Introduction to the Numerical Modeling of Groundwater · PDF fileIntroduction to the Numerical Modeling of Groundwater and Geothermal Systems Fundamentals of Mass, Energy and Solute

Introduction to the NumericalModeling of Groundwaterand Geothermal SystemsFundamentals of Mass, Energy and SoluteTransport in Poroelastic Rocks

Jochen Bundschuh

University of Applied Sciences, Institute of Applied Research,Karlsruhe, GermanyRoyal Institute of Technology (KTH), Stockholm, Sweden

Mario Cesar Suarez Arriaga

Department of Applied Mathematics and Earth Sciences,Faculty of Physics and Mathematical Sciences,Michoacdn University UMSNH, Morelia, Michoacdn, Mexico

Ltfi) CRC PressV; J Taylor & Francis Group

Boca Raton London New York Leiden

CRC Press is an imprint of theTaylor St Francis Group, an informa business

A BALKEMA BOOK

Table of contents

About the book series VII

Editorial board of the book series IX

Dedications XI

The Pioneers of fluid flow and thermoporoelasticity XIII

Preface . . XXXIII

Foreword XXXV

Authors' prologue XXXVII

About the authors XXXIX

Acknowledgements XLI

1 Introduction 1

1.1 The water problem—The UN vision 1

1.2 The energy problem—Vision of the Intergovernmental

Panel of Climate Change 3

1.3 Multiphysics modeling of isothermal groundwater and geothermal systems 5

1.4 Modeling needs in the context of social and economic development 6

1.4.1 The role of groundwater for drinking, irrigation,

and other purposes 7

1.4.2 Geothermal resources 9

1.5 The need to accelerate the use of numerical modeling of isothermalaquifers and geothermal systems 10

2 Rock and fluid properties 13

2.1 Mechanical and thermal properties of porous rocks 13

2.1.1 Absolute permeability 13

2.1.2 The skeleton: Bulk, pore and solid volumes; porosity 15

2.1.2.1 The variation of the fluid mass content 17

2.1.2.2 The advective derivative of the density 17XV

XVI Table of contents

2.1.3 The principle of conservation of mass in porous rocks 17

2.1.4 Thermal conductivity of porous rocks 19

2.1.5 Heat conduction, Fourier's law and thermal gradient 21

2.1.6 Heat capacity and enthalpy of rocks 21

2.1.7 Rock heat capacity and geothermal electric power 26

2.1.8 Thermal diffusivity and expansivity of rocks 27

2.1.8.1 Thermal diffusivity 27

2.1.8.2 Volumetric thermal expansivity 28

2.1.9 Mechanical parameters of rocks 29

2.1.9.1 Stress and strain 29

2.1.9.2 Young's modulus 29

2.1.9.3 Poisson's modulus 30

2.1.9.4 Bulk modulus 30

2.1.9.5 Rock compressibility 30

2.1.9.6 Rigidity and Lame moduli 30

2.1.9.7 Volumetric strain 30

2.1.10 Elasticity equations for Hookean rocks 31

2.2 Linear thermoporoelastic rock deformation " 32

2.2.1 Effects of the fluid on porous rock properties 32

2.2.2 A simple model for the collapse of fractures in poroelastic rocks 33

2.2.3 Linear deformation of rocks containing isothermal fluid 35

2.2.3.1 Differential relationships between porosity and volumes 36

2.2.4 Poroelastic rock parameters: Drained and undrained conditions 36

2.2.4.1 Drained bulk compressibility 37

2.2.4.2 Undrained bulk compressibility 37

2.2.4.3 Compressibility of the solid phase 38

2.2.4.4 Compressibility of the pore volume 38

2.2.5 The Biot-Willis coefficient 39

2.2.6 Biot's classical poroelasticity theory 40

Table of contents XVII

2.2.6.1 Fundamental concepts and coefficients in

Biot's poroelastic theory 40

2.2.6.2 The fundamental parameters of poroelasticity 41

2.2.6.3 Relationships among the bulk moduli

and other poroelastic coefficients 43

2.2.7 Porosity and the low-frequency Gassmann-Biot equation 44

2.2.8 Numerical values of the poroelastic coefficients 47

2.2.9 Tensorial form of Biot's poroelastic theory in 4D 48

2.2.9.1 Terzaghi effective stresses in poroelastic rocks 49

2.2.9.2 Vectorial formulation of the poroelastic equations 50

2.2.10 Mathematical model of the fluid flow in poroelastic rocks 52

2.2.10.1 Dynamic and static poroelastic equations

for Hookean rocks 53

2.2.11 Diffusion equations for consolidation 56

2.2.12 Basic thermodynamics of porous rocks 57

2.2.12.1 The first and second laws of thermodynamics

for porous rocks 57

2.2.12.2 Differential and integral forms of the first and second law 59

2.2.12.3 The Helmholtz free energy: A thermoelastic potentialfor the matrix 60

2.2.12.4 The Gibbs free enthalpy: Skeleton thermodynamics

with null dissipation 63

2.2.12.5 Thermodynamics of the fluid mass content 66

2.2.12.6 Numerical values of the thermal expansivity coefficients 69

2.2.12.7 Tensorial form of the thermoporoelastic equations 69

2.3 Mechanical and thermodynamical water properties 70

2.3.1 Practical correlations for aquifers and low-enthalpy geothermal systems 71

2.3.2 A brief history of the equation of state for water 74

2.3.3 The IAPWS-95 formulation for the equation of state of water 75

2.3.4 Exact properties of low-enthalpy water (0 to 150°C) 77

2.3.4.1 Density and enthalpy of the liquid 77

2.3.4.2 Isobaric heat capacity and thermal conductivity 77

XVIII Table of contents

23 A3 Compressibility arid expansivity 78

23 A A Dynamic viscosity and speed of sound 78

2.3.5 Exact properties of high-enthalpy water (150 to 350°C) 79

2.3.6 Properties of two-phase geothermal water (100 to 370°C) 81

2.3.6.1 Thermodynamic range of validity of the code AquaG370.For 82

2.3.6.2 Temperature of saturation (subroutine Tsat) 82

2.3.6.3 Saturation pressure (subroutine Fsat) 82

23.6 A Density and enthalpy of liquid and steam (subroutines

Likid and Vapor) 82

2.3.6.5 Dynamic viscosity of two-phase water (subroutine Visf) 83

2.3.6.6 Thermal conductivity of two-phase water (subroutine Terk) 83

2.3.6.7 Specific heat of two-phase water (subroutines CPliq

and CPvap) 84

2.3.6.8 Surface tension of two-phase water (subroutine Tensa) 84

2.3.6.9 Practical correlations for two-phase flow 84

2.3.7 Capillary pressure 85

2.3.8 Practical correlations for capillary pressures 88

2.3.8.1 Correlation of Van Genuchten 88

2.3.8.2 Correlation of Schulz and Kehrwald 89

2.3.8.3 Correlation of Li and Home 89

2.3.8.4 Correlation of Brooks-Corey 90

2.3.8.5 Correlation of Li and Home for geothermal reservoirs 90

2.3.8.6 The Li-Home general fractal capillary pressure model 92

2.3.9 Relative permeabilities 92

2.3.10 Practical correlations for relative permeabilities 94

2.3.10.1 Constant functions for perfectly mobile phases 94

2.3.10.2 Linear functions 94

2.3.10.3 Functions of Purcel 95

2.3.10.4 Functions of Corey 95

2.3.10.5 Functions of Brooks-Corey 95

Table of contents XIX

2.3.10.6 Functions of Schulz-Kehrwald 96

2.3.10.7 Functions for three-phase relative permeabilities 96

2.3.10.8 Li-Home universal relative permeability functions based

'•'•• on fractal modeling of porous rocks 96

2.3.10.9 Linear X-functions for relative permeability in fractures 97

2.3.10.10 Relative permeabilities in fractures:The Honarpour-Diomampo model 97

2.3.11 Observed effects of dissolved salts (NaCl)and non-condensible gases (CO2) 98

3 Special properties of heterogeneous aquifers 101

3.1 The problem of heterogeneity in aquifers 101

3.2 The concept of multiple porosity in heterogeneous aquifers 102

3.3 The triple porosity-permeability concept in geothermics 103

3.4 Averages of parameters at different interfaces 103

3.4.1 Permeability and thermal conductivity 103

3.4.2 Special average for thermal conductivity in dry rock 104

3.4.3 Heat capacity of the rock-fluid system 104

3.4.4 Linear Lagrange interpolation for densities 105

3.5 Averages for systems with two and three components: General

models of mixtures • 105

3.5.1 Parallel and serial models 105

3.5.2 Geometrical model 106

3.5.3 Model of Griethe 106

3.5.4 Model of Budiansky 106

3.5.5 Model of Hashin-Shtrikman 106

3.5.6 ModelofBrailsford-Major 107

3.5.7 Model of Waff 107

3.5.8 Model of Walsh-Decker 107

3.5.9 Model of Maxwell 107

3.5.10 Maxwell's dispersive model 107

3.5.11 Model of Russel 108

XX Table of contents

3.6 Some applications to field data 108

3.6.1 Application to data from rocks of the Los Azufres and

Los Humeros geothermal fields (Mexico) 108

3.7 Discontinuities of parameters when crossing heterogeneous interfaces 110

3.8 Examples of heterogeneous non-isothermal aquifers—Petrophysical properties

in Mexican geothermal fields 112

3.8.1 Cerritos Colorados (La Primavera), Jalisco 112

3.8.2 Los Humeros, Puebla 113

3.8.3 Los Azufres, Michoacdn 114

4 Fluid flow, heat and solute transport 115

4.1 The conservation of mass for fluids 115

4.2 General model of fluid flow: The Navier-Stokes equations 117

4.2.1 Flow of fluids at the scale of the pores 119

4.3 Darcy's law: pressure and head 119

4.3.1 Pressure formulation of the general groundwater flow equation 122

4.3.2 Darcy's law in terms of hydraulic head and conductivity 122

4.3.3 The hydraulic head governing equation of groundwater flow 124

4.3.3.1 Storativity and transmissivity 1254.3.3.2 Two-dimensional groundwater flow—The Boussinesq

approximation . 126

4.3.4 Reservoir anisotropy in two dimensions 128

4.4 Flow to wells in homogeneous isotropic aquifers 129

4.4.1 Simple geometries for isothermal stationary groundwater flow 129

4.4.1.1 Radial flow 129

4.4.1.2 Linear flow 130

4.4.1.3 Spherical flow 130

4.4.2 Darcy's law and the equation of state of slightly compressible water 131

4.4.3 Transient flow of slight compressible fluids, Theis solution 132

4.4.4 Flow to a well of finite radius, wellbore storage 135

4.4.5 The Brinkman equation and the coupled flow to wells 136

Table of contents XXI

4.4.5.1 Coupling the Darcy-Brinkman-Navier-Stokes equations

in the flow to wells 138

4.5 Pumping test fundamentals 139

4.5.1 Stationary flow towards a well—Dupuit and Thiem well equations 140

4.5.1.1 Confined aquifer 140

4.5.1.2 Unconfined aquifer 141

4.5.2 Transient flow—explicit Theis equation for confined aquifers 142

4.5.3 Transient flow—Hantush equation (semiconfined aquifer) 145

4.5.4 Turbulence and the Forchheimer's law 146

4.6 Heat transport equations 147

4.6.1 Heat conduction , 148

4.6.2 Heat convection 149

4.7 Flow of mass and energy in two-phase reservoirs 150

4.7.1 Darcy's law for two-phase systems 150

4.7.2 Flow of energy in reservoirs with single-phase fluid 151

4.7.3 Flow of energy in reservoirs with two-phase fluid 152

4.7.3.1 The Garg's model for two-phase fluid 153

4.7.4 Heat pipe transfer in two-phase reservoirs 153

4.7.5 The general heat flow equation 154

4.8 Solute transport equations ' 154

4.8.1 Fick's law of diffusion 155

4.8.2 Fick's law with advection and dispersion 157

4.8.3 General solute transport equations 159

5 Principal numerical methods 165

5.1 The finite difference method 166

5.1.1 Fundamentals : 166

5.1.2 Stationary two-dimensional groundwater flow 168

5.1.2.1 Difference method for model nodes and centers 168

5.1.2.1.1 Forward difference method 169

XXII Table of contents

5.1.2.1.2 Centered difference method 170

5.1.2.1.3 Backward difference method 170

5.1.2.2 Difference method for boundary nodes 171

5.1.3 Transient groundwater flow 172

5.1.3.1 Time discretization 172

5.1.3.2 Explicit difference method (e = 1) 173

5.1.3.3 Implicit difference method (s = 0) 174

5.1.3.4 Crank-Nicholson difference method (e = 0.5) 175

5.1.3.5 Difference method for an inhomogeneous, anisotropic,

confined aquifer 175

5.1.4 Calculating the groundwater flow velocity (average pore velocity) 178

5.1.5 Solute and heat transport 179

5.1.6 Stability and accuracy criteria 180

5.1.6.1 Courant criterion 181

5.1.6.2 Neumann criterion 181

5.2 Introduction to the finite element method (FEM) 182

5.2.1 Brief description of the method fundamentals 183

5.2.2 Finite elements using linear Lagrange interpolation polynomials 183

5.2.3 Numerical solution of the Poisson's equation with the

FEM—Galerkin method 186

5.2.3.1 Numerical method 1: General test functions 188

5.2.3.2 Numerical method 2: Linear polynomials 190

5.2.3.3 Variable permeability in the weak form of the Galerkin method 194

5.2.3.4 The general diffusion equation in the weak formof the Galerkin method 194

5.2.4 Galerkin weighted residuals method; weak formulation of theheat equation for a stationary temperature in two dimensions 196

5.2.5 Finite elements using bilinear Lagrange interpolationpolynomials over triangles 197

5.2.6 Finite elements using bilinear Lagrange interpolationpolynomials over rectangles 199

5.2.7 Solution of the transient heat equation using finite elements in ID 201

Table of contents XXIII

'5.3 The finite volume method (FVM) 202

5.3.1 The FVM in the solution of single-phase mass flow 202

5.3.2 The FVM in the numerical solution of single-phase energy flow 205

5.3.3 The FVM in the numerical solution of two-phase mass flow 206

5.3.4 The FVM in the numerical solution of two-phase energy flow 207

5.3.5 Numerical approximations of the time-level . 209

5.3.5.1 Explicit numerical approximation of the time-level 209

5.3.5.2 Implicit numerical approximation of the time-level 210

5.3.5.3 Three time-level numerical approximations 210

5.4 The boundary element method for elliptic problems , . 211

5.4.1 The Dirac distribution (a generalized function) ' • 213

5.4.2 The fundamental solution in free space : 213

5.4.3 BEM solution of the Poisson's equation 214

5.4.4 The BEM numerical implementation: An example 215

6 Procedure of a numerical model elaboration 217

6.1 Introduction 217

6.2 Defining the objectives of the numerical model 219

6.3 Conceptual model 221

6.4 Types of conceptual models 223

6.4.1 The porous medium continuum model (granular medium) 224

6.4.2 Conceptual models of fracture flow 226

6.4.2.1 Equivalent porous medium (EPM) approach 227

6.4.2.2 Dual and multiple continuum approach 227

6.4.2.3 Explicit discrete fracture approach 228

6.4.2.4 Discrete-fracture network (DFN) approach 229

6.4.3 Simplification by using 2-dimensional horizontal models 229

6.5 Field data required for constructing the conceptual model 230

6.5.1 General evaluation of sufficiency of available field data 230

6.5.2 Types of model boundaries and boundary values 232

XXIV Table of contents

6.5.3 Aquifer geometry, type, solid and fluid properties 233

6.5.4 Sources and sinks within the model area 235

6.6 Numerical formulation of the conceptual model 237

6.6.1 From the conceptual to the mathematical and numerical model 237

6.6.2 Discretizing the model domain of an aquifer 237

6.6.3 Initial values 239

6.6.4 Ready for numerical simulations 239

6.7 Parameter estimation 239

6.7.1 Remote sensing 239

6.7.2 Field surveys of cold (non-geothermal) aquifer systems 241

6.7.2.1 Geological and hydrogeological studies 241

6.7.2.2 Hydro(geo)chemical surveys 241

6.7.2.3 Geophysical survey 243

6.7.2.4 Aquifer tests 244

6.7.2.5 Tracer tests 246

6.7.2.5.1 Overview on principalartificial tracers and their applications 246

6.7.2.5.2 Field-scale tracer tests performed

in boreholes (wells) 246

6.7.3 Field survey of geothermal systems • 247

6.7.3.1 Geological and hydrogeological surface studies 248

6.7.3.2 Hydro(geo)chemical surveys 248

6.7.3.3 Geophysical surveys 248

6.7.3.4 Tests performed in drillings 249

6.7.4 Laboratory tests and experiments 250

6.8 Selection of model type and code 250

6.8.1 Model types 251

6.8.2 Public domain and commercial software for numerical modeling 254

6.8.3 ASM (Aquifer Simulation Model) 254

6.8.4 SUTRA 255

Table of contents XXV

6.8.5 Visual MODFLOW : •• : 256

6.8.6 Processing MODFLOW for Windows (PMWIN) 257

6.8.7 FEFLOW (Finite Element Subsurface Flow

and Transport Simulation System) ••• 258

6.8.8 TOUGH, TOUGHREACT and related codes and modules 260

6.8.9 STAR—General Purpose Reservoir Simulation System 261

6.8.9.1 RIGHTS: Single-phase geothermal reservoir simulator 262

6.8.9.2 DIAGNS: Well test data diagnostics and interpretation 262

6.8.9.3 GEOSYS: Data management and visualization system 262

6.8.10 COMSOL Multiphysics 263

6.9 Calibration, validation and sensitivity analysis 263

6.9.1 Model calibration 264

6.9.2 Model validation (history matching) 265

6.9.3 Sensitivity analysis 268

6.10 Performing numerical simulations 269

6.11 How good is the model? Assessing uncertainties 271

6.12 Model misuse and mistakes 272

6.13 Example of model construction—Assessment of the

contamination of an aquifer 273

6.13.1 Situation and tasks . . 273

6.13.1.1 Existing field data and information . ..'.. 275

6.13.2 Design of the investigation program . 275

6.13.3 Preliminary investigations—Contamination assessment 275

6.13.4 Acquisition of groundwater level and contarhinant concentration 275

6.13.5 Groundwater flow and advective transport as a first approach 276

6.13.5.1 Delimitation of the area to model and aquifer geometry 276

6.13.5.2 Hydrogeological parameters 276

6.13.5.3 Simulation of groundwater flow lines and flow times 278

6.13.5.4 Results • . 278

6.13.6 Transport model with dispersion, sorption, and resulting solutions 279

XXVI Table of contents

6.13.6.1 Introduction 279

6.13.6.2 Simulation of solute propagation with a permanentinflow of contaminants 280

6.13.6.3 Simulation of solute propagation with an immediate

suspension of inflow of contaminants 282

6.13.6.4 Water works: Diagnosis and recommended solutions 282

6.13.6.5 Farm wells: Diagnosis and recommended solutions 283

7 Parameter identification and inverse problems

(by Longina Castellanos and Angel Perez) 285

7.1 Introduction 285

7.2 Ill-posedness of the inverse problem 288

7.2.1 Existence of a solution 289

7.2.2 Uniqueness of the solution (identifiability) 289

7.2.3 Continuous dependency on the data 290

7.3 Linear least-squares (LLS) 291

7.3.1 Condition number of an invertible square matrix 292

7.3.2 Linear least-squares solution: Direct method 292

7.3.3 Tikhonov's regularization method 293

7.3.4 An iterative method for solving LLS: Linear conjugate gradients 294

7.3.4.1 L-curve regularization algorithm . 295

7.3.4.2 Linear preconditioned conjugate gradient method 296

7.4 Nonlinear least-squares (NLS) 297

7.4.1 The Levenberg-Marquardt method 298

7.4.2 Using an optimization routine (TRON) to solve NLS problems 300

7.4.3 Regularization techniques in NLS 301

7.5 Application examples 302

7.5.1 Groundwater modeling 302

7.5.1.1 Multiscale optimization • 303

7.5.1.2 An elementary example 305

7.5.1.2.1 Experimental results 305

Table of contents XXVII

7.5.2 Inverse problems in geophysics 306

7.5.2.1 An elementary geophysical inverse problem 307

7.5.2.2 Formulation of the inverse problem 308

Groundwater modeling application examples 311

8.1 Periodical extraction of groundwater 311

8.1.1 Situation and tasks 311

8.1.2 Aquifer specifications 311

8.1.3 Tasks 312

8.1.4 Elaboration of a numerical model and problem solution 312

8.1.4.1 Elaboration of the numerical flow model 312

8.1.4.2 Evaluation of the piezometric pattern 314

8.1.4.3 Evaluation of the water balance of the model area 314

8.2 Water exchange between an aquifer and a surface water body by leakage 315

8.2.1 Problem and tasks 315

8.2.2 Hydrogeological setting, available data and measurements 315

8.2.3 Formulating surface water-groundwater interactions by leakage 315

8.2.4 Execution of infiltration tests along the river 316


8.3 Scenario modeling of multi-layer aquifers and distribution of groundwater

ages caused by exploitation 320

8.3.1 Problem 320


8.3.3 Objectives of the modeling 322

8.3.4 Elaboration of a numerical model 323

8.3.4.1 Aquifer properties ;..,•..- 324

8.3.4.2 Boundary conditions . 325

8.3.4.3 Groundwater extraction 325

8.3.5 Results 325

8.3.5.1 Groundwater balance in models with two or threegeological layers 325

XXVIII Table of contents

8.3.5.2 Influence of groundwater exploitation 325

8.3.5.3 Influence of clay layers 338

1 8.3.5.4 Contamination threat for deeper aquifers 339

8.4 Point source contamination and aquifer remediation 340

8.4.1 Situation 340


8.4.3 Objectives 340


8.4.4.1 Groundwater flow pattern 342

8.4.4.2 Isochrones 343

8.4.4.3 Designing the remediation well 343

8.5 Boron contamination propagation 344

8.5.1 Situation and objectives 344

8.5.2 Existing data 345


8.5.4 Results of the simulations 346

8.6 Annual temperature oscillations in a shallow stratified aquifer 347

8.6.1 Introduction and objectives 347


8.6.2.1 Geometry 349

8.6.2.2 Aquifer properties 349

8.6.2.3 Boundary conditions 350

8.6.2.4 Initial conditions 351

8.6.3 Simulations of spring water temperatures and their results

(scenarios 1-8) ' 351

8.6.3.1 Groundwater flow field 351

8.6.3.2 Annual periodic oscillations of spring water temperature 351

8.6.3.3 Results summary 351

8.6.4 Scenario 1 and 3: Simulations and results 352

8.6.4.1 Scenario 1 355

Table of contents XXIX

' 8.6.4.1.1 Aquifer configuration, flow field

and water balance 355

8.6.4.1.2 Temperature field 355

8.6.4.1.3 Spring water temperature 355

8.6.4.2 Scenario 3 355

8.6.4.2.1 Aquifer configuration, flow pattern

and water balance 355

8.6.4.2.2 Temperature field 355

8.6.4.2.3 Spring water temperature • 356

9 Geothermal systems modeling examples 357

9.1 What is geothermal energy? 357

9.1.1 Characteristics of geothermal reservoirs in Mexico as examples

of heterogeneous non-isothermal aquifers 360

9.1.1.1 Cerro Prieto, Baja California 361

9.1.1.2 Los Azufres, Michoacan 362

9.1.1.3 Los Humeros, Puebla 364

9.1.1.4 Las Tres Virgenes, Baja California 366

9.1.1.5 Cerritos Colorados (La Primavera), Jalisco 366

9.2 Transient radial-vertical heat conduction in wells 367

9.2.1 Transient radial-vertical flow of heat 367

9.2.1.1 Radial portion of the model 369

9.2.1.2 Radial temperature distribution inside the cement 371

9.2.1.3 Vertical portion of the model 372

9.2.1.4 The well-rock radial flow of heat 373

9.3 The Model of Avdonin 374

9.3.1 Transient radial flow of hot water in ID 374

9.4 The invasion of geothermal brine in oil reservoirs 376

9.4.1 Geothermal aquifers and oil reservoirs: Available data 376

9.4.2 A general 3D mathematical model 378

9.4.2.1 The one-dimensional Buckley-Leverett model 380

XXX Table of contents

9 A3 Formulation and numerical solution using the finite element method 381

9.4.4 Numerical simulation of brine invasion 383

9.5 Modeling submarine geothermal systems 384

9.5.1 Brief description of submarine geothermal systems 385

9.5.1.1 Geothermal discharge chimneys and plumes 386

9.5.1.2 Models to compute the heat flux related to the plumes 387

9.5.1.3 Modeling the radial heat conduction in chimneys 388

9.5.2 Submarine geothermal potential 388

9.5.2.1 Submarine potential of the Gulf of California 388

9.5.2.2 Using the boundary element method to estimate the initialconditions of submarine geothermal systems 390

9.6 Modeling processes in fractured geothermal systems 392

9.6.1 Single porosity and fractured volcanic systems 393

9.6.2 The double porosity model 393

9.6.3 The triple porosity model 393

9.6.4 Fluid transport through faults 395

9.6.5 General equations for single, double and triple porosity models 396

9.6.6 Numerical comparison between single porosity and fractured media 396

9.6.6.1 Graphical results of the simulations 398

9.6.7 Effective thermal conductivity and reservoir natural state; Flowproblem with CO2 401

9.6.8 Simultaneous heat and mass flow in fractured reservoirs: Conclusions 405

APPENDIXES

A: Mathematical appendix 407

A. 1 Introduction to interpolation techniques 407

A. 1.1 Approximation and basis of interpolation in one dimension 408

A. 1.2 Basis of a linear functional space 409

A.l.2.1 Karl Weierstrass approximation theorem (1885) 409

A. 1.3 Support and matrix of the interpolation 409

Table of contents XXXI

A. 1.4 Numerical examples of interpolation 410

A. 1.4.1 Example 1 410

A. 1.4.2 Example 2 411

A. 1.5 The Lagrange interpolation polynomials 412

A. 1.5.1 Example 1 414

A. 1.5.2 Example 2 415

A.2 Interpolation in two and three dimensions 415

A.3 Elements of tensor analysis 416

A.3.1 Index notation for vectors and tensors 416

A.3.2 Differential operators in curvilinear coordinates 417

A.3.3 Tensorial notation for differential operators in cartesian coordinates 418

A.4 The integral theorem of Stokes 418

A.4.1 Riemann's theorem 419

A.4.2 Green's first identity 419

A.4.3 Green's second identity 420

A.4.4 The divergence theorem 420

A.4.5 The Dirac distribution 420

B: Tabulated thermal conductivities 422

Nomenclature 429

References 439

Subject index 457

Book series page 479

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Introduction to the Numerical Modeling of Groundwater · PDF fileIntroduction to the Numerical Modeling of Groundwater and Geothermal Systems Fundamentals of Mass, Energy and Solute