xii
Contents
2.6.1 Hydrodynamic Dispersion Equations in Orthogonal
Curvilinear Co ordinate Systems ........................
39
2.6.2 Extensions of Hydrodynamic Dispersion Equations .......
41
2.6.3 Initial and Boundary Conditions .......................
44
3 Analytical Solutions of Hydrodynamic Dispersion Equations ......
50
3.1 Superposition of Fundamental Solutions ....................
50
3.1.1 The Fundamental Solution of a Point Source ............
50
3.1.2 Superposition Principle and Image Method ..............
52
3.1.3 Continuous Injection in a Uniform Flow Field ...........
54
3.2 Some Canonical Problems Having Analytical Solutions .......
56
3.2.1 One-Dimensional Dispersion Problems .................
56
3.2.2 Two- and Three-Dimensional Dispersion Problems .......
61
3.2.3 Radial Dispersion Problems ...........................
66
3.2.4 Dispersion Problems in Fractured Rock .................
69
4 Finite Difference Methods and the Method of Characteristics for
Solving Hydrodynamic Dispersion Equations ..................
73
4.1 Finite Difference Methods ................................
73
4.1.1 Finite Difference Approximations of Derivatives ..........
73
4.1.2 Finite Difference Solutions of One-Dimensional Dispersion
Problems ...........................................
75
4.1.3 Numerical Dispersion and Overshoot ...................
80
4.1.4 Finite Difference Solutions of Two- and Three-Dimensional
Dispersion Problems .................................
83
4.2 The Method of Characteristics ............................
84
4.2.1 Basic Idea of the Method of Characteristics ..............
84
4.2.2 Computation of the Advection Part .....................
85
4.2.3 Computation of the Dispersion Part ....................
88
4.2.4 Treatment of Boundary Conditions and SinkjSource Terms
89
4.2.5 The Random-Walk Model ............................
91
5 Finite Element Methods for Solving Hydrodynamic Dispersion
Equations ................................................
97
5.1 Finite Element Methods for Two-Dimensional Problems ......
97
5.1.1 The Weighted Residual Method ........................
97
5.1.2 Finite Element Discretization and Basis Functions ........
101
5.1.3 High-Order Elements and Hermite Elements .............
105
5.1.4 Isoparametric Finite Elements .........................
109
5.1.5 Treatment of Boundary Conditions .....................
113
5.2 The Multiple Cell Balance Method .........................
115
5.2.1 Governing Equations ................................. 115
5.2.2 An Algorithm Based on Multiple Cell Balance ............
116
5.2.3 Comparing with the Finite Element Method .............
122
5.2.4 The Test of Numerical Solutions .......................
123