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MRST说明书.pdf
Introduction
Petroleum production
Reservoir simulation
Outline of the book
The first encounter with MRST
Part I Geological Models and Grids
Modelling Reservoir Rocks
Formation of sedimentary rocks
Creation of crude oil and natural gas
Multiscale modelling of permeable rocks
Macroscopic models
Representative elementary volumes
Microscopic models: The pore scale
Mesoscopic models
Modelling rock properties
Porosity
Permeability
Other parameters
Property modelling in MRST
Homogeneous models
Random and lognormal models
10th SPE Comparative Solution Project: Model 2
The Johansen Formation
SAIGUP: shallow-marine reservoirs
Grids in Subsurface Modeling
Structured grids
Unstructured grids
Delaunay tessellation
Voronoi diagrams
Other types of tessellations
Using an external mesh generator
Stratigraphic grids
Corner-point grids
2.5D unstructured grids
Grid structure in MRST
Examples of more complex grids
Part II Single-Phase Flow
Mathematical Models and Basic Discretizations
Fundamental concept: Darcy's law
General flow equations for single-phase flow
Auxiliary conditions and equations
Boundary and initial conditions
Injection and production wells
Field lines and time-of-flight
Tracers and volume partitions
Basic finite-volume discretizations
Two-point flux-approximation
Discrete div and grad operators
Time-of-flight and tracer
Incompressible Solvers
Basic data structures in a simulation model
Fluid properties
Reservoir states
Fluid sources
Boundary conditions
Wells
Incompressible two-point pressure solver
Upwind solver for time-of-flight and tracer
Simulation examples
Quarter five-spot
Boundary conditions
Structured versus unstructured stencils
Using Peaceman well models
Single-Phase Flow and Rapid Prototyping
Implicit discretization
A simulator based on automatic differentiation
Model setup and initial state
Discrete operators and equations
Well model
The simulation loop
Pressure-dependent viscosity
Non-Newtonian fluid
Thermal effects
Part III Multiphase Flow
Mathematical Models for Multiphase Flow
New physical properties and phenomena
Saturation
Wettability
Capillary pressure
Relative permeability
Flow equations for multiphase flow
Single-component phases
Multicomponent phases
Black-oil models
Model reformulations for immiscible two-phase flow
Pressure formulation
Fractional flow formulation in phase pressure
Fractional flow formulation in global pressure
Fractional flow formulation in phase potential
Richards' equation
The Buckley–Leverett theory of 1D displacements
Horizontal displacement
Gravity segregation
Front tracking: semi-analytical solutions
Solvers for Incompressible Immiscible Flow
Fluid objects for multiphase flow
Sequential solution procedures
Pressure solvers
Saturation solvers
Simulation examples
Buckley–Leverett displacement
Inverted gravity column
Homogeneous quarter five-spot
Heterogeneous quarter five-spot: viscous fingering
Buoyant migration of CO2 in a sloping sandbox
Water coning and gravity override
The effect of capillary forces – capillary fringe
Norne: simplified simulation of a real-field model
Numerical errors
Splitting errors
Grid-orientation errors
Part IV Reservoir Engineering Workflows
Flow Diagnostics
Flow patterns and volumetric connections
Volumetric partitions
Time-of-flight per tracer region: improved accuracy
Well-allocation factors
Measures of dynamic heterogeneity
Flow and storage capacity
Lorenz coefficient and sweep efficiency
Summary of diagnostic curves and measures
Case studies
Tarbert formation: volumetric connections
Layers of SPE10: heterogeneity and sweep improvement
Interactive flow diagnostics tools
Simple 2D example
SAIGUP: flow patterns and volumetric connections
The MATLAB Reservoir Simulation Toolbox
Getting started with the software
Core functionality and add-on modules
Downloading and installing
Exploring the functionality and getting help
Release policy and version numbers
Software requirements and backward compatibility
Terms of usage
Public data sets and test cases
More about modules and advanced functionality
Operating the module system
What characterizes a module?
List of modules
Rapid prototyping using MATLAB and MRST
Automatic differentiation in MRST
References
Knut-Andreas Lie
An Introduction to Reservoir
Simulation Using MATLAB
User Guide for the Matlab Reservoir Simulation
Toolbox (MRST)
December 13, 2016
SINTEF ICT, Departement of Applied Mathematics
Oslo, Norway
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Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
5
1.1 Petroleum production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Reservoir simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.3 Outline of the book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 The first encounter with MRST . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Part I Geological Models and Grids
2 Modelling Reservoir Rocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1 Formation of sedimentary rocks . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Creation of crude oil and natural gas . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Multiscale modelling of permeable rocks . . . . . . . . . . . . . . . . . . . . 29
2.3.1 Macroscopic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.2 Representative elementary volumes . . . . . . . . . . . . . . . . . . 32
2.3.3 Microscopic models: The pore scale . . . . . . . . . . . . . . . . . . 33
2.3.4 Mesoscopic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4 Modelling rock properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.4.1 Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4.2 Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.4.3 Other parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5 Property modelling in MRST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.5.1 Homogeneous models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.5.2 Random and lognormal models . . . . . . . . . . . . . . . . . . . . . . 42
2.5.3 10th SPE Comparative Solution Project: Model 2 . . . . . . 44
2.5.4 The Johansen Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.5.5 SAIGUP: shallow-marine reservoirs . . . . . . . . . . . . . . . . . . 48
3 Grids in Subsurface Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.1 Structured grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2 Unstructured grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
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3.2.1 Delaunay tessellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2.2 Voronoi diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.2.3 Other types of tessellations . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.2.4 Using an external mesh generator . . . . . . . . . . . . . . . . . . . 75
3.3 Stratigraphic grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.3.1 Corner-point grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.3.2 2.5D unstructured grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.4 Grid structure in MRST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.5 Examples of more complex grids . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Part II Single-Phase Flow
4 Mathematical Models and Basic Discretizations . . . . . . . . . . . 115
4.1 Fundamental concept: Darcy’s law . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.2 General flow equations for single-phase flow . . . . . . . . . . . . . . . . . 117
4.3 Auxiliary conditions and equations . . . . . . . . . . . . . . . . . . . . . . . . 122
4.3.1 Boundary and initial conditions . . . . . . . . . . . . . . . . . . . . . 122
4.3.2 Injection and production wells . . . . . . . . . . . . . . . . . . . . . . 123
4.3.3 Field lines and time-of-flight . . . . . . . . . . . . . . . . . . . . . . . . 127
4.3.4 Tracers and volume partitions . . . . . . . . . . . . . . . . . . . . . . . 129
4.4 Basic finite-volume discretizations . . . . . . . . . . . . . . . . . . . . . . . . . 130
4.4.1 Two-point flux-approximation . . . . . . . . . . . . . . . . . . . . . . . 131
4.4.2 Discrete div and grad operators . . . . . . . . . . . . . . . . . . . . 135
4.4.3 Time-of-flight and tracer . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5
Incompressible Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.1 Basic data structures in a simulation model . . . . . . . . . . . . . . . . . 144
5.1.1 Fluid properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.1.2 Reservoir states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.1.3 Fluid sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.1.4 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.1.5 Wells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.2 Incompressible two-point pressure solver . . . . . . . . . . . . . . . . . . . . 150
5.3 Upwind solver for time-of-flight and tracer . . . . . . . . . . . . . . . . . . 153
5.4 Simulation examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
5.4.1 Quarter five-spot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.4.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
5.4.3 Structured versus unstructured stencils . . . . . . . . . . . . . . . 165
5.4.4 Using Peaceman well models . . . . . . . . . . . . . . . . . . . . . . . . 170
6 Consistent Discretizations on Polyhedral Grids . . . . . . . . . . . . 175
6.1 The mixed finite-element method . . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.1.1 Continuous formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.1.2 Discrete formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
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6.1.3 Hybrid formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
6.2 Consistent methods on mixed hybrid form . . . . . . . . . . . . . . . . . . 185
6.3 The mimetic method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
6.3.1 General family of inner products . . . . . . . . . . . . . . . . . . . . 190
6.3.2 General parametric family . . . . . . . . . . . . . . . . . . . . . . . . . . 192
6.3.3 Two-point type methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
6.3.4 Raviart–Thomas type inner product . . . . . . . . . . . . . . . . . 195
6.3.5 Default inner product in MRST . . . . . . . . . . . . . . . . . . . . . 196
6.3.6 Local-flux mimetic method . . . . . . . . . . . . . . . . . . . . . . . . . 197
6.3.7 Monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
7
Single-Phase Flow and Rapid Prototyping . . . . . . . . . . . . . . . . . 201
7.1 Implicit discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
7.2 A simulator based on automatic differentiation . . . . . . . . . . . . . . 203
7.2.1 Model setup and initial state . . . . . . . . . . . . . . . . . . . . . . . . 203
7.2.2 Discrete operators and equations . . . . . . . . . . . . . . . . . . . . 205
7.2.3 Well model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
7.2.4 The simulation loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
7.3 Pressure-dependent viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
7.4 Non-Newtonian fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
7.5 Thermal effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
Part III Multiphase Flow
8 Mathematical Models for Multiphase Flow . . . . . . . . . . . . . . . . 231
8.1 New physical properties and phenomena . . . . . . . . . . . . . . . . . . . . 232
8.1.1 Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
8.1.2 Wettability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
8.1.3 Capillary pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
8.1.4 Relative permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
8.2 Flow equations for multiphase flow . . . . . . . . . . . . . . . . . . . . . . . . 243
8.2.1 Single-component phases . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
8.2.2 Multicomponent phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
8.2.3 Black-oil models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
8.3 Model reformulations for immiscible two-phase flow . . . . . . . . . . 248
8.3.1 Pressure formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
8.3.2 Fractional flow formulation in phase pressure . . . . . . . . . 249
8.3.3 Fractional flow formulation in global pressure . . . . . . . . . 254
8.3.4 Fractional flow formulation in phase potential . . . . . . . . . 255
8.3.5 Richards’ equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
8.4 The Buckley–Leverett theory of 1D displacements . . . . . . . . . . . 258
8.4.1 Horizontal displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
8.4.2 Gravity segregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
8.4.3 Front tracking: semi-analytical solutions . . . . . . . . . . . . . . 267
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9 Discretizing Hyperbolic Transport Equations . . . . . . . . . . . . . . 275
9.1 A new solution concept: entropy-weak solutions . . . . . . . . . . . . . 276
9.2 Conservative finite-volume methods . . . . . . . . . . . . . . . . . . . . . . . . 278
9.3 A few classical schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
9.3.1 Centered schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
9.3.2 Upwind or Godunov schemes . . . . . . . . . . . . . . . . . . . . . . . 282
9.3.3 Comparison of centered and upwind schemes . . . . . . . . . . 283
9.3.4 Implicit schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
9.4 Convergence of conservative methods . . . . . . . . . . . . . . . . . . . . . . 291
9.5 High-resolution schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
9.5.1 Flux-limiter schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
9.5.2 Slope-limiter schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
9.5.3 Semi-discrete schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
9.6 Discretization on unstructured polyhedral grids . . . . . . . . . . . . . 302
10 Solvers for Incompressible Immiscible Flow . . . . . . . . . . . . . . . . 305
10.1 Fluid objects for multiphase flow . . . . . . . . . . . . . . . . . . . . . . . . . . 306
10.2 Sequential solution procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
10.2.1 Pressure solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
10.2.2 Saturation solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
10.3 Simulation examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
10.3.1 Buckley–Leverett displacement . . . . . . . . . . . . . . . . . . . . . 315
10.3.2 Inverted gravity column . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
10.3.3 Homogeneous quarter five-spot . . . . . . . . . . . . . . . . . . . . . . 320
10.3.4 Heterogeneous quarter five-spot: viscous fingering . . . . . 324
10.3.5 Buoyant migration of CO2 in a sloping sandbox . . . . . . . 327
10.3.6 Water coning and gravity override . . . . . . . . . . . . . . . . . . . 330
10.3.7 The effect of capillary forces – capillary fringe . . . . . . . . . 336
10.3.8 Norne: simplified simulation of a real-field model . . . . . . 340
10.4 Numerical errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
10.4.1 Splitting errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
10.4.2 Grid-orientation errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
11 Compressible Multiphase Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
Part IV Reservoir Engineering Workflows
12 Flow Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
12.1 Flow patterns and volumetric connections . . . . . . . . . . . . . . . . . . 360
12.1.1 Volumetric partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
12.1.2 Time-of-flight per tracer region: improved accuracy . . . . 364
12.1.3 Well-allocation factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
12.2 Measures of dynamic heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . 365
12.2.1 Flow and storage capacity . . . . . . . . . . . . . . . . . . . . . . . . . . 366
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12.2.2 Lorenz coefficient and sweep efficiency . . . . . . . . . . . . . . . 368
12.2.3 Summary of diagnostic curves and measures . . . . . . . . . . 370
12.3 Case studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
12.3.1 Tarbert formation: volumetric connections . . . . . . . . . . . . 372
12.3.2 Layers of SPE10: heterogeneity and sweep improvement 376
12.4 Interactive flow diagnostics tools . . . . . . . . . . . . . . . . . . . . . . . . . . 381
12.4.1 Simple 2D example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
12.4.2 SAIGUP: flow patterns and volumetric connections . . . . 388
13 Grid Coarsening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
13.1 Partition vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
13.1.1 Uniform partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
13.1.2 Connected partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
13.1.3 Composite partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
13.2 Coarse grid representation in MRST . . . . . . . . . . . . . . . . . . . . . . . 399
13.2.1 Subdivision of coarse faces . . . . . . . . . . . . . . . . . . . . . . . . . . 400
13.3 Coarsening of realistic reservoir models . . . . . . . . . . . . . . . . . . . . . 403
13.3.1 The Johansen aquifer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
13.3.2 The SAIGUP model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406
13.4 General advice and simple guidelines . . . . . . . . . . . . . . . . . . . . . . . 410
14 Upscaling Petrophysical Properties . . . . . . . . . . . . . . . . . . . . . . . . 413
14.1 Upscaling for reservoir simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 415
14.2 Upscaling additive properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
14.3 Upscaling absolute permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
14.3.1 Averaging methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420
14.3.2 Flow-based upscaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
14.4 Upscaling transmissibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
14.5 Global and local-global upscaling . . . . . . . . . . . . . . . . . . . . . . . . . . 434
14.6 Upscaling examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
14.6.1 Flow diagnostics quality measure . . . . . . . . . . . . . . . . . . . . 437
14.6.2 Model with two rock types . . . . . . . . . . . . . . . . . . . . . . . . . 438
14.6.3 SPE10 with six wells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
14.6.4 General advice and simple guidelines . . . . . . . . . . . . . . . . . 445
A The MATLAB Reservoir Simulation Toolbox . . . . . . . . . . . . . . 447
A.1 Getting started with the software . . . . . . . . . . . . . . . . . . . . . . . . . . 448
A.1.1 Core functionality and add-on modules . . . . . . . . . . . . . . . 448
A.1.2 Downloading and installing . . . . . . . . . . . . . . . . . . . . . . . . . 451
A.1.3 Exploring the functionality and getting help . . . . . . . . . . 452
A.1.4 Release policy and version numbers . . . . . . . . . . . . . . . . . . 455
A.1.5 Software requirements and backward compatibility . . . . 456
A.1.6 Terms of usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
A.2 Public data sets and test cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458
A.3 More about modules and advanced functionality . . . . . . . . . . . . . 460
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Contents
A.3.1 Operating the module system . . . . . . . . . . . . . . . . . . . . . . . 460
A.3.2 What characterizes a module? . . . . . . . . . . . . . . . . . . . . . . 461
A.3.3 List of modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462
A.4 Rapid prototyping using MATLAB and MRST . . . . . . . . . . . . . . 469
A.5 Automatic differentiation in MRST . . . . . . . . . . . . . . . . . . . . . . . . 472
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481
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