Mathematical Modeling of Groundwater Pollution

Springer Science+Business Media, LLC

Ne-Zheng Sun

Mathematical Modeling of Groundwater Pollution With 104 Illustrations

Translation by Fan Pengfei and Shi Dehong

Originally published by Geological Publishing House, Beijing, People's Republic of China

Springer

Ne-Zheng Sun Civil and Environmental Engineering Department University of California Los Angeles, CA 90024 USA

Originally published as ;tt!rf*flJ~-~'*'tl:l\l!.'fIltj(fm:1J$ (Groundwater Pollution-Mathe­matical Models and Numerical Methods). © 1989 Geological Publishing House, Beijing, People's Republic of China. Translators: Fan Pengfei and Shi Dehong. Editor: Zhu Xiling.

Library ofCongress Cataloging-in-Publication Data Sun, Ne-Zheng.

Mathematical modeling of groundwater pollution / Ne-Zheng Sun. p. cm.

Inc1udes bibliographical references and index. ISBN 978-1-4757-2560-5 ISBN 978-1-4757-2558-2 (eBook) DOI 10.1007/978-1-4757-2558-2 1. Groundwater-Pollution-Mathematical models. I. Title.

TD426.s86 1995 628.1'68'015118-dc20 94-10680

Printed on acid-free paper.

© 1996 Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc. in 1996 Softcover reprint of the hardcover 1 st edition 1996

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher, Springer Science+Business Media., LLC except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrievaI, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

Production managed by Laura Carlson; manufacturing supervised by Jeffrey Taub. Typeset by Asco Trade Typesetting Ltd., Hong Kong.

9 8 7 6 5 4 3 2 1

ISBN 978-1-4757-2560-5

To the memory of my father Chi-Peng Sun

and my mother Xin-Ru Wang

Preface

Groundwater is one of the most important resources in the world. In many areas, water supplies for industrial, domestic, and agricultural uses are de­pendent on groundwater. As an "open" system, groundwater may exchange mass and energy with its neighboring systems (soil, air, and surface water) through adsorption, ion-exchange, infiltration, evaporation, inflow, outflow, and other exchange forms. Consequently, both the quantity and quality of groundwater may vary with environmental changes and human activities.

Due to population growth, and industrial and agricultural development, more and more groundwater is extracted, especially in arid areas. If the groundwater management problem is not seriously considered, over­extraction may lead to groundwater mining, salt water intrusion, and land subsidence. In fact, the quality of groundwater is gradually deteriorating throughout the world. The problem of groundwater pollution has appeared, not only in developed countries, but also in developing countries. Ground­water pollution is a serious environmental problem that may damage human health, destroy the ecosystem, and cause water shortage.

In the protection and improvement of groundwater quality, two chal­lenging problems have been presented: for uncontaminated aquifers, it is required to assess the potential dangers of pollution; for contaminated aquifers, it is required to draw up remediation projects. In both situations, we need a tool to predict the pollutant distribution in groundwater. Obviously, field experiments cannot serve this purpose. The only tool that we can use is mathematical modeling. In the past two decades, mathematical modeling techniques were extensively used in the study of mass and heat transport in groundwater and soil. Presently, we can simulate a three-dimensional multi­component transport in a multi-phase flow using a computer without any essential difficulty. Since simulation models can provide forecasts of future states of groundwater systems, the optimal protection or rehabilitation strat­egy may be found by incorporating a simulation model into a management model.

There are, however, several difficult problems in groundwater quality

vii

viii Preface

modeling that have not yet been adequately solved. The first one is called the "scale effect problem." The identified dispersivities may vary with the scale of experiment and the size of element of numerical discretization. The second one is called the "numerical dispersion problem." Sharp concentration fronts are difficult to simulate accurately using a numerical method. The third difficulty is caused by the "uncertainty enlargement problem." The uncer­tainties associated with hydraulic conductivities, porosities and head distri­butions may be enlarged and propagated to the calculation of velocity fields through Darcy's Law. Incorrect velocity distributions may cause large com­putational error in the determination of both advection and dispersion com­ponents of contaminant transport and fate. The fourth difficulty is caused by the "data insufficient problem." Tracer tests can only be carried out on a small region and it is difficult to observe concentration plumes in three­dimensional space. Generally, we do not have enough data for calibrating the mass transport model of regional problems. These difficulties make the modeling of groundwater quality more challenging than the modeling of groundwater flow. Although the importance of mathematical modeling in the study of groundwater quality problems is significant, the accuracy of a mass transport model and, thus, the reliability of management decisions derived from the model, are often questionable.

This book introduces all primary aspects of groundwater quality model­ing. The emphasis, however, is on numerical techniques. Besides introducing basic concepts, theories, methods and applications, special attentions are paid to three-dimensional models, model selection criteria, tracer test design, dispersion parameter identification, and reliability analysis. The overall purpose is to develop an applicable methodology for groundwater quality modeling. This book is designed to provide a course text at the graduate level. The materials are presented in such a manner that the book can also be used as a reference for hydrogeologists, geochemists and environmental engineers.

Chapter 1 is an introduction, in which the problem ofmodeling groundwa­ter pollution is depicted, and the relationships between simulation, parame­ter identification, and groundwater quality management are explained.

Advection-Dispersion equations (ADE) that can simulate multi-compo­nent transport in multi-phase flow are derived in Chapter 2. Hydrodynamic dispersion coefficients and other parameters in the ADE are defined. Various combinations of sink/source terms, initial conditions, and boundary condi­tions are listed.

Chapter 3 gives analytical solutions for some one-, two-, and three­dimensional advection-dispersion problems. Several often-used techniques for finding analytical solutions are introduced.

In Chapter 4, conditions of convergency and stability of finite difference methods for solving the ADE are presented. The phenomena of overshoot and numerical dispersion associated with finite difference solutions for prob-

Preface ix

lems with large Peclet numbers are then analyzed. The method of character­istics and the random walk method are also discussed.

Chapter 5 introduces the family of finite element methods and their varia­tions. In the solution of three-dimensional problems, the Galerkin finite element method, the mixed finite difference and finite element method, and the multiple cell balance method are discussed and compared.

Chapter 6 is devoted to advection dominated problems. After a Fourier analysis of numerical dispersion, various Eulerian, Lagrangian, and Eulerian­Lagrangian methods are given in detail.

Chapter 7 considers how to select and build a model for a practical ground­water quality problem. Models of simulating tracer tests in different scales are presented for the purpose of parameter identification. A coupled inverse problem of groundwater flow and mass transport is then presented and solved by a conjugate gradient method or a modified Gauss-Newton meth­od. The final section of this Chapter is a short introduction to the statistic theory of mass transport in porous media which is a rapidly developing field. Methods of evaluating the reliability of groundwater quality models are also included.

Chapter 8 discusses the main application fields of groundwater quality models. Prediction of groundwater pollution in saturated and unsaturated zones, simulation of mass transport in fractured aquifers, sea water intrusion, optimal design for aquifer remediation, and groundwater resources manage­ment are also discussed.

At the end of the text, there is a short conclusion which lists some open problems in this field. A FORTRAN program is given in Appendix B, which can be used to simulate contaminant transport either in steady-state or transient-flow fields.

The original edition of this book was written in Chinese and published by the Geological Publishing House of China in 1989. It was translated into English in 1991 by Mr. Fan Pengfei and Mr. Shi Dehong at the Hydrogeology Institute, Ministry of Geology and Mineral Resources of China. The English translation was corrected by Ms. Zhu Xiling, who was the director of the international exchange division, the Geological Publishing House of China, and modified by the author for teaching the course "Mathematical Modeling of Contaminant Transport in Groundwater," in the Civil Engineering De­partment, University of California, Los Angeles. At that time, exercises were added to each chapter. In 1994, the English manuscript was again revised to include some late developments in this field.

The author wishes to express heartfelt thanks to Dr. Susan D. Pelmulder, who read the entire manuscript and helped the author revise it for publica­tion. The manuscript was also read, in part, by Dr. Marshall W. Davert, Dr. William A. Moseley, Dr. Suresh Lingineni, Dr. Ming-Chin Jeng, and Dr. Ming-Jame Horng. They suggested many improvements. The author also wishes to thank Mrs. Cathy Jeng, secretary of the Civil Engineering Depart-

x Preface

ment, VCLA, who typed the first six chapters of the manuscript. My son, Yi-Shan Sun, typed Chapters 7 and 8, and checked all of the equations in the book. The author also acknowledges the work of Mr. Bi Lijun, editor of the Geological Publishing House of China, and Mr. Bernd Grossmann, the Vice President of Springer-Verlag, and the contributions of Ms. Elizabeth Sheehan, Editor, in the production of this book.

Los Angeles, California August 1995

NE-ZHENG SUN

Contents

Preface .................................................... vii

1 Introduction .............................................. 1 1.1 Groundwater Quality ..................................... 1 1.2 Groundwater Quality Management ......................... 2 1.3 Groundwater Modeling ................................... 4

2 Hydrodynamic Dispersion in Porous Media .................... 9 2.1 Physical Parameters ...................................... 9

2.1.1 Spatial Average Method ............................... 9 2.1.2 Fluid, Medium and State Parameters .................... 12

2.2 Phenomena and Mechanism of Hydrodynamic Dispersion ...... 20 2.2.1 Hydrodynamic Dispersion Phenomena ................... 20 2.2.2 Mechanisms of Hydrodynamic Dispersion ................ 21 2.2.3 Mass Transport in Porous Media ....................... 23

2.3 Mass Conservation and Convection-Diffusion Equations in a Fluid Continuum ........................................ 24

2.3.1 Diffusive Velocities and Fluxes .......................... 24 2.3.2 Mass Conservation Equation of a Component ............ 25 2.3.3 Convection-Diffusion Equations in a Fluid Continuum ..... 26

2.4 Hydrodynamic Dispersion Equations ....................... 27 2.4.1 The Average of Time Derivatives ........................ 27 2.4.2 The Average of Spatial Derivatives ...................... 29 2.4.3 Advection-Dispersion Equations in Porous Media ......... 30 2.4.4 The Integral Form of Hydrodynamic Dispersion Equations. 32

2.5 Coefficients of Hydrodynamic Dispersion .................... 33 2.5.1 Coefficients of Longitudinal Dispersion and Transverse

Dispersion ........................................... 33 2.5.2 Coefficients of Mechanical Dispersion and Molecular

Diffusion ............................................ 36 2.6 Extensions and Subsidiary Conditions of the Hydrodynamic

Dispersion Equation ...................................... 39

xi

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

Contents xiii

5.3 Finite Element Methods for Three-Dimensional Problems ...... 125 5.3.1 The Galerkin Finite Element Method .................... 125 5.3.2 Triangular Prism Elements and Associated Basis Functions . 132 5.3.3 A Mixed Finite Element-Finite Difference Method ......... 135 5.3.4 The Multiple Cell Balance Method ...................... 138

5.4 The Solution of Finite Element Systems ..................... 143 5.4.1 Features of Finite Element Systems and Direct Solutions ... 143 5.4.2 Point Iteration Methods ............................... 144 5.4.3 Block and Layer Iteration Methods ..................... 146

6 Numerical Solutions of Advection-Dominated Problems .......... 149 6.1 Advection-Dominated Problems. . . . . . .. . . . . . . . . . . . . . . . . . . .. 149

6.1.1 Fourier Analysis ofNumerical Errors .................... 149 6.1.2 Eulerian and Lagrangian Reference Frames ............... 154

6.2 Upstream Weighted Methods .............................. 155 6.2.1 Upstream Weighted Finite Difference Methods ............ 155 6.2.2 Upstream Weighted Finite Element Methods ............. 158 6.2.3 The Upstream Weighted Multiple Cell Balance Method .... 163

6.3 Moving Coordinate System and Moving Point Methods ....... 172 6.3.1 Moving Coordinate System Methods .................... 172 6.3.2 Element Deformation Methods ......................... 175 6.3.3 Moving Point Methods ................................ 176

6.4 The Modified Methods of Characteristics .................... 177 6.4.1 The Single Step Reverse Method ........................ 177 6.4.2 The Hybrid Single Step Reverse-Moving Point Method ..... 180 6.4.3 The Hybrid Moving Point-Characteristics Finite Element

Method ............................................. 182

7 Mathematical Models of Groundwater Quality ................ 187 7.1 The Classification of Groundwater Quality Models ............ 187

7.1.1 Hydrodynamic Dispersion Models ...................... 187 7.1.2 Coupled Equations ofGroundwater Flow and Mass

Transport ........................................... 191 7.1.3 Pure Advection Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 193 7.1.4 Lumped Parameter Models ............................ 196 7.1.5 Criteria of Model Selection ............................. 199

7.2 Model Calibration and Parameter Estimation ................ 201 7.2.1 Parameter Identification of Advection-Dispersion Equations 201 7.2.2 Field Experiments for Determining Dispersivities .......... 204 7.2.3 The Relationship Between Values ofDispersivity and Scales

of Experiment ........................................ 213 7.2.4 Determination of Mean Flow Velocities .................. 217 7.2.5 Identification of Retardation Factor and Chemical Reaction

Parameters .......................................... 218 7.2.6 Identification of Pollutant Sources ....................... 219 7.2.7 Computer Aided Design for Field Experiments ............ 220

xiv Contents

7.3 Coupled Inverse Problems of Groundwater Flow and Mass Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

7.3.1 Definition of Coupled Inverse Problems ................. 223 7.3.2 Variational Sensitivity Analysis ......................... 226 7.3.3 Identifiability ........................................ 231 7.3.4 Experimental Design .................................. 234

7.4 Statistic Theory and Uncertainty Analysis ................... 235 7.4.1 The Statistic Theory of Mass Transport in Porous Media ... 235 7.4.2 The Heterogeneity of Natural Formations ................ 236 7.4.3 Stochastic Advection-Dispersion Equations and

Macrodispersivities ................................... 237 7.4.4 Uncertainties of Groundwater Quality Models ............ 238 7.4.5 Conditional Simulations and Stochastic Inverse Problems .. 243

8 Applications of Groundwater Quality Models .................. 247 8.1 Simulation and Prediction of Groundwater Pollution ......... 247

8.1.1 The General Procedure of Studying Groundwater Pollution Problems ........................................... 247

8.1.2 Groundwater Pollution of Saturated Loose Aquifers ....... 250 8.1.3 Groundwater Pollution of Saturated-Unsaturated Aquifers . 253 8.1.4 Groundwater Pollution of Fractured Aquifers ............ 256

8.2 Seawater Intrusion ....................................... 262 8.2.1 The Problem ofSeawater Intrusion ..................... 262 8.2.2 Fresh Water-Sea Water Interfaces ...................... 264 8.2.3 Numerical Methods for Determining the Location of

Interfaces ........................................... 269 8.2.4 Determination of the Transition Zones .................. 271

8.3 Groundwater Quality Management Models .................. 275 8.3.1 Groundwater Hydraulic Management Models ............ 275 8.3.2 Management of Groundwater Pollution Sources .......... 280 8.3.3 Project Models of Groundwater Quality Management ..... 284 8.3.4 Conjunctive Use and Water Resources Planning .......... 287 8.3.5 Remediation of Polluted Aquifers ....................... 288 8.3.6 The Reliability of Management Models .................. 289 8.3.7 Economic and Political Models in Groundwater

Management ........................................ 291

Conclusions ................................................ 295

Appendix A The Related Parameters in the Modeling Mass Transport in Porous Media ......................... 297

Appendix B A FORTRAN Program for Solving Coupled Groundwater Flow and Contaminant Transport Problems ....................................... 300

B.1 Features and Assumptions .............................. 300 B.2 Program Structure .................................... 301

Contents xv

B.3 Input Data Files ....................................... 301 B.4 Output Data Files ..................................... 305 B.5 Floppy Disk and Demo Problem ......................... 306 B.6 Source Programs ...................................... 307

References .................................................. 355

Index ...................................................... 370