AppliedMathematicsin Hydrogeology

Tien-Chang LeeDepartment of Earth Sciences

University of CaliforniaRiverside, California

LEWIS PUBLISHERSBoca Raton London New York Washington, D.C.

Contents

PREFACE

1 CONSERVATION EQUATIONS 11.1 Heat Conduction 1

1.1.1 Heat Energy 11.1.2 Fourier's Law 3

1.2 Groundwater Flow 51.2.1 Equation of Continuity 51.2.2 Darcy's Law . . 61.2.3 Flow Equation 8

1.3 Advective Heat Transfer 11'1.4 Dispersion Equation ; 121.5 Boundary and Initial Conditions 131.6 Problems, Keys, and Suggested Readings 151.7 Notations 19

2 SOURCE FUNCTIONS AND CONVOLUTION 212.1 Heat Sink and Source 21

2.1.1 Instantaneous Point Source 212.1.2 Instantaneous Line Source 222.1.3 Steady Line Source 23

2.2 Convolution 242.2.1 Concept 252.2.2 Experimental Impulse Response 262.2.3 Numerical Convolution 272.2.4 Transfer Function 282.2.5 Fast Fourier Transform 292.2.6 Z-Transform 30

2.3 Theis Well Function 312.3.1 Assumptions 312.3.2 Derivation 332.3.3 Determination of Transmissivity and Storativity . . . . 352.3.4 Semilog Method 362.3.5 Radius of Pumping Influence 372.3.6 Recovery Test 37

2.4 Linear Superposition of Well Functions 382.4.1 Pumping Near a Constant Head Boundary 382.4.2 Pumping Near an Impermeable Boundary 402.4.3 Pumping Near Two Dissimilar Media 40

2.5 Evaluation of Exponential Function 402.6 Problems, Keys, and Suggested Readings 432.7 Notations 54

3 LAPLACE A N D HANKEL TRANSFORMS 553.1 Fundamentals of the Laplace Transform 55

3.1.1 Transform Pairs 553.1.2 Basic Transform Formulas 56

3.2 Contour Integration 603.2.1 Singularity 603.2.2 Residue Theorem : 633.2.3 Multivalued Function 643.2.4 Example 1: Branch Cut, No Singularity 653.2.5 Example 2: Branch Cut, Singularity 693.2.6 Example 3: Convolution 1 723.2.7 Example 4: Convolution 2 733.2.8 Example 5: Rationalization 753.2.9 Example 6: A Series Representation 753.2.10 Example 7: Roots, No Branch Cut 763.2.11 Example 8: Branch Cut, Logarithm 773.2.12 Example 9: Bessel Functions - Delta Function 78

3.3 Numerical Inversion 793.3.1 Gaver-Stehfest Method 80

3.4 Hankel Transform 813.4.1 Bessel Functions 813.4.2 Example 1H: Hankel Transform - Using a Series . . . . 853.4.3 Example 2H: Hankel Transform of l/(a2 + q2) 86

3.4.4 Example 3H: Hankel Transform of J0[ar'}/ (a2 + q2) . . 873.4.5 Example 4H: Hankel Transform of aJ^ar'j/ia2 + q2) . . 883.4.6 Example 5H: Inverse Hankel Transform 1 893.4.7 Example 6H: Inverse Hankel Transform 2 923.4.8 Example 7H: Integration Involving l/-\/l — z2 933.4.9 Numerical Hankel Transform 97

3.5 Problems, Keys, and Suggested Readings 1003.6 Notations 112

DRAWDOWN IN CONFINED AQUIFERS 1134.1 Theis Well function 1144.2 Steady-State Solution 116

4.2.1 Jacob Solution 1164.2.2 Thiem's Formula 116

4.3 Full-Penetration Pumping Well, rw ^ 0 1174.3.1 Formulation 1174.3.2 Laplace-Domain Solution 1184.3.3 Time-Domain Solution 119

4.4 Hantush's Leaky Aquifer 1214.4.1 Assumptions ; 1224.4.2 Mass Balance Equation 1224.4.3 Full-Penetration Pumping Well, rw = 0 1244.4.4 Full-Penetration Pumping Well, rw ^ 0 1274.4.5 Partial-Penetration Pumping Well, rw = 0 1294.4.6 Partial-Penetration Pumping Well, rw =fi 0 132

4.5 Leakage as Boundary Flux 1344.5.1 Full-Penetration Pumping Well, rw = 0 1354.5.2 Partial-Penetration Pumping Well, rw = 0 1414.5.3 Partial-Penetration Pumping Well, rw ^ 0 144

4.6 Slug Test 1454.6.1 Full-Penetration Well 146

4.7 Problems, Keys, and Suggested Readings 1484.8 Notations 157

DRAWDOWN IN UNCONFINED AQUIFERS 1595.1 Dupuit-Forchheimer Theory 160

5.1.1 Lateral Flow 1605.1.2 Steady Radial Flow 161

5.2 Pumping Wells of Infinitesimal Diameter 1615.2.1 Full-Penetration Pumping Well, rw = 0 1625.2.2 Partial-Penetration Pumping Well, rw = 0 169

5.3 Pumping Well of Nonzero Diameter 1715.3.1 Full-Penetration Pumping Well, rw ^ 0 1725.3.2 Partial-Penetration Pumping Well, rw ^ 0 . . . . . . . 1785.3.3 Effect of Finite Diameter 179

5.4 Well Storage and Skin Effect 1805.4.1 Full-Penetration Well 1805.4.2 Partial-Penetration Well 184

5.5 Problems, Keys, and Suggested Readings 1895.6 Notations 194

HEAT TRANSFER AND GROUNDWATER FLOW 1976.1 Advective Heat Transfer 198

6.1.1 Steady Vertical Flow 2006.1.2 Steady Horizontal Flow 2036.1.3 Lateral Gradient dT/dx = Constant 205

6.2 Heat Sources in Regional Groundwater Flow 2076.2.1 Instantaneous Point Source 2076.2.2 Continuous Point Source .2096.2.3 Influence of Ground Surface : ; 210

6.3 Topography-Controlled Flow 2126.3.1 Conservation Equations 2126.3.2 Steady Groundwater Flow . . . 2146.3.3 Temperature Distribution 216

6.4 Heating and Pressurization 2206.4.1 Equation for Fluid Pressurization 2216.4.2 Pressurization 223

6.5 Problems, Keys, and Suggested Readings 2266.6 Notations 235

SOLUTE TRANSPORT 2377.1 Formulation of Equations 237

7.1.1 Adsorption 2397.2 One-Dimensional Problems 241

7.2.1 Example 1: A Step Input 2417.2.2 Example 2: Exponentially Decaying Input 243

7.2.3 Example 3: A System with Retardation 2457.2.4 Example 4: A Reactive System 2467.2.5 Example 5: A Reactive and Retardative System . . . . 2487.2.6 .Example 6: Advection at Boundary, A ^ a 2487.2.7 Example 7: Advection at Boundary, X = a 2537.2.8 Example 8: A Pulse Input 2547.2.9 Example 9: Advection, Transfer Function 255

7.3 Two-Dimensional Problems 2577.3.1 Example 10: A Plane Dispersion Model 2577.3.2 Example 11: One Impermeable Boundary 2627.3.3 Example 12: Bounded Flow 263

7.4 Three-Dimensional Problems 2647.4.1 Green's Function 2657.4.2 Integral Transforms 265

7.5 Radial Dispersion 2677.5.1 Formulation 2677.5.2 Solution 268

7.6 Simulation by Z-Transform 2717.6.1 Example 13: Numerical Simulation 271

7.7 Problems, Keys, and Suggested Readings 2727.8 Notations 278

8 SOLVING Ax = b 2798.1 Elementary Matrix Operations 2798.2 Eigenproblems 281

8.2.1 Eigenmatrix 2818.2.2 Least-Squares Method 283

8.3 x = A-Xb 2848.3.1 Gaussian Elimination and Backsubstitution 2848.3.2 LU Decomposition 2868.3.3 Iteration Methods 287

8.4 Singular Value Decomposition 2898.5 Examples 294

8.5.1 Example 1: SVD 2948.5.2 Example 2: 111-Conditioned Matrix 2968.5.3 Example 3: Ill-Conditioned Matrix (continued) . . . . 2988.5.4 Example 4: A Well-Behaved Matrix 298

8.6 Problems, Keys, and Suggested Readings 299

9 FINITE ELEMENT ANALYSIS 3059.1 ID Finite Element Method 305

9.1.1 Formulation of a Problem .. . 3059.1.2 Galerkin Weighted Residual 3079.1.3 Elementary Matrices . . . .* 3109.1.4 Finite-Element Equation 3119.1.5 Differential Equation in Time 3149.1.6 Lumped Finite-Element Formulation 3169.1.7 Nature of Coefficient Matrix 3179.1.8 Initial and Boundary Conditions 3179.1.9 Source Term 3189.1.10 Numerical Instability and Oscillation 319

9.2 2D Finite Element Method 3229.2.1 Procedures 3229.2.2 Remarks 326

9.3 Transport Equations 3289.3.1 Advective Heat Transfer 3289.3.2 Solute Transport 330

9.4 Problems, Keys, and Suggested Readings 333

10 INVERSE PROBLEMS 33710.1 Linear Inversion ' 338

10.1.1 Example 1: A Linearizable System 33910.1.2 Example 2: Partially Linearizable 342

10.2 Constrained Linear Inversion 34310.2.1 Constraint of Parameters 34310.2.2 More on Biased Linear Inversions 34810.2.3 Goodness of Fit 349

10.3 Nonlinear Inversion 35310.3.1 Gauss-Newton Method 35410.3.2 Resolution 35610.3.3 Ridge Regression . 356

10.4 Example: A ID Finite-Element Problem 35710.5 Suggested Readings 360

Appendix: Notes on Equation Solving 361A.I Integral Transforms 361A.2 Separation of Variables 363

A.3 Series Solutions 363A.4 Linear Superposition 364A.5 Numerical Methods 364A.6 Integration 365

BIBLIOGRAPHY 367

INDEX 375