Introduction to the numerical modeling of groundwater and geothermal systems: fundamentals of mass, energy, and solute transport in poroelastic rocks
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton, Fla. [u.a.]
CRC Press
2010
|
| Series: | Multiphysics modeling
2 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XLII, 478 S. Ill., graph. Darst. |
| ISBN: | hardcover 9780415401678 |
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| 100 | 1 | |a Bundschuh, Jochen |d 1960- |e Verfasser |0 (DE-588)1064353835 |4 aut | |
| 245 | 1 | 0 | |a Introduction to the numerical modeling of groundwater and geothermal systems |b fundamentals of mass, energy, and solute transport in poroelastic rocks |c Jochen Bundschuh ; Mario César Suárez Arriaga |
| 264 | 1 | |a Boca Raton, Fla. [u.a.] |b CRC Press |c 2010 | |
| 300 | |a XLII, 478 S. |b Ill., graph. Darst. | ||
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| _version_ | 1804137307440152576 |
|---|---|
| adam_text | 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 isothermal
aquifers 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
17
XV
XVI Table of
contents
2.1.3
The principle of
conservation
of mass in porous rocks 1
7
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
Lamé
moduli
30
2.1.9.7 Volumetriestrain 30
2.1.10
Elasticity equations for Hookean rocks
3
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
З8
2.2.4.4
Compressibility of the pore volume
З8
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 potential
for 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
(Oto 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
2.3.4.3
Compressibility and expansivity
78
2.3.4.4
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°
С
) 81
2.3.6.1
Thermodynamic range of validity of the code AquaG3
70.
For
82
2.3.6.2
Temperature of saturation (subroutine rsat)
82
2.3.6.3
Saturation pressure (subroutine Psat)
82
2.3.6.4
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
Horne
89
2.3.8.4
Correlation of Brooks-Corey
90
2.3.8.5
Correlation of Li and
Horne
for geothermal reservoirs
90
2.3.8.6
The Li-Horne 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-Horne universal relative permeability functions based
on fractal modeling of porous rocks
96
2.3.10.9
Linear
Х
-functions for relative permeability in fractures
97
2.3.10.10
Relative permeabilities in fractures:
The Honarpour-Diornampo 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
Model of Brailsford-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
1 2
3.8.1
Cerritos
Colorados (La Primavera),
Jalisco
112
3.8.2
Los Humeros, Puebla
ИЗ
3.8.3
Los Azufres, Michoacán
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
125
4.3.3.2
Two-dimensional groundwater flow
—
The Boussinesq
approximation
126
4.3.4
Reservoir anisotropy in two dimensions
128
4.4
Flow to wells in homogeneous
isotropie
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
(ε
= 1 ) 173
5.1.3.3
Implicit difference method
(ε
= 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 form
of the Galerkin method
194
5.2.4
Galerkin weighted residuals method; weak formulation of the
heat equation for a stationary temperature in two dimensions
196
5.2.5
Finite elements using bilinear
Lagrange
interpolation
polynomials over triangles
197
5.2.6
Finite elements using bilinear
Lagrange
interpolation
polynomials over rectangles
199
5.2.7
Solution of the transient heat equation using finite elements in
1
D
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
Hydroťgeoichemical
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 principal
artificial 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
Hydroigeojchemical 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 contaminant 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 permanent
inflow 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
Pérez)
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 (identiflability)
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
Tíkhonov s regularizaron
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
31
1
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
8.2.5
Elaboration of a numerical model and problem solution
319
8.3
Scenario modeling of multi-layer aquifers and distribution of groundwater
ages caused by exploitation
320
8.3.1
Problem
320
8.3.2
Aquifer specifications
321
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 three
geological layers
325
XXVIII Table of
contents
8.3.5.2
Influence
of groundwater exploitation
325
8.3.5.3
Influence of clay layers
338
8.3.5.4
Contamination threat for deeper aquifers
339
8.4
Point source contamination and aquifer remediation
340
8.4.1
Situation
340
8.4.2
Aquifer specifications
340
8.4.3
Objectives
340
8.4.4
Elaboration of a numerical model and problem solution
342
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
8.5.3
Elaboration of a numerical model
346
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
Elaboration of a numerical model
349
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, Michoacán
362
9.1.1.3
Los Humeros, Puebla
364
9.1.1.4
Las Tres Vírgenes, 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
9.4.3
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 initial
conditions 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; Flow
problem 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
Α.
1.4
Numerical examples of interpolation
410
A.
1.4.1
Example
1 410
A.
1.4.2
Example
2 411
A.I.
5
The
Lagrange
interpolation polynomials
412
A.
1.5.1
Example
1 414
A.l.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
|
| adam_txt |
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 isothermal
aquifers 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
17
XV
XVI Table of
contents
2.1.3
The principle of
conservation
of mass in porous rocks 1
7
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
Lamé
moduli
30
2.1.9.7 Volumetriestrain 30
2.1.10
Elasticity equations for Hookean rocks
3
\
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
З8
2.2.4.4
Compressibility of the pore volume
З8
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 potential
for 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
(Oto 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
2.3.4.3
Compressibility and expansivity
78
2.3.4.4
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°
С
) 81
2.3.6.1
Thermodynamic range of validity of the code AquaG3
70.
For
82
2.3.6.2
Temperature of saturation (subroutine rsat)
82
2.3.6.3
Saturation pressure (subroutine Psat)
82
2.3.6.4
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
Horne
89
2.3.8.4
Correlation of Brooks-Corey
90
2.3.8.5
Correlation of Li and
Horne
for geothermal reservoirs
90
2.3.8.6
The Li-Horne 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-Horne universal relative permeability functions based
on fractal modeling of porous rocks
96
2.3.10.9
Linear
Х
-functions for relative permeability in fractures
97
2.3.10.10
Relative permeabilities in fractures:
The Honarpour-Diornampo 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
Model of Brailsford-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
1 ' 2
3.8.1
Cerritos
Colorados (La Primavera),
Jalisco
112
3.8.2
Los Humeros, Puebla
ИЗ
3.8.3
Los Azufres, Michoacán
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
125
4.3.3.2
Two-dimensional groundwater flow
—
The Boussinesq
approximation
126
4.3.4
Reservoir anisotropy in two dimensions
128
4.4
Flow to wells in homogeneous
isotropie
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
(ε
= 1 ) 173
5.1.3.3
Implicit difference method
(ε
= 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 form
of the Galerkin method
194
5.2.4
Galerkin weighted residuals method; weak formulation of the
heat equation for a stationary temperature in two dimensions
196
5.2.5
Finite elements using bilinear
Lagrange
interpolation
polynomials over triangles
197
5.2.6
Finite elements using bilinear
Lagrange
interpolation
polynomials over rectangles
199
5.2.7
Solution of the transient heat equation using finite elements in
1
D
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
Hydroťgeoichemical
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 principal
artificial 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
Hydroigeojchemical 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 contaminant 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 permanent
inflow 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
Pérez)
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 (identiflability)
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
Tíkhonov's regularizaron
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
31
1
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
8.2.5
Elaboration of a numerical model and problem solution
319
8.3
Scenario modeling of multi-layer aquifers and distribution of groundwater
ages caused by exploitation
320
8.3.1
Problem
320
8.3.2
Aquifer specifications
321
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 three
geological layers
325
XXVIII Table of
contents
8.3.5.2
Influence
of groundwater exploitation
325
8.3.5.3
Influence of clay layers
338
8.3.5.4
Contamination threat for deeper aquifers
339
8.4
Point source contamination and aquifer remediation
340
8.4.1
Situation
340
8.4.2
Aquifer specifications
340
8.4.3
Objectives
340
8.4.4
Elaboration of a numerical model and problem solution
342
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
8.5.3
Elaboration of a numerical model
346
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
Elaboration of a numerical model
349
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, Michoacán
362
9.1.1.3
Los Humeros, Puebla
364
9.1.1.4
Las Tres Vírgenes, 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
9.4.3
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 initial
conditions 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; Flow
problem 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
Α.
1.4
Numerical examples of interpolation
410
A.
1.4.1
Example
1 410
A.
1.4.2
Example
2 411
A.I.
5
The
Lagrange
interpolation polynomials
412
A.
1.5.1
Example
1 414
A.l.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 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Bundschuh, Jochen 1960- Suárez Arriaga, Mario César |
| author_GND | (DE-588)1064353835 |
| author_facet | Bundschuh, Jochen 1960- Suárez Arriaga, Mario César |
| author_role | aut aut |
| author_sort | Bundschuh, Jochen 1960- |
| author_variant | j b jb a m c s amc amcs |
| building | Verbundindex |
| bvnumber | BV023066525 |
| classification_rvk | RB 10351 |
| classification_tum | ERG 710f GEO 325f |
| ctrlnum | (OCoLC)315718016 (DE-599)BVBBV023066525 |
| discipline | Energietechnik, Energiewirtschaft Geowissenschaften Geographie |
| discipline_str_mv | Energietechnik, Energiewirtschaft Geowissenschaften Geographie |
| format | Book |
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| id | DE-604.BV023066525 |
| illustrated | Illustrated |
| index_date | 2024-07-02T19:31:07Z |
| indexdate | 2024-07-09T21:10:13Z |
| institution | BVB |
| isbn | hardcover 9780415401678 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016269718 |
| oclc_num | 315718016 |
| open_access_boolean | |
| owner | DE-703 DE-91S DE-BY-TUM |
| owner_facet | DE-703 DE-91S DE-BY-TUM |
| physical | XLII, 478 S. Ill., graph. Darst. |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | CRC Press |
| record_format | marc |
| series | Multiphysics modeling |
| series2 | Multiphysics modeling |
| spelling | Bundschuh, Jochen 1960- Verfasser (DE-588)1064353835 aut Introduction to the numerical modeling of groundwater and geothermal systems fundamentals of mass, energy, and solute transport in poroelastic rocks Jochen Bundschuh ; Mario César Suárez Arriaga Boca Raton, Fla. [u.a.] CRC Press 2010 XLII, 478 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Multiphysics modeling 2 Geothermik (DE-588)4020285-9 gnd rswk-swf Modellierung (DE-588)4170297-9 gnd rswk-swf Hydraulik (DE-588)4026288-1 gnd rswk-swf Porengrundwasserleiter (DE-588)4353455-7 gnd rswk-swf Stoffübertragung (DE-588)4057696-6 gnd rswk-swf Grundwasserstrom (DE-588)4121396-8 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Wärmeübertragung (DE-588)4064211-2 gnd rswk-swf Geothermische Energie (DE-588)4020286-0 gnd rswk-swf Numerisches Modell (DE-588)4338132-7 gnd rswk-swf Fels (DE-588)4436005-8 gnd rswk-swf Grundwasser (DE-588)4022369-3 gnd rswk-swf Grundwasserstrom (DE-588)4121396-8 s Fels (DE-588)4436005-8 s Numerisches Modell (DE-588)4338132-7 s DE-604 Geothermik (DE-588)4020285-9 s Grundwasser (DE-588)4022369-3 s Hydraulik (DE-588)4026288-1 s Stoffübertragung (DE-588)4057696-6 s Modellierung (DE-588)4170297-9 s Numerisches Verfahren (DE-588)4128130-5 s 1\p DE-604 Porengrundwasserleiter (DE-588)4353455-7 s Geothermische Energie (DE-588)4020286-0 s Wärmeübertragung (DE-588)4064211-2 s 2\p DE-604 Suárez Arriaga, Mario César Verfasser aut Erscheint auch als Online-Ausgabe 978-0-203-84810-4 Multiphysics modeling 2 (DE-604)BV036583449 2 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016269718&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Bundschuh, Jochen 1960- Suárez Arriaga, Mario César Introduction to the numerical modeling of groundwater and geothermal systems fundamentals of mass, energy, and solute transport in poroelastic rocks Multiphysics modeling Geothermik (DE-588)4020285-9 gnd Modellierung (DE-588)4170297-9 gnd Hydraulik (DE-588)4026288-1 gnd Porengrundwasserleiter (DE-588)4353455-7 gnd Stoffübertragung (DE-588)4057696-6 gnd Grundwasserstrom (DE-588)4121396-8 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Wärmeübertragung (DE-588)4064211-2 gnd Geothermische Energie (DE-588)4020286-0 gnd Numerisches Modell (DE-588)4338132-7 gnd Fels (DE-588)4436005-8 gnd Grundwasser (DE-588)4022369-3 gnd |
| subject_GND | (DE-588)4020285-9 (DE-588)4170297-9 (DE-588)4026288-1 (DE-588)4353455-7 (DE-588)4057696-6 (DE-588)4121396-8 (DE-588)4128130-5 (DE-588)4064211-2 (DE-588)4020286-0 (DE-588)4338132-7 (DE-588)4436005-8 (DE-588)4022369-3 |
| title | Introduction to the numerical modeling of groundwater and geothermal systems fundamentals of mass, energy, and solute transport in poroelastic rocks |
| title_auth | Introduction to the numerical modeling of groundwater and geothermal systems fundamentals of mass, energy, and solute transport in poroelastic rocks |
| title_exact_search | Introduction to the numerical modeling of groundwater and geothermal systems fundamentals of mass, energy, and solute transport in poroelastic rocks |
| title_exact_search_txtP | Introduction to the numerical modeling of groundwater and geothermal systems fundamentals of mass, energy, and solute transport in poroelastic rocks |
| title_full | Introduction to the numerical modeling of groundwater and geothermal systems fundamentals of mass, energy, and solute transport in poroelastic rocks Jochen Bundschuh ; Mario César Suárez Arriaga |
| title_fullStr | Introduction to the numerical modeling of groundwater and geothermal systems fundamentals of mass, energy, and solute transport in poroelastic rocks Jochen Bundschuh ; Mario César Suárez Arriaga |
| title_full_unstemmed | Introduction to the numerical modeling of groundwater and geothermal systems fundamentals of mass, energy, and solute transport in poroelastic rocks Jochen Bundschuh ; Mario César Suárez Arriaga |
| title_short | Introduction to the numerical modeling of groundwater and geothermal systems |
| title_sort | introduction to the numerical modeling of groundwater and geothermal systems fundamentals of mass energy and solute transport in poroelastic rocks |
| title_sub | fundamentals of mass, energy, and solute transport in poroelastic rocks |
| topic | Geothermik (DE-588)4020285-9 gnd Modellierung (DE-588)4170297-9 gnd Hydraulik (DE-588)4026288-1 gnd Porengrundwasserleiter (DE-588)4353455-7 gnd Stoffübertragung (DE-588)4057696-6 gnd Grundwasserstrom (DE-588)4121396-8 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Wärmeübertragung (DE-588)4064211-2 gnd Geothermische Energie (DE-588)4020286-0 gnd Numerisches Modell (DE-588)4338132-7 gnd Fels (DE-588)4436005-8 gnd Grundwasser (DE-588)4022369-3 gnd |
| topic_facet | Geothermik Modellierung Hydraulik Porengrundwasserleiter Stoffübertragung Grundwasserstrom Numerisches Verfahren Wärmeübertragung Geothermische Energie Numerisches Modell Fels Grundwasser |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016269718&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV036583449 |
| work_keys_str_mv | AT bundschuhjochen introductiontothenumericalmodelingofgroundwaterandgeothermalsystemsfundamentalsofmassenergyandsolutetransportinporoelasticrocks AT suarezarriagamariocesar introductiontothenumericalmodelingofgroundwaterandgeothermalsystemsfundamentalsofmassenergyandsolutetransportinporoelasticrocks |