Introduction to the Numerical Modeling of Groundwater and Geothermal Systems Fundamentals of Mass, Energy and Solute Transport in Poroelastic Rocks Jochen Bundschuh University of Applied Sciences, Institute of Applied Research, Karlsruhe, Germany Royal 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 Press V; J Taylor & Francis Group Boca Raton London New York Leiden CRC Press is an imprint of the Taylor St Francis Group, an informa business A BALKEMA BOOK
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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