Models and Modeling – An Introduction for Earth and Environmental Scientists

<p>About the companion website, xi</p>
<p>Introduction, 1</p>
<p>1 Modeling basics, 4</p>
<p>1.1 Learning to model, 4</p>
<p>1.2 Three cardinal rules of modeling, 5</p>
<p>1.3 How can I evaluate my model?, 7</p>
<p>1.4 Conclusions, 8</p>
<p>2 A model of exponential decay, 9</p>
<p>2.1 Exponential decay, 9</p>
<p>2.2 The Bandurraga Basin, Idaho, 10</p>
<p>2.3 Getting organized, 10</p>
<p>2.4 Nondimensionalization, 17</p>
<p>2.5 Solving for , 19</p>
<p>2.6 Calibrating the model to the data, 21</p>
<p>2.7 Extending the model, 23</p>
<p>2.8 A numerical solution for exponential decay, 26</p>
<p>2.9 Conclusions, 28</p>
<p>2.10 Problems, 29</p>
<p>3 A model of water quality, 31</p>
<p>3.1 Oases in the desert, 31</p>
<p>3.2 Understanding the problem, 32</p>
<p>3.3 Model development, 32</p>
<p>3.4 Evaluating the model, 37</p>
<p>3.5 Applying the model, 38</p>
<p>3.6 Conclusions, 39</p>
<p>3.7 Problems, 40</p>
<p>4 The Laplace equation, 42</p>
<p>4.1 Laplace s equation, 42</p>
<p>4.2 The Elysian Fields, 43</p>
<p>4.3 Model development, 44</p>
<p>4.4 Quantifying the conceptual model, 47</p>
<p>4.5 Nondimensionalization, 48</p>
<p>4.6 Solving the governing equation, 49</p>
<p>4.7 What does it mean?, 50</p>
<p>4.8 Numerical approximation of the second derivative, 54</p>
<p>4.9 Conclusions, 57</p>
<p>4.10 Problems, 58</p>
<p>5 The Poisson equation, 62</p>
<p>5.1 Poisson s equation, 62</p>
<p>5.2 Alcatraz island, 63</p>
<p>5.3 Understanding the problem, 65</p>
<p>5.4 Quantifying the conceptual model, 74</p>
<p>5.5 Nondimensionalization, 76</p>
<p>5.6 Seeking a solution, 79</p>
<p>5.7 An alternative nondimensionalization, 82</p>
<p>5.8 Conclusions, 84</p>
<p>5.9 Problems, 85</p>
<p>6 The transient diffusion equation, 87</p>
<p>6.1 The diffusion equation, 87</p>
<p>6.2 The Twelve Labors of Hercules, 88</p>
<p>6.3 The Augean Stables, 90</p>
<p>6.4 Carrying out the plan, 92</p>
<p>6.5 An analytical solution, 100</p>
<p>6.6 Evaluating the solution, 109</p>
<p>6.7 Transient finite differences, 114</p>
<p>6.8 Conclusions, 118</p>
<p>6.9 Problems, 119</p>
<p>7 The Theis equation, 122</p>
<p>7.1 The Knight of the Sorrowful Figure, 122</p>
<p>7.2 Statement of the problem, 124</p>
<p>7.3 The governing equation, 125</p>
<p>7.4 Boundary conditions, 127</p>
<p>7.5 Nondimensionalization, 128</p>
<p>7.6 Solving the governing equation, 132</p>
<p>7.7 Theis and the well function , 134</p>
<p>7.8 Back to the beginning, 135</p>
<p>7.9 Violating the model assumptions, 138</p>
<p>7.10 Conclusions, 139</p>
<p>7.11 Problems, 140</p>
<p>8 The transport equation, 141</p>
<p>8.1 The advection dispersion equation, 141</p>
<p>8.2 The problem child, 143</p>
<p>8.3 The Augean Stables, revisited, 144</p>
<p>8.4 Defining the problem, 144</p>
<p>8.5 The governing equation, 146</p>
<p>8.6 Nondimensionalization, 148</p>
<p>8.7 Analytical solutions, 152</p>
<p>8.8 Cauchy conditions, 165</p>
<p>8.9 Retardation and dispersion, 167</p>
<p>8.10 Numerical solution of the ADE, 169</p>
<p>8.11 Conclusions, 173</p>
<p>8.12 Problems, 174</p>
<p>9 Heterogeneity and anisotropy, 177</p>
<p>9.1 Understanding the problem, 177</p>
<p>9.2 Heterogeneity and the representative elemental volume, 179</p>
<p>9.3 Heterogeneity and effective properties, 180</p>
<p>9.4 Anisotropy in porous media, 187</p>
<p>9.5 Layered media, 188</p>
<p>9.6 Numerical simulation, 189</p>
<p>9.7 Some additional considerations, 191</p>
<p>9.8 Conclusions, 192</p>
<p>9.9 Problems, 192</p>
<p>10 Approximation, error, and sensitivity, 195</p>
<p>10.1 Things we almost know, 195</p>
<p>10.2 Approximation using derivatives, 196</p>
<p>10.3 Improving our estimates, 197</p>
<p>10.4 Bounding errors, 199</p>
<p>10.5 Model sensitivity, 201</p>
<p>10.6 Conclusions, 206</p>
<p>10.7 Problems, 207</p>
<p>11 A case study, 210</p>
<p>11.1 The Borax Lake Hot Springs, 210</p>
<p>11.2 Study motivation and conceptual model, 212</p>
<p>11.3 Defining the conceptual model, 213</p>
<p>11.4 Model development, 215</p>
<p>11.5 Evaluating the solution, 224</p>
<p>11.6 Conclusions, 229</p>
<p>11.7 Problems, 230</p>
<p>12 Closing remarks, 233</p>
<p>12.1 Some final thoughts, 233</p>
<p>Appendix A A heuristic approach to nondimensionalization, 236</p>
<p>Appendix B Evaluating implicit equations, 238</p>
<p>B.1 Trial and error, 239</p>
<p>B.2 The graphical method, 239</p>
<p>B.3 Iteration, 240</p>
<p>B.4 Newton s method, 241</p>
<p>Appendix C Matrix solution for implicit algorithms, 243</p>
<p>C.1 Solution of 1D equations, 243</p>
<p>C.2 Solution for higher dimensional problems, 244</p>
<p>C.3 The tridiagonal matrix routine TDMA, 244</p>
<p>Index, 247</p>