Accelerated Perturbation Boundary Element Model for Flow Problems in Heterogeneous Reservoirs

ACCELERATED PERTURBATION BOUNDARY ELEMENT MODEL FOR FLOW PROBLEMS IN HETEROGENEOUS RESERVOIRS a dissertation submitted to the department of petroleum engineering and the committee on graduate studies of stanford university in partial fulfillment of the requirements for the degree of doctor of philosophy By Kozo Sato June 1992 c Copyright 1992 by Kozo Sato All Rights Reserved ii I certify that I have read this thesis and that in my opin- ion it is fully adequate, in scope and in quality, as a dissertation for the degree of Doctor of Philosophy. Roland N. Horne (Principal Adviser) I certify that I have read this thesis and that in my opin- ion it is fully adequate, in scope and in quality, as a dissertation for the degree of Doctor of Philosophy. Khalid Aziz I certify that I have read this thesis and that in my opin- ion it is fully adequate, in scope and in quality, as a dissertation for the degree of Doctor of Philosophy. Henry J. Ramey, Jr. Approved for the University Committee on Graduate Studies: Dean of Graduate Studies iii Abstract The boundary element method (BEM), successfully applied to fluid flow problems in porous media, owes its elegance, the computational efficiency and accuracy, to the existence of the fundamental solution (free-space Green’s function) for the governing equation. If there is no such solution in a closed form, the BEM cannot be applied, and, unfortunately, this is the case for flow problems in heterogeneous media. To overcome this difficulty, the governing equation is decomposed into various order perturbation equations, for which the free-space Green’s function can be found. At each level of perturbation, the solution is computed by the BEM and the sum- mation of various order perturbation solutions gives the complete solution for the original governing equation. The convergence of the perturbation series depends on the distance to the nearest singularity inherent in the equation. A greater magnitude of heterogeneity makes this distance shorter, and, hence, the rate of convergence becomes slower and eventually the series diverges. However, it is possible to elicit a significant amount of information from the perturbation series and to recover an accurate approximation to the exact solution. To this end, Padé approximants are employed to accelerate the rate of convergence of a slowly convergent series and to convert a divergent series into a convergent series. Two kinds of perturbation boundary element models are developed: one for steady-state flow problems, associated with the Laplace operator and the other for transient flow problems, associated with the modified Helmholtz operator in Laplace space. These models are verified against analytical solutions for simplified problems and are utilized to solve various application problems. iv The perturbation BEM shows its utility in streamline tracking and well testing problems in heterogeneous media. The analytical nature of the solution is well pre- served through the free-space Green’s function methodologically and through the singularity programming technically. The durability of the model to rapid spatial variability in rock properties is established by using Padé approximants. Through the verification and application problems, it is observed that if the av- erage property value within a drainage area is not much different from the near-well property value, heterogeneity has little effect on pressure responses. The perturbation forms for steady-state and transient flow equations derived in this study should be of value in formulating any semi-analytical scheme. v Acknowledgement This dissertation leaves me greatly indebted to Professor Roland N. Horne, my principal advisor, for his advice, guidance, and encouragement during the course of this research. I wish to express my appreciation to Professor Khalid Aziz, my M.S. research supervisor, who encouraged me to come back to Stanford. Sincere thanks are due to Professor Henry J. Ramey, Jr., who served on the reading committee, and Professors Thomas A. Hewett and William E. Brigham, who participated in the examination committee. Appreciation is extended to Professor Joseph B. Keller of the Department of Mathematics who helped me to broaden my understanding of perturbation methods. I appreciate Dr. Michael F. Riley for sharing his insights and knowledge of perturbation methods. Chick Wattenbarger, Santosh Verma, and Cesar Palagi helped me in preparing this dissertation. Many other colleagues were more helpful than they realized. Being a teaching assistant, I gained a flexible thinking about boundary element methods through their questions. Financial support for this work was provided by Teikoku Oil Company (TOC) and Japan National Oil Corporation. My special gratitude goes to my colleagues of TOC for helping me to get academic leave. vi This dissertation is dedicated to my parents, Hisashi and Shizu Sato vii Contents Abstract iv Acknowledgement vi 1 Introduction 1 2 Review of Literature 5 2.1 Boundary Element Method ........................ 5 2.1.1 Indirect and Direct Methods . ................. 6 2.1.2 Applications to Flow in Homogeneous Media .......... 7 2.1.3 Applications to Flow in Heterogeneous Media . ...... 8 2.2 Perturbation Method ........................... 10 3 Mathematical Preliminaries 12 3.1 Fluid Flow Equation ........................... 12 3.1.1 Dimensionless Equations ..................... 14 3.1.2 Laplace Space Formulation . ................. 15 3.2 Green’s Function Method ........................ 16 3.2.1 Laplace Operator ......................... 17 3.2.2 Modified Helmholtz Operator . ................. 19 3.3 Boundary Element Method ........................ 21 3.3.1 Boundary Discretization ..................... 22 3.3.2 Boundary Solutions ........................ 25 3.3.3 Interior Solutions ......................... 27 viii 3.4 Perturbation Method ........................... 27 3.4.1 Steady-State Flow Problems . ................. 29 3.4.2 Transient Flow Problems ..................... 30 3.4.3 Convergence of Perturbation Series ............... 33 3.5 Improvement of Perturbation Series . ................. 34 3.5.1 Analysis of Series ......................... 34 3.5.2 Padé Approximants ........................ 36 4 Development of a Steady-State Flow Model 40 4.1 Boundary Integral Equations ....................... 40 4.2 Treatment of Wells ............................ 43 4.2.1 Well Singularity .......................... 43 4.2.2 Pressure Specified Wells ..................... 44 4.3 Evaluation of Integrals .......................... 45 4.3.1 Boundary Integrals ........................ 45 4.3.2 Domain Integrals ......................... 46 4.4 Evaluation of Padé Approximants . ................. 50 4.5 Streamline Tracking ............................ 51 4.5.1 Euler and Modified Euler Methods ............... 52 4.5.2 Displacement Performance . ................. 54 5 Development of a Transient Flow Model 56 5.1 Boundary Integral Equations ....................... 56 5.2 Treatment of Wells ............................ 59 5.2.1 Well Singularity .......................... 59 5.2.2 Pressure Specified Wells ..................... 60 5.3 Evaluation of Integrals .......................... 60 5.3.1 Boundary Integrals ........................ 61 5.3.2 Domain Integrals ......................... 63 5.4 Evaluation of Padé Approximants . ................. 65 5.5 Transient Pressure Tests in Wells . ................. 66 5.5.1 Well with Skin and Wellbore Storage .............. 66 ix 5.5.2 Pressure Derivatives ....................... 68 6 Computational Procedures 69 6.1 Steady-State Flow Model ......................... 70 6.2 Transient Flow Model .......................... 75 7 Results and Discussion 81 7.1 Verification of the Steady-State Flow Model .............. 81 7.1.1 Effectiveness of Padé Approximants ............... 82 7.1.2 Reservoir of Harmonically Varying Permeability . ...... 95 7.1.3 A Well in an Exponentially Heterogeneous Reservoir ..... 97 7.1.4 Effectiveness of Modified Euler Method ............. 104 7.2 Verification of the Transient Flow Model ................ 111 7.2.1 Optimal Number of Stehfest Sampling Times . ...... 112 7.2.2 Reservoir of Harmonically Varying Permeability . ...... 114 7.2.3 Reservoir of Varying Porosity . ................. 118 7.2.4 Wells in an Exponentially Heterogeneous Reservoir ...... 118 7.2.5 Importance of Well-Singularity Treatment ........... 126 7.2.6 Effects of Skin and Wellbore Storage .............. 126 7.3 Application Problems ........................... 131 7.3.1 Full-Field Problem ........................ 131 7.3.2 Repeated Five-Spot Pattern . ................. 138 7.3.3 Type Curves for Idealized Heterogeneities ........... 159 8 Conclusions and Recommendations 177 Nomenclature 181 Bibliography 185 A Dirac Delta Function 196 B Selected Theorems on Padé Approximants 200 B.1 Uniqueness Theorem ........................... 200 x B.2 Invariance Theorems ........................... 201 C Extra Evaluation of Influence Coefficients 204 C.1 Directional Derivatives .......................... 204 C.2 Singular Integrals . ............................ 206 D Exact Solutions for Verification Problems 208 D.1 Wells in a Heterogeneous Reservoir . ................. 208 D.1.1 Transient Pressure Solution . ................. 208 D.1.2 Steady-State Pressure Solution ................. 212 D.2 Equidimensional Laplace-Space Equations ............... 212 D.2.1 Reservoir of Varying Permeability ................ 213 D.2.2 Reservoir of Varying Porosity . ................. 214 E Multiple-Boundary

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