Theory of Gas Injection Processes

Theory of Gas Injection Processes Franklin M. Orr, Jr. Stanford University Stanford, California 2005 Library of Congr...

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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

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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 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 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

1 Introduction

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2 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 . . . . . . . . . . . . .

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5 5 10 12 13 14 14 15 17 17

3 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 . . . . . . . . . . . . . . . . . . . . . .

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21 21 22 24 26 27 31 34 34 35 37 38 40 40

4 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 . . . . . . . . . . . . . . . . .

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iv

CONTENTS . . . . . . . .

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62 62 63 64 67 69 70 71

5 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 . . . . . . . . . . . . . . . . . . . . . . . .

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73 75 78 81 81 87 90 90 92 97 98 99 106 110 117 117 118 119 119 120 127 129 130 131

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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 . . . . . . . . . . . . .

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6 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

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CONTENTS 6.6 6.7 6.8 6.9

Volume Change . . Summary . . . . . Additional Reading Exercises . . . . .

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172 176 176 177

7 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 . . . . . . . . . . . . . . . . . . .

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179 179 180 180 183 185 193 198 201 204 206 210 212

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213 . 213 . 213 . 215 . 221 . 230 . 237 . 238

8 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 . . . . . . . . . . . . . . . . . .

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Nomenclature

241

Bibliography

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Appendix A: Entropy Conditions in Ternary Systems

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Appendix B: Details of Gas Displacement Solutions

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Index

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vi

CONTENTS

Chapter 1

Introduction When a gas mixture is injected into a porous medium containing an oil (another mixture of hydrocarbons), a fascinating set of interactions begins. Components in the gas dissolve in the oil, and components in the oil transfer to the vapor as local chemical equilibrium is established. The liquid and vapor phases move under the imposed pressure gradient at ﬂow velocities that depend (nonlinearly) on the saturations (volume fractions) of the phases and their properties (density and viscosity). As those phases encounter the oil present in the reservoir or more injected gas, new mixtures form and come to equilibrium. The result is a set of component separations that occur during ﬂow, with light components propagating more rapidly than heavy ones. These separations are similar to those that occur during the chemical analysis technique known as chromatography, and they are the basis 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 (see references [90, 81, 118, 116] for examples of the many active projects), and a very large project is underway in the Prudhoe Bay ﬁeld in Alaska [74]. At Prudhoe Bay, dry gas is injected into the upper portion of the reservoir to vaporize light hydrocarbon liquids and remaining oil, and in other portions of the ﬁeld a gas mixture that is enriched in intermediate components is being injected to displace the oil. Large-scale gas injection is also underway in a 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 known as “miscible ﬂooding [112].” We will examine in some detail the mathematical description of the physical mechanisms that produce high local displacement eﬃciency. While the approach involves considerable mathematical eﬀort, the eﬀort expended on that analysis will pay oﬀ in the development of rigorous ways to calculate the injection gas compositions and displacement pressures required for miscible displacement and a very eﬃcient semianalytical calculation method for solving one-dimensional compositional displacement problems. While the focus here is on gas/oil displacements in porous media, the ideas, and the mathematical approaches apply to physical processes that range from ﬂow of traﬃc on a highway to chemical reactions in a tubular reactor to compressible ﬂuid ﬂow. Chapter 1 of First Order Partial 1

2

CHAPTER 1. INTRODUCTION

Diﬀerential Equations: Vol. I by Rhee, Amundson and Aris [106] describes these and other physical systems for which the equations solved have many similarities to those considered here. For ﬂow in porous media, the approach applies to many physical systems in which the convection of one or more phases dominates the ﬂow, and the eﬀects of dispersive mixing can be neglected. The basis for the theory is the description of chromatography, in which components in a mixture separate as they ﬂow through a column because the components adsorb (and subsequently desorb) with diﬀerent aﬃnities onto a stationary phase [108, 30]. In chromatography, however, only the carrier ﬂuid moves, and hence there is no nonlinearity that results 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 CO2 that would otherwise be released to the atmosphere into subsurface formations such as deep saline aquifers or coalbeds [139]. 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 that lower interfacial tension between oil and water. Eﬀects of volume change as components transfer between phases were not considered in that work, a completely reasonable assumption for the liquid/liquid phase equilibria of surfactant/oil/water mixtures. In gas/oil systems, however, some components can change volume quite substantially as they move between liquid and vapor phases. Dumore et al. [22] worked out the extension of the three-component theory to include the eﬀects of volume change. Monroe et al. [82] reported the ﬁrst solutions for four-component gas/oil displacements. Many other investigators contributed to the development of the full theory for three and four component systems. A detailed review by Johansen [50] summarizes the relevant papers published through 1990. Lake’s [62] comprehensive description of enhanced oil recovery also cites the large body of work related to polymer and surfactant ﬂooding processes. This book applies the one-dimensional theory of multicomponent, multiphase ﬂow to gas/oil displacements. In Chapter 2, the appropriate material balance equations are derived, and the assumptions that lead to the limiting cases explored in detail are stated. An introduction to the representation of phase equilibria with an equation of state is given in Chapter 3. Chapter 4 considers two-phase ﬂow of two components that are mutually soluble. When eﬀects of volume change are ignored, a modest generalization of the familiar Buckley-Leverett solution [10] results. That simple two-phase ﬂow reappears in more complex ﬂows involving more components, and hence its description is the basis for understanding multicomponent systems. The most important eﬀects of volume change as components transfer between phases are also illustrated in Chapter 4. The 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 that cannot be represented by three-component descriptions of the phase behavior. Chapter 6 describes the construction of solutions for four-component displacements and explores the resulting implications for multicontact miscible displacements known as

3 condensing/vaporizing gas drives, which turn out to be relevant to many gas injection projects now underway in ﬁeld applications. Chapter 7 extends the theory to systems with an arbitrary number of components in the oil or the injection gas. Chapter 7 also describes how the one-dimensional theory can be applied to create a rigorous method for calculating the so-called minimum miscibility pressure, the displacement pressure required to achieve high displacement eﬃciency, for multicomponent systems. Thus, all the mathematical eﬀort does pay oﬀ with a calculation method of considerable practical value. Eﬀects of dispersive mixing are ignored in the development of the theory presented in Chapters 4-7, though, of course, some dispersion will be present in all real displacements. Furthermore, ﬁnite diﬀerence compositional simulations of gas/oil displacements normally include some eﬀects of numerical dispersion. In fact, many ﬁnite diﬀerence compositional simulations are strongly and adversely aﬀected by numerical dispersion. Chapter 8 shows that numerical solutions for the onedimensional ﬂow equations converge to the analytical solutions, with suﬃciently ﬁne grids, and it describes how displacement behavior changes when dispersion also acts. Chapter 8 also explains when and why numerical schemes that have been proposed for calculating the minimum miscibility pressure fail to give accurate estimates. Flow is never one-dimensional in actual ﬁeld-scale gas injection projects, and hence, many additional factors inﬂuence the performance of those multidimensional ﬂows: viscous instability, gravity segregation, reservoir heterogeneity, and crossﬂow due to viscous and capillary forces [62]. Even so, the one-dimensional theory can be used eﬀectively to describe the behavior of threedimensional ﬂows by coupling one-dimensional solutions with streamline representations of the ﬂow in heterogeneous reservoirs [119, 121, 8, 120, 16, 43]. The resulting compositional streamline approach can be orders of magnitude faster than conventional ﬁnite diﬀerence reservoir simulation, and it is more accurate because it is aﬀected much less by numerical dispersion [109]. Water is also always present and is often ﬂowing in addition to oil and gas. In addition, threephase ﬂow of CO2 /hydrocarbon mixtures is also observed at temperatures about 50 C and pressures near the critical pressure of CO2 [25, 94, 86, 61]. The approach used here to study the mechanisms of gas/oil displacements has also been applied to the ﬂow of three immiscible phases [137, 24, 27]. LaForce and Johns obtained solutions for three-phase ﬂow for ternary systems with composition variation in the two-phase regions that bound the three-phase region on a ternary phase diagram. In any real displacement, of course, all these physical mechanisms interact with the chromatographic separations that occur in both one-dimensional and 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.

4

CHAPTER 1. INTRODUCTION

Chapter 2

Conservation Equations The fundamental principle that underlies any description of ﬂow in a porous medium is conservation of mass. The amount of a component present at any location is changed by the motion of ﬂuid with varying composition through the porous medium. Thus, the ﬁrst issue to be faced in constructing a model of a ﬂow process is to deﬁne and describe the ﬂow mechanisms that contribute to the transport of each component. For gas/oil systems, the physical mechanisms that are most important are: 1. Convection – the ﬂow of a phase carries components present in the phase along with the ﬂow, 2. Diﬀusion – the random motions of molecules act to reduce any sharp concentration gradients that may exist, and 3. Dispersion – small-scale random variations in ﬂow velocity also cause sharp fronts to be smeared (when transversely averaged concentrations are calculated or measured). Dispersion during ﬂow in a porous medium is always modeled as if it were qualitatively like diﬀusion. For a detailed discussion of the relationship between diﬀusion and dispersion, see [100] or [5]. In this chapter, we derive the diﬀerential equations solved in subsequent chapters, and we state the assumptions required to reduce the general material balance equations to the special cases considered in detail below. Eﬀects of chemical reactions are not included in the ﬂow problems considered here, nor are eﬀects of adsorption or temperature variation. Derivations that include such eﬀects are given by Lake [62].

2.1

General Conservation Equations

Consider an arbitrary volume, 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 i in V changes is exactly balanced by the net inﬂow of component i carried with the ﬂow of each phase (often referred to as convection) and the net inﬂow that arises from the diﬀusion-like process of hydrodynamic dispersion. Accumulation Terms The amount of phase j present in a diﬀerential element of volume, dV is ⎧ ⎫ ⎨Moles of⎬

(2.1.1) phase j ⎭ = φρj Sj dV, in dV where φ is the porosity, and ρj and Sj are the molar density and saturation (volume fraction) of phase j. The amount of the ith component present in phase j is ⎩

⎧ ⎫ ⎨Moles of com-⎬ ⎩

ponent i in = φxij ρj Sj dV, ⎭ phase j in dV

(2.1.2)

where xij is the mole fraction of component i in phase j. The total amount of component i present in dV is obtained by summation over the np phases present, which gives ⎧ ⎫ Total moles⎪ ⎪ ⎪ ⎪ ⎨ ⎬

np

of compo=φ xij ρj Sj dV. (2.1.3) ⎪ ⎪ ⎪ j=1 ⎩nent i in⎪ ⎭ 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⎪ ⎪ ⎭

=

V (t)

φ

np

xij ρj Sj dV,

(2.1.4)

j=1

and hence the rate of accumulation of component i in V is np Rate of change of d φ xij ρj Sj dV. moles of compo- = dt V j=1 nent i in V

(2.1.5)

Convection Terms Part of the accumulation of component i in V is due to the 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 j ﬂowing per 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, dS, and the negative sign gives positive accumulation for ﬂow in the opposite direction of the normal vector, which is ﬂow into the control volume. The net rate of convective inﬂow of component i is obtained by summing the contributions for ﬂow of each phase and integrating over the full surface, S, to obtain

np Net rate of inﬂow = − n · xij ρj vj dS, of component i by S(t) j=1 convection

(2.1.8)

Dispersion Terms Diﬀusion and hydrodynamic dispersion are independent physical mechanisms. Diﬀusion can occur due to concentration gradients in the absence of any ﬂow, and the local ﬂuctuations in ﬂow velocity that cause hydrodynamic dispersion [26] would occur even if diﬀusion were absent. Nevertheless, the mathematical representations of the two mechanisms are essentially the same, and both contributions are generally lumped together [100]. Hence the ﬂux due to diﬀusion and dispersion is taken to be given by [5] ⎧ ⎫ ⎨Dispersive ﬂux⎬ ⎩

of component i = −φK ij · ∇ρj xij , ⎭ in phase j

(2.1.9)

where the dispersion tensor, K ij , for component i in phase j includes contributions due to diﬀusion and dispersion. The diﬀusion contribution is represented by a molecular diﬀusion coeﬃcient, Dij , and the dispersion contribution is usually taken to be linear in the local displacement velocity. Typical forms for the longitudinal and transverse dispersion coeﬃcients are given by Bear [5]: (αj − αtj ) αtj |vj | (v )2j + , φSj |vj | φSj (αj − αtj ) |(v )(vt)|, φSj |vj |

(K )ij = Dij +

(2.1.10)

(Kt)ij =

(2.1.11)

where α is a material constant known as the dispersivity, and the subcripts and t refer to the longitudinal and transverse ﬂow directions. By arguments similar to those given for convection, the net rate of transfer of component i due to dispersion is

8

CHAPTER 2. CONSERVATION EQUATIONS

Net rate of innp

ﬂow of component = n · φ K ij · ∇ρj xij dS. i due to hydrodyS j=1 namic dispersion

(2.1.12)

Continuity Equation The accumulation, convection, and dispersion terms can be combined to yield an integral material balance for component i, d dt

V (t)

φ

np

xij ρj Sj dV = −

j=1

S

np

n ·

xij ρj vj dS +

S

j=1

n · φ

np

ρj K ij · ∇xij dS.

(2.1.13)

j=1

Eq. 2.1.13 is a balance on component i in the full control volume, V (t). If information about the spatial distribution of component i is needed, then a diﬀerential material balance known as the continuity equation must be derived. To do so, we make use of the Reynolds transport and divergence theorems. For some scalar quantity G that is conserved, and a control volume, V (t), that moves with velocity, vs , the Reynolds transport theorem states that d dt

V (t)

GdV =

V (t)

∂G dV + ∂t

S(t)

n · vs GdS.

(2.1.14)

The left side of Eq. 2.1.14 is the total rate of change of G in V (t). The ﬁrst term on the right side of Eq. 2.1.14 represents the change due to the local rate of change of G, and the second term represents the rate of change due to the surface, S(t), overtaking G as it moves. In the examples considered here, we will choose the velocity, vs , to be zero. Hence for a stationary control volume, Eq. 2.1.14 reduces to d dt

GdV = V

V

∂G dV. ∂t

(2.1.15)

The divergence theorem relates volume integrals to surface integrals. For a vector quantity, H, it states that

∇ · HdV =

V

S

n · HdS.

(2.1.16)

Application of the Reynolds transport and divergence theorems gives a set of volume integral material balances, one for each component, ⎧ np np ⎨ ∂ φ xij ρj Sj + ∇ · xij ρj vj V ⎩ ∂t j=1

j=1

−∇·φ

np

K ij · ∇ρj xij

j=1

⎫ ⎬ ⎭

dV = 0,

i = 1, nc.

(2.1.17)

2.1. GENERAL CONSERVATION EQUATIONS

9

Recall that the control volume, V , was chosen arbitrarily. If the integral in Eq. 2.1.17 is to be zero for any choice of the control volume, the integrand of Eq. 2.1.17 must be identically zero everywhere. If it were negative for some portion of V and positive for the remainder, for example, it would be possible to choose a new control volume that included either the positive or negative portion of the original control volume. If so, Eq. 2.1.17 would not be satisﬁed for the new control volume. Hence, the ﬁnal form of the continuity equations for multicomponent, multiphase ﬂow is np

np

∂ φ xij ρj Sj + ∇ · xij ρj vj ∂t j=1 j=1

−∇ · φ

np

K ij · ∇ρj xij = 0,

i = 1, nc.

(2.1.18)

j=1

To complete speciﬁcation of the ﬂow problem, a number of additional functions and conditions must be available. The ﬂow velocity is the most important part of Eq. 2.1.18 yet to be determined, because it controls the convective part of the ﬂow. In principle, we could derive and solve a set of balance equations for momentum, which must also be conserved. In practice, however, the solution of the resulting Navier-Stokes equations for the detailed velocity distributions within the pores of the medium would be intractably and unnecessarily complex. Instead, an averaged version of the momentum equation is used. For single-phase ﬂow, volume averaging of the momentum equations yields a form equivalent to Darcy’s law [37, 111], which states that the local ﬂow velocity is proportional to the pressure gradient. Flow of more than one phase is always assumed to be similarly related to the pressure gradient, and hence, the ﬂow velocity of a phase, vj , is assumed to be given by vj = −

kkrj (∇Pj + ρmj g) , µj

(2.1.19)

where k is the permeability, and krj , µj , ρmj , and Pj are the relative permeability, viscosity, mass density, and pressure of phase j. The phase subscript, j, on the pressure in Eq. 2.1.19 implies that pressures are diﬀerent in diﬀerent phases, as they must be if the phases are separated by curved interfaces with nonzero interfacial tension. The relationships between those pressures are always assumed to be represented by capillary pressure functions of the form, Pj − Pk = Pckj ,

j = 1, np,

k = 1, np,

k = j.

(2.1.20)

These equilibrium capillary pressure functions are usually assumed to be functions of the saturations of the phases (and sometimes, they are scaled with respect to interfacial tension), and they are taken to be properties of the ﬂuids and the porous medium that can be measured in independent experiments. The use of equilibrium capillary pressure functions is really an implicit assumption that ﬂow in a porous medium can be represented in terms of phases in local capillary equilibrium and that ﬂow can be driven by departures from that equilibrium. A similar assumption can be made about chemical equilibrium . The time required for diﬀusion of components over the length scale of a pore is often small enough compared to the time required for ﬂow to change the compositions signiﬁcantly within the pore that it is reasonable to assume that ﬂuids present are in chemical

10

CHAPTER 2. CONSERVATION EQUATIONS

equilibrium. If so, then the statement of chemical equilibrium gives an additional set of relations between the compositions of the phases, µij = µik ,

j = 1, np,

k = 1, np,

k = j.

(2.1.21)

Eq. 2.1.21 states that the chemical potential of component i in phase j equals its chemical potential in all the other phases present (note that according to standard usage, the chemical potential of component i in phase j is µij , while the viscosity of phase j is µj ). The calculation of chemical potential of a component in a phase given the composition of that phase is reviewed in Section 3.3. In addition, the following auxiliary relations hold. The volume fractions of the phases must sum to one, np

Sj = 1,

(2.1.22)

j=1

as must the mole fractions in each of the phases, nc

xij = 1,

j = 1, np.

(2.1.23)

i=1

Finally, the functions that describe the properties of each phase and its relative permeability must be given: ρj = ρj (x1j , x2j , . . . , xnc −1,j , Pj , T ),

(2.1.24)

= µj (x1j , x2j , . . . , xnc −1,j , Pj , T ),

(2.1.25)

µj

krj = krj (S1 , S2 , . . . , Snp−1 ),

(2.1.26)

and appropriate initial and boundary conditions must be stated for solution of Eq. 2.1.18. Eqs. 2.1.18–2.1.26 provide enough information to determine the solution to a ﬂow problem that models the eﬀects of convection, dispersion, and phase equilibrium, as the inventory of equations and unknowns in Table 2.1 indicates. The unknowns are the phase compositions, saturations, pressures and velocities. The equations are the material balance equations and the auxiliary relations that specify capillary and phase equilibrium, Darcy’s law, and the mole fraction and saturation summations. As Table 2.1 demonstrates, the number of equations exactly equals the number of unknowns. That equality is required if a solution is to exist, though it does not guarantee that a solution exists or is unique.

2.2

One-Dimensional Flow

Eq. 2.1.18 is complex enough that it must be solved numerically unless additional simplifying assumptions are made. In the remainder of this book, we will consider ﬂow in one space dimension, and we will assume that the eﬀects of pressure diﬀerences between phases can be neglected. For one-dimensional ﬂow in a Cartesian coordinate system, Eq. 2.1.18 reduces to np

np

∂ ∂ρj xij ∂ φ xij ρj Sj + xij ρj vj − φKij ∂t j=1 ∂x j=1 ∂x

= 0,

i = 1, nc.

(2.2.1)

2.2. ONE-DIMENSIONAL FLOW

11

Table 2.1: Inventory of Equations and Unknowns Unknowns Variable Compositions, xij Saturations, Sj Pressures, Pj Velocities, vj

Total

Number np nc np np np

np (nc + 3)

Equations Equation Eq. Material balances 2.1.18 Phase equilibrium 2.1.21 Capillary pressure 2.1.20 Darcy’s law 2.1.19 Saturation sum 2.1.22 Mole fraction sum 2.1.23 Total

Number nc nc (np − 1) np − 1 np 1 np np (nc + 3)

If capillary pressure diﬀerences are neglected, Eq. 2.2.1 can be simpliﬁed by eliminating the pressure gradient from the expressions for the ﬂow velocities. Phase ﬂow velocities can then be written easily in terms of fractional ﬂow functions , fj deﬁned by vj = fj v = fj

np

vk ,

j = 1, np.

(2.2.2)

k=1

In Eq. 2.2.2 v is the total ﬂow velocity , deﬁned as the sum of the phase ﬂow velocities, vj . In one-dimensional ﬂow in the absence of capillary pressure Darcy’s law (Eq. 2.1.19) becomes kkrj vj = − µj

∂P + ρmj g sin θ , ∂x

j = 1, np.

(2.2.3)

In Eq. 2.2.3, θ is the dip angle measured as the angle between the ﬂow direction and a horizontal line. An expression for the fractional ﬂow function, fj , is obtained by eliminating the pressure gradient from Eqs. 2.2.2 and Eq. 2.2.3. The expression for the pressure gradient can be obtained from any one of the np expressions for the phase ﬂow velocities (Eq. 2.2.3). For the j th phase, for example, µj vj ∂P =− − ρmj g sin θ. ∂x kkrj

(2.2.4)

Substitution of Eq. 2.2.4 into Eq. 2.2.3 written for the nth phase gives vn =

krn µj kkrn vj + g sin θ(ρmj − ρmn ). krj µn µn

(2.2.5)

Substitution of Eq. 2.2.5 into Eq. 2.2.2 gives the expression for the fractional ﬂow of phase j [62] ,

np krj /µj kg sin θ krn 1− (ρmj − ρmn ) . fj = np v µ n=1 (krn/µn ) n=1 n

(2.2.6)

Substitution of Eq. 2.2.2 into Eq. 2.2.1 gives the one-dimensional version of the convectiondispersion equation for multicomponent, multiphase ﬂow,

12

CHAPTER 2. CONSERVATION EQUATIONS

np

np

∂ ∂ρj xij ∂ φ xij ρj Sj + xij ρj fj v − φKij ∂t j=1 ∂x j=1 ∂x

2.3

= 0,

i = 1, nc.

(2.2.7)

Pure Convection

If the eﬀects of dispersion can be neglected, then Eq. 2.2.7 reduces to a set of equations that describes the interaction of pure convection with equilibrium phase behavior, np

np

∂ ∂ φ v xij ρj Sj + xij ρj fj = 0, ∂t j=1 ∂x j=1

i = 1, nc .

(2.3.1)

It is also convenient to write Eq. 2.3.1 in dimensionless form based on the following scaled variables:

τ

vinj t , φL x , L v , vinj ρj . ρinj

=

ξ = vD = ρjD =

(2.3.2) (2.3.3) (2.3.4) (2.3.5)

where vinj and ρinj are the ﬂow velocity and density of the injected ﬂuid, and L is the length of the one-dimensional ﬂow system. In Eq. 2.3.2, the time scale is the length of time required to displace one pore volume of ﬂuid at the ﬂow velocity and density of the injected ﬂuid (the volume per unit of area available for ﬂow over length L is φL, and the volumetric ﬂow rate per unit area is vinj , so the time required to ﬂow length L is φL/vinj ). Thus, τ is a dimensionless time measured in pore volumes. For simplicity, we also assume that the porosity, φ, is constant, though that assumption can be relaxed easily. The result is np np ∂ ∂ vD xij ρjD Sj + xij ρjD fj = 0, ∂τ j=1 ∂ξ j=1

i = 1, nc.

(2.3.6)

The notation of Eq. 2.3.6 can be simpliﬁed by deﬁning two additional functions, Gi and Hi, as Gi =

np

xij ρjD Sj ,

(2.3.7)

j=1

and Hi = vD

np

xij ρjD fj .

(2.3.8)

j=1

Gi is an overall concentration (in moles per unit volume) of component i. Hi is an overall molar ﬂow of component i . The ﬁnal version of the equations for multicomponent, multiphase convection is, therefore,

2.4. NO VOLUME CHANGE ON MIXING

13

∂Gi ∂Hi + = 0, ∂τ ∂ξ

i = 1, nc.

(2.3.9)

The local ﬂow velocity , vD , in the deﬁnition of Hi (Eq. 2.3.8) can vary with spatial location because volume is not conserved if components change volume as they transfer between phases. If, for example, CO2 displaces oil at modest pressure, it often occupies much less volume when dissolved in a liquid hydrocarbon phase than it does in a vapor phase. In those systems, the local ﬂow velocity can vary substantially over the displacement length [22, 82, 19]. Thus, for some gas displacements, it will be important to include the eﬀects of volume change on mixing .

2.4

No Volume Change on Mixing

If the displacement pressure is high enough, then the volume occupied by a component in the gas phase may not change greatly when that component transfers to the liquid phase. Components in the liquid/liquid systems that describe surfactant ﬂooding processes also exhibit minimal volume change on mixing. In such systems, it is reasonable to assume that the partial molar volume of each component is a constant (independent of composition or phase) and hence that ideal mixing applies. In other words, the volume occupied by a given amount of a component is constant no matter what phase the component appears in . Under the assumption that each component has a constant molar density, ρci , in any phase Eq. 2.3.9 can be simpliﬁed further. The local ﬂow velocity is constant everywhere and equal to the injection velocity, so vD = 1. Furthermore, the volume occupied by component i in one mole of phase j is xij /ρci, and the volume fraction of component i in phase j is xij /ρci . cij = nc k=1 xkj /ρck

(2.4.1)

The molar density of a phase is ρj =

n c

−1

xij /ρci

.

(2.4.2)

i=1

Comparison of Eqs. 2.4.1 and 2.4.2 indicates that ρci cij = ρj xij .

(2.4.3)

Division of Eq. 2.4.3 by ρinj followed by substitution of Eq. 2.4.3 into Eq. 2.3.6, with vD = 1, and division by ρci /ρinj yields the set of conservation equations for pure convection with no volume change on mixing, ∂Ci ∂Fi + = 0, ∂τ ∂ξ

i = 1, nc − 1.

(2.4.4)

where Ci is an overall volume fraction of component i given by Ci =

np j=1

cij Sj ,

(2.4.5)

14

CHAPTER 2. CONSERVATION EQUATIONS

and Fi is an overall fractional volumetric ﬂow of component i given by Fi =

np

cij fj .

(2.4.6)

j=1

2.5

Classiﬁcation of Equations

The convection-dominated conservation equations, Eqs. 2.3.6 and 2.4.4 are systems of ﬁrst order partial diﬀerential equations. Those equations have the general form, ∂z ∂z + Q(x, t, z) = R(x, t, z). (2.5.1) ∂t ∂x Equations like 2.5.1 are called quasilinear. If P , Q, and R are independent of z, the equation is strictly linear. It is called linear if R depends linearly on z and semilinear if R is a nonlinear function of z. In the problems considered here, R(x, t, z) = 0. Such equations are called homogeneous. In Eqs. 2.3.9 and 2.4.4 the dependent variables that correspond to z in Eq. 2.5.1 are the overall concentrations Gi or Ci . Because the phase saturations, Sj , densities, ρj , and fractional ﬂows, fj , all depend nonlinearly on those concentrations, Eqs. 2.3.9 and 2.4.4 are homogeneous, quasilinear systems of ﬁrst order equations. P (x, t, z)

2.6

Initial and Boundary Conditions

Before Eqs. 2.3.9 and 2.4.4 can be solved, initial and boundary conditions must be imposed. In the chapters that follow, solutions will be derived for initial compositions that are constant throughout a semi-inﬁnite domain, Gi (ξ, 0) = Ginit i ,

0 < ξ < ∞,

i = 1, nc ,

(2.6.1)

Ci (ξ, 0) = Ciinit ,

0 < ξ < ∞,

i = 1, nc.

(2.6.2)

or

The only boundary condition required is the composition of the injected ﬂuid, Gi(0, τ ) = Ginj i ,

τ > 0, i = 1, nc ,

(2.6.3)

Ci (0, τ ) = Ciinj ,

τ > 0, i = 1, nc.

(2.6.4)

or

Thus, at time τ = 0, the composition of the ﬂuid at the inlet changes discontinuously from the initial value to the injected value. Problems in which the initial state (sometimes referred to as the right state) is constant and the upstream boundary condition (sometimes called the left state) is also constant are known as Riemann problems. Such problems can be viewed as a description of the propagation of a discontinuity, initially placed at ξ = 0, between constant initial states for −∞ < ξ < 0, the

2.7. CONVECTION-DISPERSION EQUATION

15

injection composition, and for 0 < ξ < ∞, the initial composition. Given the fact that the ﬂow problem begins with the propagation of a discontinuity, it is no surprise that the solutions may also display discontinuities known as shocks. At a shock, the diﬀerential material balances derived in this chapter must be replaced by integral balances across the shock. The properties and behavior of shocks are considered in some detail in Chapter 4 for two-component ﬂow problems, and again in Chapter 5 for multicomponent problems.

2.7

Convection-Dispersion Equation

If only two components and one phase are present, and the assumptions of constant porosity and no volume change on mixing apply, then Eq. 2.2.1 simpliﬁes considerably to ∂ 2C ∂C v ∂C + − K 2 = 0, ∂t φ ∂x ∂x

(2.7.1)

where C is the volume fraction of one component, and K is the longitudinal dispersion coeﬃcient, assumed here to be independent of composition. If Eq. 2.7.1 is made dimensionless with the scaled length and time given in Eqs. 2.3.2 and 2.3.3, the result is ∂C 1 ∂2C ∂C + − = 0, ∂τ ∂ξ P e ∂ξ 2

(2.7.2)

where P e = vL/φK is the Peclet number . The Peclet number is a ratio of a characteristic time for dispersion, L2 /K, to a characteristic time for convection, φL/v. When the Peclet number is large, the eﬀects of dispersion are small, and convection dominates. Thus, Eqs. 2.3.9 and 2.4.4 can be viewed as applicable in the limit of large Peclet number. If Kl is a constant (independent of composition) then the Peclet number is a constant as well, and Eq. 2.7.2 can be solved easily by Laplace transforms. If the domain is chosen to be 0 ≤ ξ ≤ ∞, the initial concentration is C(ξ, 0) = 0 for 0 ≤ ξ ≤ ∞, and ﬂuid with concentration C(0, τ ) = 1 is injected for τ > 0, the solution is [12] √ √ 1 P e(ξ − τ ) P e(ξ + τ ) 1 √ √ + exp(P eξ)erfc . (2.7.3) C(ξ, τ ) = erfc 2 2 τ 2 2 τ The ﬁrst term on the right side of Eq. 2.7.3 is usually signiﬁcantly larger than the second term. The second term is signiﬁcant only at early times near the inlet when the Peclet number is small. For large Peclet number (say P e > 1000), however, the second term can be neglected. Hence, many investigators have used the approximate solution, √ P e(ξ − τ ) 1 √ . (2.7.4) C(ξ, τ ) = erfc 2 2 τ Fig. 2.1 is a plot of Eq. 2.7.3 for three Peclet numbers (P e = 10, 100, and 1000) at times, τ = 0.25 and 0.75. Fig. 2.1 shows that at each Peclet number, a transition zone from the injected composition (c = 1) to the initial composition (c = 0) moves downstream and increases in length as the ﬂow proceeds. The width of the transition zone increases as the Peclet number is reduced. At τ = 0.75, for example, detectable amounts of the injected ﬂuid have reached the outlet for P e = 10 and 100 but have not yet done so for P e = 1000.

16

CHAPTER 2. CONSERVATION EQUATIONS 1.0

Concentration, C(ξ,τ)

0.8

0.6

10 10 0.4

100 0.2

100 1000 0.0 0.0

0.2

1000

0.4

0.6

0.8

1.0

Dimensionless Distance, ξ

Figure 2.1: Solutions to the convection-dispersion equation at times τ = 0.25 and 0.75 for P e = 10, 100, and 1000. The solution for P e = 1000 is nearly symmetric about the location ξ = τ . Eq. 2.7.4 describes a concentration distribution that is exactly symmetric about ξ = τ , with c = 0.5 at ξ = τ . The full solution for P e = 1000 is very close to Eq. 2.7.4, but the solutions at lower P e display some asymmetry that reﬂects the contribution of the second term on the right side of Eq. 2.7.3. In the limit, P e → ∞, the solution approaches piston-like displacement of single-phase, dispersion-free ﬂow. In many ﬂows of practical interest, Peclet numbers can be large enough that the dispersion-free limit is appropriate. For example, displacement behavior of gas injection processes is commonly tested in what are known as slim tube displacements [138, 93]. In these experiments, gas displaces oil from a long, small diameter tube ﬁlled with sand or glass beads. Gardner et al. [25] reported a Peclet number of 1000 for slim tube displacements in a tube of length 610 cm. With a more typical length of 1250 cm, a typical average interstitial ﬂow velocity of 4 × 10−2 cm/s [93] (which just allows the experiment to be completed in an eight hour working day), and a typical dispersion coeﬃcient of 2 × 10−2 cm2 /s [100, 113], the Peclet number is 2500. Evaluation of Eq. 2.7.4 at

2.8. ADDITIONAL READING

17

τ = 1 for Pe = 2500 indicates that the length of the transition zone (deﬁned as the distance between the 90 and 10 percent concentrations) is only about 7 percent of the displacement length. Hence, at those ﬂow conditions dispersion has a relatively small eﬀect on the displacement. Still larger Peclet numbers are likely in many ﬁeld-scale ﬂows. If dispersion coeﬃcients are taken to be a property of the porous medium that is independent of scale, then the Peclet number grows linearly with the displacement length. Measurements of ﬁeld-scale dispersion coeﬃcients suggests that they grow linearly with the length scale of the system in which the displacement takes place [2]. That result is due to the fact that ﬂows that are multidimensional in heterogeneous porous media are being represented with a one-dimensional model [2][62, pp 163-168]. Flows at the larger scales sample volumes with greater variability in permeability, and the ﬂow within those volumes is nonuniform. At the scale at which ﬂuids are mixed and at chemical equilibrium, however, the magnitudes of dispersion coeﬃcients should be in the range of 10−3 -10−4 cm2 /s (depending on the pressure through the diﬀusions coeﬃcient). If so then Peclet numbers for a displacement with a spacing between wells as short as 100 m and a ﬂow velocity of 3 × 10−4 cm/s would be in the range, 3000 to 30,000. Thus, the convection-dominated theory presented here is relevant in many physical situations of practical interest.

2.8

Additional Reading

Derivations of the conservation equations used here are available from a variety of sources. See Lake [62] for another version of many of the limiting cases considered here. For application of similar ideas to the problem of reacting geochemical ﬂows in which the local equilibrium assumption can be made, see Lake et al. [63].

2.9

Exercises

1. Consider the ﬂow velocity of a compressible gas in a cylindrical region around a well-bore with radius rw . The mass density of the gas is a known function of pressure, ρg (P ). Derive a diﬀerential equation for the ﬂow velocity and express it in cylindrical polar coordinates. Note that the gradient and divergence operators in cylindrical polar coordinates are:

∂ 1 ∂ ∂ T , , ∇ = ∂r r ∂θ ∂z ∂az 1 ∂ 1 ∂aθ (rar ) + + , ∇ · a, = r ∂r r ∂θ ∂z a = (ar , aθ , az )T 2. A core sample has been obtained for measurement of the permeability. The porosity of the sample is φ. Consider the following experiment: (a) Evacuate the core so that P = 0. (b) Open the valve at the upstream end of the core at time t = 0 to ﬂow helium into the core. At that time, the pressure at the upstream end of the core jumps instantaneously to P1 .

18

CHAPTER 2. CONSERVATION EQUATIONS (c) Measure the pressure, P , somewhere in the middle of the core, for t > 0. To analyze the experiment, calculate how the pressure changes with time after the upstream valve is opened. Assume that the core is indeﬁnitely long. Also assume that the permeability in the core has constant value k, that the viscosity of the helium is a constant, µ, and that the helium is an ideal gas P V = nRT. Derive a diﬀerential equation for the pressure in the core as a function of time and space. 3. Hot water (temperature Th ) is being injected into a cold core (temperature Tc ) of length L. The ﬂow velocity in the core is v, and it is constant at all times during the displacement. The heat capacities of the water and rock are cw and cr respectively, and are constant [units = watt/(gram · ◦ K)]. Also the mass density of the rock, ρr , and the mass density of the water, ρw , are constant. (a) Consider a volume element of rock, dV . If the reference temperature is T = 0, what is the heat energy stored in the volume dV of rock ﬁlled with water with constant porosity, φ? Note that the water and rock are at the same temperature. (b) What is the net inﬂow of heat due to convection only? Note: only water ﬂows. (c) What is the net inﬂow of heat due to conduction only? The thermal conductivity of the rock is α [units = joules/(cm · sec · K)], and the thermal conductivity of the ﬂuid is zero. (d) Write an integral energy balance for the ﬂow in the core. (e) Write the diﬀerential energy balance. (f) Use the following scaled variables: T

= (T − Tc )/(Th − Tc ),

xD = x/L, tD = t/t∗ . Make the diﬀerential equation dimensionless, and choose t∗ appropriately. What dimensionless groups are obtained? 4. Certain heavy metal pollutants are carried in the sediment load of the Sacramento River. When the river reaches San Francisco Bay, the sediment is deposited on the ﬂoor of the bay at a rate R (mass/area/time). The organometallic pollutants partition between the water and the solid particles according to the relation ωw = K(T )ωsed . where ωw and ωsed are weight fractions, and K(t) is a known coeﬃcient that describes the partitioning.

2.9. EXERCISES

19

The bottom of the bay is at temperature Tb while the river water, warmed by eﬄuent power plant cooling water, is at temperature Ts . As sediment depth increases, the porosity decreases from the value at the sediment surface, φs , according to the relation φ = φs [1 − β (P − Ps )] , where φs , β and Ps are constants. The heat capacity of the solid and water, Csed and Cw , can be assumed to be independent of temperature and pressure over the range of conditions present, as can the respective mass densities, ρsed and ρw . The thermal conductivity, α, of sediment containing water in the pore space is constant throughout the sediment bed. Use appropriate integral balance equations to derive a set of diﬀerential equations that could be solved to ﬁnd the concentration distribution of organometallic pollutants in the sediment bed at any time. Also state the appropriate boundary conditions. (a) Assume that the concentration of pollutants in the river is constant, so the concentration at the sediment bed surface is also constant. (b) Assume that water ﬂows only in the vertical direction in the sediment bed and neglect dispersion in the ﬂow of pollutants in the water phase. (c) Assume that Darcy’s law describes the diﬀerence in the ﬂow velocities between the water and sediment, vw − vsed

k =− µ

∂P + ρg . ∂z

Hints: Think about what quantities are conserved and construct a balance equation for each conserved quantity. Recall that the energy required to change the temperature of a mass is Cm∆T . Consider the eﬀect of compaction. Is the sediment stationary if it is compacting? Consider how the total depth of sediment present at any time varies. Is a diﬀerential or an integral balance appropriate for that calculation? 5. A displacement experiment, in which the injected ﬂuid can be mixed in any proportion with the ﬂuid in place in the core without forming a second phase, is performed in a long cylinder core (L = 1 m). The experiment is performed by injecting ﬂuid at a velocity of 10 cm/day. From the results given in the Table 2.2, estimate the pore volume of the core, and the dispersion coeﬃcient.

20

CHAPTER 2. CONSERVATION EQUATIONS

Table 2.2: Data for Problem 5 Injected Volume (cc) 60.0 65.0 70.0 80.0 90.0 100.0 110.0 120.0 130.0 140.0 150.0

Eﬄuent Concentration 0.010 0.015 0.037 0.066 0.300 0.502 0.685 0.820 0.906 0.988 0.997

Chapter 3

Calculation of Phase Equilibrium The representation of equilibrium phase behavior is a key part of the models of convectiondominated ﬂows described by Eqs. 2.3.9 and 2.4.5, because those conservation equations are based on the assumption of local chemical equilibrium. Under that assumption, the compositions of the phases that form at a particular location in the porous medium are determined by the pressure and temperature at that location and, of course, the overall composition of the ﬂuid present. In general, the pressure at each spatial position will diﬀer, because a pressure gradient is required if ﬂow is to take place. In many situations, however, the magnitude of the pressure drop due to ﬂow is small compared to the pressure level. For example, in a slim tube displacement, the pressure drop over the full length of the slim tube is often less than 350 kP a (50 psi), while the pressure maintained at the outlet of the slim tube is typically in the range of 7–35 M P a (1000–5000 psi). In ﬁeld-scale displacements, it is commonly observed that pressure gradients near injection and production wells are signiﬁcant, but there are large portions of the reservoir for which gradients are small. Under those circumstances it is reasonable to evaluate equilibrium phase behavior at a single pressure (and temperature, already assumed to be constant), a step that also makes possible the construction of analytical solutions to Eqs. 2.4.1 and 2.4.5. In this section, we review calculation of phase equilibrium for gas/oil systems with an equation of state (EOS), and we describe brieﬂy the phase diagrams used to report equilibrium phase compositions. We make use here of the remarkable fact, shown originally by J. Willard Gibbs, that if the relationship between pressure, temperature, volume, and composition can be speciﬁed, as it can by an equation of state, then the composition of equilibrium phases can be calculated. To see why that statement is true, we need to review ﬁrst how thermodynamic functions are calculated from volumetric data.

3.1

Thermodynamic Background

The calculation of phase equilibrium solely from information about the volumetric behavior of a mixture is a remarkable demonstration of the power of thermodynamics. The fundamental idea of all thermodynamics is that energy is conserved. A second key assumption, completely consistent with experience, is that once we set the temperature, pressure and composition of a mixture, all other properties (internal energy, molar density, viscosity, heat capacity, thermal conductivity, etc.) are also determined. In other words, the state of a mixture of given composition is ﬁxed by setting its temperature and pressure. 21

22

CHAPTER 3. CALCULATION OF PHASE EQUILIBRIUM

Internal energy, U , is one thermodynamic property that is set when the temperature, pressure and composition of a mixture is speciﬁed. It represents the energy associated with molecular motions and vibrations. If we neglect kinetic and potential energy contributions, any change in U for a mixture of ﬁxed composition must be the result of addition of heat or removal of energy as work is done by the ﬂuid, a statement that can be written dU = Q − W

(3.1.1)

Heat that is transferred to or from a system, Q, is not a state function, but Q/T is, so it is convenient to deﬁne that state function as the entropy, dS = Q/T . Work is obtained by displacement through some distance against a force. For the expansion of a ﬂuid against a pressure, the work done is W = P dV , where V is the molar volume, and P is the pressure. Hence the statement of conservation of energy, also known as the fundamental property relation, is dU = T dS − P dV.

(3.1.2)

In the discussion that follows, it will be convenient to make use of other related thermodynamic functions, the enthalpy, H, Helmholtz function, A, and Gibbs function, G, deﬁned by

dH =

T dS + V dP

(3.1.3)

dA = −SdT − P dV

(3.1.4)

dG = −SdT + V dP

(3.1.5)

Each of these functions can be useful in particular situations. For example, enthalpy is often used when pressure is being held constant, and the Gibbs function has a convenient form for problems in which temperature is constant. Because each function can be obtained from the others, it is a matter of convenience which is used.

3.1.1

Calculation of Thermodynamic Functions

It is important to recognize that the change in any of the thermodynamic state functions can be calculated if enough volumetric and heat capacity data are available. (For more detailed derivations of the expressions given in this section, see Chapter 2 of Classical Thermodynamics of Nonelectrolyte Solutions by van Ness and Abbott [123]). For example, the enthalpy can be evaluated as follows for a mixture of ﬁxed composition. According to the assumption that the state of the mixture is determined if T and P are ﬁxed, H = H(T, P ), and we can write

dH =

∂H ∂T

dT + P

∂H ∂P

dP.

(3.1.6)

T

The derivative of enthalpy with respect to temperature (at constant pressure) is a property known as the constant pressure heat capacity, CP , which can be measured,

CP ≡

∂H ∂T

. P

(3.1.7)

3.1. THERMODYNAMIC BACKGROUND

23

An expression for ∂H ∂P T in terms of measurable quantities is also needed. It is obtained by manipulating the deﬁnition of enthalpy, Eq. 3.1.3. Rearrangement of Eq. 3.1.3 restricted to constant temperature gives

∂H ∂P

=T T

∂S ∂P

+ V.

(3.1.8)

T

The entropy, S, associated with a mixture is not an easily measured quantity, so it is useful to express Eq. 3.1.8 in terms of variables such as volume, temperature, and pressure. It can be shown from the properties of state functions [123, pp. 21-22] that

∂S ∂P

T

∂V =− ∂T

.

(3.1.9)

P

Eq. 3.1.9 is one of the Maxwell relations. (See [123, Appendix A-1] for a derivation.) Substitution of Eqs. 3.1.7–3.1.9 into Eq. 3.1.6 gives an expression for dH in terms of measurable quantities,

dH = CP dT + V − T

∂V ∂T

dP.

(3.1.10)

P

Because H is a state function that depends only on the endpoint values of T and P , any convenient path of integration can be used. An approach that is frequently used is to integrate at constant temperature to a pressure where heat capacity data are available (usually P at or near zero, where the ideal gas law applies), to integrate then in temperature at constant pressure to the new temperature, and ﬁnally to integrate at constant temperature to the new pressure. That sequence of operations produces H(P2 ,T2 )

H = H(P2 , T2) − H(P1 , T1) = 0

= P1

V −T

∂V ∂T

H(P1 ,T1)

dP +

T2

T =T1

P

T1

P2

+

V −T

0

dH {CP }P =0 dT

∂V ∂T

dP.

(3.1.11)

T =T2

P

The entropy function can be calculated by similar manipulations. The entropy change can be written

dS =

∂S ∂T

dT + P

∂S ∂P

dP,

and Eq. 3.1.3 can be rearranged to give an expression for

∂S ∂T

= P

1 T

∂H ∂T

= P

T

∂S ∂T P ,

Cp . T

Substitution of Eqs. 3.1.13 and 3.1.9 into Eq. 3.1.12 and integration yields S = S(P2 , T2 ) − S(P1 , T1) =

S(P2 ,T2) S(P1 ,T1 )

dS

(3.1.12)

(3.1.13)

24

CHAPTER 3. CALCULATION OF PHASE EQUILIBRIUM =

0 ∂V P1

∂T

P

dP + T =T1

T2 C P

T

T1

+

dT

P =0

P2 ∂V

∂T

0

P

dP.

(3.1.14)

T =T2

Once values of H and S have been calculated, the other thermodynamic functions can be obtained from the deﬁnitions, = H − P V,

(3.1.15)

A = U − T S,

(3.1.16)

G = U + P V − T S.

(3.1.17)

U

The integrations in Eqs. 3.1.11 and 3.1.14 require that volumetric data be available over the full range of temperatures and pressures, a requirement that is rarely satisﬁed by experimental data alone. Instead, an equation of state is used to provide the required relationship between temperature, pressure, and composition. Thus, any thermodynamic function can be calculated if enough information about heat capacity and volumetric behavior is available.

3.1.2

Chemical Potential and Fugacity

The discussion of thermodynamic functions so far has dealt only with closed systems with ﬁxed composition. For phase equilibrium calculations, however, each phase is an open system, because components can transfer between phases. If G is the Gibbs energy per mole of a mixture, then the total Gibbs energy is nG, where n is the number of moles of mixture. Any change in the total Gibbs energy for the j th phase must be given by d(nGj ) = −nSj dT + nVj dP +

nc ∂(nGj ) i=1

∂ni

dni ,

(3.1.18)

T ,P,nk=i

where ni is the number of moles of component i. The chemical potential of component i in phase j is deﬁned as

µij =

∂(nGj ) ∂ni

.

(3.1.19)

T ,P,nk=i

The chemical potential measures how much Gibbs energy is added to a mixture by an inﬁnitesimal amount of component i. The chemical potential can also be deﬁned in terms of any of the other commonly used thermodynamic functions:

µij =

∂(nHj ) ∂ni

= nS,P,nk=i

∂(nUj ) ∂ni

= nS,nV,nk=i

∂(nAj ) ∂ni

(3.1.20) nV,T ,nk=i

Eqs. 3.1.19 and 3.1.20 indicate that the chemical potential can be calculated by diﬀerentiating any of the thermodynamic functions. For example, the ﬁrst expression of Eq. 3.1.20 can be used to determine µij by diﬀerentiation of Eq. 3.1.11.

3.1. THERMODYNAMIC BACKGROUND

25

The chemical potential is of fundamental importance because the requirements for phase equilibrium can be stated concisely in terms of the chemical potential . Standard thermodynamic arguments [123, Section 2-2] lead to the statement of chemical equilibrium, µij = µik ,

i = 1, nc ,

j = 1, np,

k = 1, np,

k = j.

(3.1.21)

where µij is the chemical potential of component i in phase j. Eq. 3.1.21 states that at a given temperature and pressure, the compositions of any equilibrium phases that form must be such that the chemical potential of each component is the same in all the phases present. Chemical potential has units of energy per mole, units that carry much less physical information for most people than does pressure. As a result, phase equilibrium calculations frequently make use of fugacity, which is scaled to have units of pressure. The fugacity function is deﬁned by analogy with the behavior of an ideal gas. The Gibbs function, Gideal , for an ideal gas at constant temperature is given by dGideal = Videal dP =

RT dP = RT d ln P, P

(constant T)

(3.1.22)

where R is the gas constant. For a pure component that is not ideal, however, a quantity known as fugacity, f , is deﬁned by dG = RT d ln f.

(constant T)

(3.1.23)

Thus, the fugacity function takes whatever values are required to reproduce the nonideal Gibbs function. At constant temperature and composition, Eq. 3.1.18 indicates that dG = V dP . Substitution for dG in Eq. 3.1.23, subtraction of RT dP/P from both sides, and integration gives

f ln P

f = ln P

ref

1 − RT

P Pref

RT −V P

dP,

(constant T)

(3.1.24)

where the subscript ref indicates a reference state. To complete the deﬁnition of fugacity, therefore, a reference state must be chosen. Again, the limit of zero pressure is useful because gas behavior becomes ideal, and therefore,

lim

P →0

f P

= 1.

(3.1.25)

If zero pressure is selected as the reference state, Eq. 3.1.24 reduces to

f ln P

1 = ln φ = − RT

P RT 0

P

−V

dP,

(constant T)

(3.1.26)

where φ is the fugacity coeﬃcient, a dimensionless version of the fugacity scaled by the pressure. The integrand of Eq. 3.1.26 is just the departure from ideal gas volumetric behavior at a given pressure. Hence the integral is a summation of the departures from ideality. If the gas is ideal, the fugacity equals the pressure, and ln(f /P ) = 0. Eq. 3.1.26 gives the fugacity for a single component or a mixture as a whole. For individual components in a phase, however, the quantity of interest is the partial fugacity, fˆij , which is deﬁned by

26

CHAPTER 3. CALCULATION OF PHASE EQUILIBRIUM

dµij = RT d ln fˆij .

(3.1.27)

Correspondingly, the partial fugacity coeﬃcient , φˆij , is deﬁned by fˆij φˆij = , xij P

(3.1.28)

where xij is the mole fraction of component i in phase j, and

lim

P →0

fˆij xij P

= 1.

(3.1.29)

Thus, the partial fugacity of a component is a kind of corrected partial pressure, and the dimensionless partial fugacity coeﬃcient, φˆij , is scaled by the partial pressure, xij P . The partial fugacity of a component is important because the statement of chemical equilibrium (Eq. 3.1.21) can easily be rewritten in terms of component partial fugacities, as integration of Eq. 3.1.27 indicates: fˆij = fˆik ,

i = 1, nc,

j = 1, np,

k = 1, np,

k = j

(3.1.30)

Thus, the task that remains is to calculate the partial fugacity of each component in each phase from a description of the volumetric behavior of the mixture.

3.2

Calculation of Partial Fugacity

Equations of state in common use for gas/oil systems give the pressure as a function of temperature, volume and composition, a representation that is known as a pressure–explicit equation of state, P = P (T, V, n1, n2 , · · ·).

(3.2.1)

For equations of state of the form of Eq. 3.2.1, it is convenient to diﬀerentiate the Helmholtz function to obtain the chemical potential or partial fugacity. According to Eq. 3.1.18, the chemical potential of phase j is given by

µij =

∂(nAj ) ∂ni

T ,nV,nk=i

fˆij = RT ln xij P

.

(3.2.2)

µij dni .

(3.2.3)

For a mixture, the Helmholtz function is deﬁned by d(nAj ) = −nSj dT − P d(nVj ) +

nc i=1

For ﬁxed temperature and composition, Eq. 3.2.3 reduces to d(nAj ) = −P d(nVj ).

(3.2.4)

Eq. 3.2.4 shows why use of the Helmholtz function is convenient. If P is given as a function of volume, temperature, and composition, as it is in a pressure-explicit equation of state, that function can be inserted directly in Eq. 3.2.4. Integration of Eq. 3.2.4 gives

3.3. PHASE EQUILIBRIUM FROM AN EQUATION OF STATE

n(Aj − Aref ) = − Addition and subtraction of −

Vj

∞

P d(nV ) −

P d(nV ) = Vj

Vref

27

∞

P d(nV ).

(3.2.5)

Vref

∞ RT V V d(nV ) on the right side of Eq. 3.2.5 yields

n(A − Aref ) = n

∞ V

RT P− V

d(V ) + nRT

∞ dV

V

V

−n

∞

P dV.

(3.2.6)

Vref

If Vref is chosen to be large enough that the mixture behaves as an ideal gas, then the pressure in the last integral on the right side of Eq. 3.2.6 is P = RT /V . Substitution for P in Eq. 3.2.6 gives the ﬁnal expression for the Helmholtz function, n(A − Aref ) = n

∞ V

RT P− V

d(V ) + nRT ln

RT . PV

(3.2.7)

If the variable of integration is changed to the total volume, Vt = nV , and the diﬀerentiation in Eq. 3.2.2 is carried out, the result, after some rearrangement, is

fˆi RT ln xi P

=

∞ ∂P

∂ni

Vt

T ,Vt ,nj=i

RT − Vt

dVt − RT ln z,

(3.2.8)

where z = P V /RT is the compressibility factor. According to Eq. 3.2.8 the partial fugacity of ∂P component i can be evaluated if an expression can be obtained for the derivative, ∂ni . In other words, an equation of state of the form of Eq. 3.2.1 is needed to provide the information required to evaluate the partial fugacity of a component in the mixture. Michelsen and Mollerup [80, p. 60] have pointed out that a more convenient version of Eq. 3.2.8 can be obtained by interchanging the order of integration and diﬀerentiation, which yields

fˆi RT ln xi P

=

∂ ∂ni

∞ Vt

P−

nRT Vt

dVt − RT ln z.

(3.2.9)

Here again the relationship between P , V , T , and composition given by the equation of state provides the information needed to evaluate component partial fugacity coeﬃcients.

3.3

Phase Equilibrium from an Equation of State

The use of an equation of state for phase equilibrium calculations has its roots in the work of van der Waals [122] , who proposed in his 1873 PhD dissertation the following equation of state for a pure component i: P =

ai RT − 2. V − bi V

(3.3.1)

In Eq. 3.3.1, the parameters ai and bi represent in a simple way the forces of attraction and repulsion between molecules. Repulsion is represented by bi. As the molar volume, V , approaches bi, the pressure increases rapidly, so bi can be viewed as an estimate of molecular volume. Attractive forces between molecules are represented by ai . The negative sign of the second term on the right side of Eq. 3.2.7 indicates that the pressure is reduced by attractions between molecules. Attraction forces

28

CHAPTER 3. CALCULATION OF PHASE EQUILIBRIUM

must be present if a ﬂuid is to form a liquid phase. In the limit of large molar volume, molecules are too far apart for repulsions or attractions to have any eﬀect, and van der Waals equation reduces to the appropriate limit, the ideal gas law. Eq. 3.3.1 is the simplest equation of state that represents the limiting behavior of both liquid and vapor phases [1]. The attraction and repulsion parameters can be evaluated from the behavior of the equation of state at the critical point, the pressure and temperature at which vapor and liquid phases have identical compositions and properties. In addition, a ﬂuid at its critical pressure, Pc , and critical temperature, Tc , satisﬁes

∂P ∂V

= T

∂ 2P ∂V 2

= 0.

(3.3.2)

T

Application of Eq. 3.3.2 to Eq. 3.3.1 gives expressions for ai and bi in terms of the critical properties:

ai = bi =

27R2 Tci2 , 64Pci RTci . 8Pci

(3.3.3) (3.3.4)

Eq. 3.3.1 is written for a pure component, but it can be applied to a mixture if suitable average values, amj and bmj , can be calculated for the j th phase, for example. For van der Waals equation, the mixing rules always used are [126]: √

amj

=

bmj =

nc i=1 nc

√ xij ai ,

(3.3.5)

xij bi,

(3.3.6)

i=1

where xij is the mole fraction of component i in phase j, and ai , and bi are the attraction and repulsion parameters for component i. Phase equilibrium calculations can be performed with van der Waals equation of state, and while it gives qualitatively reasonable predictions, it does not give good quantitative predictions of phase compositions and densities. As a result, many investigators have proposed modiﬁcations to Eq. 3.3.1 designed to improve agreement with experimental observations [126]. Most of the equations of state in widespread use for gas/oil systems are based on van der Waals equation, but with more complex representations of the attraction terms. An example of a widely used equation of state is the Peng-Robinson equation [99] , P =

(aα)mj RT − 2 , V − bmj V + 2bmj V − b2mj

(3.3.7)

where the attraction and repulsion parameters, amj and bmj , for a mixture with the composition of phase j are given by the mixing rules,

3.3. PHASE EQUILIBRIUM FROM AN EQUATION OF STATE

(aα)mj

=

nc nc

xi xk (aα)ik ,

i=1 k=1

(3.3.8)

(aα)ik = (1 − δik ) (aiαi )(ak αk ), δii = 0, bmj =

29

(3.3.9) (3.3.10)

nc

xij bi.

(3.3.11)

i=1

where δik is a binary interaction parameter chosen to improve the agreement between calculated and measured phase compositions. In Eqs. 3.3.9 and 3.3.11, the attraction and repulsion parameters for the individual components, ai and bi, are given by R2Tci2 , Pci RTci = 0.07780 , Pci

ai = 0.45724

(3.3.12)

bi

(3.3.13)

where the numerical coeﬃcients in Eqs. 3.3.12 and 3.3.13 are obtained by evaluation of Eq. 3.3.7 at the critical point with the conditions of Eq. 3.3.2. In the Peng-Robinson equation, the attraction term has been modiﬁed to include temperature dependence and information about the vapor pressure of a component through the function αi ,

αi = 1 + α∗i 1 −

2

T /Tci

,

(3.3.14)

where α∗i is given by α∗i = 0.37464 + 1.54226ωi − 0.26992ωi2

(3.3.15)

The acentric factor of component i, ωi , in Eq. 3.3.14 provides the information about vapor pressure through its deﬁnition,

ωi = log10

Pci 10Pisat

,

(3.3.16)

Tr =0.7

where Pisat is the vapor pressure of component i at a reduced temperature, Tr = T /Tci = 0.7. For simple components, such as the noble gases or methane, the vapor pressure at Tr = 0.7 is about 0.1Pc . For those components, ωi is close to zero. Table 3.1 reports values of Tci , Pci , and ωi for a variety of components commonly present in gas/oil systems. For the hydrocarbons ethane and larger, Pci decreases and Tci increases as the size of the molecule increases. ωi also increases with the size of the molecule. All the values of ωi are positive because 10Pisat < Pci . The positive values of ωi indicate that attractions between pairs of hydrocarbon molecules increase with the size of the molecule. Values of ai and bi for the normal alkanes are also reported in Table 3.1. The repulsion parameter, bi, can be viewed as an excluded volume, which increases with the size of the molecule.

30

CHAPTER 3. CALCULATION OF PHASE EQUILIBRIUM

Table 3.1: Critical Properties and Acentric Factors [105]

Component Methane Ethane Propane n-Butane n-Pentane n-Hexane n-Heptane n-Octane n-Nonane n-Decane n-Dodecane n-Hexadecane n-Eicosane Nitrogen Carbon dioxide

Sym. CH4 C2 C3 C4 C5 C6 C7 C8 C9 C10 C12 C16 C20 N2 CO2

Mol. Wt. 16.043 30.070 44.097 58.124 72.151 86.178 100.205 114.232 128.259 142.286 170.340 226.448 282.556 28.013 44.010

Pc (atm) 45.4 48.2 41.9 37.5 33.3 29.3 27.0 24.5 22.8 21.8 18.0 14.0 11.0 33.5 72.8

Tc (K) 190.6 305.4 369.8 425.2 469.6 507.4 540.2 568.8 594.6 615.0 658.3 717.0 767.0 126.2 304.2

ω 0.008 0.098 0.152 0.193 0.251 0.296 0.351 0.394 0.444 0.491 0.562 0.742 0.907 0.040 0.225

ai cm6 gmol 6

×10 2.464 5.957 10.05 14.84 20.39 27.05 33.27 40.66 47.74 53.41 74.12 113.1 164.7 1.464 3.913

bi cm3 gmol 26.8 40.4 56.3 72.4 90.0 110.6 127.7 148.2 166.5 180.1 233.5 327.0 445.1 24.0 26.7

The attraction parameter, ai , also increases with the size of the hydrocarbon molecule. The eﬀect of acentric factor on the magnitude of the attraction term can be seen by examining Eq. 3.3.15, which is the equation of a parabola with a maximum at ωi ≈ 2.9. The increase in ωi with the size of the molecule, which arises from the estimates of attractions between pairs of molecules in the pure saturated liquid, also increases the magnitude of the attraction term. The temperature dependence of the attraction term can be seen easily in Eq. 3.3.14, which indicates that the value of αi decreases with increasing temperature and is largest for temperatures far below the critical temperature. As temperature increases, molecules move at higher velocities, and the attractive forces have smaller eﬀect. In the expression for the attraction parameter of the mixture, Eq. 3.3.8, (aα)m includes one additional parameter, known as a binary interaction parameter, δik , for each pair of components (although values for pairs of hydrocarbon components that have similar size are often taken to be zero). The use of binary interaction parameters is an acknowledgement that the simple mixing rules do not represent fully the attractions between dissimilar molecules. The introduction of binary interaction parameters provides additional empirical constants that can be used to improve the accuracy of phase equilibrium calculations (if enough phase behavior data are available to determine all the additional parameters). Calculated phase compositions are often quite sensitive to the values of δij (particularly for pairs of large and small components), and hence, those values can be selected to make the equation of state predictions match measured phase equilibrium data for two-component systems. The resulting representation of attractions based on binary systems can then be used to predict behavior of systems containing more than two components. Table 3.2 reports values of δij used in the phase equilibrium calculations performed for a variety of systems

3.4. FLASH CALCULATION

31

Table 3.2: Peng-Robinson Binary Interaction Parameters [99, 18] Component CH4 C2 C3 C4 C5 C6 C7 C8 C9 C10 C12 C16 C20

CH4

C2

C3 0.010

0.020 0.020 0.025 0.025 0.035 0.035 0.035 0.035 0.035 0.035

0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010

0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010

CO2 0.100 0.150 0.134 0.130 0.125 0.119 0.100 0.112 0.100 0.102 0.095 0.105 0.093

N2 0.120 0.120 0.120 0.120 0.120 0.120 0.120 0.120 0.120 0.120 0.120 0.120 0.120

in subsequent chapters. Interaction parameters for CO2 are those recommended by Deo et al. [18]. Otherwise Table 3.2 reports values of δij recommended by Peng and Robinson [99]. Eqs. 3.3.7–3.3.16 provide the description of volumetric behavior needed to complete the calculation of the partial fugacity of a component in a mixture. Diﬀerentiation of Eq. 3.3.7 gives an ∂P expression for ∂ni . That expression is then substituted into Eq. 3.2.8, and the integration is performed. The result, after considerable algebraic manipulation, is

fˆij ln xij P

am + √ 2 2bm RT

3.4

bi bm = ln φˆij = (z − 1) − ln z 1 − bm V

√ nc 1 + ( 2 + 1) bVm 2 bi √ − xkj (aα)ik ln . bm am k=1 1 − ( 2 − 1) bVm

(3.3.17)

Flash Calculation

The result of the thermodynamic analysis and the use of an equation of state to describe volumetric behavior is a set of nonlinear equations (Eqs. 3.1.30 in which each value of fˆij is given by an expression like Eq. 3.3.17) that must be solved for the compositions of the phases. The following procedure can be used: 1. Estimate composition, xij , of each of the phases present. 2. Solve Eq. 3.3.7 for the molar volume, V , of each phase. 3. Use Eq. 3.3.17 to calculate the partial fugacity, fˆij , for each component in each phase.

32

CHAPTER 3. CALCULATION OF PHASE EQUILIBRIUM 4. Check to see if component partial fugacities are equal in all of the phases (Eq. 3.1.30). If not, then adjust the estimates of phase compositions and return to step 2.

While this procedure will give the compositions of phases that satisfy the requirement that component partial fugacities (or equivalently, chemical potentials) be equal at equilibrium, it should be noted that it is sometimes possible to ﬁnd solutions for which the resulting phases are not stable [3]. In other words, the phase compositions make component chemical potentials equal but do not minimize free energy. For such situations, the stability of phases can be tested directly. See Baker et al. [3] or Michelsen [78, 80] for examples and details. Given the nonlinearity of Eq. 3.3.17, it is easy to see why ﬂash calculations with van der Waals equation did not catch on in the 1880’s when van der Waals [122] performed the ﬁrst phase equilibrium calculations for binary systems. In fact, equation-of state calculations of phase equilibrium did not come into widespread use until the late 1960’s when availability of computing resources made solution of the set of nonlinear equations reasonable. Step 1 requires that some sort of initial guess of phase compositions be made. For ﬂash calculations for two-phase systems, information about phase compositions is often expressed in terms of equilibrium ratios (also known as K-values), Ki =

xi1 , xi2

(3.4.1)

where xi1 and xi2 are the mole fractions of component i in phases 1 and 2, typically vapor and liquid. The Wilson equation [136],

Ki =

xi1 Pci Tci exp 5.37(1 + ωi ) 1 − = xi2 P T

,

(3.4.2)

is frequently used to estimate equilibrium K-values from which phase compositions can be estimated by the following manipulations. From the estimated or updated K-values, the phase compositions can be obtained from a material balance on each component. Consider one mole of a mixture in which the overall mole fraction of component i is zi . A material balance for component i gives zi = xi1 L1 + xi2 (1 − L1 ),

i = 1, nc .

(3.4.3)

where L1 is the fraction of the one mole of mixture that is phase 1. Elimination of xi1 from Eq. 3.4.3 gives xi2 =

zi , 1 + L1 (Ki − 1)

i = 1, nc .

(3.4.4)

i = 1, nc .

(3.4.5)

Similar elimination of xi2 using Eq. 3.4.2 gives xi1 =

Kizi , 1 + L1 (Ki − 1)

An equation for L1 is obtained by noting that Eqs. 3.4.4 and 3.4.5 each sum to unity, so that nc i=1

xi1 −

nc i=1

xi2 =

nc i=1

zi (Ki − 1) = 0. 1 + L1 (Ki − 1)

(3.4.6)

3.4. FLASH CALCULATION

33

Eq. 3.4.6, which is known as the Ratchford-Rice equation [103], can be solved for L1 by a NewtonRaphson iteration [80, p. 220]. Given the value of L1 , Eqs. 3.4.4 and 3.4.5 give the phase compositions consistent with the K-values. The equilibrium K-values deﬁned in Eq. 3.4.2 are related to the equilibrium partial fugacity coeﬃcients. The equilibrium relations, Eqs. 3.1.30, can be written using Eq. 3.1.28 as fˆi1 = φˆi1 xi1 P = φˆi2 xi2 P = fˆi2 ,

i = 1, nc .

(3.4.7)

Rearrangement of Eq. 3.4.7 shows that the equilibrium K-value is just the ratio of partial fugacity coeﬃcients, Ki =

xi1 φˆi2 = , xi2 φˆi1

i = 1, nc .

(3.4.8)

The form of Eq. 3.4.8 suggests a simple successive approximation scheme for updating K-values in step 4 of the ﬂash calculation [98, p. 103][80, p. 245]. If the component partial fugacites are not equal at the kth iteration, then new K-values can be estimated from Kik+1 = Kik

k fˆi2 , fˆk

i = 1, nc .

(3.4.9)

i1

Eq. 3.4.9 modiﬁes the K-values in the appropriate direction if fˆi2 = fˆi1 . While iteration with Eq. 3.4.9 usually converges to solutions that satisfy the equilibrium relations even when the guess of initial phase compositions is relatively poor, convergence will be very slow for two-phase mixtures near a critical point (a pressure, temperature and composition for which the two phases become identical). For such situations, more sophisticated iterative schemes can and should be used [79, 75]. Negative Flash The ﬂash calculation can be performed whether a mixture forms one or two phases. Whitson and Michelsen [135] pointed out that Eq. 3.4.6 can be solved for L1 equally well when only one phase forms, a calculation that is known as a negative ﬂash. When the iteration for L1 has converged for a single-phase system, the resulting value will be in the range L1 < 0 or L1 > 1. The phase compositions calculated with Eqs. 3.4.4 and 3.4.5 will be equilibrium compositions that can be combined to make the single-phase mixture. In other words, the single-phase mixture is a linear combination of the phase compositions, which means that the single-phase composition must lie on the extension of the line that connects the equilibrium compositions on a phase diagram. We will make repeated use of the negative ﬂash to ﬁnd that line, known as a tie line, for displacement calculations. Whitson and Michelsen showed that their negative ﬂash calculation converges as long as L1 lies in the range 1 1 < L1 < , 1 − Kmax 1 − Kmin

(3.4.10)

where Kmax and Kmin are the largest and smallest K-values. If the single-phase composition is far enough from the two-phase region that the condition 3.4.10 is not satisﬁed, a modiﬁed negative ﬂash suggested by Wang [128] can be used. It is based on the

34

CHAPTER 3. CALCULATION OF PHASE EQUILIBRIUM

idea that while L1 can vary over wide ranges, the equilibrium phase compositions, xij , are restricted to lie between zero and one. The mole fraction of phase 1, L1 can be eliminated by solving Eq. 3.4.4 written for component 1, L1 =

z1 − x12 . (K1 − 1) x12

(3.4.11)

Substitution of Eq. 3.4.11 into Eq. 3.4.4 gives an expression for the phase compositions in phase 2 in which the only unknown is x12 , xi2 =

zi x12 (K1 − 1) . z1 (Ki − 1) + x12 (K1 − Ki )

(3.4.12)

The revised version of Eq. 3.4.6 is nc

(Ki − 1) x12 =

i=1

nc i=1

zi x12 (Ki − 1) (K1 − 1) = 0. z1 (Ki − 1) + x12 (K1 − Ki)

(3.4.13)

Eq. 3.4.13 can be solved for x12 by a Newton-Raphson iteration.

3.5

Phase Diagrams

It will be convenient for many of the ﬂow problems considered here to represent the solutions as a collection of compositions on a phase diagram. Accordingly, we review brieﬂy the terminology and properties of binary, ternary, and quaternary phase diagrams.

3.5.1

Binary Systems

Fig. 3.1 is a typical phase diagram for a binary mixture at some ﬁxed temperature above the critical temperature of component 1. At pressure, P , liquid phase (phase 2) with mole fraction x12 of component 1 is in equilibrium with vapor phase (phase 1) containing mole fraction x11 of component 1. Those equilibrium compositions are connected by a tie line, along which a tie line material balance like Eq. 3.4.3 applies. There is one tie line in Fig. 3.1 for each pressure. A mixture with an overall mole fraction z1 of component 1 between x12 and x11 forms two phases. Mixtures with z1 < x12 are all liquid, while those with z1 > x11 form only vapor. For a given overall composition, the mole fraction of phase 1 present is easily determined by rearrangement of Eq. 3.4.3 to be L1 =

z1 − x12 . x11 − x12

(3.5.1)

Eq. 3.5.1 is a lever rule, which states that the mole fraction vapor is proportional to the distance from the overall composition to the liquid composition locus divided by the length of the tie line. A similar statement applies to systems with any number of components. Eq. 3.5.1 indicates that L1 ≤ 0 for mixtures that form only liquid, and L1 ≥ 1 for mixtures that are all vapor. At the top of the two-phase region in Fig. 3.1 is a critical point, at which the liquid and vapor phases, as well as all phase properties, are identical. The critical point can be thought of as a tie line with zero length. Because phase compositions are equal at a critical point, Eq. 3.4.2 indicates that all K-values must be equal to one for a critical mixture.

3.5. PHASE DIAGRAMS

35

2000

a 1800

Critical Point

1600

Pressure(psia)

1400 1200

Liquid Phase

Tie

Line

1000

Two Phase Region

800 600

Vapor Phase

400 200 0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Mole Fraction Component 1

Figure 3.1: Pressure-composition phase diagram for a two-component mixture. A phase diagram like Fig. 3.1 can be calculated with an equation of state. Fig. 3.2 shows such a diagram calulated with the Peng-Robinson equation for a two-component system, CO2 /decane (C10 ) system. Also shown in Fig. 3.2 are experimental data of Reamer and Sage [104] along with the phase diagram calculated with van der Waals equation of state. Critical properties used for CO2 and C10 are given in Table 3.1, and the binary interaction parameter used in the Peng-Robinson ﬂash calculations was δ12 = 0.102 (Table 3.2). Fig. 3.2 shows that the Peng-Robinson phase compositions agree much better with the experimental observations than do the van der Waals predictions. They should, of course, because the value of δ12 was chosen to minimize the disagreement between calculated phase compositions and measured data. Even so, with the average value of δij chosen by matching experimental data at several temperatures [18], there is some disagreement between experiment and calculation near the critical point.

3.5.2

Ternary Systems

A typical vapor/liquid phase diagram for a three-component system is shown in Fig. 3.3, which displays phase behavior information at ﬁxed pressure and temperature. Because the mole fractions at any composition point in the diagram always sum to one, it is useful to plot equilibrium phase compositions on an equilateral triangle. On such a diagram, the three mole (or volume or mass) fractions are read from the perpendicular distances from the composition point to the three sides. The corners of the diagram represent 100% of the component with which the corner is labeled, and the opposite side represents the zero fraction. The sides of the ternary diagram represent binary mixtures of the two components that lie on that side. For gas/oil systems, the component at the top corner of the diagram is usually the lightest component, and the heavies component is usually

36

CHAPTER 3. CALCULATION OF PHASE EQUILIBRIUM

2000

2000

b

1800

b

1800

b

1600

1600

b

Pressure(psia)

1400

Experimental Data Van der Vaals Peng-Robinson

1400

b

b

1000

b 800

b

600

b 600

b

400

b 400

b

0.1

1200

b 1000

b

800

0 0.0

b

b

1200

200

b

b 200

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 1.0

Mole Fraction CO2

Figure 3.2: Phase behavior of the CO2 /C10 system at 160 F . Experimental data shown are from Reamer and Sage [104]. placed at the bottom left corner. Any mixture of two overall compositions, whether or not either is single phase, must lie on a straight line that connects those compositions. For example, if M moles of a mixture with overall compositions, zi1 , are mixed with N moles of mixture with compositions, zi2 , the overall composition of the resulting mixture is given by zi =

1 (M zi1 + N zi2 ) M +N

(3.5.2)

Eq. 3.5.2 is the equation of a straight line (the dilution line) that connects the two overall compositions. The region of overall compositions in Fig. 3.3 that form two phases is enclosed by the loci of liquid and vapor phase compositions, which taken together are known as the binodal curve. Here again, equilibrium liquid and vapor compositions are connected by tie lines. In contrast with binary systems, in which there is only a single tie line at ﬁxed temperature and pressure, a ternary system has an inﬁnite number of tie lines. A critical point on a ternary phase diagram is often referred to as a plait point. An example of a calculated (Peng-Robinson) ternary phase diagram that includes a plait point is shown in Fig. 3.4 for mixtures of CO2 , butane (C4 ), and decane (C10 ) at 160 F (71 C) and 1250 psia (85 atm). Mixtures containing no C4 lie on the CO2 /C10 side of the diagram. Hence, that side shows one tie line, at 1250 psia, from the pressure composition phase diagram in Fig. 3.2. Also shown in Fig. 3.4 are experimental data reported by Metcalfe and Yarborough [76] and by Orr and Silva [87]. Again, the Peng-Robinson equation of state shows reasonable agreement with

3.5. PHASE DIAGRAMS

37 Comp.1

80

30

us oc

rL

20

po

Va

10

90

70

40

a

Pla

60

it P

50

Liquid Locus

oin

t

50

60

40

70

Tie Lines

30

80

20

90

10

90

80

70

60

50

40

30

20

Comp.2 10

Comp.3 Ternary Diagram

Figure 3.3: Ternary phase diagram.

the experimental observations. Calculated vapor compositions agree quite well with experimental observations, while calculated liquid phase compositions show slightly lower CO2 mole fractions. Tie line slopes are represented quite well by the equation of state. Tie line slopes turn out to be important in the analysis of ﬂows in systems with three or more components, so it is useful that the equation of state captures that behavior well.

3.5.3

Quaternary Systems

When four components are present, an equilateral tetrahedron can be used to plot phase compositions in much the same way an equilateral triangle is used for ternary systems. Fig. 3.5 is an example of a quaternary diagram for mixtures of CO2 , methane CH4 , butane (C4 ), and decane (C10 ) at 160 F (71 C) and 1600 psia (109 atm). A ternary diagram like Fig. 3.4 is one of the faces of the quaternary diagram. At that temperature and pressure, there are plait points in the CO2 /C4 /C10 and CO2 /CH4 /C4 ternary faces, and there is a locus of critical points in the interior of the quaternary diagram that connects those plait points. The two-phase region on the CO2 /CH4 /C10 face is a band of tie lines that spans that face. The locus of liquid compositions is now a surface in the interior of the diagram, as is the locus of vapor compositions. Those surfaces meet at the locus of critical points, on which the compositions of liquid and vapor are identical. Each composition point on the surface of vapor compositions is connected to another composition point on the surface of liquid compositions by a tie line. Thus, the space enclosed by the binodal surface is densely packed with tie lines. The geometry of tie lines in the four-component diagram of Fig. 3.5 is worth a bit of study because displacement behavior is intimately connected with the geometry of surfaces of tie lines, as the analysis of subsequent chapters will show.

38

CHAPTER 3. CALCULATION OF PHASE EQUILIBRIUM CO2 10

b a abb ab90 bbba

20

80

30

70

40

b

a

a

bbbba bbb a b bb

60

50

50

60

40

70

30

80

Peng-Robinson binodal curve Peng-Robinson tie lines Metalfe and Yarborough experiment Orr and Silva experiment

90

a b

20 10

90

80

70

60

50

40

30

20

C4 10

C10

Figure 3.4: Ternary phase diagram (in mole fractions) calculated with the Peng-Robinson equation of state for the CO2 /C4 /C10 system at 1250 psia (85 atm) and 160 F (71 C). Measured phase compositions reported by Metcalfe and Yarborough [76] and by Orr and Silva [87] are also shown.

3.5.4

Constant K-Values

Eq. 3.3.17 shows clearly that φˆij can depend quite strongly on the composition of phase j through the parameters, am and bm , and it follows, therefore, from Eq. 3.4.8 that K-values are also functions of composition. For example, equilibrium K-values for a phase diagram that includes a critical point must be strong functions of composition (and, of course, temperature and pressure), because Kvalues are all equal to one at the critical point and diﬀer signiﬁcantly from one for the remainder of the two-phase region. If the pressure is low enough for many gas/oil systems, however, a critical point will not be present, and K-values will depend only weakly on composition. For those systems, it is reasonable to assume that K-values are constant. The constant K-value limit is a useful one, because the solutions for pure convection simplify considerably, and hence, we will pause for a moment to explore the properties of the phase diagrams that result from the assumption of constant K-values. If K-values are constant, then the liquid and vapor portions of the binodal curve are straight lines on a ternary diagram. Consider Eq. 3.4.6 with L1 = 0, which gives an expression for compositions on the liquid locus, 3 i=1

zi (Ki − 1) =

3 i=1

zi Ki −

3 i=1

zi =

3

zi Ki − 1 = 0.

(3.5.3)

i=1

Because the Ki are constants, Eq. 3.5.3 is the equation of a straight line. (It is a bit easier to see that it is a straight line if z3 = 1 − z1 − z2 is eliminated from Eq. 3.5.3.) Similarly, when L1 = 1,

3.5. PHASE DIAGRAMS

39

CO2 •

•

CH4 C4

C10

Figure 3.5: Quaternary phase diagram calculated with the Peng-Robinson equation of state for mixtures of CO2 , methane, butane and decane at 160 F (71 C) and 1600 psia (109 atm). The dashed lines are tie lines that lie on the four ternary faces of the diagram. the equation for the vapor phase compositions is 3 zi (Ki − 1) i=1

Ki

=

3 i=1

zi −

3 zi i=1

Ki

=1−

nc zi i=1

Ki

= 0.

(3.5.4)

Again, Eq. 3.5.4 is the equation of a straight line. Hence, for a ternary system with constant Kvalues, the two-phase region must be a band across the diagram that intersects the two sides with one K-value greater than one and the other K-value less than one. Two phases cannot form on a side where both K-values are greater than one or both are less than one. Fig. 3.6, for example, shows a phase diagram for K1 = 1.5, K2 = 2.6, and K3 = 0.01. Those K-values roughly reproduce the behavior of the CO2 /CH4 /C10 face of Fig. 3.5, for example. The intersection points on the binary sides of the diagram can be easily obtained from Eq. 3.5.3 written for two components with K-values above and below one. For example, on the component 1/component 3 side of the diagram (CO2 /C10), x12 =

1 − K3 , K1 − K3

x11 = K1

1 − K3 . K1 − K3

(3.5.5)

For four–component systems, Eqs. 3.5.3 and 3.5.4 indicate that the surfaces of vapor and liquid compositions are planes. Fig. 3.7 is an example of such a phase diagram for K1 = 2.5, K2 = 1.5, K3 = 0.5, and K4 = 0.05. Those values are roughly equivalent to K-values for the CO2 /CH4 /C4 /C10 system at 160 F (71 C) and 1600 psia (109 atm) in regions of the phase diagram far from the critical locus. As Fig. 3.7 shows, the liquid and vapor phase planes intersect all four ternary sides of the diagram when two K-values are greater than one and two are less than one. If only one

40

CHAPTER 3. CALCULATION OF PHASE EQUILIBRIUM

Comp.1 10

90

20

80

30

70

40

60

50

50

60

40

70

30

80

20

90

10

90

80

70

60

50

40

30

20

Comp.2 10

Comp.3

Figure 3.6: Ternary phase diagram for a system with constant K-values, K1 = 1.5, K2 = 2.6, and K3 = 0.01. K-value is less than one (or only one is greater than one), the two–phase region does not intersect the ternary face on which all K-values are greater than (or less than) one.

3.6

Additional Reading

Thermodynamic Background, State Functions, Chemical Potential and Fugacity. van Ness and Abbott’s text, Classical Thermodynamics of Nonelectrolyte Solutions [123], gives concise but readable derivations of the material in Sections 3.1 and 3.2. The various equations of state in widespread use are reviewed in considerable detail by Walas in Phase Equilibria in Chemical Engineering [126]. A shorter description of phase equilibrium calculations with an equation of state is given by Lake in Chapter 4 of Enhanced Oil Recovery [62]. A description of ﬂash calculations and K-values can also be found there. A detailed account of the fundamental thermodynamic structure of phase equilibrium as well as the many computational issues that arise in solving practical problems is given by Michelsen and Mollerup [80]. Phase Diagrams. Binary and ternary phase diagrams are described in detail by Walas [126, Chapter 5]. Properties of ternary diagrams as well as approximate representations for two-phase regions in ternary diagrams are described by Lake [62, Chapter 4]. Examples of phase diagrams for systems with constant K-values are reported by Dindoruk [19].

3.7

Exercises

1. Use the Peng-Robinson equation of state to calculate the molar volume and molar density of CH4 and C10 at 160 F and 1600 psia.

3.7. EXERCISES

41

2(CO2)

3(C4) 1(CH4)

4(C10 ) Figure 3.7: Quaternary phase diagram for a system with constant K-values, K1 = 2.5, K2 = 1.5, K3 = 0.5, and K4 = 0.05. Solid lines outline the planes of liquid and vapor compositions. Dashed lines are tie lines in three of the four faces. 2. Consider a binary system in which x1L = 0.6 and x1V = 0.99. The overall composition is z1 = 0.75. The mass density of the liquid phase is ρL = 0.7 g/cm3, and the vapor phase density is ρV = 0.25 g/cm3 . The molecular weights of components 1 and 2 are 44 and 142 respectively. Calculate the mole fractions of the liquid and vapor phases. Also calculate the volume fractions of the liquid and vapor phases. 3. Consider a two-phase mixture in a ternary system in which the vapor phase has composition, y1 = 0.95, y2 = 0.04, and y3 = 0.01. The overall composition of the mixture is z1 = 0.8, z2 = 0.1, z3 = 0.1. Calculate the composition of the liquid phase. 4. Calculate the phase compositions and volume fractions for a binary system with K-values (based on volume fractions), K1 = 1.5 and K2 = 0.1. The overall volume fraction is 0.6. 5. Use the Peng-Robinson equation of state to calculate the compositions of the equilibrium liquid and vapor phases for a two-phase mixture of CH4 and C10 at 160 F and 1600 psia. 6. A ternary mixture has overall composition (mole fractions), z1 = 0.6, z2 = 0.2, z3 = 0.2. The K-values (based on mole fractions) are K1 = 3.0, K2 = 1.5, K3 = 0.1. Calculation the compositions and volume fractions for the equilibrium liquid and vapor phases. 7. Derive the fugacity expression for the Redlich-Kwong equation of state: P =

am RT − V − bm T 1/2V (V + bm )

(3.7.1)

42

CHAPTER 3. CALCULATION OF PHASE EQUILIBRIUM

am =

n c

√

2

xi ai

(3.7.2)

1

bm =

nc

xi bi

(3.7.3)

1 5/2

ai =

0.42748R2Tc Pc

(3.7.4)

0.08664RTc Pc

(3.7.5)

bi =

Chapter 4

Two-Component Gas/Oil Displacement In this chapter, we use a simple two-phase ﬂow problem, displacement of an oil by a gas, to illustrate application of the method of characteristics and to develop many of the themes that recur, with variations, in the more complex multicomponent ﬂows considered subsequently. The introduction given here to the method of characteristics is by no means comprehensive. For a much more careful development of the ideas behind the mathematical techniques applied here, see Volumes I and II of First Order Partial Diﬀerential Equations by Rhee, Aris and Amundson [106, 107] (see Chapters 5 and 7 especially). Their description of the mathematical structure of many closely related problems of chromatography and packed bed reactors weaves a rich tapestry of mathematics and engineering science that should be high on the reading list of students of ﬁrst order equations. Much useful material can also be found in Courant and Hilbert’s Methods of Mathematical Physics [14] and in Jeﬀrey’s Quasilinear Hyperbolic Systems and Waves [40]. It is with those sources as companions that this version of the theory is oﬀered. In the ﬂow considered in this chapter, two components are present. Consider a one-dimensional ﬂow in which a gas, composed primarily of a light component such as methane (CH4 ) or CO2 , displaces an oil containing some liquid hydrocarbon, say decane (C10 ). The components are assumed to have limited mutual solubility: some of the C10 vaporizes into the gas phase, and some of the CH4 or CO2 dissolves in the liquid phase. This problem is a modest generalization [32] of the well-known problem of displacement of oil by water (solved ﬁrst by Buckley and Leverett [10], see Lake [62, pp 130-142] for a complete description of the problem in a form similar to that used here, or see Dake [17, pp. 356-372] for the conventional description). The two-component problem is relatively straightforward because the entire displacement takes place on a single tie line. (Recall that the pressure at which the phase equilibrium is evaluated is taken to be constant – see Fig. 3.1.) The solution to the two-component ﬂow reappears as a portion of the solution in multicomponent problems, and hence, it is outlined in some detail here. We begin by solving Eq. 2.4.4 for ﬂow of two mutually soluble components with no volume change on mixing. The basic solution is derived in Section 4.1. The properties of shocks, discontinuities in composition and saturation that arise in the solutions, are examined in Section 4.2. Section 4.3 describes how the behavior of the solution depends on the initial and injection conditions. Finally, Section 4.4 examines how displacement behavior changes when components change volume as they transfer between phases. 43

44

4.1

CHAPTER 4. TWO-COMPONENT GAS/OIL DISPLACEMENT

Solution by the Method of Characteristics

For the two-component problem, Eqs. 2.4.4 reduce to a single quasilinear equation, ∂C1 ∂F1 + = 0, ∂τ ∂ξ

(4.1.1)

where the overall composition, C1 , in the two-phase region is given by Eq. 2.4.5, C1 = c11 S1 + c12 (1 − S1 ),

(4.1.2)

and the overall fractional ﬂow of component 1 in the two-phase region is Eq. 2.4.6, F1 = c11 f1 + c12 (1 − f1 ).

(4.1.3)

The equilibrium phase compositions, c11 and c12 are known and ﬁxed by the assumption of local chemical equilibrium. For any mixture that forms only a single phase (C1 < c12 or C1 > c11 ), F1 = C1 .

(4.1.4)

Rearrangement of Eq. 4.1.2, which is really just another tie-line material balance like Eq. 3.4.3 or Eq. 3.5.1, shows that S1 =

C1 − c12 . c11 − c12

(4.1.5)

Thus, the saturation is a function only of the overall composition, C1 . The overall fractional ﬂow, F1 , is also a function of C1 only. For horizontal ﬂow, for example, Eq. 2.2.6 reduces to f1 =

kr1 /µ1 . kr1 /µ1 + kr2 /µ2

(4.1.6)

The phase viscosities are ﬁxed, because the phase compositions are ﬁxed, and the phase relative permeabilities are generally assumed to be functions of saturation only. Therefore, the value of C1 determines the value of F1 , because F1 depends only on f1 , which depends only on S1 , which in turn depends only on C1 . Because F1 is a function of C1 only, Eq. 4.1.1 can be written dF1 ∂C1 ∂C1 + = 0, ∂τ dC1 ∂ξ

(4.1.7)

Because C1 is a function of ξ and τ , we can also write an expression for the ordinary derivative of C1 , ∂C1 dτ ∂C1 dξ dC1 = + dη ∂τ dη ∂ξ dη

(4.1.8)

where η is a parameter. Term by term comparison of Eq. 4.1.8 with Eq. 4.1.7 indicates that

4.1. SOLUTION BY THE METHOD OF CHARACTERISTICS

dC1 dη dτ dη dξ dη

45

= 0,

(4.1.9)

= 1,

(4.1.10)

=

dF1 . dC1

(4.1.11)

Eqs. 4.1.9–4.1.11 are known as the characteristic equations. Integration of Eqs. 4.1.9–4.1.11 gives expressions for C1 , τ , and ξ in terms of the parameter η. In other words, τ (η) and ξ(η) are characteristic curves along which the derivative of C1 is dC1 /dη. Eq. 4.1.9 indicates that the derivative is zero, so C1 does not change as η changes along the curves, and therefore, the characteristic curves are curves along which C1 is constant. Elimination of η from the expressions for τ and ξ obtained by integrating Eqs. 4.1.10 and 4.1.11 gives the trajectory in time and space for some constant value of C1 . If the value of C1 is ﬁxed, then the value of dF1 /dC1 is also ﬁxed. Hence, Eqs. 4.1.10 and 4.1.11 can be integrated easily, and η can be eliminated to obtain a relationship between τ and ξ, ξ=

dF1 τ + ξ0 , dC1

(4.1.12)

dF1 is evaluated. Eq. 4.1.12 is the where ξ0 is the initial position of the composition C1 for which dC 1 equation of a straight line, and hence, the characteristic curves are straight lines. The velocity at which the composition C1 propagates along a characteristic curve is

dF1 dξ = , dτ dC1

(4.1.13)

and that velocity remains constant for a given value of C1 . The velocity at which a given composition propagates is often called the wave velocity of that composition . When two phases with diﬀerent compositions are ﬂowing simultaneously, the wave velocity of the overall composition of the twophase mixture diﬀers from the physical ﬂow velocity of any of the phases . The wave velocity indicates how fast that overall composition moves, not how fast the individual phases, which have compositions quite diﬀerent from the overall composition, are moving. Eqs. 4.1.12 and 4.1.13 give the impression that the construction of a solution for a twocomponent gas displacement problem is quite simple. For each initial composition C1 , the propadF1 is evaluated, and all subsequent spatial positions of that composition are given gation velocity dC 1 by Eq. 4.1.12. So far, however, we have not considered how the discontinuity between the initial and injection compositions, present initially at the inlet, propagates through the porous medium. To make the illustration more concrete, we consider the following speciﬁc problem: pure component 1 (C1inj = 1) displaces a single-phase mixture of components 1 and 2 containing 5 volume percent component 1 (C1init = 0.05). The equilibrium vapor phase contains 95 percent component 1 (c11 = 0.95), and the equilibrium liquid phase contains 20 percent component 1 (c12 = 0.20). We also assume that the relative permeability functions are given by the following simple relations:

46

CHAPTER 4. TWO-COMPONENT GAS/OIL DISPLACEMENT

S1 − Sgc 1 − Sgc − Sor

kr1 =

2

,

kr1 = 0, kr1 = 1,

1 − S1 − Sor 1 − Sgc − Sor

kr2 =

2

,

Sgc < S1 < 1 − Sor ,

(4.1.14)

S1 < Sgc ,

(4.1.15)

S1 > 1 − Sor ,

(4.1.16)

Sgc < S1 < 1 − Sor ,

(4.1.17)

kr2 = 0,

1 − S1 < Sor ,

(4.1.18)

kr2 = 1,

1 − S1 > 1 − Sgc .

(4.1.19)

Relative permeability functions that are functions of phase saturations raised to a power are known as Corey type curves [13]. In Eqs. 4.1.14–4.1.19, Sgc represents a critical gas saturation, below which the vapor phase has zero relative permeability. Similarly, Sor is a residual oil saturation. When the liquid saturation is less than Sor , the liquid phase relative permeability is zero. For this example, we assume that Sor = 0.10 and Sgc = 0.05. Substitution of Eqs. 4.1.14-4.1.19 into Eq. 4.1.6 gives f1 = 0, f1 = f1

(S1 − Sgc = 1,

)2

(S1 − Sgc , + (1 − S1 − Sor )2 /M

)2

S1 < Sgc ,

(4.1.20)

Sgc < S1 < 1 − Sor ,

(4.1.21)

S1 > 1 − Sor ,

(4.1.22)

where M is the viscosity ratio, µ2 /µ1 . For this example, M = 2. Fig. 4.1 is a plot of F1 as a function of C1 (see Fig. 4.12 for a detailed description of the key points on this plot). Within the two-phase region in Fig. 4.1, the fractional ﬂow function has the typical S-shape often seen in measured fractional ﬂow curves. Diﬀerentiation of Eqs. 4.1.3 and 4.1.5 with respect to C1 shows that df1 df1 dS1 df1 dF1 = (c11 − c12 ) = (c11 − c12 ) = . (4.1.23) dC1 dC1 dS1 dC1 dS1 Eqs. 4.1.20-4.1.22 can be diﬀerentiated easily to obtain expressions for df1 /dS1 . The resulting values of dF1 /dC1 , plotted in Fig. 4.2, are dF1 = 1, dC1 dF1 = 0, dC1 df1 dF1 = , dC1 dS1 dF1 = 0, dC1 dF1 = 1, dC1

0 < C1 < c12 ,

S1 = 0, (4.1.24)

c12 < C1 < Sgc (c11 − c12 ) + c12 ,

0 < S1 < Sgc , (4.1.25)

Sgc (c11 − c12 ) + c12 < C1 < Sor (c12 − c11 ) + c11 ,

Sgc < S1 < 1 − Sor , (4.1.26)

Sor (c12 − c11 ) + c11 < C1 < c11 ,

1 − Sor < S1 < 1, (4.1.27)

c11 < C1 < 1,

S1 = 1. (4.1.28)

4.1. SOLUTION BY THE METHOD OF CHARACTERISTICS

47

Overall Fractional Flow of Component 1, F1

1.0

0.8

0.6

0.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Overall Volume Fraction of Component 1, C1

Figure 4.1: Overall fractional ﬂow function, F1 (C1 ). The wave velocity is one in the single-phase regions at high and low C1 concentration, is zero in the small regions at either end of the two-phase region where the relative permeability of one of the phases is zero, and is a nonlinear function of composition within the two-phase region. Fig. 4.2 indicates that more than one composition can have a given wave velocity, a fact that will have important implications for the construction of solutions. The behavior of characteristics is commonly plotted on t-x (or in this case, τ -ξ) diagrams. Fig. 4.3 shows the characteristics associated with the initial composition (C1 = 0.05). They emanate from the τ = 0 axis for ξ > 0, and they all have the same unit slope, because the initial composition is a constant value in the single-phase region. For compositions in the single-phase region, diﬀerentiation of Eq. 4.1.4 according to Eq. 4.1.13 leads immediately to the conclusion that single phase compositions propagate with unit velocity. The characteristics associated with the injected composition are shown in Fig. 4.4. Again, that composition is in the single-phase region, and hence, the characteristic lines are parallel, and the compositions on them all have unit wave velocity. The stage is now set for evaluating the eﬀect of the step change in composition that occurs at the inlet, and we ask next how the compositions associated with that discontinuity propagate through the porous medium. The characteristics associated with the discontinuity initially present at the origin are shown in Fig. 4.5. That ﬁgure is drawn based on the assumption that all the compositions that lie between the upstream and downstream compositions at the discontinuity

48

CHAPTER 4. TWO-COMPONENT GAS/OIL DISPLACEMENT 2.5

Wave Velocity, dF1/dC1

2.0

1.5

1.0

0.5

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Overall Volume Fraction of Component 1, C1

Figure 4.2: Derivative of the overall fractional ﬂow function, dF1 /dC1 . propagate, each with the wave velocity appropriate to the speciﬁc intermediate composition. In other words, the discontinuity can be viewed as the limit as → 0 of a change in composition from the upstream value, C1 = 1, to the downstream value, C1 = 0.05 over a distance . In the limit the characteristics for values of C1 between 0.05 and 1.0 all emanate from the origin. Construction of the solution is made more diﬃcult by two facts illustrated in Figs. 4.2 and 4.5. First, all but one of the characteristics associated with the discontinuity have wave velocities diﬀerent from one, and hence the characteristics associated with the initial discontinuity intersect those of the initial and injection compositions. Second, there are at least two compositions that have any given wave velocity between zero and the maximum value, as Figs. 4.2 and 4.5 illustrate. Both conditions seem to imply that more than one composition will exist at a single location, a physical impossibility. The inconsistency is resolved by the propagation of discontinuities known as shocks, the subject of the next section.

4.2

Shocks

When characteristics cross on a t-x diagram, a shock must form . The next task then is to ﬁnd the compositions that form on either side of the shock and to determine how the shock propagates. The trajectory of a shock is found from a material balance equation similar to Eq. 4.1.1, except that the balance is an integral one, because the diﬀerential version of the material balance is no

4.2. SHOCKS

49 1.0

0.8

0.

0.

C

05

1=

0.

C

05

1=

0.

05

C

1=

0.4

0.

C

05

1=

τ

C

05

1=

0.

05

0.6

C

1=

0.

C

05

1=

0.2

0.0 0.0

0.4

0.2

0.6

0.8

1.0

ξ

Figure 4.3: Characteristics for the initial composition.

1=

1.

0

C

1=

1.

0

C

1=

1.

0

C

0.8

1=

1.

0

C

1=

1.

0

1.0

1=

1. 0

C

τ

1=

1. 0

C

1=

1.

0

C

0.6

C

0.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

ξ

Figure 4.4: Characteristics for the injection composition.

50

CHAPTER 4. TWO-COMPONENT GAS/OIL DISPLACEMENT

1

.845

1

=0

.37

7,0

.82

.326, 0

C

1=

τ

0.

44

5,

0.

82

C1

0.6

C1 =0

0.8

C1=0.282,0. 862

1.0

0.4

628

0. C 1= 0.2

0.0 0.0

0.4

0.2

0.6

0.8

1.0

ξ

Figure 4.5: Characteristics for the discontinuity present initially at the inlet. longer appropriate because the spatial derivatives that appear in it are not deﬁned. Consider the ﬂow situation illustrated in Fig. 4.6. At time, τ , the shock is located at distance, ξ, and it moves during a period, ∆τ , to a new position, ξ + ∆ξ. The composition of the ﬂuid on the upstream side of the shock is Ci = CiII , and the composition on the downstream side is Ci = CiI . Now we write a material balance for component i for a control volume that includes the porous medium between ξ and ξ + ∆ξ. Because there is no accumulation of material at the shock, the change in the amount of component i present in the control volume (per unit area of porous medium) is exactly balanced by the net inﬂow of component i, ∆ξ(CiII − CiI ) = ∆τ (FiII − FiI ).

(4.2.1)

In the limit as ∆ξ and ∆τ go to zero, the velocity of the shock is obtained by rearrangement of Eq. 4.2.1, Λ=

F II − FiI dξ = iII . dτ Ci − CiI

(4.2.2)

Eq. 4.2.2, called a jump condition or a Rankine-Hugoniot relation, is an integral version of the

4.2. SHOCKS

51

Ci=CΠ i

shock at τ

shock at τ+∆τ

Ci=CΙi ξ

ξ+∆ξ

Figure 4.6: Motion of a shock. original conservation equation. In other words, it is a statement that volume is conserved across the shock, just as Eq. 4.1.1 states that volume is conserved at locations where all the derivatives exist. Eq. 4.2.2 says that the velocity at which the shock propagates is set by the slope of a line that connects the two states on either side of the shock on a plot of F1 against C1 such as that shown in Fig. 4.1. Now we apply the jump condition to determine what happens at the leading edge of the displacement zone, where fast characteristics (the characteristics in Fig. 4.5 that have high values of dF1 /dS1 ) intersect the characteristics for the initial composition. Point a in Fig. 4.7 is the initial composition, which is the composition on the downstream side of the leading shock, and points b, c, d, e, f, and g are possible composition points for the ﬂuid on the upstream side of the shock. Any of the shock constructions shown in Fig. 4.7 satisﬁes Eq. 4.2.2. Hence some additional reasoning is required to select which shock is part of a unique solution to the ﬂow problem. Two physical ideas play a role in that reasoning. The ﬁrst is simply an observation that compositions that make up the downstream portion of the solution must have moved more rapidly than compositions that lie closer to the inlet. If not, slow-moving downstream compositions would be overtaken by faster compositions upstream. The idea is frequently stated [31] as a Velocity Constraint: Wave velocities in the two-phase region must decrease monotonically for zones in which compositions vary continuously as the solution composition path is traced from downstream compositions to upstream compositions . When the velocity constraint is satisﬁed, the solution will be single-valued throughout. Composition variations that satisfy the velocity constraint are sometimes described as compatible waves, and the velocity constraint may also be called a compatibility condition. The second idea is that a shock can exist only if it is stable in the sense that it would form again if it were somehow smeared slightly from a sharp jump, as might happen if a small amount of physical dispersion were present, for example. That idea can be stated in terms of wave velocities [67, 83, 106] as an Entropy Condition: Wave velocities on the upstream side of the shock must be greater

52

CHAPTER 4. TWO-COMPONENT GAS/OIL DISPLACEMENT than (or equal to) the shock velocity and wave velocities on the downstream side must be less than (or equal to) the shock velocity. (In the examples considered here, the wave velocity can be equal to the shock velocity on only one side of the shock at a time.)

For the application considered here, the condition really has nothing to do with the thermodynamic entropy function, but the name has been universally used in descriptions of solutions to hyperbolic conservation laws since the ideas behind entropy conditions were ﬁrst derived for compressible ﬂuid ﬂow problems in which entropy must increase across the shock. Consider what would happen to a shock that was slightly smeared if the entropy condition were not satisﬁed. Slow-moving compositions upstream of the shock would be left behind by fast-moving compositions downstream of the shock, and as a result, the shock would pull itself apart. Hence, the entropy condition must be satisﬁed if a shock is to be stable. A shock that does satisfy the entropy condition is said to be self-sharpening. For a detailed discussion of the various mathematical forms in which entropy conditions can be expressed, see the review given by Rhee, Aris and Amundson [106, pp. 213–220 and pp. 341–348]. We now apply the velocity constraint and the entropy condition to obtain a unique solution for two-component displacement. Fig. 4.8 illustrates possible solutions for the leading shocks indicated in Fig. 4.7. Consider, for example, a shock that connects downstream composition a and upstream composition b. The top left panel of Fig. 4.8 shows the location of the shock at some ﬁxed time and also shows how the solution would behave if the concentration of C1 increased smoothly upstream of the shock. The wave velocity, Λ, of the shock (Eq. 4.2.2) is given by the slope of the chord that connects points a and b on Fig. 4.7. That velocity is clearly less than one, and hence the a→b shock moves more slowly than the single-phase compositions downstream of the shock, which have unit velocity. The wave velocity of the composition just upstream of the shock is given by dF1 /dC1 at point b. That velocity is lower still than the wave velocity of the shock. Thus, the a→b shock violates the entropy condition. As the C1 concentration upstream of the shock increases, however, the wave velocities increase to values greater than the shock velocity, a variation that produces compositions that violate the velocity constraint. Hence, a solution that includes a shock from a to b followed by a continuously varying composition violates both the velocity constraint and the entropy condition and can be ruled out, therefore. The a→c, a→d, and a→e shocks all satisfy the entropy condition, but all three violate the velocity constraint, as the proﬁles in Fig. 4.8 show. The a→g satisﬁes the velocity constraint, but it violates the entropy condition because the wave velocity of the upstream composition is lower than the shock velocity. Hence, the only remaining possible solution is that shown for the a→f shock. The point f is the point at which the chord drawn from point a is tangent to the overall fractional ﬂow curve. The a→f shock does satisfy the entropy condition, but it does so in a special way. The wave velocity of the composition C1 of point f is equal to the shock velocity, because the shock velocity is given by the slope of the tangent a–f, and that chord slope is the same as dF1 /dC1 at point f. A shock in which the shock velocity equals the wave velocity on one side of the shock is sometimes called a semishock [106, pp. 217–219] , an intermediate discontinuity [40], or a tangent shock [82]. Because the leading shock must be a semishock if it is to satisfy the velocity constraint and the entropy condition, the composition of the ﬂuid on the upstream side of the shock can be found easily by solving

4.2. SHOCKS

53 1.0

a

Overall Fractional Flow of Component 1, F1

g f 0.8

a

a

e

0.6

0.4

a

b

0.2

a a c

d

a

a

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Overall Volume Fraction of Component 1, C1

Figure 4.7: Possible shocks from the initial state at point a.

F II − FiI dF1 II | = iII . dC1 Ci − CiI

(4.2.3)

The tangent construction described in Eq. 4.2.3 and shown in Fig. 4.7 is equivalent to the wellknown Welge tangent construction [133] used to solve the problem of Buckley and Leverett [10] for water displacing oil. Just as a shock was required in order to make the solution single-valued at the leading edge of the transition zone, another shock is required at the trailing edge. The characteristics in Fig. 4.4 for the injection composition intersect the characteristics in Fig. 4.5 for slow moving compositions, C1 , greater than the shock composition. Reasoning similar to that for the leading shock shows that the trailing shock also is a semishock, this time with the wave velocity on the downstream side of the shock equal to the shock velocity. In fact, similar arguments indicate that a shock must form any time the number of phases changes for the fractional ﬂow relation used here. Fig. 4.9 shows the resulting tangent constructions for the leading (a→b) and trailing shock (c→d). Fig. 4.10 gives the completed solution proﬁles of S1 and C1 . Each proﬁle includes a zone of constant state with the initial composition ahead of the leading shock, a zone of continuous variation of overall composition and saturation between the leading shock and the trailing shock, and ﬁnally another zone of constant state with the injection composition behind the trailing shock. The solution in Fig. 4.10 is reported as a function of ξ/τ , which is the wave velocity of the corresponding

54

CHAPTER 4. TWO-COMPONENT GAS/OIL DISPLACEMENT 1

C1

1

a-b

a-e

0

0 0

1

0

ξ

1

ξ 1

C1

1

a-c

a-f

0

0 0

1

0

ξ

1

ξ 1

C1

1

a-d

a-g

0

0 0

1

ξ

0

1

ξ

Figure 4.8: Composition proﬁles for the leading shocks to various two-phase compositions. value of C1 . In this homogeneous, quasilinear problem, the wave velocity of any composition is constant, and hence the position of any composition that originated at the inlet must be a function of ξ/τ only. In fact, Lax [67] showed that the solution to a quasilinear Riemann problem is always a function of ξ/τ only. The spatial position of a given composition C1 can be obtained simply by multiplying the corresponding value of ξ/τ by the value of τ at which the solution is desired. Another version of the solution is shown in Fig. 4.11, which includes a τ -ξ diagram and a plot of the C1 proﬁle at τ = 0.60. Shown in the τ -ξ portion of Fig. 4.11 are the trajectories of the leading and trailing shocks and a few of the characteristics. The locations, ξ, of the shocks and the compositions associated with speciﬁc characteristics can be read directly from the t-x diagram for a particular value of τ , as Fig. 4.11 illustrates. From Fig. 4.11 it is easy to see that as the ﬂow proceeds, the solution retains the shape shown in the proﬁles of Figs. 4.10 and 4.11, but the entire solution stretches as fast-moving compositions pull away from slow-moving ones. That behavior is typical of problems in which convective phenomena dominate the transport. Fig. 4.11 also illustrates the point that when the entropy condition is satisﬁed for a particular

4.2. SHOCKS

55 da

1.0

a

Overall Fractional Flow of Component 1, F1

c b

a

0.8

0.6

0.4

0.2

a

a

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Overall Volume Fraction of Component 1, C1

Figure 4.9: Leading and trailing shock constructions.

shock, characteristics on either side of the trajectory of a shock either impinge on the shock trajectory or are at least parallel to the shock trajectory. In the case of the leading shock, for example, the characteristics of the initial composition, which lies downstream of the shock, intersect the shock trajectory, while characteristic just upstream of the shock overlaps the shock trajectory. The reverse is true at the trailing shock. Between the trajectories of the shocks is the fan of characteristics associated with the continuous variation of composition, which is known as a spreading wave, a rarefaction wave, or an expansion wave. Because the characteristics all emanate from a single point, the origin, they are also referred to as a centered wave. The change in slope of the characteristics in the spreading wave reﬂects the fact that the slope of the fractional ﬂow curve drops rapidly over a fairly narrow range of composition (see Fig. 4.2). As a result, the wave velocity declines signiﬁcantly during the relatively small composition change between the leading and trailing shocks. In the solution shown in Fig. 4.10 the overall compositions and saturations vary in the two-phase region, but the phase compositions do not. They are ﬁxed by the speciﬁed phase equilibrium. It is the diﬀering amounts of the two phases present and ﬂowing that change the overall composition and fractional ﬂow.

56

CHAPTER 4. TWO-COMPONENT GAS/OIL DISPLACEMENT 1.0

d

C1

c

b

0.5

a

0.0 0.0

0.5

1.0

1.5

ξ/τ 1.0

d

S1

c

b

0.5

0.0 0.0

a 0.5

1.0

1.5

ξ/τ

Figure 4.10: Solution composition and saturation proﬁles.

4.3

Variations in Initial or Injection Composition

In the binary gas/oil displacement problem, the leading shock forms because some two-phase mixtures of injected ﬂuid with initial ﬂuid move rapidly and overtake the initial composition. The trailing shock forms because some Component 2 can evaporate into the unsaturated injected vapor. How fast the leading and trailing shocks move depends on the initial and injection compositions. In this section we examine brieﬂy the sensitivity of the solution to the binary displacement problem to changes in the initial and injection compositions. Fig. 4.12 shows a set of key points on the fractional ﬂow curve. Points a, b, c, and d are from the solution discussed in the previous section. Points a and d are the initial and injection values, and points b and c are the tangent points for the leading and trailing semishocks. Point e is the saturated vapor phase, and point i is the saturated liquid. Point f corresponds to S1 = 1 − Sor , and point h is that at which S1 = Sgc . Point g is the intersection of the F1 = C1 line with the overall fractional ﬂow curve. Point j is the inﬂection point in the fractional ﬂow curve. It corresponds to the maximum in dF1 /dC1 shown in Fig. 4.2. Fig. 4.13 shows examples of the solutions that result when the initial composition is ﬁxed at a C1 and the injection composition is varied. The six panels in Fig. 4.13 illustrate changes in the appearance of the composition proﬁles for injection compositions with decreasing volume fractions, inj C1 . Fig. 4.14 shows the corresponding characteristic (τ -ξ) diagrams. The following observations can be made for injection compositions in the regions bounded by the key points in Fig. 4.13: d to e: For injection compositions in the range 1 > C1inj > C1e , the solution still

4.3. VARIATIONS IN INITIAL OR INJECTION COMPOSITION

57

.71

0. 68

Trailing Shock

1.0

=

=0

0.8

1

C

1

C

a

a

a

a

τ

0.6

Leading Shock

0.4 0.2 0.0 0

1

1.0

C1

a

a

a

a

0.5

0.0 0 0.0

0.5

1.0

ξ

Figure 4.11: Evaluation of the solution at a speciﬁc time, τ , from the τ -ξ diagram. includes leading and trailing semishocks connected by a spreading wave (see the top left panel of Fig. 4.13), and the τ -ξ diagram shown in the corresponding panel in Fig. 4.14 is qualitatively similar to Fig. 4.11. As C1inj is decreased, the trailing shock speed decreases, reaching zero when the injected ﬂuid is vapor saturated with component 2 (C1inj = C1e ). The leading portion of the solution is unchanged, however. e to f: Compositions in the range C1e > C1inj > C1f have zero wave velocity because dF1 /dC1 = 0. There is a trailing shock from the injection composition to C1e , but it has zero wave velocity, and hence the fan of characteristics in Fig. 4.14 extends all the way to the ξ = 0 axis. The leading portion of the solution remains unchanged. f to b: An injection composition between f and b has nonzero wave velocity, and as a result, the solution in the lower left panel of Fig. 4.13 shows a zone of injection compositions at the upstream end that all propagate with the same wave velocity. That portion of the solution has a set of parallel characteristics in Fig. 4.14 that emanate from the ξ = 0 axis. The fan of characteristics that represents the spreading wave terminates at the characteristic that represents that propagation of the injection composition. When C1inj = C1b , the entire solution upstream of the leading shock is that zone of constant

58

CHAPTER 4. TWO-COMPONENT GAS/OIL DISPLACEMENT da

1.0

a

Overall Fractional Flow of Component 1, F1

c

a

a

f

e

b

a

0.8

0.6

a j ag

0.4

i

a a

0.2

h a

a

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Overall Volume Fraction of Component 1, C1

Figure 4.12: Composition ranges for variation of injection and initial compositions. composition with C1 = C1b . The leading shock velocity is still unchanged, however. b to g: A leading semishock is no longer possible for C1b > C1inj > C1g , because continuous variation from the tangent shock point to the injection composition is prohibited by the velocity rule. A leading shock to the injection composition followed by a set of constant compositions at the injection composition satisﬁes the entropy condition and velocity constraint. The top right panel in Fig. 4.14 indicates, for example, that the characteristics associated with the injection composition intersect the leading shock trajectory, as do the characteristics associated with the initial composition, an indication that the entropy condition is satisﬁed. The leading shock velocity is given by Eq. 4.2.2 with the known compositions C1init and C1inj . The leading shock velocity is now lower than that of the leading semishocks that form for C1inj > C1b . When C1inj = C1g , the leading shock has unit velocity. g to h: A leading shock directly from the injection composition to the initial composition is no longer possible because it would violate the entropy condition. A shock from the initial composition to the injection composition would have a velocity less than one. The characteristics of the initial composition (see the middle right panel of Fig. 4.14)would not intersect the shock trajectory, and hence, the entropy condition would not be satisﬁed. The only path available that does not violate the entropy condition is

4.3. VARIATIONS IN INITIAL OR INJECTION COMPOSITION 1

59

1 Cinj 1 =0.560

d-e

b-g

C1

Cinj 1 =0.975

0

0 0

1

0

ξ

1

ξ

1

1 Cinj 1 =0.920

Cinj 1 =0.320

C1

e-f

g-h

0

0 0

1

0

ξ

1

ξ

1

1 Cinj 1 =0.770

Cinj 1 =0.215

C1

f-b

h-i

0

0 0

1

ξ

0

1

ξ

Figure 4.13: Eﬀect of changes in injection composition. a leading shock with unit velocity to the saturated liquid composition, C1i , followed by a slower trailing shock to the injection composition. The characteristics of the injection composition intersect the trailing shock, and the characteristics of the initial composition intersect the leading shock. The zone between the two shocks is what is known as a zone of constant state. The composition i has two wave velocities: one is the leading shock velocity, and the other is the trailing shock velocity. h to i: The situation is the same as for g-h, except that the trailing shock has zero velocity. The trailing shock is a jump from the injection composition to the saturated liquid composition, C1i . Fig. 4.13 indicates that the form of the solution changes signiﬁcantly as the injection composition changes. Solution behavior also changes if the initial composition is changed. Fig. 4.15 illustrates what happens if the injection composition is ﬁxed at C1inj = C1d = 1, and the initial composition is varied in ranges bounded by the key points shown in Fig. 4.12. Four composition intervals are

60

CHAPTER 4. TWO-COMPONENT GAS/OIL DISPLACEMENT d-e

b-g

1

τ

1

0

0 0

1 e-f

1 g-h

1

τ

1

0

0

0 0

1 f-b

1 h-i

1

τ

1

0

0

0 0

1

ξ

0

1

ξ

Figure 4.14: τ -ξ diagrams for displacements illustrated in Fig. 4.13. important: a to i: As the amount of component 1 in the initial mixture increases, the velocity of the leading semishock also increases slightly (the slope of the tangent drawn from the initial composition to the fractional ﬂow curve increases), and the composition on the upstream side of the leading shock decreases slightly below point b. The remainder of the composition proﬁle upstream of the leading shock is unaﬀected. i to j: Increasing C1 from i toward j causes the leading shock speed to increase substantially and the shock composition to approach j. j to c: For C1init greater than the inﬂection composition, there is no leading shock. The leading portion of the solution is simply a spreading wave. c to e: When C1init > C1c , the trailing shock is no longer a semishock. Instead that evaporation shock is what is known as a genuine shock, a jump from C1init to C1inj , with

4.4. VOLUME CHANGE

61

1

C1

1

a-i

j-c

Cinit 1 =0.18

Cinit 1 =0.62

0

0 0

1

0

1

ξ

ξ 1

C1

1

i-j

c-e

Cinit 1 =0.30

Cinit 1 =0.80

0

0 0

1

0

1

ξ

ξ

Figure 4.15: Eﬀect of changes in initial composition. velocity given by Eq. 4.2.2. The trailing shock velocity increases as C1init is increased, reaching unit velocity when C1init = C1e . Figs. 4.13 and 4.15 indicate that solutions for binary gas/oil displacement show considerable variation as the injection and initial conditions are changed. Many of the features of these binary solutions reappear in the multicomponent solutions that are considered in subsequent chapters, and hence a detailed understanding of the binary solutions is useful underpinning for the analysis of more complex multicomponent ﬂows.

4.4

Volume Change

When components change volume as they transfer from one phase to another, volume is not conserved, and the appropriate balance equation on moles of component i is Eq. 2.3.9 written for the two components, ∂G1 ∂H1 + ∂τ ∂ξ ∂G2 ∂H2 + ∂τ ∂ξ

= 0,

(4.4.1)

= 0,

(4.4.2)

where Gi = xi1 ρ1D S1 + xi2 ρ2D (1 − S1 ),

(4.4.3)

62

CHAPTER 4. TWO-COMPONENT GAS/OIL DISPLACEMENT Hi = vD (xi1 ρ1D f1 + xi2 ρ2D (1 − f1 )).

4.4.1

(4.4.4)

Flow Velocity

The local ﬂow velocity, vD , appears in both balance equations in the deﬁnition of Hi . That velocity changes when components change volume as they transfer between phases or as the composition of a phase changes. When compositions change along a tie line, however, the local ﬂow velocity remains constant [22]. When a composition variation remains on a single tie line within the two-phase region (as it must for this binary problem where there is only one tie line), the phase composition, xi1 and xi2 , and the dimensionless molar phase densities, ρ1D and ρ2D , remain constant at the values for the equilibrium phases. As a result, substitution of the deﬁnitions for Gi and Hi (Eqs. 4.4.3 and 4.4.4) into Eqs. 4.4.1 and 4.4.2 followed by rearrangement gives ∂S1 ∂ + ∂τ ∂ξ

vD f1 +

x12 ρ2D x11 ρ1D − x12 ρ2D

= 0,

(4.4.5)

= 0.

(4.4.6)

and ∂S1 ∂ + ∂τ ∂ξ

vD f1 +

x22 ρ2D x21 ρ1D − x22 ρ2D

Subtraction of Eq. 4.4.6 from Eq. 4.4.5 yields an expression for the spatial derivative of the velocity,

x22 ρ2D x12 ρ2D − x11 ρ1D − x12 ρ2D x21 ρ1D − x22 ρ2D

∂vD = 0. ∂ξ

(4.4.7)

It is convenient to rewrite Eq. 4.4.7 in terms of the equilibrium K-values, x11 = K1 x12 and x21 = K2 x22 , which gives

ρ2D ρ2D − K1 ρ1D − ρ2D K2 ρ1D − ρ2D

∂vD = 0. ∂ξ

(4.4.8)

D As long as K1 = K2 , which must be true if two phases are to form, Eq. 4.4.8 shows that ∂v ∂ξ = 0. Hence, the local ﬂow velocity is constant for composition variations in the two-phase region along a single tie line. This behavior results from the fact that the phase densities remain constant for mixtures on a single tie line in the two-phase region.

4.4.2

Characteristic Equations

The characteristic equations can now be obtained just as they were in Section 4.1. Arguments similar to those given in Section 4.1 indicate that H1 is a function of G1 only, and hence, ∂G1 dH1 ∂G1 + = 0. ∂τ dG1 ∂ξ

(4.4.9)

G1 is a function of ξ and τ , and therefore, ∂G1 dτ ∂G1 dξ dG1 = + . dη ∂τ dη ∂ξ dη Comparison of Eqs. 4.4.9 and 4.4.10 gives the characteristic equations,

(4.4.10)

4.4. VOLUME CHANGE

63

dG1 dη dτ dη dξ dη

= 0,

(4.4.11)

= 1,

(4.4.12)

=

dH1 . dG1

(4.4.13)

Comparison of Eqs. 4.4.11–4.4.13 with the corresponding equations for constant volume ﬂow, Eqs. 4.1.9–4.1.11, indicates that within the two-phase region, at least, the solutions for ﬂow with and without volume change have similar structure. The similarity can be seen more clearly if dH1 /dG1 is evaluated. Diﬀerentiation of Eq. 4.4.4 gives df1 dH1 = vD (x11 ρ1D − x12 ρ2D ) . dG1 dG1

(4.4.14)

Eq. 4.4.3 can be rearranged to show that S1 is a function of G1 only, and therefore, df1 dS1 1 df1 df1 = = . dG1 dS1 dG1 x11 ρ1D − x12 ρ2D dS1

(4.4.15)

Substitution of Eq. 4.4.15 into Eq. 4.4.14 shows that df1 dH1 = vD . dG1 dS1

(4.4.16)

Hence for compositions within the two-phase region, the wave velocity is simply the wave velocity for constant volume ﬂow scaled by the appropriate local ﬂow velocity within the two-phase region. The distinction between ﬂow velocity and wave velocity is an important one . The ﬂow velocity is the total volumetric ﬂow rate of all the phases per unit area. The wave velocity is the speed at which a given composition propagates. The two are very diﬀerent. When volume is not conserved, the ﬂow velocity does not change when the composition varies along a single tie line, but it does change at shocks that enter or leave the two-phase region. Hence, the next step is to determine how ﬂow velocity varies across the shocks.

4.4.3

Shocks

Consider the trailing shock from the injection composition, Gd1 , to composition, Gc1 , in the two-phase region. A shock balance indicates that the shock wave velocity, Λcd , is given by Λcd =

Hic − Hid , Gci − Gdi

i = 1, 2.

(4.4.17)

Eqs. 4.4.17 can be written for either component. When volume is not conserved, the two equations are independent. As a result, the shock balances can be solved for both the downstream c . To show how that is done, it is convenient to write composition, Gci , and the ﬂow velocity, vD Hi = vD αi = vD

np j=1

xij ρjD fj .

(4.4.18)

64

CHAPTER 4. TWO-COMPONENT GAS/OIL DISPLACEMENT

c , we write Eqs. 4.4.17 for components 1 and 2 and eliminate Λcd , which gives To ﬁnd vD d αd − v c αc d αd − v c αc vD vD 1 2 D 1 D 2 = . Gd1 − Gc1 Gd2 − Gc2

(4.4.19)

d d is the injection ﬂow velocity, and by deﬁnition (Eq. 2.3.2), vD = 1. αd1 and Gd1 are the injection vD c once Gc and αc are determined, data, so Eq. 4.4.19 can be solved for vD 2 2 c vD =

αd2 (Gd1 − Gc1 ) − αd1 (Gd2 − Gc2 ) . αc2 (Gd1 − Gc1 ) − αc1 (Gd2 − Gc2 )

(4.4.20)

Application of the entropy condition and velocity constraint shows that if injection composition is single-phase vapor, the trailing shock is a semishock that satisﬁes c vD

αd − v c αc df1 = i d D ci , dS1 Gi − Gi

i = 1 or 2.

(4.4.21)

c can be eliminated from Eq. 4.4.21, and When component 2 is not present in the injection ﬂuid, vD it can then be solved to ﬁnd the composition at point c. Otherwise, Eqs. 4.4.20 and 4.4.21 can be c and point c. solved simultaneously for vD Similar manipulations give expressions that can be solved for the composition at the leading a , ahead of the shock. The wave velocity of the leading shock is shock, Gbi, and the ﬂow velocity, vD

Λab =

Hia − Hib , Gai − Gbi

(4.4.22)

where Gai and Hia are the initial values. For an initial composition in the single-phase region, the leading shock is a semishock determined by a αb1 (Ga2 − Gb2 ) − αb2 (Ga1 − Gb1 ) vD , = c vD αa1 (Ga2 − Gb2 ) − αa2 (Ga1 − Gb1 )

(4.4.23)

and αa (v a /v b ) − αb df1 = i Da D b i , dS1 Gi − Gi

i = 1 or 2.

(4.4.24)

b is known, as it is from the solution for the trailing shock because v b = v c , Therefore, when vD D D a b vD and Gi can be obtained by solving Eqs. 4.4.23 and 4.4.24. Thus, in a binary displacement, only three ﬂow velocities exist: the known injection velocity behind the trailing shock, a ﬁxed ﬂow velocity in the two-phase region, and a diﬀerent ﬂow velocity ahead of the leading shock.

4.4.4

Example Solution

To illustrate how volume change aﬀects ﬂow behavior, we consider displacement of a hydrocarbon, decane (C10 ), by a gas, carbon dioxide (CO2 ), at 500 psia (34 atm) and 160 F (71 C). Table 4.1 reports Peng-Robinson equilibrium phase compositions and the initial, injection and phase molar densities. Table 4.2 gives compositions, wave velocities, and ﬂow velocities for the solution with volume change, and Table 4.3 reports the corresponding values for the solution without volume change. In both cases, the fractional ﬂow curves were assumed to be Eqs. 4.1.20-4.1.22, with

4.4. VOLUME CHANGE

65

Table 4.1: Equilibrium Phase Compositions and Fluid Properties at 500 psia (34 atm) and 160 F (71 C) Fluid

xCO2

xC10

Initial Oil Equil. Liq. Equil. Vap. Injected Gas

0. 0.2733 0.9976 1.

1. 0.7267 0.0024 0.

ρ (gmol/l) 4.829 5.988 1.378 1.375

ρ (g/cm3) 0.6869 0.6910 0.0610 0.0605

µ (cp) 0.333 0.018 -

Sor = Sgc = 0. Overall mole fractions shown in Table 4.2 were calculated from the values of Gi by noting that zi =

Gi np

.

(4.4.25)

ρjD Sj

j=1

Overall compositions in Table 4.3 were calculated from volume fractions using the pure component densities in Table 4.1 according to ρci {ci1 S1 + ci2 (1 − S1 )}

zi = S1

nc

i=1

ρci ci1 + (1 − S1 )

nc

,

(4.4.26)

ρci ci2

i=1

where the phase volume fractions are given by Eq. 2.4.1 with the phase mole fraction data in Table 4.1. Fig. 4.16 and the wave velocity (dξ/dτ ) data in Tables 4.2 and 4.3 show that the composition proﬁles have similar appearances in the displacements with and without volume change, but the ﬂow proceeds more slowly when volume is not constant. In particular, the velocity of the leading shock is much lower when eﬀects of volume change are included. In fact, it has a wave velocity less than one, which means that more than one pore volume must be injected for the leading shock to reach the outlet (at ξ = 1) when volume is variable. The change in ﬂow velocity occurs because CO2 occupies much less volume when it is dissolved in the liquid phase than it does in the vapor phase [19]. When CO2 saturates the C10 present in the two-phase region, therefore, signiﬁcant volume is lost, and the ﬂow slows accordingly. As Table 4.2 and Fig. 4.16 show, the ﬂow velocity, vD , ahead of the leading shock is only about half the injection velocity. In both displacements there is a slow-moving trailing evaporation shock. It moves slowly because the solubility of C10 in CO2 is small. In other words, a large amount of CO2 must be injected to evaporate the remaining C10 . The velocity change at the trailing shock is small, however. The concentration of C10 in the vapor phase is so low that the volume change associated with the transfer of C10 to the vapor has minimal eﬀect on the ﬂow velocity. The values of vD in Table 4.2 indicate again that vD remains constant for compositions in the two-phase region. The wave velocities in Fig. 4.16 and Tables 4.2 and 4.3 also indicate that the displacement of C10 by CO2 is relatively ineﬃcient. While the leading shock moves with appreciable velocity,

66

CHAPTER 4. TWO-COMPONENT GAS/OIL DISPLACEMENT

1.0 d

z1

c

c

No volume change Volume change

0.5

b b

0.0 0.0

a 1.0

0.5

a 1.5

ξ/τ

S1

1.0

0.5

0.0 0.0

0.5

1.0

1.5

1.0

1.5

ξ/τ

vD

1.0

0.5

0.0 0.0

0.5

ξ/τ

Figure 4.16: Displacement of C10 by CO2 , with and without volume change as components transfer between phases.

4.5. COMPONENT RECOVERY

67

Table 4.2: Displacement of C10 by CO2 with Volume Change Label a b c d

zCO2 0.0000 0.3676 0.4088 0.5264 0.7020 1.0000

S1 0.0000 0.3941 0.5000 0.7000 0.8630 1.0000

dξ dτ

0.5097 0.9147 0.3710 0.0662 0.0063 1.0000

vD 0.5097 0.9999 0.9999 0.9999 0.9999 1.0000

τ < 1.0932 1.0932 2.6956 15.097 158.75 > 158.75

RC10 0.5574 0.6503 0.8087 1.0000 1.0000

Table 4.3: Displacement of C10 by CO2 without Volume Change Label a b c d

zCO2 0.0000 0.7214 0.7842 0.8703 0.9294 1.0000

S1 0.0000 0.3539 0.5000 0.7000 0.8375 1.0000

dξ dτ

1.0000 1.2967 0.3710 0.0662 0.0118 1.0000

vD 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

τ < 0.7712 0.7712 2.6954 15.096 85.092 > 85.092

QC10 0.7712 0.8336 0.9137 1.0000 1.0000

somewhat higher CO2 concentrations (and saturations) move much more slowly. As a result, C10 is recovered much more slowly after the arrival of the leading shock at the outlet.

4.5

Component Recovery

The amount of component i recovered at the outlet can be calculated from the composition and saturation proﬁles obtained as part of the solution to the Riemann problem. Just as the spatial distribution of compositions is found by solving a diﬀerential material balance, the recovery of individual components is obtained from an integral balance over the ﬂow length. The amount of any component recovered from the porous medium is simply the amount present initially plus the amount of that component injected during the time the ﬂow has taken place minus the amount of that component currently present in the porous medium. When volume change is neglected, for a porous medium of dimensionless length ξ = 1, the resulting expression for Q1 , the volume of component 1 recovered, is Q1 = C1init + F1inj τ −

1 0

C1 dξ.

(4.5.1)

Prior to the arrival of the leading shock, ﬂuid leaves the porous medium with the fractional ﬂow of the initial mixture. Because the initial composition is constant, the recovery of each component is just τ Fiinit . Breakthrough of injected ﬂuid occurs at τBT , when the leading shock arrives at the outlet, where ξ = 1. Accordingly,

68

CHAPTER 4. TWO-COMPONENT GAS/OIL DISPLACEMENT

1 . Λab After breakthrough, the integral in Eq. 4.5.1 can be evaluated as τBT =

1 0

τ Λcd

C1 dξ =

0

(4.5.2)

1

C1 dξ +

= C1inj Λcd τ +

1

τ Λcd

τ Λcd

C1 dξ

C1 dξ.

(4.5.3) (4.5.4)

The integral in Eq. 4.5.4 is evaluated through integration by parts, which gives 1 τ Λcd

C1 dξ = C1 ξ

]1τ Λcd

−

C out 1 C1c

ξdC1 ,

(4.5.5)

where C1out is the overall composition at ξ = 1 at time τ . Evaluation of the ﬁrst term and substitution of Eq. 4.1.12 for ξ in 4.5.5 followed by integration gives 1 τ Λcd

C1 dξ =

C1out

− C1c τ Λcd

−

C out 1 C1c

τ

dF1 dC1 , dC1

= C1out − C1c τ Λcd − τ (F1out − F1c ).

(4.5.6) (4.5.7)

where F1out is the fractional ﬂow at the outlet at time τ . Substitution of Eqs. 4.5.4 and 4.5.7 and the deﬁnition of Λcd (Eq. 4.2.2) into Eq. 4.5.1 gives the ﬁnal expression for the recovery of component 1, Q1 = C1init − C1out + τ F1out .

(4.5.8)

Similar reasoning leads to the expression for the recovery of component 2, Q2 = C2init − C2out + τ F2out .

(4.5.9)

It is also easy to show that when volume is conserved, the diﬀerence between the total amount of ﬂuid injected and the total volume of component 1 produced must be the volume of component 2 recovered, Q2 = τ − Q1 .

(4.5.10)

Similar integral balances apply when eﬀects of volume change are included. The resulting expressions for recovery of component i are − Gout + τ Hiout Ri = Ginit i i

(4.5.11)

Fig. 4.17 compares recovery of C10 for the example solutions displayed in Fig. 4.16. Values reported in Tables 4.2 and 4.3 under the columns labeled τ are the arrival times of the corresponding compositions at the outlet at ξ = 1. Also given are the values of recovery of C10 , RC10 or QC10 , reported as a fraction of the amount of C10 initially present.

4.6. SUMMARY

69 1.0

Fraction of C10 Recovered

0.8

0.6

0.4

0.2

No volume change Volume change

0.0 0

1

τ

2

3

Figure 4.17: Recovery of component 2, C10 in displacements with and without volume change. Fig. 4.17 and Tables 4.2 and 4.3 indicate again that breakthrough of injected CO2 occurs at about 0.77 pore volumes injected (PVI) without volume change, but when more than one pore volume has been injected, at 1.09 PVI, when account is taken of volume change. The eﬀect of volume change is largest when the displacement pressure is high enough that there is appreciable solubility of CO2 in the oil but low enough that there is signiﬁcant diﬀerence between the partial molar volumes of CO2 in the vapor and liquid phases. At still higher pressures, where the partial molar volumes can diﬀer much less, the assumption of no volume change is often quite reasonable. Recovery is lower when volume change is considered because the dissolved CO2 present in the liquid phase upstream of the leading shock occupies less volume. Hence, more C10 remains in the undisplaced liquid in the transition zone when volume change is signiﬁcant. In both displacements, however, recovery of C10 is slow after breakthrough of injected CO2 . In the terminology in widespread use in the oil industry, both displacements are immiscible. There is a large region of two-phase ﬂow, and large amounts of gas must be injected to recover small amounts of oil after breakthrough. Even so, all of the C10 could eventually be recovered by evaporation. However, the arrival times of the trailing shock (see Tables 4.2 and 4.3), 85 and 158 pore volumes injected, are so long that a recovery process based on evaporation of large amounts of undisplaced oil would be unattractively slow. As the theory developed in the next two chapters shows, however, more eﬃcient displacements can be designed for systems that contain more than two components.

4.6

Summary

In this chapter we develop the basic ideas of the method of characteristics: by calculating how fast a particular composition propagates through the one-dimensional porous medium, we can work out

70

CHAPTER 4. TWO-COMPONENT GAS/OIL DISPLACEMENT

the behavior of a displacement of an oil mixture by a gas mixture. That basic idea will be applied several times more in subsequent chapters as systems with more components are described. For displacements in binary systems, the following key ideas carry over into systems with more than two components: • The propagation (or wave) velocity for a composition inside the two-phase region is df /dS1 (when volume change as components change phase is neglected). • Any solution must satisfy a velocity constraint , which requires, for regions in which compositions are varying continuously, that compositions with high wave velocity lie downstream of compositions with lower wave velocity. • A shock is required if the number of phases present changes (as the solution compositions are traced upstream or downstream). • A shock must satisfy an entropy condition , which requires that the shock be self-sharpening. This means that compositions on the upstream side of the shock must travel at wave velocities greater than or equal to the shock speed, and compositions on the downstream side of the shock must move at wave velocities that are less than or equal to the shock speed. • Displacement of a single-phase oil mixture by a single-phase gas mixture includes a leading shock from the oil composition to a mixture composition in the two-phase region and a shock from the gas composition to a diﬀerent mixture composition in the two-phase region. Both shocks are semishocks in which the wave speed of the shock matches the composition wave speed on the two-phase side of the shock. The two shock compositions are connected by a continuous composition variation along the equilibrium tie line. • Adding the eﬀects of volume change as components change phase to the analysis does not change the patterns of displacement behavior, but the wave velocities of all the compositions do change.

4.7

Additional Reading

Method of Characteristics. Volume I of First Order Partial Diﬀerential Equations by Rhee, Aris, and Amundson [106] gives an excellent introduction to the method of characteristics in Chapter 2. The method of characteristics is applied to chromatography problems closely related to the binary displacement problem in Chapter 5, and the Buckley-Leverett problem for waterﬂooding is also solved there. The behavior of shocks and entropy conditions is discussed in some detail in Chapter 5 and again in Chapter 7. Binary Displacement without Mutual Solubility. The original solution for a binary displacement was that of Buckley and Leverett [10] for displacement of oil by water. Many authors have subsequently discussed the solution to that problem. Welge [133] derived the tangent construction used to determine the leading shock velocity and composition. For reviews of the theory of water/oil displacement that are closely linked to the approach taken here, see the discussions of Lake [62, Section 5-2], Rhee, Amundson and Aris [106, Section 5.6], and Bedrikovetsky [6, Chapter 1]. Dake summarizes the conventional approach to the problem [17, pp. 356-372].

4.8. EXERCISES

71

Binary Displacement with Mutual Solubility. The discussion of the application of the velocity constraint and entropy condition to eliminate nonphysical solutions follows that of Johns [54, Chapter 3] for the Buckley-Leverett problem. Examples of solutions for displacement of C10 by CO2 are given by Pande [95, Chapter 4]. The description of the dependence of solutions on initial and injection conditions was given ﬁrst by Helﬀerich [32]. Eﬀects of Volume Change on Mixing. A comparison of binary solutions with and without volume change as components change phase is given for CO2 /C10 displacements by Dindoruk [19, Chapter 6].

4.8

Exercises

1. Characteristic curves. Consider the equation ∂C ∂C + C2 = 0. ∂t ∂x

(4.8.1)

The initial composition is Cinit = 0.1, and the injection composition is Cinj = 0.8. Derive expressions for the characteristic curves. Plot the appropriate characteristic curves on a t-x diagram. Determine whether shocks would occur in this displacement. 2. Gas dissolution. Consider a laboratory core which contains initially water that is saturated with CO2 in equilibrium with gas at the critical gas saturation, Sgc = 0.05. The equilibrium volume fraction of CO2 dissolved in the saturated water phase is 0.03, and the volume fraction of water in the gas phase is 0.001. At time τ = 0, injection of pure water into the core begins. Assuming that eﬀects of volume change as components change phase can be neglected, calculate the saturation proﬁle at τ = 0.5 pore volumes injected. Determine how much pure water would have to be injected to remove all the gas present initially in the core. 3. Calculate the saturation proﬁle at τ = 0.5 and 1.0 pore volumes injected for the relative permeability functions of Eqs. 4.1.14-4.1.19 with Sgc = 0.1, Sor = 0.3, and M = 10 for a displacement in which gas displaces oil. The volume fraction of the light component required to saturate the liquid phase is 0.4, and the volume fraction of light component in the equilibrium vapor phase is 0.95. The initial composition is a single-phase mixture in which the light component has a volume fraction of 0.2. The volume fraction of the light component in the injection gas is 0.98. Also calculate a recovery curve for the heavy component. How many pore volumes of gas must be injected to recover all of the oil initially in place? 4. Displacement with two-phase initial and injection mixtures. Consider the ﬂuid system of problem 3. Calculate the saturation proﬁles at the same times and calculate a recovery curve for a displacement in which the core initially contains a two-phase mixture in which the volume fraction of the light component is 0.5, and the injection gas is also a two-phase mixture with a light component volume fraction of 0.9. 5. Eﬀect of volume change on shock speed. Consider the situation outlined in problem 2. Determine the shock speed for a situation in which the density of the water does not change as it moves between phases, but CO2 that dissolves in the water phase occupies only half the

72

CHAPTER 4. TWO-COMPONENT GAS/OIL DISPLACEMENT

Table 4.4: Equilibrium Phase Compositions and Fluid Properties Fluid

xCH4

xC10

Initial Oil Equil. Liq. Equil. Vap. Injected Gas

0. 0.3519 0.9964 1.

1. 0.6481 0.0036 0.

ρ (gmol/l) 4.881 6.481 4.316 4.298

ρ (g/cm3) 0.6945 0.6342 0.0712 0.0690

µ (cp) 0.262 0.015 -

volume of CO2 in the vapor phase. Assume that the CO2 in the vapor phase has a component density of 1.1364 x 10−3 gmol/cm3 and that water in the liquid phase has a component density of 5.5555 x 10−2 gmol/cm3. 6. CH4 displacing C10 . Consider the ﬂuid property data given in Table 4.4 for the CH4 /C10 system at 160 F and 1600 psia. Calculate and plot the composition proﬁle as a function of ξ/τ for two diﬀering assumptions about density behavior: (1) when the volume occupied by each component is the pure component volume no matter what phase the component appears in, and (2) when the equilibrium phases assume the densities given in Table 4.4. Assume that the phase compositions in mole fractions given in Table 4.4 are correct for both cases. Calculate and plot recovery curves as a function of pore volumes of methane injected for the two density assumptions.

Chapter 5

Ternary Gas/Oil Displacements In this chapter, we consider the behavior of displacements in which three components and two phases are present. Much of the original work on gas drives was done for ternary systems, which display essential features of displacement behavior but are simple enough to analyze. The basic physical mechanisms of gas drives were outlined by Hutchinson and Braun [38], who considered what would happen if a porous medium were represented as a series of mixing cells. Figs. 5.1 and 5.2 summarize their arguments. (For another version of the mixing cell argument, see Lake [62].) Suppose that oil with composition O1 is displaced by gas with composition G1 . Mixtures of oil O1 with gas G1 in the ﬁrst mixing cell lie on the dilution line that connects O1 to G1 on the ternary diagram in Fig. 5.1. Suppose that after mixing some oil with gas in the ﬁrst cell, the overall composition is M1 . That mixture splits into two phases with compositions V1 and L1 . Now assume that the less viscous vapor phase with composition V1 moves to the next downstream cell, where it mixes with fresh oil. Those mixtures lie on the dilution line that connects V1 with the oil composition, O1 . If the new overall composition is M2 , then the phases that form in the second cell have compositions L2 and V2 . But when the vapor V2 moves to the next cell and mixes with fresh oil, the dilution line does not pass through the two-phase region. Instead, the mixtures are “miscible” after multiple contacts, even though the original gas and oil do not form only one phase when mixed in any proportions. This displacement is what is known as a vaporizing gas drive because the crucial transfer of components that leads to miscibility is the vaporization of the intermediate component from the oil into the fast-moving vapor phase. Mixture V1 is richer in component 2 than the original injection gas is, and mixture V2 is richer still. Oil O1 is rich enough in component 2 that miscibility develops. If, however, the oil had had composition L2 (or any mixture on the extension into the single-phase region of the tie line that connects V2 and L2 ), mixture of V2 with fresh oil would have given another mixture on the same tie line. In that case, the enrichment of the vapor phase with component 2 ceases to change with further contacts in downstream mixing cells. Such a vaporizing gas drive is said to be “immiscible.” In vaporizing gas drives, the mixing cell argument indicates that miscibility develops if the original oil composition does not lie within the region of tie line extensions on the ternary diagram. Fig. 5.2 summarizes a similar argument for a displacement known as a condensing gas drive in which gas G2 displaces oil with composition O2 . Mixtures of original oil with gas in the ﬁrst mixing cell give composition M1 . That mixture splits into phases with compositions V1 and L1 . Here again, the vapor phase is assumed to move ahead and contact fresh oil, but this time we focus 73

74

CHAPTER 5. TERNARY GAS/OIL DISPLACEMENTS C1 aG1 aV1 M1 a

M2 a

L1

C3

aV a2 a

L2

a

O1

C2

Figure 5.1: Mixing cell representation of a vaporizing gas drive. on what happens in the ﬁrst mixing cell. There, liquid phase with composition L1 mixes with new injected gas. Those mixtures lie on the dilution line that connects L1 with G2 . If the new composition after mixing in the ﬁrst cell is M2 , then the resulting phase compositions are V2 and L2 . When liquid phase L2 mixes with new injection gas, the mixtures are single-phase. Here again, multiple contacts of the gas with the oil have created mixtures that are miscible. This displacement is called a condensing gas drive because it relies on transfer of the intermediate component from the injected gas phase to the oil. Ternary condensing gas drives are multicontact miscible if the injection gas composition lies outside of the region of tie line extensions on a ternary phase diagram. If, on the other hand, the injection gas had had some composition on the extension of a tie line (say the L2 -V2 tie line), the enrichment of the oil in the ﬁrst cell with component 2 condensing from the gas would have stopped when that tie line was reached, because additional mixtures of fresh gas with liquid L2 would fall on the same tie line again. The mixing cell argument is necessarily a qualitative one because it assumes that only the vapor phase moves from cell to cell. In a displacement in a porous medium, the phases would move according to their fractional ﬂows. Our task for this chapter, then, is to put the qualitative argument of Hutchinson and Braun on a ﬁrm mathematical footing. The Riemann problem we will consider is illustrated in Fig. 5.3: for a given pair of initial and injection compositions, ﬁnd the set of compositions that form between the injection composition at the upstream end of the transition zone and the initial composition at the downstream end. When three components are present, the ﬂow is no longer constrained to take place along a single tie line, as it is in binary displacements. Thus, for three-component ﬂows, an essential part of the problem is to ﬁnd the collection of tie lines (and their associated phase compositions) that are traversed during a displacement. Three-component ﬂows have been considered for systems that range from alcohol displacements [125] to surfactant ﬂooding [35] to gas/oil systems [134, 22]. The ideas developed in this and the next chapter come from many sources reviewed at the end of the chapter. The development given here draws heavily from the work of Johns [54, Chapter 3], Dindoruk [19, Chapters 3, 4, and 6], and Wang [128].

5.1. COMPOSITION PATHS

75

C1

aV1 aV2

M1 a M2

O2 a

a

L1

aG2 aa

L2

C3

C2

Figure 5.2: Mixing cell representation of a condensing gas drive. We begin by formulating the eigenvalue problem that determines wave velocities and allowed composition variations, and we develop the idea of a composition path. Next we consider the behavior of shocks, which play important roles in the behavior of solutions. In Sections 5.3 and 5.4, example solutions are described that show in detail the patterns of ﬂow behavior associated with vaporizing and condensing gas drives. Section 5.5 shows that the key patterns of shocks and rarefactions (continuous composition variations) for ternary systems can be catalogued in a simple way based on the lengths of two key tie lines and whether tie lines intersect on the vapor side or the liquid side of the two-phase region. Section 5.6 introduces the important concept of multicontact miscibility. Eﬀects on ternary systems of volume change as components transfer between phases and calculation of component recovery are reviewed in the remaining sections.

5.1

Composition Paths

The conservation equations for a three-component system without volume change are ∂F1 ∂C1 + ∂τ ∂ξ ∂F2 ∂C2 + ∂τ ∂ξ

= 0,

(5.1.1)

= 0,

(5.1.2)

where Ci = ci1 S1 + ci2 (1 − S1 ),

i = 1, 2,

(5.1.3)

Fi = ci1 f1 + ci2 (1 − f1 ),

i = 1, 2.

(5.1.4)

and

76

CHAPTER 5. TERNARY GAS/OIL DISPLACEMENTS C1 Gas Composition

•B • •

• •

•A

Oil Composition

C3

C2

(a) Composition Path

Injection Gas Composition B

Oil Composition A

Produced Fluids

ξ

(b) Initial Condition Figure 5.3: Riemann problem for displacement of a three-component oil by gas. The analysis of this chapter will show that much of the behavior of solutions to Eqs. 5.1.1 and 5.1.2 is controlled by the properties of tie lines (in particular, two key tie lines that extend through the injection gas and initial oil compositions), and hence it is convenient to parametrize the ﬂow problem based on the equation of a tie line. The equation of any tie line can be written C2 = α(η)C1 + φ(η),

(5.1.5)

where α is the slope of the tie line, φ is the intercept at C1 = 0, and η is a parameter that determines which tie line is being represented. For example, η could be chosen to be the volume fraction of component 1 in the vapor phase, c11 , [54], or it could be taken to be the slope of the tie line [52]. Any convenient parameter is suitable as long as it uniquely identiﬁes a tie line. Eq. 5.1.5 is just another version of Eqs. 5.1.3, so expressions for α and φ in terms of the phase compositions can be obtained by eliminating S1 from Eqs. 5.1.3 written for components 1 and 2, α=

c21 − c22 , c11 − c12

φ=

c22 c11 − c21 c12 . c11 − c12

(5.1.6)

5.1. COMPOSITION PATHS

77

Similarly, elimination of f1 from Eqs. 5.1.4 indicates that F2 = α(η)F1 + φ(η).

(5.1.7)

Substitution of the expressions for C2 and F2 into Eq. 5.1.2 gives

dα dφ ∂η dα dφ ∂η + + F1 + = 0. (5.1.8) dη dη ∂τ dη dη ∂ξ In the original form of the conservation equations, F1 and F2 are functions of C1 and C2 only. Given a value of C1 and C2 , a ﬂash calculation can be performed to ﬁnd the equilibrium phase compositions, and the phase saturations can then be calculated (Eq. 5.1.3). From the phase saturations, relative permeabilities of the phases can be evaluated. Phase viscosities can be calculated from phase compositions. All the information required to calculate phase fractional ﬂows is then available, and Eqs. 5.1.4 can be used to evaluate them. Similarly, in the parametrization of the problem based on tie lines, F1 is a function of C1 and η. Setting η determines the tie line, and ﬁxing C1 gives the location of the overall composition on the tie line, from which all other properties can be obtained. Hence the derivatives of F1 in Eq. 5.1.1 can be written in terms of C1 and η, and Eq. 5.1.1 becomes C1

∂F1 ∂C1 ∂F1 ∂η ∂C1 + + = 0. ∂τ ∂C1 ∂ξ ∂η ∂ξ

(5.1.9)

Eqs. 5.1.8 and 5.1.9 can be rearranged into the form, ∂u ∂u + A(u) = 0, ∂τ ∂ξ

(5.1.10)

where u = (C1 , η)T and ∂F 1

A(u) =

∂C1

0

∂F1 ∂η F1 +p C1 +p

,

(5.1.11)

with p given by p=

dφ dη dα dη

=

dφ . dα

(5.1.12)

The physical interpretation of p can be seen by considering two adjacent tie lines (labeled A and B). The equations of the tie lines are C2 = αA C1 + φA ,

(5.1.13)

C2 = αB C1 + φB ,

(5.1.14)

and where α and φ are deﬁned by Eqs. 5.1.6. If the tie lines are adjacent to each other then we can write αB = αA +

dα ∆η, dη

(5.1.15)

78

CHAPTER 5. TERNARY GAS/OIL DISPLACEMENTS

and dα ∆η, (5.1.16) dη where again, η is a parameter that determines the tie line. The point at which the two tie lines intersect, (C1x , C2x ) can be found by solving Eqs. 5.1.13 and 5.1.14 using 5.1.15 and 5.1.16. The result is φB = φA +

C1e

=

dφ dη − dα dη

= −p.

(5.1.17)

Thus, C1e is a point on the locus of intersections of inﬁnitesimally separated tie lines. That curve is known as an envelope curve [19] . Fig. 5.4 shows an envelope curve for a ternary system with constant K-values. Any tie line can be constructed by drawing a tangent to the envelope curve that intersects the liquid and vapor loci. Thus, we have shown that −p is the overall volume fraction, C1e , on the envelope curve. The relationship between p and the envelope curve suggests that the geometry of tie lines will play an important role in the behavior of solutions. The remainder of this chapter will show that statement to be true.

5.1.1

Eigenvalues and Eigenvectors

To solve Eq. 5.1.10 we calculate the trajectory of a ﬁxed composition (ﬁxed values of C1 and η), just as we did for binary displacements. As before, we ask how constant values of C1 and η propagate, and hence we set dC1 and dη to zero in ∂C1 dτ + ∂τ ∂η dτ + ∂τ

dC1 = dη =

∂C1 dξ = 0, ∂ξ ∂η dξ = 0. ∂ξ

(5.1.18) (5.1.19)

Eqs. 5.1.18 and 5.1.19 can be rearranged to give ∂C1 ∂τ ∂η ∂τ

∂C1 dξ ∂C1 = −λ , ∂ξ dτ ∂ξ ∂η dξ ∂η = − = −λ , ∂ξ dτ ∂ξ = −

(5.1.20) (5.1.21)

where λ = dξ/dτ is the wave velocity of some ﬁxed composition deﬁned by the values of C1 and η. Substitution of Eqs. 5.1.20 and 5.1.21 into Eq. 5.1.10 gives ∂F 1 ∂C − λ 1

0

∂F1 ∂η F1 +p C1 +p −

∂C 1

λ

∂ξ ∂η ∂ξ

= 0.

(5.1.22)

Eq. 5.1.22 is an eigenvalue problem that has nontrivial solutions if and only if det

∂F 1 ∂C − λ 1

0

∂F1 ∂η F1 +p C1 +p − λ

=

∂F1 −λ ∂C1

F1 + p − λ = 0. C1 + p

(5.1.23)

5.1. COMPOSITION PATHS

79

C1 •

•

• • •

•

• •

C3

• •

•

•

•

C2

Envelope Curve

•

•

Figure 5.4: Tie-line construction from an envelope curve for a ternary system with constant Kvalues: K1 = 2.5, K2 = 1.5, K3 = 0.05. In Eq. 5.1.23, values of λ, the eigenvalues, give the wave velocities at which a given overall composition propagates, and the associated eigenvectors, (∂C1 /∂ξ , ∂η/∂ξ)T are directions in composition space along which compositions can vary while also satisfying the material balance equations. Expansion of the determinant in Eq. 5.1.23 gives a quadratic equation, which has the obvious solutions

λt =

∂F1 , ∂C1

λnt =

F1 + p . C1 + p

(5.1.24)

(The simple form of the eigenvalue problem , Eq. 5.1.23, is the reason for the parametrization of the problem in terms of tie line slope and intercept [52, 54].) The fact that each composition can have two possible wave velocities means that we will have to select which velocity applies at a given composition. To do that, we must consider the eigenvectors associated with each of the eigenvalues. Substitution of the eigenvalues into Eq. 5.1.22 shows that the associated eigenvectors are

80

CHAPTER 5. TERNARY GAS/OIL DISPLACEMENTS

et =

⎛

1 0

,

ent = ⎝

1

λnt −λt ∂F1 ∂η

⎞ ⎠.

(5.1.25)

The ﬁrst entry in each eigenvector corresponds to changes in C1 , and the second to changes in η. The magnitude of an eigenvector is arbitrary – it indicates only a direction in composition space. Hence, the unit value in the ﬁrst position of each eigenvector is selected for convenience. For compositions in the single-phase region, F1 = C1 everywhere. As a result, λt = λnt = 1, and Eq. 5.1.23 is satisﬁed for any composition variation. Therefore, there are no discrete composition directions in the single-phase region, and composition variations in any direction are allowed. For compositions in the two-phase region, however, there are two separate composition directions associated with et and ent . Next we consider the properties of those eigenvector directions. The existence of discrete directions given by the eigenvectors indicates that arbitrary composition variations cannot satisfy the conservation equations. Instead variations in composition that are consistent with the material balance equations must lie on curves in composition space that are obtained by integrating along the eigenvector directions. Those curves are known as composition paths [31]. In other words, the expressions for the eigenvectors are ordinary diﬀerential equations for the allowed composition paths. Self-Similarity and Coherence The idea of a composition path is quite important, because a solution to a Riemann problem must lie on a sequence of such paths if the conservation equations are to be satisﬁed. To see why that statement is true, consider again Eq. 5.1.10, and assume for the moment that a solution to it is a function of ξ/τ only, u(ξ, τ ) = w(ξ/τ ). Such solutions are said to be self-similar (the papers of Lax [67, 68] and Isaacson [39] show that problems of this sort are indeed self-similar). Substitution of the derivatives, ξ ∂u = −w 2 , ∂τ τ

(5.1.26)

1 ∂u = w , ∂ξ τ

(5.1.27)

and

in Eq. 5.1.10 gives

ξ A(w) − I w = 0. τ

(5.1.28)

As long as w = 0, ξ/τ must be an eigenvalue of A and w must be an eigenvector. At any value of ξ/τ , therefore, the solution u must lie on a curve in composition space that is tangent to an eigenvector. The idea that a given overall composition, u = (C1 , η)T , translates with a wave velocity, λ = ξ/τ is what some investigators refer to as coherence [31] . To show that the solution, u(ξ, τ ), depends only on ξ/τ , and hence that λ = ξ/τ , we note that if u(ξ, τ ) is a solution to Eq. 5.1.10, then u(aξ, aτ ) is also a solution for any a > 0 [39], as direct evaluation of the partial derivatives shows.

5.1. COMPOSITION PATHS

81

We are free to choose a = 1/τ , which gives u = u(ξ/τ, 1). Thus, u is a function of ξ/τ only, and coherence follows from the diﬀerential equations. The manipulations of this section have shown that the solution to Eq. 5.1.10 can be obtained as a set of compositions, u(ξ, τ ) = (C1 , η)T , that lie on composition paths. The next step, therefore, is to examine in more detail the behavior of those paths.

5.1.2

Tie-Line Paths

The change in η for composition variations along the direction of the eigenvector, et , is zero (see Eq. 5.1.25). Because η is a parameter that determines on which tie line the composition point lies, zero variation in η means that the eigenvector direction is that of the tie line. Furthermore, it is easy to show that for ﬁxed η, the eigenvalue, λt = ∂F1 /∂C1 reduces to df1 /dS1 . In fact the manipulations are identical to those performed in the derivation of Eq. 4.1.23 for a binary system, as they must be because the composition variation occurs along a single tie line. For the tie-line eigenvector , the path integration is simple. Stepwise integration in the direction, et , gives additional compositions that lie on the same tie line. Thus, integration of the tie line eigenvector gives a composition path that is just a straight line that coincides with the tie line. So far, we have shown that 1. a tie line is a composition path, and 2. the wave speed, λt , for variation along a tie line composition path is simply the familiar Buckley-Leverett wave velocity, df1 /dS1 . Those statements hold no matter how many components are present. The fact that a tie line is a composition path means that it is possible to have solutions in which compositions vary along a single tie line. Indeed, such variations must be possible if threecomponent solutions are to reduce smoothly to binary solutions. Consider, for example, the Riemann problem illustrated in Fig. 5.5 where initial and injection compositions lie on extensions of the same tie line. If so, the entire solution remains on that tie line. Such a displacement is really the equivalent of a binary displacement, even though three components are present [32]. For such pseudobinary systems, the theory of Chapter 4 applies.

5.1.3

Nontie-Line Paths

If the initial and injection conditions lie on diﬀerent tie lines (or tie-line extensions), then composition variations that move between the two tie lines are required, so we consider now the properties of composition variations associated with the second eigenvalue and eigenvector direction. To simplify the arguments and the notation, we choose η to be the vapor phase composition, η = c11 = y1 , and we deﬁne the liquid phase composition to be x1 = c12 = y1 /K1 , where K1 is the equilibrium K-value for component 1 (see Section 3.4). It is easy to evaluate λnt and ent at three points on any tie line, when the overall composition lies on either the saturated liquid or vapor loci or when F1 = C1 inside the two-phase region. We now ﬁnd λnt and ent for each of those situations. When the overall composition lies on the vapor phase portion of the binodal curve, C1 = y1 , and F1 = C1 . Eq. 5.1.24 shows that λnt = 1. According to Eq. 5.1.22 with λnt = 1,

82

CHAPTER 5. TERNARY GAS/OIL DISPLACEMENTS

C1 Injection

• Gas • •

•

Initial Oil

•

C3

C2

Figure 5.5: Riemann problem for displacement of a three-component oil by gas lying on the same tie-line extension.

(λt − 1)

dC1 ∂F1 + = 0. dy1 ∂y1

(5.1.29)

When the overall composition is the saturated vapor phase composition, f1 = 1, and hence, F1 = y1 f1 + x1 (1 − f1 ) = y1 .

(5.1.30)

Therefore, ∂F1 /∂y1 = 1. Also, λt = ∂F1 /∂C1 = df1 /dS1 = 0, and hence, substitution of those values into Eq. 5.1.29 gives dC1 = 1. dy1

(5.1.31)

Eq. 5.1.31 indicates that y1 = C1 on the nontie-line composition path that passes through the saturated vapor composition. Hence, the composition path is the vapor locus of the binodal curve. Similarly, for compositions on the liquid portion of the binodal curve, λnt = 1. The overall composition is C1 = x1 , λt = 0, and diﬀerentiation of Eq. 5.1.30 indicates that ∂F1 /∂y1 = dx1 /dy1 . Eq. 5.1.29 then reduces to dx1 dC1 = . dy1 dy1

(5.1.32)

Integration of Eq. 5.1.32 indicates that C1 = x1 . Therefore, the liquid portion of the binodal curve is also a composition path. If F1 = C1 in the two-phase region, then λnt = 1 (Eq. 5.1.24). Comparison of Eqs. 5.1.3 and 5.1.4 shows that F1 = C1 when f1 = S1 . Points at which f1 = S1 lie on what is known as the equivelocity curve (C1 = C1EV , because at such points the ﬂow velocities of the two phases are equal. There is one equivelocity point on every tie line, and the equivelocity curve, the locus of equivelocity points on the full set of tie lines, must pass through the critical point if one exists.

5.1. COMPOSITION PATHS

83

To show that the equivelocity curve (f1 = S1 ) is also a path, we evaluate the derivatives ∂F1 /∂y1 and dC1 /dy1 and show that Eq. 5.1.29 is satisﬁed. First, we write F1 and C1 in terms of y1 ,

F1 = (y1 − x1 )f1 + x1 = y1

1 1 (1 − )f1 + , K1 K1

C1 = (y1 − x1 )S1 + x1 = y1 (1 −

1 1 )S1 + K1 K1

(5.1.33)

.

(5.1.34)

For this exercise, we assume that phase viscosities are independent of composition. (The manipulations are similar but algebraically more complex if the viscosities in f1 depend on composition, for example). Diﬀerentiation of Eq. 5.1.33 gives

1 1 ∂F1 = (1 − )f1 + ∂y1 K1 K1

+ y1 (1 −

1 ∂f1 1 dK1 ) + (f1 − 1) K1 ∂y1 K12 dy1

(5.1.35)

Because f1 is a function of S1 only, df1 ∂S1 ∂f1 = , ∂y1 dS1 ∂y1

(5.1.36)

and ∂S1 /∂y1 can be obtained by diﬀerentiating Eq. 5.1.34 (with C1 held constant), which gives

y1

1 1− K1

1 1 y1 dK1 ∂S1 =− 1− S1 − + (1 − S1 ) 2 . ∂y1 K1 K1 K1 dy1

(5.1.37)

Finally, diﬀerentiation of Eq. 5.1.34 gives an expression for dC1 /dy1 along the curve f1 = S1 (note that S1 is constant along that curve),

1 1 y1 dK1 dC1 = 1− S1 + + 2 (f1 − 1) . dy1 K1 K1 K1 dy1

(5.1.38)

Substitution of Eqs. 5.1.35, 5.1.36, 5.1.37, and 5.1.38 into Eq. 5.1.29 shows that it is, indeed, satisﬁed. Hence the equivelocity curve is a path. Fig. 5.6 is a plot of F1 vs. C1 for the middle tie line in Fig. 5.4. It has the familiar S shape, though the range of C1 is now restricted to remain within the ternary diagram. Fig. 5.7 shows the behavior of the two eigenvalues on the same tie line. If the overall composition lies in the single-phase region, F1 = C1 , and both eigenvalues are one. The S shape of the overall fractional ﬂow curve requires that its slope be greater than one in the neighborhood of C1E . Therefore, λt > 1, and hence, λt > λnt for compositions near C1E , where λnt = 1. If the relative permeability functions (see Eqs. 4.1.14-4.1.19) vary smoothly in the neighborhood of the binodal curve, and the relative permeability of a phase with zero saturation is also zero, both physically reasonable assumptions, then λt = df1 /dS1 will go smoothly to zero as either phase boundary is approached. Therefore, λt < λnt in the neighborhood of either phase boundary. As a result, for smoothly varying functions F1 , λt = λnt at two points on each tie line. At those points, called equal eigenvalue points, Eq. 5.1.25 indicates that et = ent . Thus, integral curves of ent are tangent to tie lines at the equal eigenvalue points, a fact that will play an important role when rarefactions between tie lines are constructed. Thus, for the nontie-line paths, we have shown that

84

CHAPTER 5. TERNARY GAS/OIL DISPLACEMENTS

Overall Fractional Flow of Component 1, F1

1.0

0.5

FE 1

•

CE 1 0.0 0.0

0.2

0.4

0.6

0.8

1.0

Overall Volume Fraction of Component 1, C1

Figure 5.6: Fractional ﬂow function for a ﬁxed tie line. 1. the liquid and vapor branches of the binodal curve are composition paths associated with λnt = 1, 2. the equivelocity curve is also a composition path along which λnt = 1, and 3. nontie-line eigenvectors are tangent to tie line paths at two distinct equal eigenvalue points on each tie line. The remaining portions of the nontie-line paths must be obtained by integration of the nontieline eigenvector. The complexity of that integration depends on the form of the equation of state. For the Peng-Robinson equation, ﬁnite diﬀerences can be used to evaluate the derivative, ∂F1 /∂C1 and ∂F1 /∂η, that appear in the nontie-line eigenvector. Nontie-line Paths for Constant K-Values If the phase behavior is simpler than that represented by the Peng-Robinson equation, it is sometimes possible to perform the integration analytically. If equilibrium K-values are constant and phase viscosities are independent of composition, an explicit expression for the nontie-line paths can be obtained [128]. The ﬁrst step is to derive a diﬀerential equation for the nontie-line path. To do that it is convenient to write the deﬁnitions of the overall composition, Ci , and the overall fractional ﬂow, Fi , in terms of the liquid phase composition, xi ,

5.1. COMPOSITION PATHS

85

Tie-line and Nontie-line Eigenvalues, λt and λnt

3.0

λt

2.5

2.0

1.5 λnt 1.0

0.5

0.0 0.0

0.4

0.2

0.6

0.8

1.0

Overall Volume Fraction of Component 1, C1

Figure 5.7: Eigenvalue variation along a tie line.

Ci = xi {1 + (Ki − 1)S1 } ,

(5.1.39)

Fi = xi {1 + (Ki − 1)f1 } .

(5.1.40)

An equation for the liquid portion of the binodal curve is obtained from the fact that the liquid locus satisﬁes x1 + x2 + x3 = 1,

(5.1.41)

y1 + y2 + y3 = K1 x1 + K2 x2 + K3 x3 = 1.

(5.1.42)

and the vapor locus

Elimination of x3 from Eq. 5.1.42 gives an expression for the liquid locus, x2 =

1 − K3 K1 − K3 − x1 K2 − K3 K2 − K3

Eqs. 5.1.39 and 5.1.40 can then be written as functions of S1 and x1 ,

(5.1.43)

86

CHAPTER 5. TERNARY GAS/OIL DISPLACEMENTS

C1 = C1 = x1 {1 + (K1 − 1)S1 } , K1 − K3 1 − K3 C2 = − x1 {1 + (K2 − 1)S1 } , K2 − K3 K2 − K3 F1 = x1 {1 + (K1 − 1)f1 } , 1 − K3 K1 − K3 − x1 {1 + (K2 − 1)f1 } . F2 = K2 − K3 K2 − K3

(5.1.44) (5.1.45) (5.1.46) (5.1.47)

Substitution of these expressions into Eq. 5.1.22 with choice of S1 and x1 as the primary dependent variables gives a convenient form of the eigenvalue problem, ⎛ ⎝

df1 dS1

⎞

1 dp (f1 −S1 ) 2 dx1 C1 +p F1 +p C1 +p

−λ 0

⎠

−λ

dS1 dx1

= 0,

(5.1.48)

where p=

K1 − K2 K1 − K3 2 x21 , x = K2 − 1 1 − K3 1 γ

(5.1.49)

1 − K3 K2 − 1 . K1 − K2 K1 − K3

(5.1.50)

where γ=

The diﬀerential equation for the nontie-line path is obtained by evaluating Eq. 5.1.48 for λnt = (F1 + p)/(C1 + p), which yields dx1 + P (S1 )x1 = Q(S1 ), dS1

(5.1.51)

where P (S1 ) =

df1 dS1

−1

f1 − S1

,

(5.1.52)

γ {1 + (K1 − 1)S1 } (1 − Q(S1 ) = γ(K1 − 1) + f1 − S1

df1 dS1 )

.

(5.1.53)

Eq. 5.1.51 has the solution x1 = e−

P (S1 )dS1

Q(S1 )e

P (S1 )dS1

dS1 + c0 ,

(5.1.54)

where c0 is a constant of integration. Evaluation of Eq. 5.1.54 gives x1 =

f0 x01 1 f1

− S10 +γ − S1

+

γ(K1 − 1) 0 0 f10 − S10 −1 + (S1 f1 − S1 f1 ) f1 − S1 f1 − S1

2γ(K1 − 1) f1 − S1

S1 S10

f1 dS1 ,

(5.1.55)

5.1. COMPOSITION PATHS

87

where x01 , f10 , and S10 refer to some point that lies on the nontie-line path in question. Eq. 5.1.55 indicates that if the fractional ﬂow expression is simple enough, the integration can be performed and an explicit solution obtained. If we use the relative permeability functions stated in Eqs. 4.1.14 - 4.1.19, with phase viscosities independent of composition and Sgc = Sor = 0, the resulting expression for the integral in Eq. 5.1.55 is S1 S10

⎧

⎛

1 1 ⎨ S1 S1 − S10 M M −1 ⎝ f1 dS1 = + arctan 1 2 ⎩ 1 1 1+ M M 1+ M

⎛

− arctan ⎝

+

1 M 1 M

+1

2 ln

⎧ ⎨

S10

1 M

−

1 M

+1 −

1 M

1 M

2 M S1

+ 1+

1 M

+1 −

1 M

1 M

⎞ ⎠

⎞⎫ ⎬ ⎠ ⎭ 1 M

⎫

S12 ⎬

. ⎩ 1 − 2 S 0 + 1 + 1 (S 0 )2 ⎭ 1 M M 1 M

(5.1.56)

If Sgc = 0 or Sor = 0, similar but slightly more complex expressions result (see Problem 23). Fig. 5.8 shows the geometry of composition paths that result for constant K-values, K1 = 2.5, K2 = 1.5, and K3 = 0.05. While only a few tie lines are shown, the two-phase region is actually ﬁlled completely with the two sets of paths, the linear tie-line paths, and the curved nontie-line paths. On each tie line there are two equal-eigenvalue points, indicated in Fig. 5.8 by dots, at which the nontie-line path is tangent to the tie-line path. To construct a solution for given initial and injection compositions, a sequence of composition paths that connect the initial and injection compositions must be selected from that doubly inﬁnite set. To do that, we must consider what happens at a switch from one path to another.

5.1.4

Switching Paths

To construct a solution to the Riemann problem illustrated in Fig. 5.3, we must ﬁnd a set of composition paths that connects the initial and injection compositions. If the initial and injection compositions lie on the extensions of diﬀerent tie lines, then the solution may require a switch from a tie-line path to a nontie-line path and vice versa. Next, we consider when path switches are permitted. A path switch is allowed if and only if it satisﬁes the velocity constraint. It will do so if the wave velocity at the switch point decreases (or stays constant) as the combination of paths is traced from downstream to upstream compositions. Fig. 5.9 shows examples of three possible path switches from a tie-line path to a nontie-line path. Point b in Fig. 5.9 is the equal eigenvalue point, and Points a and c are additional switch points to be considered. Fig. 5.10 shows the corresponding eigenvalues for the initial tie line. Fig. 5.11 shows sketches of the composition proﬁles that would result for switches at the three points. Suppose that the tie-line path is being traced from some downstream composition upward along the initial tie line. Suppose also that a path switch occurs at point a, and the nontie-line path is then traversed toward the injection tie line. Fig. 5.10 shows that at point a, the wave velocity drops as the path is traced through the switch point, and Fig. 5.11 shows the zone of constant

88

CHAPTER 5. TERNARY GAS/OIL DISPLACEMENTS

C1

Nontie-Line Path

a a

a a

a

C3

Tie-Line Path a a

C2

Equivelocity Curve Figure 5.8: Tie-line and nontie-line paths for a ternary system with constant K-values: K1 = 2.5, K2 = 1.5, K3 = 0.05. The dots indicate the locations of equal-eigenvalue points where a nontie-line path is tangent to a tie line path. state that results. (Note that the proﬁles in Fig. 5.11 are plotted against wave velocity, ξ/τ . To obtain a location, ξ, of some composition at time, τ , simply multiply the wave velocity by the dimensionless time, τ .) At point a, a single composition has two wave velocities, a situation that is physically acceptable as long as the resulting composition variation is single-valued. In other words, two spatial positions for the same composition are allowed, though two compositions at the same position are not. The top panel of Fig. 5.11 shows that the composition proﬁle for point a is single-valued. At point b, the two eigenvalues are equal, so that point is certainly an acceptable switch point, as Fig. 5.11 shows. A switch at point c, however, would not satisfy the velocity constraint . The wave velocity would increase at the switch point (see Fig. 5.10), which would cause the multivalued composition proﬁle sketched in Fig. 5.11. Thus, of the three points considered, only points a and b would satisfy the velocity constraint. Next we consider whether legal path switches could occur at the intersection of a given nontieline path with the injection gas tie line. Points d, e, f, and g in Fig. 5.9 are the points to be considered. Because point c is not allowed as a switch point, the nontie-line path from c to g cannot be traced, so point g cannot be reached, and it need not be considered further. If the nontie-line path from point a, a legal switch point, to point d were traced upstream, a switch at point d would be allowed if λdnt > λdt. Fig. 5.10 shows, however, that the reverse is true. Point d lies below the equal-eigenvalue point on the injection gas tie line, and therefore λdnt < λdt . A path switch at point d would create a multivalued solution, which is not permitted. Thus, the nontie-line

5.1. COMPOSITION PATHS

89

C1 •

Injection Gas

g a f a ea d a

•

a

C3

•

•b

•c

C2 Initial Oil

Figure 5.9: Possible path switch points.

path from a to d is not an acceptable solution route because the path switch at d is prohibited by the velocity constraint, even though the path switch at point a is allowed. A path switch at point b is allowed, so the nontie-line path could be traced from b to e. However, a path switch at point e can be eliminated from further consideration by exactly the same argument used for point d. As Fig. 5.10 shows, λent < λet , and hence the path segment from b to e can be ruled out as well. The only remaining possibility is the segment from b to f. Point f lies above the equal-eigenvalue point, and as Fig. 5.10 shows, λfnt > λft . Therefore, point f is a switch point that does satisfy the velocity constraint. Thus, if a nontie-line path is traced from the initial oil tie line to the injection gas tie line in a displacement with tie lines as shown in Fig. 5.9, the upper branch of the nontieline path through point b must be used. Similar arguments show that if the positions of the injection gas and initial oil tie lines were exchanged, the lower branch of the nontie-line path with an equal-eigenvalue point on the injection tie line would be the only acceptible solution route. In this section we have examined the properties of composition paths. They are essential to the construction of a solution to a Riemann problem like that shown in Fig. 5.3 because composition paths are traced whenever continuous variations occur. However, rarefactions along paths are sometimes prohibited by the velocity constraint, and when that happens, shocks/indexshock are required.

90

CHAPTER 5. TERNARY GAS/OIL DISPLACEMENTS

Tie-line and Nontie-line Eigenvalues, λt and λnt

3.0

λt

2.5

ad aa ae

2.0

1.5

a aa

1.0

•

b

f ac

a

aa

0.5

0.0 0.0

λnt

0.2

0.4

0.6

0.8

1.0

Saturation, S1

Figure 5.10: Wave velocities at the switch points.

5.2

Shocks

Two kinds of shocks occur in ternary systems: (1) shocks that have a single-phase mixture on one side of the shock and a two-phase mixture on the other, and (2) shocks with two phases present on both sides of the shock. The ﬁrst type arises when the solution path enters or leaves the two-phase region, and the second appears when compositions on diﬀerent tie lines are connected by a shock. In this section we consider the two types in turn.

5.2.1

Phase-Change Shocks

Reasoning similar to that for binary displacements indicates that the solution route can enter or leave the two-phase region only via a shock. Consider, for example, a composition point somewhere on the liquid portion of the binodal curve. Such a composition might form if there were a continuous variation in composition from the initial oil composition at downstream locations to the injection gas composition upstream. To enter the two-phase region from that point, the solution route would have to follow the tieline path through that composition. (The nontie-line path at that point is just the binodal curve, and following it would give other compositions just at the edge of but not inside the two-phase region.) However, the tie-line wave velocity, λt, is zero at the binodal curve, as Fig. 5.7 indicates. Thus, a continuous composition variation across the binodal curve would cause a discontinuous

5.2. SHOCKS

91

1

C1

Switch at a

a

a

λant

λat

0 0

1

2

3

ξ/τ 1

C1

Switch at b

a

λbt =λbnt 0 0

1

2

3

ξ/τ 1

C1

Switch at c

a

a

λct

λcnt

0 0

1

2

3

ξ/τ

Figure 5.11: Composition proﬁles for path switches. Points a, b, and c are those shown in Fig. 5.9. Path switches at points a and b satisfy the velocity constraint, but a switch at point c violates it and gives a solution proﬁle that is multivalued.

92

CHAPTER 5. TERNARY GAS/OIL DISPLACEMENTS

drop in wave velocity from unit velocity in the single-phase region to zero velocity on the tie line just inside the two-phase region. In fact, any continuous variation would place compositions with zero wave velocity downstream of faster moving two-phase compositions upstream, which would violate the velocity constraint. Therefore, a shock must form, just as it did in the the binary displacements of Chapter 4. If a shock forms, then it must satisfy the jump condition derived in Section 4.2 (Eq. 4.2.2), which actually applies for any number of components. It states that Λ=

FiII − FiI , CiII − CiI

i = 1, nc .

(5.2.1)

If the ﬂuid on one side of the shock is single-phase, then FiI = CiI , and hence, Eq. 5.2.1 can be rearranged to give CiI =

FiII − ΛCiII , 1−Λ

i = 1, nc.

(5.2.2)

Substitution of the deﬁnitions of CiII and FiII in terms of the phase compositions (Eqs. 5.1.3 and 5.1.4) yields (Helﬀerich [31])

CiI = yi

f1 − ΛS1 1−Λ

+ xi

(1 − f1 ) − Λ(1 − S1 ) , 1−Λ

i = 1, nc.

(5.2.3)

Eq. 5.2.3 indicates that the overall volume fraction of component i in the single-phase mixture is a linear combination of the volume fractions of component i in the equilibrium liquid and vapor. Hence, the overall composition of the single-phase mixture must lie on a straight line determined by the equilibrium phase compositions. That line is a tie line, of course. Thus, we have shown that a shock that connects a single-phase composition with a two-phase composition must occur along the extension of a tie line. The fact that a shock can enter or leave the two-phase region only along a tie line extension identiﬁes two key tie lines. They are the injection tie line and the indextie line!initial oil initial tie line, which have extensions that pass through the injection and initial compositions. The same two tie lines also turn out to be important in problems with more than three components. It frequently happens (see Appendix A) that application of the entropy condition and the velocity constraint requires that a phase-change shock be a semishock in which the shock velocity matches the tie line eigenvalue at the landing point in the two-phase region. In such cases, the shock velocity is given by λII t =

5.2.2

df1II FiII − FiI = Λ = i dS1II CiII − CiI

i = 1, nc.

(5.2.4)

Shocks and Rarefactions between Tie Lines

A shock that connects two points within the two-phase region arises when continuous variation along a nontie-line path is not possible because eigenvalues increase as the path is traced from upstream compositions to downstream ones (a shock along a tie line connecting two points within

5.2. SHOCKS

93

the two-phase region is possible, but not unless the initial or injection composition is in the twophase region). Continuous variation along such a path would violate the velocity constraint, and hence a shock must form. Such shocks are sometimes called self-sharpening waves. To determine when self-sharpening waves occur, we consider how eigenvalues vary along a nontie-line path. Let η be a parameter that varies monotonically along the nontie-line path. Differentiation of the expression for λnt in Eq. 5.1.24 with respect to η gives dλnt 1 = dη C1 + p =

dp dF1 + dη dη

1 C1 + p

−

F1 + p (C1 + p)2

dp dC1 + , dη dη

dp dp dF1 dC1 + − λnt + dη dη dη dη

.

(5.2.5)

The derivative dF1 /dη can be related to λt and ent by the following manipulations: ∂F1 dC1 ∂F1 dy1 dF1 = + , dη ∂C1 dη ∂y1 dη

∂F1 dy1 = λt + ∂y1 dC1

dC1 . dη

(5.2.6)

The value of dy1 /dC1 is given by ent (Eq. 5.1.25), λnt − λt dy1 = ∂F 1 dC1 ∂y

(5.2.7)

1

Substitution of Eqs. 5.2.6 and 5.2.7 into Eq. 5.2.5 gives the desired result, C1 − F1 dp F1 − C1 dC1e dλnt = = , dη (C1 + p)2 dη (C1 − C1e )2 dη

(5.2.8)

where C1e denotes the overall volume fraction of component 1 on the envelope curve. Nontie-line paths do not cross the equivelocity curve (where C1 = F1 ), so the sign of dλnt/dη on a particular path is determined by the sign of dC1e /dη. Eq. 5.2.8 makes it easy to determine when a shock must connect two tie lines. Whether λnt increases or decreases as a nontie-line path is traced can be determined easily if the envelope curve can be drawn (or even just sketched) by ﬁnding whether C1e increases or decreases as the path is traced. The following example describes the patterns of shock and rarefaction behavior that are possible in ternary systems. Constant K-Values To illustrate how these ideas apply to a simple system, we assume that K-values are independent of composition. Also, we choose η = y1 . For constant K-values, the slope and intercept of each tie line are obtained by inserting the deﬁnitions of the K-values into the deﬁnitions of α and φ (Eq. 5.1.6), which gives α= and

K2 − 1 x2 , K1 − 1 x1

(5.2.9)

94

CHAPTER 5. TERNARY GAS/OIL DISPLACEMENTS

φ=

K1 − K2 x2 . K1 − 1

(5.2.10)

Expressions for C1e and dC1e /dη can now be obtained from Eq. 5.1.12 by diﬀerentiating Eqs. 5.2.9 and 5.2.10 and using the equation for the liquid portion of the binodal curve (Eq. 3.5.3), K1 x1 + K2 x2 + K3 (1 − x1 − x2 ) = 1.

(5.2.11)

The resulting expression for C1e is C1e

y2 = 12 K1

K1 − K2 1 − K2

K1 − K3 , 1 − K3

(5.2.12)

and the expression for dC1e /dη is dp 2y1 dC1e =− = 2 dη dη K1

K1 − K2 1 − K2

K1 − K3 . 1 − K3

(5.2.13)

Eq. 5.2.12 is an explicit expression for the envelope curve , a segment of parabola when Kvalues are constant. Fig. 5.12 shows envelope curves for the two possible situations, one in which the intermediate component (C2 ) partitions preferentially into the vapor phase (K2 > 1) and one in which the intermediate component prefers the liquid phase (K2 < 1). If K-values are found from an equation of state, an expression for the envelope curve is not easy to ﬁnd, but the curve can be sketched easily if a few tie lines are known. The argument that follows applies whether K-values are constant or not. Consider the two tie lines shown in Fig. 5.12a, in which K2 < 1. Suppose that tie line A is the initial oil tie line, and tie line B is the injection gas tie line. For gas displacing oil, some displacement composition route must connect the two tie lines. If a nontie-line path is traced from tie line A to tie line B, C1e decreases as the composition changes from the downstream (oil) tie line to the upstream (injection gas) tie line. Therefore, dC1e /dη < 0, and Eq. 5.2.8 indicates that λnt decreases as the nontie-line path is traced upstream. Such a variation satisﬁes the velocity rule, and hence a nontie-line rarefaction is permitted. If, on the other hand, tie line B is the initial oil tie line, and tie line A is the injection gas tie line, a shock is required. As the nontie-line path is traced upstream from tie line B, C1e increases as does η, dC1e /dη > 0, and according to Eq. 5.2.8, λnt increases as the path is traced upstream. That composition variation would violate the velocity rule, which requires that wave velocities of compositions upstream be lower than those downstream, and therefore composition variations along the nontie-line path are self-sharpening. Hence a shock is required. Similar reasoning for tie lines C and D in Fig. 5.12b reveals that when K2 > 1, nontie-line paths are self-sharpening when tie line D is the injection gas tie line, and a rarefaction occurs when C is the injection tie line. Table 5.1 summarizes those patterns. While the examples in Fig. 5.12 are for constant K-values, the patterns described are similar when K-values depend on composition. If the nontie-line path is self-sharpening, then a shock must connect the two tie lines. Table 5.1 indicates that whether a shock connects the initial oil and injection tie lines can be determined easily from the magnitude of K2 or equivalently, from the location of the envelope curve for systems with constant K-values. The vast majority of gas/oil systems described by one of the equations of state in common use have envelope curves like those

5.2. SHOCKS

95

C1

C D

C3

C2 b. High volatility intermediate component

C1

A

B

C3

C2 a. Low volatility intermediate component

Figure 5.12: Envelope curves for ternary systems with constant K-values: (a) low volatility intermediate (LVI) component (K2 < 1), (b) high volatility intermediate (HVI) indexHVI component (K2 > 1).

96

CHAPTER 5. TERNARY GAS/OIL DISPLACEMENTS

Table 5.1: Nontie-Line Shocks and Rarefactions for Constant K-Value Systems Envelope Curve and Tie Line Intersections Vapor Side Vapor Side Liquid Side Liquid Side

Intermediate K-Value K2 < 1 K2 < 1 K2 > 1 K2 > 1

Path Direction A→B B→A C→D D→C

Composition Variation Rarefaction Shock Shock Rarefaction

C1 a

(y1 ,y2 )

(x1 ,x2 ) a

C3

I1

C2

a

Figure 5.13: Tie lines that extend through a point. shown in Fig. 5.12 in which the envelope curve moves away from the tie line as the tie-line length increases, and for those systems, Table 5.1 is appropriate. However, it is possible to imagine systems in which the envelope curve moves toward the tie line as the length of the tie lines increases. In such systems, the patterns of shocks and rarefactions are reversed. The more general classiﬁcation of the two types of envelope curves is included in the discussion of multicomponent systems in Chapter 7. Hand’s Rule Fig. 5.13 shows another simple approximation of tie line behavior, known as Hand’s Rule , that is frequently used in models of surfactant ﬂooding [62] and is occasionally used for gas/oil systems [84, 127]. In Fig. 5.13, the tie lines all meet at a single point, I1 . In a representation of phase behavior like that shown in Fig. 5.13, the binodal curve is speciﬁed independently, and equilibrium phase compositions are found as the intersection of a tie line with the binodal curve. The slope of a tie line in Fig. 5.13 through a vapor composition point (y1 , y2 ) is

5.2. SHOCKS

97

α(η) =

y2 , y1 − I1

(5.2.14)

and the intercept is φ(η) = −

y2 I1 = −I1 α(η). y1 − I1

(5.2.15)

According to Eq. 5.1.12 then, dφ dη dα dη

p=

=

dα dη −I1 dα dη

= −I1 .

(5.2.16)

As a result, dp/dη = 0, and therefore, dλnt /dη = 0 as well (Eq. 5.2.8). Hence, when tie lines meet at a point, λnt remains constant as a nontie-line path is traversed (Cer´e and Zanotti [11]). A composition variation along such a path is neither spreading nor self-sharpening. Instead it is known as an indiﬀerent wave, but it has the same appearance as a shock on composition proﬁles because all the compositions in the wave move at the same velocity.

5.2.3

Tie-Line Intersections and Two-Phase Shocks

We now prove an important result that will be useful in ternary displacements and will be even more useful in systems with more than three components. If a shock connects two compositions, A and B, in the two-phase region then its wave velocity is given by ΛAB =

FiA − FiB , CiA − CiB

i = 1, nc .

(5.2.17)

FiA − ΛAB CiA = FiB − ΛAB CiB .

(5.2.18)

Rearrangement of Eq. 5.2.17 gives

Each side of Eq. 5.2.18 is some overall composition expressed as a volume fraction of each component, which we call JiX , so that JiX = FiA − ΛAB CiA ,

(5.2.19)

JiX = FiB − ΛAB CiB ,

(5.2.20)

and

We now ask what conditions JiX must satisfy if Eqs. 5.2.19 and 5.2.20 are to be consistent with the original shock balance, Eq. 5.2.17. Eqs. 5.2.19 and 5.2.20 can be written in terms of the phase compositions on the two tie lines by substituting the deﬁnitions of FiA , FiB , CiA and CiB (Eqs. 5.1.3 and 5.1.4). The result is

(1 − f1A ) − ΛAB (1 − S1A ) , JiX = yiA f1A − ΛAB S1A + xA i and

i = 1, nc,

(5.2.21)

98

CHAPTER 5. TERNARY GAS/OIL DISPLACEMENTS

(1 − f1B ) − ΛAB (1 − S1B ) , JiX = yiB f1B − ΛAB S1B + xB i

i = 1, nc .

(5.2.22)

Eq. 5.2.21 indicates that the mixture composition, JiX lies on the straight line that connects the A liquid phase composition, xA i , with the vapor phase composition, yi , because the right side of Eq. 5.2.21 is a linear combination of the phase compositions. That straight line is the tie line, of course. Similarly, Eq. 5.2.22 shows that the same mixture composition must also lie on the extension of B the tie line that connects xB i to yi . If the same overall composition is to lie on the extension of both tie lines, the tie line extensions must intersect. We have shown, therefore, that extensions of tie lines connected by a shock must intersect. Tie lines can intersect only at a composition point that lies in the single-phase region. Suppose that overall composition is CiX and consider the following shock balances constructed as if the shocks were from the single-phase point, CiX , to the composition points A and B on the tie lines being considered, ΛAB =

CiX − FiA , CiX − CiA

ΛAB =

CiX − FiB , CiX − CiB

i = 1, nc .

(5.2.23)

Each shock balance can be rearranged to give

FiA − ΛAB CiA = CiX (1 − ΛAB ),

FiB − ΛAB CiB = CiX (1 − ΛAB ),

i = 1, nc .

(5.2.24)

Setting the left sides of the two expressions in Eq. 5.2.24 equal and rearranging gives the original shock balance, Eq. 5.2.17. Thus, the shock velocity can be found easily if the intersection point of the tie lines is known from Eq. 5.2.23. The derivations of Eqs. 5.2.21 and 5.2.22 were not restricted to three components. Instead, they apply to systems with any number of components. In ternary systems, all tie lines intersect (unless they are parallel, an unlikely event in gas/oil systems), so the intersection of two tie lines can be found easily by solving two simultaneous equations of the form of Eq. 5.1.5 written for tie lines A and B. In systems with four or more components, the tie-line intersection requirement will be used to identify key tie lines in addition to ﬁnding shock velocities.

5.2.4

Entropy Conditions

When more than two components are present, the entropy condition used to determine unique shock compositions are more complex. The entropy conditions for the leading and trailing shocks, which occur along tie lines are similar to those for the binary displacements of Chapter 4. See Appendix A for a statement of those entropy conditions. Lax [67] stated the appropriate requirements for shocks like those that connect two tie lines in strictly hyperbolic multicomponent systems (a problem is strictly hyperbolic if the eigenvalues are everywhere distinct). The displacements considered here are not strictly hyperbolic, however, because there is a pair of points on every tie line where the eigenvalues are equal. The basic idea of the entropy condition for two-component conservation equations like those solved in this chapter is that one set of characteristics is self-sharpening and the other is not [107, p. 55]. For nonstrictly hyperbolic problems, Isaacson’s statement of the entropy condition [39] is appropriate (see also Keyﬁtz and Kranzer [58] for more discussion of

5.3. EXAMPLE SOLUTIONS: VAPORIZING GAS DRIVES

99

entropy conditions): for ternary systems, the entropy condition is satisﬁed for left and right states indicated by L and R if, for eigenvalues λt and λnt : L λR nt < Λ < λnt ,

and

R λL t < Λ = λt .

(5.2.25)

L λR nt < Λ < λnt ,

and

R Λ = λL t < λt .

(5.2.26)

or

The entropy condition for one example of a ternary displacement is derived in Appendix A and is applied to show that shocks between tie lines are semishocks at which the shock velocity is equal to the tie-line eigenvalue on one side of the shock.

5.3

Example Solutions: Vaporizing Gas Drives

In this section, we describe solutions for a gas displacement situation known as a vaporizing gas drive. The name arises from the fact that vaporization of the intermediate component from the oil into the fast ﬂowing vapor phase can lead to very eﬃcient displacement if enough of the intermediate component is present in the oil (see Section 5.6). In typical ternary phase diagrams for gas/oil systems, component 1 is chosen to be the component with the highest K-values, and it is plotted at the top vertex of the ternary diagram. Component 3 is generally chosen to be the component with the smallest K-value and is plotted at the lower left vertex of the ternary diagram. On such phase diagrams, displacements in which the injection gas tie line lies to the left of the initial oil tie line are known as vaporizing gas drives in the standard terminology. A second type of displacement known as a condensing gas drive /indexcondensing gas drive is described in Section 5.4. Condensing gas drives derive their name from the transfer of intermediate component from the injected gas to the liquid phase being contacted by the injected gas. Condensing gas drives occur when the injection tie line lies to the right of the initial tie line on a ternary diagram with components ordered as described for a vaporizing gas drive. Whether the displacement is a condensing or vaporizing gas drive, the unique composition route that connects the initial and injection compositions is selected by applying the velocity constraint, the entropy condition, and a requirement that the solution be continuous with respect to variations in the initial and injection data. High Volatility Intermediate Component In the examples in this section, we ignore eﬀects of volume change as components transfer between phases, and we assume that equilibrium K-values are independent of composition. For the ﬁrst example, we set K1 = KCH4 = 2.5, K2 = KCO2 = 1.5, and K3 = KC10 = 0.05. We will use the term high volatility intermediate (HVI) component to highlight the fact that K2 > 1. Phase viscosities are also assumed to be independent of composition with µoil /µgas = 5. Phase relative permeability functions are assumed to have the form of Eqs. 4.1.13–4.1.19 with Sgc = Sor = 0. The injected ﬂuid is assumed to be pure CH4 , and several initial oil compositions listed in Table 5.2 are considered. Fig. 5.14 shows the solution composition route and the corresponding saturation and composition proﬁles for displacement of mixture a by pure CH4 . Compositions, saturations and wave velocities of key points in the solution are reported in Table 5.3.

100

CHAPTER 5. TERNARY GAS/OIL DISPLACEMENTS

Table 5.2: Initial and Injection Compositions Fluid Injection Gas Initial Oil a Initial Oil A1 Initial Oil A2

Volume Fractions CCH4 CCO2 CC10 1.000 0.000 0.000 0.150 0.291 0.559 0.000 0.211 0.789 0.050 0.550 0.400

Table 5.3: Displacement of Several Initial Oil Mixtures by pure CH4 Point f e d c b a b1 A1 f2 e2 d2 c2 b2 A2

CCH4 1.0000 0.8178 0.7865 0.3466 0.3312 0.1500 0.3416 0. 1.0000 0.8302 0.8302 0.0927 0.0879 0.0500

CCO2 0.0000 0.0000 0.0000 0.3947 0.3866 0.2908 0.3920 0.2115 0. 0. 0. 0.6987 0.6821 0.5500

CC10 0.0000 0.1822 0.2135 0.2587 0.2822 0.5592 0.2665 0.7885 0. 0.1698 0.1698 0.2086 0.2300 0.4000

S1 1.0000 0.7395 0.6856 0.4886 0.4372 0. 0.4715 0. 1.0000 0.7607 0.7607 0.4778 0.4191 0.

ξ/τ 1.0000 0.2454 0.3593-1.1020 1.1800 1.5180 1.0000 1.2879 1.0000 1.0000 0.2468 0.2468-1.0797 1.2476 1.6474 1.0000

5.3. EXAMPLE SOLUTIONS: VAPORIZING GAS DRIVES

101

CH4 f a

ea d a

Dilution Line

aac

b

a

a

C10

CO2 fa

1

Sg

ea a d 0 1

da

CH4

da

a a d

ac

b

ca

b

CO2

0 1

f ea da

C10

0 1

0 0.0

f

ea da

a

ab aa

fa e

a

c

da

daca 1.0

ξ/τ

a a

a

a a

a

a

a

ab 2.0

Figure 5.14: Composition route and saturation proﬁle for a vaporizing gas drive with a high volatility intermediate component. The injection gas is pure CH4 , and the composition of the oil is C1 = 0.15, C2 = 0.29 and C3 = 0.46. K-values are assumed to be K1 = 2.5, K2 = 1.5, and K3 = 0.05, and the viscosity ratio is constant at µoil /µgas = 5.

102

CHAPTER 5. TERNARY GAS/OIL DISPLACEMENTS

The solution route enters the two-phase region with a shock from the initial composition a across the equivelocity curve (where F1 = C1 ) to point b. As Fig. 5.14 shows, point b lies on the tie line that extends through point a. That shock is a semishock, so the shock velocity, Λab, equals the tie-line eigenvalue, df1b /dS1 . Upstream of the leading shock is a continuous variation (also called a rarefaction) along the initial tie line. Thus, the leading portion of the solution is qualitatively the same as would be observed for a binary displacement like those described in Chapter 4. At point c, the solution route switches to the nontie-line path. Point c is the equal-eigenvalue point , where the nontie-line path is tangent to the initial tie line. Point c is the only point where a path switch is allowed. As Section 5.1.4 showed, compositions that lie above point c are not allowable path switch points because they violate the velocity rule. Points below c are acceptable switch points on the initial oil tie line, but the corresponding switch point at the injection gas tie line would violate the velocity rule. Hence the switch must occur at point c, the equal-eigenvalue point, and the upper branch of the nontie-line path must be traced if there is to be an allowable path switch at the intersection of the nontie-line path and the injection gas tie line. According to the arguments of Section 5.2 summarized in Table 5.1 ( See also Eqs. 5.2.8 and 5.2.13), eigenvalues decline as the nontie-line path is traced toward the injection tie line, and hence a rarefaction connects the injection gas and initial oil tie lines. The saturation proﬁle of Fig. 5.14 shows that while the saturation changes appreciably as the nontie-line path is traced the corresponding wave spreads only slightly because λnt changes only slightly along the nontie-line path (see also Table 5.3). Point d is the point at which the nontie-line path intersects the injection tie line. At that point, the solution route switches to the injection tie line, which creates a zone of constant state (see Section 5.3). The composition at point d has two wave velocities, a fast one given by the nontie-line eigenvalue, λdnt , and a slow one given by the tie-line eigenvalue, λdt . Therefore point d, which is a single point on the ternary diagram, appears as a region in the proﬁle. Upstream of point d the solution is identical to the trailing portion of a binary displacement. As the middle panel of Fig. 5.14 shows, CO2 is not present at point d on the injection tie line, so the portion of the path on the injection tie line must be the same as that of a binary displacement. Between points d and e, there is a continuous variation along the injection tie line, and there is a semishock from point e to point f, the injection composition. The trailing shock moves relatively rapidly in this example because the K-value of C10 is large enough that the capacity of pure CH4 to vaporize pure C10 is signiﬁcant. The concentration proﬁles of Fig. 5.14 show that the CH4 concentration increases steadily as the solution path is traced from the initial composition upstream to the injection composition in a proﬁle that mirrors that of the gas saturation, S1 , in Fig. 5.14. The proﬁle pattern for C10 is similar to an inverted version of the proﬁle for CH4 . The C10 concentration declines monotonically as the solution path is traced, and the C10 concentration drops to zero behind the trailing evaporation shock. The CO2 proﬁle, however, diﬀers qualitatively from those of CH4 and C10 . There is a bank of ﬂuid enriched with CO2 between points b and d. The CO2 that enriches that bank was separated chromatographically from the two-phase mixtures that lie upstream of the CO2 bank (between d and e). Another view of that enrichment is given by the composition route shown in the ternary diagram of Fig. 5.14. The overall compositions in the leading portion of the transition zone (a→b–c) contain larger fractions of CO2 , the intermediate component, than do mixtures that lie on the dilution line that connects the initial mixture with the injection mixture. In contrast, mixtures in the trailing

5.3. EXAMPLE SOLUTIONS: VAPORIZING GAS DRIVES

103

portion (d–e→f) contain less CO2 than dilution mixtures. The CO2 present in the leading portion of the transition zone must have been vaporized from the initial oil, because there is no CO2 present in the injection ﬂuid. That enrichment of the ﬂowing vapor phase is what is meant by the term vaporizing gas drive. Now consider how the solution would change if the initial composition were changed along the extension of the same tie line that extends through point a. If all the CH4 were removed, the initial composition would be that of point A1 in Table 5.3 and Fig. 5.15. The resulting composition path and proﬁles are also shown in Fig. 5.15. Comparison of Figs. 5.14 and 5.15 and the wave velocities reported in Table 5.3 indicates that the velocity of the leading semishock, Λab changes. Removing the CH4 slows the leading shock slightly, as Table 5.3 shows. Upstream of point b1 , however, the solution is identical to that for initial composition a. Thus, the overall pattern of the displacement is controlled by the initial and injection tie lines, while details such as leading and trailing shock velocity are determined by the location of the initial or injection composition on the tie-line extension. If the amount of CO2 in the initial mixture is increased, as it is for initial mixture A2 for example, an additional changes to the composition proﬁles occur. The leading shock now moves signiﬁcantly faster, and the nontie-path is displaced toward the CH4 /CO2 side of the ternary diagram. When the mixture A2 is displaced, the nontie-line path lands on the injection tie line at point d2 , which lies just above the tangent point for a trailing shock, e1 . A rarefaction from d2 to e1 would violate the velocity constraint, because λt increases for decreasing CH4 along the injection tie line. A shock from d2 to e1 , followed by a semishock from e1 to f is also not permitted because the shock from d2 to e1 would be slower than the trailing semishock, another violation of the velocity constraint. Therefore, the ﬁnal segment of the solution must be a genuine shock directly from d2 to the injection composition f. Λ

df

=

F1f − F1d2 C1f − C1d2

=

1 − F1d2 1 − C1d2

.

(5.3.1)

Such a shock forms whenever the intersection of the nontie-line path with the injection tie line lies above the semishock point on that tie line, and it moves with a higher wave velocity than the trailing semishock because there is less C10 to be evaporated. Low Volatility Intermediate Component If the K-value of the intermediate component is less than one (K2 < 1), (LVI, low volatility intermediate) a rarefaction along the nontie-line path between the initial and injection tie lines is not permitted: λnt increases as the nontie-line path is traced upstream (see Section 5.2.2, Table 5.1). Consider a displacement with K1 = 2.5, K2 = 0.5, K3 = 0.05. Those K-values might represent a system of CH4 , C4 , and C10 , for example. Figure 5.16 shows the solution for a displacement of an oil mixture with composition C1 = 0.1, C2 = 0.5, and C3 = 0.4 by pure CH4 with a ﬁxed viscosity ratio of ﬁve. The leading segment of the displacement is just as in the previous example: a tangent shock along the tie line that extends through the initial composition (point a), followed by a rarefaction along the initial tie line. That segment ends at another tangent shock, one that connects the initial tie line to the injection tie line. In Section 5.2.3, we proved that two tie lines connected by a shock

104

CHAPTER 5. TERNARY GAS/OIL DISPLACEMENTS

CH4 f a

e1 aad2 d1 a

aa c1 b1

C10

Sg

1

A1

a

fa ad2a e1 d

1

b2 c2

ad a2 d1 a ca2a c1

b1

aA1

0

a a a

CO2

ba2

aA2

aa

CH4

1

aa

A2 a

a a a a

0

a a

1

CO2

a

0 1

a a

aa

a a a

a a

C10

a

0 0.0

a a a

a aa a a 1.0

ξ/τ

a a 2.0

Figure 5.15: Composition path and proﬁles for displacement of oils A1 and A2 by pure CH4 .

5.3. EXAMPLE SOLUTIONS: VAPORIZING GAS DRIVES

105

a

CH4 f a

d a

ac a b

a a

C10 1

C4 fa

a

Sg

d

0 1

ad ca

ab aa

a a

a

CH4

a

a a

C4

0 1

a a

a a

a

C10

0 1

0 0.0

a a

a a

a 1.0

ξ/τ

2.0

Figure 5.16: Composition path and proﬁles for a vaporizing gas drive with low volatility intermediate component. K1 = 2.5, K2 = 0.5, K3 = 0.05, and M = 5. The injection gas is pure CH4 , and the initial oil has composition C1 = 0.1, C2 = 0.5, and C3 = 0.4.

106

CHAPTER 5. TERNARY GAS/OIL DISPLACEMENTS

must intersect, and that the velocity of the shock can be calculated easily using the intersection composition (Eq. 5.2.23): Λcd =

CiX − Fic df c c , = λ = t dS CiX − Cic

i = 1, nc .

(5.3.2)

Eq. 5.3.2 depends only on a single variable, the saturation on the initial tie line, and solving it is no more diﬃcult than solving the tangent construction for the leading shock. The landing point on the injection tie line is determined from CiX − Fid = λct , X d Ci − Ci

i = 1, nc .

(5.3.3)

Again, this equation depends only on the saturation on the injection tie line, so it can be solved easily as well. Figure 5.16 shows that there is no rarefaction present on the injection tie line: the landing point of the intermediate shock lies above the tangent point for a trailing shock. As a result, the trailing shock is a genuine shock from the landing point on the injection tie line (point d) to the injection composition (point f). Thus, only one rarefaction is present in this displacement, and the remainder of the displacement behavior is determined by the velocities of the three shocks. No integration is required to solve this problem. Instead, three relatively simple nonlinear equations in a single variable are solved to determine the shock compositions and velocities.

5.4

Example Solutions: Condensing Gas Drives

Condensing gas drives are so named because some of the intermediate component present in the injection gas stream condenses into the liquid phase present in the transition zone. Just as in the vaporizing gas drives, there are two versions of the solution. In HVI systems, there is a shock between the injection and initial tie lines in condensing gas drives . In LVI systems, on the other hand, a rarefaction connects those tie lines. High Volatility Intermediate Component To illustrate the features of a condensing drive with a self-sharpening nontie-line path, we consider an initial mixture that contains only CH4 and C10 displaced by a mixture of CH4 and CO2 . Here again we use constant equilibrium K-values with the same fractional ﬂow function used in Section 5.3. Fig. 5.17 shows the composition route, saturation proﬁle, and composition proﬁles. Composition e is the injection composition, in this case CCH4 = 0.6, CCO2 = 0.4. In a condensing gas drive the solution path is traced from the injection composition to the initial composition. From the injection composition, the solution composition route enters the two-phase region with a shock to a composition on the tie line that extends through the injection composition. The trailing shock is a semishock with velocity Λde = df1d /dS1 . Downstream of the trailing shock is a rarefaction along the injection tie line. Evaluation of dλnt /dη with Eqs. 5.2.8 and 5.2.13 indicates that λnt increases as nontie-line paths are traced from the injection tie line to the initial tie line. A rarefaction along one of those paths would violate the velocity constraint, and hence, a shock is required. Application of the entropy

5.4. EXAMPLE SOLUTIONS: CONDENSING GAS DRIVES

107

CH4

ae

b a

a ac

d

a a

C10

1

ea

a

ca

d

Sg

a

CO2

a

b

0

ba

aa

CH4

1

a a

a a

a a

a

a a

0

CO2

1

a

0 1

a

C10

a

0 0.0

a a

a a 1.0

ξ/τ

a 2.0

Figure 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 = 5. 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.

108

CHAPTER 5. TERNARY GAS/OIL DISPLACEMENTS

condition (see Section 5.2.4) shows that this shock is also a semishock, for which the wave velocity is Λbc =

Fib − Fic df1c = . dS1 Cib − Cic

(5.4.1)

Eq. 5.4.1 holds for any component, and because the concentration of CO2 is zero at point b, it is convenient to write Eq. 5.4.1 for CO2 , Λbc =

c FCO df1c 2 = . c CCO dS1 2

(5.4.2)

Eq. 5.4.2 can be solved easily for the overall composition of point c, which must lie on the injection tie line. Eq. 5.4.1 written for CH4 , b b c c − Λbc CCH = FCH − Λbc CCH , FCH 4 4 4 4

(5.4.3)

can then be solved for the overall composition at point b. In this example, the landing point b is below the overall composition at which a leading semishock lands on the initial tie line, and hence there is no rarefaction along the initial tie line. Instead, there is a genuine shock from point b to the initial composition, point a. The leading shock velocity is given by the shock material balance, Λab =

b a FCH − FCH 4 4 . b a CCH − C CH4 4

(5.4.4)

A rarefaction and semishock on the initial tie line can occur, but only if the fraction of CO2 in the injection mixture is low, in this example below about 5%. In this condensing gas drive, then, the solution consists of a single rarefaction wave along the injection tie line and three shocks. Because the rarefaction occurs on a single tie line, it is identical to the rarefaction that occurs in the binary displacements of Chapter 4. The trailing and intermediate shocks are both semishocks, each of which is determined by solving a single nonlinear equation for the composition on one side of the shock. Thus, the solution for this ternary displacement can be found by a procedure that is similar to and not much more diﬃcult than that used to solve binary displacement problems. Fig. 5.17 shows that the trailing portion of the composition route is richer in CO2 , the intermediate component, than mixtures that form on the dilution line that connects the initial and injection compositions, while the downstream portion of the route is leaner in CO2 . Another representation of the same idea is given in the CH4 proﬁle shown in Fig. 5.17, which indicates that there is a CH4 bank between the locations of the a→b and b→c shocks. Thus, mixtures ﬂowing at the leading edge of the transition zone are depleted in CO2 because it has condensed into the liquid phase present at the upstream end of the transition zone. The CO2 present there must have condensed from the injection gas because no CO2 is present in the initial ﬂuid. It is that transfer of intermediate component from the injection gas that is the source of the name, condensing gas drive. Here again, the components present in the injection and initial ﬂuids have separated chromatographically due to the interaction of phase equilibrium with multiphase ﬂow. Low Volatility Intermediate Component

5.4. EXAMPLE SOLUTIONS: CONDENSING GAS DRIVES

109

CH4 ae ad ac

b a

a a

C10 1

C4 ea

a

Sg

d

CH4

0 1

ca

a

ab

b

a a

a

aa

a

a a

C4

0 1

0 1

a a

a a

a

C10

a a 0 0.0

a a

a

a 1.0

ξ/τ

2.0

Figure 5.18: A condensing gas drive (LVI) with a nontie-line rarefaction. K1 = 2.5, K2 = 0.5, K3 = 0.05, and M = 5. The injection gas has composition, C1 = 0.8, C2 = 0.2, and C3 = 0., and the initial oil has composition, C1 = 0.3, C2 = 0, and C3 = 0.7.

110

CHAPTER 5. TERNARY GAS/OIL DISPLACEMENTS

Fig. 5.18 shows the last of the four types of displacement that occur in ternary gas drives indexLVI!condensing. When K2 < 1, tie line intersections occur on the vapor side of the twophase region, and a rarefaction occurs along the nontie-line path that connects the injection gas tie line to the initial oil tie line. Otherwise, the pattern of composition and saturation variations is qualitatively similar to the self-sharpening condensing gas drive in Fig. 5.17. There is a trailing semishock (e→d) preceded by a rarefaction along the injection gas tie line (d–c), a switch to the nontie-line path at point c, a rarefaction along which λnt increases (as it must as the solution route is traced toward the initial composition), and a leading genuine shock (b→a).

5.5

Structure of Ternary Gas/Oil Displacements

The analysis of Sections 5.1 and 5.2 and the examples of vaporizing and condensing gas drives in Sections 5.3 and 5.4 reveal patterns of displacement behavior that can be catalogued easily in terms of the lengths of the two key tie lines and the location of tie line intersections with respect to the two-phase region (see Table 5.4) for systems that exhibit envelope curves with shapes like those shown in Fig. 5.12. Similar patterns reappear in displacements with more than three components, and hence it is worth reviewing them. Most gas/oil systems do show envelope curves in which the envelope curve approaches the tie line as the tie-line length decreases, but the possibility that systems exist in which the envelope curve approaches the tie line as tie-line length increases cannot be ruled out. That situation is considered in Chapter 7. For typical ternary systems in which a single-phase vapor displaces a single-phase oil, the patterns of Table 5.4 are all that arise. To see how the patterns arise, it is useful to summarize how a solution can be constructed for any ternary gas drive. The solution to a ternary gas drive problem can be constructed with the following steps: 1. Identify the tie lines that extend through the injection gas and initial oil compositions. If, as is commonly the case, the injection gas and initial oil compositions lie in the single-phase region, the key tie lines can be found by a negative ﬂash [135, 129]. A negative ﬂash is just a ﬂash calculation for a mixture that is single-phase. If it converges, it will give the compositions of the phases on the tie line that extends through the single-phase composition. 2. Construct semishocks from the injection gas and initial oil compositions to determine the tangent shock points on the injection gas and initial oil tie lines. 3. Determine whether a rarefaction or a shock connects the two key tie lines. Table 5.4 summarizes the possible situations. Or, Eq. 5.2.13 can be used. It gives the correct variation of λnt even if K-values are not constant. 4. Identify the shortest key tie line. Solution construction begins with that tie line (see the argument below for the reasoning behind this statement). 5. If a rarefaction connects the two tie lines, integrate the nontie-line path from the equaleigenvalue point on the shortest tie line to determine where it intersects the other tie line. 6. If an intermediate shock connects the two tie lines, construct a semishock for which the shock velocity equals λt on the shortest tie line.

5.5. STRUCTURE OF TERNARY GAS/OIL DISPLACEMENTS

111

7. Construct a rarefaction along the shortest tie line that connects the equal-eigenvalue point or the intermediate shock point to the tangent shock point for the single-phase composition (oil or gas) associated with the shortest tie line. 8. Determine whether a tie-line rarefaction can occur on the longest tie line. A rarefaction can occur if variation along the tie line from the landing point of the intermediate shock or the intersection of the nontie-line path with the tie line to the tangent shock point satisﬁes the velocity rule. If so, the shock from the single-phase composition to the longest tie line is a semishock. If not, a genuine shock is constructed from the landing or intersection point to the single-phase composition. The fact that solution construction must begin on the shortest tie line arises from two observations about the geometry of composition paths and shocks. The ﬁrst observation applies to displacements in which a rarefaction connects the two key tie lines. In that case, there are two points at which the appropriate nontie-line path intersects the two key tie lines . Both points must be switch points at which the velocity rule is satisﬁed. The argument given in Section 5.3 indicates that one of the two points must be the equal-eigenvalue point. The geometry of nontieline paths (see Fig. 5.9) indicates that the point of tangency of the nontie-line path to a tie line (which is the point at which eigenvalues are equal) occurs on the shorter of the two tie lines. If the equal-eigenvalue point on the longer tie line were selected instead, the paths traced would not reach the shorter tie line. Solution construction can proceed by the steps outlined above once the equal-eigenvalue point is found on the shorter of the two key tie lines. The second observation applies to displacements in which the two key tie lines are connected by a shock. In that case, the nontie-line rarefaction is replaced by a semishock with a wave speed that matches the tie-line eigenvalue on the same tie line that includes the equal-eigenvalue point for the rarefaction path, again, the shorter of the two key tie lines. Figure 5.19 illustrates the construction of a semishock between tie lines (the example shown is the c→d shock in Fig. 5.16). The shock balance for the intermediate shock (written for component 1) is Λcd =

F1c − F1d C1X − F1c C1X − F1d ∂F1 = = = X c ∂C1 C1c − C1d C1 − C1 C1X − C1d

(5.5.1)

Fig. 5.19 shows the appropriate tangent construction: a chord drawn from point X, the intersection point of the two tie lines, to a tangent point on the fractional ﬂow curve for the shorter tie line locates point c, the point that satisﬁes Eq. 5.5.1. The intersection of the same chord with the fractional ﬂow curve for the longer tie line gives point d. The fractional ﬂow curves shown in Fig. 5.19 are typical of systems in which y1c < y1d and c M < M d , both reasonable physical assumptions. In such systems, it is possible to construct a tangent to the fractional ﬂow curve for the shorter tie line that also intersects the fractional ﬂow curve for the longer tie line. If, on the other hand, the tangent had been drawn to the fractional ﬂow curve for the longer tie line, to point d∗ in Fig. 5.19, it would not intersect the curve for the shorter tie line. In that case, there would be no solution to Eq. 5.5.1. Hence, the tangent must be constructed to the shorter tie line, and therefore it is appropriate to start solution construction with the shorter tie line.

112

CHAPTER 5. TERNARY GAS/OIL DISPLACEMENTS

a

Overall Fractional Flow of Component 1, F1

1.2

X

1.0

a

d*

a

0.8

af

d

ca

b a 0.6

0.4

0.2

aa 0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

Overall Volume Fraction of Component 1, C1

Figure 5.19: Tangent constructions for a shock between tie lines.

5.5. STRUCTURE OF TERNARY GAS/OIL DISPLACEMENTS

113

C1

f

C

1 fa

a

d a

e a d a

aabc aac

aa

C3

C2

fa

1

0

da

a

c

ab

c a

ba

0

aa

ξ

C1

C1

a

ac d

a a

C2

C3

ea

1

a

ca

d

a

b

ξ

ae a adc

b a

ae

C3

Sg

ad

LVI Vaporizing Drive

a a

0

C2

HVI Vaporizing Drive

b a

1

a

d

aa

ξ

fa

Sg

Sg

e aa d

a a

C3

C2 ea

a

b

a aa

HVI Condensing Drive

d

Sg

1

b

0

ca

b

a

a

b

ξ

aa

LVI Condensing Drive

Figure 5.20: Structure of solutions for condensing and vaporizing gas drives.

114

CHAPTER 5. TERNARY GAS/OIL DISPLACEMENTS

Table 5.4: Nontie-Line Shocks and Rarefactions Envelope Curve and Tie Line Intersections Vapor Side Vapor Side Liquid Side Liquid Side

Intermediate K-Value K2 < 1 K2 < 1 K2 > 1 K2 > 1

Process Name LVI Condensing LVI Vaporizing HVI Condensing HVI Vaporizing

Shortest Tie Line Injection Gas Initial Oil Injection Gas Initial Oil

Composition Variation Rarefaction Shock Shock Rarefaction

Tie-line length also indicates whether a displacement is condensing or vaporizing. When the initial oil tie line is the shorter of the two key tie lines, the displacement is a vaporizing gas drive. When the injection gas tie line is the shorter tie line, the displacement is a condensing gas drive. The steps outlined above determine the following segments of the solution for displacements in which a single-phase vapor displaces a single-phase oil, listed in order from the downstream to upstream locations. Fig. 5.20 illustrates the important composition variations and proﬁle segments. 1. Leading Shock, a→b. A leading shock is always present if the initial composition is a single-phase liquid. In a vaporizing gas drive (initial oil tie line is shorter than the injection gas tie line), it will always be a semishock. In a condensing gas drive (injection gas tie line is shorter than the initial oil tie line), it may be a semishock or a genuine shock. 2. Tie-Line Rarefaction, b-c. In a vaporizing gas drive, this rarefaction along the initial oil tie line is always present. It connects the landing point of the leading semishock with the point at which the nontie-line composition variation begins, either the equal-eigenvalue point or the semishock point of the intermediate shock. In a condensing gas drive, this segment is missing if the leading shock is a genuine shock, as it often is. 3. Composition Variation between Tie Lines, c-d or c→d. If the composition variation is a rarefaction, the wave velocity on the nontie-line path will match the tie line eigenvalue on the shorter tie line (at the equal-eigenvalue point), and there will be a zone of constant state associated with the point at which the nontie-line path intersects the longer tie line. If the composition variation is a shock, the shock velocity will match the tie-line eigenvalue on the shorter tie line, and there will be a zone of constant state associated with the shock landing point on the longer tie line. 4. Tie-Line Rarefaction, d-e. In a condensing gas drive, this segment, which connects the nontie-line path switch point on the injection gas tie line (point d to the trailing shock point (point e), is always present. That shock is always a semishock. In a vaporizing drive, this segment is present only if the trailing shock is a semishock. Otherwise, this segment is missing. 5. Trailing Shock, e→f. A trailing shock is always present as long as the injection gas is a single-phase mixture. In a condensing drive, it is always a semishock. In a vaporizing drive it may be a semishock or a genuine shock.

5.5. STRUCTURE OF TERNARY GAS/OIL DISPLACEMENTS

115

C

C

1 fa

1 fa

d a

d a

ac ab2

aabc

aa2 a1 a

C3 fa

a

d

1.0

ad ca

Sg 0.0 0.0

C3

0.5

aa1

1.5

fa

a

d

ba1

1.0

C2

2.0

0.0 0.0

2.5

ad ca

Sg

1.0

C2

ab2 aa2

0.5

1.0

ξ

C

C

1 fa

d4 a

d a

a4 a

ac aa3

C3

C2

Sg

d

0.0 0.0

C3 1.0

ad

fa

a

d4

ca

0.5

C2

aa3

1.0

1.5

2.0

ξ (c) Initial mixture a3

2.5

Sg

a

2.5

(b) Initial mixture a2

1 fa

fa

2.0

ξ

(a) Initial mixture a1

1.0

1.5

0.0 0.0

0.5

a a

a4

1.0

1.5

2.0

2.5

ξ (d) Initial mixture a4

Figure 5.21: Eﬀect of variations in initial composition. All initial compositions lie on the same initial tie line or its extension. (a) a1 is a single phase liquid. (b) a2 has a gas saturation of 5 percent. (c) a3 has a gas saturation of 30 percent. (d) a4 has an initial gas saturation of 80 percent.

116

CHAPTER 5. TERNARY GAS/OIL DISPLACEMENTS

CO2

aa aa aa aa

C10

A

B

a a

1.0

a

C

C4 a a

a a

a

a B

a C

aa A a a a

Sg

a

a

0.0 0.0

0.2

0.4

0.6

0.8

a

aa a

1.0

1.2

ξ

Figure 5.22: Changes in solution composition route as the initial mixture is enriched in the intermediate component for displacements conducted at 1600 psia (109 atm) and 160 F (71 C). Phase behavior was calculated with the Peng-Robinson equation of state. Initial compositions are Point A B = 0, CCA4 = 0.195950, CCA10 = 0.804050, Point B: CCO = 0, CCB4 = 0.373735, CCB10 = A: CCO 2 2 C C C 0.626265, and Point C: CCO2 = 0, CC4 = 0.475887, CC10 = 0.524113.

5.6. MULTICONTACT MISCIBILITY

5.5.1

117

Eﬀects of Variations in Initial Composition

The patterns shown in Fig. 5.20 may change if the initial composition changes. To illustrate the variations in patterns of shocks and rarefactions as the initial composition varies, we consider the displacement of Fig. 5.16, and vary the initial composition along the extension of the initial tie line and the tie line itself. Fig. 5.19 shows the fractional ﬂow curves for the initial and injection tie lines. Fig. 5.21 shows composition paths and gas saturation proﬁles for the four patterns of displacement behavior. If the initial composition lies between points a and b in Fig. 5.19, the variations in patterns all involve the leading shock and rarefaction along the initial tie line. The discussion of Figs. 4.12 and 4.15 describes the changes that are observed for initial composition variations between a and b. For example, for initial compositions between a and the inﬂection point on the fractional ﬂow curve for the initial tie line (point j in Fig. 4.12), the patterns of shocks do not change, though the wave velocity of the leading shock increases as the initial composition moves from a to j. Comparison of the positions of the leading shocks in Figs. 5.21a and 5.21b shows that the leading shock speeds up signiﬁcantly as the composition is moved from a1 in the single-phase region to a2 with a gas saturation of 5 percent in the two-phase region. The intermediate and trailing shock compositions are unchanged, however. For initial compositions between the inﬂection point on the initial tie line (Point j on Fig. 4.12) and the tangent shock point for the intermediate shock (Point c in Figs. 5.16 and 5.19), the leading shock is missing, and the leading portion of the proﬁle begins with a rarefaction along the initial tie line, but the trailing portions of the displacement are unchanged. Fig. 5.21c illustrates that situation for an initial gas saturation of 30 percent. For initial compositions between the semishock point for the intermediate shock (Point c on Figs. 5.16 and 5.19) and the vapor composition on the initial tie line, a tangent intermediate shock is no longer possible. Variation along the initial tie line from the initial composition to Point c would violate the velocity rule. Instead, there is an immediate shock to the injection tie line, but that shock is a genuine shock. Fig. 5.21d shows that the entire displacement consists of an evaporation shock from the initial composition to the injection composition when the initial gas saturation is 80 percent. This behavior is typical of condensate systems, in which the saturation of liquid is quite low. Thus, for initial compositions in the two-phase region, one or more of the segments that appear in the displacment of a liquid phase by a vapor phase may be missing.

5.6

Multicontact Miscibility

In this section we consider how displacement eﬃciency in a gas injection process changes depending on the relative locations of the injection gas and initial oil tie lines. Displacement eﬃciency depends quite strongly on the relative locations of the key tie lines with respect to the plait point, as we now show. To illustrate the important ideas behind developed miscibility or multicontact miscibility, consider ﬁrst a vaporizing gas drive that is fully self-sharpening (in other words, the two key tie lines are connected by a shock).

118

5.6.1

CHAPTER 5. TERNARY GAS/OIL DISPLACEMENTS

Vaporizing Gas Drives

Figure 5.22 shows what happens as the initial mixture is enriched in Component 2 in a vaporizing gas drive. Fig. 5.22 shows the compositions of the three initial mixtures a, b, and c, the composition routes, and the resulting saturation proﬁles for the three displacements. Those solutions show that as the initial mixture is enriched in the intermediate component, the leading shock slows, and the intermediate and trailing shocks speed up, and displacement eﬃciency improves, because the amount of oil phase present in the transition zone decreases. To see why this behavior occurs, consider the shock balances for the leading, intermediate, and trailing shocks are = Λlead i

Fia − Fib , Cia − Cib

Λint i =

Fic − Fid , Cic − Cid

Λtrail = i

Fid − Fie Cid − Cie

i = 1, nc .

(5.6.1)

Now consider what happens in the limit as the initial mixture is enriched enough in Component 2 that the initial tie line becomes the critical tie line that is tangent to the binodal curve at the plait point. The plait point terminates the equivelocity curve, where f1 = S1 . As a result, Point b, the landing point of the leading shock, lies on the equivelocity curve, and Fib = Cib . Note also that Fia = Cia because the initial mixture is single-phase. Therefore, it must be that = Λlead i

Fia − Fib Cia − Cib = = 1, Cia − Cib Cia − Cib

i = 1, nc.

(5.6.2)

Because the critical tie line has zero length, Point c must also be the plait point, and hence Fic = Cic . And as the initial mixture is enriched in the intermediate component enough to approach the critical tie line, Point d on the injection gas tie line approaches the vapor locus of the binodal curve, reaching it when the initial tie line is the critical tie line. If a nontie-line path connects the initial and injection tie lines, it becomes the binodal curve in the limit, and all compositions on that path have λnt = 1. If a shock connects the two tie lines, it is replaced in the limit by an indiﬀerent wave with unit velocity, because a path switch at the equal-eigenvalue point (the critical point) followed by a rarefaction along the binodal curve now does not violate the velocity constraint. However, shock velocities go smoothly to the limit because exactly the same velocity that is obtained for a shock from the critical point to Point d on the binodal curve on the injection tie line. At that point, S1 = f1 and therefore, Fid = Cid . Also, the injection mixture is single-phase, so Fie = Cie . Thus, it must also be that Λint i =

Fic − Fid Cic − Cid = = 1, Cic − Cid Cic − Cid

= Λtrail i

Fid − Fie Cid − Cie = = 1, Cid − Cie Cid − Cie

i = 1, nc ,

(5.6.3)

and i = 1, nc .

(5.6.4)

Thus, we have proved that in the limit as the inital tie line becomes the critical tie line, all three shocks coalesce into a single shock and move with unit velocity, and the resulting displacement is piston-like. That displacement moves all of the oil ahead of the shock, and hence all of the initial oil is recovered at one pore volume injected when the shock reaches the outlet. Similar arguments apply when the intermediate shock is replaced by a rarefaction. In that case, the rarefaction along

5.6. MULTICONTACT MISCIBILITY

119

the nontie-line path follows the vapor portion of the binodal curve, which has unit velocity, when the initial tie line is a critical tie line. Here again, the entire transition zone between injected ﬂuid and initial oil moves with unit velocity, and no oil is left undisplaced. It is this perfectly eﬃcient displacement that is meant by the term multicontact miscibility. While the intial and injection mixtures are not miscible in the sense that they could be mixed in any proportions and form only one phase, the combination of two-phase ﬂow and phase equilibrium causes chromatographic separations that lead to a composition route that avoids the two-phase region.

5.6.2

Condensing Gas Drives

The analysis of condensing gas drives is similar. In condensing gas drives, the injection gas is enriched with the intermediate component, and it is the injection gas tie line that becomes the critical tie line with suﬃcient enrichment. The appropriate shock balances are again those given in Eqs. 5.6.1. When the injection tie line is the critical tie line, Point d is the plait point, where f1 = S1 , and as a result, Fid = Cid . Eq. 5.6.4 indicates, therefore that the trailing shock has unit velocity. If a rarefaction connects the injection and initial tie lines, it becomes, in the limit as the injection tie line reaches the critical tie line, the equivelocity curve. All compositions on that curve have unit velocity, so that indiﬀerent wave is equivalent to a unit velocity shock. If, on the other hand, the nontie-line path is self-sharpening, the shock from the injection tie line to the initial tie line connects the plait point, where Fic = Cic to the intersection of the equivelocity curve and the initial tie line, where Fib = Cib . The intermediate shock velocity is then = Λtrail i

Fib − Fic Cib − Cic = = 1, Cib − Cic Cib − Cic

i = 1, nc.

(5.6.5)

Finally, the leading shock connects a point on the equivelocity curve where Fib = Cib , and at the initial composition, Fia = Cia , and therefore, Eq. 5.6.2 shows that this shock also has unit velocity. Thus, when the injection gas tie line is a critical tie line in a condensing gas drive, all the compositions present in the solution route move with unit velocity, and the displacement is piston-like.

5.6.3

Multicontact Miscibility in Ternary Systems

The analysis of this section indicates that multicontact miscible displacement occurs in ternary systems when either the initial tie line or the injection tie line is a critical tie line, a tie line of zero length that is tangent to the binodal curve at the critical point. Condensing gas drives are multicontact miscible when the injection gas tie line is the critical tie line, and vaporizing gas drives are multicontact miscible when the initial oil tie line is critical. It is relatively easy to adjust composition of the injection gas in ﬁeld applications, so it is common to determine a minimum enrichment for miscibility (MME) for condensing gas drives. It is important to note, however, that the MME depends on the displacement pressure, because the size of the two-phase region for a given ternary system as well as the locations of the plait point and the critical tie line depend on pressure. Thus, as reservoir pressure changes during the life of a displacement process, the MME will also change. While reservoir temperature is usually taken to be ﬁxed, any change due, for example, to injection of cold water, would also change the MME.

120

CHAPTER 5. TERNARY GAS/OIL DISPLACEMENTS

In oil ﬁeld settings it is generally impossible to adjust the composition of the initial oil, and hence in vaporizing gas drives, it is the pressure that is adjusted to ﬁnd the minimum miscibility pressure (MMP) for a given injection gas composition. In ternary systems, the MMP in a vaporizing gas drive does not depend on the injection gas composition. In systems with more components, however, it is possible for the MMP to depend on injection gas composition. Thus, in ﬁeld displacements it is often reasonable to consider both adjustment of reservoir pressure and injection gas composition as ways to achieve the eﬃcient displacement that results from multicontact misibility.

5.7

Volume Change

When components change volume as they transfer from one phase to another, the appropriate balance equation on moles of component i is Eq. 2.3.9 written for the three components, ∂G1 ∂H1 + ∂τ ∂ξ ∂G2 ∂H2 + ∂τ ∂ξ ∂G3 ∂H3 + ∂τ ∂ξ

= 0,

(5.7.1)

= 0,

(5.7.2)

= 0,

(5.7.3)

where Gi = xi1 ρ1D S1 + xi2 ρ2D (1 − S1 ),

(5.7.4)

Hi = vD (xi1 ρ1D f1 + xi2 ρ2D (1 − f1 )).

(5.7.5)

Manipulations similar to those of Sections 4.4 and 5.1 yield the eigenvalue problem, {[H(u)] − λ[G(u)]} u = 0,

(5.7.6)

where ⎡ ∂H1 ∂z1 ⎢ 2 H(u) = ⎣ ∂H ∂z1 ∂H3 ∂z1

⎡ ∂G 1 ∂z ⎢ ∂G12 [G(u)] = ⎣ ∂z1 ∂G3 ∂z1

∂H1 ∂z2 ∂H2 ∂z2 ∂H3 ∂z2 ∂G1 ∂z2 ∂G2 ∂z2 ∂G3 ∂z2

∂H1 ∂z3 ∂H2 ∂z3 ∂H3 ∂z3 ∂G1 ∂z3 ∂G2 ∂z3 ∂G3 ∂z3

∂H1 ∂vD ∂H2 ∂vD ∂H3 ∂vD

⎤ ⎥ ⎦,

(5.7.7)

⎤

0 ⎥ 0 ⎦, 0

(5.7.8)

and uT = [dz1 , dz2, vD ].

(5.7.9)

In these expressions, zi is the overall mole fraction of component i. Only two of the mole fractions are independent, because they sum to one, but all three of the conservation equations are independent, and they are needed because the ﬂow velocity, vD , must also be determined.

5.7. VOLUME CHANGE

121

The analysis of paths and shocks for ternary systems with volume change follows the same reasoning as the analysis for ﬂow without volume change. Tie lines are still paths, and there are also nontie-line paths that are quite similar in appearance to those for systems without volume change [19, 20]. The composition route for a solution is assembled using shocks and rarefactions that are selected using the velocity constraint and the entropy condition as tools. Additional complexity arises, of course, from the fact that ﬂow velocity is not constant. It changes across phase-change shocks, and it varies along a nontie-line path, though ﬂow velocity remains ﬁxed for composition variations that remain in the single-phase region along a single tie line (see Section 4.4). In this section, we consider what happens in fully self-sharpening ternary displacements. In these ﬂows, all ﬂow velocity changes occur at shocks, because the only rarefactions present occur along tie lines. The full problem, with nontie-line rarefactions along which ﬂow velocity changes, is considered in Chapter 6 for systems with four components. As in problems without volume change, two kinds of shocks are important, those across which the number of phases changes, and those that occur between tie lines in the two-phase region. We begin by showing that the key results of the analysis for ﬂow without volume change involving tie line extensions and intersections also hold when there is volume change on mixing. In Section 5.2 we showed that a shock that connects a single-phase composition with a two-phase composition must occur along the extension of a tie line. We now consider whether that statement is also true when components change volume as they transfer between phases. The appropriate shock balance is Eq. 4.4.17 Λ=

HiI − HiII , GIi − GII i

i = 1, nc ,

(5.7.10)

where I and II refer to the single-phase and two-phase sides of the shock. On the single-phase side of the shock, I I I I xi ρD = vD GIi HiI = vD

i = 1, nc.

(5.7.11)

Substitution of Eq. 5.7.11 into 5.7.10 gives xIi =

HiII − ΛGII i , I − Λ) ρID (vD

i = 1, nc .

(5.7.12)

Substitution of the deﬁnitions for HiII and GII i shows that II xIi = AxII i + Byi

i = 1, nc,

(5.7.13)

where A=

II II II ρII LD [vD (1 − fV ) − Λ(1 − SV )] I − Λ) ρID (vD

(5.7.14)

II II I ρII LD [vD fV ) − Λ(1 − SV I)] . I − Λ) ρID (vD

(5.7.15)

and B=

122

CHAPTER 5. TERNARY GAS/OIL DISPLACEMENTS

Thus, each single-phase mole fraction is a linear combination of the equilibrium phase mole fractions, and the statement that phase-change shocks occur along tie line extensions holds true when eﬀects of volume change are present. Hence, the two key tie lines identiﬁed previously, the injection gas and initial oil tie lines, are the important tie lines for a displacement whether or not there is volume change. That result means that the discussion of Section 4.4 provides considerable guidance for the behavior of multicomponent systems with volume change. Flow velocities can change substantially across phase-change shocks, but they do not change for variations along a single tie line. Hence, ﬂow velocities do not change for the portions of the displacements that occur along the initial oil and injection gas tie lines. They may change for the leading shock, the trailing shock, and for a rarefaction along the nontie-line path or the corresponding shock between the initial and injection tie lines if a rarefaction is not possible. It is commonly observed, however, that the only signiﬁcant change in ﬂow velocity occurs at the leading shock, for exactly the reasons described in Section 4.4. The important result that extensions of tie lines connected by a shock must intersect also holds when volume is not conserved [19]. The argument is similar to that given in Section 5.2.3 for ﬂows without volume change. Eq. 5.7.10, written for a shock that connects two tie lines containing points c and d, can be rearranged (with the addition of GX i to both sides) to give

c d cd d GX Hic + Λcd GX i − Gi = Hi + Λ i − Gi ,

i = 1, nc .

(5.7.16)

As before, we set each side of Eq. 5.7.16 equal to HiX , and write the resulting expressions in the standard form of a shock balance, Λcd =

HiX − Hic HiX − Hid = , c d GX GX i − Gi i − Gi

i = 1, nc.

(5.7.17)

X If the composition point associated with GX i and Hi is restricted to lie in the single-phase X X X region, then Hi = vD Gi . The left side of Eq. 5.7.16 can be expanded using the deﬁnitions of Hic and Gci to obtain and expression for GX i , c c c c GX i = DρLD xi + EρV D yi ,

i = 1, nc,

(5.7.18)

with the coeﬃcients D and E deﬁned as D=

c (1 − f c ) − Λcd (1 − S c ) vD V V , X − Λcd vD

(5.7.19)

c f c − Λcd S c vD V V . X − Λcd vD

(5.7.20)

and E=

In these expressions, the superscript c indicates that the quantities with the superscript are evaluated at point c or on the tie line that contains that point. Eq. 5.7.17 is the equation of a straight line in a space of molar concentration (measured in moles per unit volume). It states that the molar concentration GX i is a linear combination of the molar concentrations of the equilibrium phases. lies on the extension a line that connects the equilibrium concentrations on one side of Thus, GX i the shock. Identical manipulations of the right side of Eq.5.7.16 give a similar expression,

5.7. VOLUME CHANGE

123

d d d d GX i = JρLD xi + KρV D yi ,

i = 1, nc ,

(5.7.21)

with

J=

d 1 − f d − Λcd 1 − S d vD V V

X − Λcd vD

,

(5.7.22)

and K=

d f d − Λcd S d vD V V . X − Λcd vD

(5.7.23)

Eq. 5.7.21 states that GX i also lies on the extension of a second line in molar concentration space associated with the equilibrium compositions on the tie line on the other side of the shock. Hence we have shown that the idea of intersecting tie lines applies to a tie line represented in terms of molar concentrations when components change volume as they change phase. Eqs. 5.7.16 and 5.7.21 also imply that the extensions of the tie lines themselves must intersect. To see why this must be true, consider a line in concentration space, ρzi = (1 − β)ρL xi + βρV yi ,

i = 1, nc ,

(5.7.24)

where β is a parameter that determines position along the line. If the concentration point in question is in the single-phase region, then β will be greater than one (all vapor) or less than zero (all liquid). Summation of Eq. 5.7.24 over the nc components gives an expression for ρ, ρ = (1 − β)ρL + βρV .

(5.7.25)

Thus, ρ is a molar density that is an average of the phase molar densities. Substitution of Eq. 5.7.25 into Eq. 5.7.24 gives an expression that can be solved for β, β=

ρL (zi − xi ) . ρL (zi − xi ) + ρV (zi − yi )

(5.7.26)

Eq. 5.7.26 actually indicates that β is a volume fraction, speciﬁcally the vapor saturation, S1 = SV . Consider a tie line molar balance, zi = yi V + xi (1 − V ).

(5.7.27)

Eq. 5.7.27 can be rearranged to solve for the mole fraction vapor, V , V =

zi − xi . yi − xi

(5.7.28)

The volume fraction of vapor is given by SV = and substitution of 5.7.28 into 5.7.29 yields

V ρV (1−V ) V ρV + ρL

,

(5.7.29)

124

CHAPTER 5. TERNARY GAS/OIL DISPLACEMENTS

SV =

ρL (zi − xi ) = β. ρL (zi − xi ) + ρV (zi − yi )

(5.7.30)

Finally, substitution of Eqs. 5.7.25 and 5.7.26 into 5.7.24 gives zi =

(1 − SV ) ρL SV ρ V xi + yi . (1 − SV ) ρL + SV ρV (1 − SV ) ρL + SV ρV

(5.7.31)

Thus, we have shown that an equation of the form of 5.7.24, which places a molar concentration on a line connecting equilibrium concentrations, implies that the overall mole fraction, zi , lies on the extension of a tie line that connects the equilibrium mole fractions. Eqs. 5.7.17 and 5.7.21 have exactly the form of Eq. 5.7.24, so they require also that tie lines intersect in mole fraction space. This argument demonstrates that the extensions of tie lines connected by a shock must intersect, whether or not components change volume as the transfer between phases[19]. The intersection shock balances given in Eq. 5.7.17 are complicated expressions, but they can be reduced to remarkably simple forms [19] that are much more convenient for use in constructing solutions. Consider a shock that connects two tie lines, designated A and B. The appropriate shock balances are Λcd =

HiA − GX HiB − GX i i = , X X GA GB i − Gi i − Gi

i = 1, nc.

(5.7.32)

We now deﬁne a shock velocity scaled by the ﬂow velocity at tie line A and make use of the notation of Eq. 4.4.18, X

αA − v A GX Λ Λ = A = i A v Xi , v Gi − Gi ∗

i = 1, nc.

(5.7.33)

Eq. 5.7.33 can be rearranged to solve for the ratio of ﬂow velocities,

X Λ ∗ GA αA vX i − Gi i = − , vA GX GX i i

i = 1, nc.

(5.7.34)

Next, we eliminate the velocity ratio, v X /v A , using versions of Eq. 5.7.34 written for components 1 and 2. The result is Λ∗ =

X A X Λ αA 2 G1 − α1 G2 = . X A X vA GA 2 G1 − G1 G2

(5.7.35)

Substition of the deﬁnitions of αi (Eq. 4.4.18) and Gi (Eq. 4.4.3) [23] yields the ﬁnal expression, Λ∗ =

Λ f1A − S1XA = , vA S1A − S1XA

(5.7.36)

and similar manipulations with the shock balance for tie line B (Eq. 5.7.32) gives Λ∗ =

Λ v B f1B − S1XB = , vA v A S1B − S1XB

(5.7.37)

5.7. VOLUME CHANGE

125

In these expressions, S1XA refers to the vapor saturation at the intersection of the two tie lines measured on tie line A, while S1XB is the intersection saturation measured on tie line B. Because tie lines do not intersect inside the two-phase region, the values of S1XA and S1XB must be negative or greater than one. Eqs. 5.7.36 and 5.7.37 allow straightforward calculation of shock velocities given the composition of the intersection point. Solution construction follows the sequence described in Section 5.5, with the addition of calculation of ﬂow velocities. In a vaporizing gas drive, like that illustrated in Fig. 5.23 for example, the solution is obtained as follows. Manipulations similar to those used to derive Eq. 5.7.36 yield an expression that can be used to determine the leading shock, Λ∗ =

Λ f1b − S1a λbt df1b = = = , b b dS1 vD S1b − S1a vD

(5.7.38)

where S1a is the saturation measured along the tie line that contains point b. S1b must be negative or greater than, because point a lies in the single-phase region. The intermediate shock is determined by solving Eq. 5.7.36 set equal to λct . The landing point of the intermediate shock on the injection tie line can be found by writing shock balances for components 1 and 3 and eliminating the ratio of ﬂow velocities across the shock. The result is αc αd − αd αc

λc

df c

1 3 1 3 ' ( ' ( = ct = 1 . vD dS1 αd3 Gc1 − Gd1 − αd1 Gc3 − Gd3

(5.7.39)

d /v c , can then be found from Eq. 5.7.37 with v c = v c . The scaled The ratio of ﬂow velocities, vD D D D velocity of the trailing shock is given by an expression like Eq. 5.7.38. It is equal to the scaled tie d , if the trailing shock is a semishock. If it is a genuine shock, the speed is line eigenvalue, λet /vD set by the saturation and compositions at point d. Once all the composition points are known, the d a and vD can be determined from ﬂow velocities, vD

αei 1 = − Λ∗ef f vd Gi

Gei f

Gi

−1 ,

i = 1, nc,

(5.7.40)

i = 1, nc .

(5.7.41)

and a αbi vD = − Λ∗ab b Gai vD

Gbi −1 , Gai

An example of the eﬀect of volume change in a ternary displacement is shown in Fig. 5.23. Fig. 5.23 compares composition routes for displacements saturation proﬁles and ﬂow velocities for displacements with and without volume change. The solution shown was obtained using the Peng-Robinson equation of state to calculate phase behavior at 1000 psia (68 atm) and 160 F (71 C). The ternary diagrams in Fig. 5.23 show that in both cases, the leading shock occurs along the tie-line extension through the initial oil composition, and the trailing shock does so along the tie line that extends through the injection composition. In this LVI vaporizing gas drive, both composition routes include an intermediate shock from the initial tie line to the injection tie line. In this example, rarefactions occur on both the initial oil and injection gas tie lines. The two key tie lines are the same, but the composition points that make up the solutions with and without volume change diﬀer.

126

CHAPTER 5. TERNARY GAS/OIL DISPLACEMENTS

CO a2

CO a2

f

a a

e d

d

aa

C4

aa aa aa

C4

(b) No Volume Change

0.2

0.4

a a

0.6

ξ

0.8

a

a

a

a

1.0

(c) Saturation Profiles

1.2

vD

1.2

a a

Sg 0.0 0.0

aa

C10

(a) Volume Change 1.0

f

aa

a abc

ac ab

C10

e

0.0 0.0

Volume change No volume change 0.2

0.4

0.6

ξ

0.8

1.0

1.2

(d) Flow Velocity

Figure 5.23: Composition routes and saturation and ﬂow velocity proﬁles for displacement of a C4 /C10 mixture by CO2 at 1000 psia (68 atm) and 160 F (71 C).

5.8. COMPONENT RECOVERY

127

The saturations shown in Fig. 5.23c demonstrate that the proﬁles are similar in structure, whether or not there is volume change. The last panel of Fig. 5.23 shows the local ﬂow velocity. It is constant when volume is conserved, but it changes across each shock when volume is not conserved. In this example, however, the only signiﬁcant change in ﬂow velocity occurs across the leading shock. The dissolution of CO2 in undisplaced oil in the transition zone leads to a substantial reduction in volume because dissolved CO2 occupies much less volume at these conditions than does CO2 vapor. The result is a leading shock velocity that is actually less than one: the leading shock arrives at the outlet after one pore volume of pure CO2 has been injected. Experimental observations conﬁrm that when the displacement pressure is low enough that the CO2 density is relatively low but high enough that there is appreciable solubility of CO2 in the oil, breakthrough after one pore volume injection is observed [89]. Similar behavior is observed when the injected gas is CH4 , though the change in ﬂow velocity is smaller because the solubility of CH4 in the oil is lower and the volume change is also smaller. In N2 displacements the eﬀects are smaller still because the solubility of nitrogen in the oils is also lower. For solutions that conﬁrm those statements see Dindoruk [19].

5.8

Component Recovery

Calculation of component recovery for ternary systems is based on exactly the same material balance that was used in Section 4.5 for binary systems. The volume of component i recovered is the volume of component i present initially plus the volume of component i injection through time, τ , minus the volume of component i present at time τ . Hence, the recovery is given by Qi = Ciinit + Fiinit τ −

1 0

Ci dξ.

(5.8.1)

Integration by parts gives Qi = Ciinit − Ciout + τ Fiout ,

(5.8.2)

the same expression obtained previously (Eq. 4.5.8). The corresponding expression when components change volume as they transfer between phases is Eq. 4.5.11, − Gout + τ Hiout . Ri = Ginit i i

(5.8.3)

Given the solutions obtained in the previous sections, it is quite easy to determine recovery at the times at which the various key points in the composition proﬁles arrive at the outlet (ξ = 1). Consider, for example, the vaporizing gas drive illustrated in Figs. 5.23. Because the shock velocities and composition points are all determined as the solution is found, the recovery calculation is straightforward. Table 5.5 reports component recoveries at the times of arrival at the outlet of the key segments of the composition proﬁles in Fig. 5.23. Recovery curves for the examples in Fig. 5.23 are shown in Fig. 5.24 for the two components present in the initial oil mixture. The recovery of each component is scaled by Cia , the initial volume fraction or Gai, the initial concentration of each component. In that form, the dimensionless recovery of each component is proportional to τ until the leading shock reaches the outlet. It is equal to τ when there is no volume change, but when eﬀects of volume change are included, the recovery is scaled by the ﬂow velocity at the outlet. In the displacement of Fig. 5.23, breakthrough of

128

CHAPTER 5. TERNARY GAS/OIL DISPLACEMENTS

Table 5.5: Component Recovery at Selected Times τ 1 Λab 1 λct 1 λdnt 1 λdt 1 Λef

Arrival at ξ = 1

No Volume Change

Volume Change

Fib Cia − Cib + Λab Fc Cia − Cic + λic t Fid a d Ci − Ci + λd nt Fd Cia − Cid + λid t Fif Cia − Cif + Λef

Gai − Gbi + Λabi Hc Gai − Gci + λci

Leading Shock Intermediate Shock Zone of Constant State Trailing Rarefaction Trailing Shock

Hb

t

Gai

− Gdi

+

Gai − Gdi + Gai − Gfi +

Hid λdnt Hid λdt Hif Λef

1.0

Fraction of C4 and C10 Recovered

C4 C10

0.8

C4 C10

0.6

0.4

0.2

No Volume Change Volume Change

0.0 0

1

τ

2

3

Figure 5.24: Component recovery for the displacements shown in Fig. 5.23.

5.9. SUMMARY

129

injected CO2 occurs just after one pore volume injected because CO2 loses volume as it dissolves in undisplaced oil. After breakthrough, the recovery curves for the two components diﬀer, because the chromatographic separations that occur during the displacement cause the two components to propagate at diﬀerent speeds toward the outlet. In this example, in which pure CO2 displaces a mixture of C4 and C10 , all of the C4 present initially is recovered when the intermediate shock arrives at the outlet. Recovery of C10 is complete when the trailing shock arrives, though in this example, the slow evaporation of C10 into the ﬂowing CO2 requires 17.4 and 26.4 pore volumes for the ﬂows without and with volume change. While recovery is slow in this immiscible case, much faster recovery and correspondingly higher displacement eﬃciency is possible when the pressure is high enough that multicontact miscibility occurs.

5.9

Summary

A number of key ideas have been developed in this chapter that will be the basis for understanding displacements with more than three components. The analysis of ternary as well as multicomponent displacements is built on the following concepts: • Eigenvalues represent the velocity at which a given overall composition propagates. • Eigenvectors represent directions in composition space along which compositions vary in a way that satisﬁes the conservation equations. • Composition paths are integral curves of the eigenvectors. Tie lines are paths. In addition, there is a set of nontie-line paths. Every composition point in the two-phase region lies on a single tie-line path and a single nontie-line path. • The composition route that is the solution to any displacement problem must lie on paths that connect the initial oil composition to the injection gas composition. • A composition route can enter or leave the two-phase region only via a shock. A shock from a single-phase composition into the two-phase region must land on the tie line that extends through the single-phase composition. • The extensions of tie lines connected by a shock must intersect. • The shocks and rarefactions that are assembled to build the solution for a given set of injection and initial compositions must satisfy a velocity rule (fast compositions lie downstream of slow ones in regions of continuous composition and saturation variation), and shocks must satisfy an entropy condition (wave velocities of compositions are faster than the shock and those downstream are slower: shocks that are perturbed slightly must sharpen again into a shock). • The solution composition route may switch from a tie-line path to a nontie-line path (or vice versa) as long as the velocity rule is satisﬁed. If the wave velocities associated with the tie-line and nontie-line paths are not the same, that composition point appears as a zone of constant state in the associated solution proﬁle. • The important features of a ternary gas displacement are determined by the two key tie lines: the tie line that extends through the initial oil composition and the tie line that extends through the injection gas composition.

130

CHAPTER 5. TERNARY GAS/OIL DISPLACEMENTS

• Multicontact miscibility occurs if either of the key tie lines is a critical tie line (a tie line tangent to the binodal curve at the plait point). If the injection gas tie line is critical, the displacement is a condensing gas drive. If the initial oil tie line is the critical tie line, the displacement is a vaporizing gas drive. • Eﬀects of volume change as components transfer between phases do not change the basic patterns of displacement behavior observed for ﬂows without volume change. The same key tie lines are important, though speciﬁc compositions on those tie lines do change. The rate of recovery can be strongly inﬂuenced by volume change when it changes local ﬂow velocity appreciably. All of these ideas apply still in displacements with more than three components. While the geometry of tie lines is more complex and will require some study, the approach to the problem of multicomponent displacements draws heavily on the description of ternary ﬂows. Thus, a detailed understanding of ternary displacements is required background for understanding the multicomponent ﬂows considered in subsequent chapters.

5.10

Additional Reading

The ﬁrst analytical solution for a gas/oil displacement was published in a remarkable 1961 paper by Welge et al. [134]. That paper considered ﬂow of ternary mixtures (condensing gas drives), and it included the eﬀects of volume change as components transfer between phases. While the mathematical development is a challenge to follow, it sets the stage for much work to follow. Wachmann [125] solved the somewhat simpler problem of ternary ﬂow for oil/water/alcohol systems in which eﬀects of volume change were ignored. Work on ternary theory then paused until the 1970s when a group of investigators at Shell Development Company set out to describe the behavior of surfactant/oil/water systems. The analysis was aimed at understanding how phase behavior and resulting low interfacial tensions could be used to recover oil left behind by waterﬂooding. That theory was reported in a series of papers by Larson and Hirasaki [66], Larson [65], Helﬀerich [31], and Hirasaki [35]. Helﬀerich’s 1981 paper [31] states the theory in a general form that makes clear the fact that the underlying mathematical description of the interaction of phase behavior and two-phase ﬂow is common to a variety of displacement processes, despite diﬀerences in the details of the phase equilibrium. The ﬁrst proof that wave velocities remain constant along nontie-line paths when tie lines meet at a point was obtained by Cer´e and Zanotti [11]. Application to gas/oil systems came again with the work of Dumor´e et al. [22], who considered three-component gas injection processes with eﬀects of volume change as components transfer between phases. Detailed reviews of related work on surfactant and polymer ﬂooding systems are given by Pope [101] and Johansen [50]. Bedrikovetsky [6, pp. 295-313] gives a version of the theory that emphasizes graphical constructions involving the fractional ﬂow curves for the two key tie lines. Detailed descriptions with numerous examples of ternary displacements without volume change are given by Pande [95], Johns [54] and Dindoruk [19]. The proof that tie lines connected by a shock must intersect (for systems with no volume change) is due to Johns [54, 56], though a proof for a special case was reported by Orr et al. [92]. Johns also described the classiﬁcation of the four types of ternary displacements (LVI and HVI, self-sharpening and spreading nontie-line paths) and showed how the intermediate K-value determines self-sharpening behavior. Dindoruk [19] used

5.11. EXERCISES

131

the properties of envelope curves to determine when self-sharpening behavior occurs. Dindoruk [19] and Wang [128] derived the closed form analytical solutions for constant K-values. Dindoruk [19] proved that tie lines connected by a shock must intersect when eﬀects of volume change are included in the model.

5.11

Exercises

1. Calculate and plot the liquid locus and the vapor locus for a ternary system in which the K-values are constant at K1 = 3.0, K2 = 1.2, and K3 = 0.01. 2. Show that the liquid phase composition of a tie line that extends through a point (C1∗ , C2∗ ) is given by x∗1

=

−b ±

√

b2 − 4ac , 2a

(5.11.1)

where a=

b=

(K1 − K3 ) (K1 − K2 ) , (K2 − K3 ) (K2 − 1)

(1 − K3 ) (K1 − K2 ) (K1 − K3 ) ∗ (K1 − 1) ∗ C1 + C2 − , (K2 − K3 ) (K2 − 1) (K2 − K3 ) (K2 − 1)

and c=−

(1 − K3 ) ∗ C . (K2 − K3 ) 1

3. Problems 3-12 construct the solution for a HVI vaporizing gas drive. Find the tie lines that extend through the injection composition, C1inj = 0.9, C2inj = 0.1 and the initial composition C1init = 0.1, C2init = 0.2 for a ternary system with the K-values of Problem 1. 4. For a viscosity ratio, µliq /µvap = 10, and the relative permeability functions of 4.1.14-4.1.19 with Sor = Sgc = 0, calculate and plot the tie-line and nontie-line eigenvalues as a function of saturation for a tie line that extends through the point C1init = 0.1, C2init = 0.2 with K1 = 3.0, K2 = 1.2, and K3 = 0.01. 5. Calculate the compositions and velocity of a leading shock from the initial composition of Problem 3 into the two-phase region for the K-values, relative permeabilities, and viscosities of Problem 4. 6. Find the composition on the tie line of Problem 5 at which the tie-line and nontie-line eigenvalues are equal. Calculate the compositions and velocities of the rarefaction along the initial tie line.

132

CHAPTER 5. TERNARY GAS/OIL DISPLACEMENTS

7. Evaluate and plot the compositions on the nontie-line path that connects the initial and injection tie lines of Problem 3. Find the composition at which the nontie-line path intersects the injection tie line. 8. Determine the compositions and velocity of a trailing semishock for the injection tie line (with the data of Problem 4). 9. Determine whether a rarefaction occurs along the injection tie line. If so, calculate the compositions and wave velocities for that rarefaction. If not, calculate the velocity of the trailing shock. 10. Plot a ternary diagram that shows the composition route determined in Problems 3-9. 11. Plot a saturation proﬁle for the solution of Problems 3-9. Also plot composition proﬁles for each of the components. 12. Plot the recovery of components 2 and 3 as a function of pore volumes injected. 13. Problems 13-20 construct the solution for a HVI condensing gas drive. Find the tie lines that extend through the injection composition, C1inj = 0.7, C2inj = 0.3 and the initial composition C1init = 0.1, C2init = 0.2 for a ternary system with the K-values of Problem 1. 14. Calculate the compositions and velocity of a trailing shock from the injection composition of Problem 13 into the two-phase region. Use the K-values of Problem 1 and the relative permeabilities and viscosity ratio of Problem 4. 15. Find the composition on the upstream side of the intermediate shock and evaluate the compositions and saturation for the rarefaction on the injection tie line. 16. Find the composition on the downstream side of the intermediate shock. Determine whether a rarefaction exists on the initial tie line, and if so, evaluate the compositions and saturations that make up the rarefaction. 17. Determine the composition on the upstream side of the leading shock, and ﬁnd the shock velocity. 18. Plot a ternary diagram that shows the composition route determined in Problems 13-17 19. Plot saturation and composition proﬁles for each component for the solution obtain in Problems 13-17. 20. Plot recovery curves for each of the components for the solution of Problems 13-17. 21. LVI vaporizing gas drive. Calculate and plot composition path and saturation and composition proﬁles and recovery curves for a constant K-value system with K1 = 3.0, K2 = 0.9, and K3 = 0.01. The injection composition is C1inj = 0.9, C2inj = 0.1 and the initial composition is C1init = 0.1, C2init = 0.2. Use the relative permeabilities and viscosity ratio of Problem 4.

5.11. EXERCISES

133

22. LVI condensing gas drive. Calculate and plot composition path and saturation and composition proﬁles and recovery curves for a constant K-value system with K1 = 3.0, K2 = 0.9, and K3 = 0.01. The injection composition is C1inj = 0.7, C2inj = 0.3 and the initial composition is C1init = 0.1, C2init = 0.2. Use the relative permeabilities and viscosity ratio of Problem 4. 23. Show that the solution for a nontie-line path in a ternary system with constant K-values and Sgc = 0 and Sor = 0 is given by

x1 =

f0 x01 1 f1

− S10 +γ − S1

γ(K1 − 1) 0 0 f10 − S10 −1 + (S1 f1 − S1 f1 ) f1 − S1 f1 − S1

2γ(K1 − 1) + f1 − S1

S1 S10

f1 dS1 ,

(5.11.2)

with S1 S10

1 M

+

1 M

S1max M

1+

⎧

−1 ⎨ 1 M

2

⎩

⎛

− arctan ⎝

f1 dS1 =

⎛

arctan ⎝

S1 − S10 1 1+ M

(S1 − Sgc )

(S10 − Sgc )

1 M

S1max

1 M

S1max

+1 −

1 M

+1 −

1 M

S1max M

S1max M

⎞ ⎠

⎞⎫ ⎬ ⎠ ⎭

⎧ ⎫ ⎨ 1 (S1max )2 − 2 (S1max)(S1 − Sgc ) + 1 + 1 (S1 − Sgc )2 ⎬ M M M + , 2 ln ⎩ 1 (S max)2 − 2 (S max)(S 0 − S ) + 1 + 1 (S 0 − S )2 ⎭ 1 gc gc + 1 1 1 1 1 M M M M S1max M

(5.11.3)

where S1max = 1 − Sor − Sgc . 24. Find and plot the composition path and the saturation and composition proﬁles for a system containing CH4 , C4 , and C10 at 160 F and 1600 psia, including the eﬀects of volume change as components change phase. The injection gas is a binary mixture of CH4 and C4 containing 90 mol percent CH4 , and the initial mixture has composition C1 = 0.1, C2 = 0.4, C3 = 0.5.

134

CHAPTER 5. TERNARY GAS/OIL DISPLACEMENTS

25. Consider a displacement in a ternary system in which volume change is not important. The fractional ﬂow functions for the initial and injection tie lines have the form shown in Fig. 5.19. Assume that the initial composition is ﬁxed in the liquid portion of the single-phase region, but the injection gas composition can take any value along the injection tie line. Consider injection compositions that lie inside the two-phase region as well as in the singlephase regions on the liquid and vapor sides of the two-phase region. Sketch the solution composition paths and proﬁles for each of the possible patterns of displacement behavior and identify the transition points between them. Explain why the transition points mark the change in displacement pattern. 26. Show that the equal eigenvalue pints, λt = λnt observed as the saturation varies along a ﬁxed tie line occur at a miximum or a minimum in λnt . 27. Show that for a ﬁxed tie line in an LVI system, the following statements hold: C1L < C1 < C1E , λnt > 1, C1E < C1 < C1V , λnt < 1. In these expressions, C1L is the volume fraction of component 1 on the liquid locus, C1E is the composition of the intersection of the equivelocity curve with the tie line, and C1V is the composition on the vapor locus. 28. Show that for a ﬁxed tie line in an HVI system, the following statements hold: C1L < C1 < C1E , λnt < 1, C1E < C1 < C1V , λnt > 1.

Chapter 6

Four-Component Displacements The development of the theory for four-component displacements follows directly from the ideas presented in Chapters 4 and 5 but with some important additions. As in the simpler two- and threecomponent systems, we will formulate an eigenvalue problem. Here again, the eigenvalues represent propagation velocities for a given overall composition, and eigenvectors are allowable directions of composition variation in three-dimensional composition space. When four components are present, however, there are three eigenvalues and three eigenvectors for each point in the composition space. Thus, the problem is to ﬁnd the unique composition path that connects the injection gas and initial oil compositions in a three-dimensional composition space. Much of the machinery needed to solve the four-component problem is already in place, so the development here will focus on the new features that arise in the more complex displacements. The most important diﬀerence between ternary and quaternary displacements is that a new key tie line appears. The initial oil and injection gas tie lines are still important, but parts of the solution behavior depend on a third tie line known as the crossover tie line. We begin by considering the eigenvalue problem and the composition paths that result. Here again, the special case of constant equilibrium K-values will prove useful. Next we use four-component solutions to understand displacement behavior that has come to be know as condensing/vaporizing drives (Zick [140]), and we consider how miscibility develops in four-component systems. Finally, we consider again the eﬀects of volume change on mixing.

6.1 6.1.1

Eigenvalues, Eigenvectors, and Composition Paths The Eigenvalue Problem

When four components are present, there are three independent equations (when eﬀects of volume change as components change phase are not considered, four independent equations if volume change is included): ∂F1 ∂C1 + ∂τ ∂ξ ∂F2 ∂C2 + ∂τ ∂ξ 135

= 0,

(6.1.1)

= 0,

(6.1.2)

136

CHAPTER 6. FOUR-COMPONENT DISPLACEMENTS ∂F3 ∂C3 + ∂τ ∂ξ

= 0,

(6.1.3)

where the deﬁnitions of Ci and Fi are given by Eqs. 5.1.3 and 5.1.4. Because Fi is a function that depends only on the three independent compositions, C1 , C2 , and C3 , we can write ∂Fi ∂C1 ∂Fi ∂C2 ∂Fi ∂C3 ∂Fi = + + . ∂ξ ∂C1 ∂ξ ∂C2 ∂ξ ∂C3 ∂ξ

(6.1.4)

Substitution of Eq. 6.1.4 into Eqs. 6.1.1-6.1.3 gives ∂F1 ∂C1 ∂F1 ∂C2 ∂F1 ∂C2 ∂F1 ∂C3 ∂C1 + + + + = 0, ∂τ ∂C1 ∂ξ ∂C2 ∂ξ ∂C2 ∂ξ ∂C3 ∂ξ ∂C2 ∂F2 ∂C1 ∂F2 ∂C2 ∂F2 ∂C2 ∂F2 ∂C3 + + + + = 0, ∂τ ∂C1 ∂ξ ∂C2 ∂ξ ∂C2 ∂ξ ∂C3 ∂ξ ∂F3 ∂C1 ∂F3 ∂C2 ∂F3 ∂C2 ∂F3 ∂C3 ∂C3 + + + + = 0, ∂τ ∂C1 ∂ξ ∂C2 ∂ξ ∂C2 ∂ξ ∂C3 ∂ξ

(6.1.5) (6.1.6) (6.1.7)

which can be written in matrix form as ∂u ∂u + A(u) = 0, ∂τ ∂ξ

(6.1.8)

with u = [C1 , C2 , C3 ]T and ⎡ ∂F 1 ∂C ⎢ ∂F21 A(u) = ⎣ ∂C1 ∂F3 ∂C1

∂F1 ∂C2 ∂F2 ∂C2 ∂F3 ∂C2

∂F1 ∂C3 ∂F2 ∂C3 ∂F3 ∂C3

⎤ ⎥ ⎦.

(6.1.9)

As in the binary and ternary problems, we now ask what is the trajectory in space and time of a ﬁxed composition, for which ∂Ci ∂Ci dτ + dξ = 0. ∂τ ∂ξ

(6.1.10)

dξ ∂Ci ∂Ci ∂Ci =− = −λ , ∂τ dτ ∂ξ ∂ξ

(6.1.11)

∂u ∂u = −λ ∂τ ∂ξ.

(6.1.12)

dCi = Rearrangement of Eq. 6.1.10 gives

which means that

Substitution of Eq. 6.1.12 into Eq. 6.1.8 gives the ﬁnal form of the eigenvalue problem, −λ

∂u du ∂u + A(u) = {A(u) − λI} = 0. ∂ξ ∂ξ dξ

Eq. 6.1.13 has solutions only if det[A(u) − λI] = 0.

(6.1.13)

6.1. EIGENVALUES, EIGENVECTORS, AND COMPOSITION PATHS

6.1.2

137

Composition Paths

Expansion of the determinant det[A(u) − λI] indicates that there are three eigenvalues and three eigenvectors at every point in the composition space. We show now that tie lines are paths when four components are present, just as they are in ternary systems. In fact, tie lines are paths no matter how many components are present. To show that tie lines are paths we make use of a parametrization of the problem similar to that used in Section 5.1 for ternary systems [49, 53]. C2 = α(ξ, η)C1 + φ(ξ, η),

(6.1.14)

C3 = β(ξ, η)C1 + θ(ξ, η),

(6.1.15)

where α and β are slopes of the tie line, and φ and θ are intercepts with the C1 = 0 face. ξ and η are new dependent variables that uniquely specify some tie line in the composition space (the volume fractions of two of the components on the surface of liquid compositions, for example). Substitution of the deﬁnitions in Eqs. 6.1.14 and 6.1.15 into Eqs. 6.1.1-6.1.3 yields another equation that has the form of Eq. 6.1.8, this time with u = [C1 , ξ, η]T and ⎛ ∂F1 ∂C1 ⎜ A(u) = ⎝ 0

∂F1 ∂ξ G D (C1 −F1 ) M D

0

∂F1 ∂η (C1 −F1 ) L D H D

⎞ ⎟ ⎠.

(6.1.16)

The entries in A(u) are given by

∂θ ∂β + G = C1 ∂η ∂η

∂α ∂φ + H = C1 ∂ξ ∂ξ

∂α ∂φ + D = C1 ∂ξ ∂ξ

∂α ∂φ ∂α ∂φ + + F1 − C1 ∂ξ ∂ξ ∂η ∂η

∂θ ∂β + F1 ∂η ∂η

∂θ ∂β + C1 ∂η ∂η

L=

∂β ∂θ + − C1 ∂ξ ∂ξ

∂β ∂θ + − C1 ∂ξ ∂ξ

∂θ ∂β + F1 , ∂ξ ∂ξ

(6.1.17)

∂α ∂φ + F1 , ∂η ∂η

(6.1.18)

∂α ∂φ + C1 , ∂η ∂η

(6.1.19)

∂β ∂φ ∂α ∂θ − , ∂η ∂η ∂η ∂η

M=

(6.1.20)

∂α ∂θ ∂β ∂φ − . ∂ξ ∂ξ ∂ξ ∂ξ

(6.1.21)

This version of the problem, when cast in the form of Eq. 6.1.13, reveals that ∂F1 df1 = . (6.1.22) ∂C1 dS1 Because ξ and η, which specify a tie line, are held constant as the derivative in Eq. 6.1.22 is evaluated, the tie line also remains ﬁxed, and hence the argument given in the derivation of Eq. 4.1.23 applies here as well. Thus, the eigenvalue is df1 /dS1 , just as it was in the binary and ternary problems. The eigenvector associated with λ1 points in the direction of the tie line, and hence the tie line is also a path when four components are present. The other two eigenvalues (and eigenvectors) are associated with nontie-line paths that pass through the composition point in question. λ1 =

138

CHAPTER 6. FOUR-COMPONENT DISPLACEMENTS 2(CO2) a

a

a a a a

1(CH4)

a

a

b

h

f a a a a

3(C4)

a a a a

X

a a

a

a

a

a

a

4(C10 )

Figure 6.1: A planar surface of tie lines that intersect a generating tie line in the C2 = 0 face. Constant K-Values The special case of equilibrium K-values that are independent of the composition of the mixture is useful for understanding the structure of four-component systems as well. Johansen et al. [53] showed that when K-values are constant, several special properties arise that make construction of solutions relatively simple (and provide considerable guidance about how solutions behave when K-values are not constant). The ﬁrst important property is the geometry of tie lines. Consider two of the tie lines shown in Fig. 6.1. Tie line b is in the C1 /C3 /C4 face that forms the base of the quaternary tetrahedron, while tie line f is in the front face of the diagram, the C2 /C3 /C4 face, and the two tie lines are chosen so that their extensions intersect. Also shown is tie line h, a tie line in the interior of the quaternary diagram whose extension also intersects tie line b in the base. Each of these tie lines deﬁnes a direction in composition space. These directions can be expressed as three vectors: eb , the direction of tie line b in the C2 = 0 face, ef , the direction of tie line f in the C1 = 0 face, and eh , the direction of tie line h in the interior. Those vectors can be expressed in terms of the K-values as eb = (K1 − 1)xb1e1 + (K3 − 1)xb3e3 ,

(6.1.23)

ef = (K2 − 1)xf2 e2 + (K3 − 1)xf3 e3 ,

(6.1.24)

eh = (K1 − 1)xh1 e1 + (K2 − 1)xh2 e2 + (K3 − 1)xh3 e3 .

(6.1.25)

Tie line b can be chosen by setting xb1 , and xb3 can be found from the equation of the liquid locus in the C1 /C3 /C4 face (an expression like Eq. 5.1.43). The compositions, xf2 and xf3 , can be found for tie line f by solving for the tie line in the C2 /C3 /C4 face that extends through point X,

6.1. EIGENVALUES, EIGENVECTORS, AND COMPOSITION PATHS

139

2(CO2)

a

a

a

aa a a a a

a 1(CH4)

h a

a

a

a

a

3(C4)

4(C10 )

Figure 6.2: A planar surface of tie lines that intersect a generating tie line in the C3 = 0 face. where the extension of tie line b intersects the C3 /C4 edge of the bottom face, and then writing an expression for the liquid locus in the C2 /C3 /C4 face. Any tie line h can be selected by choosing an intersection point on the extension of tie line b and ﬁnding the tie line in the interior of the diagram that intersects that extends through that point. For any tie line in the interior of the diagram that extends to the C2 = 0 face, the saturation measured on that tie line when C2 = 0 is 1 , K2 − 1 and the liquid phase compositions for the interior tie line are given by S1h = −

xhi =

CiX , S1h (Ki − 1) + 1

i = 1, nc , i = 2,

(6.1.26)

(6.1.27)

where CiX is the composition at the intersection point in the C2 = 0 plane. Tie lines b and f deﬁne a plane, and the direction of the normal vector to that plane is eb × ef . Direct evaluation of the algebraically complex expression for eh · (eb × ef ) indicates that it is identically equal to zero. This result means that the vector eh associated with any tie line h is perpendicular to the normal to the plane deﬁned by tie lines b and f. Because tie lines h and f both intersect tie line b, they must also lie in the same plane. Thus, tie line b deﬁnes a plane of tie lines the extensions of which all intersect the extension of tie line b. Fig. 6.1 shows the plane of tie lines that intersect tie line b. The extensions of those tie lines are tangent to an envelope curve like that described in Chapter 5. A tie line in any of the faces of the quaternary diagram can be used in a similar way to generate a plane of tie lines. Hence we can ﬁnd a tie line in the C1 /C2 /C4 face, for example, that generates a plane of tie lines that also includes tie line h. Fig. 6.2 shows that plane of tie lines, all of which intersect tie line a. Tie line h lies in both planes. Hence, we have shown that each tie line in the interior of the quaternary diagram lies at the intersection of two planes of tie lines.

140

CHAPTER 6. FOUR-COMPONENT DISPLACEMENTS

The fact that planes of tie lines exist gives rise to the second important property of constant K-value systems: the nontie-line paths that intersect a given tie line lie in those planes, as do shocks that connect two tie lines that lie in the same plane [53]. This property means that a composition route that connects an injection gas composition with an initial oil composition will traverse planes that are associated with the injection gas tie line and the initial oil tie line. The portion of the displacement that occurs in each of the planes is much like the ternary displacements of the previous chapter, and the catalog of shocks and rarefactions developed there will apply here, too. For constant K-values, a closed form expression for the nontie-line path can be obtained for any planes [19, 128] by the following argument. Any plane can be described as αp C1p + β p C2p + γ pC3p = 1,

(6.1.28)

where α, β, and γ are constants that determine the position and orientation of the plane. Among the points that lie on the plane are the phase compositions. Hence the liquid phase compositions must satisfy αp xp1 + β p xp2 + γ pxp3 = 1,

(6.1.29)

In addition, the liquid phase compositions must sum to one, xp1 + xp2 + xp3 + xp4 = 1,

(6.1.30)

as must the vapor phase compositions, K1 xp1 + K2 xp2 + K3 xp3 + K4 xp4 = 1.

(6.1.31)

Eqs. 6.1.29-6.1.31 can be used to eliminate xp3 and xp4 , which yields xp2 = σxp1 + δ,

(6.1.32)

where σ=

α (K3 − K4 ) − γ (K1 − K4 ) , γ (K2 − K4 ) − β (K3 − K4 )

(6.1.33)

δ=

γ (1 − K4 ) − (K3 − K4 ) . γ (K2 − K4 ) − β (K3 − K4 )

(6.1.34)

and

As long as compositions are constrained to lie in a single plane, only two of the three material balance equations 6.1.1-6.1.3 are independent (see pp. 42-44 of [128] for a proof). Manipulations of Eqs. 6.1.1 and 6.1.2 similar to those used to derive 5.1.48 yield the following eigenvalue problem (see Appendix B of [128] for details) ⎛ ⎝

where

df1 dS1

−λ 0

1 dp (f1 −S1 ) 2 dx1 C1 +p F1 +p C1 +p

−λ

⎞ ⎠

dS1 dx1

= 0,

(6.1.35)

6.1. EIGENVALUES, EIGENVECTORS, AND COMPOSITION PATHS

p = x21

141

σ K1 − K2 . δ 1 − K2

(6.1.36)

Integration of the diﬀerential equation for the nontie-line path gives the ﬁnal result, x1 =

f0 x01 1 f1

− S10 δ (K2 − 1) + − S1 σ (K2 − K1 )

δ (K2 − 1) (K1 − 1) + σ (K2 − K1 ) f1 − S1

f10 − S10 −1 f1 − S1

(S10f10

− S1 f1 ) + 2

S1 S10

f1 dS1 ,

(6.1.37)

where x01 , f10 , and S10 refer to some point that lies on the nontie-line path in question, and the compositions of points on the nontie-line path are given by C1 = {1 + (K1 − 1) S1 } x1 ,

(6.1.38)

C2 = {1 + (K2 − 1) S1 } {σx1 + δ} ,

(6.1.39)

C3 =

1 {1 − αC1 − βC2 } , γ

(6.1.40)

C4 = 1 − C1 − C2 − C3 .

(6.1.41)

Eq. 6.1.37 is similar in form to the expression obtained for a ternary system (Eq. 5.1.55), though the constants diﬀer in a way that speciﬁes the plane of tie lines in which the nontie-line path lies. For the fractional ﬂow function given in Eqs. 4.1.20-4.1.22 with Sor = Sgc = 0, the integral in Eq. 6.1.37 is given by Eq. 5.1.56 (see Problem 5.23 for the result when Sor and Sgc are nonzero). Consider now a tie line B in the interior of the quaternary diagram (see Fig. 6.3). The constants α, β, and γ can be found for the two planes that intersect in tie line B by the following procedure. Extension of tie line B to the C2 = 0 face gives an intersection point in that face. The tie line in the C2 = 0 face that extends through that point, tie line A, must also lie in a plane associated with tie line B. The compositions of three points on tie lines A and B, for example the liquid and vapor phase compositions on tie line A and the liquid phase composition on B, can be used in Eq. 6.1.28 to determine the plane constants. For the plane determined by tie lines A and B, the constants are αAB =

β AB

1 = B x2

1 − K3 1 , K1 − K3 xA 1

xB 1 − K3 xB K1 − 1 1 − 1A − 3A x1 K1 − K3 x3 K1 − K3

(6.1.42)

,

(6.1.43)

and γ AB =

K1 − 1 1 . K1 − K3 xA 3

(6.1.44)

142

CHAPTER 6. FOUR-COMPONENT DISPLACEMENTS 2(CO2)

a a

a

B a

C a

1(CH4)

A

a

3(C4)

a a

4(C10 )

Figure 6.3: Nontie-line paths in the AB plane. Extension of tie line B to the C3 = 0 face gives an intersection point that can be used to determine tie line C, and the phase compositions for tie lines B and C give the information required to deﬁne the BC plane. The resulting expressions are αBC =

1 − K2 1 , K1 − K2 xC 1

(6.1.45)

β BC =

K1 − 1 1 , K1 − K2 xC 2

(6.1.46)

an γ BC

1 = B x3

1 − K2 xB K1 − 1 xB 1 2 1− − K1 − K2 xC K1 − K2 xC 1 2

.

(6.1.47)

Fig. 6.3 shows a few of the nontie-line paths that lie in the AB plane and Fig. 6.4 shows nontieline paths in the BC plane. While the patterns of nontie-line paths are qualitatively similar to those of ternary systems (see Fig. 5.8), there are some important diﬀerences. Each plane is divided by an equivelocity curve (not shown in Figs. 6.3 and Fig. 6.4 but equivalent to that shown in Fig. 5.8), with similar nontie-line paths on either side (though only paths for saturations greater than the saturation on the equivelocity curve are shown). There are now two nontie-line paths through every composition point in the two-phase region, and there are four points along each tie line at which two of the three eigenvalues (λt and one λnt ) are equal. Fig. 6.5 shows the eigenvalues for tie line B. At a point where λt equals λnt , the nontie-line path is tangent to the tie line. As in the ternary problems, equal-eigenvalue points are ones at which a path switch from a tie line to a nontie-line path can occur without violating the velocity rule.

6.1. EIGENVALUES, EIGENVECTORS, AND COMPOSITION PATHS

2(CO2)

a a

a

B

C a

1(CH4)

3(C4)

a a

A

a a

4(C10 )

Figure 6.4: Nontie-line paths in the BC plane.

Tie-line and Nontie-line Eigenvalues, λt and λnt

3.0

λt

2.5

2.0

1.5 λAB nt 1.0 λBC nt 0.5

0.0 0.0

0.2

0.4

0.6

0.8

Saturation, S1

Figure 6.5: Eigenvalues on tie line B.

1.0

143

144

CHAPTER 6. FOUR-COMPONENT DISPLACEMENTS 2(CO2)

E

a

1(CH4)Ga

a

3(C4)

a a

a

a aH

a

aF 4(C10 )

Figure 6.6: Composition paths for ternary displacements.

6.2

Solution Construction for Constant K-values

The solution route for a given set of initial and injection compositions must be constructed as a sequence of rarefactions and shocks that connect the initial composition to the injection composition. To see how to construct the composition path for a four-component displacement, consider ﬁrst the ternary displacements shown in Fig. 6.6. Displacement of oil F by gas E is a condensing gas drive much like that illustrated in Fig. 5.17. Displacement of oil H by gas G, on the other hand, is a vaporizing gas drive like that of Fig. 5.16. (Note that oil H lies on the extension of tie line A in Figs. 6.3 and 6.4, and gas E is on the extension of tie line C). The composition route for a displacement of oil H by gas E must be some combination of the composition route elements present in the ternary displacements, because the four-component route must reduce to the route for one of the ternary displacements in the limit as CO2 is removed from the injection gas or C4 is removed from the oil. A third key tie line, the crossover tie line, provides the connection between the condensing and vaporizing route segments that make up the four-component solution. Figs. 6.3 and 6.4 indicate that there are no composition paths that connect compositions on the injection tie line directly with compositions on the initial tie line. Instead, the composition route for the solution must traverse the paths that lie on the planes associated with the initial and injection tie lines. Those planes intersect in the crossover tie line, and the solution route must include that tie line if rarefactions connect the three key tie lines. If, on the other hand, a shock connects two of the tie lines, then the tie-line extensions must intersect. The extensions of the initial oil and injection gas tie lines do not intersect (except in special cases), so again, a third tie line is required if shocks connect the tie lines. The crossover tie line is the tie line in the interior of the diagram that, when extended, intersects the extensions of both the injection gas and initial oil tie lines. When K-values are constant, the crossover tie line is the same whether shocks or rarefactions connect the tie lines.

6.2. SOLUTION CONSTRUCTION FOR CONSTANT K-VALUES

145

For a given set of initial oil and injection gas compositions, the solution composition route can be constructed by the following steps for a system with constant K-values: 1. Identify the injection gas and oil tie lines, the tie lines that extend through the injection gas and initial oil compositions. This step is identical for problems with any number of components. With constant K-values, the required tie lines can be found analytically by solving a cubic equation in the saturation or, more easily, by a negative ﬂash [135, 129]. 2. Find the crossover tie line. Expressions like Eqs. 6.1.26 and 6.1.27 can be used to ﬁnd any interior tie line associated with the injection and initial tie lines. Once two tie lines are known for each plane, then an equation for that plane can be written (see Eqs. 6.1.42-6.1.47). The equations for the two planes can then be solved simultaneously for the crossover tie line. Or, a crossover tie line can be chosen as some interior tie line associated with either the injection tie line or the initial tie line, and the corresponding initial or injection tie line can be found with expressions like Eqs. 6.1.26 and 6.1.27. For a given pair of initial and injection tie lines, there are actually two tie lines that lie at the intersection of planes through the initial and injection tie lines. For example, in Fig. 6.1, tie line b lies in the vertical plane shown, but it also lies in a second plane, the horizontal C2 = 0 ternary base of the diagram. Similarly, tie line a in Fig. 6.2 lies in the plane of tie lines shown and also in the C3 = 0 ternary face. Thus, in addition to tie line h, the intersection of the planes illustrated in Figs. 6.1 and 6.2, the C1 -C4 axis is also a potential crossover tie line because it is the intersection of planes through the initial and injection tie lines. However, it is a straightforward matter to show that composition variations that follow the second set of planes violate the velocity rule, and hence the second potential crossover tie line can be eliminated from consideration. An example of the arguments required to eliminate the second pair of planes is given in Appendix C. 3. Identify the shortest key tie line. Solution construction begins with that tie line. The arguments of Section 5.5 that demonstrated that the shortest tie line is the starting point for the solution apply equally well to pairs of tie lines in a four-component system (in fact the argument applies to systems with any number of components). Determine how the shortest tie line is connected to adjacent tie lines. Table 6.1 summarizes the possibilities. For example, compositions on the crossover tie line lie upstream of the initial oil tie line. If the crossover tie line is shorter than the initial oil tie line, and if the intersection of the extensions of the initial and crossover tie lines lies on the liquid side of the two-phase region, then eigenvalues on the nontie-line path that connects the initial tie line to the crossover tie line increase as the path is traced upstream. A rarefaction along that path would violate the velocity rule, and therefore, a shock is required. 4. If a rarefaction connects the initial and crossover tie lines, integrate the nontie-line path from the equal-eigenvalue point on the shorter of the two tie lines to the intersection point of the nontie-line path with the longer of the two tie lines. If a shock connects the initial and injection tie lines, solve the shock balances (expressions like Eqs. 5.3.2 and 5.3.3) for the compositions on either side of the semishock that has a velocity that matches the tie-line eigenvalue on the shorter tie line.

146

CHAPTER 6. FOUR-COMPONENT DISPLACEMENTS

Table 6.1: Nontie-Line Shocks and Rarefactions in Four-Component Displacements with Constant K-Values Envelope Curve and Tie Line Intersections Vapor Side Vapor Side Liquid Side Liquid Side

Process Name LVI Condensing LVI Vaporizing HVI Condensing HVI Vaporizing

Shortest Tie Line Upstream Downstream Upstream Downstream

Composition Variation Rarefaction Shock Shock Rarefaction

5. Repeat the calculation of the rarefaction or shock compositions for the connection between the injection and crossover tie lines. 6. Determine whether a rarefaction occurs along the initial oil tie line. To do so, ﬁnd the composition on the initial oil tie line for a semishock from the initial oil composition to the point on the initial tie line where the shock velocity equals the tie-line eigenvalue. If that composition can be reached from the landing point of the rarefaction or shock from the crossover tie line that satisﬁes the velocity constraint, then a rarefaction along the tie line occurs to the semishock point. If not, there is a genuine shock from the landing point to the initial composition. 7. Determine whether a rarefaction occurs on the crossover tie line. If the crossover tie line is the shortest tie line, a rarefaction will always be present. If it is not the shortest tie line, then a rarefaction may or may not exist depending on the relative locations of the landing point of the rarefactions or shocks from the initial and injection tie lines. If a composition variation from the landing point to the appropriate equal-eigenvalue point (for rarefactions) or semishock point satisﬁes the velocity constraint, a rarefaction will occur. Otherwise, a genuine shock connects the landing point on the crossover tie line to the adjacent tie line. 8. Determine whether a rarefaction occurs along the injection tie line by constructing the appropriate semishock and determining whether a rarefaction along the tie line from the landing point of the rarefaction or shock from the crossover tie line to the semishock point would satisfy the velocity constraint. To illustrate how this procedure works, we consider two example solutions. In the ﬁrst example, we ﬁnd the solution for a displacement of an initial oil composition on the extension of tie line A by an injection gas with a composition on the extension of tie line C (see Fig. 6.3 or 6.4). Tie line B identiﬁed previously is the crossover tie line. Arguments similar to those given in Section 5.5 indicate that the construction of the solution must begin on the shortest of the three key tie lines. In this case, the crossover tie line (B) is the shortest. The ﬁrst task, then, is to determine how the crossover tie line is connected to the injection and initial tie lines. Consider, for the moment, what would happen if tie line B were the injection tie line. In that case, the displacement would be essentially equivalent to a ternary condensing gas drive, in which the injection gas tie line is shorter than the initial oil tie line (see Tables 6.1 and 5.4). As Fig. 6.3 indicates, the tie line intersections for the plane that connects tie line A and B lie on the liquid side of the two-phase

6.2. SOLUTION CONSTRUCTION FOR CONSTANT K-VALUES

147

2(CO2)

fa

e 1(CH4)

B

a

d aac

3(C4)

C

A

b

a a

a

4(C10 )

Figure 6.7: Composition route for a displacement of an oil with composition a, C1 = 0, C2 = 0, C3 = 0.492, and C4 = 0.508 by gas with composition f, C1 = 0.625, C2 = 0.375, C3 = 0, and C4 = 0. K-values are constant: K1 = 2.5, K2 = 1.5, K3 = 0.5, and K4 = .05. The viscosity ratio, µliq /µvap is 5. Tie line A is the initial oil tie line, tie line B is the crossover tie line, and tie line C is the injection gas tie line. region. Hence, the segment of the composition path that connects tie lines A and B is equivalent to a HVI condensing gas drive, and therefore, a shock connects the initial tie line to the crossover tie line. (Here again, the summary given of shocks and rarefactions in Table 6.1 refers strictly to systems with constant K-values, though it is also accurate for nearly all gas/oil systems as well. A discussion of the completely general case is given in Section 7.2) Now consider a displacement in which an injection gas on tie line C displaces an oil on the crossover tie line B. In that case, the injection gas tie line is the longer of the two, and the tie line intersections occur on the vapor side of the two-phase region. Hence, this displacement is equivalent to a ternary HVI vaporizing gas drive, and again, the two tie lines are connected by a shock. Thus, this displacement is fully self-sharpening because all the key tie lines are connected by shocks. Any rarefactions that occur in this displacement must occur along one of the three tie lines. A rarefaction must occur along the crossover tie line. Rarefactions may or may not be present on the initial and injection tie lines. Both of the shocks involving the crossover tie line are semishocks. For both shocks, the shock velocity equals λt on the crossover tie line. Those shocks can be determined once the three key tie lines are known from shock balances using the tie line intersection points (Eqs. 5.3.1 and 5.3.2, for example). Fig. 6.7 shows the composition route for the displacement, and Fig. 6.8 shows the corresponding saturation and composition proﬁles. The leading shock is a genuine shock that connects point a, the initial oil composition, with point b, which is on the tie line that extends through the initial composition. A genuine shock occurs because the composition for a semishock along the initial tie line lies above the shock landing point b, and variation toward that point along the initial tie line

148

CHAPTER 6. FOUR-COMPONENT DISPLACEMENTS a

fa e

Sg

a a

c

a a ab

d

b 0 1

1

a a

aa a

CO2

a

a

0 1

a

aa

0 1

a 1.0

ξ/τ

aa

a a aa

C4

CH4

a a

0 0.0

aa

a

a

a

C10

1

2.0

0 0.0

a a

a aa

a 1.0

2.0

ξ/τ

Figure 6.8: Saturation and composition proﬁles for the displacement of Fig. 6.7 would violate the velocity rule. The initial tie line lies in the ternary base of the quaternary diagram. No CO2 is present in the initial oil or in the mixture on the upstream side of the leading shock. Instead, there is a CH4 bank. The CH4 in that bank must come from the injected gas, because there is no CH4 in the initial oil. CH4 propagates rapidly to concentrate at the leading edge of the transition zone because it has the highest K-value, and hence it partitions preferentially into the fast moving vapor phase. CO2 ﬁrst appears upstream of the shock that connects the initial oil tie line to the crossover tie line (b → c). A short segment of rarefaction (c → d) along the crossover tie line terminates at a semishock (d → e) from the crossover tie line to the injection gas tie line. There is no C4 present upstream of the d → e shock. A combination of displacement and evaporation has removed it from the portion of the porous medium upstream of that shock. The trailing shock is a genuine shock from point e to the injection gas composition. Here again, a rarefaction along the injection gas tie line is missing because the semishock point lies below point e, and variation along the tie line to that point would violate the velocity rule. Upstream of the trailing shock, all of the remaining C10 has been evaporated. Thus, the d → e and e → f shocks are relatively slow-moving evaporation shocks at which components C4 and C10 disappear. Similar behavior is frequently observed in systems with more components: there is a sequence of slow evaporation shocks ordered by the K-values of the components. In the second example, we consider displacement of an oil that is a mixture of all four of the components on the extension of tie line B by an injection gas that is pure CH4 . Fig. 6.9 shows the composition path, and the corresponding saturation and composition proﬁles are shown in Fig. 6.10. The initial tie line is the shortest, and solution construction begins with a semishock from the oil composition a to point b on that tie line. There is also a short segment of rarefaction along

6.3. SYSTEMS WITH VARIABLE K-VALUES

149

2(CO2)

cab 1(CH4) a f

a

e

a

3(C4) a

a

d

4(C10 )

Figure 6.9: 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 = 1. K-values are constant: K1 = 2.5, K2 = 1.5, K3 = 0.5, and K4 = .05. The viscosity ratio, µliq /µvap , is 5. that tie line. A path switch occurs at the equal-eigenvalue point on the initial tie line where λt equals the eigenvalue for the nontie-line paths in the vertical direction. A rarefaction along the nontie-line path connects the initial tie line to tie line A, which is the crossover tie line. The remaining segments of the displacement lie on the ternary base of the diagram, and as might be expected, the composition path is quite similar to that shown in Fig. 5.16, but there is one diﬀerence. The shock for the ternary system in Fig. 5.16 is a semishock, but the shock in the quaternary system is a genuine shock. The semishock point on the crossover tie line lies at a lower vapor saturation than the landing point d of the nontie-line path. A rarefaction from point d to ¯ the semishock point would violate the velocity constraint, so an immediate jump from point d to the injection tie line occurs. The proﬁles shown in Fig. 6.10 indicate that all the CO2 is displaced at the upstream end of the nontie-line rarefaction, and the C4 and C10 disappear at the trailing shocks. Again, the disappearance of the three components not present in the injection gas occurs in the order of their K-values. When more components are present, a succession of slow moving evaporation shocks is commonly observed for components with the lowest K-values.

6.3

Systems with Variable K-values

When equilibrium K-values are not independent of composition, the surfaces of tie lines that contain nontie-line paths are no longer planes. Instead, they are ruled surfaces, surfaces/indexsurface!ruled that are generated by a straight line moving through the three-dimensional composition space [54, 19, 128]. A nontie-line path intersects a sequence of tie lines inﬁnitesimally separated from one another, and hence those tie lines generate a surface. The surface may twist, but it is always ﬂat in

150

CHAPTER 6. FOUR-COMPONENT DISPLACEMENTS a

fa e

a a

Sg

d

a

c

aa a a

1

a

CH4

a

a a

a

a

a

CO2

0 1

a 0 0.0

ab

C4

0 1

da

a

a

1.0

ξ/τ

a

0 1

a

a

a a

a

aa

a a

aa

a a

C10

1

a a 2.0

0 0.0

1.0

2.0

ξ/τ

Figure 6.10: Saturation and composition proﬁles for the displacement of Fig. 6.9 the direction of a tie line. Fig. 6.11a shows an example of the ruled surface of tie lines associated with nontie-line paths that intersect an initial oil tie line. The surface shown contains a family of nested nontie-line paths. For displacements in which the route segment that connects the initial tie line to the crossover tie line is a nontie-line path, any of the tie lines in that surface is a crossover tie line for some injection composition. For the two examples shown, the nontie-line paths are tangent to the crossover tie line at an equal-eigenvalue point. Wang [128] showed that no matter what equation of state is used to describe the equilibrium phase behavior, the mass balance equations require that the nontie-line path ruled surface be a developable surface, a surface in which inﬁnitesimally separated tie lines do intersect, but tie lines on the surface that have ﬁnite separation do not. Thus, in systems with variable K-values, the simple constructions of planes through the injection gas and initial oil tie lines can no longer be used to ﬁnd the crossover tie line, and nontie-line paths must be found be integrating the eigenvector directions numerically. The nontie-line paths shown in Fig. 6.11a were obtained by a stepwise integration along the nontie-line eigenvector direction. While the surfaces of tie lines mapped out by the nontie-line paths are not planar, they are only slightly curved for typical cubic equations of state such as the Peng-Robinson or Redlich-Kwong equations. Fig. 6.11b shows surfaces of tie lines for two sets of shocks that connect the crossover tie line to an injection tie line. The extensions of tie lines that are connected by a shock must intersect, whether or not K-values are constant. When K-values are variable, the surface of tie lines that intersect a given tie line (the crossover tie line in Fig. 6.11b, for example) is not planar. Instead, it twists slightly. Any of the tie lines that intersect the crossover tie lines shown are potential injection gas tie lines. Thus, the crossover tie line is a tie line in the interior of the diagram that is the intersection of the rarefaction surface of tie lines generated by the “vertical” nontie-line paths

6.3. SYSTEMS WITH VARIABLE K-VALUES

151

C3

C3

• •

• •

•• •

CH4

••

•• •

•

•• ••

C16

C6 •

(a) Vertical Ruled Surface

CH4

••

•

•• ••

C6 •

C16

(b) Horizontal Ruled Surfaces

Figure 6.11: Ruled surfaces of tie lines ([54, p. 207], used with permission), [56]. that intersect the initial tie line with the “horizontal” shock surface of tie lines that intersect the injection tie line. The sequence of steps for constructing a solution for a system with constant K-values can also be used for systems with variable K-values, except for the procedure used to determine the crossover tie line. Three situations are possible. If the crossover tie line is connected to the initial oil and injection gas tie lines by shocks, then a set of tie line intersection equations can be solved for the crossover tie line. If a shock connects either the injection gas or initial oil tie line to the crossover tie, then the crossover tie line is at the intersection of a rarefaction surface and a shock surface. Tie lines on the shock surface must intersect the injection gas or initial oil tie line. The rarefaction surface is generated by stepwise integration of some nontie-line path from the initial oil or injection gas tie line. If rarefactions connect the three tie lines, then the crossover tie line is the intersection of two rarefaction surfaces that are determined by integration. Table 6.1 can be used to determine whether a rarefaction or shock occurs for each segment. Fig. 6.12 shows the composition route for a displacement of an oil that is a mixture of CH4 , C6 , and C16 by a gas mixture that contains CH4 enriched with propane (C3 ). The geometry of the tie lines (see Table 6.1) indicates that there is a rarefaction between the initial tie line and the crossover tie line and a shock from the crossover tie line to the injection tie line. Fig. 6.13 shows the corresponding saturation and composition proﬁles. The crossover tie line is identiﬁed by integrating an arbitrary nontie-line path upwards from the initial tie line to ﬁnd the tie line that intersects the injection tie line. Once the crossover tie line is determined, the actual nontie-line path is traced from the appropriate equal-eigenvalue point on the crossover tie line to the initial tie line. The semishock from the crossover tie line to the injection gas tie line is calculated by solving the shock balance for component C6 for the semishock point on the crossover tie line. That shock balance is the simplest to use because C6 is missing on the upstream side of the shock. The landing point on the injection tie line is found from a shock balance for one of the other components.

152

CHAPTER 6. FOUR-COMPONENT DISPLACEMENTS

C3

• • •

CH4

f• e•

•

d c•

•

b

•a

C6

C16 Figure 6.12: Displacement with variable K-values. The initial oil contains 20 mole percent CH4 , and 40 mole percent each of C6 and C16 . The injection gas contains 90 mole percent CH4 and 10 mole percent C3 . The Peng-Robinson EOS was used to calculate phase equilibrium at 2000 psia (136 atm and 200 F (93 C). See Johns [54, pp. 195–197] for EOS parameters and details of the solution. In this displacement again, there is a fast-moving CH4 bank at the leading edge of the transition zone (just upstream of the a → b shock). No C3 is present in that bank, but it appears in the ﬂowing mixtures as the b - c rarefaction is traced. All of the C6 evaporates at the shock from the crossover tie line to the injection tie line (d → e). The trailing shock, across which all the C16 evaporates, moves very slowly because the solubility of C16 in pure CH4 is low. For displacements in which all the tie lines are connected by shocks (fully self-sharpening), solution construction is simpler because no integration is required. Given an injection gas composition and an initial oil composition, the tie lines that extend through those compositions can be identiﬁed by a negative ﬂash [135, 129]. The crossover tie line can then be found by solving for the tie line in the interior that intersects both the injection tie line and the initial tie line. That tie line must satisfy (see Eq. 6.1.27) an equation that speciﬁes the intersection of the crossover tie line with the injection tie line,

cr cr 1 + (Kiinj − 1)Linj = xcr xinj i {1 + (Ki − 1)L } , i

i = 1, nc ,

(6.3.1)

and must also satisfy a similar expression that determines the intersection between the initial and crossover tie lines,

cr cr 1 + (Kiinit − 1)Linit = xcr xinit i i {1 + (Ki − 1)L } ,

i = 1, nc .

(6.3.2)

In Eqs. 6.3.1 and 6.3.2, the superscripts inj, cr, and init refer to the injection, crossover, and initial tie lines, and L is the overall mole fraction on a tie line. These expressions are nonlinear because the unknown K-values on the crossover tie line depend on xcr i . They can be solved by guessing

6.3. SYSTEMS WITH VARIABLE K-VALUES

153

Sg

1

0 1

C6

CH4

1

0 1

C3

C16

0 1

0 0.0

1.0

ξ/τ

2.0

0 0.0

1.0

ξ/τ

2.0

Figure 6.13: Saturation and composition (mole fraction) proﬁles for the displacement of Fig. 6.12 [54, p. 198].

a set of K-values and then iterating. Once the crossover tie line is determined, the shocks to the injection and initial tie lines can be found as in the previous examples. Fig. 6.14 shows the composition route for a displacement that is similar to the displacement of Figs. 6.12 and 6.13 except that propane is replaced by ethane (C2 ). Replacing C3 with C2 moves the tie line intersections for the segment connecting the initial and crossover tie lines from the vapor side of the two-phase region to the liquid. As a result, the nontie-line paths become selfsharpening (Table 6.1), and the nontie-line rarefaction in Fig. 6.12 is replaced by a shock in Fig. 6.14. Otherwise, the displacement proﬁles are similar, as comparison of Figs. 6.13 and 6.15 shows. The crossover tie line is connected to the initial and injection tie lines by semishocks, and the only rarefaction present in the proﬁles is the rarefaction along the crossover tie line. Because the crossover tie line is the shortest tie line, solution construction starts there. The construction of the semishocks to the initial and injection tie lines is no more diﬃcult than shock construction for ternary systems, or even for a Buckley-Leverett problem. The key step is to ﬁnd the crossover tie line and the corresponding tie-line intersection points. Once they are known, the shock constructions require only solution of a nonlinear equation in the saturation on the crossover tie line to determine the semishock points on that tie line (c and d in Fig. 6.14) and the shock velocity. A similar nonlinear expression in the saturation on the initial or injection tie line determines the landing points on those tie lines.

154

CHAPTER 6. FOUR-COMPONENT DISPLACEMENTS

C2 • •

• g

• CH4 f e•

d c

••

•b

C6

• a

C16 Figure 6.14: Displacement with variable K-values. The initial oil contains 20 mole percent CH4 , and 40 mole percent each of C6 and C10 . The injection gas contains 90 mole percent CH4 and 10 mole percent C2 . The Peng-Robinson EOS was used to calculate phase equilibrium at 2000 psia (136 atm and 200 F (93 C). See Johns [54, pp. 195, 213, 215] for EOS parameters and details of the solution. 1.0

g

Sg

e,f

e d

b

b a 1.0

C6

CH4

0.0 1.0

c

0.0 1.0

C2

C16

0.0 1.0

0.0 0.0

1.0

ξ/τ

2.0

0.0 0.0

1.0

ξ/τ

2.0

Figure 6.15: Saturation and composition (mole fraction) proﬁles for the displacement of Fig. 6.14 [54, p. 214].

6.4. CONDENSING/VAPORIZING GAS DRIVES

6.4

155

Condensing/Vaporizing Gas Drives

In an important 1986 paper, Zick [140] used numerical compositional simulation results to demonstrate that gas displacement processes in a multicomponent system can display simultaneously features that resemble the behavior of vaporizing and condensing displacements described for ternary systems. Zick referred to such displacements as condensing/vaporizing gas drives. The analysis of four-component solutions reveals why Zick’s terminology is an accurate portrayal of the ﬂow behavior. Consider, for example, the displacement illustrated in Figs. 6.12 and 6.13. Consider also two related ternary displacements illustrated in Fig. 6.16. If pure CH4 displaces oil O1 , a mixture of CH4 , C6 , and C16 , the displacement is the ternary vaporizing gas drive shown in Fig. 6.16a. If, on the other hand, the original oil composition is a mixture of C16 and CH4 (O2 ), and the injection gas is CH4 enriched with 10 percent C3 , the displacement is the condensing gas drive shown in Fig. 6.16b. The four-component displacement is a combination of segments that are similar to the vaporizing and condensing ternary displacements. Fig. 6.17 illustrates that idea. The composition route for the ternary vaporizing gas drive for pure CH4 displacing oil O1 is shown in the base of the quaternary diagram. If 10 percent C3 is added to the injection gas, the segment of the ternary route that includes the rarefaction along the initial tie line, the intermediate shock from the initial tie line to the injection tie line, and the trailing shock move upward into the interior of the quaternary diagram. The same segments are still present: a rarefaction along the crossover tie line, a shock from the crossover tie line to the injection tie line, and a trailing shock to the injection gas composition. This vaporizing segment is connected to the initial oil tie line by a condensing segment that is essentially equivalent to the leading portions of the ternary condensing gas drive shown in Fig. 6.16b. The ternary displacement includes a shock from the initial oil composition along the initial tie line, a nontie-line rarefaction from the initial tie line to the injection tie line, and a tie-line rarefaction along the injection gas tie line. The quaternary system includes a leading shock along the initial tie line, a nontie-line rarefaction from the initial tie line to the crossover tie line, and a tie-line rarefaction along the crossover tie line. Thus, this four-component displacement consists of a leading condensing segment connected to a trailing vaporizing segment by the crossover tie line. The leading segment is similar to a ternary condensing drive, but with the crossover tie line as the injection gas tie line, and the trailing segment is closely related to a ternary vaporizing gas drive with the crossover tie line as the initial tie line. The crossover tie line links the two segments together in the quaternary system. The example shown in Figs. 6.12 and 6.17 are not the only possible combination of condensing and vaporizing segments. Fig. 6.18 illustrates why. Two surfaces of tie lines are shown in Fig. 6.18, both associated with a speciﬁc crossover tie line (the two surfaces shown are the same surfaces illustrated in Figs. 6.1 and 6.2). The crossover tie line is the tie line in the interior of the quaternary diagram that extends to point b on the C1 = 0 face and point e on the C4 = 0 face. That tie line is at the intersection of the two surfaces. The points a, b, and c all lie on extensions of tie lines in the vertical surface to the C1 = 0 face, and the points d, e, and f are on extensions of tie lines in the horizontal surface to the C4 = 0 face. Imagine that points a, b, and c are initial oil compositions and points d, e, and f are injection gas compositions. Consider what composition route patterns arise if the crossover tie line is ﬁxed but there are various combinations of injection gas and initial oil compositions. Displacement of oil a by gas

156

CHAPTER 6. FOUR-COMPONENT DISPLACEMENTS

CH4

•

CH4

•

• •

• •

• O2

•

• •

•

O1

C16

C6 C16

C3

(a) Vaporizing Segment

(b) Condensing Segment

Figure 6.16: Related ternary displacements [54, p. 204].

C3

Vaporizing Segment • • •

• CH4 • • •

Condensing Segment

•• •

•• •O1

C6

C16 Figure 6.17: Composition route for a condensing/vaporizing gas drive (modiﬁed from Johns [54, p. 196]).

6.4. CONDENSING/VAPORIZING GAS DRIVES

157

2(CO2) a

a

c aa

d aa e aa f aa

3(C4)

a

1(CH4)

a

a

ab

a

aa

4(C10 )

Figure 6.18: Combinations of injection gas and initial oil compositions. d is a condensing/vaporizing drive like those shown in Figs. 6.7, 6.12, 6.14, and 6.17. In those displacements there is a leading condensing segment followed by a trailing vaporizing segment. (Recall that whether a segment is condensing or vaporizing can be determined easily from the combination of tie line lengths summarized in Table 6.1: a segment in which the shorter tie line lies upstream of a longer one is condensing, while a segment with a longer tie line upstream of a shorter one is vaporizing). A similar composition route is obtained for any oil composition on the extension of a tie line on the vertical surface that gives an initial composition between a and b. A displacement of oil c by gas d, on the other hand, includes two vaporizing segments. The displacement of Fig. 6.9 is an example of this sort of composition route. The initial tie line is the shortest, and the segment from the initial tie line to the crossover tie line (a nontie-line rarefaction for the example shown) places the longer crossover tie line upstream of the shorter initial tie line. Similarly, the injection tie line is longer than the crossover tie line, and hence the route segment connecting those tie lines (a shock for this example) is also vaporizing. Thus, this displacement could be termed a vaporizing/vaporizing drive. Similar patterns would be observed for any injection gas composition between d and e and any initial oil composition between b and c. In fact, any combination of condensing and vaporizing segments can be obtained by choosing the appropriate pair of injection and initial compositions. Displacement of oil a by gas f is a condensing/condensing drive, for example, and displacement of oil c by gas f is vaporizing/condensing. Table 6.2 summarizes all the possibilities for the combinations of injection and initial compositions in Fig. 6.18. There are also some special cases included in the examples of Fig. 6.18 and in Table 6.2. Displacement of oil b by gas d, for example, is a pseudoternary displacement: a vaporizing drive in which the initial and crossover tie lines coincide and only two key tie lines are part of the solution route. Because there is only one route segment present, only one (vaporizing) is listed in Table 6.2. There are several other combinations of pseudoternary displacements for pairs of injection and initial oil tie lines, each describe in Table 6.2 by the appropriate single process designation.

158

CHAPTER 6. FOUR-COMPONENT DISPLACEMENTS

Table 6.2: Classiﬁcation of Four-Component Gas Drives

Initial Oil a b c

d Condensing/Vaporizing Vaporizing Vaporizing/Vaporizing

Injection Gas e f Condensing Condensing/Condensing Binary Condensing Vaporizing Vaporizing/Condensing

Displacement of oil b by gas e is a pseudobinary displacement in which the initial, injection, and crossover tie lines all coincide. The classiﬁcation of the types of composition routes possible in four-component systems is useful because it demonstrates that it is possible to have any combination of condensing and vaporizing segments of injection gas and initial oil compositions are chosen appropriately. In systems in which more components are present (see Chapter 7), similar behavior will be observed. In multicomponent systems, catalogs of the sequence of condensing and vaporizing segments will be less useful because there will be too many combinations of condensing and vaporizing segments, but it is helpful to know that a variety of sequences should be expected, as the four-component theory indicates.

6.5

Development of Miscibility

In ternary systems multicontact miscibility occurs if either of the two key tie lines (the injection gas tie line or the initial oil tie line) is a critical tie line. Arguments similar to those given in Section 5.6 indicate that the same statement holds true when more than three components are present: if the initial or injection tie line has zero length at some pressure or injection gas composition, the displacement is miscible. In four-component systems, however, there is a third tie line, the crossover tie line. We now examine whether that tie line can play a role in development of multicontact miscibility as well. Consider for example, the displacement illustrated in Figs. 6.14 and 6.15. If the injection gas mixture is enriched further by increasing the C2 concentration in the CH4 /C2 mixture, the solution route must change because the crossover tie line moves upward in the interior of the quaternary diagram to higher C2 concentrations. Fig. 6.19 shows the resulting changes in the composition route, and Fig. 6.20 shows the corresponding saturation and composition proﬁles. As the injection gas is enriched in C2 , the crossover tie line moves closer to the critical locus, which connects the plait point on the CH4 /C2 /C6 face with the plait point on the CH4 /C2 /C16 face. As the crossover tie line approaches the critical point in the surface of crossover tie lines (see Fig. 6.19), the saturation and composition proﬁles in Fig. 6.20 show signiﬁcant changes. The leading shocks slow down, the intermediate and trailing shocks speed up, and the displacement eﬃciency, as indicated by the saturation of liquid phase present upstream of the shock from the crossover tie line to the injection tie line, improves. Now consider what would happen if the injection gas mixture were enriched to composition g5 shown in Fig. 6.19. The extension of the injection gas tie line through composition g5 intersects the tie line that is tangent to the critical point shown in the vertical surface of crossover tie lines. The shock balances for this fully self-sharpening displacement are

6.5. DEVELOPMENT OF MISCIBILITY

159

C2 g6

•

• g5 (MME) g4 • g3

••

••

•

•

•• •

g2 • g1 •

CH4 •

•

•

•• •• ••

C6

a•

C16 Figure 6.19: Eﬀect of enrichment of the injection gas on composition routes ([54, p. 213] used with permission, [56]).

1.0

Sg

Injection Gas g1 Injection Gas g2 Injection Gas g3 Injection Gas g4 0.0 1.0

C6

CH4

1.0

0.0 1.0

C2

C16

0.0 1.0

0.0 0.0

1.0

ξ/τ

2.0

0.0 0.0

1.0

ξ/τ

2.0

Figure 6.20: Saturation and composition (mole fraction) proﬁles for the displacement of Fig. 6.19 ([54, p. 214], used with permission, [56]).

160

CHAPTER 6. FOUR-COMPONENT DISPLACEMENTS

= Λlead i

Fia − Fib , Cia − Cib

Λinit−X = i

Fib − Fic = λct , Cib − Cic

i = 1, nc,

(6.5.1)

Fie − Fig Cie − Cig

i = 1, nc.

(6.5.2)

and ΛX−inj = i

Fid − Fie = λdt , d e Ci − Ci

Λtrail = i

In the limit as the crossover tie line approaches the critical locus, it’s length goes to zero, and hence, points c and d coincide at the critical point. Because the critical point is on the surface that bounds the two-phase region, λct = λdt = 1. Hence the two shocks that connect the crossover tie line to the initial and injection tie lines must have unit velocity. The critical point is the termination of the equivelocity curve in the surface of crossover tie lines, and on the equivelocity curve, Fi = Ci . Therefore, Fic = Cic = Fid = Cie . Substitution of these expressions into the balance for the shock from the initial tie line to the crossover tie line (Eq. 6.5.1) gives = Λlead i

Fib − Cic = 1, Cib − Cic

i = 1, nc ,

(6.5.3)

which requires that Fib = Cib . Hence, point b is also on the equivelocity surface. The leading shock balance then becomes = Λlead i

Fia − Fib Cia − Cib = = 1, Cia − Cib Cia − Cib

i = 1, nc,

(6.5.4)

where we have made use of the fact the Fia = Cia because the initial mixture is single-phase. An equivalent argument for the trailing shock indicates that it, too, moves with unit velocity. Thus, all the shocks merge into a single shock that moves at unit velocity, and the displacement is piston-like. We have shown, therefore, that when the crossover tie line is a critical tie line, the displacement is multicontact miscible. The fact that a critical crossover tie line can also produce a multicontact miscible displacement means that a four-component displacement is multicontact miscible if any of the three key tie lines (injection gas, initial oil or crossover) is a critical tie line. If the crossover tie line becomes the critical tie line as the pressure is increased or the injection gas composition is enriched, the initial oil and injection gas compositions at the MMP or MME lie on tie lines that are not critical and therefore have length greater than zero. As a result, methods of calculating the MMP or MME that rely on identifying the pressure or enrichment that either the initial oil or injection gas tie line becomes a critical tie line will not give the correct MMP (MME) when the crossover tie line becomes the critical tie line. Which tie line controls miscibility is easily determined from tie line lengths. In systems in which the K-values maintain the same ordering as the pressure (or enrichment) is increased, it is possible to state for most, but not all, pairs of injection and initial compositions which tie line will be the shortest and therefore, which tie line will control miscibility. Table 6.3 reports the patterns that would be observed for the phase diagram of Fig. 6.19. In Table 6.3 the oil compositions a-c refer to compositions on the vertical surface of tie lines (just as in Fig. 6.18, with b on the extension of the crossover tie line), and gas compositions d-f refer to gas compositions on either side of the crossover tie line with e on the extension of the crossover tie line. If the oil tie line lies between a and b, for example, and the injection gas lies between d and e, the crossover tie line is the

6.5. DEVELOPMENT OF MISCIBILITY

161

Table 6.3: Tie Line that Controls Miscibility in Four-Component Gas Drives for the Phase Diagram of Fig. 6.19

Initial Oil a b c

d Crossover Crossover = Initial Initial

Injection Gas e Injection = Crossover Injection = Crossover = Initial Initial

f Injection Injection Initial or Injection

shortest. For these condensing/vaporizing drives (see Table 6.2), the crossover tie line is the tie line that becomes critical when the displacement is multicontact miscible. A critical crossover tie line is characteristic of condensing/vaporizing drives, in which the initial oil and injection gas tie lines have nonzero length at the MMP or MME. If, on the other hand, the injection gas lies between d and e and the oil composition is between b and c, the initial oil tie line is always shorter. Which tie line is shortest for oil compositions between b and c and gas compositions between e and f depends on the relative locations of the injection gas and initial oil tie lines with respect to the critical locus. Either can be shorter depending on the speciﬁc gas and oil compositions.

6.5.1

Calculation of Minimum Miscibility Pressure

The minimum miscibility pressure (MMP) is the pressure at which one of the three key tie lines has zero length, and therefore, the MMP can be calculated directly from the geometry of tie lines for displacements that are fully self-sharpening. In such cases, the three tie lines are connected only by shocks, which means that the crossover tie line can be found as the tie line in the interior of the quaternary diagram that extends to intersect both the initial and injection tie lines (Eqs. 6.3.1 and 6.3.2). For a given injection gas and initial oil composition (and a given temperature, of course), the MMP can be found by the following procedure: 1. Find the tie lines that extend through the initial oil and injection gas compositions by a negative ﬂash [135, 129]. 2. At some low pressure, ﬁnd a ﬁrst guess of the crossover tie line by estimating equilibrium K-values and using Eqs. 6.3.1 and 6.3.2. 3. Update the K-values by ﬂashing some mixture on the current estimate of the crossover tie line. 4. Repeat steps 2 and 3 to convergence and calculate the lengths of the initial, injection, and crossover tie lines. 5. Increase the pressure and repeat steps 1 to 4 until one of the tie lines has zero length. The pressure at which that occurs is the MMP. An analogous procedure can be used to ﬁnd the minimum enrichment for miscibility (MME). In that case, the injection gas composition is varied rather than the displacement pressure. For fully

162

CHAPTER 6. FOUR-COMPONENT DISPLACEMENTS CO2 f• e • d ← • •← c

•

CH4 • b•α

C4 a

••

A

C10

Figure 6.21: Composition route for displacement of oil a containing 10 mole percent CH4 , 20 mole percent C4 , and 70 mole percent C10 and oil α containing 30 percent CH4 , 16.24 percent C4 , and 53.76 percent C10 by pure CO2 at 1600 psia (109 atm) and 160 F (71 C). See Orr et al. [92] for details of the solution, which are also reported by Johns [54, pp. 168–172] along with the Peng-Robinson EOS parameters used. self-sharpening systems (in the absence of dispersive mixing), the MMP or MME is a property of the thermodynamic description of the equilibrium phase behavior as expressed by the geometry of tie lines in a three-dimensional composition space. For these systems, the MMP does not depend on the fractional ﬂow function. In displacements that are not fully self-sharpening, the crossover tie line is determined by the intersection of two rarefaction surfaces or the intersection of a rarefaction surface with a shock surface. The rarefaction surfaces can be determined by integration of the appropriate nontieline path, but an excellent approximation of the crossover tie line can be found more easily by solving the tie-line intersection equations as if the displacement were fully self-sharpening [129, 128]. The procedure described above can then by used to ﬁnd the MMP or MME. While the resulting MMP or MME is approximate, all the computational evidence obtained to date indicates that the approximation is a very good one.

6.5.2

Eﬀect of Variations in Initial Oil Composition on MMP

The solutions for four-component displacements can be used conveniently to investigate the dependence of displacement behavior and MMP on changes in initial oil composition or injection gas composition. Consider, for example, a four-component system representative of CO2 ﬂoods: CO2 displacing and oil that is a mixture of CH4 , C4 , and C10 . We examine ﬁrst how displacement behavior changes if the amount of CH4 present in the initial oil is increased. Fig. 6.21 compares the composition routes for displacement of two oils, shown as oil a and α in Fig. 6.21. Oil α is oil a to which CH4 has been added, but in a special way. The composition of oil α lies on the extension

6.5. DEVELOPMENT OF MISCIBILITY

163

1.0

Sg

Oil Composition a Oil Composition α

0.0 1.0

C4

CH4

1.0

0.0 1.0

C10

CO2

0.0 1.0

0.0 0.0

1.0

ξ/τ

2.0

0.0 0.0

1.0

ξ/τ

2.0

Figure 6.22: Saturation and composition (mole fraction) proﬁles for the displacements of Fig. 6.21 [92, 54]. of the same tie line that extends through oil a. As Fig. 6.21 illustrates, the composition route of the displacement is unchanged when the oil composition is changed from a to α. The leading shock occurs on the same tie line, the tie line that extends through both oil compositions. The leading intermediate shock is also unchanged. The crossover tie line, which is determined as the intersection of the extension of the initial tie line and a tie line in the CO2 /C4 /C10 face (at point A in Fig. 6.21, is also unchanged because the initial tie line is the same. Points b and c, the compositions on either side of the shock from the crossover tie line to the initial oil tie line are unchanged because they are determined by the shock balances that govern the semishock, which do not depend on the exact oil composition as long as the initial oil and crossover tie lines are ﬁxed. The remaining segments of the displacement are determined entirely by the crossover and injection gas tie lines, and hence they are unchanged as well. There is one important eﬀect of the change in initial oil composition, however. The velocity of the leading shock is given by = Λlead i

Fia − Fib , Cia − Cib

i = 1, nc .

(6.5.5)

Hence, changing the initial oil composition does change the velocity of the leading shock, as Fig. 6.22 shows. In the displacement of oil a there is a small leading CH4 bank vaporized from the undisplaced hydrocarbons in the transition zone between the injection gas and the oil. No CH4 is found in the phases upstream of the shock from the initial tie line to the crossover tie line, so any CH4 that was originally present in those ﬂuids has moved to the leading bank. In eﬀect, the high volatility of CH4 (as reﬂected by a high K-value), causes it to propagate rapidly as it partitions

164

CHAPTER 6. FOUR-COMPONENT DISPLACEMENTS

into the fast-moving vapor phase, much as high volatility components propagate rapidly in gas chromatography. Changing the amount of CH4 present in the initial oil causes the leading CH4 bank to grow and propagate more rapidly. Upstream of the CH4 bank, however, the compositions and saturations are unchanged. The MMP for this system is the lowest pressure at which one of the key tie lines is a critical tie line. As Fig. 6.21 shows, the crossover tie line is the shortest, and as the pressure is increased, it approaches the critical locus at a lower pressure than the initial or injection tie lines. The crossover tie line is the tie line in the CO2 /C4 /C10 face that intersects composition A in Fig. 6.21. Hence, the MMP is the pressure at which point A lies on the extension of the critical tie line in the CO2 /C4 /C10 ternary system. Because the crossover tie line does not change as the initial oil composition changes from a to α, the MMP also does not change. This example is a condensing/vaporizing gas drive in which the initial oil and injection gas compositions lie well within the region of tie-line extensions at the MMP. The standard deﬁnitions of multicontact miscibility derived from ternary systems, which require that either the injection gas or the initial oil tie line be a critical tie line, do not give the correct MMP for this system, therefore, because neither of those tie lines controls development of miscibility. In this case, miscibility develops because the CH4 present in the oil partitions preferentially into the vapor phase. It propagates in a leading bank, in which no CO2 appears. Upstream of the shock from the initial tie line to the crossover tie line, no CH4 is present. Thus, the injected CO2 encounters only a dead oil, and as a result, the MMP is that of the dead oil obtained by removing the CH4 from the original oil along the extension of the initial tie line. The minimal diﬀerences between the displacements of oils a and α are a consequence of the way the initial oil composition was changed. A more reasonable way to add CH4 to an oil, and the way it would be done in the laboratory, would be to add CH4 to a mixture of ﬁxed composition. In that case, mixture compositions would lie on the dilution line that connects the oil composition with pure CH4 in the CH4 /C4 /C10 face. Fig. 6.23 illustrates that process. Oil A in Fig. 6.23 contains 48 mole percent C4 and 52 mole percent C10 . This dead oil ( with no CH4 dissolved in it) lies on a critical tie line extension in the CO2 /C4 /C10 face at about 1600 psia (Fig. 6.23b), and therefore, the MMP for the dead oil is 1600 psia. Now consider what happens when CH4 is added to the dead oil along the dilution line shown in Fig. 6.23a to composition a, a mixture containing 35 mole percent CH4 that is saturated with CH4 . The tie line in the CH4 /C4 /C10 face that extends through composition a is shown in Fig. 6.23a. Its extension to the C4 /C10 edge of the ternary diagram gives composition B. The crossover tie line associated with initial oil a is the tie line in the CO2 /C4 /C10 face with an extension that intersects point B. The MMP for the live oil (composition a) is the pressure at which the crossover tie line is a critical tie line. That tie line is shown in Fig. 6.23b at about 1625 psia. Thus, the MMP does change when CH4 is added along the dilution line, but only by a small amount in this example. The weak sensitivity of the MMP to the addition of CH4 to the dead oil in this CO2 ﬂood is a consequence of the geometry of the key tie lines that determine displacement behavior. In this system, the dilution line along which CH4 is added is close to parallel to the initial oil tie lines that extend through compositions on the dilution line in the CH4 /C4 /C10 face (see Fig. 6.23a). Therefore, adding signiﬁcant quantities of CH4 to the dead oil hardly changes the initial oil tie line, which means that the crossover tie line also changes only slightly. The weak dependence of the MMP on the amount of CH4 dissolved in the oil is also observed for

6.5. DEVELOPMENT OF MISCIBILITY

165

CH4

CO2 1600 psia 1625 psia

←Tie Line

C10

← Dilution Line ••

A

(a)

1600 psia 1625 psia

A

C4

Critical → Tie Lines

•a

B

••

C4

C10

B

••

(b)

Figure 6.23: Determination of the MMP for initial composition a [92][54, p. 181].

displacements in crude oil systems. The empirical correlation for MMPs of West Texas oils oﬀered by Yellig and Metcalfe [138], for example, includes the provision that the correlation value of the MMP for the dead oil is to be used for oil containing dissolved CH4 unless the oil contains so much CH4 that the bubble point pressure of the oil exceeds the correlation MMP. In that case, Yellig and Metcalfe recommend using the bubble point pressure as the MMP. The example given here suggests that it is not strictly correct to assume that there is no dependence of MMP on the dissolved CH4 concentration, but that the change in the MMP may be quite small. The 25 psia diﬀerence for the system CO2 /CH4 /C4 /C10 system is signiﬁcantly smaller than the pressure increments often used in slim tube displacements, so changes of that magnitude would not be detected in typical experiments. While MMPs for displacements of oils containing dissolved CH4 by CO2 are insensitive to the amount of dissolved CH4 in the oil, there are other systems for which the sensitivity to initial oil composition is much greater. For example, composition routes and MMPs for nitrogen displacements are often quite sensitive to the concentration of CH4 in the initial oil [19, 21]. Fig. 6.24 shows a typical composition route for displacement of an oil that is a mixture of CH4 , C4 , and C10 by pure N2 . Saturation and composition proﬁles are shown in Fig. 6.25, along with the velocity proﬁle (this solution included the eﬀects of volume change as components change phases). The composition route shown in Fig. 6.24 is a vaporizing/vaporizing drive. The leading shock connects point a, the initial oil composition, with point b on the initial tie line. At this pressure, there is actually a second tie line that extends through the initial oil composition, a tie line in the CH4 /C4 /C10 face (also shown in Fig. 6.24. The appropriate initial tie line is the one in the interior of the quarternary diagram. That tie line must be the initial tie line if solutions are to be continuous with respect to the initial data. If, for example, the initial oil contained an inﬁnitesimal amount of N2 , the initial tie line would be in the interior of the diagram. In the limit as that N2 is removed, the composition and saturation proﬁles obtained must diﬀer only slightly from the solution with the N2 present. That would not be the case if the tie line in the CH4 /C4 /C10 face were chosen as the initial tie line. In this example, there is a short segment of rarefaction along the initial tie line, a path switch at

166

CHAPTER 6. FOUR-COMPONENT DISPLACEMENTS

CH4

h

crossover tie line

c ab

←

N2 a

ga f a

ea

ad

aa

C4

C10

Figure 6.24: Composition route for displacement of oil a containing 10 mole percent CH4 , 20 mole percent C4 , and 70 mole percent C10 by pure N2 at 1600 psia (108 atm) and 160 F (71 C). See Dindoruk [19, pp. 142, 192, 200–202] for solution parameters and details, used with permission, [21].

6.5. DEVELOPMENT OF MISCIBILITY h

VELOCITY, SATURATION AND MOLE FRACTIONS

1.0 g f

f e

167

d

d

b a

c

N2

0.0 1.0

CH4

0.0 1.0

C4

0.0 1.0 C10

0.0 1.0

Sg

0.0 1.2

vD

0.0 0.0

0.4

0.8

xD/tD

1.2

1.6

2.0

Figure 6.25: Saturation, composition, and velocity (mole fraction) proﬁles for the displacements of Fig. 6.24 [19, 21]. the equal-eigenvalue point, and a nontie-line rarefaction to the crossover tie line. There is actually a segment of rarefaction along the crossover tie line, and then there is a semishock from the crossover tie line to the injection gas tie line. The velocity of that shock matches the tie line eigenvalue for the crossover tie line at point e, as Fig. 6.24. There is also a segment of rarefaction along the injection tie line and a trailing semishock. This example demonstrates that it is possible to have rarefactions along all of the key tie lines. At higher pressures, however, the rarefactions along the crossover and injection tie lines disappear because points d and f migrate to compositions at higher N2 compositions than the semishock points, e and g, and rarefactions along the tie lines to the semishock points would violate the velocity rule. In this displacement, the initial oil tie line is the shortest, and it controls the development of miscibility. The pressure at which the phase diagram of Fig. 6.24 is drawn is too low to show a critical point. A critical point appears (according to the Peng-Robinson EOS) at the critical pressure of the CH4 /C4 binary system, 1815 psia (123.5 atm). At higher pressures, a critical locus connects a plait point in the CH4 /C4 /C10 face to another plait point in the CH4 /N2 /C4 face. As the pressure is increased further, that plait point migrates down the CH4 /N2 /C4 face, reaching base at the N2 /C4 critical pressure, 4690 psia (319 atm. At still higher pressures, the critical locus connects a critical point in the CH4 /C4 /C10 face to a critical point in the N2 /C4 /C10 face. Fig. 6.26 shows a phase diagram calculated at 4900 psia (333 atm). To simplify the discussion, we consider displacements in which the injection gas composition is ﬁxed (I3 in Fig. 6.26, 50 mole percent each of N2 and CH4 ), as is the crossover tie line, and we vary the initial oil composition. The “vertical” surface shown in Fig. 6.26 is determined in two parts. The portion below the crossover tie line consists of tie line whose extensions intersect the crossover

168

CHAPTER 6. FOUR-COMPONENT DISPLACEMENTS CH4

a3

a I3 a

critical locus

crossover tie line ‘‘horizontal’’ surface

aa2

N2

C4

aa a

‘‘vertical’’ surface

C10

a1

a a

Figure 6.26: Key tie lines for displacement of three oils a, a1 , and a2 by pure N2 at 4900 psia (333 atm and 160 F (71 C). See Dindoruk et al. for solution parameters and details [21]. tie line, because vertical paths from those tie lines to the crossover tie line are self-sharpening. The part above the crossover tie line is determined by integrating along nontie-line paths that connect the crossover tie line to tie lines above it. The surface shown, therefore, contains all the possible initial tie lines for this particular crossover tie line. Now consider initial oil compositions that lie on the liquid side of the vertical surface of tie lines. For oil compositions on tie line extensions that lie between a1 , and a2 , the displacements are condensing/vaporizing drives, and the crossover tie line is the shortest tie line. It will become a critical tie line at the lowest pressure. Because the initial mixtures were chosen in such a way that the crossover tie line is ﬁxed, the MMPs for any of these oils will be the same. For oil compositions on the vertical surface but above the crossover tie line, however, the situation diﬀers sharply. Now the displacement is a vaporizing/vaporizing drive, and the initial tie line is the shortest. For initial tie lines that lie in the interior of the quaternary diagram, the location of these initial tie lines with respect to the critical locus now depends strongly on the amount of CH4 in the initial oil mixture. Adding more CH4 moves the initial tie line closer to the critical locus, and hence the pressure increment required to make the initial tie line a critical tie line is smaller. For the highest CH4 concentrations, the sensitivity changes again, however. As the mole fraction of CH4 in the initial oil is increased, there is some composition above which there is no tie line in the interior of the quaternary diagram that intersects the initial oil composition, and therefore, the initial tie line lies in the CH4 /C4 /C10 face. Additional increases in initial CH4 concentration have no eﬀect on the MMP because the critical tie line in that face remains ﬁxed. Thus, for a given CH4 /N2 ratio in the injection gas, the MMP can be independent of initial CH4 concentration at low initial CH4 concentrations, quite sensitive to CH4 concentration for intermediate CH4 concentrations, and

6.5. DEVELOPMENT OF MISCIBILITY

169

10000

Pressure, psia

8000

Crossover Tie Line Initial Tie Line

6000 4000

Oil: C4/C10=0.10 Oil: C4/C10=0.29 Oil: C4/C10=0.80

2000 0 0.0

0.4 0.6 0.2 Mole Fraction of CH4 in Initial Oil

0.8

Figure 6.27: Eﬀect on MMP of variations in initial oil composition for displacements of oils by a 50/50 mixture of N2 and CH4 . The initial oils have ﬁxed C4 /C10 ratio but varying dilutions with pure CH4 [21]. insensitive again at high CH4 concentrations. Which of those sensitivities applies depends entirely on the geometry of the key tie lines with respect to the critical locus. The situation is similar if the CH4 is added to an oil with ﬁxed ratio of C4 and C10 , a more realistic way of varying the initial oil composition, though the patterns observed are similar to those described for the ﬁxed crossover tie line. Fig. 6.27 shows the dependence of MMP on initial CH4 concentration for three ﬁxed C4 /C10 ratios. For CH4 concentrations below about 40 mole percent, MMP is relatively insensitive to CH4 concentration. The MMP declines rapidly for intermediate CH4 concentrations and is then relatively insensitive to CH4 concentration at high initial CH4 concentrations.

6.5.3

Eﬀect of Variations in Injection Gas Composition on MMP

Minimum miscibility pressures can also show complex dependence on the composition of the injection gas. Fig. 6.28 shows the composition routes that result from enrichment of N2 with CH4 for a ﬁxed initial oil composition, and Fig. 6.29 shows the changes in displacement performance that result. When the injection gas is pure N2 , the initial oil tie line is the closest to the critical locus, though a large pressure increase over the displacement pressure of Fig. 6.28 would be required to make the initial tie line critical. For injection gas compositions between I1 and I2 , the injection gas and crossover tie lines change as the fraction of CH4 in the injection gas increases, but the initial tie line remains ﬁxed. Hence, the MMP remains unchanged for injection gas compositions between I1 and I2 . When the injection gas composition is I2 , the displacement is similar to a ternary

CH CHAPTER 6.4 FOUR-COMPONENT DISPLACEMENTS

170

Ι4

a a

Ι3

Ι2 Ι

N2 1a

a a

a

aa

a a a aa a

a a

aa

C4

C10

Figure 6.28: Composition routes for displacement of oil a containing 10 mole percent CH4 , 20 mole percent C4 , and 70 mole percent C10 by N2 /CH4 mixtures at 4900 psia (333 atm) and 160 F (71 C) [21]. displacement because there are only two key tie lines, the injection gas tie line and the initial tie line, which still controls miscibility. For injection compositions with larger fractions of CH4 than I2 , the dependence of the MMP on injection gas composition changes signiﬁcantly. Now, the displacements are condensing/vaporizing drives, and the crossover tie line lies closer to the critical locus. Furthermore, the distance from the crossover tie line to the critical locus decreases sharply as the fraction of CH4 in the injection gas rises, which means that the pressure increment required to reach the MMP (over the pressure of the phase diagram in Fig. 6.28 declines signiﬁcantly. The decrease in the size of the “horizontal” surfaces of tie lines with increasing CH4 fraction in Fig. 6.28 is another indication of this eﬀect. As the size of that surface shrinks, smaller pressure increases are required to reduce the size enough to make the crossover tie line a critical tie line. The saturation proﬁles in Fig. 6.29 also reﬂect this behavior. As the enrichment of the injection gas in CH4 is increased, the displacement eﬃciency improves. The leading shocks slow down, the trailing shocks speed up, and the amount of undisplaced liquid phase at the upstream end of the transition zone decreases. When the injection gas composition is I5 , pure CH4 , the displacement is multicontact miscible because the initial oil composition a lies on the critical tie line extension in the CH4 /C4 /C10 face at the displacement pressure (4900 psia, 333 atm). Thus, in the limit as the CH4 fraction approaches one, the crossover tie line approaches the critical tie line in the CH4 /C4 /C10 face. Fig. 6.29 shows that the result is piston-like displacement. Fig. 6.30 summarizes the eﬀect of variations of injection gas composition on MMP for the N2 /CH4 /C4 /C10 system. Results for three diﬀerent ratios of C4 to C10 in the initial oil are shown with all oils containing 10 mole percent CH4 . The middle ratio (0.29) corresponds to the composition routes and saturation proﬁles of Figs. 6.28 and 6.29. For all three C4 /C10 ratios, the MMP is unchanged for additions of small amounts of CH4 to pure N2 but drops signiﬁcantly when the tie

6.5. DEVELOPMENT OF MISCIBILITY

171

1.0

Gas Saturation

0.8

0.6

0.4 Ι1 Ι2 Ι3 Ι4

0.2

0.0 0.0

0.4

0.8

1.2

1.6

xD/tD

Figure 6.29: Saturation proﬁles for the displacements of Fig. 6.28 [21].

30000

Pressure, psia

25000 20000 15000

Initial Tie Line

Oil: C4/C10=0.10 Oil: C4/C10=0.29 Oil: C4/C10=0.80 Crossover Tie Line

10000 5000 0 0.0

0.4 0.6 0.8 1.0 0.2 Mole Fraction of CH4 in Injection Gas

Figure 6.30: Eﬀect on MMP of variations in injection gas composition for displacements of oils of ﬁxed composition. The initial oils have ﬁxed CH4 mole fraction (10 percent) but varying ratios of C4 to C10 [21].

172

CHAPTER 6. FOUR-COMPONENT DISPLACEMENTS

line that controls miscibility changes from the initial oil tie line to the crossover tie line. Enriching the initial mixture in C4 also reduces MMPs substantially because the initial and crossover tie lines move closer to the critical locus at any pressure.

6.6

Volume Change

The analyses of Sections 4.4 and 5.7 showed that local ﬂow velocity (which diﬀers from propagation velocity of a composition) varies along the displacement length if components change volume as they transfer between phases or if the partial molar volume of a component is a function of the composition of a mixture. However, the local ﬂow velocity remains ﬁxed for a composition variation along a tie line because the compositions of the phases (see Section 4.4), and therefore their densities, remain ﬁxed (see Section 4.4). Thus, local ﬂow velocity changes only across shocks that enter or leave the two-phase region, across shocks that connect two tie lines, and along rarefactions between tie lines. In view of the additional complexity of displacements with volume change, it is remarkable that the same key tie lines are part of the solution composition route, whether or not volume change plays a role. To see why that statement is true, consider the eigenvalue problem stated in Eq. 6.1.13. When the ﬂow velocity varies in space, the eigenvalue problem can be written as {[H(u)] − λ[G(u)]} u = 0,

(6.6.1)

where ⎡ ∂H 1 ∂z1 ⎢ ∂H ⎢ ∂z 2 1 H(u) = ⎢ ⎢ ∂H3 ⎣ ∂z1 ∂H4 ∂z1

⎡ ⎢ ⎢ [G(u)] = ⎢ ⎢ ⎣

∂G1 ∂z1 ∂G2 ∂z1 ∂G3 ∂z1 ∂G4 ∂z1

∂H1 ∂z2 ∂H2 ∂z2 ∂H3 ∂z2 ∂H4 ∂z2 ∂G1 ∂z2 ∂G2 ∂z2 ∂G3 ∂z2 ∂G4 ∂z2

∂H1 ∂z3 ∂H2 ∂z3 ∂H3 ∂z3 ∂H3 ∂z1

∂H1 ∂vD ∂H2 ∂vD ∂H3 ∂vD ∂H4 ∂vD

∂G1 ∂z3 ∂G2 ∂z3 ∂G3 ∂z3 ∂G4 ∂z3

0 0 0 0

⎤ ⎥ ⎥ ⎥, ⎥ ⎦

(6.6.2)

⎤ ⎥ ⎥ ⎥, ⎥ ⎦

(6.6.3)

and uT = [dz1 , dz2 , dz3 , vD ].

(6.6.4)

The deﬁnitions of G and H are given by Eqs. 4.4.3 and 4.4.4. The local ﬂow velocity, vD , can be eliminated from the ﬁrst three rows of Eq. 6.6.1 by multiplying each row in Eq. 6.6.1 by αi /vD and then subtracting the last row from each of the ﬁrst three rows (see Dindoruk [19, pp. 289-292] for details of this derivation that decouples the local ﬂow velocity from composition variations). In fact, the same manipulations apply no matter how many components are present. Recall that αi = Hi/vD and note that 1 ∂Hnc = 1. αnc ∂vD

6.6. VOLUME CHANGE

173

The result is a transformed eigenvalue problem of the form,

[E(z)] T V

0

1 vD

e dvD

=0

(6.6.5)

where the entries in the submatrix of order nc − 1 are ∂αi αi ∂αnc [E(z)] = − − λ∗ ∂zj αnc ∂zj

∂Gi αi ∂Gnc − ∂zj αnc ∂zj

, i = 1, nc − 1, j = 1, nc − 1,

(6.6.6)

, are given by and the entries in the vector, V Vj =

1 ∂αi λ∗ ∂Gi − , αi ∂zj αi ∂zj

i = nc ,

j = 1, nc.

(6.6.7)

The manipulations that led to Eq. 6.6.5 do not alter the eigenvalues and eigenvectors that are the solution to Eq. 6.6.1. The zeroes in the last column of the ﬁrst three rows of the matrix in Eq. 6.6.5 mean that the eigenvalue problem for the composition path can be solved independently of the problem for the ﬂow velocity for the scaled eigenvalues, λ∗ and the eigenvectors, eT = [dz1 , dz2 , dz3 ]. The ﬂow velocity can then be found from the eigenvectors as the solution to 1 dvD , V T e = − vD ds

(6.6.8)

where s is arc length along the nontie-line path in the direction e. If stepwise integrations along a nontie-line path are used, the approximate solution of Eq. 6.6.8 is

o T e , exp −sV vD = vD

(6.6.9)

o is the ﬂow velocity from the previous time step, and V T e is taken to be constant for small where vD steps, s. The argument given in Section 5.7, which applies equally well to systems with any number of components, indicates that ﬂow velocities can also be decoupled from the compositional part of the problem when tie lines are connected by shocks. That fact combined with the requirements that the initial and trailing shocks take place along tie line extensions and shocks between tie lines must connect tie lines with intersecting extensions means that the key tie lines for a gas oil displacement depend only on the initial oil and injection gas compositions and the equilibrium phase behavior at the given temperature and pressure, whether or not volume change is considered. Changes in ﬂow velocity induced by volume change are most apparent when the pressure is high enough that some component in the injection gas dissolves in appreciable amounts in the oil and low enough that the partial molar volumes of the component diﬀer signiﬁcantly between the vapor and liquid phases. Figs. 6.31 and 6.32 are an example of such a system. They report the solution composition route and eﬄuent composition proﬁles for a displacement of a CH4 /C4 /C10 mixture by pure CO2 . That system was studied experimentally by Heller et al. [33], who performed slim tube displacements at various pressures. Fig. 6.31 compares measured eﬄuent compositions with the calculated route. A comparison of the calculated and experimental eﬄuent compositions is given in Fig. 6.32.

174

CHAPTER 6. FOUR-COMPONENT DISPLACEMENTS CO2 f

e

ah

ANALYTICAL - VC EXPERIMENT

h

a h haad c

h

CH4

ab hh

← crossover

tie-line extension

C4 aha h

C10

Figure 6.31: Composition route for displacement of oil a containing 11.92 mole percent CH4 , 37.98 mole percent C4 , and 50.10 mole percent C10 by pure CO2 at 1407 psia (95.7 atm and 160 F (71 C). Eﬀects of volume change are included in the calculated route and compositions ([19, p. 250] used with permission.) 1

a a

COMPONENT MOLE FRACTIONS

0

a

a

volume change no volume change slim-tube data

a

a

a a a a

a

a

CO2

a

1 CH4

a

0

a

a

a

a a

a aa

a

a

a

1 C4

a

a

a

a

a a a a a

0

a

a

a

1 C10

a

a

a

0 0.0

0.4

a

a a

a aa

a

0.8 1.2 PORE VOLUMES INJECTED

a

a 1.6

2.0

Figure 6.32: Eﬄuent composition (mole fraction) proﬁles for the displacements of Fig. 6.31 ([19, p. 252] used with permission).

6.6. VOLUME CHANGE

175

The experimental compositions shown in Fig. 6.31 clearly follow the key tie lines predicted by the theory. Several measured points fall on the initial oil tie line, and the crossover tie line is also identiﬁed well. Thus, the measured compositions provide experimental conﬁrmation of the existence of the crossover tie line. The proﬁles in Fig. 6.32 also conﬁrm the main predictions of the theory, and they show the eﬀects of volume change on the ﬂow behavior, although there is also evidence of smearing of the eﬄuent compositions in the experimental results that is not present in the calculated compositions. The experimental results clearly show a leading CH4 bank that propagates ahead of the ﬂuids containing CO2 . That bank arrives at the outlet after one pore volume of pure CO2 injection, which can only happen if volume is lost as some of the CO2 dissolves in undisplaced oil in the transition zone. Fig. 6.32 also compares solutions calculated with and without volume change. The leading CH4 bank moves much more rapidly, reaching the outlet well before one pore volume injected when eﬀects of volume change are not included. Once the leading CH4 bank has left the system, the eﬄuent compositions lie on the CO2 /C4 /C10 face, as the theory predicts. An indication of the composition change from the initial tie line to the crossover tie line can also be seen in the eﬄuent proﬁles for C4 and C10 . Measurements were not continued long enough to observe the arrival of the C4 evaporation shock, though the measured C4 concentrations late in the displacement were consistent with the calculated values. While the measured compositions agree in some ways with the predictions of the theory, there are also some signiﬁcant diﬀerences. The experimental results indicate that some physical mechanisms not modeled in the theory were at work. There are several possible explanations for the diﬀerences. The most likely explanation is the existence of nonuniform ﬂow in the inlet region of the slim tube. When low viscosity CO2 is injected to displace oil with higher viscosity, the displacement is hydrodynamically unstable, and viscous ﬁngering can produce nonuniform ﬂow. In addition, some gravity segregation of the lighter CO2 in the coiled spiral tube is also possible. However, transverse dispersion and diﬀusion in the small-diameter tube will eventually create a transition zone that stabilizes the displacement. Estimates of the length required to establish such a transition zone indicate that nonuniform ﬂow is possible for about the ﬁrst 5-10 percent of the displacement length [88, 132]. Thus, at least a portion of the experimental displacement probably did not satisfy the assumption of one-dimensional ﬂow. Nonuniform ﬂow could easily produce the smearing of the shocks observed in Fig. 6.32. Some spreading could also be due to diﬀusion and dispersion and capillary pressure gradients, though those eﬀects are likely to be smaller than the eﬀects of nonuniform ﬂow. Other sources of disagreement in the timing of arrival of fronts at the outlet could be sparse sampling, inaccurate relative permeability functions, phase viscosities and densities, or phase equilibrium calculations. Finally, there is some pressure drop over the length of the slim tube (typically 10-50 psia or 0.7-3.4 atm), which is inconsistent with the evaluation of phase behavior at a constant pressure, and the assumption of local chemical equilibrium could be inaccurate. Given the possible sources of disagreement and the complexity of the physical mechanisms in play, it is perhaps surprising that the agreement between experiment and theory is as good as it is. That agreement indicates that the one-dimensional, dispersion-free theory provides a reasonable picture of the interactions of phase equilibrium and ﬂow that creates the potential for very eﬃcient gas displacement.

176

6.7

CHAPTER 6. FOUR-COMPONENT DISPLACEMENTS

Summary

Analysis of displacements in which four components are present reveals the following important ideas that are the foundation for understanding what happens in systems with an arbitrary number of components: • In four-component displacements, three key tie lines play important roles in the solution composition route: the tie lines that extend throught the initial oil and injection gas compositions (the key tie lines from ternary displacements), and a new tie line, the crossover tie line. • The three key tie lines are the same whether or not eﬀects of volume change are included in the calculation. • The solution composition route for given injection gas and initial oil compositions includes shocks in and out of the two-phase region along the extensions of the initial oil and injection gas tie lines, shocks or rarefactions that connect the initial oil tie line to the crossover tie line and the crossover tie line to the injection gas tie line. A rarefaction will always be present on the shortest of the three tie lines. Rarefactions on the other tie lines occur when they are permitted by the velocity constraint, but are frequently not present in the solution because they would violate the velocity rule. • The composition variations that connect the three key tie lines generate ruled surfaces of tie lines. Shock surfaces consist of tie lines that intersect a given generating tie line when they are extended. Rarefaction surfaces are the set of tie lines that intersect a nontie-line path. When K-values are independent of composition, these surfaces are planes, and the rarefaction and shock surfaces coincide. When K-values depend on composition, the shock and rarefaction surfaces diﬀer. • A condensing/vaporizing gas drive is a displacement in which there is a leading condensing segment connected to a trailing vaporizing segment by the crossover tie line. • A four-component displacement displacement is multicontact miscible if any of the three key tie lines is a critical tie line. The MMP or MME can be found as the pressure or enrichment at which one of the key tie lines has zero length. • In a condensing/vaporizing gas drive, it is the crossover tie line that becomes a critical tie line at the lowest pressure or enrichment.

6.8

Additional Reading

The ﬁrst solution for a four-component gas displacement was reported by Monroe et al. [82], who demonstrated the existence of the crossover tie line. The four-component solution for constant K-values was obtained by Wang [128], the demonstration that for constant K-values shock and rarefaction surfaces are planes was given by Johansen et al. [53]. Bedrikovetsky and Chumak [7] reported a solution that made use of graphical constructions for a system with constant K-values. Detailed investigations of the behavior of four-component systems in the absence of volume change as components change phase, with many examples, are given by Dindoruk [19] and Johns [54]. A

6.9. EXERCISES

177

comprehensive analysis of the eﬀects of volume change in four-component systems is reported by Dindoruk [19, 20]. The discussion of condensing/vaporizing gas drives included here draws heavily from a paper by Johns et al. [56]. The term condensing/vaporizing drive was suggested by Zick [140], based on compositional simulation results for a multicomponent displacement. The theory described here provides an explanation for Zick’s observations in terms of the geometry of key tie lines. Stalkup [114] also discussed condensing/vaporizing displacement behavior and provided useful examples for relatively simple systems. The ideas involving the development of miscibility in four-component gas drives appear in papers by Orr et al. [92], Johns et al. [56], and Dindoruk et al. [21]. The discussions of the dependence of MMP on changes in initial oil and injection gas composition includes material presented ﬁrst in papers by Orr et al. [92] and Dindoruk et al. [21]. The discussion of decoupling the ﬂow velocity and the eﬀects of volume change follows that of Dindoruk [19]. A review of the four-component theory, written for a symposium held in honor of Fred Helﬀerich, is given by Orr et al. [91].

6.9

Exercises

1. For problems 1–7, the K-values are ﬁxed: K1 = 2.5, K2 = 1.5, K3 = 0.5, and K4 = 0.05. The initial oil composition is C1 = 0.2, C2 = 0, C3 = 0.1, C4 = 0.7, and the injection gas composition is C1 = 0.1, C2 = 0.9, C3 = C4 = 0. The viscosity ratio for all tie lines is 10. Use the fractional ﬂow function of Eq. 4.1.21, with Sor = Sgc = 0. Find the tie lines that extend through the initial oil and injection gas compositions. 2. Find the crossover tie line. 3. Solve for the compositions on either side of the shock from the initial oil tie line to the crossover tie line and the shock velocity. 4. Solve for the compositions on either side of the shock from the injection gas tie line to the crossover tie line and the shock velocity. 5. Determine whether rarefactions occur along the initial and injection tie lines and ﬁnd the velocities of the leading and trailing shocks. 6. Plot the solution composition route on a quaternary diagram, identifying the key points in the solution. 7. Plot saturation and composition proﬁles, identifying the same key points shown in the quaternary plot of composition route. 8. For problems 8–13, the K-values are ﬁxed: K1 = 2.5, K2 = 1.5, K3 = 1.05, and K4 = 0.05. The initial oil composition is C1 = 0.1, C2 = 0, C3 = 0.4, C4 = 0.5, and the injection gas composition is C1 = 0, C2 = 0.9, C3 = 0.1, C4 = 0. The viscosity ratio for all tie lines is 10. Use the fractional ﬂow function of Eq. 4.1.21, with Sor = Sgc = 0. 9. Find the initial, injection, and crossover tie lines.

178

CHAPTER 6. FOUR-COMPONENT DISPLACEMENTS

10. Determine whether rarefactions connect the crossover tie to the injection gas and initial oil tie lines. For each rarefaction, determine enough composition and velocity points along the rarefaction to give smooth curves on route and proﬁle plots and determine the landing and departure points on the key tie lines. For each shock, determine the compositions on either side of the shock and the shock velocity. 11. Determine the compositions and velocities of the leading and trailing shocks. 12. Plot the solution composition route on a quaternary diagram, identifying the key points in the solution. 13. Plot saturation and composition proﬁles, identifying the same key points shown in the quaternary plot of composition route. 14. Show that a displacement with a composition route like that shown in Fig. 6.9 is multicontact miscible if the initial tie line is a critical tie line.

Chapter 7

Multicomponent Gas/Oil Displacements by F. M. Orr, Jr. and K. Jessen

In displacements of crude oils by gas mixtures, many more than the three or four components will be present. Any crude oil contains hundreds, if not thousands, of components, and injection gases often contain more than four. Thus, a comprehensive theory of compositional eﬀects that inﬂuence gas injection processes must deal with multicomponent systems. It is unlikely, of course, that displacement calculations will be performed with hundreds of components, nor is there a need to do so. There is now considerable computational experience that indicates that phase behavior can be calculated for most gas/oil systems with acceptable accuracy with somewhere between half a dozen and ﬁfteen components. The theory presented so far here suggests that lumping schemes that group together components that have similar K-values should capture how those components propagate with reasonable accuracy. Thus, the next step is to construct solutions for sytems with a modest but arbitrary number of components. That is the subject of this chapter. No matter how many components are present, the elements that are combined to construct the solution are all generalizations of the key results of the three- and four-component theory: any solution must have a unique composition route that connects the injection gas composition to the initial oil composition, any rarefactions present must lie on composition paths, and shocks must be constructed if rarefactions would violate the velocity constraint. In a displacement with nc components, tie lines are still paths, and there are now nc - 2 nontie-line paths through each composition point in the two-phase region. The most important new feature of multicomponent displacements is the number of key tie lines. In a two-component displacement, only one tie line plays a role in the solution. In three-component ﬂows, two tie lines control displacement behavior, the initial oil tie line and the injection gas tie line. In four-component displacements, there are three key tie lines. The initial oil and injection gas tie lines are still important, and there is also a crossover tie line. In a multicomponent displacement, there are nc − 3 crossover tie lines in addition to the initial oil and injection gas tie lines. The ﬁrst step in constructing a multicomponent solution is to identify the key tie lines. That topic is considered in Section 7.1. In Section 7.2, a detailed procedure for construction solutions is presented, as are some example solutions. In Section 7.3 we revisit the question of minimum miscibility pressure for multicomponent systems and consider the accuracy of other computational methods used to estimate conditions required for miscibility. 179

180

7.1

CHAPTER 7. MULTICOMPONENT GAS/OIL DISPLACEMENTS

Key Tie Lines

The derivation of the eigenvalue problem for an nc -component follows the procedure used for a four-component system. There are nc - 1 eigenvectors at every point in the two-phase region: one is the tie-line eigenvector, and the remainder are nontie-line eigenvectors. Just as in four-component systems, there are no composition paths (integral curves of the nontie-line eigenvectors) that connect the initial oil composition directly with the injection gas composition. Instead, a sequence of composition variations traverses a set of crossover tie lines. To ﬁnd the solution composition route, the crossover tie lines must be identiﬁed. If the displacement is fully self-sharpening (all the key tie lines are connected by shocks, and rarefactions occur only along tie lines), then the key tie lines are determined completely by the fact that the extensions of the crossover tie lines must intersect. The problem of ﬁnding the crossover tie lines reduces, therefore, to the problem of ﬁnding a set of intersecting tie lines. To see how that can be done, we consider ﬁrst a special case: an injection gas that is contains only a single component.

7.1.1

Injection of a Pure Component

Consider a multicomponent system in which all the components are present in the initial oil, but the injection gas contains only a single component. We consider ﬁrst systems without volume change (though the key results also apply when eﬀects of volume change are included). If all the components are present in the oil, then the phase compositions on the initial tie line (obtained by a negative ﬂash [135, 129]), also have nonzero fractions of all the components. On the injection tie line, however, there are only two components, the injection component and the heaviest (least volatile) of the components in the oil, and there are nc - 2 shocks that connect crossover tie lines to each other and the initial and injection tie lines. We show now that one component disappears across each of the shocks between tie lines. Assume, for the moment, that there are two intersecting tie lines that have nonzero fractions of an identical number of components. If that is true, then there must be nc - 3 remaining shocks across which nc - 2 components disappear, which means that two-components must disappear across one of the shocks. If two-components have zero volume fraction on a tie line on one side of the shock, then the intersection point for the two tie lines connected by the shock must also have zero fractions of those two components. However both of those components are present on the other tie line, so the extension of that tie line to the intersection point gives zero fractions for both components that disappear. Suppose that we extend the nth tie line to a point where the fractions of components j and k are zero. If so, then the tie line volume balances require that

Cj = xnj 1 + (Kjn − 1)S1n = 0,

(7.1.1)

Ck = xnk {1 + (Kkn − 1)S1n} = 0.

(7.1.2)

and

Eqs. 7.1.1 and 7.1.2 can each be solved for the saturation on the nth tie line that gives zero fractions of components j and k. The result is

7.1. KEY TIE LINES

181

S1n =

1 , 1 − Kjn

S1n =

1 . 1 − Kkn

(7.1.3)

The two expressions for S1n are inconsistent unless Kjn = Kkn , a situation that is never true (except at the critical point) as long as K-values stay ordered, a reasonable assumption for gas/oil systems with a modest number of discrete components [57] (components that are so similar that their Kvalues are close enough to become equal as the overall composition of a mixture is changed could quite reasonably be lumped together for calculation purposes). Thus, the assumption that two components disappear across a shock leads to a contradiction, and it must be true, therefore, that for an injection gas containing a single component, one component disappears across each of the shocks that connect the initial, crossover and injection tie lines. The disappearance of components is in the order of their K-values. More volatile components, with larger K-values, propagate more rapidly because they partition preferentially into the faster moving vapor phase. Low volatility components move more slowly. This behavior is consistent with chromatography, in which more volatile components elute rapidly while those that partition more strongly into the stationary phase do so more slowly. In gas/oil displacements, the liquid phase plays the role of the stationary phase in chromatography, though of course, the liquid phase is not stationary. The nonlinear relationship between the rate at which the liquid phase moves and its saturation is what distinquishes the theory of gas/oil displacements, in which pairs of eigenvalues are sometimes equal, from chromatography theory in which the eigenvalues are distinct. The fact that one component disappears across each shock simpliﬁes the problem of locating the sequence of key tie lines because in the special case of injection of a pure component, the tie lines can be found sequentially. Consider the following example of a system with nc components. Suppose that the K-values of the components are ordered (K1 > K2 > ... > Knc ) and remain similarly ordered over all the compositions in the two-phase region. Suppose further that the component with the kth K-value is being injected. The initial oil and injection gas tie lines can be identifed by performing a negative ﬂash. The ﬁrst crossover tie line is determined from the shock balances for that shock, and it is here that the requirement that tie lines connected by a shock must intersect is so useful. The intersection point for the shock from the initial tie line to the ﬁrst crossover tie line can be found easily by noting that component 1, the component with the highest K-value must disappear across that shock. If so, the intersection point for the two tie lines connected by the shock must also have zero fraction of component 1. The saturation on the initial tie line for that point is obtained by solving an equation like Eq. 7.1.1 for the saturation on tie line 1 at the intersection point: S11 =

1 , 1 − K11

(7.1.4)

where S11 refers to the saturation measured on tie line 1, the initial tie line, and K11 is the K-value of component 1 on that tie line. The volume fractions of the remaining components at the intersection point are given by

CiX2

=

x1i

Ki1 − K11 1 − K11

,

i = 2, nc .

(7.1.5)

In Eq. 7.1.5, CiX2 refers to the composition at the intersection point of the second key tie line (the

182

CHAPTER 7. MULTICOMPONENT GAS/OIL DISPLACEMENTS

ﬁrst crossover tie line). The ﬁrst crossover tie line can be found then as the tie line that extends through that intersection point, again by a negative ﬂash [135, 129]. The same calculation can then be performed to ﬁnd each of the remaining crossover tie lines. For components numbered j < k, the saturation of the zero fraction at the intersection point on the jth tie line is 1

S1j−1 =

j−1 1 − Kj−1

,

(7.1.6)

and the compositions for the other components are Xj

Ci

= 0,

i = 1, j,

and ⎧ ⎫ ⎨ K j−1 − K j−1 ⎬ X i j−1 Ci j = xj−1 i ⎩ 1 − K j−1 ⎭

i = j + 1, nc.

(7.1.7)

j−1

The kth component is present on all the tie lines, so for the kth tie line component k + 1 disappears, and the equivalent expressions are S1k = Xk+1

Ci

= 0,

1 , 1 − Kkk+1

(7.1.8)

i = 1, k − 1, i = k + 1,

and X Ci k+1

= xki

k Kik − Kk+1 k 1 − Kk+1

i = k, i = k + 2, nc .

(7.1.9)

Finally, for tie lines after the kth tie line, S1j−1 = Xj

Ci

= 0,

1 1 − Kjj−1

,

(7.1.10)

i = 1, k − 1 i = k + 1, j,

and X Ci j

= xj−1 i

⎧ ⎫ ⎨ K j−1 − K j−1 ⎬ i

⎩

j

1 − Kjj−1

⎭

i = k i = j + 1, nc .

(7.1.11)

Because each of the tie lines is found sequentially, the equilibrium K-values for each tie line are calculated as part of the iterative procedure of the negative ﬂash. Thus, for displacements in which the injection gas contains only a single component, the nc − 1 key tie lines can be found in a straightforward way.

7.1. KEY TIE LINES

7.1.2

183

Multicomponent Injection Gas

When the injection gas contains more than one component, the full set of tie-line intersection equations must be solved simultaneously for the key tie lines. Tie-line intersections can be speciﬁed in volume-fraction space or in mole-fraction space. In systems in which an equation of state is used to calculate phase equilibrium, it is more convenient to work in mole fraction space because no conversion to volume fractions is required and because the key tie lines to be found will be used to estimate minimum miscibility pressure. The key tie lines in mole fraction space are the same whether or not volume change is considered in constructing solutions for ﬂow problems. Those equations have the form

1 + Vnn+1 Kin+1 − 1 xni {1 + Vnn (Kin − 1)} = xn+1 i

, n = 1, . . . , nc − 2, i = 1, . . . , nc − 1. (7.1.12) n n In Eq. 7.1.12, xj is the liquid phase mole fraction of component j on tie line n, and Kj is the equilibrium K-value for component j on that tie line. Vnn is the vapor phase mole fraction measured on tieline n at the point of intersection between tie line n and tie line n + 1, and Vnn+1 is the vapor phase mole fraction of the intersection point measured on tie line n + 1. In addition, the initial oil and injection gas tie lines satisfy

1 + V oil Kioil − 1 zioil = xoil j

, i = 1, . . ., nc − 1,

(7.1.13)

and {1 + V gas (Kigas − 1)} , i = 1, . . ., nc − 1. zigas = xgas i

(7.1.14)

Eqs. 7.1.12-7.1.14 look simple, but they are actually quite nonlinear (because the K-values depend on composition through the equation of state), and there are a lot of them (because there is one equation for each component at each of the intersection points). Thus, solving them requires some care. Wang [128] solved Eqs. 7.1.12 by guessing a set of K-values, solving the nonlinear tie-line intersection equations by a Newton-Raphson iteration, then updating the K-values and repeating the process to convergence. Jessen and coworkers [42, 46] developed an iterative approach for solving the tie-line intersection equations that was substantially more eﬃcient. They noted that in a multicomponent system, the tie-line intersection equations are really a statement that two tie lines lie in the same plane, and hence they could write an equivalent set of equations in terms of another intersection point between lines that connect the liquid phase composition on one tie line to the vapor phase composition on the other. Fig. 7.1 illustrates the idea for a ternary system. Point I1 is the actual intersection point of the tie line extensions. The new intersection point is I2 . The formulation of the intersection equations in terms of I2 is helpful because the composition of that intersection point stays bounded, while the vapor-phase mole fractions in Eqs. 7.1.12 may not. For tie lines that are nearly parallel, for example, the values of Vnn and Vnn+1 will be very large positive or negative numbers, and they may pass through negative or positive inﬁnity as tie-lines shift orientation with changes in pressure. Use of the new intersection point eliminates the numerical diﬃculties that can arise from those variations in phase mole fractions. The equivalents of Eqs. 7.1.12 written for the tie lines that deﬁne the intersection point I2 in Fig. 7.1 are

184

CHAPTER 7. MULTICOMPONENT GAS/OIL DISPLACEMENTS

(1 − αn ) = yin+1 β n + xni (1 − β n ), yin αn + xn+1 i

n = 1, . . ., nc − 2, i = 1, . . . , nc − 1, (7.1.15)

where α and β are parameters that set the location along the intersecting lines. αn is the fractional distance from the liquid composition on tie line n + 1 to the vapor phase composition on tie line n, and β n is the fractional distance from the liquid composition on tie line n to the vapor composition on tie line n + 1. In terms of equilibrium K-values, Eqs. 7.1.15 become

Kin+1 β n + αn − 1 , xni {Kin αn + β n − 1} = xn+1 i

n = 1, . . ., nc − 2, j = 1, . . . , nc − 1. (7.1.16) Wang [128, 131] solved Eq. 7.1.12 for a given set of K-values for (xni , Vnn ) and updated the K-values based on successive substition, Kinew = Kiold

liq fˆi , fˆivap

i = 1, nc ,

(7.1.17)

where fˆiliq and fˆivap are the partial fugacities of component i in the liquid and vapor. That procedure converges slowly as the K-values approach one, as they must near a critical point when one of the key tie lines has zero length. Jessen et al. [45] solved Eqs. 7.1.16 for yin , xni , αn and β n subject to the equilibrium constraint n ˆvap + ln y n , ln φˆliq i i + ln xi = ln φi

i = 1, nc ,

(7.1.18)

and the fact that the phase compositions for each phase must sum to one, 1−

nc i=1

xni

= 1−

nc

yin ,

n = 1, nc − 1.

(7.1.19)

i=1

liq vap In Eq. 7.1.18, φˆi and ln φˆi are the fugacity coeﬃcients for the liquid and vapor. Jessen et al. [45] used a Newton-Raphson iteration to obtain the solution of the set of tie-line intersection equations. A solver was designed explicitly to take advantage of the sparse structure of the Jacobian matrix, which when combined with the quadratic convergence of the Newton iteration produced a much more eﬃcient solution procedure. Once the phase compositions for the key tie lines have been found, the actual intersection points and vapor fractions for each of the key tie lines can be calculated without diﬃculty from the tie line equations. Jessen et al. divided the solution process at a particular pressure into two parts. In the ﬁrst, the sequential procedure outlined in Section 7.1.1 is used to ﬁnd the key tie lines for injection of a pure component (typically the component with the largest mole fraction in the injection gas). In the second part, the injection gas is varied incrementally by mixing fractions of the actual injection gas with the pure component as a way to generate accurate initial guesses of tie-line compositions and K-values for the next increment. If the changes in composition of the injection gas are small enough, this procedure will converge to the key tie lines for the actual injection gas mixture. Knowledge of the key tie lines can be put to good use by using them to construct the solution for a particular displacement and by using them to determine the minimum miscibility pressure or the minimum enrichment for miscibility. The next two sections describe those uses of the key tie lines.

7.2. SOLUTION CONSTRUCTION

185

CH4

(yn1,yn2)

a

n+1 a (yn+1 1 ,y2 )

a

(xn1,xn2)

a

I2

n+1 a (xn+1 1 ,x2 )

C10

CO2

a

I1

Figure 7.1: Alternate formulation of the tie line intersection point [42].

7.2

Solution Construction

Once the key tie lines are determined, the solution for a given set of injection and initial compositions can be found by calculating compositions of the leading and trailing shocks, the points at which solution segments begin and end on the key tie lines, and any rarefactions that occur along tie lines. We begin the analysis by demonstrating that solution construction must begin on the shortest tie line, whether the tie lines are connected by a shock or a rarefaction. The next step is to determine when rarefactions connect pairs of key tie lines. Once those questions are settled, the procedures for constructing solutions with and without nontie-line rarefactions can be outlined. To see why solution construction must begin on the shortest tie line, consider ﬁrst a system in which each pair of tie lines is connected by a rarefaction. Fig. 7.2a shows an example. The arguments given in Section 5.1.4 establish that a switch to a nontie-line path must occur at an equal-eigenvalue point, no matter how many components are present. While there are nc - 2 equal eigenvalue points along any tie line, only two lie on nontie-line paths that connect to other tie line of the pair, and one of those can be excluded from further consideration because the solution route always lies on one side of the equivelocity surface, and the two equal-eigenvalue points lie on opposite sides of the intersection of the equivelocity surface with the tie line in question. Consider now the equal eigenvalue points that are candidates for switch points on the pair of tie lines to be connected by the rarefaction. As Fig. 7.2a shows, the nontie-line path that passes through the equal eigenvalue point on the longer of the two tie lines, does not reach the shorter of the two tie lines, but the path through the equal-eigenvalue point on the shorter tie line does extend to the

186

CHAPTER 7. MULTICOMPONENT GAS/OIL DISPLACEMENTS ba a a

aa

Sg

ab

ξ a. Nontie-line rarefaction

ad

Sg

ac

da a c

ξ b. Indifferent nontie-line variation

af

Sg

ae

fa a e

ξ c. Nontie-line shock

Figure 7.2: Nontie-line paths between pairs of tie lines.

longer tie line. Hence, the equal-eigenvalue point for the rarefaction that connects this pair of tie lines must lie on the shorter of the two tie lines, point a in Fig. 7.2a. The resulting saturation proﬁle is also sketched in Fig. 7.2a. Application of similar reasoning to any of the nc - 2 pairs of tie lines indicates that whenever there is a rarefaction between tie lines, the equal eigenvalue point for the nontie-line path must be on the shorter of the two tie lines. That point can be found once the tie line is identiﬁed, but the arrival point of the nontie-line path on the longer tie line must be found by integration of the nontie-line eigenvector. While it would be possible to integrate from a point on the longer tie line along a nontie-line path toward the shorter tie line, identifying the correct tie line that is tangent to the shorter tie line at the equal-eigenvalue point would require a trial-and-error procedure. Starting at the equal-eigenvalue point on the shorter tie line, however, allows direct calculation of the arrival point on the longer tie line. The arrival points on each of the key tie lines can be found sequentially

7.2. SOLUTION CONSTRUCTION

187

if the construction starts on the shortest tie line. A similar argument applies when a nontie-line shock connects two tie lines. Consider what happens if we change the orientation of the tie lines in the pair. If the solution to any displacement problem is correct, it must vary continuously with respect to changes in the problem data, and hence changing tie line slopes slightly must change the resulting solution only slightly. Suppose now that the tie lines shown in Fig. 7.2a maintain the orientation shown but are almost parallel. That system would show behavior qualitatively just like that shown in Fig. 7.2a. Changing the slope slightly to make the tie lines parallel would mean that the two tie lines would intersect at a point at inﬁnity, and all the tie lines on the nontie-line path would also intersect at the same point. Wave speeds on that nontie-line path would all be the same (see Section 5.2.2). Thus, the composition variation on the nontie-line path would be an indiﬀerent wave, one that is neither self-sharpening nor spreading, and the wave velocity would be identical to the wave velocity at the equal-eigenvalue point. Because all of the compositions on the nontie-line path have the same wave velocity, and therefore the same spatial location, the indiﬀerent wave has the same appearance as a shock. Again, however, the arrival point at the longer tie line is determined directly if the construction begins at the equal-eigenvalue point on the shorter tie line. Next we consider an additional small change in the orientation of the tie lines to create the situation shown in Fig. 7.2c. Now the nontie-line path is self-sharpening, and a shock replaces the rarefaction along the nontie-line path. But if this wave pattern is to merge continuously with that of the parallel example in Fig. 7.2b, then the wave speed of the shock must match the tie line eigenvalue on the shorter of the two tie lines. Now, the jump point on the shorter tie line is determined as a semishock, and the resulting landing point on the longer tie line can be determined directly. Repeated application of this reasoning to pairs of tie lines indicates that even when the key tie lines are all connected by shocks, the construction of those shocks must begin on the shortest of the key tie lines. We have shown, therefore, that whatever combination of rarefactions and shocks makes up the solution to a particular problem, solution construction is most straightforward if it begins on the shortest of the key tie lines. In many problems that tie line will be one of the crossover tie lines, so the solution is constructed by working downstream from that tie line through a set of crossover tie lines to reach the initial tie line and by working upstream through another set of crossover tie lines to reach the injection tie line. Once the shortest tie line is identiﬁed, the next question is: does a shock or a rarefaction connect this tie line to the next tie line upstream or downstream? That question can be answered by determining how the nontie-line eigenvalue changes as a nontie-line path is traced, just as we did in Section 5.2.2 for ternary systems. We start by obtaining a general expression for one of the nontie-line eigenvalues in a multicomponent displacement in terms of another envelope curve (see Section 5.1). For simplicity, we consider systems without volume change as components transfer between phases. For a similar derivation that includes eﬀects of volume change, see Dindoruk [19, pp. 55-63]. Consider the conservation equations for two of the nc components. Any two equations will do, so we choose the equations for components 1 and 2. They are simply Eqs. 6.1.1 and 6.1.2:

∂F1 ∂C1 + ∂τ ∂ξ

= 0,

(7.2.1)

188

CHAPTER 7. MULTICOMPONENT GAS/OIL DISPLACEMENTS ∂F2 ∂C2 + ∂τ ∂ξ

= 0,

(7.2.2)

Substitution of the deﬁnitions of Ci and Fi into Eqs. 7.2.1 and 7.2.2, and then multiplication of the ﬁrst equation by (c21 − c22 ) and the second by (c11 − c12 ) followed by subtraction of the second equation from the ﬁrst gives the expression,

∂ ∂ (c11 − c12 ) − (c11 − c12 ) ∂τ (c21 − c22 ) + (c21 − c22 ) ∂c∂τ12 − (c11 − c12 ) ∂c∂τ22 S1 (c21 − c22 ) ∂τ

∂ ∂ +f1 (c21 − c22 ) ∂ξ (c11 − c12 ) − (c11 − c12 ) ∂ξ (c21 − c22 ) + (c21 − c22 ) ∂c∂ξ12 − (c11 − c12 ) ∂c∂ξ22 = 0. (7.2.3) We are interested in what happens as the overall composition is traced along a nontie-line path. Each point on that path is intersected by a tie line, no matter how many components are present. Therefore, there must be some parameter, ψ, that determines which tie line is appropriate for a given composition point on the nontie-line path, and it must be true that ψ = ψ(ξ, τ ). Hence, the derivatives in Eq. 7.2.3 can be written in terms of ψ, which gives, after some rearrangement,

S1 +

∂c12 ∂c22 −(c11 −c12 ) ∂ψ ∂ψ ∂ ∂ (c21 −c22 ) ∂ψ (c11 −c12 )−(c11 −c12 ) ∂ψ (c21 −c22 )

(c21 −c22 )

+ f1 +

∂c12 ∂c22 −(c11 −c12 ) ∂ψ ∂ψ ∂ ∂ (c21 −c22 ) ∂ψ (c11 −c12 )−(c11 −c12 ) ∂ψ (c21 −c22 )

(c21 −c22 )

∂ψ ∂τ

(7.2.4) ∂ψ ∂ξ

= 0.

Eq. 7.2.4 can be written in a simpler form as (S1 + pk )

∂ψ ∂ψ + (f1 + pk ) = 0, ∂τ ∂ξ

(7.2.5)

where pk =

∂c22 12 (c21 − c22 ) ∂c ∂ψ − (c11 − c12 ) ∂ψ ∂ ∂ (c21 − c22 ) ∂ψ (c11 − c12 ) − (c11 − c12 ) ∂ψ (c21 − c22 )

.

(7.2.6)

The function pk applies to the kth nontie-line path along which ψ varies to set a particular tie line. Because this Riemann problem also has a self-similar solution, we can write any of the derivatives with respect to ψ in terms of η = ξ/τ , ∂ψ ∂η ∂ψ ∂η + (f1 + pk ) = 0. (7.2.7) ∂η ∂τ ∂η ∂ξ Substitution of the derivatives of η with respect to ξ and τ into Eq. 7.2.7 followed by rearrangement gives (S1 + pk )

f1 + pk ξ = λk = . (7.2.8) τ S1 + pk Eq. 7.2.8 shows that the nontie-line eigenvalue can be calculated easily if the function pk is known. Note that the form of Eq. 7.2.8 is similar in form to the expression for λnt in Eq. 5.1.24. That expression was derived for a ternary system, but Eq. 7.2.8 applies for systems with any number of components. In Eq. 5.1.24, p is related to the envelope curve for the nontie-line path. We show now that the same statement holds true for pk in the multicomponent problem.

7.2. SOLUTION CONSTRUCTION

189

In a system with nc components, the equation of a tie line can be represented as C2 − c22 Cnc − cnc 2 C1 − c12 = = ··· = . c11 − c12 c21 − c22 cnc 1 − cnc 2

(7.2.9)

An envelope curve is deﬁned as a curve tangent to a series of tie lines (the tie lines that pass through the nontie-line path, for example). Consider two tie lines that are inﬁnitesimally close to each other on the nontie-line path, with the ﬁrst described by Eq. 7.2.9. The second is described by

C1 − c12 + c11 − c12 +

∂c12 ∂ψ ∆ψ

∂c11 ∂ψ

−

∂c12 ∂ψ

= ∆ψ

C2 − c22 + c21 − c22 +

∂c21 ∂ψ

∂c22 ∂ψ ∆ψ

−

∂c22 ∂ψ

∆ψ

= ··· .

(7.2.10)

A point on the envelope curve is just the intersection of these two tie lines in the limit as ∆ψ → 0, so we can solve Eqs. 7.2.9 and 7.2.10 simultaneously for the compositions on the envelope curve, Cie . The result for component 1 is ∂c22 12 (c21 − c22 ) ∂c C1e − c12 ∂ψ − (c11 − c12 ) ∂ψ . =− ∂ ∂ c11 − c12 (c11 − c12 ) ∂ψ (c21 − c22 ) − (c21 − c22 ) ∂ψ (c11 − c12 )

(7.2.11)

The remaining compositions can be obtained from Eq. 7.2.9. Comparison of Eqs. 7.2.6 and 7.2.11 indicates that pk = −

C1e − c12 c11 − c12

(7.2.12)

Thus, pk is clearly related to the envelope curve /indexenvelope curve associated with the kth nontieline path. The left side of Eq. 7.2.11 is equally clearly a saturation (see Eq. 4.1.5, for example). Thus, pk is the negative value of the saturation at which the extension of a tie line is tangent to the envelope curve. Eq. 7.2.8 can be used to determine how λk varies as the nontie-line path is traced, just as we did in Section 5.2.2. Substitution of Eq. 7.2.12 into Eq. 7.2.8 gives an expression that is convenient for evalution of the derivative of λk , λk =

f1 − S1 −

C1e −c12 c11 −c12 C1e −c12 c11 −c12

=

f1 (c11 − c12 ) + c12 − C1e F1 − C1e Fi − Cie = = S1 (c11 − c12 ) + c12 − C1e C1 − C1e Ci − Cie

(7.2.13)

Diﬀerentiation of Eq. 7.2.13 gives 1 dλk = dψ Ci − Cie

dFi dCie − dψ dψ

Fi − Cie − (Ci − Cie )2

dCi dCie − dψ dψ

.

(7.2.14)

Because Fi depends only on Ci , we can write n c −1 ∂Fi dC1 ∂Fi dC2 ∂Fi dCj dFi = + +··· = dψ ∂C1 dψ ∂C2 dψ ∂Cj dψ j=1

Substitution of Eqs. 7.2.15 and 7.2.13 into Eq. 7.2.14 yields

(7.2.15)

190

CHAPTER 7. MULTICOMPONENT GAS/OIL DISPLACEMENTS ⎧

⎫

c ⎨ 1 λk − 1 dCie ∂Fi dCj dCi ⎬ dλk = − λ + k dψ (Ci − Cie ) ⎩ j=1 ∂Cj dψ dψ ⎭ Ci − Cie dψ

n −1

(7.2.16)

Substitution for λk using Eq. 7.2.13 gives Fi − Ci dC1e dλk = , dψ (Ci − Cie )2 dψ

(7.2.17)

which is identical to Eq. 5.2.8. Thus, the behavior of an envelope curve reveals how an eigenvalue varies along a tie line, regardless of how many components are present. Figs. 7.3 and 7.4 illustrate the full range of possible arrangements of envelope curves. Envelope curves that are concave in the same direction as the two-phase region are referred to as concave. They are sketched in Fig. 7.3. Those envelope curves that are concave in the direction opposite to that of the two-phase boundary are called convex. They are illustrated in Fig. 7.4. The situations illustrated in Fig. 7.4 are by far the most common for gas oil systems with phase behavior calculated with one of the commonly used equations of state, but the existence of systems like those shown in Fig. 7.3 cannot be ruled out, and hence we consider all the possibilities. It is convenient to rewrite Eq. 7.2.17 so that the sign of dλk /dψ is obvious. Substitution of the deﬁnitions of Ci , Cie , and Fi into Eq. 7.2.17 gives f1 − S1 dCie 1 dλk = . dψ (S1 + pk )2 (ci1 − ci2 ) dψ

(7.2.18)

Eq. 7.2.18 indicates that the sign of the derivative of λk is controlled by the signs of ci1 - ci2 and of dCie /dψ, because f1 > S1 for systems in which gas displaces oil, and the denominator of the ﬁrst fraction in Eq. 7.2.17 is always positive. If dλk /dψ is positive as the nontie-line path is traced from uptsream to downstream, then a rarefaction exists along the nontie-line path. On the other hand, if dλk /dψ is negative for the same composition variation, then a shock occurs instead. In Figs. 7.3 and 7.4, we take C1 to be measured in the vertical direction. In all the examples in Figs. 7.3 and 7.4, ci1 > ci2 , so the sign of dλk /dψ is controlled by the sign of dCie /dψ. Consider the examples illustrated in Fig. 7.3a, in which the envelope curve lies on the liquid side of the two-phase region. In the condensing composition variation shown on the left, the shorter of the two key tie lines lies upstream. As the nontie-line path is traced from upstream to downstream, the tie lines that intersect the nontie-line path are tangent to the envelope curve at increasing values of C1e . Hence, λk increases as the nontie-line path is traced, and a rarefaction is allowed. In the vaporizing segment, shown on the right side of Fig. 7.3a, however, the upstream tie line is longer, and as the nontie-line path is traced from upstream to downstream, C1e decreases. For this segment, a shock is required. The nontie-line path is shown for reference, but the shock jumps over the nontie-line path. If, on the other hand, the envelope curve lies on the vapor side of the two-phase region (see Fig. 7.3b), then the pattern of nontie-line shocks and rarefactions is reversed. As the nontie-line path is traced from upstream to downstream in the condensing segment on the left side of Fig. 7.3b for example, C1e decreases as does λk , and a shock is required. When the direction of the composition variation is reversed, as it is in the vaporizing segment in the right side of Fig. 7.3b, λk increases from upstream to downstream, and the resulting composition variation is a rarefaction.

7.2. SOLUTION CONSTRUCTION

191

Condensing

Vaporizing u a

d a

a

ad

u

e

e

__ i | <0 (ci1-ci2) dC dψ u->d

__ i | >0 (ci1-ci2) dC dψ u->d

rarefaction

shock

a. Concave envelope on the liquid side

ua da (ci1-ci2)

dC __ ei | dψ

u->d

a

a

u

<0

d

(ci1-ci2)

dC __ ei | dψ

u->d

>0

shock

rarefaction

b. Concave envelope on the vapor side Figure 7.3: Nontie-line paths with concave envelope curves.

192

CHAPTER 7. MULTICOMPONENT GAS/OIL DISPLACEMENTS

Condensing

Vaporizing ua

a

da

a

u

__ ei | <0 (ci1-ci2) dC dψ u->d

d

__ ei | >0 (ci1-ci2) dC dψ u->d

shock

rarefaction

a. Convex envelope on the liquid side

u a d a (ci1-ci2)

dC __ ei | dψ

u->d

a

a

u

>0

d

(ci1-ci2)

dC __ ei | dψ

u->d

<0

rarefaction b. Convex envelope on the vapor side Figure 7.4: Nontie-line paths with convex envelope curves.

shock

7.2. SOLUTION CONSTRUCTION

193

Table 7.1: Nontie-Line Shocks and Rarefactions in Multicomponent Displacements Envelope Curve and Tie Line Intersections Liquid Side Liquid Side Vapor Side Vapor Side Liquid Side Liquid Side Vapor Side Vapor Side

Envelope Curvature Concave Concave Concave Concave Convex Convex Convex Convex

Process Name Condensing Vaporizing Condensing Vaporizing Condensing Vaporizing Condensing Vaporizing

Shortest Tie Line Upstream Downstream Upstream Downstream Upstream Downstream Upstream Downstream

Composition Variation Rarefaction Shock Shock Rarefaction Shock Rarefaction Rarefaction Shock

Fig. 7.4 illustrates the patterns of nontie-line shocks and rarefactions when the envelope curves are convex (having curvature opposite to that of the binodal surface). The shapes of the envelope curves in Fig. 7.4 are similar to those described for systems with constant K-values (Sections 5.2.2). The change in the curvature of the envelope curves from that illustrated in Fig. 7.3 reverses the pattern of shocks and rarefactions. Table 7.1 summarizes the behavior of the examples illustrated in Figs. 7.3 and 7.4. It is a reminder that the pattern of shocks and rarefactions in a multicomponent displacement is controlled by the geometries of tie lines and the two-phase region. The upper half of Table 7.1 refers to Fig. 7.3, and the lower half describes the patterns shown in Fig. 7.4. It is worth noting again that the behavior illustrated in the lower half of Table 7.1 and Fig. 7.4 is most commonly observed for gas/oil systems.

7.2.1

Fully Self-Sharpening Displacements

In many systems, it turns out that the only rarefactions present occur along tie lines. If so, the displacement is fully self-sharpening, and the solution can be found in a relatively straightforward way. The solution for a fully self-sharpening displacement can be determined by the following steps: 1. Locate all the key tie lines by solving the appropriate tie line intersection equations (see Section 7.1.2). 2. Locate the shortest of the key tie lines. Solution construction begins on that tie line, as the previous section showed. A rarefaction always occurs on the shortest tie line. It is terminated by a semishock in the upstream direction and a second semishock in the downstream direction, where the semishock velocities match the tie line eigenvalue on the shortest tie line. Those shock constructions are equivalent to the Welge tangent construction for a Buckley-Leverett problem. For any such shock, the shock velocity can be determined easily from the tie line intersection point. If point c is the semishock point on the shortest tie line and point d is the landing point on the next tie line in either the upstream or the downstream direction, then the shock velocity is determined by

194

CHAPTER 7. MULTICOMPONENT GAS/OIL DISPLACEMENTS +

Λ

cd

F c − CiXcd f c − S1Xc df ++ = ic = = , Xcd c Xc dS +c Ci − Ci S1 − S1

(7.2.19)

and the landing point on the next tie line is given by +

Λ

cd

F d − CiXcd f d − S1Xd df ++ = id = = . Xcd d Xd dS +c Ci − Ci S1 − S1

(7.2.20)

In these expressions, CiXcd refers to the composition at the intersection of the tie lines that contain points c and d, and S1Xc is the saturation at the tie line intersection point measured on the tie line that contains point c, and S1Xd is the saturation at the intersection point measured on the landing tie line. 3. There are two possibilities for the composition variation on the landing tie line. There can be a composition variation (normally a rarefaction) along the tie line to a semishock point for the shock to the next tie line. If so, the procedure for ﬁnding the velocity and compositions for the ﬁrst pair of tie lines can be applied again. It often happens, however, that a variation along the tie line in the direction of the semishock point is prohibited by the velocity constraint. In such cases, a genuine shock connects point d to the next tie line. The velocity of that shock is given by Λde =

Fid − CiXde f d − S1Xd = , Cid − CiXde S1d − S1Xd

(7.2.21)

where CiX de refers to the intersection point for the tie lines that contain points d and e. Points d and X are known, so the shock velocity can be calculated immediately. The landing point on the next tie line can then be calculated from Λde =

Fie − CiXde f e − S1Xe = , Cie − CiXde S1e − S1Xe

(7.2.22)

where Λde is known from Eq. 7.2.21. 4. Steps 2 and 3 are then repeated for each pair of tie lines in the upstream direction from the shortest tie line and each pair of tie lines in the downstream direction from that tie line. Fig. 7.5 shows a solution for a fully self-sharpening displacement determined by the procedure outlined for a displacement of six-component oil (Oil A in Table 7.2) by pure CO2 (Gas A). The oil contains a trace of CO2 (0.5 mole percent), 35 mole percent CH4 , 25 mole percent C4 , and smaller amounts of C10 , tetradecane (C14 ), and eicosane (C20 ). Phase behavior for this system was calculated with the Peng-Robinson equation of state with the component properties given in Chapter 3 (Tables 3.1 and 3.2). With six components, there are three crossover tie lines in the solution composition route in addition to the initial oil and injection gas tie lines. The ﬁrst crossover tie line is the shortest tie line, where solution construction begins. Tracing the solution route downstream of the shortest tie line requires solving for two shocks. The shock from the ﬁrst

7.2. SOLUTION CONSTRUCTION

195

1h

g

Sg

f

f e

e c

d

b

b a

0 1

C10

CO2

0.2

0.0

1

0.2

C16

CH4

0

0.0 0.2

C4

C20

0 0.5

0.0

0.0 0

1

ξ/τ

2

0

1

ξ/τ

2

Figure 7.5: Saturation and composition (mole fraction) proﬁles for the displacement of a sixcomponent oil (Oil A in Table 7.2) by pure CO2 at 2940 psia (200 atm) and 160 F (71 C).

196

CHAPTER 7. MULTICOMPONENT GAS/OIL DISPLACEMENTS

Table 7.2: Initial Oil and Injection Gas Compositions (mole fraction) Component N2 CO2 CH4 C2 C3 C4 C5 C6 C7 C8 C9 C10 C12 C16 C20

Oil A 0.005 0.350

0.250

0.195 0.125 0.075

Oil B 0.005 0.015 0.046 0.050 0.070 0.010 0.040 0.012 0.018 0.025 0.130 0.065 0.050 0.030 0.020

Oil C 0.005

Gas A

Gas B

1.000 0.350

0.250

0.195

1.000

Gas C 0.009180 0.024010 0.801110 0.070827 0.071731 0.005816 0.011739 0.001642 0.001115 0.000706 0.001652 0.000410

0.125 0.075

crossover tie line to the initial oil tie line (b→c) is a semishock at which λct matches the shock velocity. There are four shocks upstream of the rarefaction c–d along the shortest tie line. The shock from the ﬁrst crossover tie line to the second crossover tie line (d→e) is a semishock. The wave velocity of this shock equals the the tie-line eigenvalue, λdt . The remainder of the shocks upstream of the rarefaction are genuine shocks. This pattern of slow moving evaporation shocks for the heavier components is a commonly observed feature of multicomponent displacements. As the individual component proﬁles in Fig. 7.5 show, all six components present in the oil appear on the initial tie line, the tie line that contains the composition of the zone of constant state b. CH4 disappears at the b→c shock, where the CO2 fraction increases signiﬁcantly from the low value on the initial tie line to the high concentration at the leading edge of the bank that contains the injected CO2 , on the ﬁrst crossover tie line. Thus all the CH4 from the undisplaced oil upstream of the b→c shock appears in a leading CH4 bank. All of the C4 is evaporated upstream of the d→e shock, and the C10 evaporates at the e→f shock. C14 disappears at the f→g shock, and C20 disappears at the trailing g→h shock. Because the solubility of C14 and C20 in pure CO2 is small, those shocks propagate slowly. Fig. 7.6 shows an example of a displacement with more components in both the oil and the gas. In this example system, the oil contains ﬁfteen components. The oil composition is given as Oil B in Table 7.2 along with the injection gas composition, Gas C. The injection gas is primarily CH4 , but it also contains intermediate components like CO2 , C2 , and C3 , as well as small amounts of the heavier components as well for a total of twelve components in the injection gas. In this example, there are fourteen key tie lines. The shocks shown in the saturation proﬁle in Fig. 7.6 are labeled with the number of the key tie line on which that saturation lies. The eighth crossover tie line is the shortest, and it is connected to the seventh and ninth tie lines by semishocks. As Fig.

7.2. SOLUTION CONSTRUCTION

Sg

1

14 13 12 11 109

87

197

6 5 43

2

1

0 0.1

C5

CO2

0.1

0.0 0.2

C9

CH4

0.0 1

0.0

0.1

0.1

C2

C20

0

0.0 0.0

1.0

ξ/τ

2.0

0.0 0.0

1.0

ξ/τ

2.0

Figure 7.6: Saturation and selected composition proﬁles for the displacement of a ﬁfteen-component oil (oil B in Table 7.2 by a twelve component gas (gas C) at 2940 psia (200 atm) and 160 F (344 K). The injection gas contains 80 mole percent CH4 , 7 mole percent each of C2 and C3 , 2 mole percent CO2 plus small amounts of additional components. Numbers shown in the saturation proﬁle refer to the sequence of key tie lines that contain that overall composition.

198

CHAPTER 7. MULTICOMPONENT GAS/OIL DISPLACEMENTS

7.6 shows, the gas saturation increases monotonically upstream as the key tie lines are traversed. Overall compositions of other components may increase or decrease across individual shocks. The fraction of CH4 increases signiﬁcantly as the solution is traced upstream. The C2 fraction also increases upstream, but only after a small decline across the leading shock. The fraction of CO2 , a minor component in with similar fractions in the gas and the oil, hardly changes. The overall fractions of the three heaviest components (of which only C20 is shown) all decline to zero across slow moving evaporation shocks at the upstream edge.

7.2.2

Solution Routes with Nontie-line Rarefactions

If a rarefaction occurs along a nontie-line path that connects two of the key tie lines, then the key tie line associated with that path will diﬀer slightly from the estimate determined by assuming that all the key tie lines intersect. In such systems the key tie line in question lies at the intersection of a rarefaction surface (a surface of tie lines that intersect the nontie-line path, and the next surface of tie lines, which could be a shock surface or a rarefaction surface. Because rarefaction surfaces can twist, the extensions of the pair of key tie lines connected by a rarefaction will not, in general, intersect. In this section, we consider how to ﬁnd the nontie-line path for multicomponent displacements. It is worth noting, however, that in some systems, nontie-line rarefactions are short segments, and λnt changes only slightly as the nontie-line path is traced. In such displacements, the compositions that exist along the nontie-line path have almost constant wave velocity, and hence the rarefaction is approximated well by the assumption that the rarefaction can be replaced by a shock. The fully self-sharpening solution discussed in the previous section, therefore, is often a good approximation, even if rarefactions are present. There are systems, however, for which λnt varies signiﬁcantly. Nitrogen displacements, for example, often include rarefactions that have large composition and wave velocity changes (see Jessen et al. [48] for an example in which nitrogen displaces a ﬁfteen component oil). The following steps can be used to obtain the solution for a displacement that includes one or more nontie-line rarefactions: 1. Locate all of the key tie lines by the tie-line intersection calculation. 2. Apply the envelope rule (see Table 7.1) to each neighboring pair of key tie lines. If no rarefactions are predicted, switch to the simpler sequence of steps for fully self-sharpening displacements. 3. For each predicted rarefaction, locate the appropriate equal eigenvalue point on the shorter of the two tie lines. The appropriate equal eigenvalue point is the one for which the corresponding nontie-line path to be traced from the shorter tie line passes closest to the next key tie line determined by the tie-line intersection calculation. Recall that at each point along a tie line, there are nc 1 eigenvectors. One points along the tie line, and the other nc - 2 point in the directions of the nontie-line paths. Fig. 7.7 shows an example of the how the eigenvalues for a multicomponent system vary along a tie line. Each of the curves that represent the variation of individual eigenvalues intersects the tie line eigenvalue at two points. Thus, there are 2(nc - 2) points along each tie line at which the tie line eigenvalue is equal to one of the nontie-line eigenvalues.

7.2. SOLUTION CONSTRUCTION

199

Tie-line and Nontie-line Eigenvalues, λt and λnt

3.0

λt

2.5

2.0

1.5 λ1nt 1.0 λ2nt 0.5 λ3nt λ4nt 0.0 0.0

0.2

0.4

0.6

0.8

1.0

Saturation, S1

Figure 7.7: Eigenvalues on the initial oil tie line for a displacement of Oil C (see Table 7.2) by pure nitrogen at 160 F (71 C) and 2940 psia (200 atm). Half of these points can be discarded immediately for displacements of oil by gas because they lie in the part of the two-phase region for which f1 < S1 . The remaining nc - 2 points are candidates for the point at which the composition variation along the tie line switches to the nontie-line path. The values of λnt can be found, of course, by solving the full eigenvalue problem at each point along the tie line and then ﬁnding the resulting equal eigenvalue points. A more eﬃcient approach was suggested by Dindoruk [19], however. If the eigenvalues at one point on the shorter tie line are calculated by solving the full eigenvalue problem, then the values of pk for that tie line can be obtained from Eq. 7.2.8. Those values are ﬁxed for the tie line, so the eigenvalues at any other point along the tie line can be calculated easily from Eq. 7.2.8. The equal eigenvalue points can be determined by setting the various values of λnt equal to the tie-line eigenvalue, λt and solving. The appropriate equal eigenvalue point can be identiﬁed in the following way. The appropriate nontie-line path lies quite close to the plane determined by the two key tie lines identiﬁed by the tie-line intersection calculation. At the equal eigenvalue point, the nontie-line path is tangent to the tie line, so there is no useful direction information available at that point, but for points on either side of the equal eigenvalue point, eigenvectors associated with the same nontie-line eigenvalue will point approximately in the direction of the plane determined by the two key tie lines. All other eigenvectors will point in other directions. Thus, the correct equal eigenvalue point for the path switch can be determined as the one for which the corresponding eigenvectors at points along the tie line on either side of that equal eigenvalue

200

CHAPTER 7. MULTICOMPONENT GAS/OIL DISPLACEMENTS 1 f

Sg

e

e

d

d c

b a

0

0.2

N2

C10

1

0.0

1

0.2

C16

CH4

0

0.0 0.2

C4

C20

0 0.2

0.0

0.0 0

1

ξ/τ

2

0

1

ξ/τ

2

Figure 7.8: Saturation and composition (mole fraction) proﬁles for the displacement of a sixcomponent oil (Oil A in Table 7.2, but with the small initial CO2 concentration replaced by the same mole fraction of N2 ) by pure N2 at 2940 psia (200 atm) and 160 F (71 C). point have directions that point toward the next key tie line. 4. Once the appropriate equal eigenvalue point is identiﬁed, perform a stepwise integration along the direction indicated by the corresponding nontie-line eigenvector. Terminate the integration at the point of closest approach to the next key tie line. This point is usually so close to the key tie line determined from the tie line intersections that it is not worth the eﬀort of recomputing the tie line intersections for tie lines upstream or downstream, but that could be done if desired. 5. Continue with shock constructions upstream and downstream, inserting rarefactions where needed. Fig. 7.8 shows an example of a displacement that includes a signiﬁcant rarefaction. The oil displaced is Oil C in Table 7.2, at the pressure and temperature of the displacement illustrated in Fig. 7.5. Oil C is identical to Oil A except that the CO2 has been removed and replaced with N2 . Oil C is then displaced by pure N2 instead of CO2 . Comparison of Figs. 7.5 and 7.8 reveals the following diﬀerences:

7.3. SOLUTION CONSTRUCTION: VOLUME CHANGE

201

1. The shortest tie line is now the initial tie line. 2. The leading (a→b) shock is a semishock, and there is a rarefaction along the initial tie line. 3. The intermediate shock in the displacement by CO2 is replaced by a nontie-line path rarefaction (c→d. In this case, there is a signiﬁcant change in wave velocity as the nontie-line path is traced. In both displacements, there is a CH4 bank that contains all the CH4 that was present in the oil that is left behind the leading shock and the intermediate shock or nontie-line rarefaction. The trailing portions of the two solutions are similar.

7.3

Solution Construction: Volume Change

When components change volume as they transfer between phases, ﬂow velocity does not remain constant. The results obtained in Sections 4.4, 5.7, and 6.6 for displacements with two, three and four components provide the basis for solving multicomponent problems with volume change as components transfer between phases. The previously derived results that allow solution construction for multicomponent systems with volume change on mixing are: 1. Flow velocity does not change for composition variations along a single tie line. The proof given in Section 4.4.1 applies to any tie line, not just the single tie line traversed in binary displacements. 2. A shock from a single phase composition to a composition in the two-phase region jumps to a composition that lies on the tie line that extends through the single phase composition. That statement holds for a tie line drawn in concentration space (moles per unit volume) or in mole fraction space. The proof given in Section 5.7 applies to systems with any number of components. 3. Extensions of two tie lines connected by a shock from one two-phase composition to another must intersect. Again, this statement applies to tie lines drawn in concentration space or in mole fraction space (see Section 5.7). 4. A scaled velocity of any shock can be calculated from a surprisingly simple expression like Eq. 5.7.36, Λ∗ =

Λ f1A − S1XA = , vA S1A − S1XA

(7.3.1)

where S1XA refers to the volume fraction (saturation), measured on tie line A, of equilibrium vapor mixed with equilibrium liquid that gives the composition at the intersection of the two tie lines, and v A , f1A , and S1A are the ﬂow velocity, fractional ﬂow and vapor saturation at the shock point on tie line A. The saturation at an intersection point can be calculated from the compositions and molar densities of the liquid and vapor phases (Eq. 5.7.30),

202

CHAPTER 7. MULTICOMPONENT GAS/OIL DISPLACEMENTS

S1XA =

'

X A ρA L zi − xi

(

'

X A A X A ρA L zi − xi + ρV zi − yi

(.

(7.3.2)

In addition, the shock balances imply that (Eq. 5.7.37) Λ∗ =

Λ v B f1B − S1XB = , vA v A S1B − S1XB

(7.3.3)

S1XB is the intersection saturation measured on tie line B. If the shock that connects tie lines A and B is a semishock, then A ΛAB = λA t =v

df1 , dS1

(7.3.4)

and Λ∗ =

λA df1 t = . A v dS1

(7.3.5)

If the shock is a genuine shock instead, then the shock velocity is determined from the compositions on either side of the shock. The determination of the scaled shock velocities allows construction of the complete solution. As in problems with three and four components, solution construction starts on the shortest tie line. Semishocks are constructed from that tie line to upstream and downstream tie lines. At each step, ratios of the ﬂow velocities can be eliminated, and the compositions of the shocks determined. Then, the ratio of the ﬂow velocities across each shock can be determined, though the actual velocities can be found only after all the shock compositions have been calculated and the ratios of ﬂow velocities across each shock are determined for all shocks. Then the known ﬂow velocity at the inlet can be used with the velocity ratios to calculate all the ﬂow velocities. This procedure is completely parallel to the procedure used in Chapter 5 to solve a three component problem. It relies on the fact that velocity changes can be decoupled from composition changes (see Section 6.6 for an explanation of why this is possible). If a rarefaction is present between tie lines, then an integration along the nontie-line path is required. Flow velocities and compositions change as the nontie-line path is traversed, but the stepwise integration of the eigenvectors gives the required values of composition and velocity. Here again, scaled values of the ﬂow velocity can be used until the inlet velocity can be used to calculate all the ﬂow velocities downstream. Fig. 7.9 compares solutions with and without volume change for the six-component example shown in Fig. 7.5. The two solutions display the same sequence of shocks and the single rarefaction along the shortest key tie line. The biggest diﬀerence is in the propagation speeds of the shocks. When volume change is included, the shocks all move at lower velocity. The top right panel in Fig. 7.9 shows the local ﬂow velocity. It is constant at vD = 1 for the solution without volume change, but it has lower values for the downstream shocks and the rarefaction for the solution with volume change. The biggest drop in velocity occurs at the shock at which the CO2 fraction increases substantially. Behind that shock, CH4 is missing. All of the CH4 from the oil remaining undisplaced

7.3. SOLUTION CONSTRUCTION: VOLUME CHANGE

203

1h

Sg

f

f e

1

e

vD

g

c

d

b

b a

0 1

C10

CO2

0 1

0 1

C16

CH4

0 1

0 1

C4

C20

0 1

0

0 0

1

ξ/τ

2

0

1

ξ/τ

2

Figure 7.9: Saturation and composition (mole fraction) proﬁles for the displacement of a sixcomponent oil (Oil A in Table 7.2 by pure CO2 at 2940 psia (200 atm) and 160 F (71 C). The solid lines give the solution for ﬂow with no volume change as components transfer between phases (see Fig. 7.5. The dashed lines are the solution with the eﬀects of volume change included.

204

CHAPTER 7. MULTICOMPONENT GAS/OIL DISPLACEMENTS

behind the leading shock appears in the CH4 bank shown in the CH4 proﬁle. Upstream of that bank, the CO2 in the injection gas dissolves in the undisplaced oil. Because the CO2 dissolved in the liquid phase occupies a smaller volume than does the same amount of CO2 in the injection gas, volume is lost, and the ﬂow velocity must decrease. The compositions at the shocks change slightly, the result of small changes in fractional ﬂow that arise from the diﬀerences in phase volume fractions that come from the changes in phase density. But the only signiﬁcant diﬀerence is in the ﬂow velocities. In this example, the ﬂow velocities are all less than the injection ﬂow velocity, but it is possible for volume change to cause an increase in ﬂow velocity as well. Dindoruk [19] describes several systems in which nitrogen displaces an oil containing dissolved methane. Nitrogen displaces the oil relatively ineﬃciently, leaving signiﬁcant quantities of undisplaced oil behind the leading shock. The CH4 that transfers from the undisplaced oil to the fast-ﬂowing vapor phase occupies more volume in the vapor than when it is dissolved in the oil, and hence, ﬂow velocity increases. Similarly, Ermakov [23] reports an example that shows similar behavior for a displacement in which an oil containing considerable dissolved CH4 is displaced by a CH4 -rich injection gas. Displacements in which CO2 is an important constituent in the injection gas usually show decreases in ﬂow velocity, though the presence of large amounts of CH4 in the initial oil can limit the decrease. Eﬀects of volume change are largest when the pressure is high enough that appreciable amounts of components like CO2 and CH4 can dissolve in oil, but low enough that the volume occupied by those components in the vapor phase is larger than the volume occupied in the liquid phase. At very high pressures, the diﬀerence in volume is smaller, and hence eﬀects of volume change are also smaller.

7.4

Displacements in Gas Condensate Systems

Nearly all of the gas injection processes considered so far have had an initial composition in the single-phase region, but there are situations in which the initial composition might lie in the twophase region. Condensate systems are an example. When gas reservoirs are produced, pressure declines, and in many reservoirs retrograde condensation causes a liquid phase to form as the pressure is reduced. Gas injection can be used to vaporize and recover some of the condensate, which contains valuable intermediate hydrocarbons. A relatively detailed characterization of the small amounts of the heaviest hydrocarbon components present in the reservoir is often required if dewpoint pressures and amounts of condensate dropout are to be represented accurately. Thus, multicomponent solutions are required to describe condensate displacements with reasonable accuracy. Fig. 7.10 shows the solution for a displacement of a condensate system described with 13 components. The ﬂuid characterization and the composition of the reservoir ﬂuid for this system are those reported by Jessen and Orr [44]. The characterization of the components used for this displacement is given in Table 7.3. The overall composition of the initial ﬂuid is also reported in Table 7.3. The injection gas is pure CO2 . At the initial pressure and temperature the initial ﬂuid forms two phases, with a liquid saturation of about 10.5 percent. This condensate system contains a signiﬁcant fraction of H2 S. As in the other multicomponent displacements, the solution was constructed by identifying the key tie lines and then constructing the series of shocks that make up the solution. In this displacement, no rarefactions are present. Just as in the binary displacements of Chapter 4 (see Fig. 4.15) and the ternary displacements of Chapter 5 (see Fig. 5.21), when the initial composition is close enough to the vapor locus, the rarefaction

7.4. DISPLACEMENTS IN GAS CONDENSATE SYSTEMS

205

vD

Sg

1.0

0.8 1

1.0

C2

CO2

0.1

0.0

0.5

0.05

C5

H2S

0

0.00 0.05

CH4

C12+

0.0 0.5

0.0 0.0

0.5

ξ/τ

1.0

0.00 0.0

0.5

ξ/τ

1.0

Figure 7.10: Saturation and composition (mole fraction) proﬁles for the displacement of a 13component gas condensate system by pure CO2 at 1740 psia (118 atm) and 216 F (102 C). The MMP for this system is 1857 psia (126 atm).

206

CHAPTER 7. MULTICOMPONENT GAS/OIL DISPLACEMENTS

Table 7.3: Fluid Characterization and Initial Fluid Composition for Displacement of a Condensate System by CO2 [44]. Only nonzero binary interaction parameters are listed. Critical volumes reported (in cm3 /mol) were used in the calculation of phase viscosities [70]. Comp. Tc(K) Pc (atm) Mw ω Vc kCO2 kN 2 kH2S Zinit N2 126.2 33.60 28.020 0.0400 88.46 0.0171 CO2 304.2 72.90 44.010 0.2280 92.64 0.0576 H2 S 373.5 88.50 34.080 0.0800 98.19 0.12 0.3562 CH4 190.6 45.40 16.043 0.0080 99.85 0.12 0.01 0.08 0.3631 C2 305.4 48.20 30.069 0.0980 146.59 0.15 0.06 0.07 0.0798 C3 369.8 41.90 44.096 0.1520 200.76 0.15 0.08 0.07 0.0340 C4 419.6 37.01 58.123 0.1875 254.99 0.15 0.08 0.07 0.0171 C5 465.9 33.34 72.150 0.2413 308.88 0.15 0.08 0.06 0.0300 C6 507.4 29.30 86.177 0.2960 375.96 0.15 0.08 0.06 0.0171 C7 573.6 40.47 94.000 0.2651 311.02 0.15 0.08 0.05 0.0116 C8 648.3 32.53 113.520 0.3437 425.82 0.15 0.08 0.0117 C10 630.1 30.17 114.520 0.4489 430.42 0.15 0.08 0.0126 C12+ 683.2 26.92 190.000 0.6305 489.85 0.15 0.08 0.0053

along the initial tie line is missing. That is the case here. The solution upstream of the initial composition includes a series of vaporizing shocks, with light components displaced more rapidly than heavy components. A bank of condensate (a zone of relatively low gas saturation) propagates near the leading edge of the two-phase zone. This bank arises because some components vaporized by the injected CO2 transfer into the liquid condensate phase as they propagate downstream. In this displacement, ﬂow velocities actually increase above the injection ﬂow velocity in the region of the condensate bank. That behavior is the result of the transfer of CH4 from the condensate present initially to the vapor phase, where it occupies more volume. As a result, the ﬂow velocity must increase. This displacement is relatively eﬃcient. The condensate bank moves with relatively high wave velocity, and the vaporization of the heavier components is relatively eﬃcient. In fact, the MMP for displacement of this condensate system is only slightly higher than the displacement pressure. As in other condensing/vaporizing displacements, it is a crossover tie line that approaches the critical locus. It is important to note that it is possible to have a multicontact miscible displacement even when the initial composition is inside the two-phase region, as it is for all condensate displacements. The topics of multicontact miscibility for multicomponent systems and the corresponding calculations of MMP and MME are the subject of the next section.

7.5

Calculation of MMP and MME

The ternary theory of Chapter 5 establishes that a displacement can be multicontact miscible if either the initial oil or injection gas tie line is a critical tie line, and the four-component theory (Chapter 6) shows that miscibility is also possible if the crossover tie line is a critical tie line. The extension to systems with more components is straightforward. Each additional component adds one more crossover tie line, and a multicomponent displacement is miscible if the injection gas tie

7.5. CALCULATION OF MMP AND MME

207

line, the initial oil tie line or any of the nc − 3 crossover tie lines is a critical tie line. If the key tie lines for any pair of initial oil and injection gas compositions, then the MMP or MME can be calculated by adjusting the displacement pressure or injection gas composition until one of the key tie lines is a critical tie line. Whether a tie line is critical is easily determined from it’s length. As a tie line approaches the critical locus, its length given by , - nc - Ln = . (yin − xni )2 ,

(7.5.6)

i=1

goes to zero. An algorithm that can be used to ﬁnd the MMP (MME) is: 1. At some low pressure, guess a set of equilibrium K-values (or use K-values from the negative ﬂash for the initial oil or injection gas tie line). 2. Solve the tie-line intersection equations for the key tie lines. 3. Update the equilibrium K-values and repeat step 2 to convergence. 4. Calculate tie line lengths. 5. Increase the pressure (or modify the injection gas composition) and repeat steps 1-3 until one of the key tie lines has zero length. The set of tie-line intersection equations being solved is quite nonlinear due to the equationof-state representation of phase behavior, and hence, a suitable initial guess of the key tie lines is required. An excellent initial guess at low pressure can be found from the procedure for ﬁnding the key tie lines for a single-component injection gas. If the key tie lines are found for injection of a pure gas (for the component present in the largest quantity in the actual injection gas, for example), then a starting set of key tie lines for the actual injection gas composition can be found by perturbing successively the composition of the injection gas from the pure component set. For fully self-sharpening systems the MMP or MME calculated by such a procedure is rigorously accurate for a given characterization of the ﬂuid mixtures and equation of state. In such systems, the MMP is a thermodynamic quantity. It depends only on the temperature and pressure, the compositions of the injection gas and initial oil, and, of course, on the thermodynamic characterization of the phase behavior of the system as embodied in the equation of state used to calculate phase equilibrium. Because the tie lines that determine miscibility are the same whether or not eﬀects of volume change are included in the calculation, the MMP is the same whether or not volume change is included. Because the key tie lines determine miscibility, the MMP is also independent of the fractional ﬂow function used. Thus, for fully self-sharpening systems, the calculation of key tie lines gives an unambiguous MMP that will be accurate if the phase behavior of the gas/oil system is represented accurately by the equation of state. The accuracy of MMP calculations with the key tie line approach has been investigated by Wang [128], Wang and Orr [130], Jessen [42], Jessen et al. [45, 46], and Wang and Peck [131]. To illustrate the ideas, we consider gas/oil systems described by Zick [140]. These systems are especially appropriate as examples because they were the ﬁrst for which condensing/vaporizing displacement behavior was described. The compositions of the initial oil and the injection gas

208

CHAPTER 7. MULTICOMPONENT GAS/OIL DISPLACEMENTS

Table 7.4: Initial Oil and Injection Gas Compositions for Zick’s Examples [140] Component CO2 CH4 C2 C3 C4 C5 C6 C7+(1) C7+(2) C7+(3) C7+(4) C7+(5)

Oil 1 0.0483 0.2067 0.0480 0.0408 0.0322 0.0246 0.0296 0.2515 0.1280 0.0851 0.0629 0.0425

Oil 2 0.0652 0.3542 0.0534 0.0377 0.0266 0.0192 0.0225 0.1859 0.0946 0.0629 0.0465 0.0315

Oil 3 0.0673 0.3792 0.0537 0.0370 0.0257 0.0184 0.0214 0.1755 0.0891 0.0593 0.0438 0.0296

Gas 1 0.2218 0.2349 0.2350 0.2745 0.0338

Gas 2 0.1774 0.3879 0.1880 0.2196 0.0270

Gas 3 0.1708 0.4109 0.1810 0.2114 0.0208

mixtures are reported in Table 7.4 [46]. The liquid phase compositions were obtained by ﬂashing Zick’s oil A at 185 F (85 C) and three pressures, 1515 psia (103.0 atm), 3015 psia (205.1 atm), and 3315 psia (225.6 atm). Gas 1 is Zick’s solvent A. Gases 2 and 3 are Gas 1 diluted by 20 and 23 mole percent CH4 . Jessen et al. used the characterization procedure of Pederson et al. [98] to obtain the component properties. The values shown in Table 7.5 for the twelve components diﬀer slightly from those reported by Jessen et al.. The values reported here were obtained by tuning the speciﬁc gravity of of the heaviest fraction to match the saturation pressure of the oil. Wang [128, 130] also considered this system, but used a slightly diﬀerent twelve-component characterization. The characterizations diﬀer mainly in the choice of pseudocomponents for the C7+ fractions and and the binary interaction parameters used. Fig. 7.11 shows tie line lengths obtained by Wang [128], shown in the top row, and by Jessen [42], bottom row, for displacement of Oil 1 (See Table 7.4) by Gas 1 and displacement of Oil 2 by Gas 2. In both systems, all key tie line lengths decrease as the pressure is increased, a reﬂection of the fact that the two-phase region for a gas/oil system shrinks as the pressure increases. In all the displacements, the injection gas and initial oil tie lines were much longer than many of the crossover tie lines. In the displacements of Oil 1 by Gas 1, for example, the fourth crossover tie line was found to be the tie line that had zero length at the lowest pressure, the MMP, by Wang but Crossover Tie Line 3 was slightly shorter at the MMP for the description used by Jessen. In both descriptions, Crossover Tie Lines 3 and 4 had very similar lengths over the full pressure range. In the displacements of Oil 2 by Gas 2, both Wang and Jessen found that crossover tie line 3 was the critical tie line at the MMP. In this example, Crossover Tie Lines 3 and 4 have equal length at a pressure just below the MMP in both descriptions, a situation that requires special treatment in the computational approach used by Wang but not in the formulation of Jessen. This sort of condensing/vaporizing behavior is commonly observed when the injection gas contains substantial fractions of components other than the one that has the highest K-value (CH4 in this case). Most CO2 ﬂoods fall into this category. Table 7.6 compares the MMPs obtained by Wang and Jessen with the experimental observations

7.5. CALCULATION OF MMP AND MME

209

Table 7.5: Component Properties for Zick’s Mixtures [140] Component CO2 CH4 C2 C3 C4 C5 C6 C7+(1) C7+(2) C7+(3) C7+(4) C7+(5)

Tc (K) 304.2 190.6 305.4 369.8 425.2 469.6 507.4 616.2 698.9 770.4 853.1 1001.2

Pc (atm) 72.9 45.4 48.2 41.9 37.5 33.3 29.3 28.5 19.1 16.4 15.1 14.5

Mw 44.010 16.043 30.069 44.096 58.123 72.150 86.177 137.691 243.950 347.691 481.033 735.683

ω 0.228 0.008 0.098 0.152 0.193 0.251 0.296 0.454 0.787 1.048 1.276 1.299

kCO2,j 0.000 0.105 0.130 0.125 0.115 0.115 0.115 0.115 0.115 0.115 0.115 0.115

Table 7.6: Comparison of calculated and measured MMPs (psia) for Zick’s examples [140].

Wang [128] Jessen [42] Experiment [140]

Oil 1/Gas 1 2169 2256 2215

Oil 2/Gas 2 3013 3070 3115

Oil 3/Gas3 3206 3415

210

CHAPTER 7. MULTICOMPONENT GAS/OIL DISPLACEMENTS

based on slim tube displacements reported by Zick [140]. The diﬀerences in the calculated MMPs were due to the slightly diﬀerent characterizations of the components used in the equation-ofstate calculations. All the calculated MMPs agree with the experimental observations to well within the resolution of typical slim tube displacement results. Slim tube experiments are always performed with pressure increments of hundreds of psia, increments that are signiﬁcantly larger than the pressure diﬀerences shown in Table 7.4. The results shown in Fig. 7.11 and Table 7.4 indicate clearly that the tie line intersection approach can be used to make accurate predictions of MMP (or MME) if the characterization of the ﬂuid mixture is adequate. In addition, it is clear that the tie line intersection calculation identiﬁes the correct MMP (or MME) no matter whether multicontact miscibility is controlled by a crossover tie line or the injection gas or initial oil tie line. Thus, the tie line intersection approach requires no advance knowledge of the displacement type for accurate calculation of MMP. That same statement cannot be made for some mixing cell methods for calculation of MMP (see Chapter 8 for a discussion of other approaches to MMP calculation). The tie-line intersection approach described here for calculation of the MMP or MME will be as accurate as the representation of the phase behavior of the sytem under consideration. If the system is fully self-sharpening, the calculated MMP is rigorous because it is calculated directly from the analytical solution for one-dimensional ﬂow of the multicomponent system. If the solution includes one or more rarefactions, the MMP may be an approximation, depending on whether the tie line that controls miscibility is determined by the rarefaction. If, on the other hand, a rarefaction is present, but the critical tie line is determined entirely by shocks, then the MMP is still rigorous. Even when a rarefaction determines the location of the critical tie line, the approximate MMP is quite accurate, as the examples given here and those reported by Wang et al. and Jessen et al. show. For typical gas-oil systems, the ruled surfaces of tie lines determined by a rarefaction show only slight twist, and hence, the tie lines determined by the tie-line intersection approach are an accurate approximation of the key tie lines that make up the solution. Thus, the tie-line intersection method gives an MMP that is based on the solution to the diﬀerential equations that describe the two-phase, multicomponent ﬂow.

7.6

Summary

In this chapter, the tools developed in Chapters 4-6 are used to obtain solutions for systems with more components. Extension of the analysis of ﬂow in two-, three-, and four-component systems to those with an arbitrary number of components leads to the following conclusions: • In a system with nc components there are nc - 1 key tie lines: the initial oil and injection gas tie lines and nc - 3 crossover tie lines. • In displacements that are fully self-sharpening (in which the only rarefactions occur along tie lines and pairs of tie lines are connected by shocks), the key tie lines obtained by solving the tie line intersection equations for the initial oil and the injection gas compositions are rigorously accurate. • Even in displacements that include nontie-line rarefactions, excellent estimates of the key tie lines can be obtained by assuming that all pairs of tie lines are connected by shocks and therefore that the tie lines can be calculated from the appropriate tie line intersection equations.

7.6. SUMMARY

211

Initial Tie Line Crossover Tie Line 1 Crossover Tie Line 2 Crossover Tie Line 3 Crossover Tie Line 4 Crossover Tie Line 5 Crossover Tie Line 6 Crossover Tie Line 7 Crossover Tie Line 8 Crossover Tie Line 9 Injection Tie line

Tie Line Length

0.8

0.6 Initial

0.4

Injection XO 2

XO 1

XO 4

1500

0.6

3000

2500

3500

0.4

0.0 1000

XO 2

XO 3

1500

0.4

Injection XO 1 XO 9

XO 2

Inj.

0.4

XO 1 XO 9 XO 8

XO 4

XO 7 XO 6 XO 5

XO 6 XO 3

1500

Initial

0.6

0.2

XO 7 XO 4

2000

XO 5

2500

3500

Initial Tie Line Crossover Tie Line 1 Crossover Tie Line 2 Crossover Tie Line 3 Crossover Tie Line 4 Crossover Tie Line 5 Crossover Tie Line 6 Crossover Tie Line 7 Crossover Tie Line 8 Crossover Tie Line 9 Injection Tie line

0.8

XO 8

0.2

0.0 1000

1.0

Tie Line Length

Tie Line Length

Initial

XO 2

3000

2500

(b) Oil 2, Gas 2, Wang

Initial Tie Line Crossover Tie Line 1 Crossover Tie Line 2 Crossover Tie Line 3 Crossover Tie Line 4 Crossover Tie Line 5 Crossover Tie Line 6 Crossover Tie Line 7 Crossover Tie Line 8 Crossover Tie Line 9 Injection Tie line

0.6

2000

Pressure (psia)

(a) Oil 1, Gas 1, Wang

0.8

XO 1 XO 9 XO 8 XO 7 XO 6 XO 5

XO 4

Pressure (psia)

1.0

Initial

Inj.

0.2

XO 5 XO 3

2000

Initial Tie Line Crossover Tie Line 1 Crossover Tie Line 2 Crossover Tie Line 3 Crossover Tie Line 4 Crossover Tie Line 5 Crossover Tie Line 6 Crossover Tie Line 7 Crossover Tie Line 8 Crossover Tie Line 9 Injection Tie line

0.8

XO 9 XO 8 XO 7 XO 6

0.2

0.0 1000

1.0

Tie Line Length

1.0

3000

Pressure (psia) (c) Oil 1, Gas 1, Jessen

3500

0.0 1000

XO 3

1500

2000

2500

3000

3500

Pressure (psia) (d) Oil 2, Gas 2, Jessen

Figure 7.11: Tie line lengths for condensing/vaporizing gas drive systems described by Zick [140]. (a)Crossover tie line 4 has zero length at 2169 psia (147.6 atm). (b) Crossover tie line 3 has zero length at 3013 psia (205.0 atm). (c) Crossover tie line 3 has zero length at 2256 psia (153.5 atm). (d) Crossover tie line 3 has zero length at 3070 psia (208.9 atm). (a) and (b) replotted from Wang [128], used with permission. (c) and (d) recalculated with a slightly diﬀerent ﬂuid description from Jessen [42], used with permission.

212

CHAPTER 7. MULTICOMPONENT GAS/OIL DISPLACEMENTS

• Whether or not tie line rarefactions appear in a displacement can be determined easily from the placement (on the vapor side or the liquid side of the two-phase region) and concavity of envelope curves. • Once the key tie lines are determined for a given pair of injection gas and oil compositions, the full solution for a gas displacement can be determined relatively easily. • Eﬀects of volume change as components transfer between phases are similar in multicomponent systems to those observed for systems with two, three, and four components. Use of scaled shock and rarefaction velocities allows straightforward construction of the complete solutions. • A displacement is multicontact miscible if any of the key tie lines is a critical tie line (a tie line of zero length). • In displacements in which the injection gas is a mixture or the injection gas does not have the highest equilibrium K-value, it is likely that a crossover tie line will control miscibility. Condensing/vaporizing gas drives, including most CO2 ﬂoods, are examples.

7.7

Additional Reading

The ﬁrst solutions for systems with more than four components were obtained by Dindoruk [19] and Johns [54]. Dindoruk reported solutions for a constant K-value system with ﬁve components and outlined the extension of his approach to systems with variable K-values. Johns obtained solutions with variable K-values for a ﬁve-component system and for a six-component system, both for injection of a pure component. A systematic procedure for calculating solutions for multicomponent systems with any number of components in the injection gas was reported by Jessen et al. [48]. Jessen [42] and Ermakov worked simultaneously to develop the systematic approach for ﬁnding multicomponent solutions when eﬀects of volume change as components change phase are absent or present. Numerous example solutions for systems with and without volume change are reported by Jessen [42], and additional solutions that show the eﬀects of volume change are described by Ermakov [23]. The analytical calculation of MMPs for multicomponent systems was considered ﬁrst by Johns and Orr [55]. Wang [128] and Wang and Orr [129] showed how to ﬁnd the key tie lines for injection of a single-component gas and then used the key tie lines to ﬁnd the MMP. They then considered how to ﬁnd the key tie lines and the MMP for a multicomponent injection gas [128]. Jessen et al. developed a much more eﬃcient algorithm for ﬁnding the key tie lines [45, 42] and showed that predictions of minimum miscibility pressures based on the key tie line approach agreed very well with experimental data and with MMPs calculated by other methods. Wang and Peck [131] reported results of additional tests of the accuracy of the key tie line approach. They showed that predicted MMPs and MMEs for systems with widely varying injection gas compositions agreed very well with experimental observations.

Chapter 8

Compositional Simulation by F. M. Orr, Jr. and K. Jessen The gas displacement problems considered in previous chapters can also be solved by numerical approaches, typically ﬁnite diﬀerence methods. In fact, before the analytical solutions considered here were developed, the only way to solve such problems was to do so numerically. In this chapter, we compare analytical and numerical solutions and show that the numerical solutions converge to the analytical ones. In addition, we consider the limitations of each approach to solving the ﬂow equations. Interpretation of numerical solutions requires careful attention to the eﬀects of numerical dispersion on the solutions, the subject of the next section. We also consider here numerical approaches to calculation of minimum miscibility pressure (MMP) or minimum injection gas enrichment for miscibility (MME). Several numerical schemes have been described, ranging from calculations based on mixing cells to extrapolation of numerical results from compositional simulations. The analysis of composition paths for multicomponent displacements discussed in Chapter 7 can be used to determine whether these numerical approaches yield accurate estimates of MMP or MME. That analysis shows that calculations based on a single mixing cell yield incorrect estimates of MMP and MME when a crossover tie line controls development of miscibility. Numerical approaches based on multiple mixing cells or compositional simulation can yield accurate estimates of MMP and MME if appropriate attention is paid to eﬀects of numerical dispersion. In Section 8.1, we consider how numerical dispersion arises in ﬁnite diﬀerence computations. Section 8.2 shows how calculated composition paths are modiﬁed by the presence of numerical dispersion. The impact of numerical dispersion on a particular displacement depends on the phase behavior of the gas/oil system. Section 8.3 describes how the magnitude of the eﬀect of numerical dispersion depends on phase behavior and suggests a way to determine which systems will be sensitive to the eﬀects of dispersion. Finally, numerical schemes for calculation of MMP and MME are discussed in Section 8.4.

8.1

Numerical Dispersion

Numerical solutions to material balance equations like Eqs. 2.4.4 or 4.1.1 are usually obtained by ﬁnite diﬀerence methods. Those solutions are always aﬀected to some extent by truncation error. In this section, we examine the sources of that truncation error. Eq. 4.1.1 can be solved easily with a fully explicit (forward time), backward space diﬀerence of the form 213

214

CHAPTER 8. COMPOSITIONAL SIMULATION

∆τ n n ). (8.1.1) (F − Fi,k−1 ∆ξ i,k where k indicates the grid block and n the time level. While there are several other diﬀerencing options available, Eq. 8.1.1 illustrates well the issues that arise in ﬁnite diﬀerence solutions (see Mallison et al. [73] for an evaluation of higher order computational schemes that improve accuracy for one dimensional solutions over the simple diﬀerencing scheme discussed here). It is possible to obtain more accurate ﬁnite diﬀerence solutions at a given grid resolution with the approaches Mallison describes, but the overall impacts of numerical dispersion are common to all the methods, so we use here the simplest of methods to illustrate the ideas. In a displacement calculation, the compositions in each grid block are calculated at the new time step with Eq. 8.1.1. A ﬂash calculation is then performed for the composition in each grid block to determine the phase compositions and saturations, with which the fractional ﬂows of the components can be calculated in preparation for the next time step. Lantz [64] showed that even when the conservation equation being solved is dispersion-free, the truncation error in the ﬁnite diﬀerence version of the diﬀerential equation mimics a second order term in the original diﬀerential equation. He showed that for ﬁnite diﬀerence form of Eq. 8.1.1, the numerical eqivalent of the Peclet number for single-phase ﬂow is [64, 127] n+1 n = Ci,k − Ci,k

P e−1 num

∆ξ ∆τ = 1− 2 ∆ξ

,

(8.1.2)

and for two-phase ﬂow is

∆ξ df1 ∆τ df1 = 1− . (8.1.3) 2 dS1 ∆ξ dS1 The exact form of the numerical Peclet number depends on the ﬁnite diﬀerence formulation for the time and space derivatives. Peaceman [96, pp. 74-81] reports expressions appropriate to the various implicit and explicit ﬁnite diﬀerence forms in common use. For the remainder of this discussion, we will use the simple form of Eq. 8.1.1, which permits calculation of compositions at the new time level without a matrix inversion and which illustrates in a straightforward way the impact of numerical dispersion on calculated composition paths. Eqs. 8.1.2 and 8.1.3 indicate that the inverse numerical Peclet number will be greater than zero as long as ∆τ < ∆ξ. The P e−1 num must be positive if the numerical calculation is to be stable [96]. If it is negative, the numerical solution will show nonphysical oscillations in compositions and saturations. Hence, ∆τ must be less than ∆ξ for single-phase ﬂow, and much less than ∆ξ df1 > 1, as it will be for the S-shaped fractional ﬂow curve appropriate to two-phase ﬂow in if dS 1 a porous medium, particularly when the injection gas has a viscosity that is lower than that of the oil displaced. As a result, eﬀects of numerical dispersion can be reduced by reducing ∆ξ (and therefore ∆τ ), but they cannot be eliminated entirely for Eq. 8.1.1. The limitation on time step size is a version of the Courant-Friedrichs-Levy (CFL) condition [15, 69], which states that the ﬁnite diﬀerence scheme of Eq. 8.1.1 is unstable if P e−1 num

+ + + λp∆τ + + + + ∆ξ + > 1,

p = 1, nc − 1,

(8.1.4)

for each of the p eigenvalues. Because the nontie-line eigenvalues generally have values close to one, it is the tie-line eigenvalue (df /dS) that determines the maximum stable value of the time step size.

8.2. COMPARISON OF NUMERICAL AND ANALYTICAL SOLUTIONS

215

Table 8.1: Peclet Numbers for Finite Diﬀerence Simulations Grid Blocks 20 100 1000 10000

P e−1 0.0462 0.00926 0.000926 0.0000926

Pe 21.6 108 1080 10800

Numerical dispersion arises from the way ﬂuids are treated in the ﬁnite diﬀerence representation. Consider what happens in a ﬁrst-contact miscible displacement, for example. An increment of solvent is injected into the ﬁrst grid block in the ﬁrst time step. At the end of that time step the ﬂuids are mixed so that the block has uniform composition. In the next time step, a fraction of the solvent injected in the ﬁrst time step ﬂows out of grid block 1 to grid block 2, and that happens no matter how small the amount of solvent injected into grid block 1. Thus some solvent propagates the entire length of grid block 1 in the ﬁrst time step, even though physical ﬂow over the same distance would take longer than one time step. In the second time step, some solvent leaves grid block 2 by the same sequence of events. Thus, there is a smearing of the composition proﬁle that comes from the treatment of the blocks as mixing cells with ﬁnite size. In the limit as the grid blocks become inﬁnitely small, that smearing disappears.

8.2

Comparison of Numerical and Analytical Solutions

If the analytical solutions obtained in Chapters 4-7 are correct, then numerical solutions obtained with a ﬁnite diﬀerence calculation will converge to the analytical solutions as the number of grid blocks is increased. To illustrate convergence and how numerical dispersion aﬀects compositional simulation results, we consider ﬁrst a simple ternary system with constant K-values (K1 = 2.5, K2 = 1.5, and K3 = 0.05), which might represent a CH4 /CO2 /C10 system at a relatively low pressure. In this example, the initial oil mixture contains no CH4 , and the injection gas is pure CH4 . The mobility ratio was ﬁxed at M = 5, and the simulations were performed by solving Eq. 8.1.1 with ﬁxed ratio of time step size to grid block size, ∆τ /∆ξ = 0.1. Figure 8.1 compares the composition path obtained by compositional simulation for ﬁnite difference grids of 20, 100, 1000, and 10,000 grid blocks with the composition path of the analytical solution. The corresponding saturation and composition proﬁles are also shown in Fig. 8.1, and recovery curves for CO2 and C10 are shown in Fig. 8.2. The analytical solution includes a leading semishock from the initial composition to a point on the initial tie line, a very short rarefaction along the initial tie line, a long nontie-line rarefaction that connects the initial tie line to the injection tie line, and a trailing shock to the injection composition. Comparison of the ﬁnite diﬀerence (FD) and analytical (MOC) solutions shown in Figs. 8.1 and 8.2 reveals several important points. First, the FD solutions do converge to the analytical solution, conﬁrming that the two approaches are consistent. However, the rate of convergence is not high, and very ﬁne computational grids are required for this problem if the details of the solution are to be reﬂected accurately. The FD solutions with 20 and 100 grid blocks show signiﬁcant deviations of the calculated composition paths and composition proﬁles. In this displacement, the CO2 that

216

CHAPTER 8. COMPOSITIONAL SIMULATION

CH4 a

Injection Gas

a Injection Gas Tie Line Dilution Line 20

Initial Tie Line 100

Oil

C10 1

1000 a

a

a a

CO2

a

MOC 20 Blocks 100 Blocks 1000 Blocks 10,000 Blocks

Sg

aa a a a

a

CH4

0 1

aa a

0 1

CO2

aa1000 a 0 1

a

100 20

a

C10

a 0 0.0

a a

a 1.0

ξ/τ

aa 2.0

Figure 8.1: Eﬀects of numerical dispersion on a vaporizing gas drive for a ternary system with constant K-values, K1 = 2.5, K2 = 1.5, and K3 = 0.05. The initial composition is C1 = 0, C2 = 0.3760, C3 = 0.6240, and the injection gas is pure C1 . For all simulations, ∆τ /∆ξ = 0.1, and M = 5.

8.2. COMPARISON OF NUMERICAL AND ANALYTICAL SOLUTIONS

217

Fraction of CO2 and C10 Recovered

1.0 C10

CO2

0.8

0.6

0.4

Analytical Solution 20 Grid Blocks 100 Grid Blocks 1000 Grid Blocks 10,000 Grid Blocks

0.2

0.0 0

1

τ

2

3

Figure 8.2: Component recovery for the displacements shown in Fig. 8.1. is present initially in the oil is swept up into a bank (see the CO2 proﬁle in Fig. 8.1). That bank is poorly resolved with 20 and 100 grid blocks but is much better resolved with 1000 grid blocks. With 10,000 grid blocks, however, the FD solution is nearly indistinguishable from the the MOC solution on the scale of the plots in Fig. 8.2. Thus, the FD solution does, in fact, converge to the analytical solution, but it is clear that this problem is relatively sensitive to the eﬀects of numerical dispersion (see Section 8.3). Second, the FD solutions can resolve the key tie lines, shocks and rarefactions that are important parts of the solution, but they do so only if the FD grid includes enough grid blocks. Regions of the solution where compositions are changing rapidly, the shocks and the nontie-line rarefaction, are the most diﬃcult to capture in the FD solutions. The leading and trailing shocks are resolved somewhat better at a given grid resolution because they are self-sharpening. The nontie-line rarefaction is smeared much more. The composition gradient is signiﬁcant along the nontie-line path, and numerical dispersion acts to reduce that gradient. When that rarefaction is smeared, the FD composition path follows closely a path that resembles the nontie-line paths obtained in the analytical solution. As the grid is reﬁned, the nontie-line portion of the FD solution approaches more closely the MOC nontie-line path. Calculated component recovery (Fig. 8.2) is also quite sensitive to grid resolution. Because recovery at some late time in a displacement (often 1.1 or 1.2 pore volumes injected) is often used as an indicator of multicontact miscibility, it is important that eﬀects of numerical dispersion on recovery be assessed when compositional simulations are used to estimate minimum miscibility pressure (see Section 8.4). Here it is enough to note that when numerical dispersion alters composition path signiﬁcantly, it can also have a quite signiﬁcant eﬀect on calculated recovery, particularly in multicomponent systems at pressures or enrichments near the MMP or MME.

218

CHAPTER 8. COMPOSITIONAL SIMULATION

The FD solutions reﬂect the interplay of dispersion (numerical in this case) and convection. Fig. 8.1 shows the dilution line that connects the initial oil composition to the injection gas composition. The eﬀect of dispersion is to move the solution composition path toward the dilution line. If there were no ﬂow at all, then mixtures of the initial and injection ﬂuids would lie on the dilution line. The eﬀect of convection, and the accompanying chromatographic separations of components that take place as components partition between the ﬂowing phases, is to push the composition path closer to the MOC solution, in which eﬀects of dispersion are absent. The Peclet number reﬂects the relative importance of the contributions of dispersion and convection (see Section 2.7). In this example, the Peclet number can be estimated with Eq. 8.1.2. The value of df1 /dS1 is not constant throughout the solution but for the purposes of estimating the eﬀects of dispersion, it is convenient to use the maximum value of df1 /dS1 in Eq. 8.1.2, because the resulting value of the Peclet number will also determine whether the numerical computation is stable (it is if P e−1 is positive). That value is about 2.45 for M = 5 and the relative permeability functions of Eqs. 4.1.14-4.1.19 with Sor = Sgc = 0, the values used for all the constant K-value solutions discussed in this chapter. Table 8.1 reports approximate Peclet numbers for the four FD simulations. The solutions with P e > 1000 are close to the analytical solution. The estimate given in Section 2.7 of the Peclet number appropriate to slim tube displacements (P e = 2500) suggests that about 2500 grid blocks would be required for the level of numerical dispersion in the FD calculation to approximate the physical value. With that grid resolution, the FD solution would approximate closely the dispersionfree MOC solution, another indication that the use of the analytical solutions for 1D displacements like those performed in slim tubes is a reasonable approach. Systems with K-values that depend on composition display similar behavior. To examine the impact of numerical dispersion in simulations with variable K-values, Eq. 8.1.1 was solved using the Peng-Robinson EOS with the pressure at which the phase behavior was evaluated held constant. Fig. 8.3 compares the analytical solution for a six-component displacement with no volume change(see Fig. 7.5) with FD solutions calculated with 50, 500, and 5000 grid blocks. In this system, the FD solutions converge reasonably rapidly to the analytical solutions as the grid is reﬁned. With 500 grid blocks, much more limited smearing of the shocks is observed, and with 5000 grid blocks, the FD solution is almost indistinguishable from the MOC solution. The agreement is similar when the eﬀects of volume change are included, as Fig. 8.4 shows. These examples demonstrate that FD compositional simulation can produce solutions that converge to the analytical solution if suﬃciently ﬁne grids are used. The computational cost is much higher for the FD solutions, of course. The FD solutions are needed, however, to deal with situations in which the pressure at which phase behavior is evaluated is not constant or when the injection composition is not constant, because the analytical solutions derived here are for Riemann problems only in which the initial and injection compositions are constant. Finite diﬀerences are also used for two-dimensional and three-dimensional compositional simulations, for which analytical solutions are not available. While computational cost for these one-dimensional calculations is not a problem, corresponding two- and three-dimensional simulations are often too slow to allow use of large numbers of grid blocks in each dimension. It is rare, for example, to see use of as many as twenty grid blocks between wells in ﬁeld-scale calculations. Hence, it is likely that eﬀects of numerical dispersion on calculated composition paths will be signiﬁcant in multidimensional FD compositional simulations. One approach to dealing with the diﬃculties that arise from numerical dispersion in FD calculations is to decouple the representation of the eﬀects of reservoir heterogeneity, which control where low viscosity injected gas ﬂows preferentially, from the kinds of chromatographic dtermination of

8.2. COMPARISON OF NUMERICAL AND ANALYTICAL SOLUTIONS

1

219

Sg

MOC 50 Grid Blocks 500 Grid Blocks 5000 Grid Blocks

0 1

C10

CO2

0.2

0.0 0.2

C16

CH4

0 1

0

0.0 0.2

C4

C20

0.2

0.0

0.0 0

1

ξ/τ

2

0

1

ξ/τ

2

Figure 8.3: Saturation and composition (mole fraction) proﬁles for displacement of a six-component oil (Oil A in Table 7.2 by pure CO2 at 2940 psia (200 atm) and 160 F (71C). Eﬀects of volume change as components change phase are not included in this example. The solid line is the analytical solution, and the dotted lines are the FD solution. See the discussion of Fig. 7.6 for a description of the MOC solution for this fully self-sharpening example.

220

CHAPTER 8. COMPOSITIONAL SIMULATION

1 MOC 50 Grid Blocks 500 Grid Blocks 5000 Grid Blocks

vD

Sg

1

0

0

1

C10

CO2

0.2

0 1

0.0

C16

CH4

0.2

0.0

0.2

0.2

C4

C20

0

0.0

0.0 0

1

ξ/τ

2

0

1

ξ/τ

2

Figure 8.4: Saturation and composition (mole fraction) proﬁles for the displacement of a sixcomponent oil (Oil A in Table 7.2 by pure CO2 at 2940 psia (200 atm) and 160 F (71 C). This example includes the eﬀects of volume change as components change phase. The solid line is the analytical solution, and the dotted lines are the FD solutions.

8.3. SENSITIVITY TO NUMERICAL DISPERSION

221

composition paths considered in this book through the use of streamline models [121, 8, 120, 16, 43]. The eﬀects of heterogeneity are represented by the streamlines. If spatial variations in permeability dominate the ﬂow, then it can be quite diﬃcult to move streamlines away from zones of high permeability. Indeed, one of the signiﬁcant reasons that it can be diﬃcult to achieve high overall reservoir sweep eﬃciency is that low viscosity gas ﬁnds and continues to ﬂow preferentially in high permeability zones, and gravity segregation can partially mitigate or aggravate the eﬀects of permeability variation, depending on injection rates and the spatial distribution of permeability. When permeability variations dominate the ﬂow, the locations of streamlines can be recomputed at relatively long intervals (in eﬀect FD simulations recalculate streamlines every time step). The compositional eﬀects can then be represented by one-dimensional solutions calculated for each streamline. There is no attempt to represent the eﬀects of component transport across streamlines (by diﬀusion, physical dispersion, and viscous crossﬂow), and the assumption that the displacement pressure is constant for the purposes of evaluation of phase behavior is clearly not strictly satisﬁed, however, so this approach also has limitations. For problems in which the restrictions of Riemann problems (constant initial and injection compositions) are satisﬁed, the one-dimensional analytical solutions obtained in Chapters 4-7 can be used as the one-dimensional solution along streamlines. Because those solutions are self-similar, they can be evaluated only once and applied repeatedly as the one-dimensional solutions are propagated along streamlines. The streamline approach, particularly when analytical solutions can be applied along streamlines, can be orders of magnitude faster than FD compositional simulation if it is reasonable to update streamlines relatively infrequently, and it has the advantage that it is much less subject to the adverse eﬀects of numerical dispersion [43]. For an example of application of streamlines with analytical one-dimensional solutions to a ﬁeld-scale displacement of condensate by CO2 , see Seto et al. [109]. If the initial or injection compositions are not constant, or if new wells are added, then a numerical one-dimensional solution is required[16]. In such cases, attention to the sensitivity of calculated composition paths to numerical dispersion, the subject of the next section, will be needed.

8.3

Sensitivity to Numerical Dispersion

The example shown in Fig. 8.1 is one that is relatively sensitive to the eﬀects of numerical dispersion. Fig. 8.5 compares FD and MOC solutions for a system that is less sensitive to dispersion. The only diﬀerence between the displacements in Figs. 8.1 and 8.5 is the K-value of the intermediate component, which is K2 = 1.5 in Fig. 8.1 and K2 = 0.5 in Fig. 8.5. Recovery curves for the displacement illustrated in Fig. 8.5 are reported in Fig. 8.6. This displacement is vaporizing gas drive with a low-volatility intermediate component, and it is fully self-sharpening. The only rarefaction occurs along the initial tie line, and a shock connects the initial and injection tie lines. In this example, the FD composition paths for all the grid resolutions deviate much less from the MOC solution than did the the paths, proﬁles, and recovery shown in Figs. 8.1 and 8.2. The composition paths all lie relatively close to the initial and injection tie lines and the intermediate (nontie-line) shock. The intermediate shock is smeared more than the leading and trailing shocks, because it is only weakly self-sharpening. Comparison of the results of Figs. 8.5 and 8.6 with those of Figs. 8.1 and 8.2 indicates that

222

CHAPTER 8. COMPOSITIONAL SIMULATION

CH4 a

Injection Gas

a

a a

Dilution Line

Oil

C10 a a

Sg

1

a a

a

a a

a a

CO2

0 1

0 1

MOC 20 Grid Blocks 100 Grid Blocks 1000 Grid Blocks

a a a

CH4

0 1

CO2

a

a

1000 100 20

a a

a a

C10

a 0 0.0

a a

a

a 1.0

ξ/τ

2.0

Figure 8.5: Eﬀects of numerical dispersion on a vaporizing gas drive for a ternary system with constant K-values, K1 = 2.5, K2 = 0.5, and K3 = 0.05. The initial composition is C1 = 0, C2 = 0.3760, C3 = 0.6240, and the injection gas is pure C1 . For all simulations, ∆τ /∆ξ = 0.1, and M = 5.

8.3. SENSITIVITY TO NUMERICAL DISPERSION

223

1.0

Fraction of C4 and C10 Recovered

CO2 0.8

C10

0.6

0.4

MOC Solution 20 Grid Blocks 100 Grid Blocks 1000 Grid Blocks

0.2

0.0 0

1

τ

2

3

Figure 8.6: Component recovery for the displacements shown in Fig. 8.5. the sensitivity of calculated displacement behavior to the eﬀects of numerical dispersion depends on the details of the phase behavior of the system because the two systems, which diﬀer only in the K-value of the intermediate component, show very diﬀerent rates of convergence to the MOC solution. Indeed, the FD solution with 1000 grid blocks for K2 = 0.5 achieves approximately the level of agreement with the MOC solution that requires 10000 grid blocks when K2 = 1.5. The reasons for the diﬀerences lie in the orientation of the key tie lines with respect to the dilution line. Systems in which the dilution line lies close to and nearly parallel to the key tie lines might be expected to show relatively low sensitivity to numerical dispersion, because any deviations in path toward the dilution line do not depart dramatically from no-dispersion solution path. On the other hand, systems in which the key tie lines cross the dilution line at large angles and the no-dispersion path includes portions that are well away from the dilution line, path deviations toward the dilution line caused by numerical dispersion can lead to much larger diﬀerences between the numerical and analytical paths. Because component recovery depends strongly on composition path in displacements in which signiﬁcant component transfers occur between phases, these displacements show much greater sensitivity to the eﬀects of dispersion, whether physical or numerical. Displacements that are multicontact miscible or nearly so often show signiﬁcant sensitivity to numerical dispersion. An indication of the sensitivity of a particular displacement to the eﬀects of numerical dispersion is given by the magnitude of the deviation of the composition path from the dilution line. Fig. 8.7 illustrates one way in which that deviation can be measured, in terms of a dispersive distance, for a ternary system at its minimum miscibility pressure. In this displacement, the no-dispersion composition path jumps from the initial oil composition to the critical point and then traces the vapor portion of the binodal curve to the injection gas tie line. The perpendicular distance from the

224

CHAPTER 8. COMPOSITIONAL SIMULATION

1 aInjection Gas

Dilution Line

a a

a

Critical Point Dispersive Distance

Oil

0 0

1

Figure 8.7: Schematic representation of the dispersive distance in cartesian coordinates for a ternary system. Table 8.2: Oil and Gas Compositions for Displacements with Four Components (Mole Fractions) Oil A N2 CH4 C4 C10

xoil 0.0 0.5 0.1624 0.3376

ygas,1 1.0 0.0 0.0 0.0

ygas,2 0.1 0.9 0.0 0.0

dilution line to the critical point is a measure of the diﬀerence between the dispersion-dominated path (the dilution line) and the convection-dominated path. When that distance is relatively large, the eﬀect of disperion is relatively large, as the example in Figs. 8.1 and 8.2 demonstrates. When that distance is relatively small, the impact of dispersion on composition path is smaller, as in the displacements of Figs. 8.5 and 8.6. To test whether the dispersive distance provides a reasonable quantitative measure of the eﬀect of numerical disperion in a displacement, we consider ﬁrst two four-component displacements. The oil is a ternary mixture of CH4 , C4 , and C10 (see Table 8.2). Two injection gases, pure N2 , and a mixture with 90 mole percent CH4 and 10% N2 (Table 8.2) illustrate the diﬀerences in sensitivity that are reﬂected in the dispersive distance. The composition path for displacement by pure N2 is shown in Fig. 8.8 This phase diagram is is plotted for a displacement pressure of 305 atm at 344 K. In this system, the initial oil tie line is the tie line that lies closest to the critical locus, and hence it controls development of miscibility. The initial oil tie line is quite short, and hence this displacment is nearly miscible. The MMP for this gas/oil pair is 309 atm. Also shown in Fig. 8.8 is a FD composition path computed with 100 grid blocks. This is a system with a relatively large

8.3. SENSITIVITY TO NUMERICAL DISPERSION

225

CH4

MOC FD

• Initial oil

N2

•

C4

C10 Figure 8.8: Displacement of Oil A by pure N2 at 4482 psia (305 atm) and 161 F (71 C), conditions that are quite close to the MMP, which is 4541 psia (309 atm). This displacement has a relatively large dispersive distance of 0.173 because the initial tie line, which controls miscibility, is relatively distant from the dilution line. The FD path was calculated with 100 grid blocks. dispersive distance (0.173), and the FD composition path shows signiﬁcant eﬀects of numerical dispersion. Fig. 8.9 compares oil recovery at 1.2 pore volumes injected, as a function of pressure, for the FD and analytical solutions. The recovery estimates obtained in the low resolution FD solutions diﬀer signiﬁcantly from the analytical solution, and they converge relatively slowly to the analytical results. In this system, careful reﬁnement of the FD grid would be required to obtain an accurate estimate of the MMP. Figs. 8.10 and 8.11 show similar plots for displacement of the same oil by the second gas mixture containing 90 mole percent CH4 and 10% N2 . Because the initial tie line is the same, the MMP is also unchanged, but the dispersive distance is much smaller. The composition paths shown and the corresponding recovery curves show that this system is much less sensitive to the eﬀects of numerical dispersion. The FD recovery for Oil 1 displaced by Gas 1 is 30 percent below the MOC recovery at the MMP (Fig. 8.8), but for Oil 1 displaced by Gas 2, the FD recovery is about 15 percent below the MOC value. Thus, qualitatively at least, the idea that the dispersive distance gives an indication of sensitivity to eﬀects of numerical dispersion is supported by these results for simple systems. Jessen et al. [47] investigated the quantitative response of a number of multicomponent gas/oil displacements to variations in initial oil composition and injection gas composition that induced signiﬁcant corresponding variations in dispersive distance. Table 8.3 reports the compositions of the three oils used and three gas mixtures that were then diluted with varying fractions of N2 , CH4 ,

226

CHAPTER 8. COMPOSITIONAL SIMULATION

Recovery at 1.2 PVI

1.0

0.9

0.8

0.7

100 grid blocks 500 grid blocks 1000 grid blocks 5000 grid blocks MOC

0.6

0.5 200

300

400

500

Pressure (atm) Figure 8.9: Diﬀerence in calculated recovery between ﬁnite diﬀerence solutions with with various grid resolutions and the analytical solution for displacement of Oil A by pure N2 at 4482 psia (305 atm) and 160 F (71 C). and CO2 . The characterization of Oil B and the associated separator Gas B is reported by Jessen [42]. It has a calculated bubblepoint pressure of 249 atm at 387 K. Oil C was studied by Høier [36], who reports the EOS characterization used here. At 368 K, the bubblepoint pressure of that oil is 251 atm. Oil D is one of the examples reported by Zick [140]. The characterization of the components in this oil is reported in Table 7.5. It has a calculated saturation pressure of 102 atm at 358 K. To test the use of the dispersive distance to assess a priori the sensitivity of a FD simulation to dispersion, a series of simulations was carried out with a variety of injection gases for a range of pressures above and below the MMP. For each displacement, the key tie lines were calculated, the shortest tie line was identiﬁed, and the dispersive distance was evaluated as the orthogonal distance between that tie line (of zero length) and the dilution line at the MMP. Table 8.4 summarizes the oil and gas compositions used and reports the dispersive distances. In Table 8.4, the mechanism is identiﬁed as V, for a vaporizing drive in which the initial oil tie line controls development of miscibility, or C/V, which indicates a condensing/vaporizing displacement in which one of the crossover tie lines is critical at the MMP. Fig. 8.12 shows the results of the simulations. It shows the diﬀerence in calculated recoveries at 1.2 pore volumes injected between the FD result and the MOC solution with the pressure set at the MMP. The recovery at the MMP for the MOC solution is 100 percent, of course. The FD recoveries agree better with the MOC results (see Figs. 8.10 and 8.11) at pressures well below the MMP, when composition paths are less strongly aﬀected by phase behavior, and at pressures well above the MMP, when the twophase region is smaller and the negative impact of dispersion applies over a smaller fraction of the

8.3. SENSITIVITY TO NUMERICAL DISPERSION

227

CH4 •

MOC FD

• Initial oil

N2 C4

C10 Figure 8.10: Displacement of Oil A by a gas mixture containing 10% N2 and 90% CH4 at 4482 psia (305 atm) and 161 F (71 C). This displacement has a smaller dispersive distance of 0.091 because the dilution line is now closer to the initial tie line. Table 8.3: Oil and Gas Compositions of Multicomponent Displacements

Comp N2 CO2 CH4 C2 C3 i-C4 n-C4 i-C5 n-C5 C6 C7 C11 C16 C23 C33

Oil B xoil 0.00450 0.01640 0.45850 0.07150 0.06740 0.00840 0.03110 0.01030 0.01650 0.02520 0.12440 0.06320 0.05024 0.03240 0.01996

ygas 0.0049 0.0182 0.8139 0.0915 0.0467 0.0050 0.0124 0.0020 0.0026 0.0009 0.0019

Oil C Comp xoil N2 0.00785 CO2 0.00265 CH4 0.45622 C2 0.06092 C3 0.04429 i-C4 0.00865 n-C4 0.02260 i-C5 0.00957 n-C5 0.01406 C6 0.02097 C7+1 0.04902 C7+2 0.09274 C7+3 0.09880 C7+4 0.07362 C7+5 0.03804

Oil D xoil

ygas

CO2 CH4 C2 C3

0.04400 0.20041 0.04777 0.04012

0.2218 0.2349 0.2350 0.2745

n-C4

0.02992

0.0338

n-C5 C6 C7 C13 C19 C27 C38

0.02342 0.05873 0.22013 0.13398 0.09970 0.06215 0.03916

Comp

228

CHAPTER 8. COMPOSITIONAL SIMULATION

Recovery at 1.2 PVI

1.0

0.9

0.8

0.7

100 grid blocks 500 grid blocks 1000 grid blocks 5000 grid blocks MOC

0.6

0.5 200

300

400

500

Pressure (atm) Figure 8.11: Diﬀerence in calculated recovery between ﬁnite diﬀerence solutions with with various grid resolutions and the analytical solution for displacement of Oil A by a gas mixture containing 10% N2 and 90% CH4 at 4482 psia (305 atm) and 161 F (71 C). displacement length. Hence, the diﬀerences in recoveries at the MMP are the maximum observed over the range of pressures investigated. Fig. 8.12 demonstrates a clear relationship between the magnitude of the dispersive distance and the diﬀerence in recovery that results from numerical dispersion, though there is considerable scatter about the two trend lines shown. One of those lines, indicates, approximately, the diﬀerence in recovery for FD solutions with 100 grid blocks, and the second, the same diﬀerence for solutions obtained with 1000 grid blocks. The diﬀerences are smaller for the ﬁne grid simulations, but they can still be signiﬁcant for systems with large values of the dispersive distance. Fig. 8.12 suggests that an assessment of the sensitivity of a particular gas/oil system to numerical dispersion can be made in two ways. The sensitivity can be determined directly by numerical simulation alone simply by performing ﬁne and coarse grid simulations at a series of pressures (preferably in the range of interest, including the MMP if miscible displacement is the goal) and monitoring the changes in predicted recovery as a function of grid resolution and pressure. A much more computationally eﬃcient approach is to use the analytical solutions of Chapter 7 for the key tie lines to estimate the dispersive distance. If the displacement pressure of interest is not the MMP, then the dispersive distance can be estimated by taking the orthogonal distance between the dilution line and midpoint of the section of the shortest tie line between the two equal eigenvalue points that bound the rarefaction along that tie line. In practice, simply taking the midpoint of the shortest tie line will give a quite reasonable estimate, accurate enough for the purpose of determining whether the system in question is relatively sensitive, or not, to numerical dispersion. Fig. eee can then be used to determine whether a careful grid reﬁnement study is warranted for

8.3. SENSITIVITY TO NUMERICAL DISPERSION

229

Table 8.4: Dispersive Distances for Gas Displacements.

No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

Oil A A A A A A B B B B B B B B C C C C C C C C D D D D D D D D D D D D

Injection Gas N2 75% N2 /25 % CH4 50% N2 /50 % CH4 25% N2 /75 % CH4 10% N2 /90 % CH4 CH4 Gas B CH4 CO2 25% N2 /75 % CH4 30% N2 /70 % CH4 35% N2 /65 % CH4 40% N2 /60 % CH4 N2 CH4 25% N2 /75 % CH4 30% N2 /70 % CH4 40% N2 /60 % CH4 50% N2 /50 % CH4 65% N2 /35 % CH4 75% N2 /25 % CH4 N2 Gas D 90% Gas D + 10% CH4 80% Gas D + 20% CH4 60% Gas D + 40% CH4 40% Gas D + 60% CH4 20% Gas D + 80% CH4 CH4 90% Gas D + 10% CO2 80% Gas D + 20% CO2 70% Gas D + 30% CO2 50% Gas D + 50% CO2 25% Gas D + 75% CO2

T (K) 344 344 344 344 344 344 387 387 387 387 387 387 387 387 368 368 368 368 368 368 368 368 358 358 358 358 358 358 358 358 358 358 358 358

MMP (atm) 309 309 309 309 309 309 365 371 226 380 380 380 380 380 509 537 533 533 537 537 537 537 148 171 198 260 342 456 471 159 171 184 220 329

Mechanism V V V V V V C/V V C/V V V V V V C/V V V V V V V V C/V C/V C/V C/V C/V C/V V C/V C/V C/V C/V C/V

Dispersive Distance 0.17357 0.17504 0.16269 0.16120 0.09095 0.00379 0.01470 0.04475 0.16515 0.12537 0.14961 0.16993 0.18518 0.19765 0.04307 0.17061 0.21000 0.27000 0.30161 0.31354 0.31254 0.30361 0.06930 0.06698 0.06686 0.05512 0.03463 0.02620 0.09057 0.06769 0.06618 0.06463 0.06099 0.05403

230

CHAPTER 8. COMPOSITIONAL SIMULATION

Difference in Recovery (%)

50

b c

40

(MOC-FD)/MOC, 100 grid blocks (MOC-FD)/MOC, 1000 grid blocks

b

30

b 20

10

b bb b bb bb bbb b b

0 0.0

b c

c

b

b b

b b b bbb

b c

b

b

b b

b

bb

c c c

c

0.1

0.2

0.3

Dispersive Distance Figure 8.12: Diﬀerence in recovery between ﬁnite diﬀerence solutions with 100 and 1000 grid blocks and the analytical solution as a function of dispersive distance. additional compositional simulations. Haajizadeh et al. [28] and Wang and Peck [131] described diﬀerences in sensitivity to numerical dispersion for systems with diﬀering displacement mechanisms (condensing, vaporizing, condensing/vaporizing) and suggested that the observed sensitivity was associated with the displacement mechanism. The results presented in Table 8.4 and Fig. eee suggest that it is not the displacement mechanism that controls the sensitivity of the displacement simulation to dispersion but the distance between the composition path and the dilution line that is responsible. For example, the vaporizing N2 displacements in Table 8.4 (in particular, displacements 1-4, 10-15, and 16-22) show relatively large dispersive distances and and signiﬁcant sensitivity, while some vaporizing displacements by CH4 (displacements 6, 8, and 15) show relatively small dispersive distances. One CH4 vaporizing displacement (29) shows more sensitivity, however. Thus, the sensitivity of a particular displacement should be assessed directly for the speciﬁc gas/oil system.

8.4

Calculation of MMP and MME MMP Calculation Using a Single Mixing Cell

At least three computational approaches have been proposed for calculation of MMPs and MMEs: mixing cell calculations for a single cell, multiple mixing cell calculations and compositional simulations. Next we examine the accuracy of single-cell methods. The original versions were based on the analysis of ternary systems in which miscibility occurs if either the initial oil tie line or the injection gas tie line is a critical tie line. The initial oil tie line can be identiﬁed by the following

8.4. CALCULATION OF MMP AND MME

231

sequence of calculations: 1. Mix a quantity of injection gas with the initial oil to make a mixture with an overall composition in the two-phase region that lies on the dilution line that connects the initial oil and injection gas compositions. This composition could be created physically by mixing the same quantities of oil and gas in a high pressure cell used to measure equilibrium phase compositions. 2. Calculate the equilibrium compositions of the liquid and vapor phases that form for the overall composition in the mixing cell using an equation of state and an appropriate set of parameters that describe the components in the mixture. 3. Mix the vapor phase that results from the previous contact with an additional quantity of original oil in quantities suﬃcient to make a new mixture in the two-phase region, and calculate the resulting equilibrium vapor and liquid phase compositions. 4. Repeat the mixing of fresh oil with the equilibrium vapor until the compositions of the resulting liquid and vapor phases converge. When they do, the liquid and vapor phase compositions will lie on the tie line that extends through the initial oil composition. This process of repeated mixing of an equilibrium vapor phase with fresh oil is sometimes called forward contacts. The injection gas tie line can be identiﬁed by a similar sequence of calculations in which the liquid phase that results from each mixing step is mixed with fresh injection gas (sometimes called reverse contacts. This calculation converges to the liquid and vapor compositions that lie on the tie line that extends through the injection gas composition. Once the injection gas and initial oil tie lines have been identiﬁed by the mixing cell procedure, the MMP (or MME) is calculated by changing the pressure (or injection gas enrichment) until the one of the key tie lines (injection or initial oil) has zero length. This procedure produces the correct MMP or MME for ternary systems, but it does not necessarily do so for systems with more than three components. If miscibility is controlled by the injection gas tie line or the initial oil tie line, then the result obtained will be correct. If, on the other hand, one of the crossover tie lines is responsible for the development of miscibility, the mixing cell method will give an MMP that is higher than the true MMP or an enrichment that is higher than the true MME. The four-component system, CO2 /CH4 /C4 /C10 , illustrates the reason for the error that can arise in the calculation of the MMP using a single mixing cell. Jensen and Michelsen[41] considered this system in their discussion of MMP calculations. They used the Soave-Redlich-Kwong equation of state (EOS) to describe the ﬂuid system. While the quantitative results given below diﬀer slightly from those that would have been obtained with the Peng-Robinson EOS, which was used to calculate the phase diagrams and displacement proﬁles for this ﬂuid system in Chapter 6, the patterns of MMP behavior are completely consistent for the two equations of state. The oil mixture of Jensen and Michelsen contains 40 mole percent CH4 , 5 percent CO2 , 20 percent C4 , and 35 percent C10 . Fig. 8.13 shows the composition path and the resulting sequence of tie lines for the reverse contacts calculated by mixing 80 mole percent injection gas with the original oil in the ﬁrst step and then with the liquid phase that results from each additional contact. The dot in the middle of each tie line is the overall composition at each step. Also shown with dots are the initial oil and injection gas compositions, which lie on the extensions of the oil tie line and

232

CHAPTER 8. COMPOSITIONAL SIMULATION

CO2

←

Gas Tie Line • Gas aaa a

Oil Tie Line ←

C10

Oil a

CH4

C4

Figure 8.13: Composition route and tie lines for reverse contacts of oil containing 5 mole percent CO2 , 40 mole percent CH4 , 20 mole percent C4 , and 35 mole percent C10 with injection gas containing 50 percent CO2 and 50 percent CH4 at 2557 psia and 160 F . the injection gas tie line. Fig. 8.13 conﬁrms that the reverse contacts converge to injection gas tie line, and they do so in only a few steps. Fig. 8.13 shows the ﬁrst ﬁve tie lines that result from the mixing steps, but only the ﬁrst two can be seen clearly. The others are so close to the injection gas tie line that they are diﬃcult to see. Fig. 8.14 shows the corresponding composition path and resulting tie lines for the forward contacts. The forward contacts converge much more slowly but do converge to the initial oil tie line. The ﬁrst ten tie lines and then the tie lines for contacts 12, 14, 16, 18 and 20 are shown. Again, the dots on the tie lines are the overall compositions of the mixtures created by mixing 80 mole percent equilibrium vapor from each contact with initial oil. While the exact composition paths followed by the mixtures formed by forward and reverse contacts depend on the proportions of mixing vapor and liquid at each step, the ﬁnal tie lines do not. They are the injection gas and initial oil tie lines. Figures 8.15 and 8.16 show why the single cell approach sometimes gives accurate MMPs and sometimes does not. Fig. 8.16 compares MMPs calculated by the single cell and key tie line approaches for injection gases that are mixtures of CH4 and CO2 . Fig. 8.15 shows the fourcomponent phase diagram at 1600 psia and 160 F . For injection gas mixtures that contain less that 6.6 percent CO2 (I2 in Fig. 8.16), the two methods give the same MMP. When the injection gas is nearly pure CH4 , the initial oil tie line lies closest to the critical locus, and as pressures are increased, and the two-phase region shrinks, the oil tie line is the critical tie line at the MMP. Hence, the single cell approach is correct for these injection gas mixtures. It ﬁnds the oil tie line and determines when it is a critical tie line. For gas mixtures that contain more CO2 , however, the crossover tie line is the critical tie line at the MMP. For these injection gas mixtures, the single cell approach is not accurate, as Fig. 8.16 shows. The crossover tie line lies closer to the critical locus in Fig. 8.15, and as a result, a smaller pressure increase is required (above the pressure at which

8.4. CALCULATION OF MMP AND MME

233

CO2

←

Gas Tie Line • Gas

Oil Tie Line ←

C10

Oil a

a a a a a aa aa aa aaa a

CH4

C4

Figure 8.14: Composition route and tie lines for forward contacts of oil containing 5 mole percent CO2 , 40 mole percent CH4 , 20 mole percent C4 , and 35 mole percent C10 with injection gas containing 50 percent CO2 and 50 percent CH4 at 2557 psia and 160 F . the phase diagram is plotted) to shrink the two-phase region enough to make the crossover tie line critical. The sequence of mixing operations for the forward and reverse contacts cannot identify the crossover tie line, and hence cannot determine at what pressure the crossover tie line is critical. The MMP calculated with the single cell method is 200-350 psia higher than the MMP obtained with the key tie line approach, a signiﬁcant error. Similar situations arise often in systems with more than four components. Consider, for example, the gas/oil system studied by Zick [140] (see Section 7.5 for analytical calculations of the MMP for this system). In this system one of the crossover tie lines is the critical tie line at the MMP when Gas 1 displaces Oil 1 (see Table 7.4). To examine the impact of changes in injection gas composition, Gas 1 was diluted with CH4 , and the MMP was calulated by the key tie line method. The resulting injection gas compositions for 20 mole percent and 23 mole percent dilution of Gas 1 by CH4 are reported as Gas 2 and Gas 3 in Table 7.4 Fig. 8.17 shows the results. Dilution of the injection gas, which is rich in intermediate components (CO2 , C2 , C3 , and C4 ) with CH4 aﬀects the MMP strongly. Fig. 8.17 can be used, alternately, to estimate the injection gas enrichment required for miscible displacement at a given pressure. At the highest CH4 fractions in the injection gas, the MMP does not change with changes in CH4 fraction. Just as in the example of Fig. 8.16, it is a switch in the tie line that determines miscibility that is responsible for this change in behavior. At the highest CH4 fractions, it is the initial tie line that controls miscibility, and that tie line does not depend on the composition of the injection gas. For lower CH4 fractions, however, one of the crossover tie lines controls miscibility, and those tie lines do depend on injection gas composition. This behavior is typical of displacements in which the injection gas contains signiﬁcant quantities of components other than the one that has the highest K-value. In such displacments, it is likely that a crossover tie line will control miscibility. Here again, this example demonstrates that it is important to use a technique for calculating or estimating MMP or MME that is not dependent on

234

CHAPTER 8. COMPOSITIONAL SIMULATION

CO2 I4

I3

a a

a a

Phase Envelopes Solution Routes

a

aa Vertical Surface

I2 a

CH I1a 4

a

a

a a aa a

C4

O a

C10

Figure 8.15: Composition paths and crossover tie line surface for displacement of oil O (40% CH4 , 5%CO2 , 20% C4 , and 35% C10 ) by gas mixtures I1 to I4 at 1600 psia and 160 F . Figure from Wang [128], used with permission.

4000

Pressure (psia)

3500

3000

2500

2000

1500 0.0

Mixing Cell Key Tie Line 0.2

0.4

0.6

0.8

1.0

Mole Fraction CH4 Figure 8.16: Comparison of MMPs for CO2 -CH4 mixtures calculated by the key tie line method and with a single mixing cell.

8.4. CALCULATION OF MMP AND MME

235

Minimum Miscibility Pressure

8000 7000

Zick Oil 1 Zick Oil 2 Zick Oil 3

6000 5000 4000

a

3000

a

a

2000 0.0

0.2

0.4

0.6

0.8

1.0

Mole Fraction CH4 Figure 8.17: Eﬀect of dilution of Gas 1 by CH4 on MMP calculated by the key tie line method for displacement of Zick’s Oils 1, 2, and 3 [140] (see Table 7.4 for the compositions of the oils and gases). Data points shown are for experimental measurements reported by Zick. foreknowledge of which tie line controls miscibility. The trend in MMP shown in Fig. 8.17 is conﬁrmed by experimental observations reported by Zick [140], also shown in Fig. 8.17. The point for the injection of undiluted Gas 1 is the system described in Section 7.5. It agrees well with the calculated MMP. The two points shown for CH4 dilution provide evidence of the eﬀect of changes in both the oil composition and injection gas composition because the initial oil compositions were calculated for Oils 2 and 3 in Table 7.4. Those oils were made by ﬂashing Zick’s Oil A to pressures of 3000 and 3300 psia. Hence, the initial oil compositions used in the experiments were not constant. Because the initial oil tie line does not control miscibility for those displacements, however, the eﬀect of the changes in oil composition on the calculated MMP is small. The changes in the initial oil composition induced only small changes in the crossover tie line that controls miscibility, and hence the eﬀect on the MMP is also small, as the curves for the three oil compositions in Fig. 8.17 indicate. MMPs calculated with the correct oil compositions diﬀered only slightly for the three oil compositions. Hence, the agreement between the measured MMPs provides conﬁrmation that the trend shown in Fig. 8.17 is consistent with experimental observations. MMP Calculation by Compositional Simulation Two other computational approaches have been used to calculate MMPs and MMEs: compositional simulation and multiple mixing cells. These methods are closely linked. When compositional simulation is used, recovery of the initial oil is calculated at a series of increasing displacement pressures. The MMP is estimated in much the way that it is for experimental observations of recovery

Recovery at 1.2 PVI (fraction)

236

CHAPTER 8. COMPOSITIONAL SIMULATION

1.00

1.00

0.95

0.95

0.90

0.90

2200

0.85

2000

0.80

1800

0.75

1600

MOC 100 Grid Blocks 500 Grid Blocks 1000 Grid Blocks 2500 Grid Blocks Extrap w 100, 500 Extrap w 1000,2500

0.85 0.80 0.75 0.70 2000

2500

3000

3500

0.70 0.00

3600 3200 2800 2600

0.02

0.04

2400 2300

0.06

Pressure (psia)

(Nb)

(a) Grid Resolution

(b) Extrapolation

0.08

0.10

-1/2

Figure 8.18: Extrapolation of calculated recovery to account for eﬀects of numerical dispersion. (a) Recovery as a function of pressure for several grid resolutions. (b) Extrapolation of recovery at several pressures to an inﬁnite number of grid blocks. in slim tube displacements, by interpreting the recovery versus pressure to obtain the MMP. Experimentalists often interpret slim tube displacements on the basis of recovery at 1.1 or 1.2 PVI. While deﬁnitions used to pick the MMP vary among investigators, a typical approach is to interpret recovery above 90 per cent at 1.2 PVI as multicontact miscible, especially if recovery increases only slightly in displacements at higher pressures. Because calculated recovery depends on the grid resolution used in the calculation, it is essential that a grid reﬁnement study be performed if the simulation results are to be used to estimate the MMP. Stalkup [114] suggested that extrapolation of recovery at a given pressure to an inﬁnite number of grid blocks can be used to account for the eﬀects of dispersion. Figs. 8.18 shows one way that can be done. Fig. 8.18a shows calculated recovery as a function of pressure for several grid resolutions. At a given pressure, recovery increases as the grid is reﬁned, though the magnitude of the increase also depends on pressure. Also shown in Fig. 8.18a is the recovery calculated analytically. In this example, the MMP calculated by the key tie line approach is 2250 psia. Fig. 8.18a demonstrates that considerable grid reﬁnement is required if recovery calculated directly by compositional simulation is to be used to estimate the MMP, but the extrapolated recovery lies much closer to the recovery curve obtained analytically. The dispersive distance for this example is 0.1020. Fig. 8.12 suggests that this system is only moderately sensitive to the eﬀects of numerical dispersion. In Fig. 8.18b, calculated recovery at several pressures is plotted versus the inverse of the square root of the number of grid blocks (the use of the square root leads to a more nearly linear curve connecting points at a given pressure). Extrapolation of each of the curves to the zero value on the abscissa gives an estimate of the calculated recovery at an inﬁnite number of grid blocks for

8.5. SUMMARY

237

displacement at a given pressure. The estimated MMP is then the lowest pressure at which the extrapolated recovery is 100 per cent. Fig. 8.18b shows extrapolations of the recovery for the three pressures around the estimated MMP. The curve for 2300 psia extrapolates to a value close to one, and the curve for 2400 psia extrapolates to a value just above one at inﬁnite resolution. Fig. 8.18a compares the recovery calculated by compositional simulation with that from the MOC solutions, and it also shows the extrapolated recovery based on the two coarsest grids and the two ﬁnest grids. Clearly, extrapolation gives an estimate of the of the recovery that is closer to the MOC solution, but even with the ﬁnest grids, there is still some diﬀerence between the MOC solution and the numerical estimate. In this example, the estimated MMP is between 2300 and 2400 psia, based on the extrapolation of the points at the two ﬁnest grid resolutions, a value that is in reasonable agreement with the value obtained by the key tie line approach, 2250 psia. If the two coarsest grid resolutions had been used instead for the extrapolation, the estimated MMP would be about 2400 psia. Thus, compositional simulation can be used to estimate an MMP. The accuracy of that estimate depends on the resolution of the grids used to perform the extrapolation: the ﬁner those grids are, the more accurate the estimate will be. Even in this example system that shows only moderate sensitivity to numerical dispersion, quite ﬁne grids are required to produce an accurate estimate of the MMP. The computation time required to do the compositional simulations is much greater, of course, than that required for calculation of the MMP by the key tie line method. Some investigators have used multiple mixing cells to perform a calculation that is closely related to compositional simulation [77, 97]. In these methods, gas is injected into the ﬁrst of a sequence of mixing cells. An equilibrium ﬂash calculation is performed and liquid and vapor volumes in excess of the cell volume are then moved to the next cell. The volumes of the phases are moved according to the ratio of mobilities of the phases. At each step, excess volumes are moved from each cell to the next downstream cell. Thus, this sequence of calculations is very similar to the calculation performed in a ﬁnite diﬀerence compositional simulation, and it is subject to the same issues with respect to numerical dispersion, which arises from the ﬁnite volume of the mixing cells. In this approach, the MMP is identiﬁed as the minimum pressure at which compositions in cells suﬃciently far downstream approach a critical composition. Because this approach is essentially a ﬁnite diﬀerence solution to the conservation equations, it can identify the tie line that controls miscibility, again if suﬃcient care is taken to deal with the eﬀects of numerical dispersion.

8.5

Summary

This chapter considered solution by numerical methods of the problems attacked analytically in Chapters 4-7. Comparison of ﬁnite-diﬀerence solutions to the conservation equations with the analytical solutions, along with comparison of numerical and analytical methods for calculating minimum miscibility pressure leads to the following conclusions: • Any ﬁnite diﬀerence representation of the conservation equations used to calculate composition paths in a gas/oil displacement will include eﬀects of numerical dispersion, which acts to smear shocks and, to a lesser extent, rarefactions. • Reﬁning the computational grid on which the problem is solved reduces the magnitude of the eﬀects of numerical dispersion. For the simplest ﬁnite diﬀerence representation of the

238

CHAPTER 8. COMPOSITIONAL SIMULATION conservation equations using single-point upstream weighting, very ﬁne grids may be required to resolve composition paths in some gas/oil displacements.

• The impact of numerical dispersion depends on the phase behavior of the gas/oil system. Composition paths for some systems are quite sensitive to dispersion and others are less so. Whether a system is sensitive to eﬀects of numerical dispersion can be estimated by calculating a dispersive distance. • Estimates of MMP or MME obtained from calculations with a single mixing cell are accurate only if the initial oil or injection gas tie line is responsible for development of miscibility. The values obtained will be incorrect if a crossover tie line controls development of miscibility. • Accurate estimates of MMP or MME can be obtained using multiple mixing cells or compositional simulation if careful attention is paid to the eﬀects of numerical dispersion.

8.6

Additional Reading

Many investigators have considered the impact of numerical dispersion on calculated composition paths. For a representative discussion for ternary gas/oil systems that demonstrates that the sensitivity of a particular system to the eﬀects of numerical dispersion depends on the phase behavior of the system, see Walsh and Orr [127]. Haajizadeh et al. [28] and Wang and Peck [131] described a variety of displacements that displayed varying levels of sensitivity to numerical dispersion. A detailed analysis of the reasons for the sensitivity to dispersion for multicomponent systems, with additional examples, is reported by Jessen [42] and by Jessen et al. [47]. A large literature describes the development and application of streamline simulation to prediction of ﬂow in three-dimensional heterogeneous reservoirs. See the papers of King and Datta-Gupta [59] and Crane et al. [16] for many references to the full range of work on streamlines. The use of compositional streamline simulation for gas injection processes was demonstrated by Marco Thiele, Rod Batycky, and Martin Blunt [119, 121, 120]. They noted that one-dimensional analytical solutions could be used in streamline simulations, but used numerical solutions for the examples reported so that direct comparisions with ﬁnite-diﬀerence solutions could be made. Batycky [4] described a three-dimensional streamline simulator. He reported solutions that made use of a variety of analytical solutions for one-dimensional ﬂow problems, but he did not attempt to make use of solutions for gas injection processes with three or more components. Jessen and Orr [43] showed how to combine the streamline approach with multicomponent analytical solutions for three-dimensional gas displacement problems, and Seto et al. [109] applied that approach to the simulation of a gas condensate recovery process. The use of a single mixing cell to calculate MMP has been studied by a number of investigators, including Kuo [60] and Luks et al. [71]. The limitations of mixing cell calculations are discussed by Wang and Orr [128] for a four-component system studied previously by Jensen and Michelsen [41]. The use of multiple mixing cells was described by Metcalfe et al. [77] and by Pedersen et al. [97]. Hearn and Whitson [29] compared MMPs for several gas/oil systems calculated by mixing cell calculations with a single mixing cell, with multiple mixing cells, and by compositional simulation. The idea of extrapolation of ﬁnite diﬀerence simulation results to determine minimum miscibility pressure or minimum enrichment for miscibility was described ﬁrst by Stalkup [114] in his study

8.6. ADDITIONAL READING

239

of the behavior of condensing/vaporizing gas drives. Stalkup [115] also examined how grid resolution and dispersion interact in two-dimensional displacements. Hearn and Whitson [29] applied anextrapolation approach to calculate ”dispersion-free” estimates of the MMP from compositional simulation.

240

CHAPTER 8. COMPOSITIONAL SIMULATION

Nomenclature Symbol

Deﬁnition

A A(u) ai amj aα bi bmj C Ci cij Cp Dij et ent f Fi fj fˆi n nc np Sj g G G Gi H H Hi K ij Ki K krj

Helmholtz function Jacobian matrix van der Waals attraction parameter Mixture attraction parameter for phase j Peng-Robinson attraction parameter van der Waals repulsion parameter Mixture repulsion parameter for phase j Volume fraction of one component in a two-component mixture Overall volume fraction of component i Volume fraction of component i in phase j Constant pressure heat capacity Diﬀusion coeﬃcient of component i in phase j Tie-line eigenvector Nontie-line eigenvector Fugacity of a pure component Overall fractional ﬂow of component i Fractional ﬂow of phase j, volumetric ﬂow rate of phase j per unit volume of total ﬂow Partial fugacity of component i in a mixture Normal vector to a surface Number of components Number of phases Saturation of phase j, volume of phase j per unit volume of pore space gravitational constant Conserved scalar quantity Gibbs function Overall molar concentration of component i, moles per unit volume Molar enthalpy Conserved vector quantity Overall molar ﬂux of component i Dispersion tensor for component i in phase j Equilibrium constant, yi /xi , Longitudinal dispersion coeﬃcient Relative permeability of phase j 241

242

Nomenclature

Lj M n ni p Pci Pe Pisat Pj Pckj R Q Qi Ri S S Sj Sgc Sor T Tci t U u V v vD vj v vs Vt vj W w x xij xi zi

Mole fraction of phase j Viscosity ratio, µliq /µvap Number of moles Number of moles of component i Point on an envelope curve Critical pressure of component i Peclet number, vL/φK Saturation pressure of component i Pressure in phase j Capillary pressure diﬀerence between phases k and j Gas constant Heat energy Volume fraction of component i recovered Mole fraction of component i recovered Control surface Entropy Saturation, volume of phase j per unit volume of pore space Critical gas saturation Residual oil saturation Temperature Critical temperature of component i Time Internal energy Solution vector Volume element Flow velocity in one-dimensional ﬂow Dimensionless ﬂow velocity, v/vinj Scalar ﬂow velocity of phase j Longitudinal ﬂow velocity Flow velocity of a control surface Transverse ﬂow velocity Vector ﬂow velocity of phase j in ﬂows in more than one dimension Work Self-similar solution Distance in the ﬂow direction Mole fraction of component i in phase j Mole fraction or volume fraction of component i in the liquid phase Overall mole fraction of component i

α(η) α(ξ, η) αi αi αn

Slope of a tie line First slope of tie line determined by parameters ξ and η Temperature dependent attraction term for the Peng-Robinson EOS Fractional ﬂow scaled by dimensionless ﬂow velocity, Hi /vD Fractional distance from the liquid composition on key tie line n + 1 to the vapor compositions on key tie line n, Eq. 7.1.15

Nomenclature αp αj αtj β(ξ, η) βn βp γ γp δ φ φ φ(η) φ(ξ, η) φˆi η Λ λt µj µij ωi θ(ξ, η) ρci ρinj ρj ρjD ρmj σ τ ξ

Coeﬃcient in the equation for a plane, Eq. 6.1.28 Longitudinal dispersivity in phase j Transverse dispersivity in phase j Second slope of tie line determined by parameters ξ and η Fractional distance from the liquid composition on key tie line n to the vapor composition on key tie line n + 1, Eq. 7.1.15 Coeﬃcient in the equation for a plane, Eq. 6.1.28 Constant used in ternary constant K-value solution, see Eq. 5.1.50 Coeﬃcient in the equation for a plane, Eq. 6.1.28 Binodal curve intercept, Eq. 6.1.34 Porosity, volume of pore space per unit volume of porous medium Fugacity coeﬃcient Tie line intercept First intercept for tie line determine by parameters ξ and η Partial fugacity coeﬃcient Parameter in solutions by the method of characteristics Shock velocity Eigenvalue Viscosity of phase j Chemical potential of component i in phase j Acentric factor of component i Second intercept for tie line determine by parameters ξ and η Molar density of component i Density of the injection gas Molar density of phase j Dimensionless density, ρj /ρinj Mass density of phase j Binodal curve slope, Eq. 6.1.34 Dimensionless time, pore volumes, vinj t/φL Dimensionless distance, x/L

243

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[122] van der Waals, J.D. On the Continuity of the Gaseous and Liquid States. In J.S. Rowlinson, editor, J.D van der Waals: On the Continuity of the Gaseous and Liquid States, pages 83–140. North-Holland Physics Publishing, 1988. [123] Van Ness, H.C. and Abbott, M.M. Classical Thermodynamics of Nonelectrolyte Solutions With Applications to Phase Equilibrium. McGraw-Hill, San Francisco, 1982. [124] Varotsis, N., Stewart, G., Todd, A.C. and Clancy, M. Phase Behavior of Systems Comprising North Sea Reservoir Fluids and Injection Gases. J. Pet. Tech., pages 1221–1233, November 1986. [125] Wachmann, C. The Mathematical Theory for the Displacement of Oil and Water by Alcohol. Society of Petroleum Engineers Journal, 231:250–266, September 1964. [126] Walas, S.M. Phase Equilibria in Chemical Engineering. Butterworth Publishers, Stoneham, MA, 1985. [127] Walsh, B.W. and Orr, F.M. Jr. Prediction of Miscible Flood Performance: The Eﬀect Of Dispersion on Composition Paths in Ternary Systems. IN SITU, 14(1):19–47, 1990. [128] Wang, Y. Analytical Calculation of Minimum Miscibility Pressure. PhD thesis, Stanford University, Stanford, CA, 1998. [129] Wang, Y. and Orr, F.M., Jr. Analytical Calculation of Minimum Miscibility Pressure. Fluid Phase Equilibria, 139:101–124, 1997. [130] Wang, Y. and Orr, F.M., Jr. Calculation of Minimum Miscibility Pressure. J. Petroleum Science and Engineering, 27:151–164, 2000. [131] Wang, Y., and Peck, D.G. Analytical Calculation of Minimum Miscibility Pressure: Comprehensive Testing and Its Application in a Quantitative Analysis of the Eﬀect of Numerical Dispersion for Diﬀerent Miscibility Development Mechanisms, SPE 59738, SPE/DOE Improved Oil Recovery Symposium, Tulsa, OK, April 2000. [132] Watkins, R.W. A Technique for the Laboratory Measurement of Carbon Dioxide Unit Displacement Eﬃciency in Reservoir Rock, SPE 7474, SPE Annual Technical Conference and Exhibition, Dallas, TX, Oct. 1-3 1978. [133] Welge, H.J. A Simpliﬁed Method for Computing Oil Recovery by Gas or Water Drive. Trans., AIME, 195:91–98, 1952. [134] Welge, H.J., Johnson, E.F., Ewing, S.P.,Jr., and Brinkman, F.H. The Linear Displacement of Oil from Porous Media by Enriched Gas. J. Pet. Tech., pages 787–796, August 1961. [135] Whitson, C.H. and Michelsen, M. The Negative Flash. Fluid Phase Equilibria, 53:51–71, 1989. [136] Wilson, G.M. A Modiﬁed Redlich-Kwong Equation of State, Application to General Physical Data Calculation. Paper 15C, presented at the 1969 AIChE 65th National Mtg., Cleveland, OH, 1969.

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[137] Wingard, J.S. Multicomponent, Multiphase Flow in Porous Media with Temperature Variation. PhD thesis, Stanford University, Stanford, CA, November 1988. [138] Yellig, W.F., and Metcalfe, R.S. Determination and Prediction of CO2 Minimum Miscibility Pressures. J. Pet. Tech., pages 160–168, January 1980. [139] Zhu, J., Jessen, K., Kovscek, A.R., and Orr, F.M., Jr. Analytical Theory of Coalbed Methane Recovery by Gas Injection. Soc. Pet. Eng. J., pages 371–379, December 2003. [140] Zick, A.A. A Combined Condensing/Vaporizing Mechanism in the Displacement of Oil by Enriched Gas, SPE 15493, SPE Annual Technical Conference and Exhibition, New Orleans, LA, October 1986.

Appendix A

255 APPENDIX A: Entropy Conditions in Ternary Systems

In this appendix we consider the entropy condition for shocks in ternary systems. The derivation of the entropy condition for the shock between tie lines follows that of Wang [128], which is based, in turn, on the approach used by Johansen and Winther [51] to study polymer displacements. The derivation given here is for a speciﬁc system with constant K-values, but the patterns of behavior are the same for systems with variable K-values. The use of the constant K-value example is an attempt to illustrate the abstract concept of an entropy condition in a concrete way. Entropy conditions are statements about the stability of a shock, written in terms of the relative magnitudes of eigenvalues of compositions on either side of the shock and the shock velocity. If a shock is stable, it must be self-sharpening. In other words, if a stable shock were to be smeared slightly by some physical mechanism, it must sharpen again into a shock in the limit as that physical mechanism is removed. Dispersion is one physical mechanism that can create a continuously varying composition in place of a jump in composition. In a binary displacement, the requirement of a stable shock can be translated easily into a statement about the eigenvalues on either side of the shock. For example, the discussion in Section 4.2 states that the eigenvalue on the upstream side of a shock must be greater than the shock velocity, and the eigenvalue on the downstream side must be less than the shock velocity. For a ternary displacement, however, there are two eigenvalues at each point in the composition space, so the statement of shock stability in terms of those eigenvalues is necessarily more complex. In this appendix we consider the statement of an entropy condition for each of the shocks that can appear in the solution for a ternary displacement, leading, trailing, and intermediate, and we show that if there is an intermediate shock, it is a semishock. Leading Shock To illustrate the statement of the entropy condition for the various shocks, we consider a speciﬁc case: constant K-values, with K1 = 2.5, K2 = 0.5, K3 = 0.05, and M = 5. The solution for this example is shown in Fig. 5.16. The behavior of the leading shock, which connects a single-phase composition with a composition on the initial tie line, is exactly the same as that described for leading shock in a binary system (see Section 4.2). The leading shock is a shock that arises because of the behavior of λt, and it occurs along the extension of the initial tie line. It is a semishock that is faster than the composition velocities on the downstream side of the shock, the right state for this shock (entropy conditions are frequently written in terms of left and right states, with the left state referring to upstream compositions and the right state to downstream compositions). Those velocities are are all one. Fig. A.1 shows the relationships between the shock velocity and the eigenvalues λt and λnt for the leading shock. The leading shock has a velocity, ΛLR equal to λL t, which is indicated by the fact that the line drawn from the right state composition, R, to the left state composition L is tangent to the fractional ﬂow curve. The tie line eigenvalue, λt, is given by the slope of the fractional ﬂow curve. The nontie-line eigenvalue is given by Eq. 5.1.24 (see Section 5.1) λnt =

F1 − C1e F1 + p = . C1 + p C1 − C1e

(A.1)

For constant K-values, the value of p, which is the negative of the volume fraction of component 1 on the envelope curve, C1e , is given by Eq. 5.1.49,

256

Appendix A

p = −C1e =

K1 − K2 K1 − K3 2 x21 , x = K2 − 1 1 − K3 1 γ

(A.2)

with γ given by Eq. 5.1.50, 1 − K3 K2 − 1 . (A.3) K1 − K2 K1 − K3 In the example considered here, an LVI vaporizing drive, K2 < 1, so γ is negative, as is p for any tie line. The point labeled CL 1e is the composition at the point at which the extension of the initial tie line is tangent to the envelope curve (see Fig. 5.12). Eq. A.1 indicates that the slope of the line L drawn from the left state composition, L, to CL 1e is the nontie-line eigenvalue, λnt . Comparison of the slopes for the leading shock and λnt indicates that the leading shock velocity, ΛLR is greater than λL nt on the upstream side of the shock. Hence, the relationships among the shock velocity and eigenvalues are γ=

1 < ΛLR = λL t,

(A.4)

λL nt

(A.5)

<1<Λ

LR

.

Thus, the leading shock is self-sharpening with respect to the tie line eigenvalue, but it is not with respect to the nontie-line eigenvalue. This is another indication that the leading shock is a tie-line shock. It must be self-sharpening for variations in the tie-line eigenvalue across the shock, but need not be self-sharpening for the nontie-line eigenvalue. Trailing Shock In a ternary vaporizing gas drive, the trailing shock may or may not be a semishock. Fig. A.2 shows the shock constructions. If the trailing shock is a semishock, then the shock velocity, ΛLR is given by the slope of the line from the injection composition L to Rt , the point at which the line is tangent to the fractional ﬂow curve. If it is a genuine shock, as it would be for a shock from Rg , then the shock velocity is greater than λR t , which is given by the slope of the fraction ﬂow curve at Rg . The point labeled CR 1e is the point at which the extension of the injection tie line is tangent to the envelope curve (see Fig. 5.12). The value of λR nt is given by the slope of the line drawn from Rg or Rt to CR . It is clear from the slopes of the trailing shock lines and the lines corresponding to the 1e nontie-line eigenvalue that the shock velocity is signiﬁcantly lower than the nontie-line eigenvalue. Here again, the shock is self-sharpening with respect to the tie line eigenvalue, but it is not with respect to the nontie-line eigenvalue. At a trailing semishock, then, the eigenvalue relationships are LR t < 1, λR t =Λ

Λ and at a trailing genuine shock, they are

LR

<

t λR nt

< 1,

(A.6) (A.7)

Appendix A

257

Overall Fractional Flow of Component 1, F1

1.2

a

1.0

CL1e

0.8 L a 0.6

0.4

0.2

a

R

0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

Overall Volume Fraction of Component 1, C1

Figure A.1: Tangent construction for the leading shock. The slope of the line from the initial (right state) composition, R, to the left state composition, L, gives the velocity, Λ, of the leading shock. The point labeled CL 1e is the point at which the extension of the initial tie line is tangent to the envelope curve (see Fig. 5.12). The slope of the line from L to CL 1e gives the value of λnt at L. 1.6

Overall Fractional Flow of Component 1, F1

a CR 1e 1.4

1.2

1.0

a

a

a

L

Rt Rg 0.8

0.8

1.0

1.2

1.4

1.6

Overall Volume Fraction of Component 1, C1

Figure A.2: Tangent and genuine shock constructions for the trailing shock. The slope of the line from the injection (left state) composition, L, to one of the right state compositions, Rg or Rt , gives the velocity, ΛLR , of the trailing shock. The slope of the line from Rg or Rt to CR 1e gives the value of λnt at R.

258

Appendix A

R

λt g < ΛLR < 1, Λ

LR

<

R λntg

(A.8)

< 1.

(A.9)

Both the leading and trailing shocks, therefore, are λt shocks in the sense that they are selfsharpening with respect to λt. That behavior is consistent with the discussion of shock stability given for binary displacements, which it must be if the ternary displacements are to reduce to the binary solution in the limit as one component disappears or when the initial and injection compositions lie on extensions of the same tie line. There is no requirement, however, that these shocks be self-sharpening with respect to λnt . The situation is reversed for nontie-line shocks that connect the injection and initial tie lines. These are self-sharpening with respect to the nontie-line eigenvalue but not with respect to the tie line eigenvalue. In the remainder of this appendix, we show why that must be true. Intermediate Shock The arguments given in Section 5.1.4 show that when variation along the nontie-line path is permitted by the velocity rule, the switch/indexpath switch from the tie line path to the nontie-line path must occur at the equal eigenvalue point. If the nontie-line eigenvalue increases as the nontieline path is traced upstream, however, then a shock replaces the variation along the nontie-line path. Next we consider possible left and right states for that shock. Again, we consider the LVI vaporizing gas drive example shown in Fig. A.3, which is the system shown also in Fig. 5.16. Fig. A.3 shows the compositions of potential left (L) and right (R) states, and it also shows the locations of the tie-line intersection point and the two envelope points on the L , and C R , through Eq. A.1. The tie-line envelope curve that provide information about λnt , C1e 1e intersection point and the two envelope points do not change with changes in the left or right state compositions, so they are ﬁxed for the purposes of the following discussion. We begin by considering possible landing points on the injection tie line, the left state. Three possible landing locations are shown in Fig. A.4, left states L1 , L2 , and L3 . The intersection of the line drawn from R to X with the fractional ﬂow curve for the injection gas tie line (which contains the left state compositions) gives possible landing compositions that satisfy the shock balance equations. We consider each of those compositions in turn and show that only one satisﬁes all the requirements. The intermediate shock satisﬁes the shock balances, Eqs. 5.2.23, ΛLR =

FiL − CiX FiR − CiX = , CiL − CiX CiR − CiX

i = 1, nc.

(A.10)

Eq. A.10 is represented in Fig. A.4 by the line drawn from R to X. For a given value of CiR , the intersection of that line with the fractional ﬂow curve for the injection gas tie line gives the value of CiL that satisﬁes Eq. A.10. LR , because the slope of the fractional ﬂow curve at L1 is greater than At left state L1 , λL t >Λ the slope of the shock line from R to X. While a shock to L1 does satisfy the shock balance, it can be ruled out as a potential landing composition. Any subsequent rarefaction along the injection gas tie line would violate the velocity rule, because the intermediate shock would be slower than the

Appendix A

259

aCL1e

X a

CH a4

aCR 1e

L a

aR a

a

C10

C4

Figure A.3: Composition path for a vaporizing gas drive with low volatility intermediate component. K1 = 2.5, K2 = 0.5, K3 = 0.05, and M = 5 (See Fig. 5.16 and the accompanying discussion for a description of the full solution and for the corresponding saturation proﬁles). Points L and R are the left and right states of the nontie-line shock. Point X is the intersection point of the tie lines R connected by the shock. Points CL 1e and C1e are the tangent points on the envelope curve for the tie lines that contain the left and right states for the shock.

260

Appendix A

Overall Fractional Flow of Component 1, F1

1.4

a

1.2

X

1.0

L3

a

L2

a

0.8

a Ra

0.6

L1

0.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Overall Volume Fraction of Component 1, C1

Figure A.4: Shock constructions for landing points of the intermediate shock on the injection gas tie line. compositions just upstream. In addition, a direct shock from L1 to the injection gas composition would violate the entropy condition for the trailing shock (see Section 4.2). Hence, a shock from R to L1 is prohibited. Point L2 is the tangent point for a line drawn from the tie-line intersection point, X to the LR . This landing point can also be fractional ﬂow curve for the injection tie line. At L2 , λL t = Λ ruled out. As the shape of the fractional ﬂow curves in Fig. A.4 show, the shock construction line (X to L2 ) does not intersect the fractional ﬂow curve for the initial tie line. (For the example of R this appendix, with constant mobility ratio, M , and xL 1 > x1 , a tangent drawn to the fractional ﬂow curve for the longer tie line does not intersect the fractional ﬂow curve for the shorter tie line. More care is required to show that a similar statement is true for more complex phase behavior and mobility ratio that is not constant.) Thus, there is no solution for a shock that lands at L2 , LR , and satisﬁes the shock balance equations. where λL t =Λ Point L3 is an acceptable landing point, however. At L3 , the slope of the fractional ﬂow curve LR . Variation along the injection gas is lower than the slope of the shock line, and hence λL t < Λ tie line to a trailing semishock point would be consistent with the velocity rule, and an immediate genuine shock to the injection composition is also allowed. Hence, we conclude that at the landing LR . point on the injection gas tie line, λL t <Λ Next we consider possible right states on the initial oil tie line. Fig. A.5 shows three possible LR , as jump points on that tie line. Point R3 can be ruled out immediately. At R3 , λR t < Λ comparison of the slope of the fraction ﬂow curve at R3 and the slope of the shock line indicates. In other words, the intermediate shock moves faster than the compositions on the rarefaction along the initial tie line, a situation that would violate the velocity rule. Point R1 is also not an acceptable right state, although more eﬀort is required to show that it

Appendix A

261

Overall Fractional Flow of Component 1, F1

1.4

a

1.2

X L3

1.0

aa

0.8

R2 a

a

L1

R1 a

0.6

a

R3

0.4

0.2

0.0 0.0

0.4

0.2

0.6

0.8

1.0

1.2

1.4

Overall Volume Fraction of Component 1, C1

Figure A.5: Shock constructions for jump points of the intermediate shock on the initial oil tie line. is not permitted. To work out the behavior of eigenvalues on either side of a shock for a right state at R1 , we consider a displacement in which a small amount of dispersion is present. For a ternary displacement with a small dispersion coeﬃcient , the conservation equations are ∂F1 ∂C1 + ∂τ ∂ξ ∂F2 ∂C2 + ∂τ ∂ξ

∂ 2 C1 , ∂ξ 2 ∂ 2 C2 = . ∂ξ 2 =

(A.11) (A.12)

We will seek a solution of Eqs. A.11 and A.12 for a shock traveling with wave velocity Λ subject to the boundary conditions that the compositions C1 = C1L and C2 = C2L on the far upstream side of the shock and C1 = C1R and C2 = C2R on the far downstream side satisfy the Rankine-Hugoniot conditions for a shock moving with velocity Λ. In addition, we will require that the derivatives of C1 and C2 be zero far upstream and far downstream of the shock. The entropy condition can be derived by requiring that the discontinuous solution (one with a shock) be the limit of a traveling wave solution as → 0. A traveling wave solution to Eqs. A.11 and A.12 has the form C1 = C1 (ζ) = C1 (

ξ − Λτ ),

C2 = C2 (ζ) = C2 (

ξ − Λτ ).

(A.13)

Application of the chain rule gives the derivatives of C1 and C2 , Λ dCi ∂Ci =− ∂τ dζ

i = 1, 2,

(A.14)

262

Appendix A

1 dCi ∂Ci =− ∂ξ dζ

i = 1, 2,

(A.15)

and 1 d2 Ci ∂ 2 Ci = − ∂ξ 2 2 dζ 2

i = 1, 2.

(A.16)

As a result Eqs. A.11 and A.12 become ∂F1 dC2 dC1 ∂F1 dC1 d2 C1 + −Λ , = 2 dζ ∂C1 dζ ∂C2 dζ dζ

(A.17)

d2 C2 ∂F2 dC2 dC2 ∂F2 dC1 + −Λ . = 2 dζ ∂C1 dζ ∂C2 dζ dζ

(A.18)

and

Integration of Eqs. A.17 and A.18 gives dC1 = F1 − ΛC1 − F1 (C1L , C2L ) − ΛC1L dζ

,

(A.19)

dC2 = F2 − ΛC1 − F2 (C1L , C2L ) − ΛC2L dζ

.

(A.20)

and

A shock that satisﬁes the Lax entropy condition is one that satisﬁes Eqs. A.19 and A.20 with C1 (−∞) = C1L , C2 (−∞) = C2L , C1 (∞) = C1R , and C2 (∞) = C2R [51, 128]. Wang showed that Eq. A.20 can be recast into an equation for the x1 , so that the solution variables are C1 and x1 (see [128, Appendix C] for a detailed derivation). That equation is a dx1 = (xL − x1)(ΛLR − λ), dζ b 1

(A.21)

where

a = (1 − K2 )(K1 − 1)x1 − (K2 − 1)(K3 − 1) 1 + (K1 − 1)S L ,

(A.22)

b = (1 − K2 )(K1 − 1)x1 − (K2 − 1)(K3 − 1) {1 + (K1 − 1)S} , F L − π(xL 1 , x1 ) , λ = 1L C1 − π(xL 1 , x1 )

(A.23) (A.24) (A.25)

and π(xL 1 , x1 ) = Eq. A.19 can be rearranged to yield

(K1 − K2 )(K1 − K3 ) x1 xL 1. (K2 − 1)(K3 − 1)

(A.26)

Appendix A

263 0.5

0.4

CL1 1

CL3 1

a

a

x1

C-1 dC1/dζ < 0

dC1/dζ > 0

a

0.4

0.2

dC1/dζ < 0

a

CR1 1

0.3

0.2 0.0

C+1

CR3 1

0.6

0.8

1.0

C1

Figure A.6: Regions of positive and negative values of dC1 /dζ and trajectories with dC1 /dζ = 0.

dC1 = (C1 − C1L )(Ω − ΛLR ), dζ

(A.27)

where Ω=

F1 − F1L . C1 − C1L

(A.28)

Ω is the slope of a line that connects any point along the fractional ﬂow curve for the initial oil tie line to point L (see Fig. A.5). In Eq. A.28, F1 and C1 lie along the solution to Eqs. A.19 and A.21. Those equations determine C1 (ζ) and x1 (ζ). Eq. A.28 indicates that dC1 < 0, dζ dC1 > 0, dζ

0 < C1 < C1− ,

(A.29)

C1− < C1 < C1+ ,

(A.30) (A.31)

and dC1 < 0, dζ

C1+ < C1 < 1.

(A.32)

In these expressions, C1− refers to a trajectory in composition space (C1 ,x1 ) along which dC1 /dζ = 0, the boundary between the zone of positive values of dC1 /dζ at low values of C1 (ζ). Correspondingly,

264

Appendix A 1.6

a

Overall Fractional Flow of Component 1, F1

CL1e 1.4

a

1.2

X

1.0

L

a

0.8

a

CR 1e

Ra

a 0.6 0.4 0.2

a 0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Overall Volume Fraction of Component 1, C1

Figure A.7: Tangent constructions for a shock between tie lines. The slope of the line from R to X gives the velocity, Λ, of the shock from R to L. The slope of the line from R to CR 1e gives the L value of λnt at R. The slope of the line from L to C1e is λnt at L. Comparison of the various slopes reveals the relative magnitudes of the shock velocity and eigenvalues. C1+ refers to a second trajectory with dC1 /dζ = 0, this time at high values of C1 (ζ). Fig. A.6 shows schematically the arrangement of zones of positive and negative dC1 /dζ. On each of the initial and injection tie lines, the three regions of negative, positive, and negative dC1 /dζ exist for low, intermediate and high values of C1 (ζ). In order for a trajectory to connect left state L1 to R1 , dC1 /dζ would have to be negative, but there is no way for the trajectory to pass through the zone in which dC1 /dζ is positive. Hence, there are no trajectories that connect L to R1 . Therefore, a LR is not permitted. shock from left state L to a right state R1 for which λR t >Λ The only remaining possibility is that there is a shock from L to R2 . That shock is allowed. It does not violate the velocity rule that prevented shocks to right state compositions for which LR LR because λR . And it does not violate the entropy statement that prohibits a shock λR t <Λ t =Λ LR . Hence, the intermediate shock must be a semishock to a right state composition with λR t > Λ R LR (see Section 7.2 for a continuity argument that conﬁrms that the intermediate with λt = Λ shock is a semishock at which λn t = Λ on the shorter of the initial or injection tie lines). As a result, the statement of the entropy condition for the tie line eigenvalue is R LR λL t < λt = Λ

(A.33)

L LR . The fractional ﬂow diagram Finally, we consider the relative magnitudes of λR nt , λnt , and Λ L R , and C1e using Eq. A.2 for the intermediate shock is shown in Fig. A.7. Direct evaluation of C1e L R L R L indicates that C1e > C1e as long as x1 > x1 . Fig. A.3 shows that x1 is larger than xR 1 for this system (see Appendix C of Wang [128]) for a detailed proof that the statement must be true for slightly dispersed shock traveling to the right).

Appendix A

265

L R > CiX > C1e . As the locations of the tie-line intersection point in Figs. A.3 and A.7 show, C1e The velocity of the intermediate shock is given by the slope of the line from R to X, and the R R nontie-line eigenvalues, λL nt and λnt are given by the slopes of the lines drawn from R and L to C1e L respectively. Comparisons of those slopes indicates that and C1e LR < λL λR nt < Λ nt .

(A.34)

Hence, the intermediate shock is self-sharpening with respect to the nontie-line eigenvalues upstream and downstream of the shock, as it should be if it replaces a nontie-line rarefaction that is prohibited by the velocity rule because λnt increases as the nontie-line path is traced upstream. Summary The example of the LVI vaporizing gas drive considered in the appendix leads to the following statement of the entropy condition: L λR nt < Λ < λnt ,

and

R λL t < Λ = λt .

(A.35)

If instead we had considered a LVI condensing gas drive, the statement of the entropy condition would diﬀer. Here again, one set of characteristics is sharpening (the nontie-line eigenvalues) and one set is not, but the semishock occurs on the injection (left state) tie line instead of the right state (initial) tie line. L λR nt < Λ < λnt ,

and

R Λ = λL t < λt .

These are the expressions given as Eqs. 5.2.25 and 5.2.26.

(A.36)

266

Appendix B APPENDIX B: Details of Gas Displacement Solutions

In this appendix, full details of all the solutions illustrated in Chapters 4-8 are reported. Unless otherwise noted, the fractional ﬂow functions used in the solutions have the form of Eqs. 4.1.204.1.22 and Sor = Sgc = 0.

Chapter 4–Binary Displacements Table B.1: Displacement details for Fig. 4.10. Binary gas displacement with no volume change, M = 2. Segment Injection Gas Trailing Shock Rarefaction Rarefaction Rarefaction Rarefaction Leading Shock Initial Oil

Point d d c c-b c-b c-b c-b b a a

C1 1.0000 1.0000 0.7794 0.7625 0.7250 0.6875 0.6500 0.6316 0.0500 0.0500

C2 0.0000 0.0000 0.2206 0.2375 0.2750 0.3125 0.3500 0.3684 0.9500 0.9500

S1 0.0000 0.0000 0.7725 0.7500 0.7000 0.6500 0.6000 0.5755 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

ξ/τ 1.0000 0.2786 0.2786 0.3552 0.5643 0.8329 1.1614 1.3409 1.3546 1.0000

Appendix B

267

Table B.2: Displacement details for Fig. 4.16. Binary gas displacement with volume change. Fluid properties and phase compositions are reported in Table 4.1. Segment Injection Gas Trailing Shock Rarefaction Rarefaction Rarefaction Rarefaction Rarefaction Rarefaction Rarefaction Rarefaction Rarefaction Rarefaction Leading Shock Initial Oil

Point d d c c-b c-b c-b c-b c-b c-b c-b c-b c-b c-b b a a

z1 1.0000 1.0000 0.7020 0.6833 0.6205 0.5692 0.5264 0.4902 0.4592 0.4323 0.4088 0.3881 0.3676 0.6316 0.0000 0.0000

z2 0.0000 0.0000 0.2980 0.3167 0.3795 0.4308 0.4736 0.5098 0.5408 0.5677 0.5912 0.6109 0.6324 0.3684 1.0000 1.0000

S1 0.0000 0.0000 0.8630 0.8500 0.8000 0.7500 0.7000 0.6500 0.6000 0.5500 0.5000 0.4500 0.3941 0.5500 0.0000 0.0000

Flow Vel. 1.0000 1.0000 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.5093 0.5093

ξ/τ 1.0000 0.0063 0.0063 0.0090 0.0217 0.0400 0.0662 0.1044 0.1605 0.2442 0.3710 0.5659 0.8329 0.9147 0.9147 0.5093

Table B.3: Displacement details for Fig. 4.16. Binary gas displacement with no volume change. Fluid properties and phase compositions are reported in Table 4.1. Compositions reported are in mole fractions. Segment Injection Gas Trailing Shock Rarefaction Rarefaction Rarefaction Rarefaction Rarefaction Rarefaction Rarefaction Rarefaction Rarefaction Leading Shock Initial Oil

Point d d c c-b c-b c-b c-b c-b c-b c-b c-b c-b b a a

zCO2 1.0000 1.0000 0.7894 0.7499 0.7011 0.6563 0.6150 0.5679 0.5415 0.5085 0.4779 0.4492 0.4243 0.0000 0.0000

zC10 0.0000 0.0000 0.2980 0.3795 0.4308 0.4736 0.5098 0.5408 0.5677 0.5912 0.6109 0.6109 0.3684 1.0000 1.0000

S1 0.0000 0.0000 0.8375 0.8000 0.7500 0.7000 0.6500 0.6000 0.5500 0.5000 0.4500 0.4000 0.3538 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

ξ/τ 1.0000 0.0118 0.0118 0.0217 0.0400 0.0662 0.1044 0.1605 0.2442 0.3710 0.5660 0.8693 1.2972 1.2972 1.0000

268

Appendix B

Chapter 5–Ternary Displacements Table B.4: Displacement details for Fig. 5.16. Composition path and proﬁles for a vaporizing gas drive with low volatility intermediate component. K1 = 2.5, K2 = 0.5, K3 = 0.05, and M = 5. The injection gas is pure CH4 , and the initial oil has composition C1 = 0.1, C2 = 0.5, and C3 = 0.4. Compositions in volume fractions. Segment Injection Gas Trailing Shock Zone of Constant State Intermediate Shock Initial Tie Line Rarefaction

Leading Shock Initial Oil

Point f f d d d d c c c-b c-b b b a a

CH4 1.0000 1.0000 0.8806 0.8806 0.8806 0.8806 0.5781 0.5781 0.5710 0.5476 0.5294 0.5294 0.1500 0.1500

C4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.2924 0.2924 0.2955 0.3057 0.3135 0.3135 0.2908 0.2908

C10 0.0000 0.0000 0.1194 0.1194 0.1194 0.1194 0.1295 0.1295 0.1335 0.1468 0.1570 0.1570 0.5592 0.5592

S1 1.0000 1.0000 0.8473 0.8473 0.8473 0.8473 0.5651 0.5651 0.5500 0.5000 0.4614 0.4614 0. 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

ξ/τ 1.0000 0.2878 0.2878 0.2878 0.7707 0.7707 0.7707 0.7707 0.8415 1.1111 1.3546 1.3546 1.3546 1.0000

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 = 5. 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. 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 = 5. 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. 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. 0.

C4 0. 0. 0. 0. 0. 0.0301 0.0306 0.0310 0.0313 0.1035 0.1035

C10 0. 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. 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. 0.

C4 0. 0. 0. 0. 0. 0.0577 0.0577 0.0579 0.0583 0.0586 0.2204 0.2204

C10 0. 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. 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. 0.

C4 0. 0. 0. 0. 0. 0.0763 0.0763 0.0764 0.0764 0.3008 0.3008

C10 0. 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. 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. 0.

C4 0. 0. 0. 0. 0. 0. 0. 0.1207 0.1229 0.1282 0.1330 0.1352 0.2868 0.2868

C10 0. 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. 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. 0.

C4 0. 0. 0. 0. 0. 0. 0. 0.1073 0.1095 0.1118 0.1140 0.1196 0.1219 0.2868 0.2868

C10 0. 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. 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 = 0. K-values are constant: K1 = 2.5, K2 = 1.5, K3 = 0.5, and K4 = .05. µliq /µvap = 5. 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 = 1. K-values are constant: K1 = 2.5, K2 = 1.5, K3 = 0.5, and K4 = .05. The viscosity ratio, µliq /µvap , is 5. 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 9 Shock 8 Shock 7 Shock 6 Shock 5 Shock 4 Shock 3 Shock 2 Shock 1 Initial Oil

Point

14 14 13 13 12 12 11 11 10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 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, 8 convection, 5 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, 9 Cer´e, 130 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, 8 Dumor´e, 74, 130 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, 2 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, 3 Lake, 2, 17, 40, 70, 73 Lantz, 214 Larson, 130 Lax, 80 lever rule, 34 Leverett, 53, 70 local equilibrium assumption, 9 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, 1 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, 8 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