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1、 Reservoir Simulation CB77_Chenfm_A.qxp 9/7/2007 12:29 PM Page 1 CBMS-NSF REGIONAL CONFERENCE SERIES IN APPLIED MATHEMATICS A series of lectures on topics of current research interest in applied mathematics under the direction of the Conference Board of the Mathematical Sciences, supported by the Na
2、tional Science Foundation and published by SIAM. GARRETTBIRKHOFF, The Numerical Solution of Elliptic Equations D. V. LINDLEY, Bayesian Statistics, A Review R. S. VARGA, Functional Analysis and Approximation Theory in Numerical Analysis R. R. BAHADUR, Some Limit Theorems in Statistics PATRICKBILLINGS
3、LEY, Weak Convergence of Measures: Applications in Probability J. L. LIONS, Some Aspects of the Optimal Control of Distributed Parameter Systems ROGERPENROSE, Techniques of Differential Topology in Relativity HERMANCHERNOFF, Sequential Analysis and Optimal Design J. DURBIN, Distribution Theory for T
4、ests Based on the Sample Distribution Function SOLI. RUBINOW, Mathematical Problems in the Biological Sciences P. D. LAX, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves I. J. SCHOENBERG, Cardinal Spline Interpolation IVANSINGER, The Theory of Best Approximation an
5、d Functional Analysis WERNERC. RHEINBOLDT, Methods of Solving Systems of Nonlinear Equations HANSF. WEINBERGER, Variational Methods for Eigenvalue Approximation R. TYRRELLROCKAFELLAR, Conjugate Duality and Optimization SIRJAMESLIGHTHILL, Mathematical Biofluiddynamics GERARDSALTON, Theory of Indexing
6、 CATHLEENS. MORAWETZ, Notes on Time Decay and Scattering for Some Hyperbolic Problems F. HOPPENSTEADT, Mathematical Theories of Populations: Demographics, Genetics and Epidemics RICHARDASKEY, Orthogonal Polynomials and Special Functions L. E. PAYNE, Improperly Posed Problems in Partial Differential
7、Equations S. ROSEN, Lectures on the Measurement and Evaluation of the Performance of Computing Systems HERBERTB. KELLER, Numerical Solution of Two Point Boundary Value Problems J. P. LASALLE, The Stability of Dynamical Systems - Z. Artstein, Appendix A: Limiting Equations and Stability of Nonautonom
8、ous Ordinary Differential Equations D. GOTTLIEB ANDS. A. ORSZAG, Numerical Analysis of Spectral Methods: Theory and Applications PETERJ. HUBER, Robust Statistical Procedures HERBERTSOLOMON, Geometric Probability FREDS. ROBERTS, Graph Theory and Its Applications to Problems of Society JURISHARTMANIS,
9、 Feasible Computations and Provable Complexity Properties ZOHARMANNA, Lectures on the Logic of Computer Programming ELLISL. JOHNSON, Integer Programming: Facets, Subadditivity, and Duality for Group and Semi-Group Problems SHMUELWINOGRAD, Arithmetic Complexity of Computations J. F. C. KINGMAN, Mathe
10、matics of Genetic Diversity MORTONE. GURTIN, Topics in Finite Elasticity THOMASG. KURTZ, Approximation of Population Processes CB77_Chenfm_A.qxp 9/7/2007 12:29 PM Page 2 JERROLDE. MARSDEN, Lectures on Geometric Methods in Mathematical Physics BRADLEYEFRON, The Jackknife, the Bootstrap, and Other Res
11、ampling Plans M. WOODROOFE, Nonlinear Renewal Theory in Sequential Analysis D. H. SATTINGER, Branching in the Presence of Symmetry R. TEMAM, Navier-Stokes Equations and Nonlinear Functional Analysis MIKLSCSRGO, Quantile Processes with Statistical Applications J. D. BUCKMASTER ANDG. S. S. LUDFORD, Le
12、ctures on Mathematical Combustion R. E. TARJAN, Data Structures and Network Algorithms PAULWALTMAN, Competition Models in Population Biology S. R. S. VARADHAN, Large Deviations and Applications KIYOSIIT, Foundations of Stochastic Differential Equations in Infinite Dimensional Spaces ALANC. NEWELL, S
13、olitons in Mathematics and Physics PRANABKUMARSEN, Theory and Applications of Sequential Nonparametrics LSZLLOVSZ, An Algorithmic Theory of Numbers, Graphs and Convexity E. W. CHENEY, Multivariate Approximation Theory: Selected Topics JOELSPENCER, Ten Lectures on the Probabilistic Method PAULC. FIFE
14、, Dynamics of Internal Layers and Diffusive Interfaces CHARLESK. CHUI, Multivariate Splines HERBERTS. WILF, Combinatorial Algorithms: An Update HENRYC. TUCKWELL, Stochastic Processes in the Neurosciences FRANKH. CLARKE, Methods of Dynamic and Nonsmooth Optimization ROBERTB. GARDNER, The Method of Eq
15、uivalence and Its Applications GRACEWAHBA, Spline Models for Observational Data RICHARDS. VARGA, Scientific Computation on Mathematical Problems and Conjectures INGRIDDAUBECHIES, Ten Lectures on Wavelets STEPHENF. MCCORMICK, Multilevel Projection Methods for Partial Differential Equations HARALDNIED
16、ERREITER, Random Number Generation and Quasi-Monte Carlo Methods JOELSPENCER, Ten Lectures on the Probabilistic Method, Second Edition CHARLESA. MICCHELLI, Mathematical Aspects of Geometric Modeling ROGERTEMAM, NavierStokes Equations and Nonlinear Functional Analysis, Second Edition GLENNSHAFER, Pro
17、babilistic Expert Systems PETERJ. HUBER, Robust Statistical Procedures, Second Edition J. MICHAELSTEELE, Probability Theory and Combinatorial Optimization WERNERC. RHEINBOLDT, Methods for Solving Systems of Nonlinear Equations, Second Edition J. M. CUSHING, An Introduction to Structured Population D
18、ynamics TAI-PINGLIU, Hyperbolic and Viscous Conservation Laws MICHAELRENARDY, Mathematical Analysis of Viscoelastic Flows GRARDCORNUJOLS, Combinatorial Optimization: Packing and Covering IRENALASIECKA, Mathematical Control Theory of Coupled PDEs J. K. SHAW, Mathematical Principles of Optical Fiber C
19、ommunications ZHANGXINCHEN, Reservoir Simulation: Mathematical Techniques in Oil Recovery CB77_Chenfm_A.qxp 9/7/2007 12:29 PM Page 3 CB77_Chenfm_A.qxp 9/7/2007 12:29 PM Page 4 ZHANGXIN CHEN University of Calgary Calgary, Alberta, Canada Reservoir Simulation Mathematical Techniques in Oil Recovery SO
20、CIETY FOR INDUSTRIAL AND APPLIED MATHEMATICS PHILADELPHIA CB77_Chenfm_A.qxp 9/7/2007 12:29 PM Page 5 Copyright 2007 by the Society for Industrial and Applied Mathematics. 10 9 8 7 6 5 4 3 2 1 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored
21、, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 Market Street, 6th floor, Philadelphia, PA 19104-2688. Trademarked names may be used in this book without the inclusion of a trademark sy
22、mbol. These names are used in an editorial context only; no infringement of trademark is intended. Library of Congress Cataloging-in-Publication Chen, Zhangxin, 1962 Reservoir simulation : mathematical techniques in oil recovery / Zhangxin Chen. p. cm. (CBMS-NSF regional conference series in applied
23、 mathematics ; 77) Includes bibliographical references and index. ISBN 978-0-898716-40-5 (alk. paper) 1. Oil reservoir engineeringMathematical models. 2. Oil reservoir engineeringSimulation methods. 3. Porous materialsPermeabilityMathematical models. 4. Transport theoryMathematical models. I. Title
24、II. Series. TN871.C465 2007 622.3382015118dc22 2007061749 is a registered trademark. CB77_Chenfm_A.qxp 9/7/2007 12:29 PM Page 6 This book is dedicated to my parents, wife, and children. ? CB77_Chenfm_A.qxp 9/7/2007 12:29 PM Page 7 CB77_Chenfm_A.qxp 9/7/2007 12:29 PM Page 8 chenb 2007/9 page ix i i i
25、 i i i i i Contents List of Figuresxiii List of Tablesxv List of Notationxvii Prefacexxvii 1Introduction1 1.1Petroleum Reservoir Simulation . . . . . . . . . . . . . . . . . . . . .1 1.2Classical Reservoir Engineering Methods . . . . . . . . . . . . . . . .1 1.2.1Material Balance Methods . . . . . .
26、 . . . . . . . . . . .1 1.2.2Decline Curve Methods . . . . . . . . . . . . . . . . . .2 1.2.3Statistical Methods . . . . . . . . . . . . . . . . . . . . .2 1.2.4Analytical Methods. . . . . . . . . . . . . . . . . . . .2 1.3Reservoir Simulation Methods. . . . . . . . . . . . . . . . . . . . .3 1.3.1R
27、eservoir Simulation Stages . . . . . . . . . . . . . . . .3 1.3.2 Reservoir Simulator Classifi cations . . . . . . . . . . . .4 1.3.3Reservoir SimulationApplications. . . . . . . . . . . .4 1.4SI Metric Conversion Factors . . . . . . . . . . . . . . . . . . . . . .6 2AGlossary of Petroleum Terms7 2.
28、1Reservoir Rock Properties . . . . . . . . . . . . . . . . . . . . . . . .7 2.2Reservoir Fluid Properties . . . . . . . . . . . . . . . . . . . . . . . .9 2.3Wettability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12 2.4Fluid Displacement Processes . . . . . . . . . . . . . . . .
29、. . . . . .13 2.5Reservoir Rock/Fluid Properties . . . . . . . . . . . . . . . . . . . . .13 2.5.1Two-Phase Relative Permeability . . . . . . . . . . . . .15 2.5.2Three-Phase Relative Permeability. . . . . . . . . . . .17 2.6Terms Used in Numerical Simulation . . . . . . . . . . . . . . . . . .20 3S
30、ingle-Phase Flow and Numerical Solution23 3.1Basic Differential Equations . . . . . . . . . . . . . . . . . . . . . . .23 3.1.1Mass Conservation . . . . . . . . . . . . . . . . . . . . .23 3.1.2Darcys Law . . . . . . . . . . . . . . . . . . . . . . . .25 ix chenb 2007/9 page x i i i i i i i i xConte
31、nts 3.1.3Units . . . . . . . . . . . . . . . . . . . . . . . . . . . .26 3.1.4Different Forms of Flow Equations . . . . . . . . . . . .26 3.2AnAnalytic Solution . . . . . . . . . . . . . . . . . . . . . . . . . . .31 3.3Finite Difference Methods . . . . . . . . . . . . . . . . . . . . . . . .34 3.3.
32、1First Difference Quotients . . . . . . . . . . . . . . . . .34 3.3.2Second Difference Quotients. . . . . . . . . . . . . . .36 3.3.3Grid Systems . . . . . . . . . . . . . . . . . . . . . . . .38 3.3.4Treatment of Boundary Conditions . . . . . . . . . . . .39 3.3.5Finite Differences for Stationary P
33、roblems . . . . . . . .41 3.3.6Finite Differences for Parabolic Problems . . . . . . . . .42 3.3.7Consistency, Stability, and Convergence. . . . . . . . .44 3.3.8Finite Differences for Hyperbolic Problems . . . . . . . .48 3.4Numerical Solution of Single-Phase Flow. . . . . . . . . . . . . . .51 3.4
34、.1Treatment of Initial Conditions. . . . . . . . . . . . . .52 3.4.2Time Discretization. . . . . . . . . . . . . . . . . . . .52 3.4.3Spatial Discretization. . . . . . . . . . . . . . . . . . .53 3.4.4Treatment of Block Transmissibility . . . . . . . . . . . .53 3.4.5SolutionApproaches in Time. . .
35、. . . . . . . . . . . .56 3.4.6Material BalanceAnalysis . . . . . . . . . . . . . . . . .63 4Well Modeling67 4.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .67 4.2Analytical Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . .68 4.3Single-Layer Well Models . .
36、. . . . . . . . . . . . . . . . . . . . . .69 4.3.1Square Grids . . . . . . . . . . . . . . . . . . . . . . . .69 4.3.2Extensions . . . . . . . . . . . . . . . . . . . . . . . . .71 4.4Multilayer Well Models . . . . . . . . . . . . . . . . . . . . . . . . .74 4.5Coupling of Flow and Well Equations .
37、 . . . . . . . . . . . . . . . . .75 4.6Coupling of Wellbore-Hydraulics and Reservoir Models . . . . . . . .78 4.6.1Single-Phase Flow . . . . . . . . . . . . . . . . . . . . .78 4.6.2Multiphase Flow . . . . . . . . . . . . . . . . . . . . . .79 5Two-Phase Flow and Numerical Solution83 5.1Basic Diffe
38、rential Equations . . . . . . . . . . . . . . . . . . . . . . .83 5.1.1Mass Conservation . . . . . . . . . . . . . . . . . . . . .83 5.1.2Darcys Law . . . . . . . . . . . . . . . . . . . . . . . .84 5.1.3Alternative Differential Equations . . . . . . . . . . . . .85 5.1.4Boundary Conditions. . . . .
39、 . . . . . . . . . . . . . .89 5.2AnAnalytic Solution . . . . . . . . . . . . . . . . . . . . . . . . . . .91 5.2.1Analytic Solution Before Water Breakthrough. . . . . .91 5.2.2Analytic Solution at the Water Front . . . . . . . . . . . .92 5.2.3Analytic SolutionAfter Water Breakthrough. . . . . . .9
40、3 5.3Numerical Solution of Two-Phase Flow . . . . . . . . . . . . . . . . .94 5.3.1Treatment of Initial Conditions. . . . . . . . . . . . . .95 5.3.2Source/Sink Terms . . . . . . . . . . . . . . . . . . . . .95 5.3.3Spatial Discretization. . . . . . . . . . . . . . . . . . .96 chenb 2007/9 page x i
41、i i i i i i i Contentsxi 5.3.4Treatment of Block Transmissibility . . . . . . . . . . . .97 5.3.5SolutionApproaches in Time. . . . . . . . . . . . . . .99 6The Black Oil Model and Numerical Solution103 6.1Basic Differential Equations . . . . . . . . . . . . . . . . . . . . . . . 103 6.1.1Mass Conser
42、vation and Darcys Law . . . . . . . . . . . 103 6.1.2Rock/Fluid Properties . . . . . . . . . . . . . . . . . . . 106 6.1.3Fluid Properties. . . . . . . . . . . . . . . . . . . . . . 107 6.1.4Phase States . . . . . . . . . . . . . . . . . . . . . . . . 108 6.2Numerical Solution of the Black Oil Model
43、 . . . . . . . . . . . . . . . 109 6.2.1Treatment of Initial Conditions. . . . . . . . . . . . . . 110 6.2.2Simultaneous Solution Techniques. . . . . . . . . . . . 112 6.2.3Sequential Solution Techniques . . . . . . . . . . . . . . 120 6.2.4Iterative IMPES Solution Techniques . . . . . . . . . . . 1
44、24 6.2.5Adaptive Implicit Techniques . . . . . . . . . . . . . . . 127 6.2.6Well Coupling . . . . . . . . . . . . . . . . . . . . . . . 128 7Transport of Multicomponents in a Fluid and Numerical Solution131 7.1Basic Differential Equations . . . . . . . . . . . . . . . . . . . . . . . 132 7.2Computat
45、ion of Fluid Viscosity . . . . . . . . . . . . . . . . . . . . . 133 7.3Equations of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 7.4Diffusion, Dispersion, and Tortuosity . . . . . . . . . . . . . . . . . . 136 7.4.1Ficks Law . . . . . . . . . . . . . . . . . . . . . . . . . 136
46、 7.4.2Impact of Tortuosity on Diffusion . . . . . . . . . . . . . 137 7.4.3Soret Effects and Gravity Segregation . . . . . . . . . . . 146 7.4.4Isothermal Gravity/Chemical Equilibrium. . . . . . . . 147 7.5Numerical Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . 148 7.5.1AModel Proble
47、m. . . . . . . . . . . . . . . . . . . . . 148 7.5.2Finite Difference Equations . . . . . . . . . . . . . . . . 148 7.6Nonisothermal Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 7.7Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 7.7.1Forced Convection
48、 . . . . . . . . . . . . . . . . . . . . . 152 7.7.2Forced Convection Plus Dispersion . . . . . . . . . . . . 154 8Compositional Flow and Numerical Solution157 8.1Basic Differential Equations . . . . . . . . . . . . . . . . . . . . . . . 157 8.1.1Mass Conservation and Darcys Law . . . . . . . . . .
49、. 157 8.1.2Equations of State . . . . . . . . . . . . . . . . . . . . . 159 8.2Numerical Solution of Compositional Flow . . . . . . . . . . . . . . . 161 8.2.1Choice of Primary Variables . . . . . . . . . . . . . . . . 162 8.2.2Finite Difference Equations . . . . . . . . . . . . . . . . 164 8.3Solution of Equilibrium Relations . . . . . . . . . . . . . . . . . . . . 170 8.3.1Successive Substitution Method . . . . . . . . . . . . . . 170 8.3.2NewtonRaphson Flash Calculation . . . . . . . . . . . . 171 8.3.3 Derivatives of Fugacity Coeffi cients. . . . . . . . . . . 17