Models and Modeling

Dr. Jerry P. Fairley received his PhD in Earth Resources Engineering from the University of California, Berkeley. He was the Chief Hydrologist for Site Characterization on the US- DOE's Yucca Mountain Project (1993-1995), and worked as a modeler for the Earth Sciences Division of Lawrence Berkeley National Laboratory. He is currently a Professor of Geology at the University of Idaho, Department of Geological Sciences.

Introduction 1 1 Modeling Basics 5 1.1 Learning to Model 5 1.2 Three Cardinal Rules of Modeling 6 1.2.1 Rule 1: Know Your Model Objective 6 1.2.2 Rule 2: Make Your Model Appropriate for Your Data 7 1.2.3 Rule 3: Start Simple and Build Complexity 7 1.3 How Can I Evaluate My Model? 8 1.3.1 Test Model Behavior in the Limits 8 1.3.2 Look for Behavior Congruent with the Governing Equations 8 1.3.3 Non-Dimensionalization 9 1.4 Conclusions 9 2 A Model of Exponential Decay 11 2.1 Exponential Decay 11 2.2 The Bandurraga Basin, Idaho 12 2.3 Getting Organized 12 2.3.1 Observable Quantities 13 2.3.2 Stating the Model Objective 13 2.3.3 What Data are Available? 14 2.3.4 What Can We Assume? 14 2.3.5 Finding an Approach 15 2.3.6 Executing the Plan 17 2.4 Non-Dimensionalization 19 2.5 Solving for 22 2.6 Calibrating the Model to the Data 23 2.6.1 Semi-Log Plots 24 2.6.2 Curve Matching 24 2.7 Extending the Model 25 2.8 A Numerical Solution for Exponential Decay 29 2.9 Conclusions 31 2.10 Problems 32 3 A Model of Water Quality 35 3.1 Oases in the Desert 35 3.2 Understanding the Problem 36 3.3 Model Development 36 3.3.1 Model Formulation 37 3.3.2 Non-Dimensionalization 38 3.3.3 Solving the Equation 40 3.4 Evaluating the Model 41 3.5 Applying the Model 42 3.6 Conclusions 43 3.7 Problems 44 4 The Laplace Equation 47 4.1 Laplace s Equation 47 4.2 The Elysian Fields 48 4.3 Model Development 49 4.4 Quantifying the Conceptual Model 51 4.5 Non-dimensionalization 54 4.6 Solving the Governing Equation 55 4.7 What Does it Mean? 56 4.7.1 Considering the Solution 56 4.7.2 The Flux, and its Meaning 57 4.7.3 Meanwhile, Back in the Real World... 58 4.8 Numerical Approximation of the Second Derivative 59 4.8.1 Iterative Solution Methods 60 4.8.2 Direct Solution Methods 62 4.9 Conclusions 63 4.10 Problems 64 5 The Poisson Equation 67 5.1 Poisson s Equation 67 5.2 Alcatraz Island 68 5.2.1 Early History of Alcatraz Island 68 5.2.2 The American Indian Occupation 69 5.3 Understanding the Problem 70 5.3.1 Developing an Approach 70 5.3.2 Questions About Coordinates 72 5.3.3 A Digression About Boundary Conditions 73 5.3.4 Considerations of Symmetry 78 5.4 Quantifying the Conceptual Model 79 5.5 Non-Dimensionalization 82 5.6 Seeking a Solution 84 5.6.1 An Approximate Analytical Solution 84 5.6.2 A 2-Dimensional Finite Difference Operator 86 5.7 An Alternative Non-Dimensionalization 88 5.8 Conclusions 90 5.9 Problems 90 6 The Transient Diffusion Equation 93 6.1 The Diffusion Equation 93 6.2 The Twelve Labors of Hercules 94 6.2.1 The Twelve Labors 95 6.3 The Augean Stables 96 6.3.1 Developing an Approach 96 6.4 Carrying Out the Plan 98 6.4.1 Mass Balance and the Control Volume 98 6.4.2 A Brief Digression on Storage Coefficients 101 6.4.3 Completing the Governing Equation 104 6.4.4 Nondimensionalization 105 6.5 An Analytical Solution 107 6.5.1 Separation of Variables: The Basic Idea 107 6.5.2 Initial Preparations 108 6.5.3 Separation of Variables: The Method 109 6.5.4 Solving for Spatial Dependence 110 6.5.5 Solving for Temporal Dependence 112 6.5.6 Summing Up 113 6.5.7 Joseph Fourier and the Augean Stables 113 6.5.8 Orthogonality 114 6.6 Evaluating the Solution 116 6.6.1 Behavior in the Limits 116 6.6.2 Evaluating the Solution 117 6.6.3 Dimensionless Time 120 6.7 Transient Finite Differences 121 6.7.1 The Explicit Scheme 121 6.7.2 Fully Implicit Finite Differences 122 6.7.3 A Generic Difference Formulation 124 6.8 Conclusions 125 6.9 Problems 125 7 The Theis Equation 129 7.1 The Knight of the Sorrowful Figure 129 7.1.1 The Groundwater of La Mancha 130 7.2 Statement of the Problem 131 7.3 The Governing Equation 132 7.4 Boundary Conditions 134 7.5 Non-Dimensionalization 135 7.5.1 Normalizing Independent Variables the Usual Way 136 7.5.2 Finding Similarity 137 7.5.3 The Auxillary Conditions 138 7.6 Solving the Governing Equation 139 7.7 Theis and the Well Function 141 7.7.1 Evaluating the Exponential Integral 141 7.8 Back to the Beginning 142 7.8.1 Equilibrium: Are We There Yet? 143 7.8.2 The Shape of Time and Space 144 7.9 Violating the Model Assumptions 145 7.10 Conclusions 146 7.11 Problems 147 8 The Transport Equation 149 8.1 The Advection-Dispersion Equation 149 8.2 The Problem Child 151 8.3 The Augean Stables, Revisited 152 8.4 Defining the Problem 152 8.5 The Governing Equation 154 8.6 Non-Dimensionalization 156 8.6.1 The Dependent Variable 156 8.6.2 The Independent Variables 157 8.6.3 The Velocity 158 8.6.4 Making the Substitutions 159 8.7 Analytical Solutions 160 8.7.1 Steady-State Solutions 161 8.7.2 Transient Solutions 164 8.7.3 Qualitative Behavior of the ADE 170 8.8 Cauchy Conditions 175 8.9 Retardation and Dispersion 177 8.10 Numerical Solution of the ADE 178 8.11 Conclusions 182 8.12 Problems 183 9 Heterogeneity and Anisotropy 185 9.1 Understanding the Problem 185 9.2 Heterogeneity and the REV 187 9.3 Heterogeneity and Effective Properties 188 9.3.1 Averaging Conductivity 189 9.3.2 Some Statistical Definitions 192 9.3.3 Other Effective Conductivity Results 194 9.4 Anisotropy in Porous Media 195 9.5 Layered Media 196 9.6 Numerical Simulation 198 9.7 Some Additional Considerations 200 9.8 Conclusions 200 9.9 Problems 201 10 Approximation, Error, and Sensitivity 203 10.1 Things We Almost Know 203 10.2 Approximation Using Derivatives 204 10.2.1 Another Example 205 10.3 Improving our Estimates 206 10.4 Bounding Errors 207 10.4.1 Data Uncertainty 208 10.4.2 Model Uncertainty 208 10.5 Model Sensitivity 210 10.5.1 Defining Sensitivity 210 10.5.2 Example 1 212 10.5.3 Example 2 212 10.5.4 What Good is it? 213 10.6 Conclusions 215 10.7 Problems 216 11 A Case Study 219 11.1 The Borax Lake Hot Springs 219 11.2 Study Motivation and Conceptual Model 221 11.3 Defining the Conceptual Model 222 11.4 Model Development 225 11.4.1 Fluid Flow 225 11.4.2 Heat Transport 227 11.4.3 Solving the Equation 231 11.4.4 Finding Permeability 232 11.5 Evaluating the Solution 234 11.5.1 Estimating Permeability 235 11.5.2 Checking the Limits 236 11.5.3 What Permeability is Calculated? 237 11.5.4 What are the Model Sensitivities? 237 11.6 Conclusions 239 11.7 Problems 240 12 Closing Remarks 243 12.1 Some Final Thoughts 243 A A Non-dimensional Heuristic 247 B Evaluating Implicit Equations 249 B.1 Trial and Error 250 B.2 The Graphical Method 250 B.3 Iteration 251 B.4 Newton s Method 252 C Matrix Solution for Implicit Algorithms 253 C.1 Solution of 1-Dimensional Equations 253