Theory of Gas Injection Processes Franklin M Orr, Jr Stanford University Stanford, California 2005 Library of Congress Cataloging-in-Publication Data Orr, Franklin M., Jr Theory of Gas Injection Processes / Franklin M Orr, Jr Bibliography: p Includes index ISBN xxxxxxxxxxx Enhanced recovery of oil I Title XXXXX XXXXX c 2005 Franklin M Orr, Jr All rights reserved No part of this book may be reproduced, in any form or by an means, without permission in writing from the author To Susan i Preface This book is intended for graduate students, researchers, and reservoir engineers who want to understand the mathematical description of the chromatographic mechanisms that are the basis for gas injection processes for enhanced oil recovery Readers familiar with the calculus of partial derivatives and properties of matrices (including eigenvalues and eigenvectors) should have no trouble following the mathematical development of the material presented The emphasis here is on the understanding of physical mechanisms, and hence the primary audience for this book will be engineers Nevertheless, the mathematical approach used, the method of characteristics, is an essential part of the understanding of those physical mechanisms, and therefore some effort is expended to illuminate the mathematical structure of the flow problems considered In addition, I hope some of the material will be of interest to mathematicians who will find that many interesting questions of mathematical rigor remain to be investigated for multicomponent, multiphase flow in porous media Readers already familiar with the subject of this book will recognize the work of many students and colleagues with whom I have been privileged to work in the last twenty-five years I am much indebted to Fred Helfferich (now at the Pennsylvania State University) and George Hirasaki (now at Rice University), working then (in the middle 1970’s) at Shell Development Company’s Bellaire Research Center They originated much of the theory developed here and introduced me to the ideas of multicomponent, multiphase chromatography when I was a brand new research engineer at that laboratory Gary Pope and Larry Lake were also part of that Shell group of future academics who have made extensive use of the theoretical approach used here in their work with students at the University of Texas I have benefited greatly from many conversations with them over the years about the material discussed here Thormod Johansen patiently explained to me his mathematician’s point of view concerning the Riemann problems considered in detail in this book All of them have contributed substantially to the development of a rigorous description of multiphase, multicomponent flow and to my education about it in particular Thanks are also due to many Stanford students, who listened to and helped me refine the explanations given here in a course taught for graduate students since 1985 Their questions over the years have led to many improvements in the presentation of the important ideas Much of the material in this book that describes flow of gas/oil mixtures follows from the work of an exceptionally talented group of graduate students: Wes Monroe, Kiran Pande, Jeff Wingard, Russ Johns, Birol Dindoruk, Yun Wang, Kristian Jessen, Jichun Zhu, and Pavel Ermakov Wes Monroe obtained the first four-component solutions for dispersion-free flow in one dimension Kiran Pande solved for the interactions of phase behavior, two-phase flow, and viscous crossflow Jeff Wingard considered problems with temperature variation and three-phase flow Russ Johns and Birol Dindoruk greatly extended our understanding of flow of four or more components with and without volume change on mixing Yun Wang extended the theory to systems with an arbitrary number of components, and Kristian Jessen, who visited for six months with our research group during the course of his PhD work at the Danish Technical University, contributed substantially to the development of efficient algorithms for automatic solution of problems with an arbitrary number of components in the oil or injection gas Kristian Jessen and Pavel Ermakov independently worked out the first solutions for arbitrary numbers of components with volume change on mixing Jichun Zhu and Pavel Ermakov contributed substantially to the derivation of compact versions of key proofs Birol Dindoruk, Russ Johns, Yun Wang, and Kristian Jessen kindly allowed me to use example solutions ii and figures from their dissertations This book would have little to say were it not for the work of all those students Marco Thiele and Rob Batycky developed the streamline simulation approach for gas injection processes Their work allows the application of the one-dimensional descriptions of the interactions of flow and phase to model the behavior of multicomponent gas injection processes in three-dimensional, high resolution simulations All those students deserve my special thanks for teaching me much more than I taught them Kristian Jessen deserves special recognition for his contributions to teaching this material with me and to the completion of Chapters and He contributed heavily to the material in those chapters, and he constructed many of the examples I am indebted to Chick Wattenbarger for providing a copy of his “gps” graphics software All of the figures in the book were produced with that software I am also indebted to Martin Blunt at the Centre for Petroleum Studies at Imperial College of Science, Technology and Medicine for providing a quiet place to write during the fall of 2000 and for reading an early draft of the manuscript I thank my colleagues Margot Gerritsen and Khalid Aziz, Stanford University, for their careful readings of the draft manuscript They and the other faculty of the Petroleum Engineering Department at Stanford have provided a wonderful place to try to understand how gas injection processes work The students and faculty associated with the SUPRI-C gas injection research group, particularly Martin Blunt, Margot Gerritsen, Kristian Jessen, Hamdi Tchelepi, and Ruben Juanes, and our dedicated staff, Yolanda Williams and Thuy Nguyen, have done all the useful work in that quest, of course It is my pleasure to report on a part of that research effort here And finally, I thank Mark Walsh for asking questions about the early work that caused us to think about these problems in a whole new way I also thank an anonymous proposal reviewer who said that the problem of finding analytical solutions to multicomponent, two-phase flow problems could not be solved and even if it could, the solutions would be of no use That challenge was too good to pass up The financial support for the graduate students who contributed so much to the material presented here was provided by grants from the U.S Department of Energy, and by the member companies of the Stanford University Petroleum Research Institute Gas Injection Industrial Affiliates program That support is gratefully acknowledged Lynn Orr Stanford, California March, 2005 Contents Preface i Introduction Conservation Equations 2.1 General Conservation Equations 2.2 One-Dimensional Flow 2.3 Pure Convection 2.4 No Volume Change on Mixing 2.5 Classification of Equations 2.6 Initial and Boundary Conditions 2.7 Convection-Dispersion Equation 2.8 Additional Reading 2.9 Exercises 5 10 12 13 14 14 15 17 17 Calculation of Phase Equilibrium 3.1 Thermodynamic Background 3.1.1 Calculation of Thermodynamic Functions 3.1.2 Chemical Potential and Fugacity 3.2 Calculation of Partial Fugacity 3.3 Phase Equilibrium from an Equation of State 3.4 Flash Calculation 3.5 Phase Diagrams 3.5.1 Binary Systems 3.5.2 Ternary Systems 3.5.3 Quaternary Systems 3.5.4 Constant K-Values 3.6 Additional Reading 3.7 Exercises 21 21 22 24 26 27 31 34 34 35 37 38 40 40 Two-Component Gas/Oil Displacement 4.1 Solution by the Method of Characteristics 4.2 Shocks 4.3 Variations in Initial or Injection Composition 4.4 Volume Change 43 44 48 56 61 iii iv CONTENTS 62 62 63 64 67 69 70 71 Ternary Gas/Oil Displacements 5.1 Composition Paths 5.1.1 Eigenvalues and Eigenvectors 5.1.2 Tie-Line Paths 5.1.3 Nontie-Line Paths 5.1.4 Switching Paths 5.2 Shocks 5.2.1 Phase-Change Shocks 5.2.2 Shocks and Rarefactions between Tie Lines 5.2.3 Tie-Line Intersections and Two-Phase Shocks 5.2.4 Entropy Conditions 5.3 Example Solutions: Vaporizing Gas Drives 5.4 Example Solutions: Condensing Gas Drives 5.5 Structure of Ternary Gas/Oil Displacements 5.5.1 Effects of Variations in Initial Composition 5.6 Multicontact Miscibility 5.6.1 Vaporizing Gas Drives 5.6.2 Condensing Gas Drives 5.6.3 Multicontact Miscibility in Ternary Systems 5.7 Volume Change 5.8 Component Recovery 5.9 Summary 5.10 Additional Reading 5.11 Exercises 73 75 78 81 81 87 90 90 92 97 98 99 106 110 117 117 118 119 119 120 127 129 130 131 135 135 135 137 144 149 155 158 161 162 169 4.5 4.6 4.7 4.8 4.4.1 Flow Velocity 4.4.2 Characteristic Equations 4.4.3 Shocks 4.4.4 Example Solution Component Recovery Summary Additional Reading Exercises Four-Component Displacements 6.1 Eigenvalues, Eigenvectors, and Composition Paths 6.1.1 The Eigenvalue Problem 6.1.2 Composition Paths 6.2 Solution Construction for Constant K-values 6.3 Systems with Variable K-values 6.4 Condensing/Vaporizing Gas Drives 6.5 Development of Miscibility 6.5.1 Calculation of Minimum Miscibility Pressure 6.5.2 Effect of Variations in Initial Oil Composition on MMP 6.5.3 Effect of Variations in Injection Gas Composition on MMP CONTENTS 6.6 6.7 6.8 6.9 Volume Change Summary Additional Reading Exercises v 172 176 176 177 Multicomponent Gas/Oil Displacements by F M Orr, Jr and K Jessen 7.1 Key Tie Lines 7.1.1 Injection of a Pure Component 7.1.2 Multicomponent Injection Gas 7.2 Solution Construction 7.2.1 Fully Self-Sharpening Displacements 7.2.2 Solution Routes with Nontie-line Rarefactions 7.3 Solution Construction: Volume Change 7.4 Displacements in Gas Condensate Systems 7.5 Calculation of MMP and MME 7.6 Summary 7.7 Additional Reading 179 179 180 180 183 185 193 198 201 204 206 210 212 213 213 213 215 221 230 237 238 Compositional Simulation by F M Orr, Jr and K Jessen 8.1 Numerical Dispersion 8.2 Comparison of Numerical and Analytical Solutions 8.3 Sensitivity to Numerical Dispersion 8.4 Calculation of MMP and MME 8.5 Summary 8.6 Additional Reading Nomenclature 241 Bibliography 244 Appendix A: Entropy Conditions in Ternary Systems 255 Appendix B: Details of Gas Displacement Solutions 266 Index 280 vi CONTENTS Appendix B 269 Table B.5: Displacement details for Fig 5.17 Composition route, saturation, and composition profiles for a self-sharpening (HVI) condensing gas drive K1 = 2.5, K2 = 1.5, K3 = 0.05, and M = The injection gas has composition, C1 = 0.6, C2 = 0.4, and C3 = 0, and the initial oil has composition, C1 = 0.3, C2 = 0, and C3 = 0.7 Compositions reported are in volume fractions Segment Injection Gas Trailing Shock Injection Tie Line Rarefaction Intermediate Shock Constant State Leading Shock Initial Oil Point e e d d-c d-c d-c d-c d-c c b b a a CH4 0.6000 0.6000 0.4907 0.4769 0.4595 0.4420 0.4246 0.4071 0.4058 0.5916 0.5916 0.3000 0.3000 CO2 0.4000 0.4000 0.3590 0.3538 0.3472 0.3407 0.3341 0.3276 0.3271 0.0000 0.0000 0.0000 0.0000 C10 0.0000 0.0000 0.1503 0.1693 0.1933 0.2173 0.2413 0.2653 0.2671 0.4084 0.4084 0.7000 0.7000 S1 1.0000 1.0000 0.7395 0.7000 0.6500 0.6000 0.5500 0.5000 0.4963 0.3505 0.3505 0 Flow Vel 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 ξ/τ 1.0000 0.2454 0.2454 0.3255 0.4554 0.6247 0.8415 1.1111 1.1334 1.1334 1.4833 1.4833 1.0000 Table B.6: Displacement details for Fig 5.18 A condensing gas drive (LVI) with a nontie-line rarefaction K1 = 2.5, K2 = 0.5, K3 = 0.05, and M = The injection gas has composition, CCH4 = 0.8, CC4 = 0.2, and CC10 = 0., and the initial oil has composition, CCH4 = 0.3, CC4 = 0, and CC10 = 0.7 Compositions reported are in volume fractions Segment Injection Gas Trailing Shock Injection Tie Line Rarefaction Equal Eig Point Nontie-line Rarefaction Constant State Leading Shock Initial Oil Point e e d d-c d-c d-c c c-b c-b c-b c-b b b a a CH4 0.8000 0.8000 0.6543 0.6359 0.6127 0.5894 0.5883 0.5695 0.5579 0.5572 0.5737 0.6009 0.6009 0.3000 0.3000 C4 0.2000 0.2000 0.2661 0.2744 0.2850 0.2956 0.2960 0.2987 0.2816 0.2325 0.1271 0.0000 0.0000 0.0000 0.0000 C10 0.0000 0.0000 0.0796 0.0896 0.1023 0.1151 0.1156 0.1317 0.1605 0.2103 0.2992 0.3991 0.3991 0.7000 0.7000 S1 1.0000 1.0000 0.7395 0.7000 0.6500 0.6000 0.5977 0.5500 0.5000 0.4500 0.4000 0.3665 0.3665 0 Flow Vel 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 ξ/τ 1.0000 0.2454 0.2454 0.3255 0.4554 0.6247 0.6336 0.6429 0.6750 0.7267 0.7930 0.8418 1.5015 1.5015 1.0000 270 Appendix B Table B.7: Displacement details for Initial Oil Composition A in Fig 5.22 The viscosity ratio on the initial tie line is M = 3.115, and on the injection tie line, M = 4.586 The molar volumes used to convert mole fractions to volume fractions were CO2 , 150.978 cm3 /gmol, C4 , 101.886, C10 , 215.013 Phase compositions on the initial tie line (mole fractions): xCO2 = 0.7030, xC4 = 0.0436, xC10 = 0.2534, yCO2 = 0.9566, yC4 = 0.0220, yC10 = 0.0215 Phase compositions on the injection tie line (mole fractions): xCO2 = 0.6554, xC4 = 0., xC10 = 0.3446, yCO2 = 0.9817, yC4 = 0., yC10 = 0.0183 Compositions reported in the table are in volume fractions Segment Injection Gas Trailing Shock Constant State Intermediate Shock Initial Tie Line Rarefaction Leading Shock Initial Oil A Point e e d d d c c-b c-b b a a CO2 1.0000 1.0000 0.9208 0.9208 0.9208 0.8608 0.8552 0.8501 0.8468 0 C4 0 0 0.0301 0.0306 0.0310 0.0313 0.1035 0.1035 C10 0 0.0792 0.0792 0.0792 0.1091 0.1142 0.1189 0.1219 0.8965 0.8965 S1 1.0000 1.0000 0.8131 0.8131 0.8131 0.6223 0.6000 0.5800 0.5669 0 Flow Vel 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 ξ/τ 1.0000 0.1482 0.1482 0.1482 0.8048 0.8048 0.9106 1.0125 1.0825 1.0825 1.0000 Table B.8: Displacement details for Initial Oil Composition B in Fig 5.22 The viscosity ratio on the initial tie line is M = 1.957, and on the injection tie line, M = 4.586 The molar volumes used to convert mole fractions to volume fractions were CO2 , 150.978 cm3 /gmol, C4 , 101.886, C10 , 215.013 Phase compositions on the initial tie line (mole fractions): xCO2 = 0.7636, xC4 = 0.0777, xC10 = 0.1586, yCO2 = 0.9237, yC4 = 0.0478, yC10 = 0.0285 Phase compositions on the injection tie line (mole fractions): xCO2 = 0.6554, xC4 = 0., xC10 = 0.3446, yCO2 = 0.9817, yC4 = 0., yC10 = 0.0183 Compositions reported in the table are in volume fractions Segment Injection Gas Trailing Shock Constant State Intermediate Shock Initial Tie Line Rarefaction Leading Shock Initial Oil B Point e e d d d c c-b c-b c-b b a a CO2 1.0000 1.0000 0.9557 0.9557 0.9557 0.8710 0.8709 0.8693 0.8677 0.8661 0 C4 0 0 0.0577 0.0577 0.0579 0.0583 0.0586 0.2204 0.2204 C10 0 0.0443 0.0443 0.0443 0.0714 0.0714 0.0727 0.0740 0.1753 0.7796 0.7796 S1 1.0000 1.0000 0.9202 0.9202 0.9202 0.6703 0.6700 0.6600 0.6500 0.6398 0 Flow Vel 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 ξ/τ 1.0000 0.4243 0.4243 0.4243 0.8860 0.8860 0.8876 0.9372 0.9880 1.0408 1.0408 1.0000 Appendix B 271 Table B.9: Displacement details for Initial Oil Composition C in Fig 5.22 The viscosity ratio on the initial tie line is M = 1.268, and on the injection tie line, M = 4.586 The molar volumes used to convert mole fractions to volume fractions were CO2 , 150.978 cm3 /gmol, C4 , 101.886, C10 , 215.013 Phase compositions on the initial tie line (mole fractions): xCO2 = 0.8235, xC4 = 0.0874, xC10 = 0.0891, yCO2 = 0.8833, yC4 = 0.0719, yC10 = 0.0285 Phase compositions on the injection tie line (mole fractions): xCO2 = 0.6554, xC4 = 0., xC10 = 0.3446, yCO2 = 0.9817, yC4 = 0., yC10 = 0.0448 Compositions reported in the table are in volume fractions Segment Injection Gas Trailing Shock Constant State Intermediate Shock Initial Tie Line Rarefaction Leading Shock Initial Oil B Point e e d d d c c c-b b a a CO2 1.0000 1.0000 0.9557 0.9557 0.9752 0.8663 0.8663 0.8660 0.8658 0 C4 0 0 0.0763 0.0763 0.0764 0.0764 0.3008 0.3008 C10 0 0.0443 0.0443 0.0248 0.0574 0.0574 0.0576 0.0578 0.6992 0.6992 S1 1.0000 1.0000 0.9799 0.9799 0.9799 0.7162 0.7162 0.7100 0.7074 0 Flow Vel 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 ξ/τ 1.0000 0.7366 0.7366 0.7366 0.9647 0.9647 0.9647 0.9981 1.0120 1.0120 1.0000 272 Appendix B Table B.10: Displacement details for Fig 5.23a Composition routes and saturation and flow velocity profiles for displacement of a C4 /C10 mixture by CO2 at 1000 psia and 160◦ F with volume change The viscosity ratio on the initial tie line is M = 8.319, and on the injection tie line, M = 10.68 The molar volumes used to convert mole fractions to volume fractions were CO2 , 308.685 cm3 /gmol, C4 , 104.509, C10 , 216.278 Phase compositions on the initial tie line (mole fractions): xCO2 = 0.4884, xC4 = 0.1651, xC10 = 0.3464, yCO2 = 0.9454, yC4 = 0.0514, yC10 = 0.0033 Phase compositions on the injection tie line (mole fractions): xCO2 = 0.4976, xC4 = 0., xC10 = 0.5024, yCO2 = 0.9964, yC4 = 0., yC10 = 0.0036 Compositions in the table are reported in mole fractions Segment Injection Gas Trailing Shock Injection Tie Line Rarefaction Constant State Intermediate Shock Initial Tie Line Rarefaction Leading Shock Initial Oil Point f f e d-e d-e d d c c-b c-b c-b b a a CO2 1.0000 1.0000 0.8620 0.8490 0.8104 0.7982 0.7982 0.6670 0.6580 0.6368 0.6175 0.6085 0 C4 0 0 0 0.1207 0.1229 0.1282 0.1330 0.1352 0.2868 0.2868 C10 0 0.1380 0.1510 0.1896 0.2018 0.2018 0.2124 0.2191 0.2350 0.2495 0.2562 0.7132 0.7132 S1 1.0000 1.0000 0.8658 0.8500 0.8000 0.7829 0.7829 0.6197 0.6000 0.5500 0.5000 0.4754 0 Flow Vel 1.0000 1.0000 0.9823 0.9823 0.9823 0.9823 0.9823 1.0054 1.0054 1.0054 1.0054 1.0054 0.6318 0.6318 ξ/τ 1.0000 0.0379 0.0379 0.0447 0.0710 0.0820 0.3535 0.3535 0.4033 0.5560 0.7704 0.8981 0.8981 0.6318 Appendix B 273 Table B.11: Displacement details for Fig 5.23b Composition routes and saturation and flow velocity profiles for displacement of a C4 /C10 mixture by CO2 at 1000 psia and 160◦ F with no volume change The viscosity ratio on the initial tie line is M = 8.319, and on the injection tie line, M = 10.68 The molar volumes used to convert mole fractions to volume fractions were CO2 , 308.685 cm3 /gmol, C4 , 104.509, C10 , 216.278 Phase compositions on the initial tie line (mole fractions): xCO2 = 0.4884, xC4 = 0.1651, xC10 = 0.3464, yCO2 = 0.9454, yC4 = 0.0514, yC10 = 0.0033 Phase compositions on the injection tie line (mole fractions): xCO2 = 0.4976, xC4 = 0., xC10 = 0.5024, yCO2 = 0.9964, yC4 = 0., yC10 = 0.0036 Compositions in the table are reported in mole fractions Segment Injection Gas Trailing Shock Injection Tie Line Rarefaction Constant State Intermediate Shock Initial Tie Line Rarefaction Leading Shock Initial Oil Point f f e d-e d-e d d c c-b c-b c-b c-b b a a CO2 1.0000 1.0000 0.8974 0.8831 0.8721 0.8566 0.8566 0.7208 0.7119 0.7028 0.6937 0.6713 0.6619 0 C4 0 0 0 0.1073 0.1095 0.1118 0.1140 0.1196 0.1219 0.2868 0.2868 C10 0 0.1026 0.1169 0.1279 0.1434 0.1434 0.1719 0.1786 0.1855 0.1923 0.2091 0.2161 0.7132 0.7132 S1 1.0000 1.0000 0.8260 0.8000 0.7800 0.7513 0.7513 0.5593 0.5400 0.5200 0.5000 0.4500 0.4288 0 Flow Vel 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 ξ/τ 1.0000 0.0573 0.0573 0.0723 0.0855 0.1076 0.5242 0.5242 0.5941 0.6753 0.7663 1.0429 1.1833 1.1833 1.0000 274 Appendix B Chapter 6–Four-Component Displacements Table B.12: Displacement details for Fig 6.7 and 6.8 (no volume change) Composition route (in volume fractions) for a displacement of an oil with composition a, C1 = 0, C2 = 0, C3 = 0.491852, and C4 = 0.508148 by gas with composition f, C1 = 0.625, C2 = 0.375, C3 = 0, and C4 = K-values are constant: K1 = 2.5, K2 = 1.5, K3 = 0.5, and K4 = 05 µliq /µvap = Segment Injection Gas Trailing Shock Zone of Constant State Intermediate Shock Crossover Tie Line Rarefaction Intermediate Shock Zone of Constant State Leading Shock Initial Oil Point f f e e e e d d d-c d-c d-c d-c d-c d-c d-c c c b b b b a a C1 0.6250 0.6250 0.5442 0.5442 0.5442 0.5442 0.3695 0.3695 0.3680 0.3650 0.3620 0.3590 0.3560 0.3530 0.3500 0.3489 0.3489 0.4904 0.4904 0.4904 0.4904 0.0000 0.0000 C2 0.3750 0.3750 0.3477 0.3477 0.3477 0.3477 0.2784 0.2784 0.2778 0.2768 0.2757 0.2746 0.2735 0.2724 0.2713 0.2709 0.2709 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 C3 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.2316 0.2316 0.2324 0.2340 0.2356 0.2372 0.2388 0.2404 0.2421 0.2427 0.2427 0.3034 0.3034 0.3034 0.3034 0.4918 0.4918 C4 0.0000 0.0000 0.1081 0.1081 0.1081 0.1081 0.1205 0.1205 0.1218 0.1242 0.1267 0.1292 0.1316 0.1341 0.1366 0.1375 0.1375 0.2062 0.2062 0.2062 0.2062 0.5081 0.5081 S1 1.0000 1.0000 0.8304 0.8304 0.8304 0.8304 0.5651 0.5651 0.5600 0.5500 0.5400 0.5300 0.5200 0.5100 0.5000 0.4963 0.4963 0.3550 0.3550 0.3550 0.3550 0.0000 0.0000 Flow Vel 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 ξ/τ 1.0000 0.2741 0.2454 0.2454 0.7707 0.7707 0.7707 0.7707 0.7940 0.8415 0.8911 0.9429 0.9968 1.0529 1.1111 1.1334 1.1334 1.1334 1.1334 1.2421 1.2421 1.2421 1.0000 Appendix B 275 Table B.13: Displacement details for Fig 6.9 and 6.10 (no volume change) Composition route for a displacement of an oil with composition C1 = 0.1, C2 = 0.1809, C3 = 0.3766, and C4 = 0.3425 by pure CH4 , C1 = K-values are constant: K1 = 2.5, K2 = 1.5, K3 = 0.5, and K4 = 05 The viscosity ratio, µliq /µvap , is Segment Injection Gas Trailing Shock Zone of Constant State Intermediate Shock Nontie-Line Rarefaction Tie-line Rarefaction Leading Shock Initial Oil Point f f e e e e d d d-c d-c d-c d-c d-c d-c d-c d-c d-c d-c d-c d-c d-c d-c d-c d-c d-c c c c-b c-b c-b b b a a CH4 1.0000 1.0000 0.8788 0.8788 0.8788 0.8788 0.6370 0.6370 0.6358 0.6025 0.5719 0.5437 0.5179 0.4942 0.4725 0.4528 0.4349 0.4188 0.4043 0.3915 0.3803 0.3706 0.3625 0.3558 0.3506 0.3472 0.3472 0.3440 0.3410 0.3380 0.3360 0.3360 0.1000 0.1000 C2 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0011 0.0356 0.0674 0.0962 0.1225 0.1462 0.1677 0.1869 0.2040 0.2190 0.2321 0.2432 0.2524 0.2598 0.2652 0.2688 0.2705 0.2703 0.2703 0.2692 0.2681 0.2670 0.2663 0.2663 0.1809 0.1809 C3 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.2471 0.2471 0.2470 0.2451 0.2435 0.2420 0.2408 0.2398 0.2390 0.2384 0.2381 0.2379 0.2379 0.2381 0.2384 0.2391 0.2399 0.2409 0.2422 0.2436 0.2436 0.2453 0.2469 0.2485 0.2496 0.2496 0.3766 0.3766 C4 0.0000 0.0000 0.1212 0.1212 0.1212 0.1212 0.1160 0.1160 0.1160 0.1166 0.1173 0.1180 0.1188 0.1198 0.1208 0.1218 0.1230 0.1243 0.1257 0.1272 0.1288 0.1305 0.1324 0.1345 0.1367 0.1389 0.1389 0.1415 0.1440 0.1465 0.1481 0.1481 0.3425 0.3425 S1 1.0000 1.0000 0.8442 0.8442 0.8442 0.8442 0.6603 0.6603 0.6600 0.6500 0.6400 0.6300 0.6200 0.6100 0.6000 0.5900 0.5800 0.5700 0.5600 0.5500 0.5400 0.5300 0.5200 0.5100 0.5000 0.4908 0.4908 0.4800 0.4700 0.4600 0.4533 0.4533 0.0000 0.0000 Flow Vel 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 ξ/τ 1.0000 0.2850 0.2850 0.2850 0.7840 0.7840 1.1088 1.1088 1.1090 1.1140 1.1191 1.1241 1.1290 1.1337 1.1382 1.1426 1.1467 1.1506 1.1541 1.1573 1.1600 1.1624 1.1643 1.1657 1.1665 1.1668 1.1668 1.2337 1.2978 1.3638 1.4085 1.4085 1.4085 1.0000 276 Appendix B Chapter 7–Multicomponent Displacements Table B.14: Displacement details for Fig 7.5 (no volume change) Composition route for a displacement of an oil with composition CO2 = 0.005, CH4 = 0.350, C4 = 0.250, C10 = 0.195, C16 = 0.125, and C20 = 0.075 by pure CO2 Phase behavior calculated with the Peng-Robinson EOS (see Tables 3.1 and 3.2) Segment Injection Gas Trailing Shock Zone of Constant State Intermediate Shock Zone of Constant State Intermediate Shock Zone of Constant State Intermediate Shock Tie-Line Rarefaction Intermediate Shock Zone of Constant State Leading Shock Initial Oil Point h h g g g g f f f f e e e e d d c c b b b b a a CO2 1.0000 1.0000 0.9804 0.9804 0.9804 0.9804 0.9615 0.9615 0.9615 0.9615 0.9224 0.9224 0.9224 0.9224 0.8011 0.8011 0.7789 0.7789 0.0055 0.0055 0.0055 0.0055 0.0050 0.0050 CH4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.4399 0.4399 0.4399 0.4399 0.3500 0.3500 C4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1048 0.1048 0.1119 0.1119 0.2211 0.2211 0.2211 0.2211 0.2500 0.2500 C10 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0398 0.0398 0.0398 0.0398 0.0481 0.0481 0.0554 0.0554 0.1648 0.1648 0.1648 0.1648 0.1950 0.1950 C16 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0246 0.0246 0.0246 0.0246 0.0236 0.0236 0.0236 0.0236 0.0289 0.0289 0.0336 0.0336 0.1055 0.1055 0.1055 0.1055 0.1250 0.1250 C20 0.0000 0.0000 0.0196 0.0196 0.0196 0.0196 0.0139 0.0139 0.0139 0.0139 0.0141 0.0141 0.0141 0.0141 0.0172 0.0172 0.0200 0.0200 0.0633 0.0633 0.0633 0.0633 0.0750 0.0750 S1 1.0000 1.0000 0.9148 0.9148 0.9148 0.9148 0.8548 0.8548 0.8548 0.8548 0.7476 0.7476 0.7476 0.7476 0.5074 0.5074 0.4292 0.4292 0.1359 0.1359 0.1359 0.1359 0.0000 0.0000 Flow Vel 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 ξ/τ 1.0000 0.0091 0.0091 0.0091 0.0188 0.0188 0.0188 0.0188 0.1083 0.1083 0.1083 0.1083 0.6755 0.6755 0.6755 0.6755 1.2046 1.2046 1.2046 1.2046 2.0237 2.0237 2.0237 1.0000 Appendix B 277 Table B.15: Displacement details for Fig 7.6 (no volume change) Composition route for a displacement of an oil B in Table 7.2 by Gas C in Table 7.2 Phase behavior calculated with the Peng-Robinson EOS (see Tables 3.1 and 3.2) Volume fractions reported are those for the components shown in Fig 7.6 Segment Injection Gas Trailing Shock Shock 14 Shock 13 Shock 12 Shock 11 Shock 10 Shock Shock Shock Shock Shock Shock Shock Shock Shock Initial Oil Point 14 14 13 13 12 12 11 11 10 10 9 8 7 6 5 4 3 2 1 CO2 0.0240 0.0240 0.0238 0.0238 0.0237 0.0237 0.0235 0.0235 0.0232 0.0232 0.0230 0.0230 0.0229 0.0229 0.0229 0.0228 0.0228 0.0228 0.0226 0.0226 0.0226 0.0226 0.0228 0.0228 0.0233 0.0233 0.0160 0.0160 0.0160 0.0160 0.0150 0.0150 CH4 0.8011 0.8011 0.7895 0.7895 0.7787 0.7787 0.7511 0.7511 0.7237 0.7237 0.6995 0.6995 0.6916 0.6916 0.6827 0.6784 0.6737 0.6737 0.6554 0.6554 0.6508 0.6508 0.6365 0.6365 0.6423 0.6423 0.6460 0.6460 0.6367 0.6367 0.4600 0.4600 C2 0.0708 0.0708 0.0709 0.0709 0.0712 0.0712 0.0719 0.0719 0.0725 0.0725 0.0732 0.0732 0.0734 0.0734 0.0736 0.0737 0.0738 0.0738 0.0744 0.0744 0.0745 0.0745 0.0760 0.0760 0.0468 0.0468 0.0469 0.0469 0.0469 0.0469 0.0500 0.0500 C5 0.0117 0.0117 0.0129 0.0129 0.0139 0.0139 0.0166 0.0166 0.0194 0.0194 0.0219 0.0219 0.0227 0.0227 0.0237 0.0241 0.0245 0.0245 0.0187 0.0187 0.0195 0.0195 0.0239 0.0239 0.0256 0.0256 0.0258 0.0258 0.0261 0.0261 0.0400 0.0400 C9 0.0017 0.0017 0.0025 0.0025 0.0033 0.0033 0.0060 0.0060 0.0119 0.0119 0.0331 0.0331 0.0336 0.0336 0.0354 0.0366 0.0380 0.0380 0.0457 0.0457 0.0485 0.0485 0.0655 0.0655 0.0724 0.0724 0.0734 0.0734 0.0748 0.0748 0.1300 0.1300 C20 0.0000 0.0000 0.0072 0.0072 0.0037 0.0037 0.0033 0.0033 0.0033 0.0033 0.0037 0.0037 0.0040 0.0040 0.0043 0.0045 0.0047 0.0047 0.0059 0.0059 0.0064 0.0064 0.0094 0.0094 0.0106 0.0106 0.0108 0.0108 0.0110 0.0110 0.0200 0.0200 S1 1.0000 1.0000 0.9463 0.9463 0.9105 0.9105 0.8249 0.8249 0.7391 0.7391 0.6474 0.6474 0.6156 0.6156 0.5789 0.5606 0.5405 0.5405 0.4576 0.4576 0.4340 0.4340 0.3261 0.3261 0.2952 0.2952 0.2911 0.2911 0.2839 0.2839 0.0000 0.0000 Flow Vel 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 ξ/τ 1.0000 0.1323 0.1323 0.1819 0.1819 0.2616 0.2616 0.3103 0.3103 0.3439 0.3439 0.4175 0.4175 0.4899 0.4899 0.5719 0.5719 0.6555 0.6555 0.7632 0.7632 0.8923 0.8923 1.0263 1.0263 1.1088 1.1088 1.4111 1.4111 1.7069 1.7069 1.0000 278 Appendix B Table B.16: Displacement details for Fig 7.7 (no volume change) Composition route for a displacement of an oil with composition N2 = 0.005, CH4 = 0.350, C4 = 0.250, C10 = 0.195, C16 = 0.125, and C20 = 0.075 by pure CO2 Phase behavior calculated with the Peng-Robinson EOS (see Tables 3.1 and 3.2) Segment Injection Gas Trailing Shock Zone of Constant State Intermediate Shock Zone of Constant State Intermediate Shock Zone of Constant State Intermediate Shock Zone of Constant State Rarefaction Rarefaction Leading Shock Initial Oil Point f f f f e e e e d d d d d-c d-c c c b b a a N2 1.0000 1.0000 0.9808 0.9808 0.9808 0.9808 0.9475 0.9475 0.9475 0.9475 0.8855 0.8855 0.8855 0.8855 0.7907 0.7907 0.7907 0.7907 0.3839 0.1502 0.0145 0.0145 0.0142 0.0142 0.0050 0.0050 CH4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.4049 0.6306 0.7295 0.7295 0.7221 0.7221 0.3500 0.3500 C4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0964 0.0964 0.0964 0.0964 0.1031 0.1097 0.1240 0.1240 0.1265 0.1265 0.2500 0.2500 C10 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0581 0.0581 0.0581 0.0581 0.0572 0.0572 0.0572 0.0572 0.0554 0.0567 0.0689 0.0689 0.0711 0.0711 0.1950 0.1950 C16 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0365 0.0365 0.0365 0.0365 0.0389 0.0389 0.0389 0.0389 0.0384 0.0384 0.0384 0.0384 0.0364 0.0365 0.0438 0.0438 0.0456 0.0456 0.1250 0.1250 C20 0.0000 0.0000 0.0192 0.0192 0.0192 0.0192 0.0160 0.0160 0.0160 0.0160 0.0175 0.0175 0.0175 0.0175 0.0173 0.0173 0.0173 0.0173 0.0164 0.0164 0.0196 0.0196 0.0204 0.0204 0.0750 0.0750 S1 1.0000 1.0000 0.9344 0.9344 0.9344 0.9344 0.8558 0.8558 0.8558 0.8558 0.7614 0.7614 0.7614 0.7614 0.6954 0.6954 0.6954 0.6954 0.6500 0.5518 0.3849 0.3849 0.3626 0.3626 0.0000 0.0000 Flow Vel 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 ξ/τ 1.0000 0.0003 0.0003 0.0003 0.0011 0.0011 0.0011 0.0011 0.0190 0.0190 0.0190 0.0190 0.4622 0.4622 0.4622 0.4622 1.1980 1.1980 1.2999 1.3998 1.5047 1.5047 1.7323 1.7323 1.7323 1.0000 Appendix B 279 Table B.17: Displacement details for Fig 7.9 (with volume change) Composition route for a displacement of an oil with composition CO2 = 0.005, CH4 = 0.350, C4 = 0.250, C10 = 0.195, C16 = 0.125, and C20 = 0.075 by pure CO2 Phase behavior calculated with the Peng-Robinson EOS (see Tables 3.1 and 3.2) Segment Injection Gas Trailing Shock Zone of Constant State Intermediate Shock Zone of Constant State Intermediate Shock Zone of Constant State Intermediate Shock Tie-Line Rarefaction Intermediate Shock Zone of Constant State Leading Shock Initial Oil Point h h g g g g f f f f e e e e d d c c b b b b a a CO2 1.0000 1.0000 0.9714 0.9714 0.9714 0.9714 0.9452 0.9452 0.9452 0.9452 0.8969 0.8969 0.8969 0.8969 0.7826 0.7826 0.7595 0.7595 0.0054 0.0054 0.0054 0.0054 0.0050 0.0050 CH4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.4299 0.4299 0.4299 0.4299 0.3500 0.3500 C4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1108 0.1108 0.1182 0.1182 0.2243 0.2243 0.2243 0.2243 0.2500 0.2500 C10 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0523 0.0523 0.0523 0.0523 0.0542 0.0542 0.0619 0.0619 0.1681 0.1681 0.1681 0.1681 0.1950 0.1950 C16 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0350 0.0350 0.0350 0.0350 0.0318 0.0318 0.0318 0.0318 0.0329 0.0329 0.0378 0.0378 0.1076 0.1076 0.1076 0.1076 0.1250 0.1250 C20 0.0000 0.0000 0.0286 0.0286 0.0286 0.0286 0.0198 0.0198 0.0198 0.0198 0.0190 0.0190 0.0190 0.0190 0.0196 0.0196 0.0226 0.0226 0.0646 0.0646 0.0646 0.0646 0.0750 0.0750 S1 1.0000 1.0000 0.9052 0.9052 0.9052 0.9052 0.8432 0.8432 0.8432 0.8432 0.7434 0.7434 0.7434 0.7434 0.5332 0.5332 0.4496 0.4496 0.1576 0.1576 0.1576 0.1576 0.0000 0.0000 Flow Vel 1.0000 1.0000 0.9979 0.9979 0.9979 0.9979 0.9954 0.9954 0.9954 0.9954 0.9808 0.9808 0.9808 0.9808 0.9798 0.9798 0.9798 0.9798 0.8879 0.8879 0.8879 0.8879 0.8572 0.8572 ξ/τ 1.0000 0.0061 0.0061 0.0061 0.0127 0.0127 0.0127 0.0127 0.0750 0.0750 0.0750 0.0750 0.5376 0.5376 0.5376 0.5376 1.0228 1.0228 1.0228 1.0228 1.9081 1.9081 1.9081 1.0000 Index volume change, 61 multicomponent no volume change, 13 volume change, 12 quaternary no volume change, 135 ternary no volume change, 75, 77 volume change, 120 constant K-value, 37–40, 78, 79, 131 quaternary, 41, 138–149 ternary, 40, 84–87, 93–96, 132, 133 continuity equation, convection, convection-dispersion equation, 15, 16 Courant, 43 Crane, 238 critical locus, 37 critical point, 36–38 critical tie line, 118, 119, 130, 206–208, 210, 212 Abbot, 40 Abbott, 22 Amundson, 2, 43, 70 Aris, 2, 43, 70 Aziz, ii Batycky, ii, 238 Bedrikovetsky, 70, 130, 176 binodal curve, 36 binodal surface, 37 Blunt, ii, 238 boundary condition, 14 Braun, 74 Buckley, 53, 70 capillary pressure, Cer´, 130 e characteristics, 47 chemical potential, 10, 25 Chumak, 176 coherence, 80 component recovery, 67–69, 127–129 composition path, 75, 80, 83, 84 geometry, 87 nontie line, 84 nontie-line liquid locus, 82 vapor locus, 82 tie-line, 81 compositional simulation, 155, 176, 213–230, 235–237 condensing gas drive, 2, 73–75, 106–114, 119, 130, 132, 133 condensing/vaporizing gas drive, 3, 135, 155– 158, 161, 164, 168, 170, 176, 212 conservation equations, 5–13 binary no volume change, 44 Dake, 43, 70 Datta-Gupta, 238 diffusion, 5, 7, 9, 175 dilution line, 36, 73, 74, 102, 108, 218, 223, 225–228, 230 Dindoruk, i, 40, 70, 74, 127, 130, 172, 176, 187, 199, 212 dispersion, 3, 5–7, 12, 51, 175 numerical, 3, 213, 216–218, 221–230 dispersive distance, 223–230, 236, 238 divergence theorem, Dumor´, 74, 130 e eigenvalue nontie-line, 79, 188, 255, 256, 258 tie-line, 79, 193, 255, 256, 258 280 Index eigenvalue problem quaternary no volume change, 136 volume change, 171 ternary, 86 no volume change, 78 volume change, 120 eigenvector nontie-line, 80 tie-line, 80, 81 enhanced oil recovery, enthalpy, 22, 23 entropy, 22, 23 entropy condition, 52, 54, 58, 64, 69, 92, 98– 99, 108, 121, 255–265 envelope curve, 78, 79, 93–95, 187–193, 256 equal eigenvalue point, 83, 87, 88, 102, 111, 142, 149, 151, 185, 186, 198, 199, 258 equation of state, 21, 24, 26–28, 30, 31, 40, 150 Peng-Robinson, 28, 36, 39, 84, 150, 152, 154, 162, 218, 231 Redlich-Kwong, 41, 150, 231 van der Waals, 27, 28, 32, 35 equivelocity curve, 82, 83, 102, 119, 160 Ermakov, i, 212 flow velocity, 13 fractional flow functions, 11 fugacity, 25, 41 coefficient, 25 partial, 25–27, 31 coefficient, 26, 27, 33 Gerritsen, ii Gibbs, 21 Gibbs function, 22 gravity segregation, 175 Haajizadeh, 230, 238 Hand’s rule, 96 Hearn, 238 Helfferich, i, 70, 80, 130, 176 Helmholtz function, 22, 26 high volatility intermediate, 99, 101 Hilbert, 43 281 Hirasaki, i, 74, 130 Hutchinson, 74 HVI, 99 condensing, 106 vaporizing, 101, 147 vaporizing drive, 99 Høier, 226 initial condition, 14 Isaacson, 80 Jeffrey, 43 Jessen, i, 184, 212, 225, 238 Johansen, i, 130, 176, 255 Johns, i, 3, 70, 74, 76, 130, 176, 212 jump condition, see shock, balance King, 238 Kuo, 238 LaForce, Lake, 2, 17, 40, 70, 73 Lantz, 214 Larson, 130 Lax, 80 lever rule, 34 Leverett, 53, 70 local equilibrium assumption, low volatility intermediate, 103, 105, 221, 259 Luks, 238 LVI, 95 condensing, 109, 265 vaporizing, 103, 105, 258 Mallison, 214 Maxwell relations, 23 Metcalfe, 36, 38, 238 methane bank, 148, 152, 163, 174, 196, 201, 204 Michelsen, 27, 33, 238 minimum enrichment for miscibility, 119, 161 minimum miscibility pressure, 3, 120, 213, 217 calculation of compositional simulation, 235–238 key tie line method, 161–171, 206–210, 236 mixing cell, 230–233, 238 282 miscibility development of, see miscibility, multicontact multicontact, 2, 117–120, 158–161, 206 condensing, 119 condensing/vaporizing, 164 multicomponent, 206, 212 vaporizing, 118–119 mixing cell, 73, 210, 213, 230–233 MME, 119, 160, 161, 207, 213, 217, 230 MMP, 120, 160, 207, 213, 217, 226, 230, 233 molar density, 13 Mollerup, 27 Monroe, i, 176 negative flash, 33, 110, 152 no volume change on mixing, 13–14, 43 overall fractional flow, 14 overall molar concentration, 12 overall molar flow rate, 12 overall volume fraction, 13 Pande, i, 70, 130 path, 80 equivelocity curve, 83 geometry, 87 nontie-line, 87, 88, 93, 122, 137, 142, 149, 150, 185–193, 198 differential equation, 86 liquid locus, 82 vapor locus, 82 tie-line, 81, 87, 88, 137 path switch, 87–91, 102, 110, 114, 142, 185 Peaceman, 214 Peck, 230, 238 Peclet number, 15–17, 214, 215, 218 Pedersen, 238 plait point, 36, 37 Pope, i pressure gradient, 11, 21 Prudhoe Bay, Rankine-Hugoniot condition, see shock, balance rarefaction, 55, 75, 89, 93, 96, 221 Index nontie-line, 83, 94, 102, 103, 106, 109–111, 114, 118, 119, 122, 145, 146, 149, 155, 187, 198 tie line, 201 tie-line, 94, 102, 103, 106, 108, 110, 114, 117, 125, 147–149, 155, 185, 193, 196 Ratchford-Rice equation, 33 Reamer, 35 relative permeability functions, 45, 175 Reynolds transport theorem, Rhee, 2, 43, 70 Riemann problem, 54, 74, 80–82, 87, 89 Sage, 35 self similarity, 80 self-sharpening, 52, 69, 93, 94, 97, 98, 106, 107, 110, 117, 119, 130, 187, 210, 221, 255, 256, 258 fully, 121, 147, 152, 158, 180, 193–198 self-similar, 221 semishock, 52, 53, 56–58, 60, 64, 70, 92, 99, 102, 103, 106, 108, 110, 111, 114, 117, 125, 132, 147, 193, 196, 201, 256, 257 nontie-line, 151 Seto, 221, 238 shock, 43, 48 balance, 50, 63, 92, 103, 111, 145, 151, 158, 160, 163, 181, 202 evaporation, 56, 60, 65, 148, 152, 196 genuine, 103, 106, 108, 110, 111, 114, 117, 146–148, 196, 257 intermediate, see shock, nontie-line nontie-line, 92–99, 114, 119, 145, 146, 155, 160, 163, 187 phase-change, 48–55, 90–92, 121, 122 stability, 51, 52, 255 tangent, see semishock, 257 tie-line, 48–55, 64, 90–92, 121, 146–148, 152, 196 two-phase, 90, 92–98, 110, 112 velocity, 52, 106, 119 Silva, 36, 38 slim tube, 218 slim tube displacement, 16, 165, 173, 174, 210, 218 Index spreading wave, 55 Stalkup, 176, 239 streamline simulation, 238 surface developable, 150 equivelocity, 160 rarefaction, 140, 150, 151 ruled, 151 shock, 140, 151 Thiele, ii, 238 tie line, 33, 34, 36, 37 critical, 118, 119, 130, 160, 164 crossover, 135, 144–148, 150–153, 155, 157, 158, 160–164, 167, 168, 170, 174, 175, 179–181, 187, 194, 196, 206, 208, 210– 212 equation of, 32 extension, 73, 92, 110, 121, 122 initial oil, 110, 152, 181, 194, 210 injection gas, 92, 110, 152, 181, 194, 210 intercept, 76, 93, 137 intersection, 96–98, 122, 147, 151, 152, 198, 258 key, 92, 98, 110, 111, 135, 144–147, 157, 158, 160, 161, 164, 167, 168, 171, 173, 175–177, 179, 198, 207, 212 calculation of, 180–184 parametrization, 76, 77 shortest, 110, 111, 145, 146, 148, 153, 157, 160, 161, 164, 167, 168, 175, 185, 187, 193, 194, 196, 201, 202, 226, 228 slope, 37, 76, 93, 137 van der Waals, 32 van Ness, 22, 40 vaporizing gas drive, 2, 73, 74, 99–106, 221 velocity constraint, 51, 52, 58, 69, 87–89, 91–94, 99, 102, 103, 106, 111, 117, 118, 121, 129, 142, 146, 148, 149, 175, 258 flow, 11, 45, 62–64, 121, 122 rule, see velocity constraint shock, 51, 63, 70 wave, 45, 51, 57, 59, 63–65, 69, 75, 78–81, 88, 90, 102 283 indifferent, 187 velocity constraint, 64 viscous fingering, 175 volume change on mixing, 13, 122, 125 binary, 61–65 multicomponent, 201–204, 218, 220 quaternary, 135, 171–175 ternary, 120–127 Wachman, 74, 130 Walas, 40 Walsh, 238 Wang, i, 74, 131, 176, 212, 230, 238, 255 Wattenbarger, ii wave centered, 55 expansion, 55 indifferent, 119 rarefaction, 55 spreading, 55, 57, 60 Welge, 53, 70, 74 Whitson, 33, 238 Wilson equation, 32 Wingard, i Winther, 255 Yarborough, 36, 38 Zanotti, 130 Zick, 155, 176, 226 zone of constant state, 53, 59, 88, 102, 114, 128, 129, 196 ...Library of Congress Cataloging-in-Publication Data Orr, Franklin M., Jr Theory of Gas Injection Processes / Franklin M Orr, Jr Bibliography: p Includes index ISBN xxxxxxxxxxx Enhanced recovery of. .. Department of Energy, and by the member companies of the Stanford University Petroleum Research Institute Gas Injection Industrial Affiliates program That support is gratefully acknowledged Lynn Orr Stanford, ... hence the rate of accumulation of component i in V is Rate of change of d moles of compo- = dt nent i in V np φ V xij ρj Sj dV (2.1.5) j=1 Convection Terms Part of the accumulation of component