Table Of ContentGroundwater Transport:
Handbook of
Mathematical
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Groundwater Transport:
Handbook of
Mathematical Models
Iraj Javandel, Christine Doughty,
and Chin-Fu Tsang
AMERICAN GEOPHYSICAL UNION
WASHINGTON, D.C.
Publishedu nder the aegis of the American Geophysical Union's Water Resources
Monograph Board: John D. Bredehoeft, Chairman; David Dawdy, Charles W.
Howe, Thomas Maddock III, Helen J. Peters, Eric Wood, members.
Groundwater Transport:
Handbook of Mathematical Models
Library of CongressC atalogingi n Publication Data
Main entry under title:
Groundwater transport.
(Water resources monograph series; 10)
Bibliography: p.
1. Water, Underground--Pollution--Mathematical models--
Handbooks, manuals, etc. 2. Groundwater flow--Mathematical
models--Handbooks, manuals, etc. I. Javandel, Iraj.
II. Doughty, Christine. III. Tsang, Chin-Fu. IV. Series:
Water resources monograph; 10.
TD426.G77 1984 628.1'68 84-6452
ISBN 0-87590-313-4
ISSN 0270-9600
Copyright 1984 by the American Geophysical Union
2000 Florida Avenue, N.W., Washington, DC 20009
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First Printing: 1984
Second Printing: 1985
Third Printing: 1987
Fourth Printing: 1990
Fifth Printing: 1995
Printed in the United States of
CONTENTS
Preface ......................................................................................................................... vii
1. INTRODUCTION ................................................................................................ 1
1.1 Statement of the Problem ........................................................................... 1
1.2 Objective and Approach ............................................................................. 1
1.3 Sources of Contamination .......................................................................... 2
1.3.1 Industrial Wastewater Impoundments ......................................... 2
1.3.2 Land Disposal of Solid Wastes ..................................................... 4
1.3.3 Waste Disposal Through Wells ..................................................... 5
1.4 Organization of This Handbook ................................................................7
2. ANALYTICAL METHODS ................................................................................. 9
2.1 Governing Equations ..................................................................................9
2.1.1 Initial and Boundary Conditions ................................................ 13
2.2 One-Dimensional Problems ..................................................................... 14
2.2.1 Specific Cases ............................................................................... 16
2.3 Two-Dimensional Problems ..................................................................... 18
2.3.1 Two-Dimensional Plane Dispersion Model .............................. 18
2.3.2 Dispersion in Radial Flow .......................................................... 20
2.3.3 Approximate Solution to Dispersion in
Radial Flow ................................................................................... 23
2.4 Applications ............................................................................................... 24
2.4.1 Example I ..................................................................................... 25
2.4.2 Example 2 ..................................................................................... 27
2.4.3 Example 3 ..................................................................................... 31
2.5 Advantageso f Analytical Methods .......................................................... 34
2.6 Limitations of Analytical Methods ..........................................................3 4
3. SEMIANALYTICAL METHODS ..................................................................... 35
3.1 Theory ........................................................................................................ 35
3.1.1 Uniform Flow ............................................................................... 36
3.1.2 Sources and Sinks ........................................................................ 37
3.1.3 Doublet ......................................................................................... 39
3.2 Combination of Uniform Flow With Point Sources and
Sinks .............................................................................................. 40
3.3 Combination of Uniform Flow With a Finite
Radius Source ............................................................................... 42
3.4 Use of a Single ProducingW ell for Monitoring
Purposes ........................................................................................ 44
3.5 Applications ............................................................................................... 46
3.5.1 Example I ..................................................................................... 46
3.5.2 Example 2 ..................................................................................... 48
3.5.3 Example 3 ..................................................................................... 57
3.5.4 Example 4 ..................................................................................... 57
3.6 Advantages of Semianalytical Methods .................................................. 65
3.7 Limitations of SemianalyticalM ethods ..................................................6 5
4. NUMERICAL METHODS ................................................................................ 69
4.1 Spatial Approximations ............................................................................ 69
4.1.1 Finite Difference Method ............................................................ 69
4.1.2 Integrated Finite Difference Method ..........................................7 1
4.1.3 Finite Element Method ............................................................... 71
4.1.4 Flow Path Network Method ...................................................... 72
4.2 Temporal Approximations .......................................................................
4.2.1 Implicit Equations ....................................................................... 73
4.2.2 Coupling Solution Schemes. .......................................................7. 4
4.3 Matrix Solvers ........................................................................................... 74
4.3.1 Direct Elimination Methods ....................................................... 74
4.3.2 Iterative Methods ......................................................................... 75
4.4 Computer Codes ........................................................................................ 76
4.5 Example of the Use of a SophisticatedN umerical
Model ............................................................................................ 77
4.6 Advantageso f Numerical Methods ........................................................8. 6
4.7 Limitations of Numerical Methods ......................................................... 87
5. A DISCUSSION ON CHOICE OF METHODS AND DATA NEEDS ......... 89
5.1 Data Needs ................................................................................................ 89
5.1.1 Analytical and Semianalytical Methods ..................................... 89
5.1.2 Numerical Methods ..................................................................... 90
5.2 Selection of Method .................................................................................. 91
6. CONCLUSIONS .................................................................................................. 93
APPENDICES:
A. Tables of Dimensionless Concentration for
One-Dimensional Contaminant Transport in Aquifers With
One-Dimensional Uniform Flow .................................................................... 95
B. ODAST: A Computer Program for Evaluation of the
Analytical Solution for One-Dimensional Contaminant
Transport ......................................................................................................... 129
C. Tables of Dimensionless Concentration for
Two-Dimensional Contaminant Transport in Aquifers With
One-Dimensional Uniform Flow .................................................................. 133
D. TDAST: A Computer Program for Evaluation of the
Analytical Solution for Two-Dimensional Contaminant
Transport ......................................................................................................... 159
E. Tables of Dimensionless Concentration for
Dispersion in Radial Flow ............................................................................. 165
F. LTIRD: A Computer Program for a Semianalytical
Solution to Radial Dispersion in Porous Media .......................................... 167
G. Tables of Error Function .................................................................................. 171
H. RESSQ: A Computer Program for Semianalytical
Contaminant Transport ................................................................................. 175
I. RT: A Computer Program for Mapping Concentration
Distribution in an Aquifer Based on a Time Series
Data Collection Concept ................................................................................ 205
J. Control of the Movement of a Fluid Plume by Injection
and Production Procedures ............................................................................ 213
NOTATION .............................................................................................................. 223
REFERENCES ..........................................................................................................
PREFACE
Concernso ver groundwaterp ollution have resulted in the passageo f legislation
during the last decade calling for pollution control and remedial measures to
ensure proper drinking water quality. There are two main types of groundwater
pollution caused by man: (1) pollution causedb y the use of pesticides,h erbicides,
and fertilizers over agricultural lands, where the source of contamination covers a
relatively large area, and (2) pollution caused by industries and municipalities,
which is generally more localized. For the secondt ype, becauset he contamination
in the groundwater is localized, the design of any remedial measure requires
knowledge of the extent of the contaminant plume. Various mathematical
methods may be used for estimating the size, shape, and development of a local-
ized contaminant plume. It is the need for these mathematical methodologiest hat
forms the main impetus for the preparationo f this monograph.
The study of solute transport in groundwater is a relatively old subject. Ini-
tially, various analytical methods were derived for very simple cases. Then a
number of semianalytical methodologiesw ere developed that could be applied
with the help of simple computers. More recently, a number of numerical
approaches have been used to code sophisticated numerical models that can be
used for more complicated situations. The present monograph attempts to put
together selected analytical solutions, semianalytical methods, and numerical
approaches and discuss their strengths and possible pitfalls in application.
Comprehensivet ables and computer program listings are included in the appen-
dices. On the one hand, we hope that the monograph can be readily used by
groundwater hydrologistst o study the extent and development of solute plumes in
aquifers. On the other hand, we hope that the monographa lso gives a brief over-
view of the subjectt o encourager eaderst o embark on further researcht o enlarge
the mathematical methodologiesa vailable for handling this important subject.
The material in this monograph was originally prepared for the U.S. Environ-
mental Protection Agency (EPA), Robert S. Kerr Environmental ResearchL abora-
tory (RSKERL), in part pursuant to Interagency Agreement AD 89F 2A 175
between the U.S. EPA and the U.S. Department of Energy and in part under U.
S. Department of Energy contract DE-AC03-76SF00098. Jack W. Keeley of
RSKERL provided technical guidance during the course of the study and Joseph
F. Keely of RSKERL reviewed the manuscript and provided constructive com-
ments. We acknowledget heir assistancea s well as their encouragementt o publish
this monograph. Although publication of this monograph has been approved by
EPA, their approval does not signify that the contents necessarilyr eflect the views
and policies of EPA, nor does the mention of trade names or commercial products
constitute endorsement or recommendation for use.
Finally, the authors gratefully acknowledgeJ . S. Y. Wang for his assistancein
preparing Chapter 4, J. Noorishad for reviewing the manuscript,P . Fuller for cal-
culations and plotting, and L. Armetta, J. Grant, and S. Kerst for manuscript
organization and word processing.
I. JAVANDEL
C. DOUGHTY
C. F. TSANG
Earth Sciences Division
Lawrence Berkeley Laboratory
University of California
Berkeley, California 94720
1 Introduction
1.1. Statement of the Problem
Enormous amounts of waste materials, potentially hazardoust o groundwater,
are stored or disposedo f on or beneath the land surface. In many instances,c on-
ruminants such as organic and inorganic chemicalsa nd bacteriologicals ubstances
are found in groundwater, indicating that many of the waste disposal sites com-
municate with undergroundw ater resources.
The overall goal of the Hazardous Substancesa nd Waste ResearchP rogram
(HSWRP) is to provide the scientific and technical expertise necessaryt o enable
the responsible personnel to discover, control, and clean up hazardous substances
and oil that have been released to the environment from various sources. In
respect to this overall plan the present handbook attempts to provide a useful
guide by which field personnel can become familiar with the state-of-the-art
methodology in modeling contaminant transport in the subsurface. This guide
will enable users to make initial estimationso f conruminantt ransport at a given
site and, if the need arises and the dam justify it, to select and to make use of
sophisticatedn umerical models.
1.2. Objective and Approach
The objective of this work is to review, select,c ompile, and demonstrates ome
of the best and most usable mathematical methods for predicting the extent of
subsurfacec ontamination in a format useful to field response personnel. The
methods presented range from simple analytical and semianalytical solutions to
complex numerical codes. Derailed discussionso f the assumptionsu nderlying
application of the methods are given. Primary emphasisi s on the use of simple
formulas and comprehensiver obles so that the handbook is practically oriented
and readily usable as a guide in the field.
Three different levels of complexity and sophisticationa re used to addresst he
prediction of conruminantt ransporti n groundwater. These levels are as follows.
1. Simple analytical methods based on the solution of applicable differential
equationsa re used, making a simplified idealization of the field and giving qualita-
tive estimateso f the extent of conruminant transport.
2. Semianalytical methods based on the concept of complex velocity potential
are used, providing the streamlinesf or steady state fluid flow and the correspond-
ing conruminant movement in the presenceo f an arbitrary number of sourcesa nd
sinks. An average geologicale nvironment is assumed and a schematic chemical
retardation factor is considered.
3. Sophisticatedn umerical models are used, accountingf or complex geometry
and beterogenousm edia, as well as dispersion,d iffusion, and chemical retardation
processes( e.g., sorption, precipitation, radioactive decay, ion exchange, degrada-
tion).
At the first two levels,a ppropriatem ethodsa re given, computerp rograml istings
and their user's guides are attached, and comprehensive robles and figures
2 Groundwater Transport: Handbook of Mathematical Models
presented. For the third level, different numerical approachesa re introduced, and
a number of presently available numerical codes are tabulated, based on recent
surveysb y various groups. These tables include model names, key characteristics,
and the personnel involved in their development. As an illustration, one of these
sophisticatedm odels is describeda nd an example of its application is demon-
strated.
1.3. Sources of Contamination
A report to Congressb y the EnvironmentalP rotectionA gencyi n 1977 conveyed
thato ver 17 X 106w aste-dispofsaacl ilitiesin the UnitedS tatesa ree mplacinagt
least6 .5 X 109m 3 of contaminatleiqdu idi ntot heg rouneda chy ear.A lthough
16.6X 106o f theses itesa red omestisce pticta nkst,h eya reo nlyr esponsibfoler
abou3t X 109m 3 of effluentT. heo ther4 00,000d isposfaal cilitieasr er esponsible
for the remaining3 .5 X 109m 3 . Thesef acilitiesd, escribeidn detaile lsewhere
[U.S. EnvironmentalP rotectionA gency,1 9771i,n volve the following:( 1) industrial
wastewateri mpoundments,( 2) land disposals itesf or solid wastes,( 3) wasted ispo-
sal through wells, (4) septic tanks and cesspools,( 5) collection, treatment, and
disposalp lants for municipal wastewater,( 6) land spreadingo f sludges(,7 ) brine
disposal from petroleum exploration and development, (8) disposal of mine
wastes,( 9) agricultural land leachate, (10) chemical spills, and (11) leaks from
undergroundc hemical storagef acilities. In the following section,w e shall briefly
discusss ome of the important sourceso f the groundwaterc ontamination listed
above.
1.3.1. Industrial WastewaterI mpoundments
Industrial wastewater impoundments are natural or artificial depressionsin the
ground used for the temporary or permanent storage and/or disposal of liquid
wastes. The surfacea rea of these impoundments varies from a few squarem eters
to several hectares. Their depth is generally small to improve evaporation effi-
ciency. It has been reported [Josephson1, 982] that about 70% of the industrial
impoundments have no impermeable liner. Therefore the hazardous waste can
easily infiltrate downward toward underlying groundwaters. For those which are
lined, the general rule has been to designt he coefficient of permeability of the liner
to be lesst han 10- 7 cm/s. The correspondirnagte o f leakagteB rs uchli nersis
about3 1,000m 3/yrf rome achs quarek ilometeor f the impoundmesnut rfaceI.t
has been reported [Anderson,1 9821t hat some of the organicc hemicalsc ontained
in such impoundments destructively increase the permeability of the clay liners,
leading to much greater leakagef rom the impoundments.
Many different potentially hazardous substancesa re available from industrial
waste and are discharged into these impoundments. Table 1 presentsa list of
componentso f wastewaterf rom different industriesh aving significantp otential for
pollution of groundwater [U.S. EnvironmentalP rotectionA gency, 1973]. Included
among the potential contaminants are chlorinated solvents,a rsenic,m ercury, lead,
cyanide, chromium, uranium, and many other toxic organic and inorganicm ateri-
als. Since the maximum allowable concentration of these substancesin drinking
water is often on the order of a few parts per billion (ppb), it is obvioust hat even
very slight leakage from industrial wastewater impoundments can lead to serious
incidents of groundwater
Introduction 3
TABLE 1. Industrial wastewaterc omponentsh aving or indicatings ignificant
groundwater contamination potential
Pulp and Paper Industry
Ammonia Heavy metals pH TDS
COD Nutrients Phenols TOC
Color (nitrogen Sulfite
and phosphorus)
PetroleumR efining Industry
Ammonia Cyanide Odor TDS
Chloride Iron pH TOC
Chromium Lead Phenols Total Phosphorus
COD Mercaptans Sulfate Turbidity
Color Nitrogen Sulfide Zinc
Copper
Steel Industries
Ammonia Cyanide Phenols Tin
Chloride Iron Sulfate Zinc
Chromium pH
OrganicC hemicalsI ndustry
COD pH TDS Total Nitrogen
Cyanide Phenols TOC Total Phosphorus
Heavy metals
Inorganic Chemicals,A lkalies, and Chlorine Industry
Acidity/ Chlorinated Fluoride Sulfate
Alkalinity Benzenoids Iron TDS
Aluminum and Polynuclear Lead Titanium
Arsenic Aromatics Mercury TOC
Boron Chromium Phenols Total Phosphorus
Chloride Cyanide
COD
PlasticM aterials and SyntheticsI ndustry
Ammonia COD Organic Phosphorus
Chlorinated Cyanide Nitrogen Sulfate
Benzenoids Mercaptans pH TDS
and Polynuclear Nitrate Phenols Zinc
Aromatics
Nitrogen Fertilizer Industry
Ammonia COD pH Sulfate
Calcium Iron, Total Phosphate TDS
Chloride Nitrate Sodium Zinc
Chromium Organic Nitrogen
PhosphateF ertilizer Industry
Acidity Fluoride Nitrogen Sulfate
Aluminum Iron pH TDS
Arsenic Mercury Phosphorus Uranium
Calcium
COD Carbon oxygen demand
TOC Total organic carbon
TDS Total dissolved solids
From U.S. EnvironmentaPlr otectioAng ency
4 Groundwater Transport: Handbook of Mathematical Models
1.3.2. Land Disposal of Solid Wastes
Solid waste land disposal occurs as a result of several types of operations:
dumps, landfills, sanitary landfills, and securedl andfills [U.S. Environmental Pro-
tection Agency, 1977]. A dump is an uncovered disposal site where solid or liquid
wastes are deposited. If the wastes are periodically covered with natural soils, a
landfill is created. Sanitary landfills are sitesw here solid wastesa re disposedo f by
compactingt he waste and coveringi t at the end of each operatingd ay to minimize
environmental hazards. If efforts are made to prohibit contaminant movement
between the waste and the surrounding environment (particularly the groundwa-
ter), it is called a secured landfill [Farb, 1978]. Secured landfills are generally
designedt o accept highly toxic waste and are supposedt o be continuouslym oni-
tored.
Disposing of waste in dumps and landfills is a very common practice while the
use of true sanitary landfills is rare [U.S. Environmental ProtectionA gency, 19771.
Securedl andfills are still at the experimental stage. Four out of five such secured
landfills, constructedd uring recent years in the state of New Jersey,h ave experi-
enced operational problems [Montage, 1982]. It is estimated that about 20,000
land disposal sites accommodate municipal wastes in the United States. Most of
these facilities are open dumps that are poorly sited and operated, yet most have
received some industrial wastes [U.S. Environmental Protection Agency, 1977].
The number of privately owned industrial land disposal sites is not accurately
known, but they are suspectedt o outnumber municipal landfills.
The mechanism of groundwater contamination by solid waste land disposal
facilities is mainly through the generationo f leachate,w ith subsequendt ownward
movement to underlying groundwaters. Leachate generation is due in part to pre-
cipitation which percolatest hrough the solid waste, dissolves,a nd carries out the
soluble components of the waste. This liquid, together with any liquid waste
placed in the fill and other liquids coming from waste decomposition,c onstitutes
the leachate. The volume of leachateg enerateda t each period of time dependso n
the availability of moisture within the waste. Since the contribution of moisture
from precipitation is generally essentiali n forming leachate, one might expect that
a dump located in a humid area would generate the most leachate. Furthermore,
the groundwater level in humid areas is generally much shallower than in arid or
semiarid locations,w hich resultsi n a greater risk of groundwaterc ontamination by
an uncoveredl and disposals ite in a humid area. Unless a site of this type is prop-
erly designeda nd located, the site receives not only all of the precipitation falling
directly on the site but additional inflow from adjacent surfacer unoff. Since the
rate of evaporation in humid areas is relatively low, the available volume of water
for leachateg enerationm ay be extremely high.
Secureda nd sanitary landfills, on the other hand, may be designeds o that liquid
wastes are not allowed, the inflow of surface runoff is not permitted, the site is
properly lined, and generated leachate is collected and removed from the site.
These features minimize the risk of groundwater contamination. Unfortunately,
the vast majority of the land disposals itesi n operation do not have thesef eatures.
For example, of the 18,500 municipal land disposal sites operating in 1974, only
about 20 sites were lined and only 60 sites had leachate treatment facilities [U.S.
Environmental Protection Agency, 1977]. In regard to industrial land disposal
sites, little information is available due to restricted accesst o sites and records,b ut
they are expectedt o differ little from municipal sites.
The composition of leachateg enerateda t an individual site is clearly a function
of the type of waste depositedi n that location. The composition of leachate
