Models and Modeling: An Introduction for Earth and Environmental Scientists

Table of Contents

About the companion website, xi

Introduction, 1

1 Modeling basics, 4

1.1 Learning to model, 4

1.2 Three cardinal rules of modeling, 5

1.3 How can I evaluate my model?, 7

1.4 Conclusions, 8

2 A model of exponential decay, 9

2.1 Exponential decay, 9

2.2 The Bandurraga Basin, Idaho, 10

2.3 Getting organized, 10

2.4 Nondimensionalization, 17

2.5 Solving for θ, 19

2.6 Calibrating the model to the data, 21

2.7 Extending the model, 23

2.8 A numerical solution for exponential decay, 26

2.9 Conclusions, 28

2.10 Problems, 29

3 A model of water quality, 31

3.1 Oases in the desert, 31

3.2 Understanding the problem, 32

3.3 Model development, 32

3.4 Evaluating the model, 37

3.5 Applying the model, 38

3.6 Conclusions, 39

3.7 Problems, 40

4 The Laplace equation, 42

4.1 Laplace’s equation, 42

4.2 The Elysian Fields, 43

4.3 Model development, 44

4.4 Quantifying the conceptual model, 47

4.5 Nondimensionalization, 48

4.6 Solving the governing equation, 49

4.7 What does it mean?, 50

4.8 Numerical approximation of the second derivative, 54

4.9 Conclusions, 57

4.10 Problems, 58

5 The Poisson equation, 62

5.1 Poisson’s equation, 62

5.2 Alcatraz island, 63

5.3 Understanding the problem, 65

5.4 Quantifying the conceptual model, 74

5.5 Nondimensionalization, 76

5.6 Seeking a solution, 79

5.7 An alternative nondimensionalization, 82

5.8 Conclusions, 84

5.9 Problems, 85

6 The transient diffusion equation, 87

6.1 The diffusion equation, 87

6.2 The Twelve Labors of Hercules, 88

6.3 The Augean Stables, 90

6.4 Carrying out the plan, 92

6.5 An analytical solution, 100

6.6 Evaluating the solution, 109

6.7 Transient finite differences, 114

6.8 Conclusions, 118

6.9 Problems, 119

7 The Theis equation, 122

7.1 The Knight of the Sorrowful Figure, 122

7.2 Statement of the problem, 124

7.3 The governing equation, 125

7.4 Boundary conditions, 127

7.5 Nondimensionalization, 128

7.6 Solving the governing equation, 132

7.7 Theis and the “well function”, 134

7.8 Back to the beginning, 135

7.9 Violating the model assumptions, 138

7.10 Conclusions, 139

7.11 Problems, 140

8 The transport equation, 141

8.1 The advection–dispersion equation, 141

8.2 The problem child, 143

8.3 The Augean Stables, revisited, 144

8.4 Defining the problem, 144

8.5 The governing equation, 146

8.6 Nondimensionalization, 148

8.7 Analytical solutions, 152

8.8 Cauchy conditions, 165

8.9 Retardation and dispersion, 167

8.10 Numerical solution of the ADE, 169

8.11 Conclusions, 173

8.12 Problems, 174

9 Heterogeneity and anisotropy, 177

9.1 Understanding the problem, 177

9.2 Heterogeneity and the representative elemental volume, 179

9.3 Heterogeneity and effective properties, 180

9.4 Anisotropy in porous media, 187

9.5 Layered media, 188

9.6 Numerical simulation, 189

9.7 Some additional considerations, 191

9.8 Conclusions, 192

9.9 Problems, 192

10 Approximation, error, and sensitivity, 195

10.1 Things we almost know, 195

10.2 Approximation using derivatives, 196

10.3 Improving our estimates, 197

10.4 Bounding errors, 199

10.5 Model sensitivity, 201

10.6 Conclusions, 206

10.7 Problems, 207

11 A case study, 210

11.1 The Borax Lake Hot Springs, 210

11.2 Study motivation and conceptual model, 212

11.3 Defining the conceptual model, 213

11.4 Model development, 215

11.5 Evaluating the solution, 224

11.6 Conclusions, 229

11.7 Problems, 230

12 Closing remarks, 233

12.1 Some final thoughts, 233

Appendix A A heuristic approach to nondimensionalization, 236

Appendix B Evaluating implicit equations, 238

B.1 Trial and error, 239

B.2 The graphical method, 239

B.3 Iteration, 240

B.4 Newton’s method, 241

Appendix C Matrix solution for implicit algorithms, 243

C.1 Solution of 1D equations, 243

C.2 Solution for higher dimensional problems, 244

C.3 The tridiagonal matrix routine TDMA, 244

Index, 247