Contents
xiii
5.3 Finite Element Methods for Three-Dimensional Problems ...... 125
5.3.1 The Galerkin Finite Element Method .................... 125
5.3.2 Triangular Prism Elements and Associated Basis Functions . 132
5.3.3 A Mixed Finite Element-Finite Difference Method ......... 135
5.3.4 The Multiple Cell Balance Method ...................... 138
5.4 The Solution of Finite Element Systems ..................... 143
5.4.1 Features of Finite Element Systems and Direct Solutions ... 143
5.4.2 Point Iteration Methods ............................... 144
5.4.3 Block and Layer Iteration Methods ..................... 146
6 Numerical Solutions of Advection-Dominated Problems .......... 149
6.1 Advection-Dominated Problems. . . . . . .. . . . . . . . . . . . . . . . . . . .. 149
6.1.1 Fourier Analysis ofNumerical Errors .................... 149
6.1.2 Eulerian and Lagrangian Reference Frames ............... 154
6.2 Upstream Weighted Methods .............................. 155
6.2.1 Upstream Weighted Finite Difference Methods ............ 155
6.2.2 Upstream Weighted Finite Element Methods ............. 158
6.2.3 The Upstream Weighted Multiple Cell Balance Method .... 163
6.3 Moving Coordinate System and Moving Point Methods ....... 172
6.3.1 Moving Coordinate System Methods .................... 172
6.3.2 Element Deformation Methods ......................... 175
6.3.3 Moving Point Methods ................................ 176
6.4 The Modified Methods of Characteristics .................... 177
6.4.1 The Single Step Reverse Method ........................ 177
6.4.2 The Hybrid Single Step Reverse-Moving Point Method ..... 180
6.4.3 The Hybrid Moving Point-Characteristics Finite Element
Method ............................................. 182
7 Mathematical Models of Groundwater Quality ................ 187
7.1 The Classification of Groundwater Quality Models ............ 187
7.1.1 Hydrodynamic Dispersion Models ...................... 187
7.1.2 Coupled Equations ofGroundwater Flow and Mass
Transport ........................................... 191
7.1.3 Pure Advection Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 193
7.1.4 Lumped Parameter Models ............................ 196
7.1.5 Criteria of Model Selection ............................. 199
7.2 Model Calibration and Parameter Estimation ................ 201
7.2.1 Parameter Identification of Advection-Dispersion Equations
201
7.2.2 Field Experiments for Determining Dispersivities .......... 204
7.2.3 The Relationship Between Values ofDispersivity and Scales
of Experiment ........................................ 213
7.2.4 Determination of Mean Flow Velocities .................. 217
7.2.5 Identification of Retardation Factor and Chemical Reaction
Parameters .......................................... 218
7.2.6 Identification of Pollutant Sources ....................... 219
7.2.7 Computer Aided Design for Field Experiments ............ 220
xiii
5.3 Finite Element Methods for Three-Dimensional Problems ...... 125
5.3.1 The Galerkin Finite Element Method .................... 125
5.3.2 Triangular Prism Elements and Associated Basis Functions . 132
5.3.3 A Mixed Finite Element-Finite Difference Method ......... 135
5.3.4 The Multiple Cell Balance Method ...................... 138
5.4 The Solution of Finite Element Systems ..................... 143
5.4.1 Features of Finite Element Systems and Direct Solutions ... 143
5.4.2 Point Iteration Methods ............................... 144
5.4.3 Block and Layer Iteration Methods ..................... 146
6 Numerical Solutions of Advection-Dominated Problems .......... 149
6.1 Advection-Dominated Problems. . . . . . .. . . . . . . . . . . . . . . . . . . .. 149
6.1.1 Fourier Analysis ofNumerical Errors .................... 149
6.1.2 Eulerian and Lagrangian Reference Frames ............... 154
6.2 Upstream Weighted Methods .............................. 155
6.2.1 Upstream Weighted Finite Difference Methods ............ 155
6.2.2 Upstream Weighted Finite Element Methods ............. 158
6.2.3 The Upstream Weighted Multiple Cell Balance Method .... 163
6.3 Moving Coordinate System and Moving Point Methods ....... 172
6.3.1 Moving Coordinate System Methods .................... 172
6.3.2 Element Deformation Methods ......................... 175
6.3.3 Moving Point Methods ................................ 176
6.4 The Modified Methods of Characteristics .................... 177
6.4.1 The Single Step Reverse Method ........................ 177
6.4.2 The Hybrid Single Step Reverse-Moving Point Method ..... 180
6.4.3 The Hybrid Moving Point-Characteristics Finite Element
Method ............................................. 182
7 Mathematical Models of Groundwater Quality ................ 187
7.1 The Classification of Groundwater Quality Models ............ 187
7.1.1 Hydrodynamic Dispersion Models ...................... 187
7.1.2 Coupled Equations ofGroundwater Flow and Mass
Transport ........................................... 191
7.1.3 Pure Advection Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 193
7.1.4 Lumped Parameter Models ............................ 196
7.1.5 Criteria of Model Selection ............................. 199
7.2 Model Calibration and Parameter Estimation ................ 201
7.2.1 Parameter Identification of Advection-Dispersion Equations
201
7.2.2 Field Experiments for Determining Dispersivities .......... 204
7.2.3 The Relationship Between Values ofDispersivity and Scales
of Experiment ........................................ 213
7.2.4 Determination of Mean Flow Velocities .................. 217
7.2.5 Identification of Retardation Factor and Chemical Reaction
Parameters .......................................... 218
7.2.6 Identification of Pollutant Sources ....................... 219
7.2.7 Computer Aided Design for Field Experiments ............ 220