Moment Analysis for Subsurface Hydrologic Applications

Table of Contents......Page 6
Preface......Page 10
1.1. Introduction to Random Variables......Page 12
1.2. Expectation......Page 18
1.3. The Charactersitic Function......Page 21
1.4. The Laplace Transform......Page 24
1.5. Probability Generating Functions......Page 25
1.6. Cumulants and Cumulant Generating Functions......Page 28
1.8. L-Moments......Page 30
1.9. Experimental and Theoretical Moments......Page 31
1.9.1. Gamma Distribution......Page 32
1.9.3. Log-Normal Distribution......Page 33
Appendix A: Exponential Distributions......Page 37
Appendix B: Maximum Likelihood Estimation......Page 38
2.1. Definition of the Transform and its Inverse......Page 40
2.2. Singularities of the Laplace Transform......Page 43
2.3. Green's Functions for Initial Value Problems......Page 45
2.4. Solute Transport by Diffusion......Page 46
2.5. Advective – Dispersive Solute Transport Model......Page 49
2.6. Role of Boundary Conditions......Page 54
2.7. The Mobile – Immobile Water Model......Page 57
2.8. The Physical Nonequilibrium Model......Page 60
2.9. The Chemical Nonequilibrium Model......Page 63
2.10. Nonequilibrium Sorption by Diffusion into Spherical Grains......Page 65
3.1. Solute Transport by Diffusion......Page 68
3.2. Fourier Transform Pair......Page 71
3.3. Fourier Transform of the Diffusion Equation......Page 72
3.4. Fourier Transforms of Derivatives......Page 74
3.5. Fourier Sine and Cosine Transforms......Page 76
3.6. Fourier Transform Solution for Advection-Dispersion Equation Over an Infinite Domain......Page 81
3.7. Fourier Sine Transform for Advection–Dispersion Equation Over Semi-Infinite Domains......Page 82
3.8. Fourier Transforms in Higher Dimensions......Page 83
4.1. Residence Time Distributions......Page 87
4.2. Models of Solute Transport......Page 88
4.3. Depth Moments of the Advection-Dispersion Equation......Page 93
4.4. Depth-Moments for Stochastic-Convective Models......Page 97
4.5. Transfer Functions for Layered Soils......Page 98
4.6. Stochastic Stream Tube Models......Page 101
4.7.1. Local Model......Page 103
5. Temporal Moment Analysis for Solute Transport in Porous Media......Page 114
5.1. Model Descriptions and Governing Differential Equations......Page 115
5.2. Temporal Moment Definitions......Page 118
5.3. Aris's Method of Moment Analysis......Page 120
5.4.1. Experimental Data......Page 125
5.4.2. Computing Moments from Observed Data......Page 126
5.4.3. Estimation Errors......Page 128
5.5.1. Estimating Parameters of the Transport Equation......Page 132
5.5.2. Effective Parameters......Page 137
5.5.3. Nonequilibrium Indices......Page 138
5.6. Summary......Page 143
5.7. Appendix: Sample BTC Data......Page 144
6.1. Introduction......Page 151
6.2. Spatial Moments......Page 153
6.3. Spatial Moments to Describe Solute Plume Behavior......Page 154
6.4. Spatial Moments for the PNE Model......Page 157
6.5. Spatial Moments for First-Order Rate Model......Page 159
7.1. Introduction......Page 162
7.2. Immobile Vapor Phase Model......Page 164
7.3. Description of Loss Fractions......Page 169
7.4. Effective Parameter Definitions......Page 170
7.5. Mobile Vapor Phase Model......Page 180
7.6. Spatial Moments for Mobile Vapor Phase Model......Page 188
8.1. Definitions of MGDEs......Page 190
8.2. Temporal MGDEs for Solute Transport in Soil......Page 192
8.2.1. Analysis with Degradation......Page 193
8.2.2. Analysis without Degradation......Page 194
8.3. Spatial MGDEs for PNE Model of Solute Transport in Soil......Page 196
8.3.1. Zeroth Moment......Page 199
8.3.2. First Moment......Page 204
8.3.3. Second Moment......Page 205
8.4. Spatial Moments for a Two-Layer Aquifer......Page 206
8.5. Perfectly Stratified Aquifer with Velocity Variation......Page 210
9.2. Governing Differential Equations......Page 213
9.3. Laplace Transforms......Page 215
9.4. Temporal Moment Analysis......Page 216
9.5. Temporal Moments for Advective Transport......Page 221
9.6. Spatial Moments for Compounds Undergoing Sequential First-Order Decay Chain......Page 223
10.1. General Remarks......Page 229
10.2. Interval Arithmetic Operations......Page 230
10.3. Interval Distribution Functions......Page 231
10.5. Application to a Remediation Example......Page 233
10.5.1. First-Order Degradation Model......Page 234
10.5.2. Statistical Distributions of Degradation Rates and Initial Concentrations......Page 235
10.5.3. Field-Scale Models......Page 237
10.5.4. Results and Discussion......Page 239
10.6. Application to Solute Transport Experiment......Page 243
10.6.2. Advective Solute Transport in Vadose Zone......Page 244
10.6.3. Field-Scale Model using Interval Computing Method......Page 246
10.6.4. Stochastic Advective Solute Transport......Page 247
10.6.5. Results and Discussion......Page 250
11.1. Introduction......Page 252
11.2. Moments For Linearized Subsurface Drainage with No Recharge......Page 254
11.3. Subsurface Drainage with Lateral Inflow......Page 257
11.4.1. Theoretical Development......Page 261
11.4.2. Moments and Experimental Results......Page 265
12.1. Problem Definition......Page 269
12.3.1. Gram-Charlier and Edgeworth Series Approach......Page 271
12.3.2. Expansion Methods Based on other Polynomials......Page 274
12.4. Maximum Entropy (Maxent) Method......Page 275
12.4.1. Geometrical Moments......Page 277
12.5. Example Calculations......Page 278
References......Page 280
D......Page 292
K......Page 293
R......Page 294
Z......Page 295