Volumetric Properties of Aqueous Sodium Chloride Solutions,” J. Phys. Chem. Ref. Data (1982) 11, No. 1, 15. 28. Sutton, R.P.: “Compressibility Factors for HighMolecular Weight Reservoir Gases,” paper SPE 14265 presented at the 1985 SPE Annual Technical Conference and Exhibition, Las Vegas, Nevada, 22–25 September. 29. Kestin, J., Khalifa, H.E., and Correia, R.J.: “Tables of the Dynamic and Kinematic Viscosity of Aqueous NaCl Solutions in the Temperature Range 20–150°C and the Pressure Range 0.1–35 MPa,” J. Phys. Chem. Ref. Data (1981) 10, No. 1, 71. 30. Collins, A.G.: “Properties of Produced Waters,” Petroleum Engineering Handbook, H.B. Bradley et al. (eds.), SPE, Richardson, Texas (1987) Chap. 24, 1–23. 31. McKetta, J.J. Jr. and Wehe, A.H.: “Hydrocarbon/Water and Formation Water Correlations,” Petroleum Production Handbook, T.C. Frick and R.W. Taylor (eds.), SPE, Richardson, Texas (1962) II, 22. 32. Katz, D.L. et al.: Handbook of Natural Gas Engineering, McGrawHill Book Co. Inc., New York City (1959). 33. Ramey, H.J. Jr.: “Correlations of Surface and IFT’s of Reservoir Fluids,” paper SPE 4429 available from SPE, Richardson, Texas (1973). 34. Standing, M.B.: Petroleum Engineering Data Book, Norwegian Inst. of Technology, Trondheim, Norway (1974). 35. Hocott, C.R.: “IFT Between Water and Oil Under Reservoir Conditions,” Trans., AIME (1939) 132, 184. 36. Firoozabadi, A. and Ramey, H.J. Jr.: “Surface Tension of WaterHydrocarbon Systems at Reservoir Conditions,” paper CIM 873830, Calgary, 7–10 June 1987. 37. Peng, D.Y. and Robinson, D.B.: “Two and Three Phase Equilibrium Calculations for Coal Gasification and Related Processes,” Thermodynamics of Aqueous Systems with Industrial Applications, ACS Symposium Series 133 (1980). 38. Peng, D.Y. and Robinson, D.B.: “A NewConstant Equation of State,” Ind. Eng. Chem. Fund. (1976) 15, No. 1, 59. 19
DTȀ
DT
Fig. 9.33—Hydrateformation conditions for CO2/water systems (adapted from Ref. 69). 39. Panagiotopoulos, A.Z. and Reid, R.C.: “New Mixing Rule for Cubic Equations of State for Highly Polar, Asymmetric Systems,” Equations of State: Theories and Applications, K.C. Chao and R.L. Robinson (eds.), ACS Symposium Series (1986) 571. 40. Kistenmacher, H. and Michelsen, M.L.: “On CompositionDependent Interaction Coefficients,” Fluid Phase Equilibria (1992). 41. Harvey, A.H. and Prausnitz, J.M.: “Thermodynamics of HighPressure Aqueous Systems Containing Gases and Salts,” AIChE J. (1989) 35, No. 4, 635. 42. Ludecke, D. and Prausnitz, J.M.: “Phase Equilibria for Strongly Nonideal Mixtures From an Equation of State with DensityDependent Mixing Rules,” Fluid Phase Equilibria (1985) 22, 1. 43. Makogon, Y.F.: Hydrates of Natural Gas, PennWell Books, Tulsa, Oklahoma (1981). 44. Hammerschmidt, E.G.: “Preventing and Removing Hydrates in Natural Gas Pipelines,” Oil & Gas J. (1939) 37, No. 8, 66. 45. Holder, G.D., Malone, R.D., and Lawsa, W.F.: “Effects of Gas Composition and Geothermal Properties on the Thickness and Depth of NaturalGasHydrate Zones,” JPT (September 1987) 1142. 46. Katz, D.L.: “Prediction of Conditions for Hydrate Formation in Natural Gases,” Trans., AIME (1945) 160, 141. 47. Carson, D.B. and Katz, D.L.: “Natural Gas Hydrates,” Trans., AIME (1942) 146, 150. 48. Unruh, C.H. and Katz, D.L.: “Gas Hydrates of Carbon Dioxide/Methane Mixtures,” Trans., AIME (1949) 83. 49. Wilcox, W.I., Carson, D.B., and Katz, D.L.: “Natural Gas Hydrates,” Ind. Eng. Chem. (1941) 33, No. 5, 662. 50. Katz, D.L. and Lee, R.L.: Natural Gas Engineering, Chemical Engineering Series, McGrawHill Book Co. Inc., New York City (1990). 51. Kobayashi, R. et al.: “Gas Hydrates Formation with Brine and Ethanol Solutions,” paper presented at the 1951 Natural Gasoline Assn. of America Annual Convention. 52. Campbell, J.M.: Gas Conditioning and Processing, sixth edition, Campbell Petroleum Series, Norman, Oklahoma (1984). 53. McLeod, H.D. Jr. and Campbell, J.M.: “Natural Gas Hydrates at Pressures to 10,000 psia,” JPT (June 1961) 590. 54. Trekell, R.E. and Campbell, J.M.: Petr. Chem. Div. (March 1966) 61. 55. van der Waals, J.H. and Platteeuw, J.C.: “Clathrate Solutions,” Adv. Chem. Phys. II, I. Prigogine (ed.), Interscience Publishers, New York City (1959) 1–58. 56. von Stackelberg, M. and Müller, H.G.: “On the Structure of Gas Hydrates,” J. Phys. Chem. (1951) 19, 1319. 57. Sloan, E.D.: “Phase Equilibria of Natural Gas Hydrates,” paper presented at the 1984 Gas Producers Assn. Annual Convention, New Orleans, 19–21 March. 58. Deaton, W.M. and Frost, E.M.: Gas Hydrates and Their Relation to the Operation of Natural Gas Pipelines, Monograph 8, U.S. Bureau of Mines, Washington, DC (1946). 20
Fig. 9.34—General effect of methanol added to water/ethane system (adapted from Ref. 71). 59. Parrish, W.R. and Prausnitz, J.M.: “Dissociation Pressures of Gas Hydrates Formed by Gas Mixtures,” Ind. Eng. Chem. Proc. Des. Dev. (1972) 11, No. 1, 26. 60. John, V.T., Papadopoulos, K.D., and Holder, G.D.: “A Generalized Model for Predicting Equilibrium Conditions for Gas Hydrates,” AIChE J. (1985) 31, No. 2, 252. 61. Schroeter, J.P., Kobayashi, R., and Hildebrand, M.A.: “Hydrate Decomposition Conditions in the System H2SMethanePropane,” Ind. Eng. Chem. Fund. (1983) 22, 361. 62. Ericksen and Sloan, E.D.: “Calculation Procedure Using vdWPlatteeuw Model,” Clathrate Hydrates of Natural Gas, Marcel Dekker, New York City (1990). 63. Technical Data Book—Petroleum Refining, third edition, API, New York City (1977). 64. Ng, H.J., Chen, C.J., and Saeterstad, T.: “Hydrate Formation and Inhibition in Gas Condensate and Hydrocarbon Liquid Systems,” Fluid Phase Equilibria (1987) 36, 99. 65. Robinson, D.B. and Mehta, B.R.: “Hydrates in the PropaneCarbon DioxideWater System,” J. Cdn. Pet. Tech. (January–March 1971) 33. 66. Starling, K.E. and Powers, J.E.: “Enthalpy of Mixtures by Modified BWR Equations,” Ind. & Eng. Chem. Fund. (1970) 9, 531. 67. Munck, J., SkjoldJ¢rgensen, S., and Rasmussen, P.: “Computations of the Formation of Gas Hydrates,” Chem. Eng. Sci. (1988) 43, No. 10, 2661. 68. Michelsen, M.L.: “The Isothermal Flash Problem. Part I. Stability,” Fluid Phase Equilibria (1982) 9, 1. 69. Song, K.Y. and Kobayashi, R.: “Water Content of CO2 in Equilibrium With Liquid Water and/or Hydrates,” SPEFE (December 1987) 500; Trans., AIME, 283. 70. Hammerschmidt, E.G.: “Formation of Gas Hydrates in Natural Gas Transmission Lines,” Ind. & Eng. Chem. (August 1934) 26, No. 8, 851. 71. Nielsen, R.B. and Bucklin, R.W.: “Why Not Use Methanol for Hydrate Control?” Hydro. Proc. (April 1983) 71.
SI Metric Conversion Factors bar 1.0* bbl 1.589 873 cp 1.0* dyne/cm 1.0* ft3 2.831 685 °F (°F*32)/1.8 °F (°F)459.67)/1.8 g mol 1.0* lbm 4.535 924 psi 6.894 757 psi*1 1.450 377
E)05 +Pa E*01 +m3 E*03 +Pa@s E)00 +mN/m E*02 +m3 +°C +K E*03 +kmol E*01 +kg E)00 +kPa E*01 +kPa*1
*Conversion factor is exact.
PHASE BEHAVIOR
Appendix A
Property Tables and Units TABLE A1A—COMPONENT PROPERTIES FOR CUSTOMARY UNITS
Compound Nitrogen
Molecular Weight
Specific
M (lbm/lbm mol)
Gravity* ăągąă
Liquid Density ò sc (lbm/ft3)
Critical Constants pc (psia)
Tc (°R)
Acentric
Normal Boiling Point
Ideal Liquid Yield
Gross Heating Value
L (gal/Mscf)
H (Btu/scf)
vc (ft3/lbm mol)
Zc
Factor ąw
Tb (°R)
N2
28.02
0.4700
29.31
493.0
227.3
1.443
0.2916
0.0450
139.3
Carbon dioxide
CO2
44.01
0.5000
31.18
1,070.6
547.6
1.505
0.2742
0.2310
350.4
Hydrogen sulfide
H2S
34.08
0.5000
31.18
1,306.0
672.4
1.564
0.2831
0.1000
383.1
672
Methane
C1
16.04
0.3300
20.58
667.8
343.0
1.590
0.2884
0.0115
201.0
1,012
Ethane
C2
30.07
0.4500
28.06
707.8
549.8
2.370
0.2843
0.0908
332.2
Propane
C3
44.09
0.5077
31.66
616.3
665.7
3.250
0.2804
0.1454
416.0
27.4
2,557
isobutane
iC4
58.12
0.5613
35.01
529.1
734.7
4.208
0.2824
0.1756
470.6
32.7
3,354
Butane
nC4
58.12
0.5844
36.45
550.7
765.3
4.080
0.2736
0.1928
490.8
31.4
3,369
isopentane
iC5
72.15
0.6274
39.13
490.4
828.8
4.899
0.2701
0.2273
541.8
36.3
4,001
Pentane
nC5
72.15
0.6301
39.30
488.6
845.4
4.870
0.2623
0.2510
556.6
36.2
4,009
Hexane
nC6
86.17
0.6604
41.19
436.9
913.4
5.929
0.2643
0.2957
615.4
41.2
4,756
Heptane
nC7
100.20
0.6828
42.58
396.8
972.5
6.924
0.2633
0.3506
668.8
46.3
5,503
Octane
nC8
114.20
0.7086
44.19
360.6
1,023.9
7.882
0.2587
0.3978
717.9
50.9
6,250
Nonane
nC9
128.30
0.7271
45.35
332.0
1,070.3
8.773
0.2536
0.4437
763.1
55.7
6,996
Decane
nC10
142.30
0.7324
45.68
304.0
1,111.8
9.661
0.2462
0.4902
805.2
61.4
7,743
28.97
0.4700
29.31
547.0
239.0
1.364
0.2910
0.0400
141.9
Air Water Oxygen
H2O
18.02
1.0000
62.37
3,206.0
1,165.0
0.916
0.2350
0.3440
671.6
O2
32.00
0.5000
31.18
732.0
278.0
1.174
0.2880
0.0250
162.2
1,783
*Water+1.
PROPERTY TABLES AND UNITS
1
TABLE A1B—COMPONENT PROPERTIES IN SI METRIC UNITS
Compound Nitrogen
Molecular Weight
Specific
M (kg/kmol)
Gravity* ągą
Liquid Density ò sc
Critical Constants
Acentric
Normal Boiling Point
Ideal Liquid Yield
Gross Heating Value
Tb (K)
L (m3/1000 m3)
H (MJ/std m3)
(kg/m3)
pc (kPa)
Tc (K)
vc (m3/kmol)
Zc
Factor ąw
N2
28.02
0.4700
469.5
3 399
126.3
0.0901
0.2916
0.0450
77.39
Carbon dioxide
CO2
44.01
0.5000
499.5
7 382
304.2
0.0940
0.2742
0.2310
194.67
Hydrogen sulfide
H2S
34.08
0.5000
499.5
9 005
373.6
0.0976
0.2831
0.1000
212.83
25.04
Methane
C1
16.04
0.3300
329.7
4 604
190.6
0.0993
0.2884
0.0115
111.67
37.71
Ethane
C2
30.07
0.4500
449.6
4 880
305.4
0.1479
0.2843
0.0908
184.56
Propane
C3
44.09
0.5077
507.2
4 249
369.8
0.2029
0.2804
0.1454
231.11
3.67
95.27
isobutane
iC4
58.12
0.5613
560.7
3 648
408.2
0.2627
0.2824
0.1756
261.44
4.37
125.0
Butane
nC4
58.12
0.5844
583.8
3 797
425.2
0.2547
0.2736
0.1928
272.67
4.20
125.5
isopentane
iC5
72.15
0.6274
626.8
3 381
460.4
0.3058
0.2701
0.2273
301.00
4.86
149.1
Pentane
nC5
72.15
0.6301
629.5
3 369
469.7
0.3040
0.2623
0.2510
309.22
4.83
149.4
Hexane
nC6
86.17
0.6604
659.7
3 012
507.4
0.3701
0.2643
0.2957
341.89
5.51
177.2
Heptane
nC7
100.20
0.6828
682.1
2 736
540.3
0.4322
0.2633
0.3506
371.56
6.20
205.0
Octane
nC8
114.20
0.7086
707.9
2 486
568.8
0.4920
0.2587
0.3978
398.83
6.80
232.9
Nonane
nC9
128.30
0.7271
726.4
2 289
594.6
0.5477
0.2536
0.4437
423.94
7.45
260.7
Decane
nC10
142.30
0.7324
731.7
2 096
617.7
0.6031
0.2462
0.4902
447.33
8.20
288.5
28.97
0.4700
469.5
3 771
132.8
0.0852
0.2910
0.0400
78.83
Air Water Oxygen
H2O
18.02
1.0000
999.0
22 105
647.2
0.0572
0.2350
0.3440
373.11
O2
32.00
0.5000
499.5
5 047
154.4
0.0733
0.2880
0.0250
90.11
66.43
*Water+1.
2
PHASE BEHAVIOR
TABLE A2—UNIVERSAL GAS CONSTANT FOR DIFFERENT UNITS Pressure
Volume
Temperature
Mass (mole)
Unit
Unit
Unit
Unit
Gas Constant R
psia
ft3
°R
lbm
10.7315
psia
cm3
°R
lbm
303,880
psia
cm3
°R
g
669.94
bar
ft3
°R
lbm
0.73991
atm
ft3
°R
lbm
0.73023
atm
cm3
°R
g
45.586
Pa
m3
K
kg
8314.3
Pa
m3
K
g
8.3143
kPa
m3
K
kg
8.3143
kPa
cm3
K
g
8314.3
bar
m3
K
kg
0.083143
bar
cm3
K
g
83.143
atm
m3
K
kg
0.082055
atm
cm3
K
g
82.055
Btu
°R
lbm
1.9858
Btu
°R
g
0.0043780
calorie
°R
lbm
500.76
calorie
°R
g
1.1040
kcal
°R
lbm
0.50076
kcal
°R
g
0.0011040
calorie
K
kg
1985.8
calorie
K
g
1.9858
erg
K
kg
8.3143 1010
erg
K
g
8.3143 107
J
K
kg
8314.3
J
K
g
8.3143
Energy Unit
TABLE A3—RECOMMENDED BIP’s FOR PR EOS AND SRK EOS FOR NONHYDROCARBON/HYDROCARBON COMPONENT PAIRS PR EOS* N2
CO2
SRK EOS** H2 S
N2
CO2
H2 S
N2
—
—
—
—
—
—
CO2
0.000
—
—
0.000
—
—
H2 S
0.130
0.135
—
0.120†
0.120
—
C1
0.025
0.105
0.070
0.020
0.120
0.080
C2
0.010
0.130
0.085
0.060
0.150
0.070 0.070
C3
0.090
0.125
0.080
0.080
0.150
iC4
0.095
0.120
0.075
0.080
0.150
0.060
C4
0.095
0.115
0.075
0.080
0.150
0.060
iC5
0.100
0.115
0.070
0.080
0.150
0.060
C5
0.110
0.115
0.070
0.080
0.150
0.060
C6
0.110
0.115
0.055
0.080
0.150
0.050
C7+
0.110
0.115
0.050‡
0.080
0.150
0.030‡
*Nonhydrocarbon
BIP’s from Ref. 1.
**Nonhydrocarbon
BIP’s from Ref. 2.
†Not
reported in Ref. 2.
‡Should
decrease gradually with increasing carbon number.
BIP+binary interaction parameter, PR EOS+PengRobinson equation of state, and SRK EOS+SoaveRedlichKwong equation of state.
PROPERTY TABLES AND UNITS
3
TABLE A4—FORTRAN PROGRAM FOR CALCULATING SPLIT OF C7+ WITH GAMMA DISTRIBUTION C C–––– C
PROGRAM GAMSPL IMPLICIT DOUBLE PRECISION (A*H,O*Z) DOUBLE PRECISION MWBL,MWBU,MWAV,MW7P OPEN(10,FILE+’GAMSPL.OUT’) WRITE(*,*) ’Input ALFA, ETA, M7+ >’ READ (*,*) ALFA,ETA,MW7P BETA+(MW7P*ETA)/ALFA MWBU+ETA S1+0.0 S2+0.0 WRITE(10,2000) ALFA,ETA,MW7P DO 100 I+1,20 MWBL+MWBU MWBU+MWBL)14.0 IF (I.EQ.20) MWBU+10000.0 CALL P0P1(ALFA,ETA,BETA,MWBL,P0L,P1L) CALL P0P1(ALFA,ETA,BETA,MWBU,P0U,P1U) Z+P0U*P0L S1+S1)Z MWAV+ETA+ALFA*BETA*(P1U*P1L)/(P0U*P0L)
S2+S2)Z*MWAV WRITE(10,2100) I,Z,MWAV 100 CONTINUE WRITE(10,2200) S1,S2/S1 2000 FORMAT (/ . ’ ALFA ........ :’,F10.3/ . ’ ETA ......... :’,F10.3/ . ’ MW7P ........ :’,F10.3/ . ’ –––––––––––––––––––––––––––––’/ . ’ Frac. Mole Molecular ’/ . ’ No. Fraction Weight ’/ . ’ ––––– –––––––––– –––––––––– ’) 2100 FORMAT (1X,I3,3X,F10.7,2X,F10.3) 2200 FORMAT (’ ––––––––– ––––––– ’/7X,F10.7,2X,F10.3) END SUBROUTINE P0P1 (ALFA,ETA,BETA,MWB,P0,P1) IMPLICIT DOUBLE PRECISION (A*H,O*Z) DOUBLE PRECISION MWB P0+0.0 P1+0.0 IF (MWB.LE.ETA) RETURN Y+(MWB*ETA)/BETA Q+DEXP(*Y)*Y**ALFA/GAMA(ALFA) TERM+1.0/ALFA S+TERM DO 100 J+1,10000 TERM+TERM*Y/(ALFA)DFLOAT(J)) S+S)TERM IF (DABS(TERM).LE.1.0D *8) GOTO 200 100 CONTINUE WRITE (*,2000) 200 CONTINUE P0+Q*S P1+Q*(S*1.0/ALFA) 2000 FORMAT (1X,’*** PR : SUM DOES NOT CONVERGE’) RETURN END DOUBLE PRECISION FUNCTION GAMA (X) IMPLICIT DOUBLE PRECISION(A*H,O*Z) DIMENSION B(8) DATA B /*0.577191652, 0.988205891,*0.897056937, . 0.918206857,*0.756704078, 0.482199394, . *0.193527818, 0.035868343 / CONST+1.0 XX+X IF (X.LT.1.0) XX+X)1.0 100 IF (XX.LE.2.0) GOTO 200 XX+XX*1.0 CONST+XX*CONST GOTO 100 200 XX+XX*1.0 Y+1.0 DO 300 I+1,8 Y+Y)B(I)*XX**I 300 CONTINUE GAMA+CONST*Y IF (X.LT.1.0) GAMA+GAMA/X RETURN END
4
PHASE BEHAVIOR
TABLE A5—GREEK ALPHABET
TABLE A6—SI SYSTEM UNITS
Upper Case
Lower Case
Name
A
a
Alpha
B
b
Beta
G
g
Gamma
D
d
Delta
E
e
Epsilon
Z
z
Zeta
H Q
h q
Eta Theta
I
i
Iota
K
k
Kappa
L
l
Lambda
M
m
N C O P
Base SI Units Used in Phase Behavior Quantity Length
Unit
Symbol
meter
m
Time
second
s
Mass
kilogram
kg
Temperature
kelvin
K
Amount of substance
mole
mol
Quantity Mass Volume
Unit
Symbol
tonne
Mg
liter
Definition 1
L
SI Term
Mg + 103 kg 1 L+1
Mg dm3
dm3
TABLE A7—SI PREFIXES Multiplication Factor
Prefix
1012
Tera
Symbol* T
Mu
109
Giga
G
n
Nu
106
Mega
M
c
Xi
103
Kilo
k
o
Omicron
102
Hecto
h
p
Pi
10
Deka
da
10*1
Deci
d
10*2
R
ò
Rho
S
s
Centi
c
Sigma
10*3
Milli
m
T
t
Tau
10*6
Micro
m
U
u
Upsilon
10*9
Nano
n
Pico
p
F
f
Phi
10*12
X
x
Chi
10*15
Femto
f
10*18
Atto
a
Y
y
Psi
W
w
Omega
PROPERTY TABLES AND UNITS
*Only the symbols T (tera), G (giga), and M (mega) are capital letters. Compound prefixes are not allowed; e.g., use nm (nanometer) rather than mmm (millimicrometer).
5
TABLE A8—PHYSICAL CONSTANTS AND VALUES (from Ref. 3) Triple point of water
273.16 exactly
K*
0.01 exactly
°C
491.688 exactly
°R
32.018 exactly
°F
0.00 exactly
K*
*273.15 exactly
°C
0.00 exactly
°R
Absolute zero
*459.67 exactly
°F
8.3143
J@mol*1@K*1*
10.731 5
psia@ft3@(lbmmol)*1@°R*1
Density of water at 60°F
999.014
kg@m*3*
[15.56°C, 288.71 K]
0.999 014
g@cm*3
62.366 4
lbm@ft*3
Gas constant, R
Standard atmosphere
1.013 2
bar
14.696 0
psia
1.223 2
kg@m*3*
1.223 2 10*3
g@cm*3
0.076 362
lbm@ft*3
9.806 650
m@s*2*
980.665 0
cm@s–2
32.174 05
ft@s*2
1.000 000
kg@m@N*1@s*2*
1.000 000
g@cm@dyne*1@s*2
32.174 05
lbm@ft@lbf*1@s*2
Earth’s gravitational acceleration, g
gc
p
Pa*
1.013 25 Density of air at 1 atm, 60°F [15.56°C, 288.71 K]
105
3.141 593 …
gAPI, °API
[141.5/g(60°F)]*131.5
*SI values. All quantities are consistent with conversion factors for the current SI system.
TABLE A9—TEMPERATURE SCALE CONVERSIONS (from Ref. 3) To Convert
To
Solve
degree Fahrenheit, TF
kelvin, TK
TK = (TF + 459.67)/1.8
degree Rankine, TR
kelvin, TK
TK = TR /1.8
degree Fahrenheit, TF
degree Rankine, TR
degree Fahrenheit, TF
degree Celsius, TC
TR = TF + 459.67 TC = (TF *32)/1.8
degree Celsius, TC
kelvin, TK
TK = TC + 273.15
The SI standard, the kelvin (K), is defined so that the triple point of water is 273.16 K exactly. The SI temperature symbol is written K, without a degree symbol. The cgs (and common) temperature unit is degree Celsius, °C; the common oilfield unit is degree Fahrenheit, °F, or degree Rankine, °R.
6
PHASE BEHAVIOR
TABLE A10—CONVERSION FACTORS USEFUL IN PHASE BEHAVIOR (from Ref. 3) To Convert From
To
Multiply By
Inverse
Area acre (acre)
square meter (m2)* square foot (ft2)
4.046 856 4.356 000**
E + 03 E + 04
2.471 054 2.295 684
E – 04 E – 05
darcy (darcy)
square meter (m2)* square centimeter (cm2) square micrometer (mm2) millidarcy (md) cm2cp@sec*1@atm*1
9.869 23 9.869 23 9.869 23 1.000 000** 1.000 000**
E – 13 E – 09 E – 01 E + 03 E + 00
1.013 25 1.013 25 1.013 25 1.000 000** 1.000 000**
E + 12 E + 08 E + 00 E – 03 E + 00
square foot (ft2)
square meter (m2)* square centimeter (cm2) square inch (in.2)
9.290 304** 9.290 304** 1.440 000**
E – 02 E + 02 E + 02
1.076 391 1.076 391 6.944 444
E + 01 E – 03 E – 03
hectare (ha)
square meter (m2)* acre
1.000 000** 2.471 054
E + 04 E + 00
1.000 000** 4.046 856
E – 04 E – 01
square mile (sq mile)
square meter (m2)* acre
2.589 988 6.400 000**
E + 06 E + 02
3.861 022 1.562 500**
E – 07 E – 03
gram per cubic centimeter (g/cm3)
kilogram/cubic meter (kg/m3)* poundmass/cubic foot (lbm/ft3) poundmass/gallon (lbm/gal) poundmass/barrel (lbm/bbl)
1.000 000** 6.242 797 8.345 405 3.505 070
E + 03 E + 01 E + 00 E + 02
1.000 000** 1.601 846 1.198 264 2.853 010
E – 03 E – 02 E – 01 E – 03
poundmass per cubic foot (lbm/ft3)
kilogram/cubic meter (kg/m3)* poundmass/gallon (lbm/gal) poundmass/barrel (lbm/bbl)
1.601 846 1.336 805 5.614 583
E + 01 E – 01 E + 00
6.242 797 7.480 520 1.781 076
E – 02 E + 00 E – 01
poundmass per gallon (lbm/gal)
kilogram/cubic meter (kg/m3)* poundmass/barrel (lbm/bbl)
1.198 264 4.200 000
E + 02 E + 01
8.345 406 2.380 952
E – 03 E – 02
dyne (dyne)
newton (N)* poundforce (lbf)
1.000 000** 2.248 089
E – 05 E – 06
1.000 000** 4.448 222
E + 05 E + 05
kilogramforce (kgf)
newton (N)* poundforce (lbf)
9.806 650** 2.204 622
E + 00 E + 00
1.019 716 4.535 924
E – 01 E – 01
poundforce (lbf)
newton (N)*
4.448 222
E + 00
2.248 089
E – 01
angstrom (Å)
meter (m)*
1.000 000**
E – 10
1.000 000**
E + 10
centimeter (cm)
meter (m)*
1.000 000**
E – 02
1.000 000**
E + 02
foot (ft)
meter (m)* centimeter (cm)
3.048 000** 3.048 000**
E – 01 E + 01
3.280 840 3.280 840
E + 00 E – 02
inch (in.)
meter (m)* centimeter (cm)
2.540 000** 2.540 000**
E – 02 E + 00
3.937 008 3.937 008
E + 01 E – 01
micron (mm)
meter (m)*
1.000 000**
E – 06
1.000 000**
E + 06
mile (U.S. statute)
meter (m)* foot
1.609 344** 5.280 000**
E + 03 E + 03
6.213 712 1.893 939
E – 04 E – 04
Density
Force
Length
*SI conversions. All quantities are current to SI standards as of 1974. **Conversion factor is exact and all following digits are zero. All other factors have been rounded. The notation E + 03 is used in place of 103, and so on.
PROPERTY TABLES AND UNITS
7
TABLE A10 (continued)—CONVERSION FACTORS USEFUL IN PHASE BEHAVIOR (from Ref. 3) To Convert From
To
Multiply By
Inverse
Mass grammass
kilogram (kg)*
1.000 000**
E – 03
1.000 000**
E + 03
ouncemass (avoirdupois)
kilogram (kg)* gram (g)
2.834 952 2.834 952
E – 02 E + 01
3.527 397 3.527 397
E + 01 E – 02
poundmass
kilogram (kg)* ouncemass
4.535 923 7** E – 01 1.600 000** E + 01
2.204 623 6.250 000**
E + 00 E – 02
slug
kilogram (kg)* poundmass (lbm)
1.459 390 3.217 405
E + 01 E + 01
6.852 178 3.108 095
E – 02 E – 02
ton (U.S. short)
kilogram (kg)* poundmass (lbm)
9.071 847 2.000 000**
E + 02 E + 03
1.102 311 5.000 000**
E – 03 E – 04
ton (U.S. long)
kilogram (kg)* poundmass (lbm)
1.016 047 2.240 000**
E + 03 E + 03
9.842 064 4.464 286
E – 04 E – 04
ton (metric)
kilogram (kg)*
1.000 000**
E + 03
1.000 000**
E – 03
tonne
kilogram (kg)*
1.000 000**
E + 03
1.000 000**
E – 03
atmosphere (atm) (Normal is 760 mm Hg)
pascal (Pa)* mm Hg (0°C) feet water (4°C) poundforce/square inch (psi) bar
1.013 25 7.600 000** 3.389 95 1.469 60 1.013 25
E + 05 E + 02 E + 01 E + 01 E + 00
9.869 23 1.315 789 2.949 90 6.804 60 9.869 23
E – 06 E – 03 E – 02 E – 02 E – 01
bar (bar)
pascal (Pa)* poundforce/square inch (psi)
1.000 000** 1.450 377
E + 05 E + 01
1.000 000** 6.894 757
E – 05 E – 02
centimeter of Hg (32°F)
pascal (Pa)* poundforce/square inch (psi)
1.333 22 1.933 67
E + 03 E – 01
7.500 64 5.171 51
E – 04 E + 00
dyne/square centimeter (dyne/cm2)
pascal (Pa)* pound force/square inch (psi)
1.000 000** 1.450 377
E – 01 E – 05
1.000 000** 6.894 757
E + 01 E + 04
feet of water (39.2°F)
pascal (Pa)* pound force/square inch (psi)
2.988 98 4.335 15
E + 03 E – 01
3.345 62 2.306 73
E – 04 E + 00
kilogramforce/square centimeter
pascal (Pa)* bar pound force/square inch (psi)
9.806 650** 9.806 650** 1.422 334
E + 04 E – 01 E + 01
1.019 716 1.019 716 7.030 695
E – 05 E + 00 E – 02
poundforce/inch2 (psi)
pascal (Pa)*
6.894 757
E + 03
1.450 377
E – 04
day (d)
second (s)* minute (min) hour (h)
8.640 000** 1.440 000** 2.400 000**
E + 04 E + 03 E + 01
1.157 407 6.944 444 4.166 667
E – 05 E – 04 E – 02
hour (h)
second (s)* minute (min)
3.600 000** 6.000 000**
E + 03 E + 01
2.777 778 1.666 667
E – 04 E – 02
minute (min)
second (s)*
6.000 000**
E + 01
1.666 667
E – 02
Pressure
Time
*SI conversions. All quantities are current to SI standards as of 1974. **Conversion factor is exact and all following digits are zero. All other factors have been rounded. The notation E + 03 is used in place of 103, and so on.
8
PHASE BEHAVIOR
TABLE A10 (continued)—CONVERSION FACTORS USEFUL IN PHASE BEHAVIOR (from Ref. 3) To Convert From
To
Multiply By
Inverse
Viscosity centipoise (cp)
pascalsecond (Pa@s)* dynesecond/square centimeter (dynes/cm2) poundmass/footsecond (lbm/ftsec) poundforcesecond/square foot (lbfsec/ft2) poundmass/foothour (lbm/fthr)
1.000 000** 1.000 000** 6.719 689 2.088 543 2.419 088
centistoke (cSt)
square meter/second (m2/s)* centipoise/gramcubic centimeter (cp/gcm3)
1.000 000** E – 06 1.000 000** E + 00
1.000 000** E + 06 1.000 000** E + 00
poise
pascalsecond (Pa@s)*
1.000 000** E – 01
1.000 000** E + 01
poundmass/footsecond (lbm/ftsec)
pascalsecond (Pa@s)*
1.488 164
E + 00
6.719 689
E – 01
poundmass/foothour (lbm/fthr)
pascalsecond (Pa@s)*
4.133 789
E – 04
2.419 088
E + 03
pascalsecond (Pa@s)*
4.788 026
E + 01
2.088 543
E – 02
acrefoot (acreft)
cubic meter (m3)* cubic foot (ft3) barrel (bbl)
1.233 482 E + 03 4.356 000** E + 04 7.758 368 E + 03
8.107 131 2.295 684 1.288 931
E – 04 E – 05 E – 04
barrel (bbl)
cubic meter (m3)* cubic foot (ft3) gallon (gal)
1.589 873 E – 01 5.614 583 E + 00 4.200 000** E + 01
6.289 811 1.781 076 2.380 952
E + 00 E – 01 E – 02
cubic foot (ft3)
cubic meter (m3)* cubic inch (in.3) gallon (gal)
2.831 685 1.728 000 7.480 520
E – 02 E + 03 E + 00
3.531 466 5.787 037 1.336 805
E + 01 E – 04 E – 01
gallon (gal)
cubic meter (m3)* cubic inch (in.3)
3.785 412 2.310 001
E – 03 E + 02
2.641 720 4.329 003
E + 02 E – 03
liter (L)
cubic meter (m3)*
1.000 000** E – 03
1.000 000** E + 03
barrel/day (B/D)
cubic meter/second (m3/s)* cubic meter/hour (m3/h) cubic meter/day (m3/d) cubic centimeter/second (cm3/s) cubic foot/minute (ft3/min) gallon/minute (gal/min)
1.840 131 6.624 472 1.589 873 1.840 131 3.899 016 2.916 667
E – 06 E – 03 E – 01 E + 00 E – 03 E – 02
5.434 396 1.509 554 6.289 810 5.434 396 2.564 750 3.428 571
E + 05 E + 02 E + 00 E – 01 E + 02 E + 01
cubic foot/minute (ft3/min)
cubic meter/second (m3/s)*
4.719 474
E – 04
2.118 880
E + 03
cubic foot/second (ft3/sec)
cubic meter/second (m3/s)*
2.831 685
E – 02
3.531 466
E + 01
(m3/s)*
6.309 020
E – 05
1.585 032
E + 04
poundforcesecond/square foot (lbfsec/ft2)
E – 03 E – 02 E – 04 E – 05 E + 00
1.000 000** 1.000 000** 1.488 164 4.788 026 4.133 789
E + 03 E + 02 E + 03 E + 04 E – 01
Volume
Volumetric rate
gallon/minute (gal/min)
cubic meter/second
*SI conversions. All quantities are current to SI standards as of 1974. **Conversion factor is exact and all following digits are zero. All other factors have been rounded. The notation E + 03 is used in place of 103, and so on.
PROPERTY TABLES AND UNITS
9
TABLE A11—ADDITIONAL CONVERSION FACTORS USEFUL IN PHASE BEHAVIOR To Convert From
To
Multiply By
Inverse
Amount of substance mole (mol)
kilomole (kmol)
poundmass mole (lbm mol)
2.204 623
E + 03
4.535 923
E – 04
gram mole (gmol)
1.000 000*
E + 00
1.000 000*
E + 00
kilomole (kmol)
1.000 000*
E – 03
1.000 000*
E + 03
mole (gmol)
1.000 000*
E + 03
1.000 000*
E – 03
gram mole (gmol)
1.000 000*
E + 03
1.000 000*
E – 03
poundmass mole (lbm mol)
4.535 923
E – 01
2.204 623
E + 00
square meter/second (m2/s)
1.000 000*
E – 04
1.000 000*
E + 04
square millimeter/second (mm2/s)
1.000 000*
E + 02
1.000 000*
E – 02
square foot/second (ft2/sec)
1.076 390
E – 03
9.290 304
E + 02
square foot/hour (ft2/hr)
3.875 000
E + 00
2.580 640
E – 01
dyne/centimeter (dyne/cm)
1.000 000*
E + 00
1.000 000*
E + 00
kiloJoule (kJ)
1.055 056
E + 00
9.478 160
E – 01
calorie (cal)
2.521 640
E + 02
3.965 660
E – 03
kilocalorie (kcal)
2.521 640
E – 01
3.965 660
E + 00
erg
1.055 056
E + 10
9.478 160
E – 11
Diffusivity square centimeter/second (cm2/s)
Surface tension milliNewton/meter (mN/m) Energy British thermal unit (Btu)
*Conversion factor is exact.
References 1. Nagy, Z. and Shirkovskiy, A.I.: “Mathematical Simulation of Natural Gas Condensation Processes Using the PengRobinson Equation of State,” paper SPE 10982 presented at the 1982 SPE Annual Technical Conference and Exhibition, New Orleans, 26–29 September.
10
2. Reid, R.C., Prausnitz, J.M., and Polling, B.E.: The Properties of Gases and Liquids, fourth edition, McGrawHill Book Co. Inc., New York City (1987). 3. Earlougher, R.C. Jr.: Advances in Well Test Analysis, SPE Monograph Series, SPE, Richardson, Texas (1977) 5.
PHASE BEHAVIOR
Appendix B
Example Problems Introduction Many of the problems presented here were introduced by Standing during his 2 years as visiting professor at the Norwegian Inst. of Technology in Trondheim during 1973–74. Some of the problems have been modified or expanded, and additional problems have been included to cover subjects presented in the monograph that were not necessarily covered in Standing’s problems. Problem 1 Problem. A lighthydrocarbon gas has the compositional analysis given in Table B1. Calculate the following properties. a. Weight composition. b. Molecular weight. c. Specific gravity. d. Density in lbm/ft3 at 20 psia and 120°F, assuming ideal gas behavior. e. Density in kg/m3 at 3.1 atm and 50°C, assuming ideal gas behavior. Solution. The problem is solved by calculating mass, mi +xi Mi , and mass (weight) fractions, as shown in Table B2. The following equations have been used. wi +
mi
n i Mi
+
ȍm ȍn M N
j
j
j+1
gg +
;
N
. . . . . . . . . . . . . . . . . . . . . . . (3.3)
j
j+1
ǒò gǓ
sc
ǒò airǓ sc
+
Mg Mg + M air 28.97
and M g + 28.97 g g ;
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.28)
ò g + pM gńZRT;
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.34)
ȍy M , i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.50a)
ci ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.50b)
ȍy p
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.50c)
a. Weight composition is given as wi in Eq. 3.3. b. The ratio of total mass, S mi , to total moles, Syi , gives the average molecular weight. M g + (24.97)ń(1.00) + 24.97 lbmńlbm mol. c. Gas specific gravity is given by g g + (24.97)ń(28.97) + 0.864 (air + 1). d. Gas density is calculated with Eq. 3.35. ò g + (20)(24.97)ń[(1)(10.732)(120 ) 460)] + 0.0801 lbmńft 3. e. Gas density in SI units is also calculated with Eq. 3.35 with the correct gas constant, R, from Table A2 in Appendix A. ò g + (3.1)(24.97)ń[(1)(0.082055)(50 ) 273)] + 2.92 kgńm 3. Problem 2 Problem. Table B3 gives the compositional analysis of a relativelyhighsulfurcontent Canadian gas. If the gasprocessing plant that treats the gas removes 100% of the H2S and converts it to elemental sulfur, how many long tons (2,200 lbm) of sulfur will result from processing 1,000 Mscf of field gas? Solution. Total mass in lbm mol of 1,000 Mscf gas is calculated from the real gas law, pV + nZRT.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.30)
N
and M +
i
i+1
ȍy T N
T pc +
i
Component
i+1 N
and p pc +
TABLE B1—GAS COMPOSITIONAL ANALYSIS (PROBLEM 1)
i
ci .
i+1
EXAMPLE PROBLEMS
Mole Fraction
Methane
0.49
Ethane
0.38
Propane
0.13 1
TABLE B2—MASS AND MASS (WEIGHT) FRACTIONS (PROBLEM 1)
Component i
Molecular Weight Mi (lbm/lbm mol)
Mole Fraction xi
C1
16.04
0.49
7.84
0.314
C2
30.07
0.38
11.40
0.456
C3
44.09
0.13
5.73
0.230
1.00
24.97
1.000
Total
Mass mi = xi Mi (lbm)
TABLE B3—GAS COMPOSITIONAL ANALYSIS (PROBLEM 2) Mole Fraction zi
Component i
0.0112
C3
0.05
0.2609
nC4
0.10
0.5575
nC5
0.15
C2
0.0760
nC6
0.70
C3
0.0433
iC4
0.0061
nC4
0.0137
iC5
0.0033
nC5
0.0052
C6
0.0053 0.0175
+ 128 and g C
7)
+ 0.780.
Note: Canadian standard pressure base is 14.65 psia. Assume that Z=1 at standard conditions.
Solving for n, n + pVńZRT + (14.65)ǒ1
10 6Ǔń[(1.0)(10.73)(60 ) 460)]
+ 2, 625 lbm mol. Moles of H2S is calculated by multiplying the total moles by the mole fraction of H2S. nH
2S
Liquid Volume Fraction xVi
Component i
H2 S
C7+ 7)
TABLE B4—LIQUID VOLUME COMPOSITION (PROBLEM 3)
CO2 C1
MC
Weight Fraction wi = mi /(Smj )
+ (0.2609)(2, 625) + 685 lbm mol.
Problem 3 Problem. At 15.56°C, a storage tank contains 1,000 m3 of gasoline with the liquid volume composition given in Table B4. Calculate the following. a. Weight (mass) composition. b. Molar composition. c. Molecular weight. d. Specific gravity. e. Oil gravity (°API). f. Moles in kilogram moles (kmol) of n C in the tank. 6 g. Gallons of n C in the tank. 5 h. Pounds of n C in the tank. 4 Note: Use component properties from Appendix A and values from Table B5. Solution. a. Weight composition from Column 5, where wi +mi /(Smj ). b. Mole composition from Column 8, where xi +ni /(Snj ). c. Molecular weight from M + (640.0 kg)ń(8.248 kmol) + 77.6 kgńkmol + 77.6 lbmńlbm mol.
There is one mole of sulfur (S) per mole of H2S, so n S + (0.2609)(2, 625) + 685 lbm mol.
d. Density from
The mass of sulfur equals the moles of sulfur times the molecular weight of sulfur (MS+32), m S + (685)(32)ńǒ2, 200 lbmńtonǓ
ò o + m ońV o + (640.0 kg)ńǒ1.0 m 3Ǔ + 640.0 kgńm 3. Specific gravity is calculated from g o + ò ońò w (Eq. 3.12), where densities are at standard conditions. g o + ǒ640.0 kgńm 3Ǔńǒ999.0 kgńm 3Ǔ + 0.640 (water + 1).
+ 9.96 long tonsń1, 000 Mscf produced gas.
TABLE B5—COMPOSITION CONVERSIONS FOR MIXTURES (PROBLEM 3) Column
Component i
1 Liquid Volume Fraction xVi
2 Liquid Volume* Vi (m3)
3 Liquid Density òi (kg/m3)
4 Mass m i + òV i (kg)
5 Weight Fraction wi
6 Molecular Weight Mi (kg/kmol)
7 Moles ni = mi /Mi k (kmol)
8 Mole Fraction xi
C3
0.05
0.05
507.2
25.36
0.040
44.09
0.575
0.070
nC4
0.10
0.10
583.9
58.39
0.091
58.12
1.005
0.122
nC5
0.15
0.15
629.5
94.43
0.148
72.15
1.309
0.159
nC6
0.70
0.70
659.8
461.86
0.722
86.17
5.360
0.650
640.04
1.000
8.248
1.000
Total
1.00
*On the basis of 1 m3.
2
PHASE BEHAVIOR
TABLE B6—SEPARATOR GAS AND SEPARATOR OIL COMPOSITIONS FOR WELLSTREAM RECOMBINATION CALCULATION (PROBLEM 4) Component i
Gas Mole Fraction yi
Liquid Volume Fraction xVi
C1
0.968
0.020
C2
0.010
0.006
C3
0.011
0.011
iC4
0.003
0.009
nC4
0.003
0.013
iC5
0.002
0.016
nC5
0.001
0.010
C6
0.002
0.038
C7+
0.000
0.877
Problem 5 Problem. A new well was completed with perforations in three separate intervals. Initial pressure at midperforations (4,650 ft subsurface) was 2,000 psig at 150°F. The first 24hour production test gave the information in Table B8. On the basis of these data, which of the following do you consider best describes the well effluent. a. Production of a single phase from a gascondensate reservoir. b. Production of separate gas and liquid phases into the well. c. Production of undersaturated liquid into the well. Explain the basis for your decision.
M C7) + 144 and g C7) + 0.775.
e. g API + (141.5)ń(0.640) * (131.5) + 89.4°API. f. Moles of n C + ǒ1000 m 3Ǔǒ5.36 kmolńm 3Ǔ + 5360 kmol. 6
g. Volume of n C + ǒ1000 m 3Ǔǒ0.15 m 3ńm 3Ǔǒ6.289 bblńm 3Ǔ 5 ǒ42 galńbblǓ + 3.962 10 4 gal. h. Mass of n C + ǒ1000 m 3Ǔǒ58.39 kgńm 3Ǔǒ2.205 lbmńkgǓ 4
+ 1.2875
10 5 lbm.
Problem 4 Problem. During a 24hour test, a well produced 463 STB oil and 5,783 Mscf of separator gas (these volumes are expressed at 14.4 psia and 60°F). Table B6 gives oil and gas compositions. Calculate the welleffluent composition in mole fraction. Use apparent liquid densities for methane and ethane of 0.30 and 0.45 g/cm3, respectively. Solution. From Eq. 3.18, the producing gas/oil ratio (GOR) is R p + q gńq o + ǒ5.783
Table B7 calculates oil molar composition and recombined wellstream composition with 1 STB oil volume as a basis. Ideal solution mixing is assumed for the stocktank oil. Also, note that the component moles in the stocktank oil are given by n oi + 5.6146 V i ò ińM i (Eq. 3.4).
10 6Ǔń(463) + 12, 500 scfńSTB,
or in terms of the producing oil/gas ratio (OGR) from Eq. 3.19, r p + 1ńR p + ǒ10 6 scfńMMscfǓńǒ12, 500 scfńSTBǓ + 80 STBńMMscf. On a basis of 1 STB, the moles of gas produced is given by solving for ng from the real gas law [ pV+nZRT (Eq. 3.30)], with Z+1, n g + [(14.4)(12, 500)]ń[(1.0)(10.73)(60 ) 460)] + 32.3 lbm mol.
Solution. The GOR of 19,000 might be descriptive of a gascondensate system (Answer a). However, at the reservoir pressure of 2,000 psi and 150°F, it would be unlikely that a 27°API liquid could dissolve in the gas phase. The reservoir gas probably has been or currently is in contact with a reservoir oil. At 2,000 psia, the K values (Ki +yi /xi ) of the heavy components that make up a 27°API crude would be extremely small (mostly t10*3) and the heaviest components would have the lowest K values. Even if the reservoir oil contacting the reservoir gas is very heavy, the resulting amounts of heavy components found in the equilibrium gas would be very small and proportionally more of the lighter fractions would be found in the reservoir gas. The condensate from such an equilibrium gas would tend to have a lower gravity (e.g., gAPIu50°API). Answer c is also wrong because it is not possible to dissolve 19,000 scf of gas in 1 STB of such a heavy crude oil. Consequently, Answer b is the best answer. Both reservoir oil (with a gravity somewhat heavier than 27°API) and reservoir gas (with a much lighter condensate gravity) are both flowing into the well simultaneously. Coning, leakage behind the casing, or multiple completion intervals are three situations that might cause the production characteristics seen in this well. Problem 6 Problem. Table B9 gives the gas composition of the Sabine field in Texas. This is a typical composition of field gases produced from primary separators. Assuming that this gas is to be compressed and reinjected into a reservoir at 200°F, calculate the compressibility factor, Z; gas formation volume factor (FVF), Bg ; and gas density, ò g , at 2,000 psig and 160°F. Make the calculations using pseudocritical properties calculated from the gas composition in Table B9 and from gas gravity.
TABLE B7—OIL MOLAR COMPOSITION AND RECOMBINED WELLSTREAM COMPOSITION (PROBLEM 4) Gas Component Mole Fraction i yi
Gas Moles ngi +ng yi (lbm mol)
Oil Volume Voi (STB)
Liquid Density òi (lbm/ft3)
Molecular Weight Mi (lbm/lbm mol)
Oil Moles noi (lbm mol)
Total Moles ni +ngi + noi (lbm mol)
Wellstream Mole Fraction zi
C1
0.968
31.266
0.020
18.73
16.04
0.131
31.398
0.9123
C2
0.010
0.323
0.006
28.09
30.07
0.031
0.354
0.0103
C3
0.011
0.355
0.011
31.66
44.09
0.044
0.400
0.0116
iC4
0.003
0.097
0.009
35.01
58.12
0.030
0.127
0.0037
C4
0.003
0.097
0.013
36.45
58.12
0.046
0.143
0.0041
iC5
0.002
0.065
0.016
39.13
72.15
0.049
0.113
0.0033
C5
0.001
0.032
0.010
39.30
72.15
0.031
0.063
0.0018
C6
0.002
0.065
0.038
41.19
86.17
0.102
0.167
0.0048
C7+
0.000
0.0000
0.877
48.33
144.00
1.653
1.653
0.0480
2.117
34.417
1.0000
Total
1.000
1.000
Compostions are calculated on the basis of 1 STB oil volume.
EXAMPLE PROBLEMS
3
TABLE B8—RESULTS OF FIRST 24HOUR PRODUCTION TEST (PROBLEM 5)
TABLE B9—GAS COMPOSITION (PROBLEM 6) Mole Fraction yi
Oil produced, STB
65
Component
Stocktankoil gravity, °API
27
C1
0.875
C2
0.083
C3
0.021
iC4
0.006
nC4
0.008
iC5
0.003
nC5
0.002
C6
0.001
C7+
0.001
Gas produced, MMscf
1.23
Gas/oil ratio, scf/bbl separator oil
19,000
Solution. Properties From Composition. M g + 28.97 g g ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.28)
ȍy M , N
M+
i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.50a)
i
i+1
ȍy T
and Z + 0.846.
N
T pc +
i
ci ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.50b)
i+1
ò g + pM gńZRT ,
ȍy p N
and p pc +
Gas density is given by
i
ci .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.50c)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.34)
which yields ò g + (2, 015)(18.83)ń[(0.846)(10.73)(160 ) 460)]
i+1
With the pseudocritical properties in Table B10, these equations give, T pc + 376°R,
+ 6.74 lbmńft 3. Properties From Specific Gravity Correlations. The Sutton3 correlations for pseudocritical properties are
p pc + 667 psia,
T pcHC + 169.2 ) 349.5g gHC * 74.0 g 2gHC . . . . . . . . . . (3.47a)
M g + 18.83 (KayȀs mixing rule),
and p pcHC + 756.8 * 131g gHC * 3.6g 2gHC ,
g g + (18.83)ń(28.97) + 0.65 (air + 1),
. . . . . . . . . (3.47b)
which give
T pr + TńT pc + (160 ) 460)ń376 + 1.65,
2
T pc + 169.2 ) 349.5(0.65) * 74.0(0.65) + 365°R,
and p pr + pńp pc + 2, 015ń667 + 3.02.
2
p pc + 756.8 * 131.0(0.65) * 3.6(0.65) + 670 psia ,
Gas Z factor is given by the HallYarborough1,2 correlation.
T pr + TńT pc + (160 ) 460)ń365 + 1.70,
Z + ap prńy, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.42)
p pr + pńp pc + 2, 015ń670 + 3.01,
where a + 0.06125 t expƪ* 1.2(1 * t) ƫ, where t + 1ńT pr . This 2
Z + 0.865, and ò g + 6.59 lbmńft 3.
gives t + 1ńT pr + 1ń1.65 + 0.606, a + (0.06125)(0.606) expƪ(* 1.2)(1 * 0.606)
2
ƫ + 0.0308,
y + 0.10996 ǒd Fńdy + 0.79798Ǔ,
Problem 7 Problem. Calculate the viscosity of the Sabine field gas of Problem 6 under reservoir conditions of 2,000 psig and 160°F. Use the Lucas4 and LohrenzBrayClark5 viscosity correlations based on gas composition.
TABLE B10—PSEUDOCRITICAL PROPERTIES (PROBLEM 6) Component
zi
Mi
pci (psia)
Tci °R
zi Mi
zi pci (psia)
zi Tci °R
C1
0.8750
16.04
667.8
C2
0.0830
30.07
707.8
343.0
14.04
584.3
300.1
549.8
2.50
58.7
C3
0.0210
44.09
45.6
616.3
665.7
0.93
12.9
14.0
iC4
0.0060
58.12
529.1
734.7
0.35
3.2
4.4
C4
0.0080
58.12
550.7
765.3
0.46
4.4
6.1
iC5
0.0030
72.15
490.4
828.8
0.22
1.5
2.5
C5
0.0020
72.15
488.6
845.4
0.14
1.0
1.7
86.17
436.9
913.4
0.09
0.4
0.9
360.6
1,023.9
0.11
0.4
1.0
18.83
666.8
376.4
C6
0.0010
C7+*
0.0010
Total
1.0000
114.0
*Use properties for nC8.
4
PHASE BEHAVIOR
TABLE B11—LOHRENZBRAYCLARK5 VISCOSITY CALCULATIONS (PROBLEM 7) Component
zi
vci (ft3/lbm mol)
Zci
zi vci (ft3/lbm mol)
zi Zci
C1
0.8750
1.590
0.2884
1.391
0.2524
C2
0.0830
2.370
0.2843
0.197
0.0236
C3
0.0210
3.250
0.2804
0.068
0.0059
iC4
0.0060
4.208
0.2824
0.025
0.0017
C4
0.0080
4.080
0.2736
0.033
0.0022
iC5
0.0030
4.899
0.2701
0.015
0.0008
C5
0.0020
4.870
0.2623
0.010
0.0005
C6
0.0010
5.929
0.2643
0.006
0.0003
C7+
0.0010
7.882
0.2587
Total
1.0000
m gńm gsc + 1 )
where A 1 +
A 2 p pr5 ) ǒ1 ) A 3 p pr4Ǔ A
A
*1
,
. . . . . . (3.66a)
A 3 + 0.272, A 4 + 1.105,
A 2 + A 1ǒ1.6553T pr * 1.2723Ǔ , A3 +
Ǔ 0.4489 expǒ3.0578T *37.7332 pr , T pr
A4 +
Ǔ 1.7368 expǒ2.2310T *7.6351 pr , T pr
A 1 + 0.0607, A 2 + 0.0886,
Ǔ 10 *3) expǒ5.1726T *0.3286 pr , T pr
(1.245
A 5 + 0.7473, m gńm gsc + 1.360, and m g + 0.0167 cp. LohrenzBrayClark Correlation. Eqs. 3.133 through 3.135 give the LohrenzBrayClark correlation.
and A 5 + 0.9425 expǒ* 0.1853T pr0.4489Ǔ ,
. . . . . . . . . . . (3.66b)
ƪǒm * m oǓz T ) 10 *4ƫ
where m gsc c + ƪ0.807T pr0.618 * 0.357 expǒ* 0.449T prǓ ) 0.340 expǒ* 4.058T prǓ ) 0.018ƫ ,
ǒ Ǔ
ȍy Z
) 0.058533ò 2pr * 0.040758ò 3pr ) 0.0093324ò 4pr ,
ò ò pr + ò
pc N
and p pc +
ci
ȍy v
i
.
. . . . . . . . . . . . . . . . . . . . . . . . (3.67)
and m o +
i ci
,
ȍ z ǸM
i
.
. . . . . . . . . . . . . . . . . . . . . . . . (3.133)
i
i+1
The Lucas correlation gives
m iz Ti + ǒ34
T pc + 376°R,
10 *5ǓT ri0.94
. . . . . . . . . . . . . . . . . . . . . (3.134a)
for Tri x1.5, and
Z pc + 0.2876,
m iz Ti + ǒ17.78
v pc + 1.752 ft 3ńlbm mol, p pc + 663 psia,
10 *5Ǔ(4.58T ri * 1.67)
for Tri u1.5, where z Ti + 5.35ǒT ci M 3ińp 4ciǓ
M + 18.83 lbmńlbm mol,
v cC
T pr + TńT pc + (160 ) 460)ń376 + 1.65,
1ń6
+ 77.3 cp
7)
+ 21.573 ) 0.015122M C ) 0.070615M C
p pr + pńp pc + 2, 015ń663 + 3.04, *1
0.618 m gscc + Ǌ0.807(1.65) * 0.357 exp[(* 0.449)(1.65)]
) 0.340 exp[(* 4.058)(1.65)] ) 0.018ǋ + 0.948, EXAMPLE PROBLEMS
i
i+1 N
i
i+1
3 4 c + 9, 490Ǌ(376)ńƪ(18.83) (663) ƫǋ
1ń6
ò v , M pc
+
ȍ z m ǸM
N
i
+ 0.10230 ) 0.023364ò pr
ǒ Ǔ
,
i+1 RT pc N
1ń4
T pc where z T + 5.35 M 3p 4pc
1ń6
T pc c + 9, 490 M 3p 4pc
0.0003 0.2876
m gsc + ǒ m gsc c Ǔńc + 0.948ń77.3 + 0.0123 cp,
Solution. Lucas Correlation With Composition. A 1 p 1.3088 pr
0.008 1.752
g . 7) C 7)
7)
5ń8
1ń6
. . . . . . . (3.134b)
.
* 27.656g C
7)
. . . . . . . . . . . . . . . (3.135)
On the basis of the data in Tables B11 and B12, this correlation yields T pc + 376°R, T pr + 1.65, p pc + 663 psia, 5
TABLE B12—LOHRENZBRAYCLARK VISCOSITY5 CALCULATIONS (PROBLEM 7) Tri
ci
mi (cp)
343.0
1.81
0.0463
707.8
549.8
1.13
44.09
616.3
665.7
0.0060
58.12
529.1
0.0080
58.12
550.7
iC5
0.0030
72.15
C5
0.0020
C6 C7+
Mi
pci (psia)
Tci (°R)
0.8750
16.04
667.8
C2
0.0830
30.07
C3
0.0210
iC4 C4
zi mi M½ i
z i M½ i
0.0125
0.0438
3.504
0.0352
0.0108
0.0049
0.455
0.93
0.0329
0.0097
0.0013
0.139
734.7
0.84
0.0322
0.0090
0.0004
0.046
765.3
0.81
0.0316
0.0088
0.0005
0.061
490.4
828.8
0.75
0.0310
0.0083
0.0002
0.025
72.15
488.6
845.4
0.73
0.0312
0.0081
0.0001
0.017
0.0010
86.17
436.9
913.4
0.68
0.0312
0.0076
0.0001
0.009
0.0010
114.00
360.6
1,023.9
0.61
0.0314
0.0068
0.0001
0.011
0.0516
4.268
Component
zi
C1
Total
1.0000
TABLE B13—ANALYSIS OF SOUR CANADIAN GAS (PROBLEM 8)
p pc +
* * eǓ p *pcǒ Tpc * T pc ) yH
2S
ǒ1 * y Ǔe
Component i
Mole Fraction yi
CO2
0.0112
H2 S
0.2609
C1
0.5575
C2
0.0760
C3
0.0433
iC4
0.0061
e + 29.8,
nC4
0.0137
T pc + 489.6 * 29.8 + 459.8°R,
iC5
0.0033
nC5
0.0052
C6
0.0053
C7+
0.0175
M C7) + 128 and g C7) + 0.780.
and e + 120
ǒ
ƪǒ
y CO ) y H 2
4 ) 15 y 0.5 H S * yH 2
2S
2S
Ǔ
Ǔ,
0.9
ǒ
* y CO ) y H 2
2S
Ǔ
1.6
ƫ
. . . . . . . . . . . . . . . . . . . . . . . (3.52c)
which (with the pseudocritical properties in Table B14) gives
p pc +
(829.5)(489.6 * 29.8) + 770 psia, (489.6) ) (0.2609)(1 * 0.2609)(29.8)
T pr + 696ń459.8 + 1.51, and p pr + 3, 065ń770 + 3.98. Where the StandingKatz8 Zfactor chart is fit by the HallYarborough1,2 correlation,
p pr + 3.04,
Z + ap prńy, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.42)
M + 18.83, v Mpc + 1.752 ft 3ńlbm mol,
where a + 0.06125 t expƪ* 1.2(1 * t) 2ƫ, where t + 1ńT pr ,
ò pr + ǒ6.74ń18.83Ǔ(1.752) + 0.627,
and F( y) + 0 + * ap pr )
Ǌ
c T + 5.35 (376)ńƪ(18.83) (663) 3
4
ƫǋ
1ń6
and m g + 0.0121 ) ƪ(0.131) * 10 *4ƫń(0.0436) + 0.0166 cp. 4
Problem 8
) ǒ90.7t–242.2t 2 ) 42.4t 3Ǔy 2.18)2.82t,
. . . . . . (3.43)
with t + 1ń1.51 + 0.6622, a + 0.06125(0.6622) expƪ* 1.2(1 * 0.6622)
2
ƫ + 0.03537,
y + 0.18088,
Problem. Table B13 gives the analysis of the sour Canadian gas of Problem 2. Use the method developed by Wichert and Aziz6,7 and calculate adjusted pseudocritical properties for use with the StandingKatz8 Zfactor chart. Then, calculate the gas FVF, Bg , at reservoir conditions of 3,050 psig and 236°F. Note that Canadian standard conditions are 14.65 psia and 60°F.
and gas FVF given by
Solution. The WichertAziz pseudocritical correction is given by
yields
T pc + T *pc * e, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.52a)
y ) y2 ) y3 * y4 (1 * y) 3
* ǒ14.76t * 9.76t 2 ) 4.58t 3Ǔy 2
+ 0.0436,
m gsc + 0.0516ń4.268 + 0.0121 cp,
6
, . . . . . . . . . . . . . . . . . (3.52b)
H 2S
and Z + 0.778,
Bg +
ǒTp Ǔ ZTp sc
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.38)
sc
B g + ǒ14.65ń520Ǔƪ(0.778)(696)ń(3, 065)ƫ + 0.00498 ft 3ńscf. PHASE BEHAVIOR
TABLE B14—PSEUDOCRITICALPROPERTY CALCULATIONS FOR A SOUR GAS (PROBLEM 8) zi
Mi
pci (psia)
Tci (°R)
zi Mi ă
CO2
0.0112
44.01
H2 S
0.2609
34.08
C1
0.5575
C2
0.0760
C3
1,070.6
547.6
0.49
12.0
6.1
1,306.0
672.4
8.89
340.7
175.4
16.04
667.8
343.0
8.94
372.3
191.2
30.07
707.8
549.8
2.29
53.8
41.8
0.0433
44.09
616.3
665.7
1.91
26.7
28.8
iC4
0.0061
58.12
529.1
734.7
0.35
3.2
4.5
C4
0.0137
58.12
550.7
765.3
0.80
7.5
10.5
iC5
0.0033
72.15
490.4
828.8
0.24
1.6
2.7
C5
0.0052
72.15
488.6
845.4
0.38
2.5
4.4
C6
0.0053
86.17
436.9
913.4
0.46
2.3
4.8
C7+*
0.0175
386.7
1,099.5
2.24
6.8
19.2
26.98
829.5
489.6
Total
128.0
1.0000
zi pci (psia)
zi Tci (°R)
*C7+ pseudocriticals from Eq. 3.51.
A 3 + * 3.57
TABLE B15—SURFACE PRODUCTION DATA (PROBLEM 9) Reservoir pressure, psia
5,200
Reservoir temperature °F
250
Separator pressure, psia
950
Separator temperature, °F
160
Primary separator gas rate, Mscf/D
4,265
Primary separator gas gravity (air = 1)
0.70
Tankoil rate, STB/D
370
Tankoil gravity, °API
45
R s) +
10 *6(45) + * 1.607
(385)(1.15) 1 * (385)(* 1.607
10 *4)
10 *4,
+ 417 scfńSTB,
10 *4(417) + 1.08 (air + 1).
and g gs1 + 1.15 * 1.607
The total GOR’s and OGR’s are given by R 1 + ǒ4.265
10 6Ǔń(370) + 11, 527 scfńSTB,
R p + 11, 527 ) 417 + 11, 944 scfńSTB, and r p + 1ńR p + 8.37
10 *5 STBńscf + 83.7 STBńMMscf.
Total gas specific gravity is given by Problem 9 Problem. Calculate the reservoir voidage, DVR , expressed as cubic feet, resulting from 1 day of production from the gascondensate reservoir with surface production data given in Table B15.
DV R + DV g + ǒDV gńD tǓD t B gd + q g(1 day) B gd + q g B gd . Surfacegas rate is q g + q o R p + q oǒR 1 ) R s)Ǔ , where R 1 is the separator gas/oil ratio (per stocktank barrel of condensate) and R s) is the solution gas/oil ratio of the separator oil. Estimating the additional gas from the separator oil (Eqs. 3.61 through 3.63), R s) + A 1g gs1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.61a) p ƪǒ18.2 ) 1.4 Ǔ10 ǒ sp1
0.0125g API*0..00091T sp1
Ǔ
ƫ
;
. . . . . . . . . . . . . . . . . . . . . . . . . . (3.62) 10 *6)g API ;
where A 2 + 0.25 ) 0.2g API and A 3 + * (3.57 and R s) +
A1 A2 . . . . . . . . . . . . . . . . . . . . . . . . . . (3.63) ǒ1 * A 1 A 3Ǔ
gives A 1 +
ƪǒ
Ǔ
ƫ
950 ) 1.4 10 (0.0125)(45)*(0.00091)(160) 18.2
A 2 + 0.25 ) 0.02(45) + 1.15, EXAMPLE PROBLEMS
. . . . . . . . . . . . . . . . . . . . . . (3.64)
which yields
+ 0.713 (air + 1). The condensate stocktankoil molecular weight is estimated from the Cragoe9 correlation (Eq. 3.59), Mo +
6, 084 , g API * 5.9
. . . . . . . . . . . . . . . . . . . . . . . . . . (3.59)
resulting in M o + 6, 084ń(45 * 5.9) + 156,
1.205
. . . . . . . . . . . . . . . . . . . (3.61b) g gs1 + A 2 ) A 3 R s) ,
g g1 R s1 ) g gs1 R s) , R s1 ) R s)
g g + [11, 527(0.70) ) 417(1.08)]ń(11, 527 ) 417)
Solution. On the basis of 1 day of production,
and A 1 +
gg +
1.205
+ 385,
which gives the wellstream specific gravity from Eq. 3.55. gw +
g g ) 4, 580 r p g o 1 ) 133, 000 r p ǒ gńM Ǔ o
.
. . . . . . . . . . . . . . . . . . (3.55)
This yields gw +
0.713 ) (4, 580)(83.7 10 *6)(0.8017) 1 ) (133, 000)(83.7 10 *6)ǒ0.8017ń156Ǔ
+ 0.963 (air + 1) . The Sutton3 pseudocritical correlations T pcHC + 169.2 ) 349.5g gHC * 74.0 g 2gHC and p pcHC + 756.8 * 131g gHC * 3.6g 2gHC
. . . . . . . . . . (3.47a) . . . . . . . . . (3.47b) 7
give T pc + 437°R and p pc + 627 psia, and reduced properties are T pr + TńT pc + 710ń437 + 1.625
VC
and p pr + pńp pc + 5, 200ń627 + 8.293. The gas volumetric properties are given by Eqs. 3.42 and 3.43, Z + ap prńy, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.42) where a + 0.06125 t expƪ* (1.21 * t) ƫ, where t + 1ńT pr , 2
and F(y) + 0 + * ap pr )
3
. . . . . (3.43)
giving Z+1.024. With Eq. 7.12, sc
sc
and C og given by C og + 133, 000
Ǔ
og r s
2
and m C
Recalling Eq. 3.95,
2)
og
+
* b ) Ǹb 2 * 4ac , 2a
. . . . . . . . . . . . . . . . . . . (3.95)
a + 0.3167(1.385) + 0.439; b + 3.40 * 0.3167(69.97) ) 15.3(1.385) + 2.43; c + * 15.3(69.97) + * 1, 071;
. . . . . . . . . . . . . . . . . . . . (7.12) 2)
+
+ 46.70 lbmńft 3 ,
. . . . . . . . . . . . . . . . . . . . . . . . . (7.13)
og
* (2.43) ) Ǹ(2.43) * 4(0.439) (* 1, 071) 2(0.439) 2
and ò C
ǒMg Ǔ
+ 69.97 lbm.
2)
2)
) ǒ90.7t * 242.2t 2 ) 42.4t 3Ǔy 2.18)2.82t,
ǒTp Ǔ ZTp ǒ1 ) C
m C + 3.40 lbm,
where a + 0.3167V C , b + m C * 0.3167 m C ) 15.3 V C , 3) 2 2) 3) and c + * 15.3m C , we calculate
* ǒ14.76t * 9.76t 2 ) 4.58t 3Ǔy 2
B gd +
+ 1.385 ft 3,
3)
òC
y ) y2 ) y3 * y4 (1 * y)
through 3.97. From Table B17 and Eqs. 3.93 and 3.94, volumes and masses needed for the calculations are
the pseudoliquid density of the C2+ mixture at standard conditions. From Eq. 3.96,
+ 133, 000ǒ0.8017ń156Ǔ + 683 scfńSTB.
VC
So with r s + 1ńR p ,
2)
+ VC
B gd + ƪǒ14.7ń520Ǔ(1.024)(160 ) 460)ń(5, 200)ƫ
3)
mC )ò 2 C
3)
)
+ VC
2
mC
2
15.3 ) 0.3167ò C
ƪ1 ) ǒ683ń11, 944Ǔƫ
,
. . . . . . . . . . . (3.96)
2)
TABLE B16—OIL COMPOSITION (PROBLEM 10)
+ 0.00395 ft 3ńscf.
Component
The initial daily reservoir voidage is then DV g + ǒDV gńD tǓǒD tǓǒB gǓ + (370)(11, 944)(0.00395) + 17, 470 ft 3 + 3, 110 bbl . Problem 10 Problem. Table B16 shows the composition of a reservoir oil in the Kabob field, Canada. Bubblepoint pressure is 3,100 psia at 236°F reservoir temperature. Calculate the density in lbm/ft3 of the reservoir oil at bubblepoint conditions using idealsolution principles according to the StandingKatz8 method. Solution. Following the calculation procedure outlined in Chap .3, pseudoliquid density, ò po, is calculated explicitly with Eqs. 3.94
Mole Fraction
CO2
0.0111
C1
0.3950
C2
0.0969
C3
0.784
iC4
0.0159
nC4
0.0372
iC5
0.0123
nC5
0.0211
C6
0.0295
C7+
0.3026
M C7) + 182 and g C7) + 0.8275.
TABLE B17—STANDINGKATZ8 DENSITY CALCULATION (PROBLEM 10) mi + zi Mi (lbm)
V i + m ińò i (ft3)
zi
C1
0.3950
16.04
6.34
C2
0.0969
30.07
2.91
CO2
0.0111
44.01
C3
0.0784
44.09
31.66
3.46
0.109
iC4
0.0159
58.12
35.01
0.92
0.026
C4
0.0372
58.12
36.45
2.16
0.059
iC5
0.0123
72.15
39.13
0.89
0.023
C5
0.0211
72.15
39.30
1.52
0.039
C6
0.0295
86.17
41.19
2.54
0.062
C7+
0.3026
182.00
51.61
55.07
1.067
Total
1.0000
Mi
òi (lbm/ft3)
Component
0.49
76.31
Note: CO2 is treated as C2.
8
PHASE BEHAVIOR
which gives VC
2)
+ 1.385 ) (3.40)ń[15.3 ) (0.3167)(46.70)] + 1.50 ft 3 .
The mass of methane and of the total mixture (C1+) are taken from Table B17. m C 1 + 6.34 lbm and m C
1)
+ 76.31 lbm.
From Eq. 3.97, the pseudoliquid density of the overall mixture is calculated at standard conditions. ò po +
*b ) Ǹb 2 * 4ac , 2a
which results in
* (27.53) ) Ǹ(27.53) * 4(0.674)(* 23.81) 2(0.674)
ò ga + 38.52
2
Pseudoliquid oil density is given by
The pressure correction is calculated with Eq. 3.98 and ò po + 41.69 lbm/ft3. Dò p + 10 *3 ƪ0.167 ) ǒ16.181 * 10 *8 ƪ0.299 ) ǒ263
10 *0.0425òpoǓƫ p 10 *0.0603òpoǓƫ p 2,
Dò p + 10 *3 Ǌ0.167 ) ƪ16.181 * 10 *8 Ǌ0.299 ) ƪ263
. . . . (3.98)
10 *0.0603(41.69)ƫǋ (3, 500)
2
10 *8Ǔ(3, 500)
2
+ 1.26 lbmńft 3. The temperature correction is calculated with Eq. 3.99 and ò po ) Dò p +41.69)1.26+42.95 lbm/ft3.
ƪ
Dò T + (T * 60) 0.0133 ) 152.4ǒò po ) Dò pǓ
*2.45
ƫ
10 *6Ǔ
ǋ
. . . . . . . . . . . (3.99)
giving Dò T + (238 * 60)ƪ0.0133 ) 152.4(42.95)
Ǌ
2 * (238 * 60) ǒ8.1
* ƪ0.0622
*2.45
ƫ
10 *6Ǔ
ǋ
10 *0.0764(42.95)ƫ + 5.85 lbmńft 3 .
Eq. 3.89 gives the oil density at 3,100 psia and 238°F. ò o + ò po ) Dò p * Dò T ,
. . . . . . . . . . . . . . . . . . . . . . (3.89)
resulting in ò o + 41.69 ) 1.26 * 5.85 + 37.10 lbmńft 3 . EXAMPLE PROBLEMS
ò po +
1 ) 0.0136ǒR s g gńò gaǓ
,
. . . . . . . . . . . . . . . . . (3.100)
62.4(0.845) ) 0.0136(900)(0.85) + 45.3 lbmńft 3. 1 ) (0.0136)ƪ(900)(0.85)ń(26.4)ƫ
ò o + ò po ) Dò p * Dò T ,
. . . . . . . . . . . . . . . . . . . . . . (3.89)
as ò o + 45.3 ) 1.1 * 3.5 + 42.9 lbmńft 3. Pressure gradient with depth (dp/dh) in psi/ft is given by dp/dh + ò oǒgńg cǓń144, where ò is in lbm/ft3, g+32 ft/sec2, and gc +32 lbmft/(lbfsec2), giving dp/dh+42.9(32/32)/144+0.298 psi/ft. Assuming that this gradient is more or less constant from 7,200 to 6,000 ft subsea, the oil pressure at a depth of 6,000 ft subsea is ǒ p RǓ
10 *0.0764ǒòpo)DòpǓƫ ,
62.4g o ) 0.0136 R s g g
The pressure correction to density, if given by Eq. 3.98 is Dò p + 1.1 lbmńft 3 and ò po ) Dò p + 45.3 ) 1.1 lbmńft 3 + 46.4 lbm/ft3. On the basis of ò po ) Dò p , the temperature correction is given by Eq. 3.99 and is Dò T + 3.5 lbmńft 3, yielding the reservoir oil density from Eq. 3.89,
10 *0.0425(41.69)ƫǋ (3, 500)
10 *3Ǔ(3, 500) * ǒ1.104
ò po +
which gives
giving
* ƪ0.0622
10 *(0.00326)(36) ) ƪ94.75 * (33.93) log (36)ƫ
log (0.85) + 26.4 lbmńft 3
+ 41.69 lbmńft 3.
Ǌ
10 *0.00326g API
) ǒ94.75 * 33.93 log g APIǓ log g g , . . . . . . . . . . . (3.101)
c + * 0.312(76.31) + * 23.81,
* (T * 60) 2 ǒ8.1
Apparent gas pseudoliquid density is given by ò ga + 38.52
b + 6.34 * 0.45(76.31) ) 0.312(1.50) + * 27.53,
+ ǒ0.441
Solution. From Eq. 3.100, pseudoliquid density, ò po, can be calculated from oil and gas surface gravities, g o and g g, respectively; solution gas/oil ratio, R s ; and apparent liquid density of separator gas, ò ga. Stocktankoil gravity is g o + 141.5ń(131.5 ) 36) + 0.845 (water + 1).
. . . . . . . . . . . . . . . . . . . . (3.97)
where a + 0.45(1.50) + 0.674,
and ò po +
Problem 11 Problem. An oil well produces at a total GOR of 900 scf/STB. Total gas gravity is 0.85 (air+1). Stocktankoil gravity is 36°API. Calculate, using idealsolution principles and apparent liquid density of the gas, the density of the reservoir oil at 3,300 psia and 190°F. If reservoir pressure is 3,300 psia at 7,200 ft subsea, what would the reservoir pressure be at a datum level of 6,000 ft subsea?
6000
+ 3, 300 * 0.298(7, 200 * 6, 000) + 2, 942 psia.
This result assumes that a continuous oil column exists from 6,000 to 7,200 ft subsea. Problem 12 Problem. For the reservoir considered in Problem 11, use the Standing10 bubblepoint correlation to estimate bubblepoint pressure. On the basis of this estimate, is it possible that a gas cap might be found between the test depth of 7,200 ft subsea and the structure top at 6,000 ft subsea? If so, at what depth? Solution. The Standing bubblepointpressure correlation, Eq. 3.78, p b + 18.2ǒ A * 1.4 Ǔ, where A + ǒR sńg gǓ 0.83 A + ǒ900ń0.85Ǔ
0.83
. . . . . . . . . . . . . . . . . . . . . . . . . (3.78) 10 ǒ0.00091T*0.0125gAPIǓ, gives 10 [0.00091(190)*0.0125(36)] + 171.2
and p b + 18.2(171.2 * 1.4) + 3, 090 psia. 9
If this bubblepointpressure estimate is accurate (even though the correlation accuracy is probably only "5%), a gas cap may be expected at a subsea depth, calculated from ǒ p RǓ
GOC
+ p b + 3, 090 + 3, 300 * 0.298ǒ7, 200 * D GOCǓ ,
where 0.298 psi/ft is the oil gradient calculated in Problem 11. Solving this relation for D GOC gives D GOC + 6, 500 ft subsea. Problem 13 Problem. If the hydrocarbon pore volume (HCPV) of the reservoir in Problem 11 is approximately 40 106 ft3/ft reservoir thickness, estimate the original oil in place, N, and original gas in place, G. The water/oil contact (WOC) is at 7,300 ft subsea, the gas/oil contact (GOC) depth is given in Problem 12 as 6,500 ft subsea, and the top of the structure is at 6,000 ft subsea. Solution. To solve this problem, oil and gas FVF’s must be estimated. The oil FVF will vary throughout the 800ft oil column. The oil is saturated at the GOC and undersaturated at depths down to the WOC. Several assumptions must be made because so little data are available. 1. Constant temperature is assumed throughout the reservoir, although a gradient of 1 to 2°F/100 ft probably exists. 2. Oil composition is assumed to be uniform in the oil column, although it would not be surprising if the GOR decreased somewhat from the GOC to the WOC (e.g., from 900 to 800 scf/STB). 3. Gas composition is assumed to be uniform in the gas cap (probably a reasonable assumption). 4. A condensate yield must be assumed for the reservoir gas. From data in the literature (or from a similar reservoir in the same geographical area), we can find a similar reservoir oil/gas system. An initial solution OGR, rsi , of 40 STB/MMscf is assumed here. 5. Surface condensate gravity of 60°API ( g og + 0.739) is also assumed. 6. The surfacegas gravity for the reservoir gas is assumed to be slightly less than the surfacegas gravity for the reservoir oil, g gg + 0.80 (see, for example, Fig. 7.12). The gas and oil column HCPV’s, V HCg and V HCo , respectively, are given by V pHCg + ǒ40
10 6Ǔ(6, 500 * 6, 000) + 20
and V pHCo + ǒ40
10 9 ft 3
6, 084 , g API * 5.9
Z + ap prńy, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.42) where a + 0.06125 t exp[* 1.2(1 * t) 2], with t + 1ńT pr, gives Z+0.808. Gas density is calculated at the GOC to obtain a gas gradient for estimating the average pressure in the gas cap. òg +
(3, 090)(28.97)(0.904) + 14.36 lbmńft 3 , (0.808)(10.73)(190 ) 460)
ǒd pńdhǓ + 14.36ń144 + 0.0997 psińft g and ǒ p RǓ g + [3, 090 * (0.0997)(6, 500 * 6, 000)]ń2 + 3, 065 psia. Pseudoreduced pressure at ( p R) g is p pr + 3, 065ń650 + 4.715, and the Z factor is 0.806. The wetgas FVF at ( p R) g is given by Bg +
ǒTp Ǔ ZTp, sc
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.38)
sc
resulting in B gw + (0.02827)(0.806)(190 ) 460)ń(3, 065) + 0.00483 ft 3ńscf and b gw + 1ńB gw + 207 scfńft 3. However, the drygas FVF is needed to calculate dry surface gas for the estimated V pHCg . Eqs. 7.12 and 7.11 are used to calculate B gd . B gd +
ZT p sc ZT ǒ 1 ) C og r sǓ + 0.02827 p ǒ1 ) C og r sǓ T sc p
+ B gwǒ1 ) C og r sǓ , . . . . . . . . . . . . . . . . . . . . . . . . (7.12) where C og + (133, 000)(0.739)(112) + 876 scfńSTB, and B gd + V gńV gg ,
10 9 bbl.
Initial gas in place represents the free gas in place plus the gas in solution in the oil column. To calculate gas FVF, a wellstream gravity, g w, must be calculated first. With g og + 0.739, Eq. 3.59 gives an estimate of the condensate molecular weight. Mo +
are Tpc +426°R and ppc +650 psia. At the GOC, reduced properties are Tpr +(190)460)/426+1.526 and ppr +3,090/650+4.754. The StandingKatz8 Zfactor correlation (Eq. 3.42),
10 9 ft 3
10 6Ǔ(7, 300 * 6, 500) + 32
+ 5.700
and p pcHC + 706 * 51.7g gHC * 11.1g 2gHC , . . . . . . . . . . (3.49b)
. . . . . . . . . . . . . . . . . . . . . . . . . . (3.59)
giving M og + 6, 084ń(60 * 5.9) + 112. From Eq. 3.55, gw +
g g ) 4, 580 r p g o , 1 ) 133, 000 r p ǒ gńM Ǔ o
gw +
0.8 ) ǒ4, 580 Ǔ (40 10 *6)(0.739) 1 ) 133, 000(40 10 *6)ƪ(0.739)ń(112) ƫ
. . . . . . . . . . . . . . . . . . (3.55)
+ 0.904 (air + 1).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.11)
giving B gd + (0.00483)ƪ1 ) (876)ǒ40
10 *6Ǔƫ + 0.00500 ft 3ńscf
and b gd + 1ńB gd + 200 scfńft 3. For the oil column, oil FVF must be estimated at an average oil pressure ( p R) o.
ǒ p RǓ + 3, 090 ) (0.298)(7, 300 * 6, 500)ń2 + 3, 209 psia. o Bubblepoint oil FVF is estimated from the Standing correlation (Eq. 3.111), B ob + 0.9759 ) ǒ12
10 *5Ǔ A 1.2,
. . . . . . . . . . . . . . . (3.111)
where A + R sǒg gńg oǓ 0.5) 1.25T, giving 0.5 A + 900ǒ0.85ń0.845Ǔ ) 1.25(190) + 1, 140
and B ob + 0.9759 ) ǒ12
10 *5Ǔ(1, 140)
1.2
+ 1.535 bblńSTB.
With g gg + g gHC + 0.80, pseudocritical properties from the Standing10 “wetgas” correlations (Eq. 3.49),
Undersaturated oil FVF can be calculated with an estimate of the undersaturated oil compressibility with Eq. 3.107 for c o at ( p R) o , and Eq. 3.105 for B o.
T pcHC + 187 ) 330 g gHC * 71.5g 2gHC . . . . . . . . . . . . . . (3.49a)
c o + Ańp, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.107)
10
PHASE BEHAVIOR
Problem 14 Problem. Estimate oil and gas viscosities at 2,500 psia and 190°F for the reservoir considered in Problems 11 through 13.
giving c o + 10 *5ƪ(5)(900) ) (17.2)(190) * (1, 180)(0.85) ) (12.61)(36) * 1, 433ƫń(3, 209) + 18.0
10 *6 psi *1.
ò o + ò ob expƪc oǒp * p bǓƫ [ ò ob ƪ1 * c oǒ p b * p Ǔƫ
. . . . . . . . . . . . . . . . . . . . (3.105a)
Solution. Gas viscosity can be estimated from the Lucas4 correlation (Eq. 3.66). m gńm gsc + 1 )
and B o + B ob expƪc oǒ p b * p Ǔƫ [ B ob ƪ1 * c o ǒp * p bǓƫ ,
. . . . . . . . . . . . . . . . . (3.105b)
which give B o + 1.535 expƪǒ18.3
However, a more exact approach uses Eq. 109, which properly accounts for the pressure dependence of oil compressibility.
ƪ
ƫ
ò ob ) 0.004347 ǒ p * p bǓ * 79.1 , (7.141 10 *4)ǒ p * p bǓ * 12.938 . . . . . . . . . . . . . . . . . . . (3.109)
resulting in A + 10 *5[(5)(900) ) (17.2)(190) * (1, 180)(0.85) ) (12.61)(36) * 1, 433] + 0.05776 0.05776 + 1.532 bblńSTB. and B o + 1.535ǒ3, 090ń3, 209Ǔ
The two approaches result in almost no difference in B o for this example of slight undersaturation. However, for higher degrees of undersaturation, the difference can be significant; therefore, in general, Eq. 3.109 is recommended. Initial oil in place in the oil column is given by (N) o + V pHCońB oi + ǒ5.700 + 3.720
10 9Ǔ(200) + 4.000
10 12 scf.
10 Ǔǒ40 12
10 Ǔ *6
10 9Ǔ(900) + 3.348
N + (N) o ) (N) g + ǒ3.720
10 12 scf .
10 9Ǔ ) ǒ0.160
10 9Ǔ
10 9 STB
and G + (G) g ) (G) o + ǒ4.000
10 12Ǔ ) ǒ3.348
10 12Ǔ
10 12 scf.
Note that significant gas reserves are found as solution gas in this oil reservoir. This is not uncommon for volatile and even moderately volatile oil reservoirs (GORu750 scf/STB). In general, in larger field developments, the economic value of solution gas cannot be ignored as both production revenue for depletion drive and lost income in waterflooding projects. EXAMPLE PROBLEMS
Ǔ 0.4489 expǒ3.0578T *37.7332 pr , T pr
A4 +
Ǔ 1.7368 expǒ2.2310T *7.6351 pr , T pr
and A 5 + 0.9425 expǒ* 0.1853T pr0.4489Ǔ .
. . . . . . . . . . . (3.66b)
Pseudocritical properties are estimated from reservoir gas (wellstream) gravity, g w. The initial wellstream gravity of 0.904 calculated in Problem 13 is somewhat higher than would be expected for the equilibrium gas at 2,500 psia (see, for example, Table 6.11). We therefore assume a current wellstream gravity of g w + g gHC + 0.85. With the Standing10 wetgas correlations (Eq. 3.49) for pseudocritical properties, T pcHC + 187 ) 330 g gHC * 71.5g 2gHC . . . . . . . . . . . . . . (3.49a) + 416°R and p pcHC + 706 * 51.7g gHC * 11.1g 2gHC ,
. . . . . . . . . (3.49b)
+ 654 psia,
The gas molecular weight is M g + (28.97)(0.85) + 24.62 lbmńlbm mol, which is used to calculate c. 4
ƫ
1ń6
+ 69.3 cp *1,
giving
Thus, the initial stocktank oil plus condensate in place, N, and the initial dry gas plus solution gas in place, G, are, respectively,
+ 7.348
A3 +
3
Initial gas in solution in the oil column is given by
+ 3.880
, . . . . . . . (3.66a)
Ǔ 10 *3) expǒ5.1726T *0.3286 pr , T pr
(1.245
c + 9, 490ƪ(416)ń(24.62) ń(654)
10 6 STB.
(G) o + NR si + ǒ3.720
*1
and p pr + pńp pc + 2, 500ń654 + 3.823.
Initial condensate in place in solution in the gas column is given by
+ 160
A
T pr + TńT pc + (190 ) 460)ń416 + 1.562
Initial (dry) gas in place in the gas column is given by
(N) g + G d r si + ǒ4.000
A
giving pseudoreduced properties
10 9Ǔń(1.532)
10 9 STB.
(G) g + V pHCo b gd + ǒ20
A 2 p pr5 ) ǒ1 ) A 3 p pr4Ǔ
A 2 + A 1ǒ1.6553T pr * 1.2723Ǔ ,
10 *6Ǔ(3, 090 * 3.209)ƫ
+ 1.532 bblńSTB.
c o + 10 *6 exp
where A 1 +
A 1 p 1.3088 pr
m gsc + ǒ m gsc c Ǔńc + 0.9046ń69.3 + 0.0131 cp, m gńm gsc + 1.601, and m g + 0.0210 cp. Use the LeeGonzalez correlation (Eq. 3.65) to calculate gas viscosity.11 mg + A1 where A 1 +
10 *4 expǒA 2 ò g 3Ǔ , A
. . . . . . . . . . . . . . . . . . (3.65a)
ǒ9.379 ) 0.01607M gǓT 1.5 209.2 ) 19.26M g ) T
,
A 2 + 3.448 ) ǒ986.4ńTǓ ) 0.01009M g , and A 3 + 2.447 * 0.2224A 2 .
. . . . . . . . . . . . . . . . . . . (3.65b)
Gas density must be calculated first with ò g + pM gńZRT.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.34) 11
Using the Chew and Connally12 correlation (Eq. 3.123),
TABLE B18—THREECOMPONENTSYSTEM COMPOSITION (PROBLEM 15)
m ob + A 1 ǒm oDǓ A2, . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.123)
Component
Mole Fraction
C1
0.20
C3
0.32
nC5
0.48
and the Bergman equations for constants A 1 and A 2 (Eq. 3.125), ln A 1 + 4.768 * 0.8359 ln(R s ) 300) . . . . . . . . . . (3.125a) and A 2 + 0.555 )
TABLE B19—COMPONENT PROPERTIES (PROBLEM 15) Component
pci (psia)
Tci (°R)
wi
Ki
C1
667.8
343.0
0.0115
9.208
C3
616.3
665.7
0.1454
1.439
nC5
488.6
845.4
0.2510
0.358
TABLE B20—CALCULATED RESULTS FROM ITERATIONS (PROBLEM 15) Iteration
Fv
h(Fv ) *1.75
dh/dFv
10*2
*0.98880
1
0.5
2
0.48227
1.48 10*4
*1.00606
3
0.48242
1.16
10*8
*1.00590
4
0.48242
7.07 10*17
*1.00590
This gives òg +
A 2 + 0.555 ) (133.5)ń(700 ) 300) + 0.6885 , and m ob + (0.3656)(1.78)
0.6885
+ 0.544 cp.
Problem 15 Problem. Table B18 gives the composition of a threecomponent system of methane, propane, and normal pentane. Use the modified Wilson13 Kvalue equation (Eq. 3.159) with a convergence pressure of 2,000 psia to estimate K values at 500 psia and 160°F. Make a flash calculation using the MuskatMcDowell14 (or RachfordRice15) algorithm given by Eqs. 4.36 through 4.40. Solution. Table B19 gives component properties taken from Appendix A needed to calculate K values from the modified Wilson Kvalue equation. A 0 + 0.7 is used in the modified Wilson Kvalue correlation, where A + 1 * ( pńp k ) 0.7 in Eq. 3.159. For example, the K value for methane is given by p K i + pci k
Ǔƫ expƪ5.37 A 1 (1 ) w i)ǒ1 * T *1 ri , p ri
A 1*1
+ 11.0 lbmńft 3 + 0.176 gńcm 3 ,
. . . . . . . . . . . . . . . . . . (3.159)
where Z+0.803 is estimated from the StandingKatz8 correlation (Eqs. 3.42 and 3.43). From Eq. 3.65b, the constants in the gas viscosity correlation are [9.379 ) 0.01607(24.62)](650) A1 + 209.2 ) 19.26(24.62) ) 650
1.5
+ 121.5,
resulting in A+1*
ǒ
and K C + 667.8 1 2, 000
Ǔ expƪ(5.214)(0.176)
+ 0.627,
1.287
ƫ + 0.0212 cp.
The oil viscosity is calculated by first estimating deadoil viscosity, m oD. With the Bergman* correlation (Eq. 3.119), ln lnǒ m oD ) 1Ǔ + A 0 ) A 1 ln(T ) 310), . . . . . . . . . (3.119)
Ǔ
0.627*1
expƪ5.37(0.627)(1 ) 0.0115)ǒ1 * 1ń1.807Ǔƫ + 9.21, 0.749
giving 10
0.7
1
and A 3 + 2.447 * 0.2224(5.214) + 1.287 , m g + ǒ121.5
* 14.7 Ǔ ǒ2,500 000 * 14.7
(T r) C + (160 ) 460)ń343 + 1.807,
A 2 + 3.448 ) ǒ986.4ń650Ǔ ) 0.01009(24.62) + 5.214,
*4
. . . . . . . . . . . . . . . . . (3.125b)
gives A 1 + expƪ4.768 * 0.8359 ln(700 ) 300)ƫ + 0.3656,
ǒ Ǔ
(2, 500)(24.62) (0.803)(10.73)(190 ) 460)
133.5 , R s ) 300
and the ci value for methane is c i + 1ń(K i * 1) + 1ń(9.21 * 1) + 0.122. With these K values, four iterations are used to solve the MuskatMcDowell equation,
2
where A 0 + 22.33* 0.194(36)) 0.00033 (36) + 15.77, and m oD + * 1 ) expǊ exp[15.77*2.534 ln(190 ) 310)] ǋ + 1.78 cp. The viscosity correction for a saturated live oil depends on the amount of gas in solution, R s . We estimate the solution gas/oil ratio using Standing’s10 bubblepoint pressure correlation (Eq. 3.78), setting the current pressure of 2,500 psia as the bubblepoint of the saturated oil and solving for R s as given by Eq. 3.87, R s + (0.85)
Ǌ[(0.055)(2, 500)10 ) (1.4)]
(0.00091)(190)
ǋ
10 (0.0125)(36)
+ 700 scfńSTB. *Personal communication with D.F. Bergman, Amoco Research, Tulsa, Oklahoma (1992).
12
ȍF N
h(F v) +
A 1 + * 3.20 ) (0.0185)(36) + * 2.534,
i+1
v
zi + 0, ) ci
. . . . . . . . . . . . . . . . . . . . . . (4.39)
where c i + 1ń(K i * 1). Table B20 summarizes the calculated results from the iterations, and Table B21 gives the final results for the flash calculation, including equilibrium vapor and liquid compositions. Problem 16 Problem. Calculate the bubblepoint pressure for the ternary system in Problem 15 at 160°F using the modified Wilson Kvalue equation. Eq. 3.165 is used to solve for bubblepoint pressure given a Kvalue correlation based on convergence pressure. Fǒ p KǓ + 1 *
ȍ z K ǒ p , p , TǓ + 0. N
i
i
K
b
. . . . . . . . . . . . (3.165)
i+1
PHASE BEHAVIOR
TABLE B21—FINAL FLASHCALCULATION RESULTS (PROBLEM 15) zi /(Fv + ci )2
zi
Ki
ci
zi /(Fv + ci )
C1
0.20
9.208
0.122
0.331
0.548
0.0403
0.3713
C3
0.32
1.439
2.278
0.116
0.042
0.2641
0.3800
nC5
0.48
0.358
*1.556
Total
*0.447 7.07 10*17
1.00
h(Fv )+7.07
xi
yi
0.416
0.6956
0.2487
1.00590
1.0000
1.0000
10*17 and hȀ(Fv )+*1.00590.
TABLE B22—PRESSUREGUESS CALCULATIONS* (PROBLEM 16) pK +1,300 psia
pK +1,375 psia (correct)
pK +1,500 psia
Component
zi
Ki
yi +zi Ki
Ki
yi +zi Ki
Ki
yi +zi Ki
C1
0.20
2.181
0.436
1.982
0.396
1.703
0.341
C3
0.32
1.003
0.321
0.996
0.319
0.988
0.316
C5
0.48
0.560
0.269
0.594
0.285
0.657
0.315
Total
1*Syi +*0.02591
1.00
1*Syi +*0.00002[0
1*Syi +0.028024
*At T+160°F.
a. Calculate the convergence pressure, p K, that matches the measured bubblepoint pressure. Use the modified Wilson Kvalue equation (Eq. 3.159) with A 0 + 0.7. b. Use the Kvalue correlation developed in Part a to make a singlestage separator flash calculation to 14.7 psia and 60°F. Report the stocktankgas and oil compositions, GOR, oil gravity in °API, and gas specific gravity.
TABLE B23—OIL COMPOSITION (PROBLEM 17) Component
Mole Fraction
CO2
0.0111
C1
0.3950
C2
0.0969
C3
0.0784
iC4
0.0159
nC4
0.0372
iC5
0.0123
Solution. Table B25 gives relevant component properties for this problem. The K values at reservoir conditions are calculated with T+236°F. The modified Wilson equation (Eq. 3.159) is
ǒ Ǔ
p K i + pci K
A 1*1
Ǔƫ expƪ5.37 A 1 (1 ) w i)ǒ1 * T *1 ri , p ri
nC5
0.0211
C6
0.0295
. . . . . . . . . . . . . . . . . . (3.159)
C7+
0.3026
where A + 1 * ( pńp K) and A 0 + 0.7 is assumed. By adjusting convergence pressure, p K, the bubblepoint condition given by Eq. 3.165 is satisfied with p K + 4, 052.8 psia. Table B26 gives the K values and incipientphase gas composition. The Kvalue correlation is then used to make a flash calculation at standard conditions p+14.7 psia and T+60°F. With K values at these conditions, the RachfordRice equation, Eq. 4.36, is solved for gasphase mole fraction ( F v + F g) where F g + 0.64241; stocktankoil and separatorgas compositions are given later. On the basis of the surfacegas composition, specific gravity g g is
MC gC
0.7
182 7)
0.8275
7)
K wC
11.79 7)
C7+ is split into three fractions; Table B24 gives mole fractions and properties.
Solution. Although an iterative procedure, such as NewtonRaphson, can be solved analytically with the modified Wilson Kvalue equation, it takes only a few guesses to locate the pressure that satisfies Eq. 3.165. Table B22 summarizes the results of the calculations for three guesses of pressure, where p K + 1, 375 psia gives a satisfactory result for bubblepoint pressure.
g g + 27.32ń28.97 + 0.943 (air + 1) and stocktank oil properties are M o + ǒSm o iǓńǒSn o iǓ + 167.8ń1.0 + 167.8 lbmńlbm mol, ò o + ǒSm o iǓńǒSV o iǓ + 167.8ń3.323 + 50.48 lbmńft 3 ,
Problem 17 Problem. Tables B23 and B24 show the composition of a reservoir oil in the Kabob field, Canada. Bubblepoint pressure is 3,100 psia at 236°F reservoir temperature.
g o + ò ońò w + 0.8094 (water + 1), and g API + 141.5ńg o * 131.5 + 43.3°API,
TABLE B24—MOLE FRACTIONS AND PROPERTIES OF C7+ COMPONENT (PROBLEM 17) C7+ Fraction
zi
Mi
Tci (°R)
pci (psia)
ąăwi ąă
F1
0.1578
114.1
1065.5
409.6
0.3255
0.7674
727.0
F2
0.1243
223.1
1356.0
235.1
0.6538
0.8403
1,029.5
F3
0.0205
455.0
1689.1
134.6
1.1489
0.9254
1,410.1
0.3026
182.0
Total
ąăgi *
Tbi (°R)
0.8275
*Water+1.
EXAMPLE PROBLEMS
13
TABLE B25—COMPONENT PROPERTIES (PROBLEM 17) Component
zi
Mi
wi
0.0111
44.01
31.18
1,070.6
547.6
0.2310
0.3950
16.04
20.58
667.8
343.0
0.0115
C2
0.0969
30.07
28.06
707.8
549.8
0.0908
C3
0.0784
44.09
31.66
616.3
665.7
0.1454
iC4
0.0159
58.12
35.01
529.1
734.7
0.1756
C4
0.0372
58.12
36.45
550.7
765.3
0.1928
iC5
0.0123
72.15
39.13
490.4
828.8
0.2273
C5
0.0211
72.15
39.30
488.6
845.4
0.2510
C6
0.0295
86.17
41.19
436.9
913.4
0.2957
F1
0.1578
114.10
47.86
409.6
1,065.5
0.3255
F2
0.1243
233.10
52.41
235.1
1,356.0
0.6538
F3
0.0205
455.00
57.72
134.6
1,689.1
1.1489
1.0000
pri
Modified Wilson Ki
1.270
2.90
1.324
0.0147
C1
2.028
4.64
1.539
0.6079
C2
1.265
4.38
1.196
0.1159
CO2
Tci (°R)
C1
TABLE B26—K VALUES AND INCIPIENTPHASE GAS COMPOSITION (PROBLEM 17)
Tri
pci (psia)
CO2
Total
Component
òi (lbm/ft3)
Incipient Phase yi +zi Ki
where n o i + x i, m o i + x i M i, and V o i + x i M ińò i . On the basis of 1 mole of feed, the surface volumes are given by V g + 379F g + 379(0.64241) + 243.5 scf and V o + ǒ1 * F gǓǒM ońò oǓ + (1 * 0.64241)ǒ167.8ń50.48Ǔ + 1.188 ft 3 + 0.2117 STB and the GOR is
C3
1.045
5.03
0.990
0.0776
iC4
0.947
5.86
0.867
0.0138
C4
0.909
5.63
0.831
0.0309
iC5
0.839
6.32
0.733
0.0090
C5
0.823
6.34
0.709
0.0150
Problem 18
C6
0.762
7.10
0.614
0.0181
F1
0.653
7.57
0.461
0.0727
F2
0.513
13.19
0.189
0.0234
F3
0.412
23.03
0.04302
Problem. Make equationofstate (EOS) calculations using the PengRobinson 16 EOS (PR EOS) for the ternary system described in Table B28. Use the cubic m term (Eq. 4.22) for wu0.4 (C10). a. Make a twophase flash calculation at 500 psia and 280°F. b. Make a Michelsen phasestability test followed by a twophase flash calculation at 1,500 psia and 280°F.
Total
0.0009 1.0000
R go + V gńV o + 243.5ń0.2117 + 1, 150 scfńSTB. Table B27 summarizes the results.
TABLE B27—SEPARATOR FLASH CALCULATION (PROBLEM 17) Modified Wilson Ki
MuskatMcDowell zi /(F g + ci )
StockTank Oil xi
Separator Gas yi
V oi + x i M ińò i (ft3)
m gi + y i M i
0.01
0.000
0.75
0.03
0.002
9.84
0.1484
0.13
0.005
4.46
0.1136
0.67
0.021
5.01
0.0204
0.45
0.013
1.19
Component
Tri
pri
CO2
0.949
0.014
51.05
0.0168
0.0003
0.0171
C1
1.515
0.022
287.94
06116
0.0021
0.6137
C2
0.945
0.021
34.28
0.1441
0.0043
C3
0.781
0.024
7.44
0.0983
0.0153
iC4
0.707
0.028
2.64
0.0127
0.0077
(lbm)
C4
0.679
0.027
1.81
0.0199
0.0244
0.0443
1.42
0.039
2.58
iC5
0.627
0.030
0.662
*0.0053
0.0157
0.0104
1.13
0.029
0.75
C5
0.615
0.030
0.493
*0.0159
0.0313
0.0154
2.26
0.057
1.11
C6
0.569
0.034
0.153
*0.0549
0.0647
0.0099
5.58
0.135
0.85
F1
0.488
0.036
1.58 x 10*2
*0.4223
0.4291
0.0068
48.96
1.023
0.77
*0.3476
0.3476
0.0000
81.02
1.546
0.00
*0.0573
0.0573
0.0000
26.08
0.452
0.00
0.0000
1.0000
1.0000
167.76
3.323
27.32
F2
0.383
0.063
9.93x 10*6
F3
0.308
0.109
4.83x 10*11
Total 14
m oi + x i M i (lbm)
PHASE BEHAVIOR
TABLE B28—TERNARY SYSTEM (PROBLEM 18) Component i
zi
Mi
Tci (°R)
pci (psia)
wi
si +ci /bi
C1
0.50
16.04
343.0
667.8
0.0115
*0.1595
C4
0.42
58.12
765.3
550.7
0.1928
*0.0675
C10
0.08
142.29
1,111.8
304.0
0.4902
0.0655
TABLE B29—CHANGES DURING ITERATIONS (PROBLEM 18) Convergence Tolerance log[S(1–fLi /fvi )2]
Iteration
Trivial Solution Indicator S(ln Ki )2
VaporPhase Mole Fraction Fv
1
0.708
30.73
0.852187
2
*2.230
15.24
0.853914
3
*4.380
14.66
0.853528
4
*6.454
14.61
0.853423
5
*8.457
14.61
0.853405
6
*15.236
14.61
0.853401
Solution. a. The flash calculation is made with five successivesubstitution iterations followed by a general dominant eigenvalue method (GDEM) promotion. Tables B29 and B30 give the results of the calculations for the six iterations required to solve the flash problem. Table B31 shows the change in convergence tolerance, the trivialsolution indicator, and vaporphase mole fraction during each iteration. The convergence tolerance indicates how close the phase fugacities of each component have come to one another. Convergence was specified as 10 *12 in this example. The trivialsolution indicator stabilizes after three iterations. Convergence toward a trivial solution is usually indicated for values S(ln K i) 2 t 10 *4. Details of the EOS calculations for the first iteration are summarized later, step by step. KValue Estimate. The Wilson13 equation is used to estimate K values.
Ki +
ƪ
ǒ
exp 5.37ǒ1 ) w i Ǔ 1 * T *1 ri
Ǔƫ .
pr i
. . . . . . . . . . . . . (4.42)
This gives (m) C + 0.3796 ) 1.485(0.4902) * 0.1644(0.4902)
expƪ5.37(1 ) 0.4902)ǒ1 * 1ń0.666Ǔƫ + 0.0108, ǒ500ń304Ǔ
and for the other components, K C1 + 24.58 and K C4 + 0.8820. Phase Split. With Kvalue estimates and the feed composition known, a phase split is made with either the RachfordRice15 or MuskatMcDowell14 algorithms. This results in vaporphase mole fraction, F v + 0.852187, and the compositions given in Table B30. EOS Constants for Each Phase Separately. EOS Constants A and B must now be calculated separately for the vapor and liquid phases on the basis Compositions y i and x i. For decane,
3
) 0.01667(0.4902) + 1.070, (T r) C + (280 ) 460)(1, 111.8) + 0.666, 10
ƪ
(a) C + 1 ) (1.070)ǒ1 * Ǹ0.666Ǔ 10
(a) C + 10
2
ƫ + 1.432,
2 2 W a R 2 T c2 (10.73) (1, 111.8) (1.432) p c a + 0.45724 304.0
+ 306, 500 psiaft 3ńlbm mol *1, RT (10.73)(1, 111.8) and (b) C + W b p c + 0.07780 c 10 304.0 + 3.053 ft 3ńlbm mol. From Eq. 4.9, A+a
p (RT)
and B + b
2
pr + 27 2 64 T r
pr p +1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.9) RT 8 Tr
at 500 psia and 280°F. EOS Constants A and B for decane are
and (B) C + (3.053) 10
500 + 2.435 2 2 (10.73) (280 ) 460)
500 + 0.1922, (10.73)(280 ) 460)
and for other components A C + 0.04906, A C + 0.4544, B C 1
+ 0.02701, and B C + 0. 07308.
4
1
4
The A i and B i constants are the same for both phases. To calculate A and B constants for the vapor phase ( A v and B v) and the liquid phase ( A L and B L ), traditional mixing rules are used (Eq. 4.16).
ȍȍy y A N
Av +
N
i j
ij,
i+1 j+1
ȍȍx x A N
N
R 2T 2 a + W oa p c a , where W oa + 0.45724; c
AL +
RT b + W ob p c , where W ob + 0.07780; c
A i j + ǒ1 * k i jǓ ǸA i A j
a + ƪ1 ) mǒ1 * ǸT rǓƫ ;
Bv +
i j
ij ,
i+1 j+1
2
ȍy B , N
and m + 0.37464 ) 1.54226 w * 0.26992 w . 2
. . . . . . . (4.21)
The modified relation for m (Eq. 4.22) is used for decane because its acentric factor is greater than 0.4, EXAMPLE PROBLEMS
2
10
10
T r + TńT c + (280 ) 640)ń(1, 111.8) + 0.666,
10
. . . . . . . . . . . . . . . . . . . . (4.22)
(A) C + (306, 500)
For decane,
and K C +
m + 0.3796 ) 1.485w * 0.1644w 2 ) 0.01667w 3 .
i
i
i+1
ȍx B . N
and B L +
i
i
i+1
15
TABLE B30—FUGACITY CALCULATION RESULTS (PROBLEM 18) Component i
yi
fvi (psia)
Ki +yi /xi
xi
fLi (psia)
fLi /fvi
85.3847
0.28650
Iteration 1 (Wilson K–Value Estimate) C1
0.58262
0.02370
C4
0.41186
0.46695
24.5823 0.882021
298.023
C10
0.00553
0.50935
0.010854
C1
0.57165
0.08117
7.04293
294.517
279.596
0.94934
C4
0.41277
0.46224
0.892986
148.515
148.117
0.99732
C10
0.01557
0.45659
0.034107
C1
0.57115
0.08542
6.6861
294.392
292.992
0.99524
C4
0.41258
0.46326
0.89059
148.363
148.288
0.99950
C10
0.01628
0.45132
0.03607
C1
0.57114
0.08583
6.6543
294.394
294.253
0.99952
C4
0.41254
0.46345
0.890144
148.344
148.332
0.99992
C10
0.01633
0.45072
0.036227
C1
0.57114
0.08587
6.6511
294.396
294.381
0.99995
C4
0.41253
0.46348
0.890073
148.342
148.34
0.99999
C10
0.01633
0.45065
149.526
151.385
1.06778
1.01243
3.35554
3.14253
Iteration 2 (Successive Substitution)
2.89097
3.05736
1.05755
Iteration 3 (Successive Substitution)
3.01459
3.02765
1.00433
Iteration 4 (Successive Substitution)
3.02324
3.02426
1.00034
Iteration 5 (Successive Substitution)
0.036239
3.02377
3.02385
1.00003
Iteration 6 (GDEM Promotion) C1
0.57114
0.08588
6.65071
294.397
294.397
1.00000
C4
0.41253
0.46349
0.890061
148.342
148.342
1.00000
C10
0.01633
0.45064
0.03624
Recall that the compositions y i and x i result from the phasesplit calculation based on feed composition z i and the current Kvalue estimates. For the initial Kvalue estimates and resulting compositions from the phasesplit calculation, EOS Constants A and B are 0.5
A L + (0.02370)(0.02370)[(0.04906)(0.04906)] (1 * 0) 0.5 ) (0.02370)(0.46695)[(0.04906)(0.4544)] (1 * 0) 0.5 ) (0.02370)(0.50935)[(0.04906)(2.435)] (1 * 0) 0.5 ) (0.46695)(0.02370)[(0.4554)(0.04906)] (1 * 0) 0.5 ) (0.46695)(0.46695)[(0.4554)(0.4554)] (1 * 0) 0.5 ) (0.46695)(0.50935)[(0.4554)(2.435)] (1 * 0) 0.5 ) (0.50935)(0.02370)[(2.435)(0.04906)] (1 * 0) 0.5 ) (0.50935)(0.46695)[(2.435)(0.4544)] (1 * 0) 0.5 ) (0.50935)(0.50935)[(2.435)(2.435)] (1 * 0) + 1.252, B L + (0.02370)(0.02701) ) (0.46695)(0.07308) ) (0.50935)(0.1922) + 0.1327, A v + 0.1725,
3.02379
ƪ(0.1812) 3 * (1 * 0.1327)(0.1812) 2ƫ ) ƪ1.252 * 3(0.1327) * 2(0.1327)ƫ(0.1812) 2
* ƪ1.252(0.1327) * (0.1327) * (0.1327) 2
and Z 3v * (1 * B v) Z 2v ) ǒA v * 3 B 2v * 2 B vǓ Z v * ǒA v B v * B 2v * B 3v Ǔ + 0. 16
3
ƫ
+ 0.0005 [ 0. Fugacity Calculations. Fugacity values of each component for each phase are calculated with Eq. 4.23, f ln p +ln f + Z * 1 * ln(Z * B) *
ln
ZFactor Calculation. With the EOS constants for each phase, the Z factor (i.e., volume solution to the cubic EOS) can be solved. Eq. 4.20 is used for each phase separately.
* ǒA L B L * B 2L * B 3LǓ + 0
1.00000
The solutions to these two equations with Constant A and B values calculated in the previous section are Z L + 0.1812 and Z v + 0.8785. We can check, for example, the liquid solution by substituting Z L + 0.1812 into Eq. 4.20 together with A L + 1.252 and B L + 0.1327.
and B v + 0.04690.
Z 3L * ǒ1 * B L Ǔ Z 2L ) ǒA L * 3 B 2L * 2 B LǓ Z L
3.02379
and ln
ƪ
Z) ǒ1 ) Ǹ2Ǔ B
Z) ǒ1 ) Ǹ2Ǔ B
ƫ
A 2 Ǹ2 B
fi B + ln f i + i (Z * 1) * ln(Z * B) B yi p )
A 2 Ǹ2 B
ǒ
Bi 2 * B A
ȍyA N
j
j+1
Ǔƪ
ij
ln
Z) ǒ1 ) Ǹ2Ǔ B
Z) ǒ1 ) Ǹ2Ǔ B
ƫ
,
. . . . . . . . . . . . . . . . . . . . (4.23) PHASE BEHAVIOR
TABLE B31—PHASE STABILITY TEST RESULTS (PROBLEM 18) Component i
yi
zi
fyi (psia)
Ki
fzi (psia)
S+fzi /fyi
Vapor–Like Stability Test: Ki +yi /zi * C1
0.66910
0.50
1.3540
1,053
1,066
1.0118
C4
0.30930
0.42
0.7450
194.7
197.0
1.0118
C10
0.02166
0.08
0.2740
2.712
2.744
1.0118
C1
0.31870
0.50
1.5430
1,048
1,066
1.0168
C4
0.47670
0.42
0.8664
193.7
197.0
1.0168
C10
0.20460
0.08
0.3846
2.699
2.744
1.0168
Liquid–Like Stability Test: Ki +zi /yi **
*Unstable; converged solution, SV =1.0118, 12 iterations. **Unstable; converged solution, SL =1.0168, 6 iterations.
TABLE B32—CONVERGED FLASH SOLUTION (PROBLEM 18) Component i
Initial K Values From Stability Test Ki +(yi )v /(yi )L
yi
xi
Ki =yi /xi
fvi (psia)
fLi (psia)
C1
2.08907
0.629843
0.330082
1.90814
1,019.52
1,019.52
C4
0.645515
0.348699
0.513307
0.67932
210.076
210.076
C10
0.10537
0.021457
0.156611
0.13701
2.26859
2.26859
giving the results in Table B30. Component fugacities are clearly not equal within an acceptable tolerance; e.g., ( f v) C 10 + 1.068 psia and ( f L) C 1 + 3.355 psia. K values are then updated with the fugacity ratio, f Lńf v, as a correction term. K i(n)1) + K i(n)
f Li(n) . f vi(n)
. . . . . . . . . . . . . . . . . . . . . . . . . . . (4.48)
This type of simple Kvalue update is called successive substitution, and for decane the second Kvalue estimate is given by + K (1) K (2) C C 10
10
f (1) L ,C f (1) v ,C
10
10
+ (0.01085) 3.355 + (0.01085)(3.142) + 0.0341. 1.068 After the first GDEM promotion, convergence was achieved, resulting in vaporphase mole fraction of F v + 0.853401. K values were K C1 + 6.65071, K C4 + 0.890061, and K C10 + 0.03624. Table B30 gives the phase compositions. b. Table B31 gives the phasestability test results at 1,500 psia and 280°F. Results from the converged solutions of the vapor and liquidlike tests are shown. Both stability tests indicated that the feed composition was unstable and would therefore split into two (or more) phases. The vaporlike test required 12 iterations to converge, including two GDEM promotions. The liquidlike stability test required six iterations to converge, including one GDEM promotion. Because two unstable solutions were found, the twophase flash calculation was initialized with K values based on the two incipientphase compositions found in the stability tests; i.e., K i + (y i) vń(y i) L . With these initial estimates, the twophase flash calculation converged in eight iterations, including one GDEM promotion. The final vaporphase mole fraction was F v + 0.566844. Note how close the final converged K values are to the initial estimates from the stability test. Table B32 gives the results. Problem 19 Problem. The following are calculated phase properties from the flash calculation at 500 psia and 280°F in Problem 18. M L + 111.7 lbmńlbm mol, M v + 35.46 lbmńlbm mol, EXAMPLE PROBLEMS
v L + 2.721 ft 3ńlbm mol, and v v + 13.837 ft 3ńlbm mol. These molar volumes include the effect of a slight shift in volume by use of volume translation. What is the phase molar volume and liquid density without volume translation? Solution. The volume shift, c i, for each component is calculated from EOS constants b i and the volume translation ratios, s i, given in Problem 18. Eq. 4.21 gives the b i values for the PR EOS.16 R 2T 2 a + W oa p c a , where W oa + 0.45724; c RT b + W ob p c , where W ob + 0.07780; c a + ƪ1 ) mǒ1 * ǸT rǓƫ ; 2
and m + 0.37464 ) 1.54226 w * 0.26992 w 2 .
. . . . . . . (4.21)
This gives b C + 0. 07780(10.7315)(343.0)ń(667.8) 1
+ 0.4288 ft 3ńlbm mol, b C + 1.160 ft 3ńlbm mol, 4
b C + 3.053 ft 3ńlbm mol, 10
c C + (* 0.1595)(0.4288) + * 0.06840 ft 3ńlbm mol 1
c C + (* 0.0675)(1.1603) + * 0.07832 ft 3ńlbm mol, 4
and c C + (0.0655)(3.053) + 0.2000 ft 3ńlbm mol. 10
From Eq. 4.25,
ȍx c N
v L + v LEOS *
i i
i+1
ȍy c . N
and v v + v vEOS *
i i
. . . . . . . . . . . . . . . . . . . . . . . (4.25)
i+1
17
TABLE B33—RECOMBINED SEPARATOR WELLSTREAM MOLAR COMPOSITION AND CONSISTENCY CHECK OF SEPARATOR K VALUES WITH THE STANDING18 LOWPRESSURE KVALUE CORRELATION (PROBLEM 20) zi xi
Reported
yi
CO2
4.01
1.12
3.84
3.84
2.087
N2
0.85
0.03
0.80
0.80
3.394
C1
89.83
10.68
85.12
85.16
2.606
8.42
8.41
C2
2.88
2.56
2.86
2.86
1.543
1.14
1.13
C3
1.30
3.86
1.45
1.45
0.811
0.289
0.337
iC4
0.32
2.60
0.46
0.45
0.346
0.121
0.123
nC4
0.43
5.31
0.72
0.72
0.180
0.0884
0.0810
iC5
0.15
3.88
0.37
0.37
*0.256
0.0390
0.0387
C5
0.11
4.16
0.35
0.35
*0.391
0.0303
0.0264
Calculated
Fi
Standing
Reported
3.18
3.58
37.0
28.3
C6
0.07
7.58
0.52
0.51
*0.859
0.0126
0.0092
C7+
0.05
58.22
3.51
3.48
*2.010
0.00145
0.00086
Total gC
100.00
100.00
100.00
100.00
0.778
0.778
7)
MC
7)
M
0.7783 135
98
135
135
100.2
18.6
23.4
23.4
With liquid compositions calculated in Problem 18 at 500 psia and 280°F, the molar volume without volume translation, v LEOS , is given by v LEOS + 2.721 ) [(0.08588)(* 0.0684) ) (0.46349) (* 0.07832) ) (0.45064)(0.2000)] + 2.721 ) (* 0.006 * 0.0363 ) 0.0901), + 2.769 ft 3ńlbm mol. The molecular weight of liquid is needed to convert from molar volume to density. M L + (0.08588)(16.04) ) (0.46349)(58.12) ) (0.45064)(142.29) + 92.44 lbmńlbm mol, ò L + 94.44ń2.769 + 33.38 lbmńft . 3
Problem 20 Problem. Separator samples were collected during a production test from the discovery well of a gascondensate reservoir. Use the HoffmannCrumpHocott 17 (HCH) Kvalue plot (Eqs. 3.155 and 3.156) to check the consistency of measured separator compositions. Plot the data together with the lowpressure Standing18 Kvalue correlation line given by Eq. 3.161. Also recombine the separator samples to check the reported wellstream composition (laboratory recombined values can be in error). Finally, calculate the Watson characterization factor of the C 7) component. Solution. Table B33 gives reported separator compositions; calculated K values from the ratio of separatorgas to separatoroil molar compositions, K i + y ińx i; and finally, the recombined wellstream composition, z i. Separator conditions are 390 psig and 52°F. The HCH variable F i is given by Eq. 3.156, with b i and T bi values given in Table 3.3. Methane, for example, has an F i value given by b i + 300 cycleń°R, T bi + 94°R,
and F i + 300 ƪ1ń94 * 1(52 ) 460)ƫ + 2.606, where modified values of b i and T bi are given by Standing (instead of values given by Eq. 3.156). The Kvalue pressure product for methane is given by K i p sp + ǒ89.83ń10.68Ǔ(390 ) 14.7) + (8.411)(404.7), + 3, 404 psia. which is plotted vs. F i + 2.606 on semilog paper (Fig. B1). The Standing lowpressure Kvalue correlation is plotted together with the measured Kvalues on Fig. B1. From Standing’s18 correlation, Slope A 0 and Intercept A 1 are K i + p1 10 ǒ A0 ) A1 Fi Ǔ ,
. . . . . . . . . . . . . . . . . . . . . . . (3.161a)
sp
F i + b iǒ1ńT bi * 1ńTǓ, . . . . . . . . . . . . . . . . . . . . . . . (3.161b) b i + logǒ p cińp scǓńǒ1ńT bi * 1ńT ciǓ ,
which gives
18
Ki
Component
A 0( p) + 1.2 ) ǒ4.5
. . . . . . . . . . . . . . (3.161c)
10 *4Ǔ p ) ǒ15
10 *8Ǔ p 2 ,
. . . . . . . . . . . . . . . . . . (3.161d) A 1ǒ pǓ + 0.890 * ǒ1.7
10
*4
Ǔ p * ǒ3.5
10 *8Ǔ p 2,
. . . . . . . . . . . . . . . . . . . (3.161e) nC
7)
+ 7.3 ) 0.0075T ) 0.0016p,
bC
7)
+ 1, 013 ) 324n C
and T bC
7)
7)
+ 301 ) 59.85n C
. . . . . . . . . . . . . (3.161f)
* 4.256n 2C 7)
7)
,
* 0.971n 2C
. . . . . . . (3.161g) ,
7)
. . . . . (3.161h)
giving A 0 + 1.2 ) ǒ4.5
10 *4Ǔ(404.7) ) ǒ15
10 *8Ǔ(404.7)
2
+ 1.407 and A 1 + 0.890 * ǒ1.7 * ǒ3.5
10 *4Ǔ(404.7)
10 *8Ǔ(404.7) + 0.8155. 2
The methane K value from the Standing correlation is, for example, K C + ǒ1ń404.7Ǔ10 [1.407)(0.8155)(2.606)] + 8.415, 1
which can be compared with the measured value of 8.411. PHASE BEHAVIOR
The Standing C 7) K value requires calculating b C and T bC 7) 7) from separator conditions. nC
+ 7.3 ) 0.0075(52) ) 0.0016(404.7)
7)
y w + y ow A g A s ,
+ 8.34 (approximate carbon number), bC
7)
+ 1, 013 ) (324)8.34 * (4.256)(8.34)
Solution. From Fig. 9.29, the temperature for hydrate formation of a 0.7gravity gas at 1,000 psia is about 69°F. From Eq. 9.23, the water content in the gas at 1,000 psia and 69°F is
* 1.117 ln p ) 16.44 ,
+ 3, 419 cycle°R, T bC
2
7)
and F C
+ 301 ) (59.85)(8.34) * (0.971)(8.34) + 732.6°R,
7)
7)
. . . . . . . . . . . . . . . . . (9.23b)
g g * 0.55
Ag + 1 )
ǒ1.55
10 4 Ǔg g T *1.446 * ǒ1.83
+ 3, 419ƪ(1)ń(732.6) * (1)ń(52 ) 460)ƫ + * 2.01.
+ ǒ1ń404.7Ǔ10
[1.407)(0.8155)(*2.01)]
+ 0.00145,
which can be compared with the measured value of K C + 0.050 7) B 58.22 + 0.00086. From Fig. B1 the measured Kvalue data plot as a straight line almost coincident with the Standing correlation. This indicates that the measured compositions are probably consistent. Recombination is made on the basis of separator gas/oil ratio, R sp with Eq. 6.8. Separatoroil density and molecular weight are both required for the recombination calculation, and most laboratories use the StandingKatz8 density correlation to estimate ò osp on the basis of separatoroil composition (oil molecular weight is calculated from Eq. 6.9). For this sample, the separator properties and gas mole fraction, F gsp, are given by M osp + 100.2 lbm/lbm mol and ò osp + 45.28 lbm/ft3.
ǒ
F gsp + 1 )
2, 130ò osp M osp R sp
Ǔ
*1
,
. . . . . . . . . . . . . . . . . . . . (6.8)
ƪ
ƫ
+ 0.94102 .
* ǒ1.83 ln y ow +
ƫǋ + 0.9972,
and y w + (0.9972)(0.000546) + 0.000544. From Eq. 9.24, the solution water/gas ratio is given by yw [ 135y w , 1 * yw
. . . . . . . . . . . . . . . . . . . (9.24)
which gives r sw + (47, 300)(0.000544)ń(1 * 0.000544) + 25.7 lbm/MMscf. At 15°F and 1,000 psia, the water content is
10 *5 ,
EXAMPLE PROBLEMS
*1.446
+ 0.000546,
y w + 8.61
Problem 21 Problem. A refrigeration/expansion process is used to reduce water and condensate content of a gas stream. The well effluent arrives at the separator at 2,000 psia and 155°F. It is cooled in the separator and heat exchanger. Separator pressure is 1,000 psia, separatorgas gravity is 0.70 (air+1), and separatorgas rate is 65 MMscf/D. What is the minimum temperature upstream of the choke to prevent hydrate formation in the separator? What is the water content of the separator gas? How much water in lbm/D must be removed from the separator gas if sales specifications call for a maximum dewpoint of )15°F at 1,000 psia?
*1.288
10 4Ǔ(0.7)(69)
* 1.117 ln(1, 000) ) 16.44, y ow
10 *5 ,
which is quite close to the reported value of 85.12% (as it should be). Occasionally, because of entry errors to the recombination computer program or possibly because of inconsistent recombination GOR used in the laboratory, reported wellstream compositions may not be the same as those calculated with Eqs. 6.7 through 6.9. In these situations, contact the laboratory about the inconsistency. It may even be worthwhile to request a preliminary report of the separator and recombined compositions before completing pressure/volume/ temperature study.
. . . . . . . . . . . . . . . . (9.23e)
(0.05227)(1, 000) ) 142.3 ln(1, 000) * 9, 625 69 ) 460
y ow + 8.61
1
,
. . . . . . . . . . . . . . . . . . (9.23d)
10 *9ǓC 1.44 . s
10 4Ǔ(69)
z i + F gsp y i ) ǒ1 * F gspǓ x i .
z C + (0.94102)(89.83) ) (1 * 0.94102)(10.68) + 85.16% ,
ǓC s ,
Ǌ
A g + 0.9996,
For methane, this is
*6
A g + 1 ) (0.7 * 0.55)ńƪǒ1.55
The wellstream composition is calculated from Eq. 6.7. . . . . . . . . . . . . . . . . . . . . . (6.7)
*1.288
This gives
which yields *1
10
and A s + 1 * ǒ3.92
r sw + 135
(2, 130)(45.28) F gsp + 1 ) (100.2)ń(15, 357)
10 4 T Ǔ
. . . . . . . . . . . . . . . . . . (9.23c) A s + 1 * ǒ2.22
This yields KC
0.05227p ) 142.3 ln p * 9, 625 T ) 460
ln y ow +
2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.23a)
and r sw + 4.0 lbmńMMscf. At a separatorgas rate of 65 MMscf/D, the water removal capacity must then be (25.7*4.0)(65)+1,445 lbm/D. Problem 22 Problem. Estimate gas solubility for the reservoir brine in Problem 21 at reservoir conditions of 4,050 psia and 255°F. Also estimate brine density, compressibility, and FVF. The reservoir gas yields 13 STB/MMscf (MMscf of separator gas) of a 69°API stocktank condensate. Brine salinity is 36,200 ppm total dissolved salts. Assume separator conditions are 1,000 psia and 80°F. Solution. The reservoir (wellstream hydrocarbon) specific gravity is given by gw +
g g ) 4, 580 r p g o . 1 ) 133, 000 r p ǒ gńM Ǔ o
. . . . . . . . . . . . . . . . . . (3.55)
However, we need to estimate the amount and specific gravity of the gas coming from separator condensate at 1,000 psia and 80°F using Eqs. 3.62 through 3.64. g g) + A 2 ) A 3 R s) ,
. . . . . . . . . . . . . . . . . . . . . . . . . . (3.62) 19
where A 2 + 0.25 ) 0.2g API and A 3 + * ǒ3.57 R s) + and g g +
10 *6Ǔg API ;
A1 A2 ; . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.63) ǒ1 * A 1 A 3Ǔ g g1 R s1 ) g gs1 R s) . R s1 ) R s)
. . . . . . . . . . . . . . . . . . (3.64)
Brine density at standard conditions is given by Eq. 9.14, with T sc + 60°F [289 K]. v woǒ p sc,T Ǔ +
1 + A 0 ) A 1w s ) A 2w 2s , ò woǒ p sc,T Ǔ
where A 0 + 5.916365 * 0.01035794T ) ǒ0.9270048
This gives A 1 + 1, 152,
10 *5ǓT 2
* 1, 127.522T *1 ) 1, 00674.1T *2 ,
A 2 + 1.63, 10 *4 ,
A 3 + * 2.46
A 1 + * 2.5166 ) 0.0111766T * ǒ0.170552
10 *4ǓT 2 ,
and A 2 + 2.84851 * 0.0154305T ) ǒ0.223982
10 *4ǓT 2 ,
g g) + 0.985 (air + 1),
. . . . . . . . . . . . . . . . . . . . (9.14)
R s) + 2, 620 scfńSTB, r p + 1ńƪ1ńǒ13
yielding A 0 + 1.00106,
10 *6Ǔ ) 2, 620ƫ + 12.6 STBńMMscf,
A 1 + * 0.7112,
and g g + 0.711 (air + 1) . From Fig. 9.2, the gas solubility of a 0.65°API gravity gas in pure water at 4,000 psia and 250°F is about 19 scf/STB. Pure methane solubility in pure water at reservoir conditions can be estimated from Eq. 9.6. x C + 10 *6 1
ƪȍǒȍ Ǔ ƫ 3
3
i+0
j+0
A i jT j p i , . . . . . . . . . . . . . . . . (9.6)
v w + 0.9756 cm 3ńg,
At reservoir temperature (397 K) and standard pressure, brine density, ò ow, is also given by Eq. 9.14, which results in A 0 + 1.0642,
x C + 10 *3ƪ * 0.0256 ) (0.00107)(4, 050) * ǒ9.59 1
(4, 050) ) ǒ3.98 2
or R osw + 7, 370Ǌǒ2.73
10 *12Ǔ(4, 050) ƫ + 2.73 3
10 *8Ǔ 10 *3
10 *3Ǔƫǋ
10 *3Ǔńƪ1 * ǒ2.73
+ 0.1813 * ǒ7.692 ) ǒ2.6614
10 *6ǓT 2 * ǒ2.612
10 *4Ǔ(255)ǒ2.6614
10 *6Ǔ(255)
2
10 *9Ǔ(255) + 0.115. 3
This can be corrected for specific gravity with Eq. 9.11, but we neglect the correction for simplicity. The resulting gas solubility of the brine is then xg R sw [ x o + 10 *kscs [ 10 *ǒ17.1 R osw g
10 *6Ǔ k sC s
c *wǒ p, T Ǔ + ǒ A 0 ) A 1 p Ǔ
10 *6Ǔ (0.115)(36,000)ƫ
.
,
where A + 10 6ƪ0.314 ) 0.58w s ) ǒ1.9 *ǒ1.45
,
. . . . . . . . . (9.9)
A 0 + 0.289
10 *4ǓT
10 *6ǓT 2ƫ
10 6 ,
A 1 + 8.656, and c *w + 3.09
10 *6 psi *1.
The FVF of brine without dissolved gas at atmospheric pressure is given by B ow +
ò wǒ p sc, T scǓ v oǒ p sc, TǓ + w , . . . . . . . . . . . . . . . . . (9.13) o v wǒ p sc, T scǓ ò wǒ p sc, TǓ
yielding B ow + 1.025ń0.9646 + 1.063 bbl/STB. From Eq. 9.18, the FVF of brine at reservoir pressure and temperature without dissolved gas is
resulting in R sw + (17.5)10 ƪ*ǒ17.1
*1
yielding
10 *9ǓT 3 ,
which gives
* ǒ2.612
v ow + 1.0367 cm 3ńg,
and A 1 + 8 ) 50w s * 0.125w sT , . . . . . . . . . . . . . . . . . (9.17)
10 *4ǓT
. . . . . . . . . . . . . . . . . . (9.10)
k s + 0.1813 * ǒ7.692
A 2 + 0.253,
Compressibility of brine without solution gas is given by
This compares with 22 scf/STB from Fig. 9.1. We assume that R osw + 17.5 for this gas in pure water (from a plot of 19 scf/STB at g g + 0.65 and 22 scf/STB at g g + 0.55). Reduction in solubility resulting from salinity can be estimated from the Setchenow correction (Eqs. 9.9 and 9.10).19 For methane, the Setchenow constant is 1*NaCl
A 1 + * 0.768,
and ò ow + 0.9646 gńcm 3.
+ 20. 2 scfńSTB.
20
10 *6 + 0.0362,
w s + 36, 200
and ò w + 0.9756 gńcm 3 .
yielding
(k s) C
A 2 + 0.2601,
B *wǒ p,
ǒ
TǓ + B owǒ p sc, TǓ 1 )
A1 p A0
Ǔ
ǒ1ńA1Ǔ ,
. . . . . . . . . . (9.18) PHASE BEHAVIOR
giving B *w + 1.063ƪ1 ) (4, 050)(8.656)ńǒ0.289
10 6Ǔƫ
ǒ*1ń8.656Ǔ
+ 1.049 bblńSTB. With the DodsonStanding20 corrections for compressibility and FVF as a function of gas solubility (Eqs. 9.19 and 9.20, respectively), the brine volumetric properties including gas solubility effect are 1.5Ǔ B wǒ p, T, R swǓ + B *wǒ p, TǓǒ1 ) 0.0001 R sw
. . . . . . . . (9.19)
and c wǒ p, T, R swǓ + c *wǒ p, TǓ ǒ1 ) 0.00877 R swǓ , . . . . . . . (9.20) which give B w + (1.049)ƪ1 ) (0.0001)ǒ17.5 1.5Ǔƫ + 1.057 bblńSTB and c w + ǒ3.09 + 3.55
10 *6Ǔƪ1 ) (0.00877)ǒ17.5 Ǔƫ 10 *6 psi *1.
References 1. Hall, K.R. and Yarborough, L.: “A New EOS for Zfactor Calculations,” Oil & Gas J. (18 June 1973) 82. 2. Yarborough, L. and Hall, K.R.: “How to Solve EOS for Z–factors,” Oil & Gas J. (18 February 1974) 86. 3. Sutton, R.P.: “Compressibility Factors for HighMolecularWeight Reservoir Gases,” paper SPE 14265 presented at the 1985 SPE Annual Technical Conference and Exhibition, Las Vegas, Nevada, 22–25 September. 4. Lucas, K.: Chem. Ing. Tech. (1981) 53, 959. 5. Lohrenz, J., Bray, B.G., and Clark, C.R.: “Calculating Viscosities of Reservoir Fluids From Their Compositions,” JPT (October 1964) 1171; Trans., AIME, 231. 6. Wichert, E. and Aziz, K.: “Compressibility Factor of Sour Natural Gases,” Cdn. J. Chem. Eng. (1971) 49, 267. 7. Wichert, E. and Aziz, K.: “Calculate Z’s for Sour Gases,” Hydro. Proc. (May 1972) 51, 119. 8. Standing, M.B. and Katz, D.L.: “Density of Natural Gases,” Trans., AIME (1942) 146, 140. 9. Cragoe, C.S.: “Thermodynamic Properties of Petroleum Products,” U.S. Dept. of Commerce, Washington, DC (1929) 97. 10. Standing, M.B.: Volumetric and Phase Behavior of Oil Field Hydrocarbon Systems, SPE, Richardson, Texas (1981).
EXAMPLE PROBLEMS
11. Lee, A.L., Gonzalez, M.H., and Eakin, B.E.: “The Viscosity of Natural Gases,” JPT (August 1966) 997; Trans., AIME, 237. 12. Chew, J.N. and Connally, C.A.: “A Viscosity Correlation for GasSaturated Crude Oils,” Trans., AIME (1959) 216, 23. 13. Wilson, G.M.: “A Modified RedlichKwong EOS, Application to General Physical Data Calculations,” paper 15c presented at the 1969 AIChE Natl. Meeting, Cleveland, Ohio. 14. Muskat, M. and McDowell, J.M.: “An Electrical Computer for Solving Phase Equilibrium Problems,” Trans., AIME (1949) 186, 291. 15. Rachford, H.H. and Rice, J.D.: “Procedure for Use of Electrical Digital Computers in Calculating Flash Vaporization Hydrocarbon Equilibrium,” JPT (October 1952) 19; Trans., AIME, 195. 16. Peng, D.Y. and Robinson, D.B.: “A NewConstant Equation of State,” Ind. & Eng. Chem. (1976) 15, No. 1, 59. 17. Hoffmann, A.E., Crump, J.S., and Hocott, C.R.: “Equilibrium Constants for a GasCondensate System,” Trans., AIME (1953) 198, 1. 18. Standing, M.B.: “A Set of Equations for Computing Equilibrium Ratios of a Crude Oil/Natural Gas System at Pressures Below 1,000 psia,” JPT (September 1979) 1193. 19. Pawlikowski, E.M. and Prausnitz, J.M.: “Estimation of Setchenow Constants for Nonpolar Gases in Common Salts at Moderate Temperatures,” Ind. Eng. Chem. Fund. (1983). 20. Dodson, C.R. and Standing, M.B.: “Pressure, Volume, Temperature and Solubility Relations for Natural GasWater Mixtures,” Drill. & Prod. Prac., API (1944) 173.
SI Metric Conversion Factors °API 141.5/(131.5)°API) +g/cm3 atm 1.013 250* E)05 +Pa bbl 1.589 873 E*01 +m3 cp 1.0* E*03 +Pa@s ft 3.048* E*01 +m E*02 +m3 ft3 2.831 685 °F (°F*32)/1.8 +°C °F (°F)459.67)/1.8 +K gal 3.785 412 E*03 +m3 lbm 4.535 924 E*01 +kg lbm mol 4.535 924 E*01 +kmol psi 6.894 757 E)00 +kPa E*01 +kPa*1 psi*1 1.450 377 °R 5/9 +K ton 9.071 847 E*01 +Mg *Conversion factor is exact.
21
Appendix C
EquationĆofĆState Applications This appendix presents two examples of fluid characterization with an equation of state (EOS). The examples treat the gas condensate and the oil discussed in Chap. 6, Good Oil Co. Wells 7 and 4, respectively. Details of developing a complete fluid characterization are given for the gascondensate fluid, including the splitting of C 7) into five fractions, determining volumetranslation coefficients for the C 7) fractions, and estimating methane through C 7) binary interaction parameters (BIP’s). The resulting characterization is the starting point for EOS predictions and, particularly, the simulation of pressure/volume/temperature (PVT) experiments. GasĆCondensateĆFluid Characterization The characterization is developed for the PengRobinson1 EOS (PR EOS) on the basis of the C 7) characterization suggested in Chap. 5 with five C 7) fractions. First, predictions are made without modifying the EOS parameters. Then, the measured dewpoint is matched by modifying the BIP between methane and all C 7) fractions. Finally, constantvolumedepletion (CVD) data are matched by modifying the characterization with three regression parameters. C7 + Molar Distribution. The first step in the C 7) characterization is to split the heptanesplus component into five fractions by use of the Gaussian quadrature model in Chap. 5. In the absence of experimental trueboilingpoint data, the following parameters are assumed: a+1, h+90, and N+5, with M C7)+143 and g C7)+0.795. The value selected for heaviest fraction molecular weight, M N, is somewhat higher than the recommended value of M N + 2.5M C7) + 2.5(143) + 358. Instead, we use M N + 500, which allows us to develop a better characterization (particularly the taillike behavior of the liquiddropout curve). The modified b * term is b *+ ǒ M N * h ǓńX N + (500 * 90)ń(12.6408) + 32.435, where X 5 is taken from Table 5.6. The d parameter is calculated from Eq. 5.30. d + exp
ǒ
a b* MC
7)
*h
Ǔ
*1 ,
. . . . . . . . . . . . . . . . . . . (5.30)
giving d + expǊ(1)(32.435)ń[(143 * 90) * 1]ǋ + 0.67840. Table C1 gives values of f(X) for each fraction, according to Eq. 5.31, together with calculated mole fractions and molecular weights based on quadrature points and weighting factors, X i and W i , respectively. EQUATIONOFSTATE APPLICATIONS
zi + zC
7)
[W i f( X i)],
Mi + h ) b* Xi , and fǒ X Ǔ +
ǒ X Ǔ a*1 ǒ1 ) ln dǓ a . G(a) dX
. . . . . . . . . . . . . . . . . (5.31)
For the first fraction, X 1 + 0.263560, W 1 + 0.52175561, (0.263560) fǒX 1Ǔ + G(1)
(1*1)
[1 ) ln(0.67840)] (0.67840)
1
0.263560
+ 0.677878,
z 1 + 6.85(0.52175561)(0.677878) + 2.4228, and M 1 + 90 ) 32.435(0.263560) + 98.55. C7 + Specific Gravities and Boiling Points. Given mole fractions and molecular weights of the fractions, specific gravities are estimated with the Søreide2 correlation.3 g i + 0.2855 ) C f ǒ M i * 66 Ǔ
0.13
,
. . . . . . . . . . . . . . . (5.44)
where the characterization factor, C f , is modified to ensure that the calculated C 7) specific gravity equals the measured value of g C 7)+0.795.
ǒgC7)Ǔ
exp
+
zC
7)
MC
7)
ȍ N
.
. . . . . . . . . . . . . . . . . . . . . (5.37)
ǒz i M ińg iǓ
i+1
By trial and error, C f + 0.28927 is found to satisfy Eq. 5.37; Table C2 gives the results. For the first fraction, g F + 0.2855 ) 0.28927(98.55 * 66) 1
0.13
+ 0.7404.
Normal boiling points are calculated from the Søreide correlation. T b + 1, 928.3 * ǒ1.695 exp ƪ * ǒ4.922 ) ǒ3.462
10 5Ǔ M *0.03522 g 3.266 10 *3Ǔ M * 4.7685 g
10 *3Ǔ Mgƫ ,
. . . . . . . . . . . . . . . . . . . . (5.45) 1
TABLE C1—GAUSSIAN QUADRATURE METHOD TO SPLIT C7+ INTO FIVE FRACTIONS FOR RESERVOIR GAS CONDENSATE C7+ Fraction i
Quadrature Point Xi
Quadrature Weight Wi
f(Xi )
Mole Fraction zi
Molecular Weight Mi
Mass mi +zi Mi
1
0.263560
0.52175561
0.677878
2
1.413403
0.39866681
1.059051
2.4228
98.55
238.8
2.8921
135.84
3
3.596426
0.07594245
2.470516
392.9
1.2852
206.65
265.6 75.7
4
7.085810
0.00361176
9.567521
0.2367
319.83
5
12.640801
0.00002337
82.58395
0.0132
500.00
6.8500
143.00*
Total
6.6 979.5
*Equals 979.5/6.85.
TABLE C2—PROPERTIES OF C7+ FRACTIONS FOR RESERVOIR GAS CONDENSATE C7+ Fraction i
Molecular Weight Mi
Mass mi +zi Mi
Specific Gravity gi *
Ideal Volume V+zi Mi /gi
Boiling Point Tb °R
1
98.55
238.8
0.7407
322.5
674.1
2
135.84
392.9
0.7879
498.6
793.9
3
206.65
265.6
0.8358
317.8
972.7
4
319.83
75.7
0.8796
86.1
1,175.5
5
500.00
6.6
0.9226
7.2
1,386.3
143.00
979.5
0.7950
1,232.1
*Water+1.
TABLE C3—TWU4 METHOD FOR CALCULATING CRITICAL PROPERTIES OF C7+ FRACTIONS FOR RESERVOIR GAS CONDENSATE Component i
Tb (°R)
TcP
a
gP *
g*
DgT
fT
Tc (°R)
1
674.1
978.7
0.3112
0.6908
0.7404
*0.2195
0.003224
1,004.3
2
793.9
1,102.9
0.2802
0.7304
0.7879
*0.2498
0.003599
1,135.1
3
972.7
1,268.7
0.2333
0.7705
0.8358
*0.2783
0.003965
1,309.6
4
1,175.5
1,434.6
0.1807
0.8005
0.8796
*0.3267
0.004754
1,490.2
5
1,386.3
1,589.5
0.1278
0.8201
0.9226
*0.4008
0.006209
1,670.5
vcP (ft3/lbm mol)
Dgv
fv
vc (ft3/lbm mol)
pcP (psia)
Dgp
fp
pc (psia)
1
6.90
*0.2471
*0.0085
6.4475
393.8
*0.0245
0.00256
441.4
2
9.33
*0.2947
*0.0114
8.5142
314.2
*0.0283
0.00294
362.7
3
14.15
*0.3424
*0.0152
12.5336
220.0
*0.0321
0.00504
266.9
4
21.69
*0.4124
*0.0217
18.2317
142.5
*0.0388
0.01028
191.2
5
32.14
*0.5103
*0.0328
24.7141
87.2
*0.0499
0.02037
140.4
*Water+1.
which, for the first fraction, gives T b + 1, 928.3 * ǒ1.695 exp ƪ * ǒ4.922 ) ǒ3.462
10 5Ǔ(98.55)
*0.03522
(0.7404)
3.266
10 *3Ǔ(98.55) * 4.7685 (0.7404)
10 *3Ǔ(98.55)(0.7404)ƫ + 674.1°R .
C7 + Critical Properties. Critical properties T c and p c are calculated from the Twu4 correlations (Eqs. 5.68 through 5.78). Table C3 shows the calculations from left to right, in the order required to solve the rather tedious Twu correlations. Acentric factor is calculated from the LeeKesler5 correlation. 2
w+
) A 3 ln T br ) A 4 T br6 * lnǒ p cń14.7Ǔ ) A 1 ) A 2 T *1 br A 5 ) A 6 T *1 ) A 7 ln T br ) A 8 T br6 br . . . . . . . . . . . . . . . . . . . . (5.60)
for reduced normal boiling points T br + T bńT c t 0.8. The KeslerLee6 correlation, w + * 7.904 ) 0.1352K w * 0.007465K 2w ) 8.359T br ) ǒ1.408 * 0.01063 K wǓT *1 br ,
. . . . . . . (5.61)
is used to calculate higher reduced boiling points [making use of the /g)]. Watson characterization factor defined by Eq. 5.34, ( K w + T 1ń3 b Table C4 shows the results. PHASE BEHAVIOR
m + 0.7941,
TABLE C4—CALCULATION OF ACENTRIC FACTOR FOR C7+ FRACTIONS OF RESERVOIR GAS CONDENSATE Component i
a + 1.4955,
Tb /Tc
Kw [(°R)1/3]
w
1
0.671
11.842
0.2864
and b + 1.8997 ft 3ńlbm mol.
2
0.699
11.752
0.3881
3
0.743
11.855
0.5754
By trial and error, the value of c that gives p+14.7 psia from Eq. 4.19 is
4
0.789
11.998
0.8313
5
0.830
12.086
1.1185
a + 1.7995
c + 0.06151 ft 3ńlbm mol or s + cńb + (0.06151)ń(1.8997) + 0.0324 .
VolumeTranslation Parameters. Volumetranslation parameters, s i, for pure components through C 6 are taken from Table 4.3. Values of s i for the C 7) fractions are determined to ensure that the EOS characterization for each separate C 7) fraction correctly calculates a density at standard conditions that is consistent with the specific gravity of that fraction. The actual molar volume at standard conditions, v + Mń(62.37g) in ft3/lbm mol, is equal to the EOScalculated molar volume, v EOS (without volume translation), less the volumetranslation parameter, c,
Table C5 gives results for the other fractions. BIP’s. The BIP’s between nonhydrocarbons and hydrocarbons are taken from Table 4.1. The modified ChuehPrausnitz7 equation,
k ij
+ 2.1340 ft 3ńlbm mol. The correct c is determined when v EOS and EOS Constants a and b in the PR EOS,
v 1ń3 ci
)
v 1ń3 cj
Ǔ ȳȴȧ ,
. . . . . . . . . . . . . . . . (5.79)
. . . . . . . . . . . . (4.19)
with v c in ft3/lbm mol. For the first fraction, v cF1 + 6.508 ft3/lbm mol. By use of the same approximate relation for methane, v c + 1.447 ft3/lbm mol and k ij for this pair is kC
Z * (1 * B)Z ) ǒ A * 3B * 2B Ǔ 2
ǒ
B
2v 1ń6 v 1ń6 ci cj
v ci [ 0.4804 ) 0.06011M i ) 0.00001076M 2i ,
calculate a pressure of 14.7 psia at T + T sc . The EOS constants are calculated from Eqs. 4.20 through 4.22. 3
ȱ + Aȧ1 * Ȳ
is used for methane/ C 7) pairs with A+0.18 and B+6. For use with this correlation, hydrocarbon critical volumes should be estimated with the following approximate correlation.
v + v EOS * c + (98.55)ń[(62.37)(0.7404)]
a , p + RT * v * b v(v ) b) ) b(v * b)
10 5 psiaft 3ńlbm mol
1ńF 1
+ 0.18 * 0.18
2
ƪ
2(1.447)
(1.447)
1ń6
1ń3
(6.508)
1ń6
) (6.508)
1ń3
ƫ
6
+ 0.0301.
Table C6 gives other methane/ C 7) BIP’s, and Tables C7 and C8 summarize the PR EOS fluid characterization.
Z * ǒAB * B 2 * B 3Ǔ + 0 and Z c + 0.3074 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.20) R 2T 2 a + W oa p c a , where W oa + 0.45724; c RT b + W ob p c , where W ob + 0.07780; c a + ƪ1 ) mǒ1 * ǸT rǓƫ ; 2
and m + 0.37464 ) 1.54226 w * 0.26992 w 2 .
. . . . . . . (4.21)
m + 0.3796 ) 1.485w * 0.1644w ) 0.01667w . 2
3
. . . . . . . . . . . . . . . . . . . . (4.22) for wu0.49. This results in T r + TńT c + (60 ) 460)ń(1, 004.3) + 0.5174,
EOS Predictions With the PR EOS characterization given in Tables C7 and C8, a dewpoint pressure of 3,535 psia is predicted at reservoir temperature of 186°F; this is approximately 500 psi lower than the measured value. Figs. C1 through C4 show calculated EOS results. The liquiddropout data are seriously overpredicted at pressures from 2,500 to 3,500 psia. Otherwise, the predictions are quite reasonable. Wellstream compositions are acceptable, being somewhat too rich at 3,500 psia and somewhat too lean at 2,900 psia. Matched Dewpoint Pressure Multiplying the BIP’s between methane and all C 7) fractions by a factor of 2.09 matches the measured dewpoint pressure of 4,015 psia. Figs. C1 through C4 present calculated CVD results. The predicted PVT data are not very good; in particular, the liquiddropout curve at 3,515 psia is overpredicted (21.2% vs. the measured value of 3.3%) and equilibriumgas C 7) compositions are severely underpredicted.
TABLE C5—CALCULATION (CHECK) OF VOLUMETRANSLATION PARAMETERS, s, FOR C7+ FRACTIONS IN RESERVOIR GAS CONDENSATE i
v + Mńò *
Tr+Tsc /Tc
m
a
b
Guess c
vEOS+v + c
pcalc (psia)
s+c/b
105
a
1
2.1340
0.5174
0.7941
1.4955
1.7995
1.8997
0.06151
2.1955
14.7
0.0324
2
2.7644
0.4578
0.9325
1.6941
3.1686 105
2.6127
0.14435
2.9087
14.4
0.0552
3
3.9644
0.3968
1.1829
2.0671
6.9940
105
4.0964
0.44038
4.4048
14.5
0.1075
4
5.8295
0.3487
1.5100
2.6189
1.6022
106
6.5089
1.00387
6.8334
14.8
0.1542
3.4744
106
9.9361
1.58485
10.2745
14.5
0.1595
5
8.6897
0.3111
1.8582
3.3189
* ò + 62.37g.
EQUATIONOFSTATE APPLICATIONS
3
parameters were chosen, and a sumofsquares (SSQ) function was minimized with a nonlinear regression algorithm. The SSQ function is defined as
TABLE C6—CALCULATION OF METHANE/C7+ BIP’s FOR RESERVOIR GAS CONDENSATE Approximate vc (ft3/lbm mol)
Component Methane
1.447
—
Fraction 1
6.509
0.0301
Fraction 2
8.844
0.0416
Fraction 3
13.362
0.0582
Fraction 4
20.806
0.0763
Fraction 5
33.225
0.0945
ȍr , M
Methane kij
F SSQ +
2 i
i
where M+total number of measured data included in the regression. The residuals, r i, are defined in terms of experimental data, d xi ; calculated data, d ci ; and weight factors, w i . For dewpoint pressure and Z factors,
ǒd
ri + Regression of CVD Data The measured CVD data, including dewpoint pressure, were then matched by modifying parameters in the original EOS (the characterization with predicted dewpoint of 3,535 psia). Three regression
xi
Ǔ
* d ci wi . d xi
For relative oil volumes, V ro , and cumulative gas produced, n pńn, r i + (d xi * d ci)w i . All weight factors, w i , are set to unity.
TABLE C7—FINAL PR EOS CHARACTERIZATION FOR RESERVOIR GAS CONDENSATE Component
z
Tc (°R)
M
pc (psia)
w
vc (ft3/lbm mol)
Zc
g*
Tb (°R)
s+c/b
N2
0.0018
44.01
547.6
1,070.6
0.2310
1.505
0.2742
0.5072
350.4
*0.0577
CO2
0.0013
28.01
227.3
493.0
0.0450
1.443
0.2916
0.4700
139.3
*0.1752
C1
0.6192
16.04
343.0
667.8
0.0115
1.590
0.2884
0.3300
201.0
*0.1651
C2
0.1408
30.07
549.8
707.8
0.0908
2.370
0.2843
0.4500
332.2
*0.1070
C3
0.0835
44.10
665.7
616.3
0.1454
3.250
0.2804
0.5077
416.0
*0.0848
iC4
0.0097
58.12
734.7
529.1
0.1756
4.208
0.2824
0.5631
470.6
*0.0686
C4
0.0341
58.12
765.3
550.7
0.1928
4.080
0.2736
0.5844
490.8
*0.0686
iC5
0.0084
72.15
828.8
490.4
0.2273
4.899
0.2701
0.6247
541.8
*0.0410
C5
0.0148
72.15
845.4
488.6
0.2510
4.870
0.2623
0.6310
556.6
*0.0410
C6
0.0179
86.18
913.4
436.9
0.2957
5.929
0.2643
0.6640
615.4
*0.0154
F1
0.024227
98.55
1,004.4
441.5
0.2864
6.447
0.2640
0.7405
674.1
0.0322
F2
0.028921
135.84
1,135.1
362.7
0.3882
8.514
0.2535
0.7879
793.9
0.0552
F3
0.012852
206.65
1,309.6
266.9
0.5756
12.535
0.2380
0.8357
972.7
0.1075
F4
0.002367
319.83
1,490.2
191.1
0.8316
18.236
0.2179
0.8796
1,175.5
0.1544
F5
0.000132
500.00
1,670.4
140.3
1.1188
24.725
0.1935
0.9224
1,386.4
0.1599
*Water+1.
TABLE C8—BIP’s FOR FINAL PR EOS CHARACTERIZATION OF RESERVOIR GAS CONDENSATE N2
4
N2
0
CO2
CO2
C1
C2
C3
iC4
0
0
C1
0.025
0.105
0
C2
0.010
0.130
0
0
C3
0.090
0.125
0
0
0
iC4
0.095
0.120
0
0
0
0
C4
iC5
C5
C6
F1
F2
F3
F4
C4
0.095
0.115
0
0
0
0
iC5
0.100
0.115
0
0
0
0
0
0
C5
0.110
0.115
0
0
0
0
0
0
0
C6
0.110
0.115
0
0
0
0
0
0
0
0
F1
0.110
0.115
0.030
0
0
0
0
0
0
0
0
F2
0.110
0.115
0.042
0
0
0
0
0
0
0
0
0
F3
0.110
0.115
0.058
0
0
0
0
0
0
0
0
0
F4
0.110
0.115
0.076
0
0
0
0
0
0
0
0
0
0
0
F5
0.110
0.115
0.095
0
0
0
0
0
0
0
0
0
0
0
F5
0
0 0 PHASE BEHAVIOR
Fig. C1—CVD liquiddropout behavior for gascondensate example comparing measured, EOSpredicted, and dewpointmatched calculations.
Fig. C2—CVD Zfactor behavior for gascondensate example comparing measured, EOSpredicted, and dewpointmatched calculations.
Fig. C3—CVD equilibriumgas C7+ behavior for gascondensate example comparing measured, EOSpredicted, and dewpointmatched calculations.
Fig. C4—CVD equilibriumgas C7+ molecularweight behavior for gascondensate example comparing measured, EOSpredicted, and dewpointmatched calculations.
The total number of data is 17 and includes one saturation pressure, six Z factors, five relative oil volumes, and five cumulative gas productions. Because the number of data is somewhat limited, only three regression parameters are used. Initially, before parameters have been changed, three data contribute most to the SSQ function: p d , Z d , and V ro at 2,915 psia. Each is approximately 25% of the total SSQ. The initial SSQ is approximately (F SSQ) i + 0.05. Regression I. The first regression uses the following three regression parameters: P 1, the multiplier to BIP’s between methane and all C 7) fractions; P 2, the multiplier to T c for all C 7) fractions; and P 3, the multiplier to p c for all C 7) fractions. Fig. C5 shows the reduction in the SSQ function at each iteration. The final SSQ value is approximately 4% of the initial value (0.002/0.05). Six iterations were required to find a minimum. Practically, however, the minimum was located after four iterations, with only small parameter adjustments made during the last two iterations. The final parameters are P 1 + 4.34, P 2 + 0.910, and P 3 + 0.849. Figs. C6 and C7 show the change in the multipliers at each iteration. The BIP multipliers increase monotonically to a value of approximately 4.3, resulting in C 1 through C 7) BIP’s ranging from 0.13 to 0.40. The large BIP values are outside the range of what is probably acceptable because they generally should not exceed approximately 0.3 for the PR EOS. C 7) critical temperatures decreased almost monotonically to approximately 10% less than the initial values. C 7) critical pressures increased during the first iterations, then finally decreased to EQUATIONOFSTATE APPLICATIONS
Fig. C5—Reduction in SSQ function for regression cases with three different sets of parameters to match measured gascondensate PVT data.
approximately 15% below the starting values. At Iteration 3, the minimum SSQ was almost reached, but the multiplier to critical pressures was approximately 1.0. During the last three iterations, the multiplier was reduced to 0.85 without any significant reduction 5
pc
Tc
Fig. C6—Variation in C1 through C7+ BIP multipliers used in Regressions I and II to match measured gascondensate PVT data.
Fig. C7—Variation in C7+ criticalproperty multipliers used in Regression I to match measured gascondensate PVT data.
Fig. C8—CVD liquiddropout behavior for gascondensate example comparing measured and EOS Regression I calculations.
Fig. C9—CVD Zfactor behavior for gascondensate example comparing measured and EOS Regression I calculations.
Fig. C10—CVD equilibriumgas C7+ behavior for gascondensate example comparing measured and EOS Regression I calculations.
in the SSQ function. This indicates that C 7) critical pressures are probably not very important when matching PVT data and that another parameter could be chosen instead. Figs. C8 through C11 show calculated results for the CVD experiment. Dewpoint pressure was overpredicted by only 8 psi 6
Fig. C11—CVD equilibriumgas C7+ molecularweight behavior for gascondensate example comparing measured and EOS Regression I calculations.
(0.2%) despite the relative low weight factor used (a factor of 10 or more is commonly used). Also, the experimental liquid dropout of 3.3% at 3,515 psia was 4.9% with the modified characterization, a very good match. PHASE BEHAVIOR
Fig. C12—Variation in the two C7+ criticaltemperature multipliers used in Regression II to match measured gascondensate PVT data.
Fig. C13—CVD liquiddropout behavior for gascondensate example comparing measured and EOS Regression II calculations.
Fig. C14—CVD Zfactor behavior for gascondensate example comparing measured and EOS Regression II calculations.
Fig. C15—CVD equilibriumgas C7+ behavior for gascondensate example comparing measured and EOS Regression II calculations.
Regression II. The second regression uses the following three regression parameters: P 1, the multiplier to BIP’s between methane and all C 7) fractions; P 2, the multiplier to T c for C 7) fractions F 1 through F 3 ; and P 3, the multiplier to T c for C 7) fractions F 4 and F 5. Fig. C5 shows the reduction in the SSQ function at each iteration. The final SSQ function value is approximately 3% of the initial value (0.017/0.05). Four iterations were required to find a minimum. The final parameters are P 1 + 2.29, P 2 + 0.932, and P 3 + 1.047. Figs. C6 and C12 show the change in the multipliers at each iteration. The BIP multipliers converged to a value of approximately 2.3, resulting in C 1 through C 7) BIP’s ranging from 0.07 to 0.22. These BIP values are reasonable for the PR EOS. C 7) critical temperatures for fractions F 1 through F 3 decreased to approximately 7% less than the initial values. C 7) critical temperatures for fractions F 4 and F 5 increased, fluctuating from 2 to 10% above the initial values, finally converging to a 5% increase. Figs. C13 through C16 show calculated results for the CVD experiment. This regression gives an excellent match of almost all measured PVT data, including the data used in the regression and equilibriumgas compositions and properties that were not included in the regression. Dewpoint pressure was overpredicted by 8 psi (0.2%), which is sufficiently close, although a larger weight factor (e.g., 10) would force the calculated dewpoint to match the measured value almost exactly. On the other hand, the experimental accuracy of dewpoint pressure is less than 0.2% and further refinement with a EQUATIONOFSTATE APPLICATIONS
Fig. C16—CVD equilibriumgas C7+ molecularweight behavior for gascondensate example comparing measured and EOS Regression II calculations.
larger weight factor is probably not justified. Finally, the measured liquid dropout of 3.3% at 3,515 psia was calculated to be 4.9%, also a very good match. 7
Fig. C17—Variation in the three C7+ criticaltemperature multipliers used in Regression III to match measured gascondensate PVT data.
Fig. C18—CVD liquiddropout behavior for gascondensate example comparing measured and EOS Regression III calculations.
Measured Calculated
Fig. C19—CVD Zfactor behavior for gascondensate example comparing measured and EOS Regression III calculations.
Fig. C20—CVD equilibriumgas C7+ behavior for gascondensate example comparing measured and EOS Regression III calculations.
Regression III. Results almost as good as those for Regression II are achieved by fitting only critical temperatures of the C7+ fractions, namely multipliers to T c(F 1, F 2), T c(F 3, F 4), and T c(F 5), with the final parameters being P 1 + 0.915, P 2 + 1.023, and P 3 + 1.239 (Fig. C17). The converged F SSQ + 0.0026 is 5% of the initial value. The C 1 through C 7) BIP’s are the same as those used in the prediction, ranging from 0.03 to 0.095. Calculated dewpoint is 4,044 psia (0.7%), and liquid dropout at 3,515 psia is 4.9% compared with the measured value of 3.3%. Figs. C18 through C21 compare calculated and measured results for the CVD experiment. Comparing Different Fluid Characterizations. More analysis is needed to determine whether any real difference exists between the fluid characterizations determined in Regressions II and III. For depletion calculations, the results are almost identical. For gas cycling, however, they may provide quite different results. When limited PVT data are available to tune an EOS (as in this example), it usually is good practice to evaluate two or three “equally good” characterizations. As in our example, different modifying parameters might be used. Alternative EOS’s can also be tried [e.g., the SoaveRedlichKwong EOS8 (SRK EOS) with the Pedersen et al.9 fluid characterization as a starting point]. Each fluid characterization can then be evaluated with the results from compositional simulation of the reservoir process being studied. 8
Fig. C21—CVD equilibriumgas C7+ molecularweight behavior for gascondensate example comparing measured and EOS Regression III calculations.
Generating Modified BlackOil PVT Data. Figs. C22 through C24 present modified blackoil PVT properties calculated with the various characterizations discussed earlier. Figs. C22 and C23 give oil properties R s (solution gas/oil ratio) and B o [oil formation PHASE BEHAVIOR
Fig. C22—Modified blackoil PVT property solution gas/oil ratio, Rs , vs. pressure for gascondensate example for three EOS models: dewpointmatch only and Regressions I and II.
Fig. C23—Modified blackoil PVT property saturatedoil FVF, Bo , vs. pressure for gascondensate example for three EOS models: dewpointmatch only and Regressions I and II.
volume factor (FVF)]. Note that these oil properties do not increase monotonically, as is usually exhibited by reservoir oils. The reason is that the first condensate that drops out just below the dewpoint is relatively “heavy” compared with the condensate that drops out at lower pressures. For example, the fluid characterization from Regression II yields a stocktankoil (STO) gravity of 45°API produced from the reservoir condensate at the dewpoint, a 50°API STO produced from the reservoir condensate at 3,515 psia, and a 53°API STO produced from the reservoir condensate at 3,000 psia. The corresponding solution gas/oil ratios at dewpoint, 3,515, and 3,000 psia are 1,500, 1,880, and 2,100 scf/STB, respectively, and oil FVF’s are 1.835, 2.109, and 2.319 bbl/STB, respectively. This behavior is typical for gas condensates with a “tail” on the liquiddropout curve; i.e., the retrograde condensation is small in a pressure interval just below the dewpoint (approximately 500 psi in this example), with the start of a more rapid increase in retrograde condensation occurring at some lower pressure (at approximately 3,500 psia in this example). In the region of the tail retrograde behavior, produced reservoir gas has only slight changes in composition during depletion because only small amounts of the heaviest components are being lost from the original reservoir gas. This should be reflected in the EOS characterization by only slight decrease in C 7) composition. The behavior should also be reflected by modified blackoil PVT property r s (solution oil/gas ratio) of the reservoir gas. Solution oil/gas ratio should decrease only slightly in the region of the taillike retrograde condensation.
ReservoirĆOil Fluid Characterization The second example treats the oil in Chap. 6, Good Oil Co. Well 4. This is a slightly volatile oil with a bubblepoint of 2,600 psi at 220°F, an initial solution gas/oil ratio of 750 scf/STB, and a bubblepoint oil FVF of 1.45 RB/STB. In this example, we look at two EOS characterizations. The first characterization uses the PR EOS with the Søreide2 and Whitson10 methods for developing three C 7) fractions. This approach is basically the same as that used for the gas condensate presented earlier. The second characterization uses the SRK EOS with the Pedersen et al.9 method for characterizing the
Fig. C24—Modified blackoil PVT property solution oil/gas ratio, rs , vs. pressure for gascondensate example for three EOS models: dewpointmatch only and Regressions I and II.
Fig. C25—CVDbased cumulative condensate recovery vs. pressure for gascondensate example for three EOS models: dewpointmatch only and Regressions I and II.
EQUATIONOFSTATE APPLICATIONS
Referring again to the fluid characterization from Regression II, calculated r s decreases only slightly from 136 STB/MMscf at the dewpoint to 122 STB/MMscf at 3,515 psia. Compared with larger decreases in r s at lower pressures (e.g., to 88 STB/MMscf at 3,015 psia), the slight decrease in r s predicted from dewpoint to 3,515 psia appears very reasonable. Calculations from Regressions I and III also show similar r s behavior. Fig. C25 summarizes the effect of treating the taillike retrograde behavior properly with an EOS fluid characterization. The figure plots cumulative stocktank condensate produced during depletion on the basis of modified blackoil PVT data ( r s) generated with the fluid characterizations discussed previously. In particular, the characterization based only on fitting the dewpoint pressure is compared with the the fluid characterizations determined in Regressions I and II. The effect on condensate recovery is clear from the comparison.
9
TABLE C9—COMPOSITIONS OF RESERVOIR OIL AND EQUILIBRIUM GAS AND K VALUES AT 2,600psia BUBBLEPOINT PRESSURE AND 220°F BubblepointOil Composition Component
PR EOS
SRK EOS
EquilibriumGas Composition PR EOS
SRK EOS
K Values at Bubblepoint PR EOS
SRK EOS
N2
0.16
0.16
0.52
0.59
3.28
3.66
CO2
0.91
0.91
1.31
1.43
1.44
1.57
C1
36.47
36.47
77.13
76.97
2.11
2.11
C2
9.67
9.67
10.16
10.57
1.05
1.09
C3
6.95
6.95
4.87
4.95
0.70
0.71
iC4
1.44
1.44
0.77
0.78
0.54
0.54
C4
3.93
3.93
1.85
1.82
0.47
0.46
iC5
1.44
1.44
0.51
0.50
0.36
0.35
C5
1.41
1.41
0.46
0.44
0.33
0.31
C6
4.33
4.33
1.00
0.94
0.23
0.22
F1
15.91
19.07
1.35
0.97
0.085
0.051
F2
14.28
9.31
0.0623
0.0358
0.0044
0.0038
F3
3.11
4.91
0.000050
0.000110
0.000016
0.000022
TABLE C10—PR EOS CHARACTERIZATION OF RESERVOIR OIL WITH SØREIDEWHITSON C7+ METHOD2,3,10 Component
M
Tc (°R)
pc (psia)
w
s+c/b
ăąg*ąă
Tb (°R)
vc (ft3/lbm mol)
Zc
N2
28.01
227.3
493.0
0.0450
*0.1930
0.4700
139.3
1.443
0.2916
CO2
44.01
547.6
1,070.6
0.2310
*0.0820
0.5072
350.4
1.505
0.2742
C1
16.04
343.0
667.8
0.0115
*0.1590
0.3300
201.0
1.590
0.2884
C2
30.07
549.8
707.8
0.0908
*0.1130
0.4500
332.2
2.370
0.2843
C3
44.10
665.7
616.3
0.1454
*0.0860
0.5077
416.0
3.250
0.2804
iC4
58.12
734.7
529.1
0.1756
*0.0840
0.5631
470.6
4.208
0.2824
C4
58.12
765.3
550.7
0.1928
*0.0670
0.5844
490.8
4.080
0.2736
iC5
72.15
828.8
490.4
0.2273
*0.0610
0.6247
541.8
4.899
0.2701
C5
72.15
845.4
488.6
0.2510
*0.0390
0.6310
556.6
4.870
0.2623
C6
86.18
913.4
436.9
0.2957
*0.0080
0.6640
615.4
5.929
0.2643
F1
120.08
1,086.6
397.1
0.3419
0.0403
0.7750
746.2
8.333
0.2838
F2
255.96
1,401.5
230.0
0.6866
0.1255
0.8618
1,070.9
17.562
0.2685
F3
545.00
1,707.3
137.0
1.2213
0.1326
0.9354
1,424.3
28.250
0.2113
*Water+1.
three C 7) fractions. Both EOS characterizations predict the measured PVT data reported in Chap. 6 (Tables 6.2 through 6.7) reasonably well. The characterizations are not modified by regression in this example (possibly an interesting exercise for the reader). The two characterizations are presented first. Calculated results are then compared with measured data reported in Chap. 6. Finally, a study of modified blackoil PVT properties is given. PengRobinson1 Characterization.The methods presented for the gas condensate in the GasCondensateFluid Characterization section (see also Sec. 5.6) were used to develop a fluid characterization for this reservoir oil. Three C 7) fractions, determined with the Gaussian quadrature approach, were used. Table C9 gives mole fractions of the reservoir oil, Table C10 gives component properties, and Table C11 provides BIP’s. Volume translation was used to ensure accurate volumetric predictions. SoaveRedlichKwong Characterization.8 For comparison, the Pedersen et al.9 characterization procedure (Sec. 5.6) was used to develop an EOS description of the same reservoir oil. The split of the C 7) fraction is made by use of an exponential distribution to C 80, then regrouping in subfractions with approximately equal mass 10
fractions. Tables C9, C12, and C13 give the resulting composition and properties. TABLE C11—BIP’s FOR PR EOS CHARACTERIZATION OF RESERVOIR OIL Component
N2
N2
0.000
CO2
C1
CO2
0.000
0.000
C1
0.025
0.105
0.000
C2
0.010
0.130
0.000
C3
0.090
0.125
0.000
iC4
0.095
0.120
0.000
C4
0.095
0.115
0.000
iC5
0.100
0.115
0.000
C5
0.110
0.115
0.000
C6
0.110
0.115
0.000
F1
0.110
0.115
0.035
F2
0.110
0.115
0.063
F3
0.110
0.115
0.092 PHASE BEHAVIOR
TABLE C12—SRK EOS CHARACTERIZATION WITH PEDERSEN et al.9 C7+ METHOD FOR RESERVOIR OIL Tc (°R)
pc (psia)
vc (ft3/lbm mol)
Zc
1.443
0.2916
s+c/b
493.0
0.0450
*0.0080
547.6
1,070.6
0.2310
0.0830
0.5072
350.4
1.505
0.2742
343.0
667.8
0.0115
0.0230
0.3300
201.0
1.590
0.2884
30.07
549.8
707.8
0.0908
0.0600
0.4500
332.2
2.370
0.2843
44.10
665.7
616.3
0.1454
0.0820
0.5077
416.0
3.250
0.2804
iC4
58.12
734.7
529.1
0.1756
0.0830
0.5631
470.6
4.208
0.2824
C4
58.12
765.3
550.7
0.1928
0.0970
0.5844
490.8
4.080
0.2736
iC5
72.15
828.8
490.4
0.2273
0.1020
0.6247
541.8
4.899
0.2701
C5
72.15
845.4
488.6
0.2510
0.1210
0.6310
556.6
4.870
0.2623 0.2643
M
N2
28.01
227.3
CO2
44.01
C1
16.04
C2 C3
ăąg*
Tc (°R)
w
Component
0.4700
139.3
C6
86.18
913.4
436.9
0.2957
0.1470
0.6640
615.4
5.929
F1
133.98
1,079.5
354.8
0.5935
0.1535
0.7899
802.3
9.561
0.2928
F2
258.05
1,307.1
232.5
0.9030
0.1422
0.8577
1,075.7
16.734
0.2774
F3
468.63
1,615.2
199.4
1.2322
*0.0422
0.9247
1,369.0
26.756
0.3078
TABLE C13—BIP’s FOR SRK EOS CHARACTERIZATION OF RESERVOIR OIL Component
N2
N2
0.000
CO2
C1
CO2 C1
0.000
0.000
0.000
0.020
0.150
0.000
C2
0.060
0.150
0.000
C3
0.080
0.150
0.000
iC4
0.080
0.150
0.000
C4
0.080
0.150
0.000
iC5
0.080
0.150
0.000
C5
0.080
0.150
0.000
C6
0.080
0.150
0.000
F1
0.080
0.150
0.000
F2
0.080
0.150
0.000
F3
0.080
0.150
0.000
Analyzing EOS Results. Fig. C26 plots oil density vs. pressure. EOS predictions are accurate at the bubblepoint, somewhat too high at undersaturated conditions, and significantly overpredicted at low pressures. Overall, the predictions are quite good, particularly in the important pressure regions. Fig. C27 shows the differential oil volume factor, B od. The EOS predictions are similar, slightly overpredicting the undersaturated oil compressibility and overpredicting the shrinkage of oil at lower pressures. A useful graphical analysis for undersaturated oil behavior is a loglog plot of oil relative volume, B odńB od, b, vs. the pressure ratio pńp b (Fig. C28). The slope of this plot yields Constant A ( A+*slope), where instantaneous oil compressibility is given by c o + Ańp ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.107)
and cumulative oil compressibility, c o (used in materialbalance equations), is given by A lnǒ p ńp Ǔ . co + p * i p i
. . . . . . . . . . . . . . . . . . . . . . . . . . (C1)
The constant A is a characteristic value for an oil reservoir with constant bubblepoint pressure. With Eqs. 3.107 and C1, the constant allows easy calculation of undersaturated oil compressibility at any reservoir pressure. For reservoirs with bubblepoint variation, A correlates well with bubblepoint pressure (approximately linear and increasing with bubblepoint). For this example, the plot in Fig. C28 indicates that measured data intercept the relative oil volume ratio, V ro + B odńB od,b, at a EQUATIONOFSTATE APPLICATIONS
Fig. C26—DLE oildensity behavior for the reservoiroil example comparing measured and EOS predictive models based on the PR and SRK EOS’s that use the SøreideWhitson method2,3,10 and the Pedersen et al.9 method, respectively, for characterizing C7+ fractions.
value of logǒ pńp bǓ + 0.015, corresponding to pńp b + 1.03. Such an intercept indicates that the reported bubblepoint pressure is approximately 3% too low. Forcing measured data through log(V ro) + 0 should be done only if it results in a linear trend through all the data. For this example, it is very difficult to honor the reported bubblepoint pressure and still have a linear plot that passes through most of the reported oil relative volume data. EOS results confirm that a linear trend with a zero intercept should be expected. The PR EOS has a slope of 0.070, slightly less than the SRK EOS slope of 0.074. The measured data have a slope (with nonzero intercept) of 0.059. Then, resulting oil compressibilities at initial pressure of 5,000 psia are (c o) meas + 0.059ń5, 000 + 11.8
10 *6 psi *1,
(c o) PREOS + 0.070ń5, 000 + 14.0 and (c o) SRKEOS + 0.074ń5, 000 + 14.8
10 *6 psi *1, 10 *6 psi *1.
Cumulative oil compressibilities are given by (c o) meas + ƪ0.059ń(5, 000 * 2, 650)ƫ ln(5, 000 * 2, 650) 11
Measured intercept implies reported pb too low.
Fig. C27—DLE differentialoil FVF behavior for the reservoiroil example comparing measured and EOS predictive models based on the PR/SøreideWhitson method2,3,10 and the SRK/Pedersen et al.9 method.
+ 15.9
10 *6 psi *1,
(c o) PREOS + 18.9 and (c o) SRKEOS + 19.9
10 *6 psi *1, 10 *6 psi *1.
Returning to Fig. C27, the SRK EOS seems to underpredict differential oil volume factors more than the PR EOS. Practically, however, the two characterizations predict nearly identical oil shrinkage. This is seen in Fig. C29, which shows the oil volume ratio, V ro + B odńB od, b, vs. pressure. This ratio gives a true measure of the reservoiroil shrinkage during depletion, whereas the ratio B od is misleadingly related to the “meaningless” residual oil volume. We highly recommend that the ratio B odńB od, b be used as “data” in regression (instead of B od directly) to ensure accurate oil shrinkage from the EOS without also having to fit the residual oil volume at standard conditions. [The residual oil is of no practical interest because it will never be produced to the surface and probably never ex
Fig. C29—DLE oilshrinkage behavior for reservoiroil example comparing measured and EOS predictive models based on the PR/Søreide2 method and the SRK/Pedersen et al.9 method.
12
Fig. C28—DLE undersaturatedoilvolume (compressibility) behavior for reservoiroil example comparing measured and EOS predictive models based on the PR/SøreideWhitson method2,3,10 method and the SRK/Pedersen et al.9 method.
isted in the reservoir. Furthermore, the experimental procedure used in reducing the pressure from the last stage of depletion (approximately 150 psi) to standard pressure and reservoir temperature involves bleeding the system down slowly. This bleeding is a nonequilibrium process that cannot really be simulated with a PVT package (it can be estimated by a series of 5 to 10 additional depletion stages, starting at the lowest reported depletion stage)]. Fig. C30 shows the differential solution gas/oil ratio vs. pressure and indicates that both EOS characterizations overpredict the measured data by 5 to 10%. Correcting this deviation from measured data may lead to unnecessary and severe changes in the EOS characterization. What is really important to predict are (1) the separator flash gas/oil ratio (GOR) and (2) the cumulative gas coming out of solution during depletion. Table C14 shows the calculated and measured separator data. Interestingly, calculated separator gas/oil ratios are 1 to 2% lower than measured data. That is, the differential GOR’s are consider
Fig. C30—DLE solution gas/oil ratio behavior for reservoiroil example comparing measured and EOS predictive models based on the PR/Søreide2 and SRK/Pedersen et al.9 methods.
PHASE BEHAVIOR
TABLE C14—MEASURED AND CALCULATED TWOSTAGE SEPARATOR TEST RESULTS FOR RESERVOIR OIL GOR (scf/STB) Measured Stage 1 (315 psia and 75°F) Stage 2 (14.7 psia and 60°F) Total or at bubblepoint
ąăgg *
Bo (bbl/STB)
gAPI (°API)
549 246 795
0.704 1.286 0.884
1.148 1.007 1.495
40.1
PR EOS Characterization Stage 1 Stage 2 Total or at bubblepoint Percent deviation
559 219 778 *2.1
0.707 1.272 0.866 *2.0
1.129 1.006 1.483 *0.8
40.1 0.0
SRK EOS Characterization Stage 1 Stage 2 Total or at bubblepoint Percent deviation
569 216 785 *1.2
0.712 1.270 0.865 *2.1
1.124 1.006 1.494 *0.1
39.0 *2.6
*Air+1.
ably overpredicted, while the separator gas/oil ratios are only slightly underpredicted. Clearly, the separator gas/oil ratio predictions are accurate enough, satisfying the first requirement given previously. However, the question is how well the EOS characterizations estimate cumulative gas coming out of solution. Fig. C31, which plots ǒR sd,b * R sdǓńB od, b vs. pressure, shows this. The figure indicates that the measured data are somewhat overpredicted by both EOS’s (the two characterizations give very similar results). Although the overprediction is not excessive, these data [ ǒR sd,b * R sdǓńB od, b] could be used in regression (together with oil shrinkage data B odńB od,b) to improve the EOS characterization. Fig. C32 shows the gas specific gravity of equilibrium gas released during the differentialliberation experiment (DLE). The EOS characterizations predict the measured data accurately, with slight underestimation at the two highest pressures. Laboratory gas specific gravities may be difficult to measure accurately because of sampling procedures that can result in loss of liquids during transfer from the PVT cell to the sampler. Such errors would tend to result in specific gravities that are too low, the opposite of what Fig. C32 shows. A problem that may arise in fitting reservoir gas specific gravities with an EOS is the choice and number of components used to describe the C 7) fraction. Often the lightest EOS C 7) fraction consti
Fig. C31—DLE cumulativereleasedgas behavior for reservoiroil example comparing measured and EOS predictive models based on the PR/Søreide2 and SRK/Pedersen et al.9 methods. EQUATIONOFSTATE APPLICATIONS
tutes most of the total C 7) material in the calculated reservoir gas phase (in certain pressure regions). If this component is too heavy or too light compared with the actual C 7) material of the reservoir gas, it will cause the EOScalculated gas specific gravity to be too heavy or too light. For this example, the SoaveRedlichKwong characterization with 12 C 7) fractions gave basically the same gas specific gravities (within 1%) for all pressures down to 200 psia. Gas specific gravity usually is not important in reservoir engineering calculations of oil reservoirs, particularly if gas Z factors are predicted accurately. However, because the equilibriumgas specific gravity indirectly reflects the gas composition (and thus the liquid yield from the reservoir gas), it may be important for gascondensate and volatileoil reservoirs where a significant amount of stocktankliquid production comes from the reservoir gas phase. Fig. C33 shows the equilibriumgasphase Z factor. At pressures just below the bubblepoint the PR EOS predicts the measured data accurately, while the SRK EOS predicts the data somewhat better at intermediate and lower pressures. Neither EOS predicts the general shape of the measured Zfactor curve. As an independent check of the EOS Z factors, the StandingKatz11 correlation (Eq. 3.42) was used with specific gravities from the PR EOS results and with the Sutton12
Fig. C32—DLE released (equilibrium) gas specificgravity behavior for reservoiroil example comparing measured and EOS predictive models based on the PR/Søreide2 and SRK/Pedersen et al.9 methods. 13
Fig. C33—DLE released (equilibrium) gas Zfactor behavior for reservoiroil example comparing measured and EOS predictive models based on the PR/Søreide2 and SRK/Pedersen et al.9 methods; StandingKatz11 Z factors calculated on the basis of PR gas compositions also shown.
Fig. C34—DLE oilviscosity behavior for reservoiroil example comparing measured and LohrenzBrayClark13 (LBC) viscosity model (regressed Vc )/predictive PR EOS/Søreide2 and SRK EOS/Pedersen et al.9 methods.
pseudocritical properties (Eq. 3.47). Fig. C33 presents the StandingKatz Z factors as open circles. The results are closest to the SRK EOS Z factors, which is not surprising. The SRK EOS usually gives better gas volumetric properties than the PR EOS for methanerich systems. Fig. C34 presents the oil viscosities. Measured values are compared with calculated values by use of the LohrenzBrayClark13 correlation, with compositions and densities from EOS results. Experimental oil viscosities are difficult to obtain with an accuracy of more than approximately 5 to 10%, so the results presented here are acceptable. To obtain these calculated results, the critical volumes of C 7) fractions were modified by regressing on measured oil viscosities and reported (calculated) gas viscosities. The default critical volumes were increased 10 to 20% to obtain the match. The modifications to C 7) critical volumes differ for every reservoir system, mainly because the LohrenzBrayClark correlation is strongly dependent on both critical volumes
and oil densities. The modifications are usually less when oil densities are accurately predicted by the EOS. A useful plot for correlating oil viscosities measured at different laboratories is oil viscosity vs. density (Fig. C35). Reservoir oils from the same reservoir should have a unique viscosity/density relationship. (One exception would be if a reservoir exhibited compositional gradients characterized by variation in relative oil paraffinicity/aromaticity.) Because most laboratories measure oil density accurately (i.e., consistently from one laboratory to another), erroneous viscosity data from a laboratory will plot parallel to the reservoir’s correct viscosity/density relation, shifted by a more or less constant amount. Reported gas viscosities, even though they are calculated with a correlation (on the basis of measured specific gravities), should be accurate within 5% or less. Therefore, including gas viscosities in the viscosity regression ensures that critical volumes of the C 7)
Oil Density, lbm/ft3 Fig. C35—DLE oilviscosity vs. density behavior for reservoiroil example comparing measured and EOS predictive models based on PR/Søreide2 and SRK/Pedersen et al.9 methods. 14
Fig. C36—DLE oil and gasviscosity behavior for reservoiroil example comparing measured and LBC viscosity model13 (regressed Vc )/predictive PR EOS/Søreide2 and SRK EOS/Pedersen et al.9 methods.
PHASE BEHAVIOR
Fig. C37—Modified blackoil PVT property solution gas/oil ratio, Rs , for reservoiroil example comparing measured/converted and EOS predictive models based on the PR/Søreide2 and SRK/ Pedersen et al.9 methods.
Fig. C38—Modified blackoil PVT property saturatedoil FVF, Bo , for reservoiroil example comparing measured/converted and EOS predictive models based on the PR/Søreide2 and SRK/ Pedersen et al.9 methods.
fractions are not modified unrealistically (i.e., to the point where gas viscosities are no longer predicted accurately). Fig. C36 shows gas and oil viscosities together for this reservoir system. Generating BlackOil PVT Data. In this section, we consider calculation of modified blackoil PVT properties (Chap. 7). We look at the problems involved in generating consistent blackoil PVT properties for a reservoir with a gas cap in equilibrium with an underlying reservoir oil and try to determine whether blackoil PVT properties are the same for the gas cap and reservoir oil. Several other questions are also raised. How accurate are reservoir phase densities calculated from blackoil PVT data? What surface gravities should be chosen? How do differential data corrected with separator flash data (Eqs. 6.32 and 6.33) compare with results from the WhitsonTorp14 method? And finally, how should modified blackoil PVT data be extrapolated for saturation conditions above the original saturation condition?
Gas Cap and ReservoirOil PVT. The WhitsonTorp method was used to develop modified blackoil PVT for the reservoir oil with the two EOS characterizations presented previously. A DLE was simulated where the equilibrium oil and equilibrium gas from each stage of depletion was passed separately through a twostage separator (300 psia at 75°F and 14.7 psia at 60°F). Figs. C37 through C40 present the results for the reservoir oil for the two characterizations as solid lines. The reservoir was then considered to have a gas cap. The gascap composition was taken from the bubblepoint calculation (Table C9). This gas was depleted by a CVD experiment, where the equilibrium gas and equilibrium oil from each stage of depletion was passed separately through a twostage separator under the same conditions as in the previous paragraph. Figs. C37 through C40 present the results for the two characterizations for the reservoir gas as dashed lines. As Figs. C37 through C39 show, significant differences in modified blackoil PVT data exist for the two characterizations. Signif
Fig. C39—Modified blackoil PVT property solution oil/gas ratio, rs , for reservoiroil example comparing EOS predictive models based on the PR/Søreide2 and SRK/Pedersen et al.9 methods.
Fig. C40—Modified blackoil PVT property dry gas FVF, Bgd , for reservoiroil example comparing EOS predictive models based on the PR/Søreide2 and SRK/Pedersen et al.9 methods.
EQUATIONOFSTATE APPLICATIONS
15
icant differences are also seen between the PVT properties generated from the reservoir oil and the reservoir gas. The difference in PVT properties calculated for the two EOS characterizations seems considerably larger than the differences in predictions of measured PVT data. This is because the differences in blackoil PVT data lie mainly in the gasphase properties, which are not welldefined by the experimental PVT data. Comparison of equilibriumgas compositions (Table C9) supports the differences seen in Figs. C37 through C39. Table C9 shows that more C 7) material is predicted by the PR EOS for the bubblepoint equilibrium gas. The significant differences in blackoil PVT properties calculated from the reservoir oil and gas cause a real dilemma. First, most reservoir simulators require that saturated R s and B o data increase monotonically with pressure. From Figs. C37 and C38, we see that only the reservoiroil PVT data satisfy this requirement. This leads to the question of how use of the reservoiroil PVT properties in the gas cap would affect reservoir performance? The answer can only be found by comparing blackoil with compositional simulations. Another concern is choosing the surface oil and gas gravities. These gravities are used together with the pressuredependent blackoil properties to calculate reservoir phase densities (Eq. 7.6). Typically only one oil gravity and one gas gravity can be provided to a reservoir simulator. If phase densities are important, then care must be taken to choose the surface gravities that give the best reservoir phase densities, particularly in the range of pressures most important to the reservoir recovery mechanisms. For this example, the oil specific gravities range from 0.715 (from the reservoir gas) to 0.825 (from the reservoir oil) and the gas specific gravities range from 0.88 to 0.91. Figs. C37 and C38 show the blackoil properties B o and R s calculated with Eqs. 6.32 and 6.33 on the basis of conversion of differentialliberation data by use of separator test results. For this particular oil, the traditional conversions are not bad, somewhat overpredicting B o and R s. For more volatile oils, the difference can be much more significant. For a reservoir system that is initially undersaturated, the fluid can become saturated at a pressure greater than the initial saturation condition. For example, the reservoir oil in this example might produce at a low flowing bottomhole pressure that results in a high gas saturation near the wellbore. During a shutin period, the pressure increase near the wellbore will saturate the free gas developed during production. If the R s vs. pressure curve increases only to the initial bubblepoint and remains constant at higher pressures, the gas would stop dissolving in the oil at the initial bubblepoint. To ensure that free gas continues to dissolve into the oil at higher pressures, the R s curve must be extrapolated to higher pressures. One approach to developing an extension to the R s curve is to add a small amount of equilibrium gas (evolved at the original bubblepoint pressure) to the original oil. A new bubblepoint is determined for the new mixture. The separator gas/oil ratio is also determined, thereby providing a new point on the R s vs. (bubblepoint) pressure curve. This procedure can be continued at increasing bubblepoint pressures until the initial reservoir pressure is reached. Alternatively, equilibrium gas from each new bubblepoint can be used to generate the next mixture. This approach is often used to estimate the PVT properties for a reservoir that exhibits bubblepoint variation with depth. Two problems may arise when generating an extension of the R s curve. Either a maximum in bubblepoint pressure may be reached that is less than the initial reservoir pressure or a dewpoint instead of a bubblepoint may be calculated, indicating that the procedure has passed through a critical condition. In either of these situations, completing the extension of the R s to the initial pressure is not possible. If successful, this method generates an extension to the original R s curve that may become flat or even exhibit a decreasing slope at higher pressures. Immiscible gas injection into an undersaturated oil reservoir defines a second situation that requires an extrapolated R s curve. The development of the extrapolated R s for this situation is somewhat different. Here, the injection gas should be used to determine mixtures with increasing bubblepoints and GOR’s. A swelling test with the injection gas can be simulated to obtain the necessary mixtures for extending the R s curve. 16
The same two problems that can occur with the equilibriumgas procedure also can occur with this method. Namely, that a maximum can be reached below the initial pressure and that transition through a critical mixture can result in a dewpoint condition. A richer injection gas tends to cause both problems, whereas leaner injection gas may avoid the problems (depending somewhat on the degree of undersaturation). Extension of the R s curve with this method usually results in a relatively steep increase in R s at increasing pressures. Leaner injection gas results in steeper curves. Caution should be used in modeling a gasinjection process with modified blackoil PVT properties, particularly when significant phase behavior effects are expected (e.g., vaporization and swelling). The Cook et al.15 method (Chap. 7) for modifying blackoil PVT properties for vaporizing immiscible gas injection processes and compositional simulation are alternatives that can be considered. References 1. Peng, D.Y. and Robinson, D.B.: “A NewConstant Equation of State,” Ind. & Eng. Chem. (1976) 15, No. 1, 59. 2. Søreide, I.: “Improved Phase Behavior Predictions of Petroleum Reservoir Fluids From a Cubic Equation of State,” Dr.Ing. dissertation, Norwegian Inst. of Technology, Trondheim, Norway (1989). 3. Whitson, C.H., Andersen, T.F., and Søreide, I.: “C7) Characterization of Related Equilibrium Fluids Using the Gamma Distribution,” C7 ) Fraction Characterization, L.G. Chorn and G.A. Mansoori (eds.), Advances in Thermodynamics, Taylor & Francis, New York City (1989) 1, 35–56. 4. Twu, C.H.: “An Internally Consistent Correlation for Predicting the Critical Properties and Molecular Weights of Petroleum and CoalTar Liquids,” Fluid Phase Equilibria (1984) No. 16, 137. 5. Lee, B.I. and Kesler, M.G.: “A Generalized Thermodynamic Correlation Based on ThreeParameter Corresponding States,” AIChE J. (1975) 21, 510. 6. Kesler, M.G. and Lee, B.I.: “Improve Predictions of Enthalpy of Fractions,” Hydro. Proc. (March 1976) 55, 153. 7. Chueh, P.L. and Prausnitz, J.M.: “Calculation of HighPressure Vapor– Liquid Equilibria,” Ind. Eng. Chem. (1968) 60, No. 13. 8. Soave, G.: “Equilibrium Constants from a Modified RedlichKwong Equation of State,” Chem. Eng. Sci. (1972) 27, No. 6, 1197. 9. Pedersen, K.S., Thomassen, P., and Fredenslund, A.: “Characterization of Gas Condensate Mixtures,” C7) Fraction Characterization, L.G. Chorn and G.A. Mansoori (eds.), Advances in Thermodynamics, Taylor & Francis, New York City (1989) 1. 10. Whitson, C.H.: “Characterizing Hydrocarbon Plus Fractions,” SPEJ (August 1983) 683; Trans., AIME, 275. 11. Standing, M.B. and Katz, D.L.: “Density of Natural Gases,” Trans., AIME (1942) 146, 140. 12. Sutton, R.P.: “Compressibility Factors for HighMolecular Weight Reservoir Gases,” paper SPE 14265 presented at the 1985 SPE Annual Technical Conference and Exhibition, Las Vegas, Nevada, 22–25 September. 13. Lohrenz, J., Bray, B.G., and Clark, C.R.: “Calculating Viscosities of Reservoir Fluids From Their Compositions,” JPT (October 1964) 1171; Trans., AIME, 231. 14. Whitson, C.H. and Torp, S.B.: “Evaluating Constant Volume Depletion Data,” JPT (March 1983) 610; Trans., AIME, 275. 15. Cook, R.E., Jacoby, R.H., and Ramesh, A.B.: “A BetaType Reservoir Simulator for Approximating Compositional Effects During Gas Injection,” SPEJ (October 1974) 471.
SI Metric Conversion Factors °API bbl ft3 °F lbm mol psi psi*1 °R
141.5/(131.5)°API) +g/cm3 1.589 873 E*01 +m3 2.831 685 E*02 +m3 (°F*32)/1.8 +°C 4.535 924 E*01 +kmol 6.894 757 E)00 +kPa 1.450 377 E*01 +kPa–1 5/9 +K PHASE BEHAVIOR
Appendix D
Understanding Laboratory Oil PVT Reports M.B. Standing
Introduction The subject of how to read and make proper use of information contained in laboratory pressure/volume/temperature (PVT) reports has not been treated adequately in course texts. This is borne out by comments of students in a basic petroleum engineering course, who find the subject one of the most difficult to understand in the whole course. I hope the following discussion of the why and wherefore of a typical PVT report will be helpful. The discussion pertains to Report RFL 10641 on the Raleigh field contained in this section. Sample pages of this report are given at the end of this appendix. Purpose of the Report First, the form of data presentation in the report developed because of its use in materialbalance calculations. Some of the tabular information is set up to satisfy that need. Second, the report should cover all past, present, and future situations that might require calculations. To do this with a minimum of tables and curves, the data are normalized to a reference state and only data for the reference state are given. The petroleum engineer must then “work back” from the reference state to a particular situation. Third, the laboratory tests are carried out on the basis of two different thermodynamic processes being under way at the same time. These are (1) flash equilibrium separation of gas and oil in the surface traps during production and (2) differential equilibrium separation of gas and oil in the reservoir during pressure decline. As a consequence, the report gives both flash and differential data and it becomes necessary to be able to shift between the two sets of data. Finally, the report gives data on the particular sample obtained. This may not be the proper “average” of all the fluid in the reservoir, and slight adjustment of the data may be necessary at a later time. Therefore, some detail is given to the manner of obtaining the sample and the conditions that exist at the sampling time. Also, the compositional analysis of the sample is given so that equilibrium calculations can be made for conditions other than those studied in the laboratory. With these generalities in mind, we now consider specific data presented in the Raleigh report. The surface flash separation data are considered first, followed by the reservoir differential data. We then consider how to convert certain differential data to equivalent flash data. Page numbers refer to the pages of report. UNDERSTANDING LABORATORY OIL PVT REPORTS
Separator Tests of Reservoir Fluid Sample (Report Page 5) As we show later, one form of the materialbalance equation is an equality between the expansion of the original reservoir oil (between the initial pressure and any subsequent pressure) and the volume voidage that has occurred down to the subsequent pressure. The separator test data on Page 5 of the report, which shows the quantity of surface gases and stocktank oil (STO) that results when 1 bbl of bubblepoint oil is flashed through certain surface trap sequence, allows computation of voidages. The tabulation also gives the oil gravity (°API) of the STO and, in some instances, the gravity of gas coming from the primary trap. Cols. 1 and 2 give the pressure/temperature condition of the surface trap tests that were investigated. These should be specified by the reservoir engineer at the time the test is planned so that they will apply to future field operations. Referring to the bottom line of data, the surface situation modeled here is a twostage separation [i.e., a primary trap operating at 200 psig and 73°F, followed by a stock tank operating at 14.7 psia (0 psig) and 73°F]. When 1 bbl of bubblepoint oil (defined in Footnote 2 as oil saturated at 3,236 psig and 258°F) is flashed (processed) through this trap arrangement, the STO amounts to 0.5974 bbl and has a quality of 48.5°API (Cols. 6 and 5). The formation volume factor (FVF) of the bubblepoint oil, B ob, is therefore 1/0.5974+1.674 bbl/bbl STO (Col. 7). Cols. 3 and 4 show the surface gas/oil ratio from the trap and tank. The primary trap ratio is 875 ft3/bbl STO, and the tank vapors amount to 134 ft3/bbl STO. The solution gas/oil ratio at bubblepoint conditions (3,236 psig and 258°F), R sb , is 875)134+1,009 ft3/bbl STO when flashed through this surface trap arrangement. As this table shows, R sb, B ob, and oil gravity all vary with the trap pressure/ temperature situation. Surfacegas gravity does also, but usually is reported only for the singlestage atmosphere flash. To calculate reservoir voidage properly, the measured STO and the produced gas have to be handled according to the information in this table. However, note that these data always refer only to the bubblepoint oil as the reference fluid. Determination of FVF’s for other reservoir fluids requires additional information. 1
FVF, volume/residual volume
Fig. D1—FVF for oil and gas and for total (oil plus gas) system as a function of pressure above and below bubblepoint.
Fluid Properties at Pressures Lower Than Bubblepoint Pressure We now consider situations at reservoir pressures greater than the bubblepoint pressure. We first look at the FVF, then at the fluid density, and then the compressibility of the fluid. Cols. 1 and 2 of the Reservoir Fluid Sample Tabular Data on Page 3 give the pressure/volume relations of the original fluid at 258°F. Note that the data are presented in terms of a unit volume at the bubblepoint condition. Col. 2 gives the volume of the system at pressure p per unit system volume at 3,236 psig and 258°F. These are listed as relative volumes (i.e., relative to the bubblepoint). Consider the FVF of the original oil in the reservoir. On Page 1, we see that the original pressure (listed as last reservoir pressure under well characteristics) was 5,783 psig at *12,650 ft. Thus, if we want the oil FVF at 5,783 psig, we obtain it by multiplying the FVF at the bubblepoint by the relative volume (to the bubblepoint), V rel + V RńV ob . We multiply because Bo +
V V VR + ob R Vo V o V ob
. . . . . . . . . . . . . . . . . . . . . . . . . . (D1)
and the reference bubblepoint oil volume cancel out. Therefore, B oi, the initial FVF, is 1.674 0.9424+1.577 when the 200psig primary trap is involved. It will be different if another trap pressure is used. Reservoir oil density at pressures greater than 3,236 psig also make use of the relativevolume data of Col. 2, Page 3. The added information is the density of the bubblepoint oil. This is always given in the summary data on Page 2 of the report. We see here that the specific volume at the bubblepoint v^ ob + 0.02772 ft3/lbm. This comes from direct weight/volume measurements on the sample in the PVT cell. If now we wish the density, ò oi, of the initial reservoir oil, we have ò o + ^1 + ^1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . (D2) v oi v ob V rel and ò oi +
1 0.02772
0.9424
+ 38.3 lbmńft 3 .
. . . . . . . (D3)
Compressibility of reservoir oil at pressures higher than the bubblepoint is also obtainable from the relativevolume data. Recall that the definition of compressibility is
ǒ Ǔ.
c o + 1 dV V dp
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (D4)
T
It makes no difference whether the volume units in the equation are relative volumes to the bubblepoint, to FVF’s, or to specific volume values. To evaluate c o at pressure p, it is only necessary to differentiate the p * V rel data in Cols. 1 and 2 graphically to get dV/d p at the 2
pressure and divide by V rel . A less accurate value can be obtained by the assumption
ǒ Ǔ
DV rel co + 1 V rel Dp
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (D5) T
For example, to get c o at 4,500 psig by use of relativevolume values of 500 psi on each side of 4,500 psig co + +
(0.9562 * 0.9781) 1 (0.9562 ) 0.9781)ń2 (5, 000 * 4, 000) 0.0219 + 22.7 1 0.9671 1, 000
10 *6 volńvolpsi.
Note that Page 2 of the report lists some compressibility numbers. These are not the same as those indicated earlier because they are changes in volume (in the pressure interval indicated) per unit volume at the lower pressure. For example, the value of 22.33 10*6 for the 5,000 to 4,000psi interval is obtained as (0.9562 * 0.9781) 1 + 22.39 0.9781 (5, 000 * 4, 000)
10 *6.
The compressibility data on Page 2 are set up in this manner because of the way they are used in one form of material balance. Total FVF of Original Oil at Less Than Bubblepoint Pressure We have seen that, to calculate the FVF of the oil at pressures higher than bubblepoint, we multiply the bubblepoint FVF times the relative volume given in Col. 2, Page 3. Obviously, if we multiply B ob by V rel at pressures less than p b, we also get an FVF. In fact, we get the total FVF, B t, of the original system. That is, at p t p b , we will have two phases and B t is the volume relation of both gas and liquid phases in equilibrium at pressure p (Fig. D1). We mentioned earlier that one form of the material balance makes use of the expansion of the original oil between the initial system pressure and any subsequent pressure. This expansion is given by E o + N(B t * B oi) ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (D6)
where N+initial stocktank barrels in the reservoir and (B t * B oi) +the expansion per unit STO; therefore, E o +expansion (in barrels) of the original oil system. Sometimes the expansion equation is written E o + N(B t * B ti). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (D7) At p u p b , whether the FVF is considered to be total FVF or an oil FVF makes no difference, it is the same thing. For example, see Eq. 4.4 of Ref. 2 or Eq. 8.17 of Ref. 3. PHASE BEHAVIOR
Initial bubblepoint oil
Fig. D2—PVT cell volume vs. pressure during differential vaporization test (showing oil shrinkage) and incremental liberated gas volumes (bȀ–b and cȀ–c) at pressures below bubblepoint.
DifferentialĆLiberation Tests Up to this point, we have considered what happens when reservoir fluid comes to the surface and is separated into surface gas and oil products. We modeled flash equilibrium conditions because we believe that the action going on in the trap is essentially one where the whole system entering the trap immediately separates into two components, trap gas and trap liquid. This constitutes the elements of a flash separation. The standard PVT report includes data referred to as “differential data.” These are gassolubility and phasevolume data taken in a manner to model what some people believe happens to the oil phase in the reservoir during pressure decline. The argument that differentialliberation tests model the subsurface behavior comes primarily from two things. 1. Reservoir pressure changes are not as violent or as large as the pressure changes that occur when entering surface traps. The subsurface changes are more gradual and might be considered to be a series of infinitesimal changes. 2. Because of the relative permeability characteristics of reservoir rock/fluid systems, the gas phase moves toward the well at a faster rate than the liquid phase. As a result, the overall composition of the entire reservoir system is changing. These two ideas promote the idea that a test procedure modeled on a differential process should be used to study subsurface behavior. Because of experimental limitations and time/cost considerations, a laboratory cannot perform a true differential procedure. Instead, it performs a series of stepwise flashes at the reservoir temperature (usually about 10) beginning at the bubblepoint. Of course, the greater the number of steps, the more closely the true differential process is modeled. The differential data are reported in the last three columns of Page 3. Supplementary differentialrelease data are given on Page 4. Note that the three columns are headed “Differential Liberation at 258°F.” The best way to understand these data is to explain how the values are obtained. The laboratory starts with a known volume of the original system in the PVT cell, which may be of the order of 100 to 200 cm3. The volume at the bubblepoint pressure (3,236 psig in this instance) is determined accurately because it is a reference for all subsequent measurements. Referring to Page 3, we see that the first pressure step was to 2,938 psig. At this pressure, the original system will be in two phases. Its volume would be at bȀ on Fig. D2. The first step in altering the overall system composition is made at 2,938 psig by removing the gas phase from the PVT cell while maintaining constant pressure. The quantity of gas removed is determined by collecting it in a calibrated container. The volume that the gas phase occupied in the cell is deUNDERSTANDING LABORATORY OIL PVT REPORTS
termined by the amount of mercury injected during the removal process. Also, the gas gravity is measured on the sample bleedoff. The volume of liquid remaining in the cell is at Point b on Fig. D2. This procedure is repeated by taking the 2,938psig saturated liquid to 2,607 psig (Point cȀ) and removing a second batch of gas at that pressure. Again the volume of the displaced gas in the cell at 2,607 psig is determined along with the gravity of the removal gas. The volume of liquid phase remaining after the second gasremoval step is illustrated by Point c in Fig. D2. This process of removing batches of equilibrium gas continues until the cell pressure at the last displacement is 0 psig. The differential data on Page 3 show 11 equilibrium removals, all at 258°F. The final volume of liquid phase remaining in the cell at 0 psig and 258°F is corrected by thermalexpansion tables (or by cooling the cell) to 0 psig and 60°F. This 0psig/60°F liquid is called residual oil. Note that residual oil and STO are not the same thing. They are both products of the original oil in the system but are developed by different pressure/temperature routes. Once residual oil has been reached, the data obtained are recalculated and presented on the basis of a unit barrel of residual oil. The cumulative amount of gas removed from the cell (liberated from solution) at each pressure step is given as a gas/oil ratio(GOR). Col. 4 shows that 183 ft3/bbl residual oil was liberated between 3,236 and 2,938 psig and 362 ft3/bbl residual oil was liberated between 2,938 and 2,607 psig. By the time 0 psig and 258°F had been reached, the original system had liberated 1,518 ft3/bbl residual oil. Col. 5 shows the amount of gas in solution at the various pressures. This is the difference of the 1,518 ft3 total liberated and the amount liberated between the original bubblepoint pressure and that pressure. For example, the solution gas/oil ratio at 2,938 psig is 1,518*183+1,335 ft3/bbl residual oil. At this point, be sure that you understand why the solution gas/oil ratio determined from surface flash and from differential removal will be different. It is because the processes for obtaining residual oil and STO from bubblepoint oil are different. The first is a multiple series of flashes at the elevated reservoir temperature, and the second is generally a one– or twostage flash at low pressures and low temperature. The quantity of gas released will be different, and the quantity of final liquid will be different. Also, the quality (gravity) of the products will be different (compare °API of residual oil with °API of STO). The only thing that will be the same for the two processes is the total weight of the end products. Col. 6 gives the relative volumes of the liquid phase measured during the differential liberation of gas. Note that these are volumes at pressure p per unit volume of residual oil. Again, these relative volumes must not be confused with FVF’s because FVF’s are specified per barrel of STO. Note on Page 3 that relative volumes start at 1.000 3
Fig. D3—Differential vaporization and flashcorrected solution gas/oil ratio vs. pressure above and below bubblepoint pressure.
at 0 psig/60°F and that the value of 1.109 at 0 psig/258°F is the thermal expansion of 42.2°API residual oil from 60 to 258°F. At pressures higher than 3,236 psig (the original bubblepoint), the system composition remained constant. Therefore, the relation of the relative oil volume at pupb to the bubblepoint value, 2.075, must be the same as the relativevolume numbers in Col. 2 (e.g., 1.948/2.075+0.9387 at 6,000 psig). The data on Page 4 are differential liberation data that refer to the oil and gas phases in the reservoir at 258°F. Col. 2 shows that the gravity of the 183 ft3/bbl residual oil liberated between 3,236 and 2,938 psig was 0.870. The next batch between 2,938 and 2,607 psig (362*183+179 ft3/bbl residual oil) was 0.846. The gas deviation (compressibility) factor of the first liberated gas was 0.886 at 2,938 psig. The oil density at 2,938 psig/258°F was 0.5905 g/cm3. Once you understand the basic difference between flash and differential data as given in the standard PVT report, proceed to calculation of flash solubilities and oil FVF’s at less than the bubblepoint from the differential data. Calculation of Flash Solubility From Differential Solubilities The laboratory report requires calculation of flash solubility data rather than providing it because the laboratory does not know what trap pressures will be used in the field during its producing life. Instead, the laboratory concentrates on providing sufficient data to handle any normal situation by simple data conversions. First, consider the solubility data we have. 1. Differential solubility data at the bubblepoint state (3,236 psig/258°F) and at 11 pressures less than the bubblepoint pressure. The bubblepoint value is 1,518 ft3/bbl residual oil. All fluids at pressures greater than pb have this amount of gas. 2. Flash solubility of the bubblepoint oil for four different surface trap situations. These vary from 1,206 ft3/bbl STO for a single flash to atmospheric pressure to 1,009 ft3/bbl STO for a 200psig primarytraptank situation. Fig. D3 shows these. We now wish to determine the “flashconverted” values (i.e., the amount of gas obtained at the surface when a unit of saturated reservoir oil at less than 3,236 psig is flashed through a surface trap setup). To illustrate, we use the 200psigprimary/0psigtank situation at the reservoir pressure of 2,301 psig. Looking at the differentialliberation data in Col. 4, Page 3, we see that 506 ft3 of gas has come out of solution per barrel of residual oil when the pressure declined from 3,236 to 2,301 psig. In other words, we can say that the 2,301psig saturated oil contains less gas by this amount. If this liquid were taken to the surface and processed through the traps, it would also show somewhat less gas solubility than the 1,009 ft3/bbl STO that the bubblepoint oil shows; however, it would not be 506 ft3 less because we are on a different oil base. 4
If we let (DR s) diff be the liberated gas/oil ratio by differential vaporization, (DR s) diff + (R sb) diff * ( R s) diff , we can convert this to a (DR s) flash as follows. Vg V g V or + , V or V ob V ob V g V ob Vg + , V ob V o Vo Vg + (DR s) flash , Vo V or + 1 , V ob 2.075 and
V ob + 1.674 RBńSTB , Vo
. . . . . . . . . . . . . . . . . . . . . . (D8)
where V g is in cubic feet and V o , V ob, and V or ar