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 1. 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 ex- planations 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 ma- terial 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 7 and 8. 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 pre- sented 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 Affili- ates program. That support is gratefully acknowledged. Lynn Orr Stanford, California March, 2005 Contents Preface i 1 Introduction 1 2 Conservation Equations 5 2.1 GeneralConservationEquations 5 2.2 One-DimensionalFlow 10 2.3 PureConvection 12 2.4 NoVolumeChangeonMixing 13 2.5 ClassificationofEquations 14 2.6 Initial and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.7 Convection-DispersionEquation 15 2.8 AdditionalReading 17 2.9 Exercises 17 3 Calculation of Phase Equilibrium 21 3.1 ThermodynamicBackground 21 3.1.1 CalculationofThermodynamicFunctions 22 3.1.2 ChemicalPotentialandFugacity 24 3.2 CalculationofPartialFugacity 26 3.3 Phase Equilibrium from an Equation of State . . . . . . . . . . . . . . . . . . . . . . 27 3.4 FlashCalculation 31 3.5 PhaseDiagrams 34 3.5.1 BinarySystems 34 3.5.2 TernarySystems 35 3.5.3 QuaternarySystems 37 3.5.4 ConstantK-Values 38 3.6 AdditionalReading 40 3.7 Exercises 40 4 Two-Component Gas/Oil Displacement 43 4.1 SolutionbytheMethodofCharacteristics 44 4.2 Shocks 48 4.3 VariationsinInitialorInjectionComposition 56 4.4 VolumeChange 61 iii iv CONTENTS 4.4.1 FlowVelocity 62 4.4.2 CharacteristicEquations 62 4.4.3 Shocks 63 4.4.4 ExampleSolution 64 4.5 ComponentRecovery 67 4.6 Summary 69 4.7 AdditionalReading 70 4.8 Exercises 71 5 Ternary Gas/Oil Displacements 73 5.1 CompositionPaths 75 5.1.1 EigenvaluesandEigenvectors 78 5.1.2 Tie-LinePaths 81 5.1.3 Nontie-LinePaths 81 5.1.4 SwitchingPaths 87 5.2 Shocks 90 5.2.1 Phase-ChangeShocks 90 5.2.2 ShocksandRarefactionsbetweenTieLines 92 5.2.3 Tie-LineIntersectionsandTwo-PhaseShocks 97 5.2.4 EntropyConditions 98 5.3 ExampleSolutions:VaporizingGasDrives 99 5.4 ExampleSolutions:CondensingGasDrives 106 5.5 StructureofTernaryGas/OilDisplacements 110 5.5.1 EffectsofVariationsinInitialComposition 117 5.6 Multicontact Miscibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.6.1 VaporizingGasDrives 118 5.6.2 CondensingGasDrives 119 5.6.3 Multicontact Miscibility in Ternary Systems . . . . . . . . . . . . . . . . . . . 119 5.7 VolumeChange 120 5.8 ComponentRecovery 127 5.9 Summary 129 5.10AdditionalReading 130 5.11Exercises 131 6 Four-Component Displacements 135 6.1 Eigenvalues,Eigenvectors,andCompositionPaths 135 6.1.1 TheEigenvalueProblem 135 6.1.2 CompositionPaths 137 6.2 SolutionConstructionforConstantK-values 144 6.3 SystemswithVariableK-values 149 6.4 Condensing/VaporizingGasDrives 155 6.5 Development of Miscibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 6.5.1 Calculation of Minimum Miscibility Pressure . . . . . . . . . . . . . . . . . . 161 6.5.2 Effect of Variations in Initial Oil Composition on MMP . . . . . . . . . . . . 162 6.5.3 Effect of Variations in Injection Gas Composition on MMP . . . . . . . . . . 169 CONTENTS v 6.6 VolumeChange 172 6.7 Summary 176 6.8 AdditionalReading 176 6.9 Exercises 177 7 Multicomponent Gas/Oil Displacements 179 byF.M.Orr,Jr.andK.Jessen 179 7.1 KeyTieLines 180 7.1.1 InjectionofaPureComponent 180 7.1.2 MulticomponentInjectionGas 183 7.2 SolutionConstruction 185 7.2.1 FullySelf-SharpeningDisplacements 193 7.2.2 Solution Routes with Nontie-line Rarefactions . . . . . . . . . . . . . . . . . . 198 7.3 SolutionConstruction:VolumeChange 201 7.4 DisplacementsinGasCondensateSystems 204 7.5 CalculationofMMPandMME 206 7.6 Summary 210 7.7 AdditionalReading 212 8 Compositional Simulation 213 byF.M.Orr,Jr.andK.Jessen 213 8.1 NumericalDispersion 213 8.2 ComparisonofNumericalandAnalyticalSolutions 215 8.3 SensitivitytoNumericalDispersion 221 8.4 CalculationofMMPandMME 230 8.5 Summary 237 8.6 AdditionalReading 238 Nomenclature 241 Bibliography 244 Appendix A: Entropy Conditions in Ternary Systems 255 Appendix B: Details of Gas Displacement Solutions 266 Index 280 vi CONTENTS [...]... theory of three-component gas/ oil displacements is developed in Chapter 5 The threecomponent theory leads directly and rigorously to the ideas of “multicontact miscible” displacement via condensing or vaporizing gas drives Extensions of the analysis to systems with more than three components are considered in Chapters 6 and 7 That treatment shows that there are important features of gas injection processes. .. when two or more phases flow Similar theory applies to ion exchange [102], diagenetic alteration of porous rocks [34, 63] and to leaching of minerals [9] Many of these ideas also apply to the area of geologic storage of carbon dioxide [85], or CO2 sequestration, as it is sometimes called These processes are intended to reduce the rate of increase of the concentration of CO2 in the atmosphere by injecting... for a variety of enhanced oil recovery processes This book describes the mathematical representation of those chromatographic separations and the resulting compositional changes that occur in such processes Gas injection processes are among the most widely used of enhanced oil recovery processes [62, 117] CO2 floods are being conducted on a commercial scale in the Permian Basin oil fields of west Texas... V (t), of the porous medium bounded by a surface, S(t) A material balance on component i in the control volume can be stated as Rate of change of amount of component i in V (t) = Net rate of inflow of component i into V (t) due to flow of phases 5 + Net rate of inflow of component i into V (t) due to hydrodynamic dispersion 6 CHAPTER 2 CONSERVATION EQUATIONS Thus, the rate at which the amount of component... ⎩ ⎭ dV Integration of Eq 2.1.3 gives the total amount of component i in the control volume, V (t), ⎧ ⎫ ⎪Total moles⎪ ⎪ of compo-⎪ ⎬ ⎪nent ⎪ ⎩ V (t) i in⎪ ⎪ ⎭ np = V (t) φ xij ρj Sj dV, (2.1.4) j=1 and 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 i in V... In the area of enhanced oil recovery, theoretical descriptions of the displacement of oil by water containing polymer and displacement of oil by surfactant solutions are closely linked to the theory described here In fact, the theory for three-component systems was developed first for applications to surfactant flooding [31, 35, 65], processes that make use of chemical constituents in the injection fluid... variety of Canadian projects [110, 72] and in the North Sea [124] In all these processes, there are transfers of components between flowing phases that strongly affect displacement performance The goal of this book is to develop a detailed description of the interactions of equilibrium phase behavior and two-phase flow, because it is those interactions that make possible the efficient displacement of oil by gas. .. multidimensional flows Hence the analysis given here of one-dimensional flow is only a first step toward full understanding of fieldscale displacements It is an important first step, however, because it reveals how and why high displacement efficiency can be achieved in gas injection processes, and thus it provides the understanding needed to design an essential part of any gas injection process for enhanced oil recovery... transport of component i in the phases that flow in and out of the control volume At any differential element of area dS, the convective molar flux (moles of component i per unit area per unit time) of component i in the j th phase is ⎧ ⎫ ⎨Molar flux of ⎩ component i in = xij ρj vj , ⎭ phase j (2.1.6) 2.1 GENERAL CONSERVATION EQUATIONS 7 where vj is the Darcy flow velocity of phase j, the volume of phase... unit area of porous medium per unit time The flux vector may or may not be normal to the surface, S(t), and hence the magnitude of the vector component of the flow crossing the element of surface is ⎧ ⎫ ⎪Rate of inflow⎪ ⎪ ⎪ ⎨ ⎬ of component ⎪i in phase j ⎪ ⎪ ⎪ ⎩ ⎭ = −n · xij ρj vj dS, (2.1.7) across dS where n is the outward-pointing normal to the surface at the location of the differential element of area, . 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. SolutionConstruction:VolumeChange 201 7.4 DisplacementsinGasCondensateSystems 204 7.5 CalculationofMMPandMME 206 7.6 Summary 210 7.7 AdditionalReading 212 8 Compositional Simulation 213 byF .M. Orr, Jr.andK.Jessen. rate of accumulation of component i in V is Rate of change of moles of compo- nent i in V = d dt  V φ n p  j=1 x ij ρ j S j dV. (2.1.5) Convection Terms Part of the accumulation of component