Wang / Anderson Introduction to Groundwater Modeling

Finite Difference and Finite Element Methods
1. Auflage 1995
ISBN: 978-0-08-057194-2
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark

E-Book, Englisch, 237 Seiten

The dramatic advances in the efficiency of digital computers during the past decade have provided hydrologists with a powerful tool for numerical modeling of groundwater systems. Introduction to Groundwater Modeling presents a broad, comprehensive overview of the fundamental concepts and applications of computerized groundwater modeling. The book covers both finite difference and finite element methods and includes practical sample programs that demonstrate theoretical points described in the text. Each chapter is followed by problems, notes, and references to additional information. This volume will be indispensable to students in introductory groundwater modeling courses as well as to groundwater professionals wishing to gain a complete introduction to this vital subject. - Systematic exposition of the basic ideas and results of Hilbert space theory and functional analysis - Great variety of applications that are not available in comparable books - Different approach to the Lebesgue integral, which makes the theory easier, more intuitive, and more accessible to undergraduate students

Autoren/Hrsg.

Wang, Herbert F.

Anderson, Mary P.

Inhaltsverzeichnis

1;Cover Page;1
2;Introduction to Groundwater Modeling;4
3;Copyright ;5
4;Table of contents ;6
5;Preface ;10
6;Chapter 1: Introduction;13
6.1;1.1 Models;13
6.1.1;Types of groundwater models ;14
6.1.2;1.2 Physics of groundwater flow;18
6.1.2.1;Darcy's law ;18
6.1.2.2;Hubbert's force potential ;20
6.1.2.3;Darcy's law in three dimensions ;22
6.1.2.4;Continuity equation for steady-state flow ;23
6.1.3;1.3 Laplace's equation;25
6.1.3.1;Boundary conditions ;25
6.1.4;1.4 Regional groundwater flow system;26
6.1.5;Notes and additional reading ;29
7;Chapter 2: Finite differences: Steady-state flow (laplace's equation);31
7.1;2.1 Introduction;31
7.2;2.2 Differences for derivatives;33
7.2.1;Finite difference expression of laplace's equation ;33
7.2.2;Example with dirichlet boundary conditions: region near a well ;35
7.3;2.3 Iterative methods;36
7.3.1;Jacobi iteration ;37
7.3.2;Gauss-seidel iteration ;38
7.3.3;Successive over relaxation (SOR);39
7.4;2.4 Gauss-seidel computer program;41
7.5;2.5 Boundary conditions;43
7.6;Notes and additional reading ;49
7.7;Problems ;49
8;Chapter 3: Finite differences: Steady-state flow (poisson's equation);53
8.1;3.1 Introduction;53
8.2;3.2 Poisson's equation;53
8.3;3.3 Island recharge;55
8.4;3.4 Finite difference models;56
8.4.1;Island recharge ;56
8.4.2;Well drawdown (confined aquifer) ;58
8.5;3.5 Unconfined aquifer with dupuit assumptions;64
8.5.1;Seepage through a dam ;65
8.5.2;Well drawdown (unconfined aquifer) ;70
8.6;3.6 Validity of a numerical solution;74
8.7;Notes and additional reading ;77
8.8;Problems ;77
9;Chapter 4: Finite differences: Transient flow;79
9.1;4.1 Transient flow equation;79
9.2;4.2 Explicit finite difference approximation;81
9.2.1;Validity of the explicit solution ;82
9.2.2;Aquifer response to sudden change in reservoir level ;82
9.3;4.3 Implicit finite difference approximation;88
9.3.1;Well drawdown (theis problem) ;91
9.4;4.4 Unconfined aquifer with dupuit assumptions;99
9.4.1;Explicit approximation ;99
9.4.2;Implicit approximation ;100
9.5;Additional reading ;100
9.6;Problems ;101
10;Chapter 5: Other solution methods;105
10.1;5.1 Introduction;105
10.2;5.2 Matrix notation;106
10.3;5.3 Tridiagonal matrices;107
10.3.1;Thomas algorithm ;108
10.3.2;Direct solution of reservoir problem ;111
10.4;5.4 Alternating direction implicit (ADI) method;111
10.4.1;Iterative ADI;118
10.5;5.5 Prickett-lonnquist and trescott-pinder-larson models;119
10.5.1;Input data ;120
10.6;5.6 Calibration and verification;121
10.6.1;Calibration ;121
10.6.2;Verification ;122
10.7;Notes and additional reading ;123
10.8;Problems ;124
11;Chapter 6: Finite elements: Steady-state flow;125
11.1;6.1 Introduction;125
11.2;6.2 Galerkin's method;126
11.2.1;Integration by parts ;128
11.3;6.3 Triangular elements;129
11.3.1;Finite element mesh ;129
11.3.2;The archetypal element ;131
11.3.3;Patch of elements ;133
11.4;6.4 Assembly of conductance matrix;133
11.4.1;Element conductance matrix ;135
11.4.2;Global conductance matrix ;136
11.5;6.5 Boundary conditions;138
11.5.1;Specified flow ;138
11.5.2;Specified head ;140
11.6;6.6 Finite element computer program;140
11.6.1;Node Specification (lines 6 to 21);141
11.6.2;Global Conductance Matrix (lines 23 to 50);141
11.6.3;Iterative Solution of Equations (lines 52 to 73);145
11.7;6.7 Region-near-a-well example;146
11.7.1;Irregular mesh ;146
11.8;6.8 Seepage through a dam;150
11.8.1;Self-consistent iterative solution ;151
11.8.2;Computer program ;152
11.9;6.9 Poisson's equation;157
11.9.1;Computer program ;159
11.10;Notes and additional reading ;160
11.11;Problems ;161
12;Chapter 7: Finite elements: Transient flow;163
12.1;7.1 Introduction;163
12.2;7.2 Galerkin's method;164
12.2.1;Trial solution ;164
12.2.2;Weighted residual ;165
12.2.3;Integration by parts ;165
12.3;7.3 Rectangular element;165
12.3.1;Local and global coordinates ;168
12.3.2;Patch of elements ;168
12.4;7.4 Assembly of matrix differential equation;168
12.4.1;Element matrices ;169
12.4.2;Gaussian quadrature ;170
12.4.3;Global matrices ;171
12.4.4;Boundary conditions ;172
12.5;7.5 Solving the matrix differential equation;172
12.6;7.6 Computer program for reservoir problem;173
12.6.1;Nodal coordinates, initial conditions, and boundary conditions (lines 11 to 28);178
12.6.2;Global conductance and storage matrices (lines 30 to 71);179
12.6.3;Time stepping (lines 73 to 114);179
12.7;Notes and additional reading ;182
12.8;Problems ;182
13;Chapter 8: Advective-dispersive transport;185
13.1;8.1 Introduction;185
13.2;8.2 Dispersion;187
13.2.1;Dispersive flux ;189
13.2.2;Dispersion coefficient (uniform flow field);189
13.2.3;Dispersivity ;190
13.2.4;Dispersion coefficient (nonuniform flow field);191
13.3;8.3 Solute transport equation;193
13.3.1;Sources, sinks, and chemical reactions ;195
13.3.2;Solving the governing equation ;195
13.4;8.4 Finite element example: Solute dispersion in uniform flow field;197
13.4.1;Finite element theory ;198
13.4.2;Finite element computer program ;200
13.5;Additional reading ;210
13.6;Problems ;210
14;Concluding remarks ;213
15;Appendix A: Anisotropy and tensors;216
15.1;A.1 Introduction;216
15.2;A.2 Hydraulic conductivity tensor;217
15.3;A.3 Coordinate system rotation;218
16;Appendix B: Variational method;221
16.1;B.1 Introduction;221
16.2;B.2 Minimum dissipation principle;221
16.3;B.3 finite elements ;223
17;Appendix C: Isoparametric quadrilateral elements;224
17.1;C.1 Introduction;224
17.2;C.2 Coordinate transformation;225
17.2.1;Element trial solution;226
17.2.2;Transformation of integrals;226
17.3;C.3 Computer program modification;229
18;Appendix D: Analogies;231
18.1;D.1 Introduction;231
18.2;D.2 Electrical analogy;231
18.3;D.3 Heat flow analogy;234
18.4;D.4 Structural mechanics analogy;235
19;Glossary of symbols ;237
20;References ;239
21;Index ;247

Leseproben

Chapter 1. Introduction

1.1. MODELS

A model is a tool designed to represent a simplified version of reality. Given this broad definition of a model, it is evident that we all use models in our everyday lives. For example, a road map is a way of representing a complex array of roads in a symbolic form so that it is possible to test various routes on the map rather than by trial and error while driving a car. A road map could be considered a kind of model (Lehr, 1979) because it is a way of representing reality in a simplified form. Similarly, groundwater models are also representations of reality and, if properly constructed, can be valuable predictive tools for management of groundwater resources. For example, using a groundwater model, it is possible to test various management schemes and to predict the effects of certain actions. Of course, the validity of the predictions will depend on how well the model approximates field conditions. Good field data are essential when using a model for predictive purposes. However, an attempt to model a system with inadequate field data can also be instructive as it may serve to identify those areas where detailed field data are critical to the success of the model. In this way, a model can help guide data collection activities.

Types of Groundwater Models

Several types of models have been used to study groundwater flow systems. They can be divided into three, broad categories (Prickett, 1975): sand tank models, analog models, including viscous fluid models and electrical models, and mathematical models, including analytical and numerical models. A sand tank model consists of a tank filled with ah unconsolidated porous medium through which water is induced to flow. A major drawback of sand tank models is the problem of scaling down a field situation to the dimensions of a laboratory model. Phenomena measured at the scale of a sand tank model are often different from conditions observed in the field, and conclusions drawn from such models may need to be qualified when translated to a field situation.

As we shall see later in the book, the flow of groundwater can be described by differential equations derived from basic principles of physics. Other processes, such as the flow of electrical current through a resistive medium or the flow of heat through a solid, also operate under similar physical principles. In other words, these systems are analogous to the groundwater system. The two types of analogs used most frequently in groundwater modeling are viscous fluid analog models and electrical analog models.

Viscous fluid models are known as Hele–Shaw or parallel plate models because a fluid more viscous than water (for example, oil) is made to flow between two closely spaced parallel plates, which may be oriented either vertically or horizontally. Electrical analog models were widely used in the 1950s before high-speed digital computers became available. These models consist of boards wired with electrical networks of resistors and capacitors. They work according to the principle that the flow of groundwater is analogous to the flow of electricity. This analogy is expressed in the mathematical similarity between Darcy's law for groundwater flow and Ohm's law for the flow of electricity. Changes in voltage in an electrical analog model are analogous to changes in groundwater head. A drawback of an electrical analog model is that each one is designed for a unique aquifer system. When a different aquifer is to be studied, an entirely new electrical analog model must be built.

A mathematical model consists of a set of differential equations that are known to govern the flow of groundwater. Mathematical models of groundwater flow have been in use since the late 1800s. The reliability of predictions using a groundwater model depends on how well the model approximates the field situation. Simplifying assumptions must always be made in order to construct a model because the field situations are too complicated to be simulated exactly. Usually the assumptions necessary to solve a mathematical model analytically are fairly restrictive. For example, many analytical solutions require that the medium be homogeneous and isotropic. To deal with more realistic situations, it is usually necessary to solve the mathematical model approximately using numerical techniques. Since the 1960s, when high-speed digital computers became widely available, numerical models have been the favored type of model for studying groundwater. The subject of this book is the use of numerical methods to solve mathematical models that simulate groundwater flow and contaminant transport.

We consider two types of models—finite difference models (Chapter 2, Chapter 3, Chapter 4 and Chapter 5) and finite element models (Chapter 6, Chapter 7 and Chapter 8). In either case, a system of nodal points is superimposed over the problem domain. For example, consider the problem shown in Figure 1.1. The problem domain consists of an aquifer bounded on one side by a river (Figure 1.1a). The aquifer is recharged areally by precipitation, but there is no horizontal flow out of or into the aquifer except along the river. Two finite difference representations of the problem domain are illustrated in Figures 1.1b and 1.1c and a finite element representation is shown in Figure 1.1d.

Figure 1.1 Finite difference and finite element representations of an aquifer region. (a) Map view of aquifer showing well field, observation wells, and boundaries. (b) Finite difference grid with block-centered nodes, where ?x is the spacing in the x direction, ?y is the spacing in the y direction, and b is the aquifer thickness. (c) Finite difference grid with mesh-centered nodes. (d) Finite element mesh with triangular elements where b is the aquifer thickness. (Adapted from Mercer and Faust, 1980a.)

The concept of elements (that is, the subareas delineated by the lines connecting nodal points) is fundamental to the development of equations in the finite element method. Triangular elements are used in Figure 1.1d, but quadrilateral or other elements are also possible. In the finite difference method, nodes may be located inside cells (Figure 1.1b) or at the intersection of grid lines (Figure 1.1c). The finite difference grid shown in Figure 1.1b is said to use block-centered nodes, whereas the grid in Figure 1.1c is said to use mesh- centered nodes. Aquifer properties and head are assumed to be constant within each cell in Figure 1.1b. In Figure 1.1c, nodes are located at the intersections of grid lines, and the area of influence of each node is defined following one of several different conventions. Regardless of the representation, an equation is written in terms of each nodal point because the area surrounding a node is not directly involved in the development of finite difference equations.

The goal of modeling is to predict the value of the unknown variable (for example, groundwater head or concentration of a contaminant) at nodal points. Models are often used to predict the effects of pumping on groundwater levels. For example, consider the aquifer shown in Figure 1.1. In this example, a model could be used to predict the effects of pumping the three wells in the well field on water levels in the four observation wells or to predict the effects of installing additional pumping wells. The model could also be used to determine how much water would be diverted from the river as a result of pumping. However, before a predictive simulation can be made, the model should be calibrated and verified. The process of calibrating and verifying a model is discussed in Chapter 5.

1.2. PHYSICS OF GROUNDWATER FLOW

Darcy's Law

Darcy set out to find experimentally what factors govern water flow through a sand filter (Figure 1.2). He measured the discharge by timing the rate at which water filled a 1 square meter basin at the outlet, and he measured the head drop across the sand. Darcy defined head to be the height, relative to the elevation of the bottom of the sand, to which water rises in each U-shaped tube. Although Darcy used mercury-filled manometers, he always reported his head data in terms of the equivalent water height. We shall demonstrate that head is proportional to the sum of the pressure potential of the mercury (or any fluid) in the U-shaped tube plus the elevation potential relative to the base level. Applying the term head to the height above sea level of water in a well is the correct field use in Darcy's original sense of the term.

Figure 1.2 Darcy's experimental sand column. (From Hubbert, 1969. © 1956, Society of Petroleum Engineers of AIME, published JPT, Oct. 1956; Trans. AIME, 1956. Facsimile of Fig. 3 in Darcy, Henry, Les Fontaines de la Ville de Dijon, Victor Dalmont, Paris, 1856.)

By a series of experiments, Darcy established that, for a given type of sand, the volume discharge rate Q is directly proportional to the head drop h2 – h1 and to the cross-sectional area A, but it is inversely proportional to the length difference ?2 – ?1. Calling the proportionality constant K the...