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Randolf Rausch; Wolfgang Schäfer; René Therrien; Christian Wagner:

Solute Transport Modelling

An Introduction to Models and Solution Strategies

2005. VI, 205 pages, 66 figures, 11 tables, 17x22cm, 450 g
Language: English

ArtNo. ES001200502, paperback

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Synopsis top ↑

Transport models have become an essential tool to investigate groundwater quality problems. This book presents the fundamental hydraulic, hydrochemical and nume rical concepts that are required for the sound and efficient application of solute transport models in groundwater studies. Advection, dispersion and diffusion, which are the main physical transport processes, are first introduced, followed by the derivation of the advection-dispersion equation. A separate chapter is dedicated to multispecies reactive transport modelling, presenting both theory and simulation examples. Special methods used to simulate transport in fractured geological material are also presented.

The authors, all of them groundwater modelling professionals, focus on the detailed presentation of numerical methods commonly used in transport models, to provide practitioners with a sound theoretical basis for transport model applications. Grid-based methods are presented, including explicit and implicit finite differences, finite elements and finite volume methods. Particle tracking techniques are also covered, among them the method of characteristics and the random-walk method.

This professional text addresses academics, scientists, engineers, hydrologists and hydrogeol ogists interested in the application of transport models in hydrogeology, geoecology, hydrology, geography, environmental engineering, hydraulic engineering and water economics.

Review: Journal of Hydrologic Engineering (Sept/Oct 2006, p. 512) top ↑

The past 4 decades have witnessed an increasing concern worldwide for environmental quality and management. This concern has been for land, water, and air resources as well as ecosystems. Concern for water resources has extended to water over and below the land surface (i.e., surface water, unsaturated water, and groundwater), atmospheric water, oceanic waters, as well as snow, ice, and glacier resources. Fundamental to sustained environmental quality and management is solute transport. As a result, a huge amount of literature, including journal papers, technical reports, conference proceedings papers, and books has been published and continues to be published. This is particularly true in the area of groundwater. This book deals with solute transport modeling in groundwater.

This is a small book comprising 7 chapters encompassing 205 pages. Chapter 1 introduces main physical and chemical transport processes, derives the transport equation, and discusses the mathematical nature of the equation along with initial and boundary conditions. The discussion in the chapter is presented clearly and is quite easy to understand, even if one has no prior background in the area. Illustrations supplementing the discussion make the reading enjoyable.

Chapter 2 presents analytical solutions of the transport equation under different conditions. For one dimensional transport, solutions are provided for instantaneous source, continuous source, and finite-duration source. Two dimensional transport solutions include instantaneous source, continuous source, steady-state plume, and finite aquifer. Analytical solutions are supplemented with a concise and meaningful discussion.

Recognizing the limitations of analytical solutions, grid-based numerical methods are presented in Chapter 3. These methods include finite-difference method, finite-volume method, and finite-element method. The chapter discusses numerical schemes for time discretization of the transport equation first, such as explicit Euler method, implicit Euler method, Crank-Nicholson method, and higher order methods, the chapter then goes on to discuss mass balance for ID as well as 2D finite difference. Criteria for numerical stability as well as precision are included in the discussion. Finite-volume method is discussed next. Triangulation and dual grids, approximation functions and mass balance, upwind stabilisation, and incorporation of boundary conditions are included in the discussion. The discussion of the finite- element method includes Galerkin method, stabilisation, boundary conditions, and adaptive "ridding. The chapter is quite easy to follow and is clear and to-the-point in presentation.

Numerical methods based on particle tracking constitutes the subject matter of Chapter 4. Discussed in this method are the Bowline and travel-time method, characteristics method, and random-walk method. The chapter is concluded with a discussion on the comparison of these methods as well as the numerical methods presented in the previous chapter.

Chapter 5 discusses methods for obtaining solutions of systems of equations that appear in the methods presented in Chapters 3 and 4. These methods include direct solution, classical linear iterative methods, conjugate-gradient method, multigrid methods, and the Newton-Raphson method. The discussion of the methods is clear and concise.

Transport and reactions are presented in Chapter 6. Retardation, dual porosity model, multispecies models, coupling of transport and reactions, and an example application of multispecies simulation are included in the chapter. The discussion in the chapter is comprehensive.

Chapter 7 deals with transport in fractured media, including flow and transport in a single fracture, equivalent porous media approach, multidomain approach, discrete fracture approach, and modeling strategies. The chapter provides a good and comprehensive discussion.

On the whole the book is well written, is easy to follow, and concise. A significant amount of the literature cited in the text is older than 10 years. It would have been desirable if it had presented the inherent difficulties in dealing with solute transport in groundwater, especially from the point of view of uncertainties associated with mathematical formulations being employed these days and the resulting errors and the reliability of different formulations for different conditions. The book will serve as a good text for a course on solute transport either at the senior undergraduate level or the beginning graduate level. It would also be useful to have this book on one's bookshelf.

Vijay P. Singh

Journal of Hydrologic Engineering, Sept/Oct 2006, p. 512

Table of Contents top ↑

1 Transport processes and equation 1
1.1 Transport model 1
1.2 The transport equation 2
1.3 Transport processes 2
Advection - Molecular diffusion - Dispersion - Reactions
1.4 Derivation of the transport equation 15
1.5 Mathematical nature of the transport equation 19
1.6 Initial and boundary conditions 20
2 Analytical solutions 23
2.1 One-dimensional transport 24
Instantaneous source - Continuous source - Finite-duration source
2.2 Two-dimensional transport 27
Instantaneous source - Continuous source - Steady-state plume -
Semi-in nite aquifer
2.3 Other transport solutions 32
3 Grid-based numerical methods 33
3.1 Time discretization 33
The explicit EULER method - The implicit EULER method - The
CRANK-NICOLSON method - Higher-order methods
3.2 The nite difference method 36
Mass balance for Finite difference - Matrix equation in one dimension
- Criteria for numerical stability - Criteria for numerical precision
- Finite difference solution of the 2D transport equation - Mass
balance for 2D nite difference - Stability and precision of the 2D
solution
3.3 Finite volume method 52
Triangulation and dual grids - Approximation functions and mass
balance - Upwind stabilization - Incorporation of boundary conditions
3.4 Finite element method 61
GALERKIN method - Stabilization - Finite element boundary conditions
3.5 Adaptive gridding 66
Example
4 Numerical methods: Particle tracking 71
4.1 The owline and travel time method 71
4.2 Method of characteristics 75
Standard method of characteristics - Modified method of characteristics
4.3 Random-walk method 80
Theoretical basis - Ca-lculation of dispersion - Generation of
particle distributions - The random-walk method in two and three
dimensions - Computation of concentration distribution - Determination
of the ow eld
4.4 Comparison of methods 88
5 Solution of systems of equations 91
5.1 Direct solution 92
5.2 Classical linear iterative methods 94
The JACOBI and GAUSS-SEIDEL methods - Underrelaxed JACOBI method and
SOR method - ILU decomposition
5.3 The conjugate gradient method 100
Conjugate gradient method for symmetric matrices - Preconditioned
CG method for symmetric matrices - The GMRES method - The BiCGStab method
5.4 Multigrid methods 106
Smoothing - Correction for coarse grids - Prolongation and restriction
- Multigrid algorithm - Grid hierarchy - Computational and memory
requirements - Algebraic multigrid
5.5 The NEWTON-RAPHSON method 116
The NEWTON method for functions of a single variable -
The NEWTON method for systems of equations
6 Transport and reactions 121
6.1 Simple reaction models 121
Retardation - First-order decay
6.2 Dual-porosity model 126
Equations and parameters - Dual-porosity behavior -
Application of the dual-porosity model
6.3 Multispecies models 131
Carbonate system equilibrium - Cation exchange - Other equilibrium
models - Microbial redox reactions - Overview of existing multispecies
models
6.4 Coupling of transport and reactions 145
6.5 Example of a multispecies simulation 147
Hypothetical column - Results - Application of multispecies models
7 Transport in fractured media 159
7.1 Flow and transport in a single fracture 160
7.2 Equivalent Porous Medium approach 164
7.3 Multi-domain approach 164
Dual-domain approach - Dual-porosity approach
7.4 Discrete fracture approach 171
Analytical solutions for discrete fracture transport - Semi-analytical
solution for fracture networks - Numerical transport models for
fracture networks - Analogy between dual-porosity and discrete
fracture model
7.5 Modelling strategies 183
A References 187
B List of symbols 197
C Index 201

NULL top ↑

The authors, all of them active groundwater modelling professionals, focus on the detailed presentation of numerical methods commonly used in transport models. They provide practitioners with a sound theoretical basis for successfully planning and conducting transport model applications. Grid-based methods are presented, including explicit and implicit finite differences, finite elements and finite volume methods. Particle tracking techniques are also covered, among them the method of characteristics and the random-walk method.