Applied Mathematics in Hydrogeology

by Lee; Tien-Chang

• ISBN13: 9781566703758
• ISBN10: 1566703751
• Edition: 1st
• Format: Hardcover
• Copyright: 1998-12-10
• Publisher: CRC Press

Summary

As introduced in Dr. Lee's 10-week class, Applied Mathematics in Hydrogeology is written for professionals & graduate students who have a keen interest in the application of mathematics in hydrogeology. Its first seven chapters cover analytical solutions for problems commonly encountered in the study of quantitative hydrogeology, while the final three chapters focus on solving linear simultaneous equations, finite element analysis, & inversion for parameter determination. Dr. Lee provides various equation-solving methods that are of interest to hydrogeologists, geophysicists, soil scientists, & civil engineers, as well as applied physicists & mathematicians. In the classroom, this same information will help students realize how familiar equations in hydrogeology are derived-an important step toward development of a student's own mathematical models. Unlike other applied mathematics books that are structured according to systematic methodology, Applied Mathematics in Hydrogeology emphasizes equation-solving methods according to topics. Hydrogeological problems & governing differential equations are introduced, including hydraulic responses to pumping in confined & unconfined aquifers, as well as transport of heat & solute in flowing groundwater. Features

Table of Contents

PREFACE
1 CONSERVATION EQUATIONS	1	(20)
1.1 Heat Conduction	1	(4)
1.1.1 Heat Energy	1	(2)
1.1.2 Fourier's Law	3	(2)
1.2 Groundwater Flow	5	(6)
1.2.1 Equation of Continuity	5	(1)
1.2.2 Darcy's Law	6	(2)
1.2.3 Flow Equation	8	(3)
1.3 Advective Heat Transfer	11	(1)
1.4 Dispersion Equation	12	(1)
1.5 Boundary and Initial Conditions	13	(2)
1.6 Problems, Keys, and Suggested Readings	15	(4)
1.7 Notations	19	(2)
2 SOURCE FUNCTIONS AND CONVOLUTION	21	(34)
2.1 Heat Sink and Source	21	(3)
2.1.1 Instantaneous Point Source	21	(1)
2.1.2 Instantaneous Line Source	22	(1)
2.1.3 Steady Line Source	23	(1)
2.2 Convolution	24	(7)
2.2.1 Concept	25	(1)
2.2.2 Experimental Impulse Response	26	(1)
2.2.3 Numerical Convolution	27	(1)
2.2.4 Transfer Function	28	(1)
2.2.5 Fast Fourier Transform	29	(1)
2.2.6 Z-Transform	30	(1)
2.3 Theis Well Function	31	(7)
2.3.1 Assumptions	31	(2)
2.3.2 Derivation	33	(2)
2.3.3 Determination of Transmissivity and Storativity	35	(1)
2.3.4 Semilog Method	36	(1)
2.3.5 Radius of Pumping Influence	37	(1)
2.3.6 Recovery Test	37	(1)
2.4 Linear Superposition of Well Functions	38	(2)
2.4.1 Pumping Near a Constant Head Boundary	38	(2)
2.4.2 Pumping Near an Impermeable Boundary	40	(1)
2.4.3 Pumping Near Two Dissimilar Media	40	(1)
2.5 Evaluation of Exponential Function	40	(3)
2.6 Problems, Keys, and Suggested Readings	43	(11)
2.7 Notations	54	(1)
3 LAPLACE AND HANKEL TRANSFORMS	55	(58)
3.1 Fundamentals of the Laplace Transform	55	(5)
3.1.1 Transform Pairs	55	(1)
3.1.2 Basic Transform Formulas	56	(4)
3.2 Contour Integration	60	(19)
3.2.1 Singularity	60	(3)
3.2.2 Residue Theorem	63	(1)
3.2.3 Multivalued Function	64	(1)
3.2.4 Example 1: Branch Cut, No Singularity	65	(4)
3.2.5 Example 2: Branch Cut, Singularity	69	(3)
3.2.6 Example 3: Convolution 1	72	(1)
3.2.7 Example 4: Convolution 2	73	(2)
3.2.8 Example 5: Rationalization	75	(1)
3.2.9 Example 6: A Series Representation	75	(1)
3.2.10 Example 7: Roots, No Branch Cut	76	(1)
3.2.11 Example 8: Branch Cut, Logarithm	77	(1)
3.2.12 Example 9: Bessel Functions -- Delta Function	78	(1)
3.3 Numerical Inversion	79	(2)
3.3.1 Gaver-Stehfest Method	80	(1)
3.4 Hankel Transform	81	(19)
3.4.1 Bessel Functions	81	(4)
3.4.2 Example 1H: Hankel Transform -- Using a Series	85	(1)
3.4.3 Example 2H: Hankel Transform of 1/(a(2) + q(2))	86	(1)
3.4.4 Example 3H: Hankel Transform of J(0)[ar(XXX)]/(a(2) + q(2))	87	(1)
3.4.5 Example 4H: Hankel Transform of aJ(1)[ar(XXX)]/(a(2) + q(2))	88	(1)
3.4.6 Example 5H: Inverse Hankel Transform 1	89	(3)
3.4.7 Example 6H: Inverse Hankel Transform 2	92	(1)
3.4.8 Example 7H: Integration Involving 1/XXX(1 - z(2))	93	(4)
3.4.9 Numerical Hankel Transform	97	(3)
3.5 Problems, Keys, and Suggested Readings	100	(12)
3.6 Notations	112	(1)
4 DRAWDOWN IN CONFINED AQUIFERS	113	(46)
4.1 Theis Well function	114	(2)
4.2 Steady-State Solution	116	(1)
4.2.1 Jacob Solution	116	(1)
4.2.2 Thiem's Formula	116	(1)
4.3 Full-Penetration Pumping Well, r(w) Not equal to 0	117	(4)
4.3.1 Formulation	117	(1)
4.3.2 Laplace-Domain Solution	118	(1)
4.3.3 Time-Domain Solution	119	(2)
4.4 Hantush's Leaky Aquifer	121	(13)
4.4.1 Assumptions	122	(1)
4.4.2 Mass Balance Equation	122	(2)
4.4.3 Full-Penetration Pumping Well, r(w) = 0	124	(3)
4.4.4 Full-Penetration Pumping Well, r(w) Not equal to 0	127	(2)
4.4.5 Partial-Penetration Pumping Well, r(w) = 0	129	(3)
4.4.6 Partial-Penetration Pumping Well, r(w) Not equal to 0	132	(2)
4.5 Leakage as Boundary Flux	134	(11)
4.5.1 Full-Penetration Pumping Well, r(w) = 0	135	(6)
4.5.2 Partial-Penetration Pumping Well, r(w) = 0	141	(3)
4.5.3 Partial-Penetration Pumping Well, r(w) Not equal to 0	144	(1)
4.6 Slug Test	145	(3)
4.6.1 Full-Penetration Well	146	(2)
4.7 Problems, Keys, and Suggested Readings	148	(9)
4.8 Notations	157	(2)
5 DRAWDOWN IN UNCONFINED AQUIFERS	159	(38)
5.1 Dupuit-Forchheimer Theory	160	(1)
5.1.1 Lateral Flow	160	(1)
5.1.2 Steady Radial Flow	161	(1)
5.2 Pumping Wells of Infinitesimal Diameter	161	(10)
5.2.1 Full-Penetration Pumping Well, r(w) = 0	162	(7)
5.2.2 Partial-Penetration Pumping Well, r(w) = 0	169	(2)
5.3 Pumping Well of Nonzero Diameter	171	(9)
5.3.1 Full-Penetration Pumping Well, r(w) Not equal to 0	172	(6)
5.3.2 Partial-Penetration Pumping Well, r(w) Not equal to 0	178	(1)
5.3.3 Effect of Finite Diameter	179	(1)
5.4 Well Storage and Skin Effect	180	(9)
5.4.1 Full-Penetration Well	180	(4)
5.4.2 Partial-Penetration Well	184	(5)
5.5 Problems, Keys, and Suggested Readings	189	(5)
5.6 Notations	194	(3)
6 HEAT TRANSFER AND GROUNDWATER FLOW	197	(40)
6.1 Advective Heat Transfer	198	(9)
6.1.1 Steady Vertical Flow	200	(3)
6.1.2 Steady Horizontal Flow	203	(2)
6.1.3 Lateral Gradient XXXT/XXXx = Constant	205	(2)
6.2 Heat Sources in Regional Groundwater Flow	207	(5)
6.2.1 Instantaneous Point Source	207	(2)
6.2.2 Continuous Point Source	209	(1)
6.2.3 Influence of Ground Surface	210	(2)
6.3 Topography-Controlled Flow	212	(8)
6.3.1 Conservation Equations	212	(2)
6.3.2 Steady Groundwater Flow	214	(2)
6.3.3 Temperature Distribution	216	(4)
6.4 Heating and Pressurization	220	(6)
6.4.1 Equation for Fluid Pressurization	221	(2)
6.4.2 Pressurization	223	(3)
6.5 Problems, Keys, and Suggested Readings	226	(9)
6.6 Notations	235	(2)
7 SOLUTE TRANSPORT	237	(42)
7.1 Formulation of Equations	237	(4)
7.1.1 Adsorption	239	(2)
7.2 One-Dimensional Problems	241	(16)
7.2.1 Example 1: A Step Input	241	(2)
7.2.2 Example 2: Exponentially Decaying Input	243	(2)
7.2.3 Example 3: A System with Retardation	245	(1)
7.2.4 Example 4: A Reactive System	246	(2)
7.2.5 Example 5: A Reactive and Retardative System	248	(1)
7.2.6 Example 6: Advection at Boundary, Lambda Not equal to Alpha	248	(5)
7.2.7 Example 7: Advection at Boundary, Lambda = Alpha	253	(1)
7.2.8 Example 8: A Pulse Input	254	(1)
7.2.9 Example 9: Advection, Transfer Function	255	(2)
7.3 Two-Dimensional Problems	257	(7)
7.3.1 Example 10: A Plane Dispersion Model	257	(5)
7.3.2 Example 11: One Impermeable Boundary	262	(1)
7.3.3 Example 12: Bounded Flow	263	(1)
7.4 Three-Dimensional Problems	264	(3)
7.4.1 Green's Function	265	(1)
7.4.2 Integral Transforms	265	(2)
7.5 Radial Dispersion	267	(4)
7.5.1 Formulation	267	(1)
7.5.2 Solution	268	(3)
7.6 Simulation by Z-Transform	271	(1)
7.6.1 Example 13: Numerical Simulation	271	(1)
7.7 Problems, Keys, and Suggested Readings	272	(6)
7.8 Notations	278	(1)
8 SOLVING Ax = b	279	(26)
8.1 Elementary Matrix Operations	279	(2)
8.2 Eigenproblems	281	(3)
8.2.1 Eigenmatrix	281	(2)
8.2.2 Least-Squares Method	283	(1)
8.3 x = A(-1)b	284	(5)
8.3.1 Gaussian Elimination and Backsubstitution	284	(2)
8.3.2 LU Decomposition	286	(1)
8.3.3 Iteration Methods	287	(2)
8.4 Singular Value Decomposition	289	(5)
8.5 Examples	294	(5)
8.5.1 Example 1: SVD	294	(2)
8.5.2 Example 2: Ill-Conditioned Matrix	296	(2)
8.5.3 Example 3: Ill-Conditioned Matrix (continued)	298	(1)
8.5.4 Example 4: A Well-Behaved Matrix	298	(1)
8.6 Problems, Keys, and Suggested Readings	299	(6)
9 FINITE ELEMENT ANALYSIS	305	(32)
9.1 1D Finite Element Method	305	(17)
9.1.1 Formulation of a Problem	305	(2)
9.1.2 Galerkin Weighted Residual	307	(3)
9.1.3 Elementary Matrices	310	(1)
9.1.4 Finite-Element Equation	311	(3)
9.1.5 Differential Equation in Time	314	(2)
9.1.6 Lumped Finite-Element Formulation	316	(1)
9.1.7 Nature of Coefficient Matrix	317	(1)
9.1.8 Initial and Boundary Conditions	317	(1)
9.1.9 Source Term	318	(1)
9.1.10 Numerical Instability and Oscillation	319	(3)
9.2 2D Finite Element Method	322	(6)
9.2.1 Procedures	322	(4)
9.2.2 Remarks	326	(2)
9.3 Transport Equations	328	(5)
9.3.1 Advective Heat Transfer	328	(2)
9.3.2 Solute Transport	330	(3)
9.4 Problems, Keys, and Suggested Readings	333	(4)
10 INVERSE PROBLEMS	337	(24)
10.1 Linear Inversion	338	(5)
10.1.1 Example 1: A Linearizable System	339	(3)
10.1.2 Example 2: Partially Linearizable	342	(1)
10.2 Constrained Linear Inversion	343	(10)
10.2.1 Constraint of Parameters	343	(5)
10.2.2 More on Biased Linear Inversions	348	(1)
10.2.3 Goodness of Fit	349	(4)
10.3 Nonlinear Inversion	353	(4)
10.3.1 Gauss-Newton Method	354	(2)
10.3.2 Resolution	356	(1)
10.3.3 Ridge Regression	356	(1)
10.4 Example: A 1D Finite-Element Problem	357	(3)
10.5 Suggested Readings	360	(1)
Appendix: Notes on Equation Solving	361	(6)
A.1 Integral Transforms	361	(2)
A.2 Separation of Variables	363	(1)
A.3 Series Solutions	363	(1)
A.4 Linear Superposition	364	(1)
A.5 Numerical Methods	364	(1)
A.6 Integration	365	(2)
BIBLIOGRAPHY	367	(8)
INDEX	375