Petroleum Reservoir Simulation

Description

The book contains a relatively complete treatment of finite-difference models of black-oil type rservoirs. CONTENTS Preface Nomenclature INTRODUCTION What is a Computer Model? Other Models What Questions Can a Computer Model Answer? Concluding Remarks FLUID FLOW EQUATIONS Introduction The Law of Mass Conservation Single-Phase Flow Multiphase Flow Darcy’s Law Single-Phase Flow Multiphase Flow The Basic Flow Equations Single-Phase Flow Multiphase Flow Use of Pseudopotential Boundary Conditions Alternative Forms of Multiphase Flow Equations Formulation in ‘Parabolic’ Form Formulation in ‘Hyperbolic’ Form Flow Equations Which Include Non-Darcy Effects High Flow Rates (Inertial and Turbulent Effects) Threshold and Slip Phenomena Non-Newtonian Flow Other Effects Fluid and Rock Properties Fluid Properties Rock Properties Concluding Remarks Exercises FLOW OF A SINGLE FLUID IN ONE DIMENSION Introduction Finite-Difference Approximations Discretisation in Space Discretisation in Time Discretisation Errors Other Selected Methods Other Explicit Methods Other Implicit Methods ODE Methods Comparison of Methods Grid Systems and Boundary Conditions Two Methods of Grid Construction Boundary Conditions Discretisation of One-Dimensional Flow Equations in Cartesian Co-ordinates Difference Equations for Irregular Grid Difference Equations in Matrix Form Treatment of Variable Coefficients Discretisation of One-Dimensional Flow Equations in Radial Cylindrical Co-ordinate Difference Equations for Irregular Grid Difference Equations in Matrix Form Treatment of Variable Coefficients Some Properties of Finite-Difference Equations Existence of Solution and Material Balance Treatment of Nonlinearities Concluding Remarks Exercises SOLUTION OF TRIDIAGONAL MATRIX EQUATIONS Introduction Methods of Solution Thomas’ Algorithm Tang's Algorithm Solution of Symmetric Tridiagonal Matrix Equations Special Cases of Non-Unique Solution Other Special Cases Exercises MULTIPHASE FLOW IN ONE DIMENSION Introduction The Simultaneous Solution (SS) Method The SS Method for Two-Phase Flow Extension of the SS Method to Three-Phase Flow Other Formulations of the SS Method The Implicit Pressure-Explicit Saturation (IMPES) Method IMPES Method for Three-Phase Flow Other Derivations of the IMPES Method Analysis of SS and IMPES Methods Stability Existence and Uniqueness of Solution Convergence Treatment of Nonlinearities Weighting of Transmissibilities Approximation to Transmissibilities in Time Nonlinearity due to P c Function Gas Percolation The Sequential Solution Method (SEQ) SEQ Method for Two-Phase Flow Other Forms and Derivations Numerical Results SEQ Method for Three-Phase Flow Discussion Treatment of Production Terms Differential Form of Boundary Conditions Discretisation of Boundary Conditions Exercises SOLUTION OF BLOCK TRIDIAGONAL EQUATIONS Introduction Methods of Solution Extension of Thomas’ Algorithm Use of Methods for Band Matrices FLOW OF A SINGLE FLUID IN TWO DIMENSIONS Introduction Classification of 2-D Problems Areal Problems ( x–y ) Cross-Sectional Problems ( x–z ) Single-Well Problems ( r–z ) Comments on Two-Dimensional Models Discretisation of Flow Equations Difference Approximations Stability of Difference Schemes Boundary Conditions No-Flow or Closed Boundaries Flow Boundaries Discretisation of Boundary Conditions Initial Conditions Treatment of Nonlinearities Treatment of Individual Wells Equations in Matrix Form Special Methods for 2-D Problems Alternating Direction Explicit (ADE) Methods Alternating Direction Implicit (ADI) and Related Methods Comparison of Methods Methods of Grid Construction Irregular Grid in 2-D Use of a Curvilinear Grid Concluding Remarks Exercises SOLUTION OF PENTADIAGONAL MATRIX EQUATIONS Introduction Direct Methods of Solution LU Factorisation Ordering of Equations Sparse Matrix Techniques Iterative Methods Point Jacobi Method Point Gauss-Seidel Method Point Successive Over Relaxation (SOR) Method Line and Block SOR Methods Additive Correction Methods Iterative Alternating Direction Implicit (ADI) Methods Strongly Implicit Method Other Methods Comparison of Iterative Methods Practical Considerations in the Use of Iterative Methods Comparison of Iterative and Direct Methods Concluding Remarks Exercises MULTIPHASE FLOW IN TWO DIMENSIONS Introduction Classification of 2-D Problems Areal Problems ( x– y ) Cross-Sectional Problems ( x–z ) Single-Well Problems ( r–z ) General Comments Methods of Solution and Their Comparison Discretisation in 2-D Stability of SS and IMPES in 2-D Comparison of Various Solution Methods and Computer Requirements Boundary Conditions Differential Formulation Compatibility Conditions and Constraints Finite-Difference Formulation Initial Conditions Simulation of Aquifers Simulation of Areal and Cross-Sectional Problems Use of Curvilinear Grid Treatment of Individual Wells Grid Orientation Phenomenon Simulation of Single-Well Problems Treatment of the Production Terms (the Well Model) Comparison of Stability and Efficiency of Various Treatments of Transmissibilities Practical Considerations Concluding Remarks Exercise SOLUTION OF BLOCK PENTADIAGONAL EQUATIONS Introduction Direct Methods Iterative Methods BSOR Method Iterative ADI Method The SIP Method Comparison of Iterative Methods Comparison of Iterative and Direct Methods Concluding Remarks Exercise THREE-DIMENTIONAL PROBLEMS AND SOLUTION TECHNIQUES Introduction Single-Phase Flow Basic Equation and Discretisation Special Methods for 3-D Problems Direct Methods of Solution Iterative Methods Comparison of Methods Multiphase Flow Basic Solution Methods and Their Work Requirements Methods for Solving the Matrix Equations Concluding Remarks Exercises SPECIAL TOPICS Introduction Pseudo-Functions Coats et al. (1971a) Vertical Equilibrium Model Other Pseudo-Functions Stream Tube and Related Models Simulation of Variable Bubble-Point Problems Simulation of Miscible Displacement Simulation of Compositional Effects History-Dependent Saturation Functions Physical Model of Hystersis Numerical Treatment of Hysteresis Simulation of Naturally Fractured Reservoirs Automatic Time-Step Control Concluding Remarks Exercise PRACTICAL CONSIDERATIONS Program Development Development of the Mathematical Model Development of the Numerical Model Developmetn of the Computer Model (program) Program Usage Steps Involved in a Simulation Study Selection and Design of the Model History Matching Concluding Remarks APPENDIX A APPENDIX B BIBLIOGRAPHY INDEX