Advanced Modelling with the MATLAB Reservoir Simulation Toolbox

Cover
Half-title
Title page
Copyright information
Contents
List of Contributors
Preface
Acknowledgments
Navigating the Book and the MRST Modules
Part I Grid Generation, Discretizations, and Solvers
	1 Unstructured PEBI Grids Conforming to Lower-Dimensional Objects
		1.1 Introduction
		1.2 Basic Introduction to PEBI Grids
			1.2.1 Delaunay Triangulation
			1.2.2 PEBI Grids
			1.2.3 Clipping PEBI Grids
		1.3 Three Approaches for Optimizing PEBI Grids
			1.3.1 Background Cartesian Grids
			1.3.2 Delaunay Optimization
			1.3.3 Minimized Centroidal Energy Function
		1.4 Internal Face Constraints
			1.4.1 First Method: Simplex Conformity
			1.4.2 Configuring the Simplex-Conformity Methods
			1.4.3 Second Method: PEBI Conformity
		1.5 Adapting Cell Centroids
		1.6 Worked Examples
			1.6.1 Complex Fault Network in 2D
			1.6.2 Statistical Fracture Distribution
			1.6.3 Adapting to Permeability (SPE10)
			1.6.4 Conforming to Triangulated Surfaces in 3D
			1.6.5 Representing a Multilateral Well Path
			1.6.6 A More Realistic 3D Case
		1.7 Concluding Remarks
		References
	2 Nonlinear Finite-Volume Methods for the Flow Equation in Porous Media
		2.1 Introduction
		2.2 Model Equations
		2.3 Nonlinear Finite-Volume Methods
			2.3.1 Construction of One-Sided Fluxes
			2.3.2 Harmonic Averaging Point
			2.3.3 Nonlinear TPFA
			2.3.4 Nonlinear MPFA
			2.3.5 Nonlinear Solver
		2.4 Numerical Examples
			2.4.1 Example 1: Homogeneous Permeability
			2.4.2 Example 2: Discontinuous Permeability
			2.4.3 Example 3: No-Flow Boundary Conditions
		2.5 Concluding Remarks
		References
	3 Implicit Discontinuous Galerkin Methods for Transport Equations in Porous Media
		3.1 Introduction
		3.2 Model Equations
		3.3 Discontinuous Galerkin Methods
			3.3.1 Weak Residual Form
			3.3.2 Basis Functions
			3.3.3 Numerical Integration
			3.3.4 Evaluating the Interface Flux
			3.3.5 Velocity Interpolation
			3.3.6 Limiters
		3.4 Numerical Examples
			3.4.1 1D Buckley–Leverett Displacement
			3.4.2 Smearing of Trailing Waves
			3.4.3 Inverted Five-Spot Pattern on a Perpendicular Bisector Grid
			3.4.4 Grid-Orientation Errors for Adverse Mobility Ratios
			3.4.5 Channelized Medium
		3.5 Concluding Remarks
		References
	4 Multiscale Pressure Solvers for Stratigraphic and Polytopal Grids
		4.1 Introduction and Background Discussion
			4.1.1 Why Do We Need Multiscale Methods?
			4.1.2 Basic Flow Model and Abstract Notation
			4.1.3 Local Upscaling
		4.2 Multiscale Finite-Volume Methods
			4.2.1 Geometric Formulation of the Original MsFV Method
			4.2.2 Algebraic Formulation of the Original MsFV Method
			4.2.3 Deficiencies and Limitations of the Original MsFV Method
			4.2.4 The Multiscale Restriction-Smoothed Basis Method
			4.2.5 Introduction to the MRST Implementation
			4.2.6 Iterative Formulation
		4.3 Numerical Examples
			4.3.1 Lack of Monotonicity
			4.3.2 Grid-Orientation Errors
			4.3.3 Coarsening Complex Meshes
			4.3.4 Multiscale Methods as an Alternative to Upscaling
			4.3.5 Incompressible Multiphase Flow in Fractured Media
			4.3.6 Gravity Segregation
			4.3.7 Compressible Black-Oil Models: Fully ImplicitMethods and CPR
			4.3.8 Compressible Black-Oil Models: Sequential Solution Methods
			4.3.9 Compositional Flow
		4.4 Concluding Remarks
		References
Part II Rapid Prototyping and Accelerated Computation
	5 Better AD Simulators with Flexible State Functions and Accurate Discretizations
		5.1 Introduction
		5.2 Numerical Models in MRST
			5.2.1 A Generic Multicomponent Flow Model
			5.2.2 Anatomy of a stepFunction
			5.2.3 Validation and Preparation
		5.3 StateFunctions: Framework for AD Functions
			5.3.1 A Crash Course in State Functions
			5.3.2 Evaluation of Properties
			5.3.3 Examples of State Functions
			5.3.4 The StateFunctionGrouping Class
		5.4 Discretization with State Functions
			5.4.1 The Simulator as a Graph
			5.4.2 The Component Implementation
			5.4.3 Temporal Discretizations
			5.4.4 Example: Fully Implicit, Explicit, and Adaptive Implicit
			5.4.5 Spatial Discretizations
		5.5 Concluding Remarks
		References
	6 Faster Simulation with Optimized Automatic Differentiation and Compiled Linear Solvers
		6.1 Introduction
		6.2 Accelerated Implementation of Automatic Differentiation
			6.2.1 Different Backends for Automatic Differentiation
			6.2.2 Motivation for Different Types of AD Backends
			6.2.3 Sparse AD Backends in MRST
			6.2.4 High Performance: DiagonalAutoDiffBackend
			6.2.5 Performance of AD Backends
		6.3 High-Performance Linear Solvers
			6.3.1 Selecting Different Linear Solvers
		6.4 Setting Up and Managing Simulation Cases
			6.4.1 Packed Problems: Storing and Running Simulation Cases
			6.4.2 Automatic Setup of ECLIPSE DataSets
		6.5 Numerical Examples
			6.5.1 Packed Problems: Simulation of an Ensemble
			6.5.2 Bringing It All Together: Running a Big Model
		6.6 Concluding Remarks
			Appendix A Compilation of MRST Extensions
			Appendix B Output from AD Benchmark
		References
Part III Modeling of New Physical Processes
	7 Using State Functions and MRST's AD-OO Framework to Implement Simulators for Chemical EOR
		7.1 Introduction
		7.2 Effective Modeling Using Black-Oil-Type Equations
			7.2.1 Immiscible Flow Models
			7.2.2 Physical Effects of Polymer
			7.2.3 Physical Effects of Surfactants
		7.3 The Surfactant–Polymer Flooding Simulator
			7.3.1 Design of Flexible Model Classes
			7.3.2 The Full Three-Phase, Five-Component Model
			7.3.3 A Generic Surfactant–Polymer Model
			7.3.4 Running the Simulator from an Input Deck
		7.4 Numerical Examples
			7.4.1 Numerical Resolution of Trailing Waves
			7.4.2 Subset from SPE10: Conformance Improvement
			7.4.3 The Dynamics of Slug Injection
			7.4.4 Validation against a Commercial Simulator
		7.5 Directions and Suggestions for Future Improvements
		References
	8 Compositional Simulation with the AD-OO Framework
		8.1 Introduction
		8.2 Governing Equations
			8.2.1 Basic Flow Equations
			8.2.2 Thermodynamics
		8.3 Solving the Flash Problem
			8.3.1 Rachford–Rice: Determination of Vapor–Liquid Equilibrium
			8.3.2 Updating the Thermodynamic Equilibrium
			8.3.3 Phase Stability Testing
			8.3.4 Equation of State
		8.4 Coupled Flow and Thermodynamics
			8.4.1 Overall Composition Formulation
			8.4.2 Natural Variables Formulation
			8.4.3 Comparison between Different Formulations
			8.4.4 Implementation as Generic Models
			8.4.5 State Functions for Compositional Models
			8.4.6 Limitations and Caveats
		8.5 Examples
			8.5.1 Validation of MRST’s Simulators
			8.5.2 Numerical Accuracy
			8.5.3 Surface Volumes and Separators
			8.5.4 Miscibility
			8.5.5 Performance of Compositional Solvers
		8.6 Concluding Remarks
		References
	9 Embedded Discrete Fracture Models
		9.1 Introduction
		9.2 Fracture Permeability
		9.3 Mathematical Formulation
		9.4 Hierarchical Fracture Model Module
		9.5 Two-Phase Flow through a Simple Fracture Network
		9.6 Upscaling a Stochastically Generated Fracture Network
		9.7 Simulation of Well Test Response in an Outcrop-Based Fracture Network
		9.8 Concluding Remarks
		References
	10 Numerical Modeling of Fractured Unconventional Oil and Gas Reservoirs
		10.1 Introduction
		10.2 Shale Module
		10.3 Compositional Flow and Modeling of Fractured Reservoirs
			10.3.1 Governing Equations for Compositional Flowin Conventional Reservoirs
			10.3.2 Modeling of Fractured Reservoirs in MRST
			10.3.3 The pEDFM Transmissibilities
		10.4 EDFM and Compositional Simulation in MRST
		10.5 Stochastic Generation of Fractures with Arbitrary Orientations in 3D
			10.5.1 Generation of Fracture Sets Using ADFNE
		10.6 Applications of 3D pEDFM to Model UOG Reservoirs
			10.6.1 Basic Model Parameters Representative of the Eagle Ford Shale
			10.6.2 Implementation Steps
			10.6.3 Eagle Ford Shale Reservoir Simulation Results
		10.7 Modeling Transport and Storage Mechanisms in Organic-Rich Source Rocks
			10.7.1 Sorption
			10.7.2 Molecular Diffusion
			10.7.3 Geomechanics Effect
		References
	11 A Unified Framework for Flow Simulation in Fractured Reservoirs
		11.1 Introduction
		11.2 Modeling and Simulation Techniques for Fractured Reservoirs
			11.2.1 Governing Equations
			11.2.2 Multicontinuum Models
			11.2.3 Discrete Fracture and Matrix Model
		11.3 Implementation in MRST
			11.3.1 Multicontinuum and Discrete Fracture and Matrix Models
			11.3.2 A Brief Note on Other Methods
			11.3.3 Description of the fractures Module
		11.4 Applications
			11.4.1 Validation of the DFM Implementation
			11.4.2 Pressure Buildup in Fractured Aquifers during CO[sub(2)] Storage Operations
			11.4.3 A Model with Explicit Fractures and Dual Porosity
			11.4.4 Multirate Transfer in Multicontinuum Model
		11.5 Summary and Conclusion
		References
	12 Simulation of Geothermal Systems Using MRST
		12.1 Introduction
		12.2 Governing Equations for Geothermal Applications
		12.3 The Geothermal Module
			12.3.1 A Simple Worked Example
			12.3.2 Utility and State Functions
		12.4 Numerical Examples
			12.4.1 Benchmark with TOUGH2
			12.4.2 Subset of SPE10 Model 2
			12.4.3 Enhanced Geothermal System
			12.4.4 Thermal Aquifer Energy Storage
		12.5 Concluding Remarks
		References
	13 A Finite-Volume-Based Module for Unsaturated Poroelasticity
		13.1 Introduction
		13.2 Governing Equations
			13.2.1 Richards’ Equation
			13.2.2 Unsaturated Poroelasticity
			13.2.3 Boundary and Initial Conditions
		13.3 Discretization and Implementation
			13.3.1 MPFA and MPSA
			13.3.2 Discretization
			13.3.3 Solving the Equations
		13.4 Numerical Examples
			13.4.1 Numerical Convergence Tests
			13.4.2 Water Infiltration in a Column of Dry Soil
			13.4.3 Desiccation of a Clayey Soil in a Petri Dish
		13.5 Concluding Remarks
		References
	14 A Brief Introduction to Poroelasticity and Simulation of Coupled Geomechanics and Flow in MRST
		14.1 Introduction
		14.2 Governing Equations
			14.2.1 Equations of Linear Elasticity
			14.2.2 Equations of Linear Poroelasticity
			14.2.3 The Linear Poroelastic Equations
		14.3 Moduli, Moduli, Moduli …
			14.3.1 The Biot–Willis Coefficient, α
			14.3.2 Drained and Undrained Moduli
			14.3.3 Specific Storage Coefficients
			14.3.4 Geertma's Uniaxial Expansion Coefficient, C[sub(m)]
			14.3.5 Automatic Computation of Poroelastic Parameters
		14.4 Coupling Strategies
			14.4.1 Fully Coupled and Sequentially Split Schemes
			14.4.2 The Fixed Stress Split Scheme
			14.4.3 The ad-mechanics Module in MRST
		14.5 Numerical Examples
			14.5.1 Compression of a Dry Sample
			14.5.2 Compression of a Wet Sample: The Terzaghi Problem
			14.5.3 Mandel’s Problem
		14.6 Concluding Remarks
		References