>EPA
           United States
           Environmental Protection
           Agency
           Robert S. Kerr Environmental
           Research Laboratory
           Ada OK 74820
Centeryor Environmental
Research Information
Cincinnati OH 45268
           Technology Transfer
                        CERi-87-28
Seminar on
Transport and Fate of
Contaminants in the
Subsurface

Background Information

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                             CONTENTS


I.      BACKGROUND INFORMATION ON GROUND-WATER MODELING

       Ground-Water Modeling:  An Overview

       Ground-Water Modeling:  Mathematical Models

       Ground-Water Modeling:  Numerical Models

       Optimizing Pumping Strategies for Contaminant Studies and
Remedial Actions

II.    QUALITY ASSURANCE IN GROUND-WATER MODELING

       Quality Assurance in Computer Simulations of Ground-Water
Contamination

       Representation of Individual Wells in Two Dimensional Ground-
Water Modeling

       Remedial Actions Under Variability of Hydraulic Conductivity

       A New Annotation Database for Ground-Water Models

III.   AVAILABILITY AND DOCUMENTATION OF MODELS

       Technology Transfer in Ground-Water Modeling:  The Role of
the International Ground-Water Modeling Center (IGWMC)

       Available Solute Transport Models for Ground Water and Soil
Water Quality Management

       The Fundamentals of Geochemical Equilibrium Models with a
Listing of Hydrochemical Models that are Documented and Available

       Price list of Publications and Services Available from IGWMC

IV.    U.S. EPA GROUND-WATER MODELING POLICY STUDY GROUP

       Report of Findings and Discussions of Selected Ground-Water
Modeling Issues

V.     THE USE OF MODELS IN MANAGING GROUND-WATER PROTECTION PROGRAMS

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Ground-Water  Modeling: An Overview
by James W-. Mercer and Charles R. Faust
                  ABSTRACT
     Ground-water modeling is a tool that can help analyze
many ground-water problems. Models are useful for
reconnaissance studies preceding field investigations, for
interpretive studies following the field program, and for
predictive studies to estimate future field behavior. In
addition to these applications, models are useful for
studying various types of flow behavior by examining
hypothetical aquifer problems. Before attempting such
studies, however, one must be familiar with ground-water
modeling concepts, model usage, and modeling limitations.

               INTRODUCTION
     The use of aquifers  is increasing as both a
source of water supply and a medium for storing
various hazardous wastes. As this usage expands, our
knowledge of ground-water systems must also
expand. Numerical ground-water modeling is a tool
that can aid in studying ground-water problems and
can help increase our understanding of ground-water
systems. Numerical models have been extensively
used for ground-water analysis since the mid-1960s,
yet confusion and misunderstanding over their
application still exists. As a result, some hydrologists
have become disillusioned and have overreacted,
concluding that models are worthless. At the other
extreme are those who have been willing to accept
any model results, regardless of whether or not
they make hydrologic sense.
     This is the first in a series of papers on ground-water
modeling.
     bGeoTrans, Inc., P.O. Box 2550, Reston, Virginia
22090.
     Discussion open until September 1, 1980.
     The purposes of this series of papers are to
introduce the basic concepts of ground-water
modeling and to show how the various types of
models can be effectively applied. We discuss
ground-water flow modeling as well as transport
modeling of both  heat and hazardous wastes. Most
of this material  is  available in the literature (e.g.,
van Poolen, et al,  1969; and Coats, 1969), but it
is usually directed toward an audience assumed to
be familiar with petroleum terminology or the
mathematical terminology of numerical analysis.
Our intent  is to consolidate  this material into a
form that is meaningful to the experienced, though
not mathematical, ground-water hydrologist. In
so doing, we hope to eliminate some of the
misunderstanding associated with ground-water
modeling.

           MODELING  APPROACHES
     Simulation of a ground-water system refers to
the construction and operation of a model whose
behavior assumes  the appearance of the actual
aquifer behavior.  The model can be physical (for
example, a laboratory sandpack), electrical analog,
or mathematical.  Other model divisions may be
found in Karplus  (1976) and Thomas (1973). A
mathematical model is simply a set of equations
which, subject to  certain assumptions, describes
the physical processes active in the aquifer. While
the model itself obviously lacks the detailed  reality
of the ground-water system, the behavior of a valid
model approximates that of the aquifer.
Mathematical models may be deterministic,
statistical, or some combination of the two.  In this
discussion, we restrict ourselves to deterministic

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               [CONCEPTUAL MODEL ]
                MATHEMATICAL MODEL
  ANALYTICAL MODEL
   SIMPLIFY EQUATION SO
   THAT SOLUTIONS MAY
   BE OBTAINED BY
   ANALYTICAL METHODS
NUMERICAL MODEL
APPROXIMATE EQUATIONS
NUMERICALLY RESULTING
M A MATRIX EQUATION
THAT MAY BE SOLVED
USING A COMPUTER
Fig. 1. Logic diagram for developing a mathematical model.
models, that is, those that define cause and effect
relationships based on an understanding of the
physical system.
     The procedure for developing a deterministic,
mathematical model of any physical system can be
generalized as shown in Figure 1. The first step is
to understand the physical behavior of the system.
Cause-effect relationships are determined and a
conceptual model of how the system operates is
formulated. For ground-water flow, these
relationships are generally well known, and are
expressed using concepts such as hydraulic gradient
to indicate flow direction. For the movement of
hazardous wastes, these relationships, especially
those involving physical-chemical behavior, are only
partially understood.
     The next step is to translate the physics into
mathematical terms, that is, make appropriate
simplifying assumptions and develop the governing
equations. This constitutes the mathematical model.
The mathematical model for ground-water flow
consists of a partial differential  equation together
with appropriate boundary and initial conditions
that express conservation of mass and that
describe continuous variables (for example,
hydraulic head) over the region of interest. In
addition, it entails various phenomenological
"laws" describing the rate processes active in the
aquifer. An example is Darcy's law for fluid flow
through porous media; this is generally used to
express conservation of momentum. Finally,
various assumptions may be invoked such as those
of one- or two-dimensional flow and artesian or
water-table conditions.
     For solute (e.g., hazardous wastes) and heat
transport, additional partial differential equations
with appropriate boundary and initial conditions
are required to express conservation of mass for
the chemical species considered, and conservation
of energy, respectively. Examples of corresponding
phenomenological relationships are Pick's law for
chemical diffusion and Fourier's law for heat
conduction.
     Once the mathematical model is formulated,
the next step is to obtain a solution using one of two
general approaches. The ground-water flow equation
can be simplified further, for example, assuming
radial flow and infinite aquifer extent, to form a
subset of the general equation that is amenable to
analytical solution. The equations and solutions
of this subset are referred to as analytical models.
The familiar Theis type curve represents the solution
of one such analytical model.
     Alternatively, for problems where the
simplified analytical models no longer describe the
physics of the situation, the partial differential
equations can be approximated numerically, for
example, with finite-difference techniques or with
the finite-element method. In so doing, one replaces
continuous variables with discrete variables that
are defined at grid blocks (or nodes). Thus, the
continuous differential equation, defining hydraulic
head everywhere in an aquifer, is replaced by a
finite number of algebraic equations that defines
hydraulic head at specific points. This system of
algebraic equations is generally solved using matrix
techniques. This approach constitutes a numerical
model, and generally, a computer program is written
to solve the equations on a digital computer.
     Probably the most frequent application of
ground-water models is that of history matching
and prediction of site-specific aquifer behavior.
Of the various types of models discussed, the
numerical model offers the most general tool for
simulating aquifer behavior. Physical models
usually offer the most intuitive insight into
aquifer behavior, but are limited in application
(once constructed), and have the difficulty of
scaling results to field level. Electric analog models
can be applied to field problems, but are usually
site-specific and expensive to construct. Deter-
ministic mathematical models (both analytical
and numerical) retain a good measure of physical
insight while permitting a larger class of problems to
be considered with the same model. Analytical
methods, such as type curve analysis, are relatively
easy to use. Numerical models, although more
difficult to apply, are not limited by many of the
simplifying assumptions necessary for the analytical
methods. Finally, purely statistical methods are
useful in classifying data and describing poorly
understood systems, but generally offer little
physical insight.
     Each type of model has both advantages and
disadvantages. Consequently, no single approach

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should be considered superior to others for all
applications. The selection of a particular approach
should be based on the specific aquifer problem
addressed. Whichever approach is taken, the final
step in modeling a ground-water flow system is to
translate trie (mathematical) results back to their
physical meanings. In addition, these results must
be interpreted in terms of both their agreement
with reality and their effectiveness in answering
the hydrologic questions that motivated the model
study.

     TYPES OF GROUND-WATER MODELS
     Four general types of ground-water models are
listed in Figure 2. The problem of water supply is
normally described by one equation, usually in
terms of hydraulic head. The resulting model
providing a solution for this equation is referred to
as a ground-water flow model. If the problem
involves water quality, then an additional
equation (s) to the ground-water flow equation must
be solved  for concentration (s) of the chemical
specie (s). Such a model is referred to as a solute
transport  model. Problems involving heat also
require an equation in addition to the ground-water
flow equation, similar to the solute transport
equation, but now in terms of temperature.
This model is referred to as a heat transport model.
Finally, a deformation  model combines a ground-
water flow model with a set  of equations that
describe aquifer deformation.
     Ground-water flow models have been most
extensively used for such problems as regional
aquifer studies, ground-water basin analysis and
near-well performance. More recently,
solute transport models have been used to aid in
understanding and predicting the  effects of problems
 WATER SUPPLY

 REGIONAL AQUIFER
 ANALYSES

 NEAR-WELL
 PERFORMANCE

 GROUND-WATER/
 SURFACE WATER
 INTERACTIONS

 DEWATEMNG
 OPERATIONS
       Applications

• SEA-WATER INTRUSION CEOTHERMAL

• LAND FILLS       THERMAL STORAGE


• WASTE INJECTION   MEAT PUMP
                                      LAND SUBSIDENCE
• RADIOACTIVE
 WASTE STORAGE
•HOLOMG PONDS
             •GROUND WATER
              POLLUTION
             THERMAL POLLUTION
Fig. 2. Types of ground-water models and typical
applications.
involving hazardous wastes. Some of the applica-
tions include: sea-water intrusion, underground
storage of radioactive wastes, movement of leachate
from sanitary landfills, ground-water contamination
from holding ponds, and waste injection through
deep wells. Heat transport models have been
applied to problems concerning geothermal energy,
heat storage in aquifers, and thermal problems
associated with high-level radioactive waste storage.
Deformation models have been used to examine
field problems where fluid withdrawal has
decreased pressures and caused consolidation. This
compaction of sediments results in subsidence at
the land surface.
     This classification of ground-water models is
by no means complete. All of the above models can
be further subdivided into those describing porous
media and those describing fractured media.
Ground-water models can be combined with
statistical techniques in an effort to characterize
uncertainty in model parameters. These models
can also be used to estimate aquifer parameters.
In addition, there are other models that deal with
multifluid flow (e.g., oil and water) and multiphase
flow (e.g., unsaturated  zone problems). Some
resource management models combine flow models
and linear programs, which are used to optimize
certain decision parameters, like pumping rates.
Other models combine some or all of the models
in Figure 2. For example, a thermal loading problem
may require that a heat transport model
be combined with a deformation model. The
type of model used will obviously depend on the
application. For further information on the various
models and their availability, the interested reader
is referred to Bachmat, et al. (1978), and Appel
and Bredehoeft (1976).
     A numerical model is most appropriate  for
general problems involving aquifers having
irregular boundaries, heterogeneities, or
highly variable pumping and recharge rates. The
remaining sections are therefore generally concerned
with numerical  ground-water models. We discuss the
use of the numerical modeling approach with respect
to the various types of ground-water models,
giving the most emphasis to ground-water flow
models and the least emphasis to deformation
models.

                 MODEL USE
     Because the number of ground-water models
available today  is large, when beginning a study
the first question that may come to mind is,  "Which
one should I use?" Actually, the first question one

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 asks should be, "Do I need a numerical model study
 for this problem?" The answers to both of these
 questions can be determined by first considering
 the following: (1) What are the study objectives?
 (2) How much is known about the aquifer system;
 that is, what data are available? and (3) Does the
 study include plans to obtain additional data?
      The study objectives may be such that a
 numerical model is unnecessary. Or, if necessary,
 objectives may require only a very simple model.
 Additionally, lack of data may not justify a
 sophisticated model-, however, if a field study is in
 its initial stages, the ideal approach is to integrate
 the data collection and analysis with a model study.
 Once it is decided that a model is necessary, the
 one used will, in part, depend on the study
 objectives. For example, if one is interested in the
 drawdowns near a well, then a regional model,
 where these local effects are lost due to the large
 spacing between nodes, should not be used.
 Instead, perhaps a radial flow model with small
 grid spacing would be sufficient.
     The application of a ground-water model to
 an aquifer involves several areas of effort. These
 are shown in Figure 3 and include: data collection,
 data preparation for the model, history matching,
 and predictive simulation. These tasks should not
 be considered separate steps of a chronological
                  DETERMINE NECESSITY
                  Of NUMERICAL MODEL
        PREPARE DATA
        FOR MODEL
        USING ESTIMATED
        PARAMETERS
       INTERPRET RESULTS
                               COMPARE RESULTS
                               WITH OBSERVED
                               DATA
                                        Compartton
                    SENSITIVITY RUNS
                    IS MORE DATA
                      NEEDED?
                                     yet
                     PREDICTIVE
                   SIMULATION RUNS
Fig. 3. Diagram showing model use.
 procedure; rather, they should be considered as a
 feedback approach. It is best to use the model not
 only as a predictive tool, but also as an aid in
 conceptualizing the aquifer behavior. For example,
 a model used in the early stages of a field study-
 can help in determining which and how much data
 should be collected.
     Data preparation for the ground-water model
 first involves determining the  boundaries of the
 region to be modeled. The boundaries may be
 physical (impermeable or no flow, recharge or
 specified flux, and  constant head) or merely
 convenient (small subregion of a  large aquifer).
 Once the boundaries of the aquifer are
 determined it is necessary to discretize the region,
 that is, subdivide it into a grid. Depending on the
 numerical procedure used, the grid may have
 rectangular or irregular polygonal subdivisions.
 Figure 4 shows typical two-dimensional gridding
"for both the  finite-difference and finite-element
 methods.
     Once the grid  is designed, it is necessary to
 specify aquifer parameters and initial data for the
 grid. For descriptive purposes, the following
 discussion refers to the finite-difference method,
 utilizing a rectangular grid. Required program input
 data include  aquifer properties for each grid block,
 such as storage coefficients and transmissivities
 (see Table 1). For solute transport (that is, programs
 used for tracking hazardous wastes) and heat
 transport, additional data are required, such as
 hydrodynamic dispersion properties and thermal
 conductivity, respectively.  Computed results
 generally consist of hydraulic  heads at each of the
 grid blocks throughout the aquifer. These spatial
 distributions of hydraulic head are determined  at
 each of a sequence  of time levels  covering the
 period of interest. For transport problems,
 computed results might also include concentrations
 and temperatures at each of the grid blocks.
     Initial estimates of aquifer parameters
 constitute the first  step in a trial-and-error
 procedure known as history matching. The
 matching procedure (often referred to as model
 calibration) is used  to refine initial estimates of
 aquifer properties and to determine boundaries
 (that is, the area! and vertical extent of the
 aquifer) and the flow conditions at the boundaries
 (boundary conditions); aquifer tests generally
 provide the initial estimates for storage coefficients
 and transmissivities. For certain ground-water
 problems, steady-state (or equilibrium) heads must
 also be determined  and used as initial or beginning
 conditions. Simulated wells in the aquifer grid

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system a..-.  -
rates, and > ;
compared v-.-
      Assurr.i
between i:Vic
initial e?vi •••••••
to modi;'  ':
and cal: •.   .
the niv  ri."
 • "   ...  pump at the observed
' v5;j-;/.li-.red> drawdowns are
"•.   ' '•;': ••.vdov/ns.
"-.'..- -'Viuc'-i is correct, comparison
      •-"":' ;';C accuracy of the
      ;" :t;;   li 1-nsy be necessary
        :- -'T ~i uuti! aii observed
          ,;• "rident'y well. In
      •    --.' '.r'.-i and err or; more

Fig. 4a. fi/iap v.ow o> fi
boundaries.
                            showing we!) field and
 Fig. 4b. Finite difiij^ii^t iiiia ioi aquifer study, where Ax is
 the spacing in the x-oiieotion, Ay is the spacing in the y-
 direction and b is th* aqi-iTer thickness.
 Fig. 4c. Finite-eieme;it -nnfiguration for aquifer study where
 b is the aquifei thickness.
                                                                  Table 'i. Dt
                                                               I. Physical Frarii :
                                                                  X. Ground-!..'
                                                                     4.1-.-?.:-.-

                                                                        bed.
                                                                     6. Map shovv':;r
 .  — :  t.  aa Considered for a
;.;:: ;£,:ti..  I'.lac.'fe, 1979)
                                                                                              .•j.-^ ..real extent, bounda-
                                                                                            •..ci.'joiir. of all aquifers.
                                                                                             i: •  .4U!t,-ce-v/ater bodies.
                                                                                            '.-.,': ', i_tio::. ar.d saturated-

                                                                                                  ;  -,':?.,- ,'.,tJ boundaries.
                                                                                            . .•.,... j.c-i..6i- .ii2.p of confining

                                                                                            . in srorage coefficient of
                                               7. Rciatiui, i>i i^,,;iaicu uuukness to D'ansmissivity.
                                               8. Reiati--... o; si.;i:i.. .no aquifer (hydraulic
                                                  connecno::.'.
                                            B. Solute 7~a~spoil «'»';; addition to above)
                                               9. Estimatei of the parameters that comprise hydro-
                                                  dynamic dispersion.
                                               10. Effective ;-orosiry distribution.
                                               11. Back?-o\:s.: ;'v;\.irir.stion on natural concentration
                                                  district.,  ',•: .,.•.: r; ou^i.tyj in aquifer.
                                               12. Estimivc - «' fluio r.cnsiry variations and relation-
                                                  ship of dcrisii'v- {-) concentration.
                                               13. Hydraulic h-ao distributions (used to determine
                                                  ground-water velocities).
                                               14. Boundary r.orto: it' •-•
                                                                  A. Ground \ ...•
                                                                     1. Typ. sr.       .:.     '.•'-•.: .i.e arras (irrigated areas,

                                                                     2. Surfact-v.v.rr  J^•••,.• .--ns
                                                                     3. GrounJ-v/r,;•;•!• pi.r,".'..;:iif (disrributed in time and
                                                                       space).
                                                                     4. Stream flow (Jisruoi-u-d in time and space).

                                                                  B. Solute Tran:;.o:: (in a;i:.iiiun tc- above)
                                                                     6. Areal and twnfi-.-va! ai.;cribution of water quality
                                                                       in aquifer.
                                                                     7. Stream '•-••'

                                                                  C.HeatT.ar.   • > •.; •.'.'•
                                                                     9. Area', aru  -....  •  .
                                                                       aquifer.
                                                                     10. Streiuribi; r:  !.:.-«t .-:•.:
                                                                                                 !.>  .I.-;, ii. time and space).
                                                                                                 OihltlOli.
                                                                                                       .-)
                                                                                                           rnperature in
                                                               III. Other Factors
                                                                  A. Ground-Water I-i.n^> ami Transport
                                                                     1. Economic ir.formarior, of 'vater supply.
                                                                     2. Legal and a.'ininisrrztivc rules.
                                                                     3. Environmentai factors.
                                                                     4. Planned chants in water and land use.

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 recently the amount of work in matching has been
 reduced somewhat by using parameter estimation
 methods that modify initial estimates of input data
 in a more objective fashion.
      No hard and fast rules exist to indicate when
 a satisfactory match is obtained. The number of
 "runs" required to produce a satisfactory match
 depends on the objectives of the analysis, the
 complexity of the flow system and length of
 observed history, as well as the patience of the
 hydrologist. Once completed, the model can be used
 to predict the future behavior of the aquifer. Of
 course, confidence in any predictive results must
 be based on (1) a thorough understanding of model
 limitations, (2) the accuracy of the match with
 observed historical behavior, and (3) knowledge of
 data reliability and aquifer characteristics.
      The main purpose of prediction is estimation
 of aquifer performance under a variety of pumping
 schemes. While the aquifer can be developed only
 once at considerable expense, a model can be
 "pumped" or run  many times at low expense
 over a short period of time. Observation of model
 performance under differing development
 schemes then aids in selecting an optimum set of
 operating conditions for utilizing the ground-
 water resource. More specifically, ground-water
 modeling allows estimates of: (a) recharge (both
 natural and induced) due to leakage from confining
 beds, (b) effects of boundaries and boundary
 conditions,  (c) the effects of well locations and
 spacing, and (d) the effects of various
 withdrawal  (or injection) rates.
     Other purposes for prediction include
 estimating the rates of movement of hazardous
 wastes from sanitary landfills and other containment
 areas. Models are used to predict the encroachment
 rate of salt water in coastal regions due to fresh-
 water withdrawal. They are also used to help
 determine what, if any, remedial  action is best to
 take in a contamination situation. Finally, heat
 transport models are used to help predict the
 behavior of geothermal reservoirs and aquifers
used for thermal storage.
     in addition to these site-specific applications,
models are also used to examine general problems.
 Hypothetical (but typical) aquifer problems may
be designed  to study various types of flow behavior,
such as ground-water/surface-water interactions
or flow around a deep radioactive waste repository.
The feasibility of certain proposed mechanisms for
observed behavior can be tested. Parameters may
be changed to learn what effect they may have on
the over-all process. This is sometimes  referred to
 as a sensitivity analysis, since results from these
 runs will indicate what parameters the computed
 hydraulic heads are most sensitive to. Sensitivity
 analysis is also useful for site-specific applications
 to indicate what additional data need to be
 determined and areas where additional data are
 needed.

                MODEL MISUSE
     There are a variety of ways to  misuse
 models (Prickett, 1979). Three common and related
 ones are:  overkill, inappropriate prediction, and
 misinterpretation. The temptation to apply the most
 sophisticated computational tool to a problem is
 difficult to resist. A question that often arises is,
 under what circumstances should simulation be
 three-dimensional as opposed to two- or even one-
 dimensional. Inclusion of flow in the third (nearly
 vertical) direction is often recommended only if
 aquifer thickness is "large" in relation to area!
 extent or if pronounced heterogeneity exists in
 the vertical direction (for example, high  stratifica-
 tion). Another type of overkill is  that grid sizes are
 used which are finer (smaller) than necessary
 considering available information about aquifer
 properties, resulting in additional work and
 expense.
     In some applications, complex models are
 used too early in the study.  For example, for a
 hazardous waste problem, one generally  should not
 begin with a solute transport model.  Rather, the
 first step is to be sure the ground-water hydrology
 (velocity in particular) can be characterized
 satisfactorily, and therefore one begins by modeling
 ground-water flow alone. Once this is done to
 satisfaction, then solute transport can be included.
 One must assess the complexities of the problem,
 the amount of data that is available, and the
 objectives of the analysis, and then determine the
 best approach for the particular situation. A
 general rule might be to start with the simplest
 model  and a coarse aquifer description and refine
 the model and data until the desired estimation of
 aquifer performance is obtained.
     One must always be aware that the  history-
 match  portion of the simulation occurred under a
given set of field conditions, and that these
conditions are subject to change during the
prediction portion. For example, during  the history-
 match  portion, the aquifer may be confined, but
 be on the verge of becoming desaturated. Using a
confined model  for prediction will give erroneous
results  since the  saturated thickness and storage
coefficient will be incorrect. Because ground-

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water models deal with the subsurface, there are
always unknown factors that could affect results.
In general, one should not predict more than about
twice the period used for matching, and only then
under similar pumping schemes.
     Perhaps the worst possible misuse of a model
is blind faith in model results. Calculations that
contradict normal hydrologic intuition almost
always are the result of some data entry mistake,
a "bug" in the computer program, or misapplication
of the model to a problem for which it was not
designed. Proper application of a ground-water
model requires an understanding of the specific
aquifer. Without this conceptual understanding
the whole exercise may become a meaningless waste
of time and  money.

   LIMITATIONS AND SOURCES OF ERROR
                IN MODELING
     In order to avoid model misuse, it is important
to know and understand the limitations and possible
sources of error in numerical models. All numerical
models are based on a set of simplifying assump-
tions, which limit their use for certain problems.
To avoid applying an otherwise valid model to an
inappropriate field situation, it is not only important
to understand the field behavior but also to under-
stand all of the assumptions that form the basis of
the model. An areal (two-dimensional) model, for
example, should be  applied with care to a three-
dimensional problem involving a series of aquifers,
hydrologically connected by confining beds, since
rhe model results may not be indicative of the
field's behavior. Errors of this type are considered
conceptual errors.
     In addition to  these limitations, there are
several potential sources of error in the numerical
model results. First, replacement of the model
differential equations by a set of algebraic
equations introduces truncation error; that is,  the
exact solution of the algebraic equations differs
somewhat from the solution of the original
differential equations. Second, the exact solution
of the algebraic equations is not obtained  due  to
round-off error, as a result of the finite accuracy
of computer calculations. Finally, and perhaps
most importantly, aquifer description data (for
example, transmissivities, storage coefficients, and
the distribution of heads within the aquifer) are
seldom known accurately or completely, thus
producing data error.
     The level of truncation error in computed
results may  be estimated by repeating runs or
portions of runs with smaller space and/or time
increments. Significant sensitivity of computed
results to changes in these increment sizes indicates
a significant level of truncation error and the
corresponding need for smaller spatial and/or time
increments. Compared to the other error sources,
round-off error is generally negligible.
     Error caused by erroneous aquifer description
data is difficult to assess since the true aquifer
description is never known. An adage used to
describe the error associated with this data is,
"Garbage in, garbage out." A combination of
core analysis, aquifer tests, and geological studies
often give valuable insight into the nature of
transmissivity, storage coefficients, and aquifer
geometry. However, much of this information
may be very local in extent and should be
regarded carefully when  used in a model  of a large
area. As discussed, the final parameters that
characterize the aquifer are usually determined by
obtaining the best agreement between calculated
and observed aquifer behavior during some
historical period.

                  SUMMARY
     Numerical ground-water models are an
important tool for the ground-water hydrologist.
They can be used to simulate the behavior of
complex aquifers including the effects of
irregular boundaries, heterogeneity, and different
processes such as ground-water flow, solute
transport and heat transport. The use of  numerical
models involves data collection, data preparation,
history matching, and prediction. The process of
constructing a model for an aquifer study forces
one to develop a conceptual understanding of
how the aquifer behaves. Models, therefore, can
be used in all phases of the aquifer study including
conceptualization and data collection, as well
as prediction. To be most effective, the hydrologist
must have a thorough understanding of the specific
aquifer studied, must be familiar with alternative
modeling techniques, and must realize the limita-
tions and sources of error in models. Upon meeting
these criteria, a successful model study will not
only improve one's understanding of the particular
hydrologic system, but should  also provide
appropriate prediction and analysis of the problem
under study.

             ACKNOWLEDGMENTS
     This effort was supported by the Holcomb
Research Institute, Butler University, which is, in
part, supported by EPA  grant number R-803713.
The authors also wish to acknowledge the National

-------
Water Well Association's contribution for drafting,
editing and publication. Finally, many of the ideas
presented in this series of papers were formulated
during the authors' participation in training courses
for the U.S. Geological Survey.
Thomas, R. G. 1973. Groundwater models. Irrigation and
     Drainage Paper 21, Food and Agriculture Organization
     of the United Nations, Rome. 192 pp.
van Poollen, H. K., H. C. Bixel, and J. R. Jargon. 1969.
     Reservoir modeling — 1: What it is, what it does. Oil
     and Gas Jour. (July), pp. 158-160.
                   REFERENCES
Appel, C. A., and J. D. Bredehocft. 1976. Status of ground-
     water modeling in the U.S. Geological Survey. U.S.
     Geological Survey Circular 737. 9 pp.
Bachmat, Y., B. Andrews, D. Holtz, and S.  Sebastian. 1978.
     Utilization of numerical groundwater models for
     water resource management. U.S. Environ. Prot.
     Agency Report EPA-600/8-78-012.178 pp.
Goats, K. H. 1969. Use and misuse of reservoir simulation
     models. J. Pet. Tech. (Nov.). pp. 1391-1398.
Karplus, W. J. 1976. The future of mathematical models of
     water resource systems. In System simulation in water
     resources (G. C. Vansteenkistc, ed.). North-Holland
     Publishing Co. pp. 11-18.
Moore, J. £. 1979. Contributional ground-water modeling
     to planning. Journ. of Hydrology, v. 43 (Oct.),
     pp. 121-128.
Prickctt, T. A. 1979. Ground-water computer models —
     state of the an. Ground Water, v. 17, no. 2, pp.
     167-173.
     James W. Mercer attended Florida State University
and University of Illinois, where be received bis Ph.D. in
Geology in 1973. Prior to graduation, be worked for Exxon
Oil Corporation and the Desert Research Institute,
University of Nevada. Most recently, be worked for the
U.S. Geological  Survey, and is currently President of
GeoTrans, Inc. Dr. Mercer has taught ground-water
modeling at the  U.S.G.S. and at George Washington
University. He has also coauthor ed several articles on
modeling transport problems in ground water.
     Charles R. Faust received bis B.S. and Ph.D. in
Geology from Pennsylvania State University in 1967 and
1976. Until recently, be was a bydrologist with the Water
Resources Division, U.S. Geological Survey. His interests
there involved thermal pollution in rivers, geo thermal
reservoir simulation, and fluid flow in fractured rocks.
Presently be is a principal with GeoTrans, Inc., where be is
interested in quantitative evaluation of hazardous waste and
radionuclide migration in fractured rocks.

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Ground-Water  Modeling:   Mathematical  Models"
by James W. Mercer and Charles R. Faustb
                  ABSTRACT
     Ground-water modeling begins with a conceptual
 understanding of the physical problem. The next step in
 modeling is translating the physical system into mathe-
 matical terms. In general, the final results are the familiar
 ground-water flow equation and transport equations.
 These equations, however, are often simplified, using
 site-specific assumptions, to form a variety of equation
 subsets. An understanding of these equations and their
 associated boundary and initial conditions is necessary
 before a modeling problem can be formulated.

                INTRODUCTION
     The variety and complexity of mathematical
 models (usually partial differential equations) used
 in ground-water applications have increased
 dramatically during the last twenty years. This
 increase has been made possible by significant
 advances in digital computers, but has also been
 enhanced by environmental and energy concerns.
 This proliferation of mathematical models is
 often bewildering to the hydrologist who is trying
 to keep up with the research literature. Although
 the number of model  types is large, only a few
 basic processes are considered.  This is somewhat
        is is the second in a series of papers on ground-
 water modeling.
     bGeoTrans, Inc., P.O. Box 2550, Reston, Virginia
 22090.
     Discussion open until November 1, 1980.

 212
ironic since the large number of "different" models
is the result of various simplifying assumptions
used to reduce a general set of equations to some
solvable form. Fortunately, if one keeps in mind
the fundamental processes being simulated, then
different or simplified forms of equations are less
confusing.
     The purposes of this paper are: (1) to review
the basic processes of interest in  ground-water
applications, (2) to show how mathematical
models are developed, and (3) to discuss some of
the more commonly used equations. In order to
apply mathematical models effectively, it is
necessary to develop an intuitive knowledge of
both the physical process and the corresponding
mathematical model. Unfortunately the mathe-
matics are often overwhelming. With this in mind, a
review of the  mathematical aspects is given in the
appendices, whereas the main part of this paper
emphasizes the physical processes. The appendices
include a review of derivatives, alternative notation,
coordinate systems, viewpoints, and a simplified
derivation of the ground-water flow equation.

              BASIC PROCESSES
     The major processes that may be considered
part of many  ground-water problems are fluid flow,
solute transport, heat transport and deformation.
Even though only four general processes have been
identified, the number of different models that can
be conceived is very large. The reason for this is
that many of the model variations are application

      Vol. 18, No. 3-GROUND WATER-May-June 1980

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dependent. For example, flow in fractured media
behaves differently from flow in porous media, and
is, hence, described by different equations. In
addition to application-dependent variations, for
convenience, equations describing the same process
are often posed in terms of different dependent
variables. Hydraulic head is usually used as the
dependent variable (also called unknown or state
variable) in the ground-water flow equation, but
drawdown or fluid pressure is sometimes used.
Table 1 provides a summary of the major processes,
dependent variables, and application-dependent
variations that are commonly encountered in
ground-water applications. Rather than discuss
each of the possible models that may be useful,
attention in this report will focus on a few
mathematical models that deal with single-phase
flow in porous media.
     The basic processes that are considered
include ground-water flow, solute transport, and
heat transport. Ground-water flow is a process that
can be, and usually is, modeled without considera-
tion of heat or solute transport.  Modeled by itself,
it has important applications in ground-water
supply and design of engineering structures, such
as dams, mines, and excavations, that interact with
the ground-water system. Both solute and heat
                   transport require either simultaneous solution with
                   or results (e.g., velocities) from a ground-water
                   flow model. This is because the movement
                   (transport) of solutes or heat is controlled partially
                   by the ground-water movement. Solute transport
                   models are used for a wide variety of ground-water
                   quality problems, such as point source pollution
                   (e.g., waste disposal wells), spread source pollution
                   (e.g., landfills) or sea-water intrusion. Heat transport
                   models are applied  to problems involving thermal
                   storage in aquifers,  geothermal reservoir engineer-
                   ing, and usage of ground-water heat pumps.

                      MATHEMATICAL MODEL DEVELOPMENT
                        The development of a mathematical model
                   begins with a conceptual understanding of the
                   physical system. Once these concepts are
                   formulated they can be translated into a mathe-
                   matical framework  resulting in equations that
                   describe the process. A variety of analytical and
                   numerical techniques can be applied to solve the
                   equations, resulting in practical tools (often
                   referred to as models) such as type curves or
                   finite-difference and finite-element computer
                   programs.
                        In this section we look at the procedure for
                   translating ground-water concepts into mathe-
                  Table 1. Major Processes with Dependent Variables and Application Variations
 Major Process
Dependent Variable
 Application Dependent Variations
Fluid Flow
 fluid pressure,
 hydraulic head,
 hydraulic potential
 or drawdown
Porous Media
Doubly Porous Media
Discrete Fractured Media
Single-phase Fluid
Two-phase Fluid
Heat Transport
temperature,
enthalpy or
internal energy
Same as for Flow plus
Convection
Conduction
Radiation
Solute Transport
concentration
Same as for Flow plus
Convection
Dispersion
Chemical Source Terms
 Nuclide Decay & Daughter Products
 Adsorption & Dcsorption
 Precipitation & Dissolution
 Complexing
Deformation
                                   strain or
                                   strain rate
                                     Elastic Media
                                     Plastic Media
                                     Visco — Elastic Media
                                     Visco — Plastic Media
                                     Discrete Fractured Media
                                                                                                  213

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matical terms, assuming that the reader already has
a good conceptual understanding of ground-water
systems and associated processes. Rigorous
development of specific equations used in ground-
water applications may be found in textbooks
(e.g., Bear,  1972). The emphasis here is on
(1) a review of basic considerations and assumptions,
(2) the procedure for obtaining the final equations,
and (3) a discussion of the equations in physical
terms. As part of this discussion, we also include
boundary ahd initial conditions as they relate to
the solution of the equations. A summary of basic
mathematical concepts and notation is given in
Appendix 1.

General Equations
     The derivations of equations used in ground-
water applications are based on the conservation
principles dealing with mass, momentum, and
energy. These principles require that the net
quantity  (mass, momentum, or energy) leaving or
entering a specified volume of aquifer during a
given time interval be equal to the change in the
amount of  that quantity stored in the volume.
The derivation of the particular conservation
equation involves representing the balance in terms
of mathematical expressions. Once the balance
equation is developed in mathematical terms, it is
necessary to specify additional relationships among
variables so that the equations can be solved.
These include thermodynamic (e.g., the effect of
fluid pressure on density) and  constitutive (e.g., the
effect of fluid pressure on porosity) relationships.
The result of the derivation is usually a set of general
partial differential equations in the three-dimen-
sional Cartesian coordinate system. As an example
of the procedure, a simplified derivation of  the
ground-water flow equation is given in Appendix 2.
General equations for ground-water  flow, solute
transport and heat transport are discussed in the
remainder of this section. It should be noted that
these equations were derived based on certain
assumptions, and even more general (and complex)
equations could be presented. The general
equations that we discuss, however,  are the  ones
most commonly used.

Ground-Water Flow
     An equation describing unsteady ground-water
flow in three dimensions can be written (see
Appendix 2 for reference) as
           Water
            Mass
          Balance
                                 Water
                              Momentum
                               Balance
                                       Darcy's
                                      Equation
                         Ground
                       Water Flow
                        Equation
        Fig. 1. Diagram of the major components of the ground-
        water flow equation.
       in which h is the hydraulic head (L); K is the
       hydraulic conductivity tensor (L/t); R is a general
       source or sink term (1/t), that is, volume of water
       injected per unit volume of aquifer per unit time;
       Ss is the specific storage (1/L); and V « a differential
       operator (1/L). For general problems, K, Ss and R
       can vary  from point to point within the aquifer
       (that is, are functions of x, y and z). Equation (1)
       is known as a diffusion-type equation and is
       derived by combining the mass conservation
       (water balance) and momentum conservation
       (Darcy's  Law) equations, as shown in Figure 1
       and outlined in Appendix 2.  In a more explicit
       form, equation (1) can be written as,
                        "
                                                 at
                                            	  (2)
           -  =   -          dh
           V - K -  Vh + R = Ss — ,
                             dt
(1)
In developing this equation it was assumed that
the principal components of the hydraulic
conductivity tensor are colinear with the Cartesian
coordinate system (that is, the directions of the
anisotropy line up with the coordinate system).
To get an intuitive feel for what equation (2)
expresses, consider a small control volume of the
aquifer. The first three terms in (2) represent
the difference in the rate of water flowing into
and out of the control volume. R represents the
rate of water gained or lost from some source or
sink within or along the boundary of the volume.
The right-hand side represents the change in the
214

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amount of water stored in the control volume
expressed as a rate.
     When it is necessary to consider the effect of
other processes (such as variable-density solute
transport or heat transport) on ground-water flow,
a more general form of equation (1) in terms of
pressure is used (Bredehoeft and Finder, 1973):
                                            (3)
In this equation, p is water density (M/L3); g is the
gravitational constant (L/t2); p is dynamic viscosity
(M/Lt); Ic is the intrinsic permeability tensor (L2);
and 0 is porosity (dimensionless). In general, p,n,
R, and 0 can depend on the fluid pressure (M/Lt),
p, as well as concentration of dissolved materials
and fluid temperature. Pressure and hydraulic head
are related by (Hubbert, 1940)

                       P dp
                h = z + / -£  ,
                      Po gP
whert z is the elevation above some datum and p0
is some reference pressure (usually atmospheric).

Solute Transport
     In general, a complete physical-chemical
description of moving ground water would include
the movement of the fluid and all species of
material dissolved in the fluid, and chemical
reactions among the various species. The
difficulties encountered in solving the set of
equations describing this interaction (for real
problems) have forced hydrologists to consider
simplified subsets of the general problem. The
ground-water flow equation describes the rate
of propagation of a pressure or head change in an
aquifer. In order to describe the transport of
dissolved chemical species in ground water,  the
transport equation and the ground-water flow
equation must be solved simultaneously.
     The ground-water flow equation may be
used to determine the velocity field of ground-
water flow. To determine the movement of the
material, an additional equation is necessary. This
equation is also obtained by taking a mass balance
of the material as shown in Figure 2 and may be
written as (see, for example, Reddell and  Sunada,
1970)
-  VC- V
                          RC* =
3(»C)
 3t
(4)
                                         and D is the dispersion tensor (L2/t). The solute
                                         transport equation (also known as the convection-
                                         diffusion equation) contains (from left to right) a
                                         dispersion term, a convection term, a source term
                                         (which can include chemical reactions), and a rate
                                         change in concentration term.
                                             Although equation (4) mathematically
                                         describes solute transport, determining the
                                         parameters used in this equation usually proves to
                                         be quite difficult. These parameters include:
                                         (1) velocity, (2) source term properties, and
                                         (3) dispersion properties. Velocities are generally
                                         determined using Darcy's equation (see Appendix
                                         2). Therefore, porosity, hydraulic conductivity,
                                         as well as the hydraulic head distribution, must be
                                         known.
                                             Although not shown in equation (4), an
                                         additional source term can be incorporated that
                                         includes the effects of the various chemical
                                         processes and reactions operating in the
                                         ground-water system. These include precipitation
                                         and solution, co-precipitation, oxidation and
                                         reduction, adsorption and desorption, ion exchange,
                                         complexation, nuclear decay, ion filtration and
                                         gas generation. Not only are many of these processes
                                         poorly understood, but in pollution problems,
                                         often the type and strength of the source is
                                         inadequately described. For example, most
                                         landfills have no detailed records describing the
                                         materials buried in them. A common attempt to
                                         quantify some aspects of the source term is
                                         through the use of distribution coefficients (K
-------
In general, a high Ka indicates a strong tendency
for sorption and therefore retards movement of
material. Back and Cherry (1976) discuss this and
other problems associated with incorporating
geochemistry into transport models.
     Dispersion refers to the spreading and
mixing caused in part by molecular diffusion and
microscopic variation in velocities within
individual pores (Anderson, 1979).  For many
field problems, molecular diffusion (described by
Pick's law) is1 small compared to mechanical mixing.
The form of the dispersion tensor is complex and is
discussed by Scheidegger (1961). A simplified
representation that illustrates the basic components
may be written as,
             = f(v,dltda)
(5)
where f indicates "a function of;" and D
-------
orations have negligible effect on fluid density and
ground-water flow is steady; (2) variations in
concentration have negligible effect on fluid
density and ground-water flow is transient; and
(3) variations in concentration  have significant
effect on fluid density and ground-water flow is
transient. Many materials dissolve in water, while
others may be carried with the water in
suspension. Often times, for practical purposes,
this material in relatively small concentrations,
does not change the density of the fluid
significantly and the ground-water flow equation
in terms of hydraulic head [equation (1)] may
be used in conjunction with the solute transport
equation (4). When steady flow is assumed
[(ah/at) = 0], it is necessary to solve the ground-
water flow equation only once to obtain the
velocity distribution for the convection term in
the transport equation. If the velocity distribution
changes with time, the ground-water flow
equation and the transport equation must be
solved simultaneously. The  negligible density
assumption is appropriate for many problems
in which contaminant concentrations are low.
Common applications include point source problems
involving disposal wells and ponds, and areal
problems such as fertilizers  and pesticides
applied to the surface.
     For those applications in which fluid
                                 Cantmng BM
                                          •II •
                                          • I
Fig. 4. A. Physical situation of an instantaneous uniform
concentration source in an aquifer. B. Typical discrepancy
between observed and predicted concentration distributions
for some later time t,.
  Water  and
  Rock  Heat
    Balance
   Water
Momentum
  Balance
                                     i
                                 Darcy's
                                Equation
                    Heat
                 Transport
                  Equation
Fig. 5. Diagram of the major components of the heat
transport equation.
density variations with contaminant concentration
are significant, the hydraulic head formulation of
the ground-water flow equation is not valid,
because mass will not be conserved. For these
problems the solute transport equation and the
ground-water flow equation in terms of pressure
[equation (3)] are often used. In general,
problems involving saline water require considera-
tions of density changes, e.g. sea-water intrusion
and upconing in stratified saline aquifers. Density
effects can be significant in situations where
contaminant concentrations are high, such as local
contamination near point sources.

Heat Transport
     The equation describing transport of heat
energy in saturated porous media has the same form
as the solute transport equation. It may be
developed by taking a heat energy balance as
shown in Figure 5, and may be written as (Mercer,
Finder and Donaldson, 1975),
                                                    V • Km- VT- pcvq •  VT + RcvT* =
                                                                           + (1- 0)prcvr] —
                                                                                         01
                                            (6)
                                                                                                 m
where p, is the density of the rock (M/L3); cv is the
specific heat (LVt'T); T is temperature (T); and R
is the medium thermal conduction/dispersion tensor
(ML/t3T). The medium thermal conduction/
dispersion tensor in equation (6) includes the effects
of thermal dispersion, which is analogous to solute

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dispersion, and heat conduction which is analogous
to molecular diffusion. In addition, unlike the solute
dispersion term, it also includes thermal conduction
in the rock. The conductive flow of heat is described
by Fourier's law, which has the same form as
Darcy's law. The derivation of equation (6)  is
based on the assumption of thermal equilibrium
between water and rock. This is generally reasonable
since the movement of water in a porous
medium is often slow while the surface area of the
water-rock contact is large.
     Although the heat and solute transport
equations are similar in form, they are different in
several important aspects. In solute transport the
velocity dependent dispersion is generally much
more important than molecular diffusion. In heat
transport, heat conduction is often as important
as thermal dispersion. Because of this, it is very
difficult to determine thermal dispersion properties
from experimental or field data. Consequently,
even less is known about thermal dispersion than
solute dispersion, and for many heat transport
applications, all effects must be lumped into one
parameter as presented in equation (6).
     The heat transport equation and the ground-
water flow equation in terms of pressure (3)
can be applied to a variety of thermal problems
such as those involving geothermal reservoir
engineering, storage of hot or cold fluid in aquifers,
and ground-water heat pumps. If the range of
                temperatures for a particular application is small,
                specific heat of the fluid can be assumed constant,
                and density can be expressed as a linear function of
                pressure and temperature. If the temperature range
                is higher, these simplifying approximations may
                not be adequate. If more complicated
                thermodynamic functions are used, the heat
                transport equation will exhibit stronger nonlinear
                behavior.

                Boundary and Initial Conditions
                     In order to obtain a unique solution of a
                partial differential equation corresponding to a
                given physical process, additional information
                about the physical state of the process is required.
                This information is described by boundary and
                initial conditions. For steady-state problems only
                boundary conditions are required, whereas for
                unsteady-state problems both boundary and initial
                conditions are required. Mathematically, the
                boundary conditions include the geometry of the
                boundary and the values of the dependent variable
                or its derivative normal to the boundary.  In
                physical terms, for ground-water applications the
                boundary conditions are generally of three
                types: (1) specified value; (2) specified flux; or
                (3) value-dependent flux, where the value is head,
                concentration or temperature, depending on the
                equation. These are shown in Table 2.
                     The initial conditions are simply the values
                                 Table 2. Ground-Water Boundary Conditions
     Type
                            Description
Specified
 Value
Values of head, concentration or temperature are specified along the boundary.
(In mathematical terms, this is known as the Diricblet condition.)
Specified
 Flux
Flow rate of water, concentration or temperature is specified along the
boundary and equated to the normal derivative. For example, the volumetric
flow rate per unit area for water in an isotropic media is given by
                                  dh
                           qn = -K-.
                                  dn
where the subscript n refers to the direction normal (perpendicular) to the
boundary. A no-flow (impermeable) boundary is a special case of this type in
which qn = 0. (When the derivative is specified on the boundary, it is called a
Neumann condition.)
Value-Dependent
 Flux
The flow rate is related to both the normal derivative and the value. For
example, the volumetric flow rate per unit area of water is related to the
normal derivative of head and head itself by

                            dh
                         -K — = qn(hb) ,
                            dn
where qn is some function that describes the boundary flow rate given the
head at the boundary (ho).
218

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 of the dependent variable specified everywhere
 inside the boundary. For example, in a confined
 aquifer for which the equations are linear, there is
 no need to impose the natural flow system since
 the computed drawdown can be superimposed on
 the natural flow system. In this case, the initial
 condition is drawdown (the dependent variable)
 equal to zero everywhere.

 Physical Interpretation
     Just as the physical ground-water system is
 idealized as continuous in deriving the differential
 equations, it is also expedient to idealize the
 conditions on the boundaries of the system in
 order that they too can be given mathematical
 expression. The boundary conditions of ground-
 water systems in nature are of several kinds,
 perhaps the most common being those describing
 the conditions at a well. Since the porous media
 stops at the well face, the aquifer not only has a
 boundary around its perimeter, but the outline
 or each well is also considered a boundary to the
 aquifer. The boundary conditions at wells are
 treated as constant or variable specified flux, or
 constant head, depending on which best describes
 the actual physical conditions. If the well is
 discharging or recharging at a given concentration
 or temperature, then these may also be specified,
 if the transport equations are being considered.
     Impermeable or nearly impermeable
 boundaries are formed by underlying or overlying
 beds of rock, by contiguous rock masses (as along
 a fault or along the wall of a  buried rock valley),
 or by dikes or similar structures (Jac°b, 1950).
 Permeable boundaries are formed by the
 bottom of rivers, canals, lakes, and other bodies
 of surface water. These permeable boundaries may
 be treated as surfaces of equal head (specified), if
 the body of surface water is large in volume, so that
 its level is uniform and independent of changes in
 ground-water flow. The uniform head on a
 boundary of this type may, however, change with
 time due to seasonal variation in the surface-water
 level. Other bodies of surface water, such as
 streams, may form boundaries with nonuniform
 distributions of head which may be either constant
 or variable with rime. A small stream, for
example, might be affected by a nearby withdrawal
of ground water if that withdrawal occurred at a
rate of the same order of magnitude as the flow in
the stream. Then the boundary condition would
not be independent of the ground-water flow;
that is, it would be a head-dependent flux.
     The body of surface water may be a holding
pond containing hazardous wastes. In solving the
solute transport equation, the boundary condition
at the holding pond must be specified. In theory,
this may be simply accomplished by specifying some
concentration at the bottom of the pond which
may vary with time as the waste in the pond is
changed. In practice, however, historical records
rarely contain the necessary data to accurately
define this boundary condition. This condition is
perhaps the most critical portion of the model
study since it describes the source of pollution.
Often, it can only be estimated, and depends on
the judgment of the hydrologist performing the
study.

VARIATIONS  OF THE GENERAL EQUATIONS
Ground-Water Flow
     As discussed, there are many subsets to the
general equations. We present several of the more
common ones used in modeling, and, by way of
example, only present the modifications to the
ground-water flow equation.

Confined, Artesian Flow
     Equation (2) may be integrated over the
thickness of an aquifer (see Pinder and Gray, 1977)
to give,
8x

              ) +    (T     )*W-S         (7)
           3x    3y   3y 3y         3t
which is an areal ground-water flow equation where
T is transmissivity; S is the storage coefficient; and
W is a source/sink term. This is the equation most
commonly used to examine ground-water flow in
confined aquifers. Because all the terms in (7) are
linear, it is a linear partial differential equation.
This is important to note in terms of obtaining a
solution. For more discussion on this equation,
the interested reader is referred to Pinder and
Bredehoeft (1968).
     In order to obtain equation (7), many
simplifying assumptions were invoked to idealize
the ground-water flow system to make a mathe-
matical treatment possible. The major assumptions
include: (1) porous media, (2) Darcy's law,
(3) slightly-compressible fluid, (4) small vertical
variation in properties and head, (S) single aquifer
with areal, confined flow, (6) linear elastic aquifer
vertical compressibility, and (7) principal compo-
nents of the transmissivity tensor are aligned with
the coordinate axes.
     The assumption of a porous medium is not
very restrictive; even fractured media over a large
area generally behave as a porous media. Darcy's

-------
law has been found to be valid for most field
applications. The slightly-compressible fluid
assumption implies that the density of water
remains almost constant. For problems involving
temperature variations or large differences in
dissolved solids, the assumption of constant density
may not be valid. For such problems, the ground-
water flow equation may be  reformulated in terms
of pressure. If vertical variations in properties and
head occur, the three-dimensional equation (1)
must be used  to' describe the flow system. If the
system is under water-table conditions, then the
equation must be modified as discussed later.
Finally, the assumption of linear vertical
compressibility is valid for most field applications,
except perhaps for problems where consolidation
and subsidence are major factors.

Leaky Artesian Conditions
     The source term in (7), W, can include well
discharge, induced infiltration from streams, steady
and/or transient leakage from confining beds,
recharge from precipitation and evapotranspiration.
These source terms are discussed in Prickett and
Lonnquist (1971) and in Trescott, et al.  (1976); as
an example, steady-state leakage is discussed.
     It is assumed that the leakage through the
confining bed into the aquifer is vertical and
proportional  to the difference in the head between
the aquifer and the head in an overlying or under-
lying source aquifer. It is also assumed that the
head in the source aquifer is constant with time,
that storage in the confining bed is neglected, and
that the aquifer head does not fall below the
bottom of the confining bed. Under these condi-
tions, equation (7) may be written as,

 3      3h     3       9h
— (Txx — )  + — (Tyv — ) +
dx     8x    3y    yy 3y
b
                                = S    ,     (8)
                                    3t
where W contains the remaining source terms;
K' is the confining bed hydraulic conductivity;
b' is the confining bed thickness; and h' is the head
in the source aquifer.

Water-Table Conditions
     In water-table aquifers, transmissivity is a
function of head. Following Bredehoeft and Pinder
(1970), equation (7) may be written as
                             where Sy is the specific yield of the aquifer; and b
                             is its saturated thickness. Because b depends on
                             head, (9) is a nonlinear partial differential equation.
                             To obtain this equation, the vertical components of
                             flow are considered negligible and the principal
                             components of the transmissibility tensor are
                             assumed colincar with the coordinate axes.
                                  Water-table conditions as well as storage
                             coefficient conversion from compressible fluid to
                             drainage conditions are discussed in Prickett and
                             Lonnquist  (1971). Related topics on free-surface
                             approximation and saturated-unsaturated flow are
                             discussed in Narasimhan and Witherspoon (1977).

                             Radial Flow
                                  Finally, equation (7) may be transformed to
                             cylindrical coordinates (see Appendix 1), and
                             assuming radial symmetry, can be reduced to
            32h   1  3h
               . I  _  _____
            3r2   r  3r
                                                      S dh
                                                     i ____ .^B™ .
                                                      T dt
(10)
which is the one-dimensional ground-water flow
equation in cylindrical-coordinate notation where
r is the radius. This equation is used (with
appropriate boundary conditions) to describe
radial flow to a well. A solution to this problem is
helpful in two respects: (1) to aid in determining
aquifer parameters, T and S (the inverse problem);
and (2) to aid in predicting drawdowns due to
pumpage (the forward problem). This equation,
with appropriate boundary conditions, may also be
used to examine well-bore storage and related
phenomena.

Miscellaneous
     In addition to these, there are several other
equation subsets. For certain problems, salt water
and fresh water can be treated  as immiscible fluids
and separated by a sharp interface. Under this
assumption, solution of the solute transport
equation is unnecessary; instead, the flow equations
for fresh water and salt water are solved (see for
example, Mercer, et d., 1980).
     Flow in fractured media is currently an
important topic. In general, two approaches are
considered:  (1) discrete fracture modeling; and
(2) dual porosity modeling. In  discrete fracture
modeling, it is assumed that fracture location and
description are known, and the remaining portion
of the media is assumed to be porous. The dual
porosity approach assumes the media is so
fractured that it behaves as a continuum; that is,
flow mainly occurs  in the fractures, with the
porous part providing leakage to the fractures.
220

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  This approach leads to an equation similar to
  equation (8), with the leakage term being
  time dependent and the functional form of the
  leakage being related to fracture geometry.
       It is also possible to develop mathematical
  models for multifluid and multiphase flow in
  porous media. Such models have found widespread
  use in petroleum reservoir engineering. Similar
  models should find increased use in analyzing local
  pollution problems such as oil and gasoline spills
  and solvent contamination of aquifers.
                    SUMMARY
      We have presented a mathematical basis for
 many existing models used for ground-water
 problems. It consists of three partial differential
 equations for (1) ground-water flow, (2) solute
 transport, and (3) heat transport. The ground-water
 flow equations with appropriate boundary and
 initial conditions are used to analyze many ground-
 water problems, such as water supply. They are
 generally well understood and have been studied
 extensively, with many solutions available. The
 solute transport equation is used with the
 ground-water flow equation to address pollution
 problems. These problems are not as well
 understood, especially the characterization of
 source terms and dispersion. For thermal problems,
 the heat transport equation is used in conjunction
 with the ground-water flow equation.
      These equations and their boundary conditions
 can be simplified and solved analytically. This is
 the subject of many textbooks (for example,
 Kruseman and DeRidder, 1976; and Walton, 1970)
 and will not be discussed in this series of papers.
 Alternatively, more complex forms of the equations
 and boundary conditions may be solved numerically.
 Numerical methods applied to the ground-water
 equations are the subject of the next paper in this
 series.
             ACKNOWLEDGMENTS
     This effort was supported by the Holcomb
Research Institute, Butler University, which is, in
part, supported by EPA grant R-803713. The
authors also wish to acknowledge the National Water
Well Association's contribution for drafting,
editing and publication. Finally, many of the ideas
presented in this series of papers were formulated
during the authors' participation in training courses
for the U.S. Geological Survey.
                   REFERENCES

 Anderson, M. P. 1979. Using models to simulate the
      movement of contaminants through ground-water
      flow systems. Critical Reviews in Environmental
      Control, v. 9, issue 2, pp. 97-156.
 Back, W. and J. A. Cherry. 1976. Chemical aspects of
      present and future hydrogeologic problems. In
      Advances in Groundwater Hydrology, Saleem, A. Z.,
      Ed., American Water Research Association,
      Minneapolis, 153 pp.
 Bear, J. 1972. Dynamics of Fluids in Porous Media. American
      Elsevier, New York. 764 pp.
 Bredehoeft, J. D., and G. F. Finder. 1970. Digital analysis
      of area! flow in multiaquifer ground water systems: A
      quasi three-dimensional model. Water Resources
      Research, v. 6, no. 3, pp. 883-888.
 Bredehoeft. J. D., and G. F. Pinder. 1973. Mass transport
      in flowing ground water. Water Resources Research.
      v. 9, no. 1, pp. 194-210.
 Davis, S. N., and R.J.M. DcWiest. 1966. Hydrogeology.
      John Wiley and Sons, Inc., New York. 463 pp.
 Domenico, P. A. 1972. Concepts and Models in Ground-
      water Hydrology. McGraw-Hill Book Co., New York.
      405pp.
 Fried, J. J. 1975. Groundwater Pollution. Elsevier Scientific,
      Amsterdam. 330pp.
 Hubbert, M. K. 1940. The theory of ground-water motion.
      Journal of Geology, v. XLVIH, no. 8, pp. 785-944.
 Jacob. C. E. 1950. Flow of groundwater. In H. Rouse (cd.),
      Engineering Hydraulics. John Wiley & Sons, Inc.,
      New York, pp. 321-386.
 Kreyszig, E. 1968. Advanced Engineering Mathematics.
      John Wiley and Sons. Inc. New York. 898 pp.
 Kruseman, G. P., and N. A. DeRidder. 1976. Analysis
      and  Evaluation of Pumping Test Data. International
      Institute for Land Reclamation and Improvement/
      ILRI, Wageningen, The Netherlands. 200 pp.
 Mercer, J.  W., G. F. Pinder, and I. G. Donaldson. 1975. A
      Galerkin-finite element analysis of the hydrothermal
      system at Wairakei, New  Zealand. Jour. Geophys.
      Research, v. 80. no. 17, pp. 2608-2621.
 Mercer, J. W.. S. P. Larson, and C. R. Faust. 1980.
      Simulation of salt-water interface motion. Submitted
      to Ground Water.
 Narasimhan, T. N., and P. A. Witherspoon. 1977. Recent
     developments in modeling groundwater systems.
     Presented at the IBM Seminar on Regional Ground-
     water Hydrology and Modeling, Venice, Italy,
     May  25-26. 1976.
Pinder, G.  F., and J. D. Bredehoeft. 1968. Application of
     the digital computer for aquifer evaluation. Water
     Resources Research, v. 4, no. 5, pp. 1069-1093.
Pinder. G.  F.. and W. G. Gray. 1977. Finite Element
     Simulation in Surface and Subsurface Hydrology.
     Academic Press, New York.  295 pp.
Prickett, T. A., and C. G. Lonnquist. 1971. Selected digital
     computer techniques for groundwater resource
     evaluation. Bull. No. 55. Illinois State Water Survey,
     Urban*. 62 pp.
Reddcll, D. L., and D. K. Sunada.  1970. Numerical
     simulation of dispersion in groundwater aquifers.
     Colorado State Univ. Hydrology Paper 41. 79 pp.
Scheidcgger, A. E. 1961. General theory of dispersion in

-------
     porous media. Jour. Geophys. Research, v. 66, no. 10,
     pp. 3273-3278.
Trescott, P. C., G. F. Pinder, and S. P. Larson. 1976.
     Finite-difference model for aquifer simulation in
     two dimensions  with results of numerical experiments.
     U.S. Geol. Survey Techniques of Water Resources
     Investigations. Book 7, Chap. Cl, 116 pp.
Walton, W. C. 1970. Groundwater Resource Evaluation.
     McGraw-Hill Book Co., New York. 664 pp.
             *       *       *    i   *
     James W. Mercer attended Florida State University
and University of Illinois, where be  received bis Ph.D. in
Geology in 1973. Prior to graduation, be worked for Exxon
Oil Corporation and the Desert Research Institute,
University of Nevada. Most recently, he worked for the
U.S. Geological Survey, and is currently President of
GeoTrans, Inc. Dr. Mercer has taught ground-water
modeling at the U.S.G.S. and at George Washington
University. He has also coauthored several articles on
modeling transport problems in ground water.
     Charles R. Faust received bis B.S. and Ph.D. in
Geology from Pennsylvania State University in 1967 and
1976. Until recently, be was a Hydrologist with the Water
Resources Division, U.S. Geological Survey. His interests
there involved thermal pollution in rivers, geothermal
reservoir simulation, and fluid flow  in fractured rocks.
Presently he is a principal with GeoTrans, Inc., where be is
interested in quantitative evaluation of hazardous waste and
radionuclide migration in fractured rocks.
     APPENDIX 1. BASIC MATHEMATICAL
                   CONCEPTS
 Review of Derivatives
     The ground-water flow and transport equations
 are partial differential equations; therefore, a review
 of derivatives is in order. By way of analogy,
 consider a simple method of measuring the
 discharge rate from a pumping well. The method
 we use is to discharge the water into a holding
 tank for which we have some method of measuring
 the volume of water that it contains. The volume
 will change with time so that V(t) designates the
 volume of water in the tank at time t after the
 pump is turned on. Then the  change in the amount
 of water from time t| to time t2 is the difference
 between the volume present at t2 and the volume
 present at ti, or V(t2) - V(t,). The average rate of
 change of the volume of water per unit of time
 during the time interval between ti and t2 (the
 average discharge rate)  is simply the difference in
 volume divided by the time interval, or
            AV_V(t2)-V(t.)
            At       t2 - t,
                                             (1.1)
If we decrease the time interval, that is, we choose
t2 closer to t,, and continue doing so, eventually
t| and t2 will be nearly equal. In the limit t2
                                                      approaches ti, and the average rate of change in
                                                      (1.1) becomes the instantaneous rate of change in
                                                      the volume of water at time t| (that is, the
                                                      instantaneous discharge rate). This instantaneous
                                                      rate of change is also the definition of the
                                                      derivative of the volume at ti with respect to time
                                                      and may be written as,
                                                              dV(t.) _ limit V(t)-V(t.)
                                                              ~~dt
t-»t,
t- t.
                            (1-2)
                                                      where t2 is replaced by any arbitrary time t.
                                                      Geometrically, equation (1.1) represents the chord
                                                      slope of the graph of the volume between t, and t2
                                                      and equation (1.2) represents the slope of the
                                                      tangent to the graph of the volume at tj (see
                                                      Figure 1.1).
                                                           In the above example, volume was considered
                                                      a function of only one independent variable, t-,
                                                      that is, the volume change does not vary with
                                                      distance throughout the tank. In the general case,
                                                      we deal with functions of several variables; for
                                                      example,  hydraulic head may be considered a
                                                      function of two space variables, h = h(x,y). The
                                                      derivatives of h with respect to x and y are called
                                                      partial derivatives since one variable is held
                                                      constant during differentiation with respect to
                                                      the other variable. The two partial derivatives
                                                      of h(x,y) may be written as,
                                                      V(I2)
                                                      V(ti)
                                                                          Time
                                                       Fig. 1.1. Graph showing the chord slope AV/At and the
                                                       tangent line dV/dt of the volume V(t).
222

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                                Table 1.1. Definitions of Tensor Quantities
Quantity
Scalar or zero-
order tensor.
Vector or first-
order tensor.
Second-order
tensor.
Definition
A quantity that is characterized by magnitude
only.
A quantity that is characterized by both
magnitude and direction.
A quantity that may be fully described only
with reference to at least three components;
it has direction, magnitude, and magnitude
changing with direction.
Example
head, concentration,
temperature.
velocity, mass flux,
heat flux.
intrinsic permeability,
hydrodynamic
dispersion, thermal
conductivity.
       dh(a,b) = limit h(x.b)-h(a.b)
         3x     x-*a      x-a

       dh(a,b) _ limit h(a.y)-h(a.b)
         8y     y-*b      y-b
                       (1.3)
 where a and b arc specific values of x and y,
 respectively. It is interesting to note that the
 definitions in (1.2) and (1.3) are similar to finite-
 difference approximations.
     The derivatives defined in equations (1.2) and
 (1.3) are called first derivatives. These first
 derivatives form new functions, which in general
 can be differentiated again to form second
 derivatives, which can be differentiated
 again to form third dervatives, and so on. Probably
 the most common example of a first derivative is
 velocity (the instantaneous change of distance with
 respect to time), which can be differentiated again
 with respect to time to give acceleration. In
 ground-water hydrology, second derivatives are
 generally the highest derivative encountered.
     There are many rules concerning differentiation
 which can be found in any calculus book.  One
 useful rule that is presented here, is the chain rule.
 Consider, for example, porosity which is a function
 of pressure, ^(p), and pressure which is a function
 of time, p(t); then to evaluate the derivative of
 porosity with respect to time we use the chain rule
 as follows:
                dt
d*  dp
dp  dt
                                           (1.4)
Mathematical Notation
     Perhaps one of the most confusing aspects of
the mathematical description of ground-water flow
is the wide variety of mathematical notations used.
Basically, these notations amount to shorthand
descriptions of the equations. Before discussing
the notations, we first discuss some of the common
nomenclature. Table 1.1 shows the definitions of a
scalar, vector and tensor quantity. As may be seen,
these quantities are used in every aspect of ground
water, each having a different physical meaning,
and therefore a different mathematical treatment.
     For example, consider head in an aquifer.
Starting at a given point we wish to determine the
rate of change in head with distance as we move
away from the starting point. Even though head is
a scalar, this rate of change will be different,
depending on the direction we choose; conse-
quently it is called a directional derivative. In
rectangular coordinates, this derivative is expressed
with the aid of the del operator (vector) as
                                                   -3h  -3h  -3h
                                      V h = grad h = i — + j — + k —  ,
                                           e         3x  J3y    3z
                                           (1.5)
where i, j and k are vectors of unit length in the
positive x-, y- and z-dircctions. This quantity is
known as the gradient of head.
     This new quantity, Vh, is a vector, and may
be used to determine the velocity vector, v. Thus, at
each point of space, there is a vector, v, and the
magnitude and direction of v may vary from point
to point. If we apply the del operator to a vector
quantity, it forms the dot product or divergence
of v:
      _         _   dvx    3vy   dvz
      V • v = div v =	  +	+ ——  .      (1.6)
                   3x    3y     dz
The divergence of Vh is called the Laplacian,
which is found in the ground-water flow equation.
There arc several different ways of writing the
Laplacian and these are summarized in Table 1.2.
All of these various notations are found in the
ground-water literature and all are equivalent.
                                                                                                 223

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            Table 1.2. Laptacian Notation
32h  32h  32h
a^dy^F?
div grad h
V2h
 9   3h
	/	\
3x;   3xj
Definition

Vector notation

del notation

Summation notation,

repeated index in a term
implies summation

 3  3h    N   3  dh
— (—}= Z  — (— ),
3xj  dxi   i=l ox;  oxj

where N is number of
dimensions — 1, 2, or 3.
Coordinate Systems
     The two most commonly used coordinate
systems in ground-water flow problems are
rectangular (Cartesian) and cylindrical. Regional
problems are described using equations in terms of
a rectangular system. In this system, the lines or
axes are perpendicular to each other, and are
generally designated x, y and z, as shown in Figure
1.2. Cylindrical coordinate systems are used in
examining well problems. Cylindrical coordinates
  Cylindrical Coordinates
  • - r cos 9
  *V : I siri 6
  t • I
                         Cx.Y.ll
                         (1,1,1}
 Fig. 1.2. Rectangular and cylindrical coordinate systems
 showing volume elements and transformation.
(r, 8, z) are also shown in Figure 1.2, as well as a
volume element and the equations for coordinate
transformation. As may be seen, z is the vertical
axis, r is the radius, and 6 is the angle in the
horizontal plane measured from the
x-coordinate. Note that cylindrical coordinates
are simply polar coordinates in the x-y plane with
z for the third variable.
     As an example of a coordinate transformation,
consider the Laplacian in two dimensions
                                                                    ,    32h  32h
                                                                   V2h = —- + —-  .
                                                                         3x2  3y2
                                            (1.7)
Transformations of differential expressions from
one coordinate system into another are frequently
required in applications. Following an example
given in Kreyszig (1968), we shall denote partial
derivatives by subscripts and h(x,y,t), as a function
of r, 8 , t, by the same letter h.
     By applying the chain rule, we obtain,

              dh
              — = hx = hrrx + h00x .
              3x
By differentiating this again with respect to x we
first have
(hrrx)x +
 (hr)xrx
                        (h0)x0x + h00
                                                                xx
                                                                      (1.8)
                                                     Now, by applying the chain rule again, we find

                                                     (hr)x = hrrrx + hr00x and  (h0)x = h0rrx + h000x .
                                                     To determine the partial derivatives rx and 0X, we
                                                     have to differentiate
                                                              r = Vx2 +y2  and 8 = arc tan — ,
                                                     finding
                                                       rv =
                                                                        '   x
                                                                      r       1 + (y/x)2    x2      r2

                                                     Differentiating these two formulas again, we obtain
                                                           r- xr,
                                                     rxx -
                                         1 _ x2 _ y2
                                        7-75-75
                          7 XX
                               2
                               r3
                                                      Substituting these expressions into (1.8), using
                                                      hrf - h0r, gives
                                                                             «»*       *r»
                                                                     xy
                           In a similar fashion we obtain
                                                                    xy
                                                                    r4
                                                                                                 (1.9)
224

-------
      y2       xy      x2      x2       xy
hw = -r nff + 2 -r hrf + — "66 + -7 "r ~ 2 — he .
 yy   r2       r3      r4      r       r

                                   	(1-10)

By adding these two expressions we see that the
Laplacian in polar coordinates is
        V2h =
              3r2
I  ^
1  3r
                             32h
(1.11)
Viewpoints
     Assuming we can identify or in some way
understand the concept of ground-water flow and
related processes of interest, the development of a
mathematical model may be approached from
(A. Klute, written comm., 1970): (1) the molecular,
(2) the microscopic, or (3) the macroscopic
viewpoint. A discussion of these viewpoints is
important since confusion often arises between
hydrologists over which viewpoint they are
considering. An example of this is a modeler
discussing chemical source terms at the macroscopic
level with a geochemist who is describing the same
chemical source terms at the molecular level.
     In the molecular point of view, theories and
explanations of the mechanisms of flow are given
in terms of the behavior of the water molecules.
For this approach, statistical mechanical concepts
might be used. Chemical reaction mechanisms are
generally described using this viewpoint.
     At an intermediate level, the microscopic, a
theory of flow may be developed treating water in
the pores as a continuum and applying the
principles of continuum mechanics, especially  fluid
mechanics, to work out the detailed behavior of
the water within the pores. An example of this
approach is using the Navicr-Stokes equation for
the flow of a viscous fluid to work out the
detailed water velocity pattern within the pores.
However,  in natural environments, the identification
of pore geometry and other conditions necessary
for the solution of the equations is virtually
impossible, and thus solutions are generally
available only for rather simple pore space
geometry, such as flow in straight capillary tubes,
or between parallel plates. This approach may be
used to describe flow in fractures assuming simple
geometry.
     From a practical viewpoint, one generally
must continue the theoretical development to the
macroscopic level in order to create a useful tool.
We cannot observe the behavior of individual
molecules, nor can we observe the velocity and
fluid pressure distributions that one might, in
principle, calculate in the microscopic approach.
All measurements in the field are generally made
at the macroscopic level; therefore, to be useful,
any theory of water movement must be developed
to the point of describing flow on the macroscopic
level.
     In the macroscopic approach, all variables are
assumed  continuous functions  of space and time,
with derivatives of as high an order as needed. The
permeable medium is treated as a superposition of
two continuous phases, solid and liquid. Velocities,
hydraulic heads and other necessary variables are
treated as point functions. That is, they are defined
at every point in the region of  interest. The way
these variables are defined is important since it
involves choosing a volume element of the
permeable medium that is representative of the
medium. For example, consider the bulk
density, or mass of solids per unit bulk volume
of permeable medium, at a point surrounded by
some volume element. From Figure 1.3, we see
that if we choose the volume element smaller
than V,, then we may cither have a volume of all
solid or a volume of all liquid,  causing the value
for bulk  density to vary. Alternatively, if we
choose a volume element larger than  V2, the value
of bulk density again varies due to heterogeneous
effects (e.g., major changes in rock type). In
                           v,
           fig. 1.3. Variation in bulk density as a function of the
           average volume, with inserts.

-------
 practices the volume element (which in some
 literature is referred to as an REV or Representative
 Elementary Volume) is taken large enough to
 contain a representative assortment of pores, but at
 the same time small enough so that the values of
 macroscopic variables are approximately constant
 within the  element. In our figure, this volume lies
 between V, and V2. In the following development,
 all equation* and coefficients are considered at
 the macroscopic level.

  APPENDIX 2. SIMPLIFIED DERIVATION OF
   THE GROUND-WATER FLOW EQUATION
     A rigorous development of the ground-water
 flow equation (or equation of motion) generally
 begins with a mass and momentum balance.

 Momentum Balance
     For the momentum balance, Darcy's equation,
 which is based on empirical evidence, is used and
 from Figure 2.1, may be written as,
              Q=KA
 or
Q
A
                     h,-h,
                      h, - h2
                              (2.1)
(2.2)
 where Q is the rate of volume flow (vol/time); K is
 a proportionality constant (length/time); q is
 specific discharge (Darcy velocity) or discharge per
 unit area; and all other variables are shown in
 Figure 2.1. Using the definition of a derivative and
 choosing smaller and smaller values of L, allows us
 to write Darcy's equation in differential form:
                                                            (Op),
                                        Fig. 2.2. Volume element of a porous material showing
                                        flow across all faces.
                                                                            dh
                                                               ah
                                                     qy =  Kyy a7  '
                                                                                              (2.3)
                                                                  qz=-K;
                                                                         zz
                                 ah
                                 3z
Fig. 2.1. Apparatus to demonstrate Darcy's equation.
                                       where we have generalized the equation to allow
                                       the Darcy velocity to change with direction (see
                                       Domenico, 1972 for details). In equation (2.3), h is
                                       hydraulic head, qx, qy, qz are the components of
                                       Darcy velocity, and KXX, Kyy, Kzz are the principal
                                       components of the hydraulic conductivity tensor.
                                       Generally, the hydraulic conductivity tensor has
                                       nine components, describing its effects in all
                                       directions. We have reduced it to  three  components
                                       by assuming that it is symmetric and that the
                                       principal components are  oriented with the
                                       x-, y-, z-directions, respectively.

                                       Mass Balance
                                            The mass balance is determined by considering
                                       changes in the mass of a small volume element of
                                       porous material over a small time interval (At).
                                       Figure 2.2 shows such a volume element with a
                                       source/sink term and  flow across all faces. In words,
                                       the mass balance equation for our problem is,

                                       (mass leaving-mass entering) + (final mass-initial mass) = 0
                                                                         	(2.4)

                                       Using the quantities shown in Figure 2.2, equation
                                       (2.4) may be rewritten to  give,

-------
  [(Qp)x+Ax + (Qp)y+Ay + (Qp)z+Azl

- [(Qp)x + (Qp)y + p)t+At -
                               At
                                          ,, ,,
                                          (2.6)
       AxAyAz
     The Darcy velocity is the flow volume across
the element face divided by the area of the face, or
         Qx
        AyAz
                qy =
                    AxAz
 Qz
AxAy
                                          (2.7)
     Using equations (2.7), and choosing smaller
and smaller values of Ax, Ay, Az and At, and using
the definition of a derivative, equation (2.6)
becomes
  3(pqx) _ 3(pqy) _ 3(pqz)
   3x      3y       3z
                            R _
                                 9t
(2.8)
Final Equations
     For a slightly compressible fluid such as water,
density changes are small. Under this condition,
the spatial derivatives of density on the left side of
the equation are generally negligible; whereas, the
time derivative on the right side may be related to
hydraulic head (see, for example, Davis and DeWiest,
1966). The resulting equation is
       _3qx_3qy_3qz+R=   3h
         3x    3y    az          at
where Ss is the specific storage.
     Substitution of equations (2.3) into (2.9) gives
        ax   ay
                     ay
                                    	(2.10)

 which is the unsteady or transient, three-dimensional
                                                   ground-water flow equation. It is sometimes called
                                                   the diffusion equation, and equations of the same
                                                   form occur in the theories of unsteady flow of heat
                                                   and electricity. In mathematics, it is classified as
                                                   a parabolic partial differential equation. Equation
                                                   (2.10) states that the flow components in the x-,
                                                   y-, and z-directions plus the source/sink term must
                                                   balance the change in storage. Equation (2.10) is
                                                   often written using a type of mathematical
                                                   shorthand as
                  _   =  -           3h
                  V •  K • Vh + R = Ss —  ,
                                     3t
                                                                                            (2.11)
         where V is the differential operator, the single bar
         indicates a vector quantity, and the double bar
         indicates a second-order tensor quantity.
              For many problems, the velocity distribution,
         and hence the hydraulic head distribution, does not
         change with time; that is, the problem is steady-
         state. Many regional ground-water flow systems
         can be represented as a steady-state boundary value
         problem. For steady flow, 3h/3t = 0, so equation
         (2.10) becomes
         where for convenience the source/sink term has
         been dropped. In mathematics, the steady-state
         equation is classified as an elliptic partial differential
         equation.
              For a homogeneous medium, equation (2.12)
         reduces to
                  32h
                                      d^h.
                                      3y2
                                                                                32h
                                                                                             (2.13)
                                                    and for an isotropic medium, Kxx = Kyy = Kzz = K,
                                                    and
                                                                                             (2,4,
                                                                                             .„,..
                                                                                             (2.15)
         or dividing (2.14) by hydraulic conductivity

                    32h   32h  32h
                    ^: + -^ + —; = 0  ,
                    ax2   ay2  3z2
         which is Laplace's equation. Note that if we had
         used the transient equation we would Jjavt obtained
                         ...   ..  ,  ,  /*JMSyV« .Or Tyff
         an SS/K term, which is called the^nydrauhc
         diffusivity.

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 Ground-Water  Modeling:  Numerical Models
 by Charles R. Faust and James W. Mercer
                   ABSTRACT
      Partial differential equations may be used to describe
 a large number of problems in ground-water hydrology.
 Without a solution, however, these equations are of little
 value. Only a simplified subset of the general equations can
 he solved by analytical means, and these often describe
 idealized situations that are limited in application.
 Numerical solution of these equations using high speed
 digital computers offers a logical alternative.
                INTRODUCTION
     Numerical models provide the most general
 tool for the quantitative analysis of ground-water
 applications. They are not subject to many of the
 restrictive assumptions required for familiar
 analytical solutions (e.g., Theis' solution for radial
 flow to a pumping well in an infinite, confined
 aquifer). In spite of the flexibility of numerical
 models, their mathematical basis is acually less
 sophisticated than that of the analytical methods.
 Unfortunately, to the would-be-model user
 numerical methods seem complex. This perception
 results from two primary causes-, the first is that the
 number of alternative methods appears to be very
 large. Actually, the number of basic alternative
 methods is few; only the number of minor variations
 is large. Each of  these variations contributes to the
 second cause, unfamiliar terminology, by
 introducing new names and jargon. In this paper
 we introduce numerical methods commonly used
     aThis is the third in a series of papers on ground-water
modeling.
     bGeoTrans, Inc., P.O. Box 2550, Reston, Virginia
22090.
     Discussion open until January 1, 1981.
 in ground-water problems, while emphasizing basic
 ideas and only necessary terminology. For easy
 reference, common terminology is listed in
 Appendix 2.
     To develop a numerical model of a physical
 system (in our case, an aquifer), it is first necessary
 to understand how that system behaves. This
 understanding takes the form of laws and
 concepts (e.g., Darcy's law and the concept of
 storage). These concepts and laws are then
 translated into mathematical expressions, usually
 partial differential equations, with boundary and
 initial conditions  (the subject of the second paper
 in this series). Numerical methods provide a means
 for solving these equations in their most general
 form.
     Numerical solution normally involves
 approximating continuous (defined at every point)
 partial differential equations with a set of discrete
 equations in time and space. Thus, the region and
 time period of interest are divided in some fashion,
 resulting in an equation or set of equations for each
 sttbregion and time step. These discrete  equations
 are combined to form a system of algebraic
 equations that must be solved for each time step.
 Finite-difference and finite-element methods are
 the major numerical techniques used in  ground-water
 applications. The  important components and steps
 of model development for the two alternative
 methods are shown in Figure 1.
     In the remainder of this paper we discuss these
general methods in more detail showing how they
lead to a matrix equation (system of algebraic
equations). Various matrix solution techniques
are reviewed, including direct and iterative methods.
This is followed by a discussion of general consider-
ations in applying models (e.g., initial and boundary
conditions) and special techniques such  as nonlinear
techniques. Finally, the pros and cons of the various
numerical techniques are presented.

-------
                  Concepts of the
                  physical system
                          Translate to
               Partial differential equa-
               tion, boundary and initial
               conditions
Subdivide region
into a grid and
apply finite-
difference approx-
imations to space
and time derivatives.
Finite-difference
approach
Finite-element
 approach

   , Transform to
«-
1
i/es.
j Integral equation I
Subdivide region
into elements
and integrate
i

[First-order differential
equations

/Apply finite-difference
approximation to
time derivative
                 System of algebraic
                 equations
1
Solve by direct or
, iterative methods
Solution

Fig. 1. Generalized model development by finite-difference
and finite-element methods.
    FINITE-DIFFERENCE METHODS  (FDM)
     One numerical approach that has been applied
successfully to the ground-water flow equation
involves finite-difference approximations. When
using FDM to solve a partial differential  equation,
a grid is first established throughout the  region of
interest.  For two-dimensional, areal problems, we
overlay a grid  system on a map view of the aquifer.
There are two common types of grids: mesh-
centered and block-centered. These are shown in
Figure 2. Associated with the grids are node points
that represent the position at which the solution of
the unknown values (head, for example) is
obtained. In the mesh-centered grid the nodes are
located on the intersection of grid lines, whereas in
the block-centered grid the nodes are centered
between grid lines. The choice of the type of grid
to use depends largely on the boundary conditions.
The mesh-centered grid is convenient for problems
where values of head are specified on the boundary,
whereas the block-centered grid has an advantage
in problems where the flux is specified across the
boundary. From a practical point of view, the
differences in the two types of grids are minor.
     Note that the grids shown in Figure 2 are
rectangular and regular; that  is, the spacing in the
x-direction, Ax, and y-direction, Ay, are constant.
Often, irregular, rectangular grids with variable
Ax and Ay are used to describe the aquifer, using
smaller spacing for areas where more detail is
required (as near a well field). Such a grid is
shown in Figure 3. In addition, grids need not be
rectangular.
                                         A.
                 NODE POINT
      Ax"                      Ax
MESH  CENTERED      BLOCK CENTERED
Fig. 2. Grids showing mesh-centered nodes and block-
centered nodes.


g
e









•
•1
ft


i
D





|


•


HI






I t
L_ T



                                                                                                      Ay:
                                                            AXJ
                                      Fig. 3. Four by five block-centered grid used in difference
                                      equation development (A) and typical connection (B) for
                                      node (i,j).

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     The integrated finite-difference method
(IFDM) utilizes an arbitrary grid. Though less
commonly applied than the conventional finite-
difference approach, the I FDM has been used to
study ground-water problems since the early
1960's, when it was known by a different name.
Tyson and Weber (1964) introduced the method to
the ground-water field, but referred to it as a
polygonal model technique. The method was
further discussed by Cooley (1971) and Thomas
(1973), and was finally given the current name by
Narasimhan and Witherspoon (1976).
     For the IFDM, the region of interest must also
be divided into smaller areas. Thomas (1973) refers
to these as nodal areas, since they each have a node
point which is used for mathematical purpses to
connect each area with its neighbor. Further, as
with other finite-difference methods, it is assumed
that all recharge or withdrawal to and from the
nodal area occurs at the node point and that water
levels in the entire nodal area are the same as at the
node point. For this reason, the polygon geometry
as well as rectangular grids should be kept to a
reasonable size to maintain accuracy.
     A typical node point, adjacent nodes, and the
polygonal zone associated with it are shown in
Figure 4. As may be seen, the triangles formed by
connecting the node  points have no interior angle
greater than 90° since the polygon sides are
perpendicular bisectors of the lines connecting the
intersects. When rectangular nodes  are used, the
interconnects form rectangles. Thomas (1973)
points out that construction errors of more than
5 to 8 degrees from the 90° limit will result in
significant computational errors which cannot be
easily identified in the results.
     Difference approximations may be developed
using truncated Taylor series (that is, only the first
few terms in the series are included). Taylor series
are used frequently in ground-water hydrology for
many purposes. For example, if water density is

                    Nodal Area with Surface S

                             node
Fig. 4. Polygon geometry (after Thomas, 1973).
a weakly dependent function of temperature, it
may be expanded about some reference density
using a Taylor series. Usually the series is truncated
after the first derivative (i.e., truncated in first-
order terms), resulting in a linear relationship
between density and temperature. For numerical
methods, truncated Taylor series may be used to
approximate the derivatives in partial differential
equations. Alternatively, a more intuitive approach
can be used  to obtain the final equations by
considering the  fluxes into and out of a finite-
difference block. This second approach  is essentially
that used to derive the equations for the IFDM.
Prickett and Lonnquist (1971) develop finite-
difference equations using the second approach and
a mesh-centered grid; whereas Freeze and Cherry
(1979) present a development based on the first
approach and a  block-centered grid. A similar
approach to Freeze and Cherry's is presented in
Appendix 1.
     Regardless of the approach, the final result is
an algebraic equation for each node in the grid
system. For a rectangular grid the form  of a typical
equation is
The notation in equation (1) refers to the nodal
locations shown in Figure 3 where h is the head at
the designated node; the explicit definitions of the
coefficients Bjj, DJJ, EJJ, FJJ and HJJ are not given
here, but may be found, for example, in Freeze and
Cherry (1979). The main reason for presenting
equation (1) is to demonstrate the form of the
algebraic equation. This equation is for an arbitrary
node (i,j) and as may be seen, it has contributions
from four adjacent nodes: north (i,j+l), east (i+l,j),
south (i,j-l), and west (i-l,j). These are evaluated
at the new time level (n) and are related to some
known quantity, Q"J ,  which is computed from
information at the old time level (n-1).
     Writing an equation similar to (1) for each node
results in N equations with N unknown head values
to be determined, where N is the total number of
nodes. This may be formulated in matrix form and
solved using matrix methods. These are discussed
in a later section.
     A partial history of the use of this approach in
solving the ground-water flow equation may be
found in Finder and Bredehoeft (1968) and  Remson
et al. (1971). Details of this approach applied to
petroleum problems similar to ground-water

-------
problems may be found in Crichlow (1977) and
Peaceman (1977).

      FINITE-ELEMENT METHODS (FEM)
     There are two fundamental problems in
calculus: (1) examining the area under a curve, i.e.,
integration; and  (2) examining the tangent of a
curve at a point, i.e., differentiation. Both of these
concepts were fairly well understood by the
seventeenth century. For example, Archimedes
demonstrated an understanding of integration by
deriving his approximation for JT. However, it was
not until 1667 that Isaac Barrow (1630-1677),
the teacher of Newton, discovered that integration
and differentiation are essentially inverse to one
another, which is the fundamental theorem of
calculus (AHendoerfer and Oakely, 1959).
     Whereas FDM approximates differential
equations by a differential approach, FEM
approximates differential equations by an integral
approach. Based on the fundamental theorem, one
would expect the two methods to be related and
to converge to the same solution, but perhaps
from different directions.
     The FEM actually refers to the numerical
method whereby a region is divided into subregions
called elements,  whose shapes are determined by a
set of points called nodes (see Figure 5). Note that
flexibility of elements enables consideration of
regions with complex geometry; for  example, a
water-table aquifer with a meandering river can
be outlined with elements fairly accurately.
     For transient problems, the time domain  may
also be approximated using finite elements. In
general, however, most studies use finite-difference
approximations for the time derivatives.
         ^Aquifer Boundary
Fig. 5. Finite-element configuration showing typical node
and element.
     Triangles were used in the grid shown in
Figure 5; however, several  other element shapes
may be used. For one-dimensional problems,
the elements are lines; for two dimensions, the
elements may be either triangles or quadrilaterals;
and for three dimensions, they are tetrahedrons or
prisms.
     The first step in applying the FEM, as shown
in Figure 1,  is to derive an  integral representation
of the partial differential equation. This may be
accomplished by several methods; two of the more
popular ones include: (1) the method of weighted
residuals and (2) the variational method. In the
method of weighted residuals (see for example,
Finlayson, 1972), one works directly with the
differential equation  and boundary conditions,
whereas in the variational method (see for
example, Zienkiewicz, 1971), one uses a functional
(a function of a function) related to  the differential
equation and boundary conditions. The mathematics
of both of these approaches is fairly  straightforward,
but not intuitive.
     The next step is to approximate the dependent
variables (head, concentration or temperature) in
terms of interpolation functions. The interpolation
functions  are called basis functions, and are chosen
to satisfy certain mathematical requirements and
for ease of computation. Although any system of
independent functions can be chosen as the basis
function, piecewise-continuous polynomial sets
are often preferred because they are  both easily
integrated and differentiated. Since the element is
usually small, the interpolation function can be
sufficiently approximated  by a low-order
polynomial, for example, linear, quadratic, or
cubic. As an example, consider a linear basis
function for a  triangular element. This basis
function describes a plane surface including the
values of the dependent variable (head) at the node
points in the element. This is illustrated in Figure 6.
For additional information on basis functions, see
Desai and Abel (1972) or Zienkiewicz (1971).
     Once the basis functions are specified and the
grid designed, the integral relationship must be
expressed for each element as a function of the
coordinates  of all node points of the element.
Next the values of the integrals are calculated for
each element. The values for all elements are
combined, including boundary conditions, to yield
a system of  first-order linear differential equations
in time. As previously mentioned, this is approxi-
mated using finite-difference techniques to produce
a set of algebraic equations. As with  finite-difference
equations, matrix methods are required for solution.

-------
                             hydraulic head surface
                              described by linear basis
                                  function
Fig. 6. Surface described by linear basis function for a
triangular element.
     For additional information on the FEM as
used in ground water, see Verruijt (1970), Remson
etal. (1971) and Finder and Gray (1977).

      MATRIX SOLUTION TECHNIQUES
     As we have seen, each numerical approximation
leads to an algebraic equation for each node point.
These are combined to  form a matrix equation,
that is, a set of N equations with N unknown, where
N is the number of nodes. The general form of
these equations, written in matrix form is
                   Ah =
(2)
where A is a matrix containing coefficients related
to grid spacing and to aquifer properties, such as
transmissivity; h is a vector containing the dependent
variables to be determined, for example, head values
at each node; and d is a vector containing all known
information, for example, specified pumpage and
boundary  condition information.
     In general, a matrix equation may be solved
numerically by one of two basic ways: (1) direct
and (2) iterative. Some solutions may involve a
combination of the two. In direct methods a
sequence of operations is performed only
once, providing a solution that is exact, except for
machine round-off error. Iterative methods attempt
solution by a process of successive approximation.
They involve making an initial guess at the matrix
solution, then improving this guess by some
iterative process until an error criterion  is attained.
Therefore, in  these techniques, one must be
concerned with convergence, and the rate of
convergence.
     Although solving the matrix equation is a
mathematical problem, the hydrologist must be
aware of some of its important aspects, since
generally the matrix solution is the most expensive
part of the computer costs. In very general terms,
iterative techniques are more efficient than direct
solution techniques for matrix equations that
contain more than 1,000 unknowns. The relative
merits of direct and iterative methods are shown
in Table 1. it should also be pointed out that for
some  problems, the matrix A does not have to be
regenerated each_time step. For a direct method,
this means that A is decomposed only once, and a
subsequent time step requires only back substitu-
tion. Since back substitution is much less
expensive than decomposition (elimination), this
improves the efficiency of direct methods
considerably.

Direct Methods
     Direct methods can be further subdivided into:
(1) solution by determinants, (2) solution by
successive elimination of the unknowns, and
(3) solution by matrix inversion. According to
Narasimhan and Witherspoon (1977), perhaps
the most widely used direct approach for transient
problems is that of successive elimination and
back substitution, which includes the Gaussian
elimination method (Scarborough, 1966) and the
Cholesky decomposition method (Weaver, 1967).
     As shown in Table  1, direct methods have
two main  disadvantages. The first problem deals
with storage requirements and computation time
for large problems. The matrix in equation (2) is
sparse (contains many zero values) and in order to
minimize computational effort, several techniques
have been proposed. Various schemes of numbering
the nodes have been studied; an efficient one for
finite-difference nodes is alternating direction  (D4)
ordering (Price and Coats, 1974). Other methods
have been attempted with the finite-element
method. However, for both finite-difference and
finite-element methods storage requirements may
still prove to be unavoidably large for three-
dimensional problems.
     The second problem with direct methods deals
with round-off errors. Because many arithmetic
operations are performed, round-off errors can
accumulate for certain types of matrices.

Iterative Methods
     Iterative schemes avoid the need for storing
large matrices, which make them attractive for
solving problems with many unknowns. Numerous
schemes have been developed; a few of the more
commonly used ones include successive over-
relaxation methods (Varga, 1962), alternating
direction implicit procedure (Douglas and
Rachford,  1956), iterative alternating direction
                                                                                               399

-------
                      Table 1. Advantages and Disadvantages of Direct and Iterative Methods
               Advantages
                                      Disadvantages
 Sequence of operations performed only once.
 No initial estimates required.
 No iteration parameters required.
 No tolerance required.


 Efficient in terms of storage and computation
  time for large problems.
                                           DIRECT METHODS
ITERATIVE METHODS
                           May be inefficient in terms of storage and
                            computation time for large problems.
                           Can have round-off errors.
                           Requires initial estimates.
                           Requires iteration parameters.
                           Requires tolerance.
                           Matrix must be well conditioned.
 implicit procedure (Wachpress and Habetler,
 1960), and the strongly implicit procedure
 (Stone, 1968).
     Since iterative methods start with an initial
 estimate for the solution, the efficiency of the
 method is dependent on this initial guess. This
 makes the iterative approach less desirable for
 solving steady-state problems (Narasimhan and
 Witherspoon, 1977). To speed up the iterative
 process, relaxation and acceleration factors are
 used. Unfortunately, the definition of best values
 for these factors is often problem dependent. In
 addition, iterative approaches require that an error
 tolerance be specified to stop the iterative process.
 This, too, may be problem dependent.
     According to Narasimhan and Witherspoon
 (1977), perhaps the greatest limitation  of the
 iterative schemes is the requirement that the
 matrix be well conditioned. An ill-conditioned
 matrix can drastically affect the rate of convergence
 or even prevent convergence. An example of an
 ill-conditioned matrix is one in which the main
 diagonal terms are much smaller than other terms
 in the  matrix. Such matrices can result  from finite-
 element applications.

             SPECIAL TECHNIQUES
     For certain types of applications serious
 numerical difficulties are encountered. Sometimes
 these difficulties manifest themselves as
 unrealistic  or inaccurate results. At other times,
 there may be no results at all—the numerical
 solution will "blow up." The first type  of
 difficulty is associated with convective-dominated
 transport problems, whereas the second type of
difficulty is associated with nonlinear problems.
 Special techniques have been developed to deal
with these problems, and some of these are
 discussed below.
            Method of Characteristics (MOC)
                 The motivation for the method of character-
            istics (also known as particle-in-cell method) lies
            in the numerical difficulties that occur when
            solving a convection-dominated transport problem.
            With conventional finite-element and finite-
            difference approaches we have a choice between
            solutions that show numerical dispersion or
            numerical oscillation. Numerical dispersion yields
            answers that are smeared out. Oscillatory solutions
            are not smeared out, but lead to physically
            unaesthetic consequences such as large negative
            concentrations for solute transport problems. To
            illustrate the two types of problems, consider the
            physical situation  of one-dimensional convection
            between a line of injection wells and a line of
            pumping wells in a confined aquifer. If the
            concentration of the injection fluid is held at some
            constant value different from that of the aquifer
            and if there is no physical dispersion or chemical
            processes active, the concentration will move out
            as a sharp front  (see Figure 7). Also shown in
            Figure 7 are typical results that demonstrate
            numerical dispersion and numerical oscillation.
            Usually numerical oscillation is associated with the
            finite-element method and numerical dispersion
1
            a>
            o
            o
            O
                                 -^
                               .A
oscillatory

<— analytical
                               numerical dispersion
                  Distance from injection well	*
            Fig. 7. Typical numerical solutions for transport problem
            with convection only.
400

-------
 is associated with the finite-difference method.
 However, depending upon the approximation used
 for the convective term, both methods can
 demonstrate either type of behavior. The method of
 characteristics was devised to alleviate these
 numerical difficulties.
      The MOC has been used in ground-water
 applications to solve the solute transport equation
 (see for example, Reddell and Sunada, 1970; or
 Finder and Cooper, 1970). Although not a require-
 ment, it is usually used in conjunction with a
 finite-difference method; that is, finite-difference
 approximations are used for the flow equation
 and the finite-difference grid is utilized in solving
 the transport equation.
      The approach is not to solve the transport
 equation directly, but rather to solve an equivalent
 system of ordinary differential equations. The
 ordinary differential equations are obtained by
 rewriting the transport equation using the fluid
 particles as the point of reference. That is, instead
 of observing how the concentration changes with
 time at a fixed position in space, we observe
 changing concentration as we move with the
 fluid. Therefore, we need to know the velocity
 distribution. In two dimensions, the end result is
 three equations for x-velocity, y-velocity, and
 concentration, the solutions of which are called
 the characteristic curves, hence the name, method
 of characteristics.
      This is accomplished numerically by intro-
 ducing a set of moving points (or reference particles)
 that can be traced within the stationary coordinates
 of a finite-difference grid (Konikow and Bredehoeft,
 1978). Points are placed in each finite-difference
 block and then allowed to move a distance propor-
 tional to the length  of the time increment and the
 velocity at that point (see Figure 8). The moving
 points effectively simulate convective transport
 because the  concentration at each node varies as
 different points having different concentrations
 enter and leave the area of that block. Once the
 convective effect is determined, the remaining parts
 of the transport equation are solved using finite-
 difference approximations and matrix methods.
     A more complete discussion of this method  is
presented by Carder, Peaceman, and Pozzi (1964);
Pinder and Cooper (1970); Reddell and Sunada
(1970); Bredehoeft and Pinder (1973); and Konikow
and Bredehoeft (1978). In addition, applications  of
this method to field problems are  presented by
Bredehoeft and Pinder (1973); Konikow and
Bredehoeft (1974); Robertson (1974); Robson
(1974); and Konikow (1977).
Nonlinear Techniques
     A differential equation is nonlinear if it
includes products of dependent variables and/or
derivatives of dependent variables. In unconfined
flow problems, the transmissivity is a function of
the saturated thickness and consequently, a function
of head. The product of transmissivity (as a function
of head) and the second derivative of head results
in a nonlinear flow equation. Some unconfined
problems, however, may be treated as linear if the
water level changes are small compared to the total
saturated thickness of the aquifer. Most confined
ground-water systems are linear.
     For transport problems, the equations are
nonlinear when changes in pressure, concentration,
and temperature cause changes in density,
viscosity, or porosity. With heat transport, this is
generally the case; however, many solute problems
may be approximated as linear systems because
concentrations are too low to affect the flow field.
     In general, linear problems are easier to solve
than nonlinear ones, and superposition  may be used
to add (or subtract) solutions.  The principle of
superposition simply states that any linear combina-
tion of solutions to a linear equation is  itself a
solution. In terms of ground-water flow, this
means that a drawdown solution can be subtracted
from the natural head distribution to give the
computed head distribution. This is why many
analytical solutions are expressed in terms of
drawdown.
     For nonlinear problems, equation  parameters,
such as density, are changing with time as pressure,
concentration, or temperature  change with time.
This means that the density at  the beginning of a
time step will be different from the density at the
  X  finite-difference
     node
  •  reference particle
  o  new location
      flow line
                              '     .-?
Fig. 8. Finite-difference grid showing reference particles
(after Konikow and Bredehoeft, 1978).
                                                                                                401

-------
end of the time step. If this change is not treated
properly,  the numerical solution may have mass
balance errors.
     In terms of numerical methods, the nonlinear
differential equations result in the matrix equation
(2) having a coefficient matrix that is a function of
the unknown values. In general this requires some
type of iterative procedure. The easiest approach is
to make an initial estimate of the unknowns,
calculate the coefficients and solve the matrix
equation for the unknowns. The solution now
becomes the new estimate and the procedure is
repeated until the computed values of the unknowns
do not change (or change only slightly). Unfortu-
nately, this simple technique does not always
work; that is, it fails to converge to a solution. This
lack of convergence is serious for problems with
severe nonlinearities; problems in which small
changes in unknowns result in large changes in
coefficients. Severe nonlinearities occur in
unsaturated flow problems and in very thin
unconfined aquifers. For these problems better
nonlinear techniques have been developed. Two of
the more general ones are quasilinearization and
extrapolation (see Peaceman, 1977, for details of
the various methods).

       NUMERICAL CONSIDERATIONS
     In applying numerical methods, we are
concerned with three general characteristics of
the solution procedure: (1) accuracy, (2) efficiency,
and (3) stability. Accuracy deals with how well
the discretized solution approximates the solution
to the continuous problem it represents. Efficiency
is a measure of how much computational work and
computer resources are required to obtain a solution.
Stability addresses the question of whether or not
a solution is possible at all. These definitions are,
of course, oversimplifications, but for practical
purposes,  sufficient. The important point to note
is that the reason we have so many variations of
numerical techniques is that each one was
developed to improve at least one of these
characteristics.
     From the study of numerical analysis, we
obtain information on accuracy, efficiency, and
stability. For example, by comparing finite-
difference or finite-element approximations with
Taylor series approximations, their order of
accuracy can be determined. Similar a priori
analysis into a particular method's stability or
efficiency can be made. From such analysis we
may learn that the FEM offers more accurate
approximations than the FDM or that quasilineari-
zation is a more stable nonlinear technique than
extrapolation.
     Even though numerical analysis provides
measures of the important characteristic of
alternative numerical methods, it does not tell us
which one is optimum for a particular application.
The reason for this is that field situations are much
more complex than the ones assumed in any
numerical analysis. For theoretical analysis it is
often assumed that the properties and grid are
uniform, whereas in applications this  is rarely the
case. Even when the theoretical analysis is more
general, it still falls short of the field complexity we
typically simulate. Consequently, the quantitative
results of numerical analysis only serve as qualitative
guides. The information from numerical analysis
must be augmented with some numerical experi-
mentation or practical experience in order to
match numerical techniques to particular
applications.

        PRACTICAL CONSIDERATIONS
Design of Grids
     One of the critical steps in applying a ground-
water model is designing the grid. Intuitively we
would expect that the finer the grid the more
accurate the solution. Numerical analysis confirms
this intuition; therefore, we should use fine grids
where we want accurate solutions, and we can use
coarse grids where details are not important. No
matter whether the FDM, I FDM, or FEM is  used,
some general guidelines should be followed
(Tresconetal., 1976):

     (1) Locate "well" nodes near the physical
location of the  pumping well or center of the well
field.
     (2) Locate boundaries accurately. For distant
boundaries the grid may be expanded, but avoid
large spacings next to small ones.
     (3) Nodes should be placed closer together in
areas where there are large spatial changes in
transmissivity or hydraulic head.
     (4) Align axes of grid with the major directions
of anisotropy (that is, orient grid with major trends).

     Each of the numerical approaches has its own
considerations as well. With the conventional FDM,
for situations where the aquifer boundary is curved
and not square  or rectangular, the node points will
generally not coincide with the boundary. This
results in errors in approximating heads near the
boundary. For aquifers, however, the subsurface
402

-------
 location of the boundaries is seldom known
 accurately enough to be concerned about these
 errors. Boundary approximation is not a serious
 problem with the IFDM or FEM. With  these
 methods, care must be taken to avoid certain node
 and element shapes. The requirements of node
 shapes for the IFDM have already been mentioned.
 The topic is more involved for finite-element
 methods, because of the large variety of element
 types. A good feature of the FEM is the flexibility
 to use higher-order elements (for example, cubic
 or parabolic) in areas where accurate solutions are
 desired. For details of the practical design of finite-
 element grids, see Finder and Gray (1977).

 Initial Conditions
     In many ground-water flow simulations, the
 important results are not the computed heads but
 the changes in head (drawdown) caused by a stress
 such as pumping wells. For this objective in a
 confined aquifer, for which the equations are linear,
 there is no need to impose the natural flow system
 as the initial condition,  since the computed
 drawdowns can be superimposed on the natural
 flow system. Therefore, the initial conditions are
 simply zero drawdown everywhere.
     For nonlinear problems (e.g. water-table
 conditions), a head distribution must be specified
 as initial  conditions. In this case, boundary  and
 initial conditions must be  compatible. To achieve
 this, the transient simulation should start from an
 equilibrium or steady-state position. To start from
 steady-state conditions in  which flow is occurring,
 a model can be used to compute the initial head
 by leaving out the new stresses or changes in
 stresses (e.g., wells) and setting all storage
 coefficients to zero.

 Choice of Time Step
     An appropriate  choice of time steps is  of
 considerable practical interest in efficiently solving
 a given problem (Narasimhan and Witherspoon,
 1977). One scheme is to progressively increase  the
 size of the time step with  advancing time. This
 scheme is particularly useful for variable pumping
 rates, where the time step  is reduced at the
 beginning of each pumping period and  allowed to
 increase (see Figure 9). Such progressive adjustments
 to the time step may be either arbitrary or
 determined by specific system behavior. While  these
 adjustments may be relatively easy to make in
 linear problems with smooth boundary conditions,
 mass balance errors may result in strongly nonlinear
problems with time-dependent boundary conditions.
                            actual pumping rate
                             pumping rate in model
            time step    pumping period
                   Time
Fig. 9. Example of idealization of variable pumping rates,
showing how time step is allowed to increase over each
pumping period (after Prickett and  Lonnquist, 1971).
Treating Heterogeneities
     Field problems in ground water almost
invariably demonstrate spatial variation in material
properties. In the FEM, integration is performed
over each element for which the material properties
are specified. If material properties do not vary
within each element, then heterogeneity is
accounted for when element integrations are
combined.  If a material property does vary spatially
within an element, Finder et a!. (1973) describe a
way of accounting for the variability using
functional coefficients.
     For FDM and  IFDM, mean values of material
properties are assigned to the node points and are
assumed constant over the block or polygon. The
fluxes between nodes, however, are evaluated at
the interfaces between blocks or polygons.
Therefore,  appropriate mean values of permeability
or transmissivity are used at the interfaces where the
fluxes are evaluated. In many finite-difference
models, this is referred to as interblock trans-
missivity. There are several ways to evaluate these
terms (see, for example, Appel, 1976), however,
one common way is to use the harmonic mean. Use
of the harmonic mean (1) insures continuity across
block boundaries at steady state even if a variable
grid is used, and (2) makes the appropriate
coefficients zero at no-flow boundaries (Trescott
etai, 1976).
     For nonlinear  problems, the interblock
transmissivity terms may contain a parameter that
is a function of head (for example, saturated
thickness in a water-table aquifer). For this case,
the upstream value  of the saturated thickness may
be used. The upstream node is located by
                                                                                                403

-------
 comparing the heads at adjacent nodes, and the
 interblock terms are evaluated using the thickness
 at the node having the greater head. Upstream
 weighting yields a lower-order approximation of
 the spatial derivatives but generally exhibits a more
 stable solution.

 Boundary Conditions
      Rigorous treatment of boundary conditions
 for FDM may be found in references such as von
 Rosenberg (1969); only a few helpful hints are
 given here. For a constant head boundary, the value
 of head at the node is known and need not be
 solved. This may be accomplished by not solving
 the finite-difference equation at the node  in
 question, but another approach is to solve the
 trivial equation
 where h*; is the prescribed head and n is the new
 time level. A trivial equation can be created from
 the usual finite-difference equation by setting the
 storage coefficient equal to infinity (that is, a very
 large number, e.g., 107) and setting h".:  equal to
 h*; on the right-hand side (known side).
     For an impermeable boundary, that is, no
 flow (constant flux of zero), the interblock trans-
 missivities along the boundaries are set to zero in
 the FDM. That is, using equation ( 1 ), for i = 1 ,
 B = 0;fori = NC, H = 0; for j = 1,  D = 0; and for
j = NR. F = 0, where NC is the number of columns
and NR is the number of rows. This eliminates the
need for an  extra boundary of blocks used  in some
models, e.g., Trescott ct al. (1976). For the FEM,
the boundary flux is incorporated in a surface
integral ; when the flux is zero, this term vanishes.
Finally, for  constant flux, not equal to zero, source
terms may be used as approximations and included
in the right-hand side of the matrix equation.
     Note that where it is impractical to include
one or more physical boundaries (e.g., an alluvial
valley that may be  extremely long), the
grid can be expanded to an artificial boundary.
The artificial boundary should be located far
enough from the project area so that it will have
negligible effect on the area of interest during the
simulation period, but can be much closer than
the physical boundary. In this case the boundary
condition is arbitrary (e.g., impermeable
conditions), but the influence of the artificial
boundary should be checked by comparing the
results of two simulation runs using different
artificial boundary conditions.
 Quality Control
      The steps in developing a numerical model
 consist of different levels of error elimination.
 The first step :.<• to compile the program to remove
 FORTRAN errors. Next, the numerical solution
 is compared vn:h analytical solutions to remove
 logic errors in solving the equation. Numerical
 solution? ;;re compared with laboratory ana field
 observe:-ans to remove logic errors in equations
 describing the physics. Finally, it is good
 programming practice to include mass and energy
 (if needed) balances as checks that the  model is
 working properly.

   PROS AMD CONS OF VARIOUS METHODS
     Many of the pros and  cons associated with
 the various methods have already been  discussed
 and some are listed in Table 2. These are
 summarized below according to the following
 categories: (i) ease in understanding the theoretical
 basis, (2) ease in programming, (3) ease in designing
 grid and preparing data input, (4) handling complex
 geometries, (.5) approximation accuracy, (6) handling
 tensors, (7) handling low dispersion, and (8) solution
 efficiency.
     The FDM is based on truncated Taylor series,
 which are similar to the definition of the derivatives
 that they approximate. The theory is therefore
 straightforward and relatively easy  to understand.
 According to Finder and Frind (1972),  "The
 theoretical development of the Galerkin method of
 approximation is possibly more abstract than finite
 difference rhc-orv. ' As for the MOC, Konikow and
 BredehccK v; •"". i) state, "Although it is difficult
 to present a rigorous mathematical  proof for this
 numerical scheme, it has been successfully applied
 to a variety of field problems." The I FDM again
 uses difference approximations and is therefore
 fairly straightforward.

   Table 2. Brief Summary of Important  Advantages
         and Disadvantages of FDM and  FEM
           (as They  Are Commonly Used)
       Advantages
Disadvantages
          FINITE-DIFFERENCE METHOD
Intuitive basis.                  Low accuracy for some
Easy data input.                 problems.
Efficient matrix techniques.       Regular grids.
Program changes easy.

            FINITE-ELEMENT METHOD
Flexible geometry.              Mathematical basis is
High accuracy easily included.       advanced.
Evaluates cross-product terms      Difficult data input.
 better.                       Difficult programming.
404

-------
     Finite-difference approximations are relatively
 easy to program; the resulting code is usually short in
 length (see for example, Prickett and Lonnquist,
 1971).  Again quoting Finder and Frind on the FEM,
 the development of an efficient computer code for
 the Galerkin procedure is a formidable task. Accord-
 ing to Finder (1973),  "Although it was simple  in
 concept, the method of characteristics proved to be
 tedious to program for the computer and was not
 suitable for several situations commonly encountered
 in the field." Because of the additional geometry
 considerations, the IFDM is  perhaps slightly more
 difficult to program than the FDM.
     Designing a finite-difference grid is not a
 difficult task if the few rules outlined previously are
 followed. It simply consists of intersecting
 perpendicular lines. As for data input, only spacings
 in the respective directions are required. Designing
 a finite-element grid requires considerably more time
 and effort than that required by the FDM.
 According to Finder (1973), the design of the
 finite-element mesh is probably the most critical
 step in applying the Galerkin, finite-element
 approach to field problems.  Furthermore, as
 Finder and Frind (1972) point out, experience has
 shown that errors in the input of nodal locations
 in the Galerkin model can lead to problems that
 are difficult to detect; this problem does not arise
 in the finite-difference model because the entire
 grid is specified by  the spacing between rows and
 columns. Since the MOC uses a finite-difference
 grid, it requires the same amount of effort as the
 FDM. In the IFDM, care must be taken to design
 the mesh so that the lines joining nodal points
 coincide with the normals to the interfaces between
 the points (Narasimhan and Witherspoon, 1976).
 This restriction, as well as the requirement for
 providing geometrical parameters as input data,
 may require added effort in the design of
 networks for complex problems (Narasimhan and
 Witherspoon, 1976).
     The FDM grid consists of rectangles and,
 therefore, approximating a complex curved
 boundary is somewhat difficult. For many ground-
 water applications,  the subsurface boundaries are
 not that well known and this difficulty is  not a
 problem. These same comments apply to the
 MOC. On the other hand, both the FEM and
 the IFDM can  be used to analyze systems with
 complex geometry. In fact, for ground-water flow
 modeling, Finder and Frind (1972) state that in
the final analysis the primary advantage of the
Galerkin approach to digital  modeling of aquifer
systems is its flexibility in application.
     A common misconception is that the finite-
element method is inherently more accurate than
the finite-difference method. By using parabolic
or cubic basis functions, high-order (higher-
accuracy) solutions can be obtained with the
finite-element method. High-order approximations
can also be derived for finite-difference methods,
but the procedure is somewhat cumbersome.
Consequently, although it is not necessary,
finite-difference models commonly use
low-order approximations, whereas finite-element
models often have convenient options for higher-
order approximations. Perhaps another way of
saying this is that a carefully designed model
using finite elements may provide the same
accuracy as a finite-difference model that uses
many more nodes (Finder and Frind,  1972).
As we will see, however, this has little correlation
with the relative costs of the two methods.
     In general, the finite-element method handles
tensor parameters that include cross-product
terms (e.g., for conductivity, Kxy) better than
the finite-difference method. These terms are not
well treated with the conventional finite-difference
approximation, because diagonal Unkings between
adjacent nodes are not considered (see Appendix 1).
Related to this are grid-orientation effects (i.e.,
different solutions depending on the orientation of
the grid).  Both of these problems can be
eliminated by including (at some extra expense)
the diagonal linkings in the finite-difference
approximation. The finite-element method does
not require any modifications because diagonal
linkings are naturally included.
     Over the past several years, many researchers
have attempted to solve transport problems using
the FDM, and for various reasons felt that the
results were inadequate. One of the difficult, but
perhaps not commonly encountered, problems in
flow through porous media involves sharp fronts.
A sharp front refers to a large change  in a
dependent variable (e.g., concentration) over a
small distance. The most common complaint
about low-order, FDM and IFDM applied to sharp
front problems is that the computed front is
"smeared  out." As previously discussed, the
process by which the front becomes smeared is
generally referred to as numerical dispersion or
diffusion (Lantz,  1971). An application of FDM
applied to both the solute and heat transport
equations, as well as a summary on numerical
diffusion may be found in INTERCOMP (1976).
In general, the FDM requires very small spacing
to obtain reasonable results for these problems.
                                                                                               405

-------
      Sharp fronts are encountered in transport
 problems if hydrodynamic dispersion is
 v.c. ghgible. This behavior is particularly  amenable
 re1 solution by the MOC. The MOC has  minimal
 numericai diffusion and is not constrained by
 sr.iaii spacing. Other attempts to more accurately
 scivrr the transport equations generally  make use
 r>i rhc FEM. In general, for linear problems the
 l:Lv. car truck srurp fronts fairly accurately,
 v/hich reduces considerably the  numerical
 diffusion problem. The FEM, however, is not
 problem-tree and for nonlinear problems can
 demonstrate numerical oscillation that  becomes
 unstable (Mercer and Faust,  1976). For a further
 discussion on this topic see Anderson (1979), and
 Pindcr and Gray (1977).
     The final topic of comparison is solution
 efficiency. This is a difficult  topic to assess since
 each computer code incorporates various
 techniques TO improve efficiency-, only very general
 concents are discussed. Generating the matrix
 ;:v;u:uicn for the FDM is fairly straightforward and
 resu'ts .r. a matrix having  properties that allow
 efficient solution (in terms of computer storage
 ;;nd time) by several different means. The FEM, on
 the other hand, requires integration over each
 element before the final matrix equation can be
 formulated. This integration  process can be very
 nme-::onsuming. Further, once the matrix is
 generated, it generally requires more storage and
 computer time to solve than  that generated by
 •he F DM. The IFDM also  requires integration and
 results in a matrix that generally requires more
 •.:--!:.:._:'j  ::..;- ?. FDM matrix. An attempt to reduce
 i'i  ume required for solution of an IFDM matrix
 tuuaur.i) is to use an explicit-implicit procedure
 (\iiiasimhan and Witherspoon, 1976). As for the
MOC, tracking points and iterating between the
 convective and dispersive parts of the transport
equation can be fairly expensive.
                   SUMMARY
     The numerical techniques commonly used in
ground-water applications are variations of two
general methods—the finite-difference method
(including the integrated finite-difference method)
and the finite-element method. Occasionally,
specialized techniques such as the method of
characteristics are also used. All of these methods
approximate the continuous partial differential
equations with discrete equarions, requiring matrix
solution.
     There are two basic ways to solve matrix

406
 equations numerically.- (1) direct and (2) iterative.
 In the direct methods, a sequence of operations is
 performed only once, and the results obtained are
 an approximation to the true results. The iterative
 methods attempt solution by a process of
 successive approximation.
     No particular combination of numerical
 technique and matrix solution procedure is best
 for all applications. For most ground-water flow
 problems, the FDM is probably adequate. For
 sharp-front problems the MOC or  FEM will
 probably give better results. For deformation
 problems, the FEM is better because of its
 treatment of tensorial parameters. For any given
 class of problems the choice of the best approach
 depends on the processes being modeled, the
 accuracy desired, and the effort that can be
 expanded on obtaining a solution. Oftentimes,
 however, the hydrologist simply uses a technique
 that he is familiar with, and a computer code
 that is well documented.

             ACKNOWLEDGMENTS
     This effort was supported by the Holcomb
 Research Institute, Butler University, which is, in
 part, supported by EPA grant R-803713. The
 authors also wish to acknowledge the National
 Water Well Association's contribution for drafting,
 editing and publication. Finally, many  of the
 ideas presented in this series of papers were
 formulated during the authors' participation in
 training courses for the U.S. Geological Survey.

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      Charles R. Faust received his B.S. and Ph.D. in
Geology from Pennsylvania State University in 1967 and
1976. Until recently, be was a Hydrologist -with the Water
Resources Division,  U.S. Geological Survey. His interests
there involved thermal pollution in rivers, geothermal
                                                                                                              407

-------
reservoir simulation, and fluid flow in fractured rocks.
Presently he is a principal with GeoTrans, Inc., where be is
interested in quantitative evaluation of hazardous waste and
radionuclide migration in fractured rocks.
     James W. Mercer attended Florida State University
and University of Illinois, where he received his Ph.D. in
Geology in 1973. Prior to graduation, be worked for Exxon
Oil Corporation and the Desert Research Institute,
University of Nevada. Most recently, be worked for the
L.'.S Geological Survey, and is currently President of
Gfn'frjns. inc. Dr. Mercer has taught ground-water
modeling at the U.S.G.S. and at George Washington
University. He has also coautbored several articles on
modeling transport problems in ground water.
       APPENDIX 1. DEVELOPMENT OF
     FINITE-DIFFERENCE EQUATION FOR
      TRANSIENT FLOW IN A CONFINED
    HOMOGENEOUS,  ISOTROPIC AQUIFER
     When using FDM to solve the ground-water
 flow equation, as shown in Figure 1.1, a network of
 ycid points is first established throughout the region
 of interest. As may be seen, the aquifer is divided
 into rectangular blocks  having a thickness equal to
 that of the aquifer, b. Each block has hydrogeologic
 properties associated with it and a node at its center
 at which the hydraulic head is defined for the entire
 block. Also note from Figure 1.1 that some of the
 blocks may be the site of wells.
     The next step is to take a water balance over a
 typical block (see Figure 1.2). To do this we
 consider one of the interior blocks and the four
 blocks connected to it.  In this case, the center
 block is labeled  1; and Q2I, for example, represents
 the volumetric flux from block 2  to block 1; Ax and
£y are the spacings in the x- and y-directions,
                                                     Fig. 1.2. Water balance over a typical finite-difference block
                                                     (after Freeze and Cherry, 1979).
                                                     respectively. The equation of continuity for
                                                     transient, saturated flow states that the net rate of
                                                     flow into any block must equal the time rate of
                                                     change of storage within that block. With reference
                                                     to Figure 1.2, the water balance for block  1 may
                                                     be written as
, +Q-
     -M
                                                                                          ,
                                                                        Qsl=Ax,Ay1S1 — - ,    (1.1)
                                                                                         dt
                                                     where S, is the storage coefficient for block 1. In the
                                                     general case, the source/sink term, W, should also
                                                     be included in equation (1.1); it was omitted in this
                                                     development for convenience.
                                                          The third step in developing a finite-difference
                                                     approximation to the ground-water flow equation
                                                     is to evaluate the volumetric fluxes using Darcy's
                                                     equation. For example, the flux Q2i may be
                                                     evaluated as
                                                                              3h
                                                                 j, =Ax,Tai ( —
                                                                                                (1.2)
                 /
   *S/y\&/~/*/
   * tf y*   - f* jf *-   *•   f
                      /./•/./
                                              b
                                             T
 123   4    5   6  ' * *
          JL
Fig. 1.1. Finite-difference grid, showing typical node
connections lafter Freeze and Cherry, 1979).
                                                     where T2i is a representative value for the trans-
                                                     missivity between nodes 1 and 2. Similar expressions
                                                     may be written for QJI, Q41 and Qs,.
                                                         Using Figure 1.3, the hydraulic head derivative
                                                     may be evaluated as
                                                                     Ax,T
                                                                             h,-h.
                                                                              Ay
                                (1.3)
                                                     which is a cord slope approximation. For simplifica-
                                                     tion, we assume the aquifer is homogeneous and
                                                     isotropic so that T2! = T3, = T4, = Ts, = T and
                                                     S, = S2 = S3 = S4 = S5 = S. In addition, we assume a
                                                     uniform grid spacing, so that Ax = Ay. Substitution
                                                     for the volumetric fluxes in (1.1) now gives
408

-------
        s
        —
        T
                    ah,
                     ot
                                   ,
                         = — Ax3 —  .     (1.4)
This leaves the time derivative on the right side to
evaluate. Using an expression similar to that in
equation (1.3) gives
ah,
 at
u;: _ u1
n,   ni
   At
                        n -1
                                          (1.5)
where n is the new time level and At is the time step.
     Substitution for the time derivative in equation
(1.4) and using general notation, a finite-difference
equation for node (i,j) may be written as
                    S  Ax2          i
                '•J ~ T  At    ''•>    l']

An equation similar to (1.6) is obtained for each
node in our grid; however, boundary nodes may
require special consideration. In summary, applica-
tion of the FDM to a ground-water flow problem
involves:  (1) divide aquifer into grid blocks with
nodes; (2) node spacing is determined by Ax and
Ay; (3) take time steps, At; (4) obtain an  algebraic
equation for each node, and (5) solve matrix
equation.

      APPENDIX  2. NUMERICAL MODEL
                TERMINOLOGY
     Numerical models are often referred to on the
basis of minor characteristics. For example, a finite-
difference model that uses the ADI method for
matrix solution may be called the "ADI model."
Because of the large number of minor variations,
the terminology is confusing. A partial list of
terminology (including those terms most commonly
used) is presented in outline form. Many  of the
                               K«-T-
                                    Potentiometric
                                    Surface
           node
Derivative Midway Between Node 2
and Model   ^  _h;.hi

Fig. 1.3. Approximation of hydraulic head derivative.
terms are defined in the text; others are self-
explanatory. For rhose terms that are not defined,
the interested reade-- is referred to standard texts
on numerics! methods, such as those listed in the
references.

I. Finite-difference methods may be classified by:
  A. Type of Grid
     1. Block-centered
     2. Mesh-centered
     3. Irregular shape (integrated finite-difference)
  B. Type of approximation
     1. Explicit in time
     2. Implicit in time
     3. Central difference in space
     4. Upstream weighting in space
  C. Type of matrix equation solution
     1. Direct methods
       a. Gauss elimination
       b. Choiesky method
       c. Sparse matrix methods
       d. Special ordering techniques (e.g. D4)
     2. Iterative methods
       a. Alternating  direction implicit (ADI)
       b. Strongly implicit procedure (SIP)
       c. Line successive over-relaxation (LSOR)
       d. Point successive over-relaxation (PSOR)
II. Finite-element methods may be classified by:
   A. Type of element
      1. Triangles, 2-D
     2. Quadrilateral,  2-D
     3. Tetrahedron, 3-D
     4. Prism, 3-D
   B. Type of approximating (basis) function
      1. Linear
     2. Quadratic
      3. Cubic
     4. Hermite
   C. Type of method used to obtain integral
      equation
      1. Variational
      2. Weighted residual
       a. Galerkin
       b. Collocation
   D. Type of time approximation
   E. Type of matrix equation solution
III. Special numerical techniques include:
   A. Method of characteristics
   B. Nonlinear techniques
      1. Newton-Raphson method (fully implicit)
      2. Semi-implicit  method
      3. Quasi-linearization
     4. Picard iteration
      5. Extrapolation

-------
                     r r	•
                                                 ies  for
                                                  Studies  and
                                                    Actions
 L->Y .:o3cphF. Keely
   One of the more common techniques for controlling
 the migration of contaminant plumes is the use of
 pumping wells to produce desired changes  in local
 flow rates and hydraulic gradients. When seeking to
 optimize an array of pumping well locations and dis-
 charge rates, it is important to consider the effects
 that non-ideal aquifer conditions, well construction
 and demographic constraints produce. Heterogeneous
 and anisotropic aquifer conditions seriously  compli-
 cate siting and discharge rate requirements forpump-
 ing wciis because of the distorted cones of depression
 that result from withdrawing water in such settings.
 Proi >er screen selection, gravel pack emplacement and
 well development are crucial  factors affecting the
 opera! ionai characteristics and economics of pumping
 wells; these factors are generally recognized,  though
 often undervalued. The impacts that well depth and
 diameter, and screen length and position have on the
 effectiveness of pumping efforts are also often under-
 valued. with detrimental consequences. Perhaps the
 most difficult problems to  overcome in designing
 pu mpi ng schemes, however, are posed by demograph ic
 constraints. Denial of property access, vandalism and
 the unpredictability of nearby water supply and irri-
 gation pumpage tend to wreak havoc with the best of
 pumping strategies.

 Introduction
   Safe storage and disposal of hazardous wastes have
 become major social issues because of the discovery
 that many sites lack proper precautions for the pre-
 vention of soil and water contamination. Ground water
contamination has received the major share of socie-
 ty's attention to these issues, primarily because the
 route of human exposure by this pathway is direct. In
 practical terms, this means that the level of cleanup of
 the damage done by contamination incidents  is often
dictated by social concerns (e.g. health risk). Plume
stabilization by interception and control with perime-
 ter wells, injection and recovery  loops, and other
pumping schemes may be chosen as the "remedial
action" appropriate fora particular plume Theaffected
plume may be held in place and treated, it maybe held
and allowed to move on after alternate public supplies
have been located for downgradient water systems, it
may be  held in  place to allow biodegradation  of
particular constituents, or it may be held until better
treatment procedures can be devised.

Factors Affecting Pumping Strategies
   Hydrodynamic control and recovery strategies vary
considerably in their efficiencies. Besides the obvious
need to choose the well locations and flow rates care-
fully, a number of other considerations demand atten-
tion (Figure  I). Non-ideal aquifer conditions are a
real i ty for virtually all real-1 ife situations: heterogeneity
is the rule rather than the exception. Three-dimen-
sional anisotropy. as expresed by the vertical vs. hori-
zontal hydraulic conductivity ratio, is a near certainty
for most strata. Less visibly pronounced, yet almost as
prevalent, is an expressed anisotropy in the horizontal
plane of many strata. These commonplace non-ideal
aquifer conditions complicate our perception of where
a given plume can go (Fetter 1981) under both natural
flow conditions and remedial action pumping schemes.
The preferential flow paths that are created by buried
lake beds, glacial outwash gravels, streambeds. coastal
deposits and the like cannot  be delineated without
expensive and time-consuming field tests. Likewise, it
is nearly impossible to accurately predict the magni-
tudes of distortion in the cones of depression created
by wells pumping from heterogeneous, anisotropic
aquifers.
  Variations in the properties of the fluid in  an
aquifer,  particularly the solution  density, also can
significantly  affect the behavior of contaminant
plumes (Jorgenson et al. 1982). Immiscible plumes
with lower density than that of the native ground
water will float at the surface  of the saturated zone.
traveling along the same general gradient, but traveling

-------
at a dillerent rate than tne underlying ground water.
Immiscible plumes with greater density than that of
the native ground water will sink through the ground
water, losing small but significant amounts of low
solubility constituents as they move. Miscible plumes
of any density, by definition, mix intimately with native
ground water. The duration of time required to achieve
a specific dilution by this mixing changes markedly,
however, and is generally inversely related to the den-
sity. For most situations, the greater the density, the
shorter the mixing period. The exception to this gen-
eral rule would be the case of a large volume of highly
dense, miscible fluid  penetrating a shallow aquifer
quickly enough to reach bedrock in a relatively undis-
turbed form.

               Considerations

Non-Ideal Aquiler Conditions:
•  Heterogeneity
• Anisotropism
• Variable density

Well Construction Elects:
•  Partial penetration
•  Partial screening
•  Incomplete development

Anthropogenic Influences:
•  Property access
• Vandalism
0  Unknown pumpage/injection

Other Factors:
•  Physiochemical attenuation
•  Biological transformations
•  Operational failures

Figure 1.  General considerations lor optimizing pump-
Ing strategies
   These complexities work against us ifwe are ignor-
ant of them. Working up an appropriate recovery sys-
tem for a contaminant plume can be compared to
designing an oil production system. What you get out
depends directly on what you put in—up to a point.
Where that break-even point comes is hard to say.
given unknowns like the source strength and timing.
and immeasurables like the dollar value of additional
cancer victims. What is abundantly clear, however, is
that there is a substantial minimum for serious play.
One does not blithely draw up plans to pump and treat
a plume until considerable manpower and funds are
expended to obtain information  on the natural flow
direction, gradient and velocity. The question is usually
one of how much to spend to reach some desired level
of detail: the level of detail is set by social concerns.
   This seems to be logical application of technology
for social need, but the logic may be shortsighted. If
social concerns (based on preliminary evaluation of a
contaminant incident) are minimal, there is no guaran-
tee that such social concerns are appropriate. Addi-
tional studies, which could delineate preferential flow
paths and quantify factors affecting contaminant
behavior,  might well generate findings that would
justify considerably greater  or lesser social  concern.
Quite often data from preliminary investigations are
limited to samples from shallow on-site wells, which
may fail to signify the potential impact of dense plumes
or seasonally-occurring ieachate plumes that have
moved off-site. Additionally, the preliminary investiga-
tion wells are not normally installed to a sufficient
depth for appreciation of the local stratigraphic and
lithologic characteristics of the aquifer.
   In  addition to a better understanding of where
contaminants might go with the natural flow, a second
powerful argument to avoid "penny-wise and pound-
foolish" investigations concerns the need to provide
the best information possible for targeting well loca-
tions and pumping strengths in remedial actions. The
occurrence of specific heterogeneities can be used to
advantage by locating wells near low permeabil ity clay
units to generate greater drawdown for a given pump-
ing rate. Likewise, knowledge of the direction of the
principal horizontal axis in anisotropic strata can help
to maximize the arrangement of the "troughs of
depression" for wells to be located in such settings;
knowledge of the magnitude of vertical anisotropy can
help determine the amount of water pumped from
strata containing contaminants vs. the amount of
"clean" water from the other strata open to the well.
   This latter factor,  vertical anisotropy, leads to
examination of some of the more controllable items to
be considered in optimizing pumping strategies—well
construction effects. For example, the impact that par-
tial penetration of a fully screened pumping well can
have on the estimate of potential for contamination of
a water supply or on the effectiveness of a remedial
action scheme is tremendous (Saines 1981). The effect
is to cause exaggerated drawdowns near the well. The
magnitude of the effect is inversely dependent on the
degree of penetration of the well into the aquifer, being
greatest for slight penetration. Naturally, partial screen-
ing of a fully penetrating well results in  the same
effect—greater drawdown fora given pumping rate as
compared with a fully screened, fully penetrating well.
   Again, knowledge of these factors can be used to
enhance a pumping scheme that is, for example, de-
signed to maintain hydrodynamlc control of a plume at
the lowest possible level of pumpage. Lack of appre-
ciation of these well  construction effects can result in
poor estimates of potential contaminant impacts on
supply wells and in poorly designed remedial action
schemes. Another effect worthy of mention is that
generated by well development practices. If a well is
properly developed, the drawdown measurable  inside
the well  will agree with the level projected by close
observation wells. More often,  however, a well  is not
perfectly efficient because the  well development pro-
cedures were not adequate to remove drilling fluid fines
and locally disturbed aquifer material resulting from
the drilling process. These materials lower the permea-
bility  of the gravel pack and  formation immediately
adjacent to  the well. The greater the  degree of  well
Inefficiency caused by lack of proper development the
greater the amount of non-productive drawdown inside
the well: this means that the well may never be able to
pump at design capacity without risk of running dry,
and it means increased operating expense due to the
additional pump lift required. What it may also portend.
for seriously Inefficient wells, is that certain strata
penetrated by the well may be effectively sealed by
drilling mud or by natural clays that were smeared over
the borehole face by the actions of the drilling operation.
Such "sealed off' strata may carry the bulk  of the
contaminants, resulting in poor recovery of the plume.
   Some of the most significant though less control-
lable, factors that should be considered when optimiz-
             
-------
 ing pumping strategies concern direct anthropogenic
 influences: denial of property access, vandalism and
 unknown pumpage all tend to wreak havoc with the
 best laid plans. Bedient et al. (1984) describe efforts to
 delineate a plume of contaminants migrating under a
 residential area from an abandoned wood creosoting
 plant in Conroe. Texas:
      "Several wells exist in the general flow direc-
   tion, but not directly downgradient from the
   waste pit locations. Access was not granted for
   installing monitoring wells.. .Approximately 50
   percent of the chloride plume has been defined
   since the monitoring well network is incomplete
   at this time... Completion of the monitoring well
   network is needed to capture the center of the
   contaminant plume. This will require more wells
   downgradient on land that has not previously
   been accessible for investigation."
    The granting of property access during investiga-
 tions of ground water  contamination incidents in
 populated areas is no trivial matter. One typically finds
 it necessary to contact not only homeowners and land-
 lords for private property access, but also to negotiate
 with company engineers, vice presidents and attorneys
 for commercial property access. It is quite normal for
 such negotiations to be involved and protracted as city
 councils, educational boards, corporate headquarters
 and other bureaucratic entities are asked to concur in
 signing  access agreements  containing provisions
 deemed necessary to ensure against incurred liability
 and potential damage.
    The role  played by  unknown pumping and/or
 injection wells operating near a remedial action pump-
 ing system is subtle but far-reaching. Such unknown
 stresses can significantly  distort the flow field and
 render remedial actions ineffective. Projections on
 plume movement made  during an investigation of a
 ground water contamination incident would also be
 in error if unknown wells are causing distortion in the
 flow field:  both the direction  and the speed of the
 plume could  be dramatically altered. The reason for
 the subtlety of the effects of many such wells is that
 their cyclic, seasonal or on-demand pumping sched-
 ules allow them to be delected only by continuous
 recording of water level changes at numerous points
 around the zone of interest. Since aquifer responsesat
 a given observation point are somewhat non-unique.
 merely detecting extraneous sources of drawdown
 does not automatically result In identification of the
 sources.
   There are a few other important factors to consider
 that also affect pumping strategies. The physlochemi-
 cal properties of the contaminant Itself can result in a
 need to pump several pore volumes from each unit
 volume of aquifer to be decontaminated. Sorption. ion
 exchange  and speciation changes can result in re-
 tarded movement  of contaminants relative to  the
average velocity of the water  with which they  are
 initially associated. Biotransformation of contami-
 nants may result  in  reaction  products  (daughter
 products) that are of greater or lesser toxicity. mobility
and persistence—in other words, uncertain contami-
 nant behavior. Unlike the aquifer propertlesof storage
coefficient, saturated thickness and hydraulic conduc-
 tivity, which can be readily determined, the current
state-of-the-science with regard to determining  the
potentials for physiochemical  attenuation and bio-
transformation is not up  to the  level of routinely
providing reliable answers on a site-specific basis.
   Finally, an obvious but often overlooked considera-
tion involved in optimizing pumping strategies is the
need to develop adequate contingencies for operational
failures. This means some intentional overdesign for
reserve capacity, total redundancy of key wells and
electronic controls, backup power systems and so on.
It also means bonding or insurance against unfore-
seen  catastrophies so that as little downtime is
expended as possible. It may also mean that an escrow
account or trust fund must be established to provide
the necessary capital for replacement of bumed-out or
inadequate  pumps, deepening or abandonment of
existing wells, or drilling of additional wells.

Capture Zones vs. Zones of Pressure
Influences
   Keely and Tsang (1983) introduced the term "cap-
ture zone" to  describe that portion of the aquifer
affected by pumping which actually yields water to the
well. They have shown  that the capture zone is gener-
ally much smaller than the zone of pressure influence
because  a balance is achieved, under  steady-state
conditions, between the pull of water back toward the
well from its downgradient side and  the tendency of
the natural flow system to move on further downgra-
dient. Figure 2 is a series of four idealized illustrations
that present conceptualizations of how the size of the
capture zone changes, relative to the zone of pressure
influence/cone of depression, as the local gradient
is increased. In Figure 2A the well is pumping from a
stagnant aquifer, indicated  by the flat pre-pumping
surface, overlaid on the theoretical cone of depression
that would  occur  during pumpage. For stagnant
aquifer conditions, the capture zone is everywhere
identical to the zone of pressure influence and flow is
radial into the well. As the successive diagrams indi-
cate, however, non-stagnant aquifer conditions lead to
smaller capture zones (Figures 2B to 2D).
   The slopes of the pre-pumping surfaces are overlaid
on the theoretical drawdown cones in each frame of
Figure 2 to emphasize the interaction of the natural
flow system with the pumping stress to yield a capture
zone smaller than the zone of pressure influence. There
is no intention to show the net surface resulting from
pumpage by subtracting theoretical drawdown values
from pre-pumping  water elevations. These  sketches
do have the cosmetic drawback of showing crossing
water level lines/curves, but the point is to illustrate
the individual  components of the net surface (cross-
sectional view) and how they interact to yield a capture
zone (three-dimensional view).
   The flow lines generated by pumping a well from an
Idealized aquifer (homogeneous, isotropic, constant
density, etc.) under different natural flow conditions
are shown In Figure 3. In Figure 3 a well pumping
1 ,OOOm3/day from a 1 Om thick aquifer having a poros-
ity of 0.10 and a hydraulic conductivity of lOOm/day
has uniform radial flow under stagnant aquifer condi-
tions (e.g. natural flow velocity equal to zero). When a
mild hydraulic gradient (O.OO01)  is imposed on the
same system (Figure 3B). the resulting natural flow
velocity (0.1 m/day) is insufficient to significantly aflect
the flow lines, and uniform radial flow is nearly main-
tained. With a more  moderate hydraulic  gradient
(O.OO1). the resulting natural flow velocity (l.Om/day)
is sufficient to sweep away many of the flow lines and
the capture zone Is clearly evident (Figure 3C). Where
a steep gradient (0.01) is present the capture zone
diminishes to a small fraction of the zone of pressure

-------
                            SATURATED ZONE
                                                       CONE OF DEPRESSION
  CKOSS-SCCI10*lli
                                                                                   SATURATED ZONE
                                                         _» OSS-SeCTIOHAL  COHCCPlUHlltttlU*
       CAPTURE ZONE - CONE OF DEPRESSION
                                                             CAPTURE ZONE < CONE OF DEPRESSION
  runic ainf»s/a * *L concipru»Lii»rian




      Figure 2A.  Stagnant aquiier conditions
 CONE O '  OtPRlSSION
                             SATURATED /ONE
                                                         H H f t P 1 ff [ H S 1 C H f L  COHCfP!UtLj:«7IDIt
            Figure 2B. Mild natural gradient
   CUDS*  s r c r i o * K i  co»ctpru*ti/*r i_u » .
                                                          '.SPSS -sfcrjo/r/ti co*Cf?'yti 1 / * r i o
       CAPIURf /ONE « CONE OF DEPRESSION
                                                             CAPTURE  ZONE <« CONE OF DEPRESSION
   T H H f f  0 1 n [ * S 1 O * * L C O » Ct P I V f i I I • ' I P *




        Figure 2C. Moderate natural gradient
Figure 2. Cicra sectional and thr^p
                                                                               COMCCP'UAl f f * ' J " "
            Figure 2D. Steep natural gradient




tualizations of capture zone vs. cone of depression

-------
 500.
    0.
-500.
        VeJ: 0.0 m/d
                   J	L.
                                       l	L.
                                                    500.
                                                      0.
                                                  -500.
   -ROO.                  0-               50°

        Figure 3A. Stagnant aquiler conditions
  -500.                 0.              500.

      Figure 3B. Mild hydraulic gradient (0.0001)
 500.
         V»l: 1.O m/d
                                                    500.
    0.
                          0.              500.

     Hguie 3C. Moderate hydraulic gradient (0.001)
-500 .

   -500.
                                                                              _L
                   0.               500.

Figure 3D. Steep hydraulic gradient (0.01)
 figure 3. Haw line plots lor a single well discharging l.OOOmVday faom cm aquiter with 10m saturated
 tliickness, lOOm/day hydraulic conductivity and 0.10 porosity
 NOTE: S
-------
 500.
  	TT	1	T	1	—]	
V«l: 0.0 m/d
                                                     500.
    0.

-500.
                  _L
                                                            V«l: 0.1 m/d
                                           i	I     -500.
   -500.                  0.               500

        Flguie 4A. Stagnant aquiler conditions
                                              -500.                 0.               500.

                                                  Figure 4B. Mild hydraulic gradient (0.0001)
         iv >l: 1 .O m/d
    0.
         r
                         *
                          ••v*
                                                     500   I—•'•' •'••••'•'—''
                                                     DUU •     l: 10
                                                0.
   -500.                  0.               500.
     Figure 4C. Moderate hydraulic gradient (0.001)
                                                    -500.
                                                                          *..-'.'•'.•'.'••'.•'.'•
                                                                          '••'•'•'.'• :*••'.'•'.'•'
                                              -500.                  0.              500

                                                  Figure 4D. Steep hydraulic gradient (0.01)
 Figure 4. Flow line plots for a line ot tiro wells, each discharging 200mVday from an aquifer with 10m
 saturated thickness, lOOm/day hydraulic conducttvlty and 0.10 porosity

 NOTE: Scale is In meters and natural flow proceeds from lower left-hand comer to upper right-hand corner ol
 each plot art the velocity indicated.
 rate well. By rearranging some of the expressions for
 capture zone dimensions given by Keely and Tsang
 (1983). it Is possible to define the maximum width of
 the capture zone upgradient of the well as: WI1VLX = Q -^
 (h<£r V,nt). Using this relationship.  It Is apparent that
 the  maximum width (Wmax) of the capture zone of a
 well is directly and linearly related to its flow rate (Q).
 and Is Inversely related to the  natural flow velocity
 (Vml).
   For the example discussed here regarding a single
 well pumping 1.000m3/day, the maximum width of
 the  capture zone is 1.000m when the natural flow
 velocity is l.Om/day. and is 100m when the natural
 flow velocity is lOm/day. Each of the five wells in the
 second example discussed pumps at a flow rate equal
 to one-fifth the flow rate of the well  in the first example
                                            (200m3/day). and each, therefore, has a capture zone
                                            the maximum width of which is one-fifth that of the
                                            single well (200m). Hence, by comparing Figure 3 with
                                            Figure 4, it is seen that the way in which the total
                                            pumpage Is distributed does directly affect the distri-
                                            bution of the capture zone(s). but does not affect the
                                            magnitude or total area of the capture zone(s). Also to
                                            be seen  in  Figures 3 and 4 is that increasing the
                                            natural flow velocity estimate can have a dramatic
                                            impact on the effectiveness of the pumping strategy.
                                            Given the  order-of-magnitude uncertainty so often
                                            associated with hydraulic conductivity estimates, it is
                                            not surprising that many seemingly acceptable reme-
                                            dial action schemes are doomed to fail miserably.
                                               A more complicated example provides further Illus-
                                            tration of these points. Assume that we have the same

-------
 500 •
                      *          *
                           *

                                                     500.

                                                   -500.
                                                            V«l: 0.1 ma
                                                                       ....*

   -500.                 0.               500

       Figure 5A. Stagnant aqutter conditions
  -500.                 0.               500.

      Figure 5B. Mild hydraulic gradient (0.0001)

                                                     500.
         V « i.  1.0 m / o
                                                            V.I: 1.0 m/d
    o    -
         -
        .

        •

    o.  r
        K
                                                                         *
                                                                       *
                                                                                    *
                            -
                                                                               *
                                                            -

                                                            -
                                                   -500
                                                                               _.
-500.  .	i- -i  .  J .—i— -L— J-- i    '   -'    J
   -500.                  0.               500.       -500-                 °-               5°°

     Figure 5C. Moderate hydraulic gradient (0.001)      	Figure 5D. Steep hydraulic gradient (0.01)

Figure 5. Flow line plots lor a circle ol eight wells, each discharging 125mVday from an aquller with 10m
saturated thickness, lOOm/day hydraulic conductivity and 0.10 porosity

NOTE: Scale is in meters and natural flow proceeds from lower left-hand comer to upper right-hand comer ol
each plol at the velocity indicated.
aquifer  conditions  and total  pumpage limitation
(1.00Cm3/day) as the preceding  examples. We will
distribute the pumpage uniformly by pumping each of
clp;ht wells at 125mVday. The eight wells are evenly
spaced around a circle of 200m radius. We are trying to
hold a plume within the circle. With stagnant aquifer
conditions to low natural flow velocities, the plume
appears to be stable: no flow lines pass through the
circle (Figures 5A and 5B). At moderate to high natural
flow velocities, however, the situation is quite different;
flow lines readily pass through the circle, indicating
that the plume stabilization attempt has failed (Fig-
ures 5C and 5D).
   A pump and treat scenario can be examined by
modifying the example shown in Figure 5 to change
the operation of the eight wells from pumping to
injecting and by adding a major pumping well in the
center of the circle. The single pumping well will with-
draw  l,OOOm3/day  from the plume. The withdrawn
water will be treated and re-injected into the eight
injections wells at 125m3/day each. At zero to low flow
velocities, the injected water flows radially toward the
central pumping well, forming a closed loop for recov-
ery and treatment of the plume (Figures 6A and 6B). At
moderate to high natural flow velocities, the recovery
loop is  broken and an increasing amount  of the
injected water and  the plume are swept away by the
regional flow (Figures 6C and 6D). It must be empha-
sized that the cones of impression or depression of the
wells overlap significantly for all of the multiwell exam-
ples discussed so far. Despite those overlaps, the net
surface resulting from the natural gradient and the

-------
 500       '       r -  -v
         v.i: 0.0 m/d
                            500.  ''
                                   V«l: O.1 m/d
                                                                                  i   r   i
    0.  -
                              0.  -
-500.
_____ I

 0.
  -500.

Figure 6A. Stagnant aquller conditions
 500               T       1
         v«l: 1.0 m/d
                                          500.
        -500.  _

           -500.
               111       I    I    I    I

                         0.              500
                                                   Figure 6B. Mild hydraulic gradient (0.0001)
                                                     500.  r
                                                     250.  -
                                                                                   *
    C.  -
                                                       0.  -
                                              *       X       *
                                                                         *      ._•:••  *
                                                                              x
                                                    -250.  -
-500.  __i

   -500.
  :
500.
Figuie 6C Moderate hydraulic gradient (0.001)
-500 .   .     	L	u	1	i.. -. .L.

   -500.    -250.       0.      250.

Figure 6D. Steep hydraulic gradient (0.01)
                                                                     500.
Figure6. now line plots lor a single well discharging l.OOOmVday. encircled by eight wells injecting 125m3/
day into an aquifer with 10m saturated thickness. lOOm/day hydraulic conductWty and 0.10 porosity

NOTE: Scale is in meters and natural flowproceeds from lower left-hand coiner to upper right-hand comer of
each plot at the velocity indicated.                                          	^^^
water level changes due to pumpage and/or Injection
Is shaped such that the streamlines are truly as pre-
sented here. For further discussion of capture zones
and velocity distribution plots, see Keely and Tsang
(1983). The detailed theoretical development and
source code listings for the models that were used to
generate the stream line plots shown here are given in
Javandel et al. (1984).

A Little More Detail
   It was quite clear in each of the preceding examples
that the pumping strategy began to fail as the natural
flow velocities became appreciable. The tendency to
fail is generally becoming evident at a natural flow
velocity of l.Om/day and  is beyond question at a
natural flow velocity of lOm/day. Figure 7 shows that
                           failure of each design is certain at S.Om/day as well;
                           the point at which the flow lines break through must
                           be at much lower natural flow velocities.
                              In Figure 8 the natural flow velocity has been
                           reduced to 0.5 and 0.4m/day for the last two examples
                           only. Breakthrough of the streamlines (failure of the
                           pumping strategy) occurs somewhere between the 0.4
                           and 0.5m/day natural flow velocities. Similar compar-
                           isons for the first two examples are not presented
                           because flow line breakthrough does not apply to the
                           first example (a single production well) and the flow
                           line did not indicate breakthrough at l.Om/day for the
                           second example (a line of five wells).
                              The presence of an unknown well is being studied
                           in Figure 9. A major pumping well (1.000m3/day) has
                           been arbitrarily located downgradient of the same line

-------
  50° •      '
         «•
                  T"
500.
                           *
                                              -
                                               •
 -500.
                                                   -500.
                                                                                     -i	r-	r
                                                            V»l: 5.0 m/d
                                                                '.-•'".-•     *
                                                                           K

                                                                              *
   -500.                 0.              500

 Figure 7A. Single well discharging l.OOOmVday

  500.
         vn: 5.0 m/d
                                             -500.                 0.              500.

                                          Figure 7B. Line ol ttve wells, each discharging 200m3/day

                                           500.  r-'  .r
                                                    250.   -
         ,

                      *         *
    0.   r
r            „        ;../:
               •

        •
-500.

   -500.
                 :
    -.
     •i



    4

    j

     I
    1
     :
    -l

	1	J

 500.
                                                       0.  -
                                                   -250.   -
                                                                            *
                                                                        *
                                                      -250.
                        :
250
,.

 500.
Figure 7C Circle  ol eight wells,  each discharging
125mVday
                                           Figure 7D. Single well discharging 1 .OOOmVday. encircled
                                           by eight wells injecting 125mVday each.
 Figure 7. Comparison of pumping aiiays in an aqulter with 10m saturated thickness, lOOm/day hydraulic
 conductivity, 0.10 poroslt7 «™1 0.005 hydraulic gradient

 NOTE: Scale is in meters emA natural flow proceeds from lower left-hand coiner to upper right-hand comer of
 each plot at the velocity indicated.
of five wells discussed in the second example. Naturally.
under stagnant aquifer conditions, the unknown well
creates a hydraulic divide by distorting the flow field,
but It does not cause breakthrough of the flow line
from across the line of five wells (Figure 9A). With a
natural flow velocity of 0.5m/day. however, flow lines
do begin to break through the line of five pumping
wells (Figure 9B). Substantial failure of the pumping
scheme occurs at 1 .Om/day natural flow velocity (Fig-
ure 9C). Contrast  the onset of breakthrough due to
unknown pumpage (Figure 9B) with the same situa-
tion In the absence of the unknown pumpage (Fig-
ure 9D). The Impact of the unknown well is staggering.
not only because flow line breakthroughs are occur-
ring, but the collective size of the capture zones of the
five pumping wells Is being substantially reduced.
                                             Another Illustration of the Impact of an unknown
                                           well on the effectiveness of a pumping scheme is shown
                                           In Figure 10, which Is the same example as discussed
                                           earlier (Figure 6) for a closed-loop aquifer rehabilita-
                                           tion system. Under stagnant aquifer conditions, the
                                           unknown well diverts flow away from two of the Injec-
                                           tion wells (Figure 10A).  At  1.Om/day natural flow
                                           velocity, the unknown well diverts flow from five of the
                                           eight Injection wells (Figure  1 OB). It also allows flow to
                                           break away from the well field entirely, as Indicated by
                                           the streamline  leaving the uppermost Injection well
                                           and heading downgradlent In Figure 10B. The re-
                                           gional flow lines were omitted from Figure 10 and
                                           some of the diagrams In previous figures because
                                           inclusion of those  flow lines would create confusion
                                           due to the excessive number of plotted points.

-------
  500    I •' '   '" " '    '
         V.I: 0.4 m/d
                                         500 .  '    .'•  -T---T-   i
                                                .V.I: O.S m/d
    0.   -
                                            0.  -
                      *          *
                            *
                                                              K          X

                                                                   X
                                                ;     -500.   _

                                           500.       -500.
                                   0.
                                                                                   500
Figure  8A. Circle  ol  eight wells, each discharging     Figure 8B. Circle ol eight  wells,  each discharging
125mVday with 0.0004 hydraulic gradient                 125m3/day, with 0.0005 hydraulic gradient
  500.
                                 '     i
         V.I: 0.4 m/d
                                         500 •   !
                                                                                :
  250.   -
                                         250.  -
                                                                                *
                                                                           *          *
    0.   -
                                                         o.   -
                                                            *       *•
                                                                           X
                                                                                      X
-250.   -
                                        -250.  -
  JOO.
                                                     -500.
    500.
-250.
                           ,
500.
                                                       -500.    -250
                                                                  :
                                          .

                                          250.
500.
Figure BC. Single well discharging l.OOOmVday, encircled
by eight wells injecting  125mVday each, with 0.0004
hydraulic gradient
                                        Figure 8D. Single well discharging 1 ,000m3/day, encircled
                                        by eight wells injecting 125m3/day each, with 0.0005
                                        hydraulic gradient
Figure 8. Detailed views of the onset of flow line breakthroughs tor two plume control strategies in an
aquifer with 10m saturated thickness, 100m/'day hydraulic conductivity and 0.10 porosity
NOTE: Scale is in meters and natural flow proceeds from lower left-hand comer to upper right-hand comer of
each plot at the velocity indicated.
Conclusions
   Heterogeneity, anisotropy. partial penetration and
so on distort drawdown patterns and associated velocity
distributions. If known, such influences can be used to
enhance recovery efficiencies for remedial actions. If
unknown, such influences may cause recovery effi-
ciencies to be substantially lowered. Similarly, predic-
tions of plume migration in non-ideal aquifers under
non-pumping/natural flow conditions will be strength-
ened by specific knowledge regarding the occurrences.
extent  and magnitude of the non-ideal condition(s).
Such predictions may be seriously in error If non-ideal
conditions are not evaluated properly.
   Denial  of property access, loss by  vandalism and
                                        unpredictable operation of nearby wells are also major
                                        sources of uncertainty in predicting contaminant
                                        migration and in designing remedial actions. Though
                                        commonly perceived to be less of an impact on opti-
                                        mizing pumping strategies than  non-ideal aquifer
                                        conditions, these factors may indeed be the most
                                        uncontrollable and  the most detrimental to opera-
                                        tional success. Other factors that have major impacts
                                        are the physiochemical attenuation and biotransfor-
                                        matlon potentials of the individual contaminant; it is
                                        not yet economically feasible to conduct adequately
                                        detailed studies of these potentials on a routine site-
                                        specific basis. Finally, a factor often overlooked that
                                        greatly impacts  optimization efforts Is the risk of

-------


          V»l: O.O m. a
         -

                                  PbO.

 Figure 9 A. Stagnant aquiler conditions



 -250.
-50C .

                                      •
0.
   -500.    -250.

Figure 9C Hydraulic gradient of 0.001
250.
                                           500.
                                                     500.
                                                     250.
                                                        :
                                                    -250.   -
                                                    -500.
                                                             v*l: 0.5 m
                                                                   T
                                                                        :* '•-.'
                                                                           *:
                                          bOO.       -500.    -250.        0.     250.     500
                                                   Figure 9B Hydraulic gradient ol 0.0005
                                                     500.          '
                                                            V.I: 0.6 m/d
                                                       0.  ~
                                                   -500.
                                                     -500.
:
500
                                                   Figure 9D. Hydraulic gradient ol 0.0005—without the
                                                   unknown well
Figure 9. Influence ol an unknown well discharging l.OOOmVdoY on flow line breakthroughs tor a line of
five wells discharging 200mYday each from an aquiler with 10m saturated thickness, lOOm/day hydraulic
conductivity and 0.10 porosity

NOTE: Scale Is In meters and natural flow proceeds from lower left-hand comer to upper right-hand comer of
each plot at the velocity Indicated
mechanical and electrical operational failure; adequate
contingency plans must provldecertaln minimal levels
of excess/reserve capacity and redundancy of key sys-
tem components.
   The capture zones of wells do not equal their asso-
ciated zones of pressure Influence (cones of depres-
sion), except for stagnant aquifer conditions. Velocity
distribution  plots  must be  constructed to  define
potentials of contaminant  migration. In  particular.
plotting the streamlines for various scenarios involv-
ing pumping and/or injection wells subject to a spe-
cific natural flow velocity can greatly assist the ground
water professional in selection of an optimal pumping
strategy.
                                                  Acknowledgments
                                                     Thanks  go to Rosemary Keely and  Christine
                                                  Doughty for (computer) drafting the illustrations.
                                                  Thanks also go  to Renae Daniels for typing  this
                                                  manuscript.

                                                  Disclaimer
                                                     Although this article was produced by an employee
                                                  of the United States Environmental Protection Agency.
                                                  it  has not  been subjected to Agency review  and
                                                  therefore does not necessarily reflect the views of the
                                                  Agency: no official endorsement should be inferred.

-------
  500.
  250.
         V»l: 0.0 m/d

                                        •
                   *       -        *. . • •
                      *
 -250.   -



 -500.  L
                                                     250.  -
                                                       0.  -
                         *.
   -500.    -250.       0.      250.
Figure IOA  Stagnant aquifer conditions
                                          500
500.
 -500.    -250.        0.      250
Figure 10B. Hydraulic gradient ol 0.001
500.
Ficauv I •  influence oi an unknown woli discharging l.OOOinVday on flowline breakthroughs lor a single
well disciiargina 1 ,OGOmJ/day that Is encircled by eight wells injecting 125m3/day into an aquifer with 1 Om
saturated :hlcjmess. lOOrn/day hydraulic conductivity rrnrf 0.10 porosity.
NOTE: Seal*. is in meters and natural flow proceeds from lower left-hand corner to upper right-hand comer of
each i-'iot cd the vwlcdty indicated.
References
Bedient.P.B..A.C.Rodgers,T.C.Bouvette.M.B.Tomson
  and T.H. Wang. 1984. Ground water quality at a
  creosote waste site. Ground Water, v. 22, no. 3. pp.
  318-329
Fetter. C.W. 1981. Determination of the direction of
  ground  water flow. Ground  Water Monitoring
  Review, v. 1. no. 3. pp. 28-31.
Javandei. 1.. C. Doughty and C.F. Tsang. 1984. Ground
  \vater transport: handbook of mathematical models.
          ;i  Geophysical  Union. Water Resources
          .•>!' 10.
Jorgi-ns-ii. :).G., T.  Gogel  and D.C.  Signer. 1982.
         • .-.ation of flow in aquifers containing vari-
       • i-i!si:v  aater. Ground Water Monitoring Re-
  vie-.-.. v. 1. no  2. pp. 4O45.
Keely J.K.  and  C.F. Tsang.  1983. Velocity plots and
  capture zones of pumping centers for ground water
  investigations. Ground Water, v. 21. no.  6. pp.
  701-714.
Saincs, M. 1981. Errors in interpretation of ground
  water level data. Ground Water Monitoring Review,
  v. 1 no. 1. pp. 56-61.
                                                   Biographical Sketch
                                                      Joseph F. Keely (Robert S. Kerr Environmental
                                                      Research Laboratory, U.S. EPA, P.O. Box 1198,
                                                      Ada, OK 7482O) received his B.S. inprofessional
                                                      chemistry and M.S. in hydrology from the Uni-
                                                      versity of Idaho (Moscow). He is employed as a
                                                      hydrologist at EPA's  Robert S.  Kerr Environ-
                                                      mental Research Laboratory, where his efforts
                                                      are directed toward geohydraulic and  hydro-
                                                      geochemical investigations ojground water con-
                                                      tamination  incidents, coordination of ground
                                                      water modeling  research and instructional
                                                      assistance.  He sits on the Policy Board of the
                                                      International  Ground Water Modeling  Center
                                                      and serves as expert witness to the U.S. Depart-
                                                      ment qfJwsticefor Superfund cases.

-------

-------
EQUI
         UALITY
ENT
Data collected  before
or after model
selection?

Spatial variability
controls

Temporal variability
controls

Sensitivity analyses
Comparability

-------
        IGWMC   GROUND*1 AT FR   MODELING   REPRINT
                    Quality Assurance in Computer Simulations of

                             Grourtdwcter  Contamination
                              Paul K.M. van der Heijde

                     Internationa"i Ground k'ater Modeling Center
                                       repri r,t

                                   ."of i^rr-.,  1987, Vol  2, Mo.  1
                                     GWKI 87-08
INTERNATIONAL   GROUND   WATER   MODELING   CENTER

                             Holcomb  Research  Institute
                                  Butler University
                             Indianapolis,  Indiana   46208

-------
                                                                 Groundwater Contamination: P.K.M  van tier Heiide
Quality  Assurance in Computer  Simulations  of
Groundwater Contamination

Paul K.M. van der Heijde
International Ground Water Modelling Center. Holcomb Research Institute. Butler University,
Indianapolis. IN 46208. USA
                                                      ABSTRACT

     In the development of policies and regulations for groundwater  protection, in permitting,  and  1n planning  monitoring
and remedial actions,  the role of mathematical  models  1s growing  rapidly.   Because  water-resource management  decisions
should be based on technically and scientifically sound Methods, quality  assurance (QA) needs to be  applied to groundwater
•odeling, both in Model  development and field studies,  and  should also play  an Important part  1n model selection.

     Important  aspects  of QA 1n  groundwater  Model  development  are  peer review,  and  verification  and  validation of the
computer code and Its underlying  theoretical principles.  This paper  discusses the  role  of review and testing as  part  of an
overall QA approach,  and addresses QA in Model selection and field application.
Key Words:  groundwater.  Mathematical models, quality assurance, model validation, pollution, model selection

                      INTRODUCTION

     The  science  of  groundwater   flow  and  contaminant
transport   1s  not  yet   an  exact  field  of   knowledge.
Although  the  physical   processes  Involved   obey   known
mathematical  and  physical principles, exact  aquifer and
contaminant  characteristics are hard  to  obtain  and  often
make even  plume definition  a  difficult  task.   However,
where  these  characteristics  have  been  reasonably  estab-
lished, groundwater  models may provide a  viable,  if not
the  only,  method  to predict  contaminant  transport,  to
locate   areas  of  potential  environmental  risk,  and  to
assess  possible remediation/corrective actions  |lj.
     Mathematical  models  are used  to help  organize  the
essential  details  of   complex  groundwater  management
problems   so   that  reliable   solutions   are  obtained.
Applications  Include  a wide  range of  technical, economic.
and  sociopolitical aspects  of  groundwater  supply  and
protection |2,  3,  4. 51.


     A  groundwater protection  policy  based on monitoring
1s  by   its very  nature always  reactive,  not preventive;
however, model-based  policies and regulations can be both
preventive and reactive.   Because adequate on-s1te moni-
toring  Is not  always  feasible due  to  costs,  available
manpower,  or  $He accessibility,  models  can provide  a
viable  and effective alternative.   An optimal approach to
the  management  of  groundwater  resources   includes  the
Integrated use of  modeling and monitoring strategies.


     The  role  of  groundwater-flow  and contaminant-trans-
port  models  In  the  development of  policies  and  regula-
tions.  In permitting,  and  1n   planning of monitoring and
remedial action.  1s continuing  to grow.  Some  of the prin-
cipal  areas  where  mathematical  models can now be used to
assist  In the management of groundwater protection pro-
grams are  16):

     •  development of  regulations and  policies

     •  planning and design of corrective  actions and waste
        storage facilities

     •  problem conceptualization and analysis

     •  development of  guidance documents

0266-9B3e/B?/01001»07 S2OO
• 1B87 Co««utation*i Mechanics Pubtcaions
 Paper received on December 8,  1986 Referee: Dr. Paolo Zannetti
     •  design  and  evaluation   of   monitoring  and  data
       collection strategies

     •  enforcement

Specifically,  groundwater  modeling  plays  or can  play  a
role In:

     •  determining or  evaluating the  need  for  regulation
       of   specific  waste   disposal,  agricultural,  and
       industrial practices

     -  analyzing  policy  Impacts such  ss   evaluating  the
       consequences  of setting  regulatory  standards  and
       banning rules, and of  dellsting actions

     •  assessing  exposure,  hazard,   damage,  and  health
       risks

     •  evaluating  reliability,   technical  feasibility  and
       effectiveness, cost,  operation  and maintenance, and
       other aspects  of  waste-disposal facility  designs
       and of alternative remedial actions

     •  providing  guidance In siting of new facilities and
       1n permit  Issuance and petitioning

     •  detecting  pollutant  sources

     •  developing aquifer or  well-head protection zones

     •  assessing    liabilities   such   as   post-closure
       liability  for disposal sites

     These  activities  can be broadly categorized as either
 site-specific or generic modeling efforts,  and these cate-
 gories can  be further  subdivided into point-source or  non-
 point-source problems.    The  success of  these  modeling
 efforts depends  on the accuracy and efficiency with which
 the natural  processes controlling  the behavior of ground-
 water,  and the  chemical  and biological species it trans-
 ports,  are simulated.   The accuracy and efficiency of the
 simulations. In  turn, depend heavily on the applicability
 of   the  assumptions  and  simplifications   adopted  in  the
 Model(s).  on the availability  of  reliable  data,  and  on
 subjective  judgments made by the Modeler and management.


     If  litigation  1s Involved, the model  code  itself and
 its theoretical  foundation  may  become  contested.  There-
                                                              ENVIRONMENTAL SOFTWARE, 1987. Vol 2, No. 1  19

-------
Grounc!wa:er Contamination, P.K.M. van tier Heijde
 fore,  oc'eou&te  guidelines  should  tie  developed for  selec-
 tion of  \iax.i iiti-jn  cooes  to  be used  under  such  circum-
 stances.    Si/en  guioeli^ei  snoi/, d cover ose  re,1ew,  vali-
 dation,  ira .locumenticion and should De wic;;i> accepted.
      : .   ;::  '}•:'  trie  highest  iiKonr idnce liisl  t,.ter  resource
 rr.andq! ";>-ni  -..--c •>;'ons  De  based  on  the  use  of  technically
 £.:•,<:  :c-." ;: ;• fr: < i y   sr:jr,j   data  collection,   information
 procesc  TK,, 4,,;  interpretation  rreihoo-,.   f.ViiMty  Assurance
 (0>';  i1'""'1 '-1»'i  I'v. fiiachanisr.'.s to ensu\e  ti-.i*. decisions  are
 bistd on  i^ ;,est available  dsta and anal j so.   This paper
 di sc.nl St. s QA :iuKic 1 ines  apclicstls  tc1 groLirv'v-ater  modeling
 and tht  role of QA in the model  selection process.
          QUALITY ASSURANCE IN SROUNOWATER MODELING

      Qua-ily  assurance   in  ground*ater  modeling  1s  the
 procedural  ano  operational  framework  put  in  place  by the
 organization  managing  the modeling  study,  to assure tech-
 nically  and  scientifically  adequate  execution of all pro-
 ject  'asks  included in  the  study, and  tt  assure that »))
 modeMn>; bdsed  analysis   is  verifiable  and  defensible [7|.
 QA  'I,  orounoviiter  modeling should t>e applied  to  both model
 development  anrf rvrdel  application and  should be an  Inte-
 gra!  pc'-t   of   ell  projects.   The  two  major elements  of
 Cyai'.tj  issuance  are  quality   control  (QC)  and quality
 assessment.   Oualily control refers to  the procedures that
 pnsurt  the  quality of  the final  product.   These  procedures
 incluoe  ttw usr of appropriate methodology,  adequate  vali-
 dation,  anc proper usage.
       io   mcr.itor   the  quality  control  procedures  and  to
  evaluate  the  quality of  the  products of  field  studies,
  quality  assessment  is applied.   It  consists  of  two  ele-
  ments:  auditing  and  technical  review.   Audits are  pro-
  cedures  Designed  to assess  the  degree of compliance  with
  QA  re^jirenient?,  commensurate with  the  level  of QA  pre-
  icrioeJ  ror the project.   Compliance  1s measured  in  terms
  of  trareaDi 1 i ty of  records,  accountability  (approvals  from
  responsible staff), and  fulfillment of commitments  1n the
  QA  plan.   Technical review consists  of independent evalua-
  tirn  of  tr>e  technical  and   scientific  basis  of  a  project
  en:!- Tr.t  usefulness of  Us results.
       f.   i.,  . •.;.  i ctoons"D\ 1' ty of  boti'  rn» project  teair.
  irj.i i • •. •:  ri.i'to*   six-'  internsl  eviluftl : J"-;   8no  the  COn-
  I'-tcliai or supervising  organization (quality assessment).
  oa  shou'lo  i.ot  crive or  manage the direction  of  a  project
  no,-  '•-.,   ^  intended  to  be  an  after-ihL'-fact  filing  of
  technical
tfves.   Major  elements of  such  t QA plan are:  (1)  formu-
lation  of  QA  objectives  and  required  quality  level  in
terms  of  validity,  uncertainty,  accuracy,   completeness,
and  comparability;  (2)  development  of operational  proce-
dures   and   standards   for   performing  adequate  modeling
studies;  (3)  establishing a paper trail for  QA  activities
1n  order to document  that  standards of quality  have  been
maintained;  and  (4)   internal  and  external  auditing  and
review   procedures.     The  OA  Plan   should  also  specify
individual  responsibilities for achieving these goals.
Model Development
      Ideally,  QA should  be  applied  to all  codes currently
 in  use and yet-to-be-Jeveloped  codes.   Relevant QA proce-
 dures   include  such  aspects  as  the  verification of  the
 mathematical   framework,   field   validation,  benchmarking,
 and   code  comparison.    A  detailed  discussion  of  model
 testing and  review is described  in the second part of this
 paper.
      QA  for   code   development  and  maintenance   should
 include complete  record-keeping of  the model  development,
 of modifications made  in  the code,  and of  the code-valida-
 tion process.  The paper  trail  for  QA in model development
 consists of  reports  and  files on  the  development  of  the
 model.  The reports should Include a description of:

      • assumptions
      • parameter values and sources
      • boundary and initial conditions
      • nature of grid  and grid  design justification
      • changes and verification of changes  made in code
      • actual input used
      • output of model runs and interpretation
      • validation (or  at  least  calibration) of model

      In  addition,  depending on the  level   of  QA  required,
 the following  files  may  be retained  (in hard-copy and,  at
 higher levels, in digital forw):

      • version of source  code  used
      • verification  input and  output
      • validation input and output
       • application  input  and output

       If  any  modifications are  made  to the model coding for
 a  specific problem,  the  code   should be  tested again; all
 QA   procedures  for  model   development   should   again  be
 applied,  including  accurate record keeping and reporting.
 All  new input and output files should be  saved for  inspec-
 tion  and possible reuse.
       Virious phases  of  quality  assessment  exist  for both
  model  development  and  application.    First,   review  and
  testiny  Is  performed  by the author, and sometimes by other
  employees  not  involved  in  the  project,  or   by   invited
  experts  trow outside  the organization.   Also  to  be con-
  sidered  li  the quality  assessment  by the organization for
  which  the  project has  been  carried  out.    Again, three
  levels  ca.i  be distinguished: project  or product review or
  testing  hv  the  p-oject officer  or  project  monitor,  by
  technical   expens  within  the   funding   or   controlling
  organization, and  by  an external  peer review group.


       Ciec is ions   by  natural  resources   and environmental
  managers  vest  on the quality  of   environmental  data  and
  data  inalysiv.   therefore,  program  managers in  regulatory
  agencies  should  be  responsible  for:   (1)  specifying  the
  quality  of  the  data  required from  environmentally  related
  measurements  and  for  the  level   of  problem-solving  data
  analysis;  and (2) providing sufficient  resources to assure
  an  adequate level  of  QA.
        QA procedures should  be contained in  a  QA  plan to be
   developed for  each  modeling study.    The  plan  lists  the
   measures  required  to  achieve  prescribed  quality  objec-
 Model Application


      QA  in  model application should  address  all  facets  of
 the model application process:

      • correct  and  clear  formulation  of  problems  to  be
        solved
      • project description  and objectives
      • modeling  approach  to the project
      • is modeling the best  available  approach  and if so,
        is   the   selected  model    appropriate   and  cost-
        effective?
      • conceptualization    of    system   and   processes.
        Including    hydrogeologic    framework,     boundary
        conditions,  stresses,  and  controls
      • explicit     description    of    assumptions    and
        simplifications
      • data acquisition and Interpretation
      • model selection,  or  justification  for  choosing  to
        develop a new model
       • model preparation  (parameter selection, data  entry
         or reformatting, gridding)
      •  the   validity  of  the  parameter values  used  in  the
        model application
       •  protocols for parameter  estimation and model  cali-
  20  ENVIRONMENTAL SOFTWARE, 1987, Vol 2, No. 1

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                                                                      Groundwater Contamination: P.K.M van der Heijde
       bration  to provide guidance,  especially  for  sensi-
       tive  parameters
      • level  of Information  1n  computer  output  (variables
       and parameters displayed;  formats;  layout)
      • identification  of calibration  goals  and  evaluation
       of how well they  have  been met
      • sensitivity analysis
      • postsimulation  analysis  (including  verification of
       reasonability    of   results.   interpretation   of
       results,  uncertainty   analysis,  and  the  use  of
       manual or automatic data  processing  techniques, as
       for contouring)
      • establishment  of  appropriate   performance  targets
       (e.g.. 6-foot  head error  should be compared  with a
       20-foot   head  gradient or drawdown,  not  with the
       250-foot aquifer  thickness!);  these  targets  should
       recognize the  limits of the data
      • presentation and  documentation  of  results
      • evaluation of   how closely   the  modeling  results
       answer the questions raised by  management

      A  major  problem  in  model   use  is model credibility.
 In  the  selection process special attention should be  given
 to  ensure  the  use of qualified  models that have undergone
 adequate review and  testing.


     As  is  the  case  with model  development  QA, all  data
files, source codes,  and  executable versions of  computer
software used in the modeling study  should be retained for
auditing or postproject re-use.
                  MOOtI  REVIEW AND TESTING

     Before  a  groundwater model  is used  as  a planning and
decision-making  tool,  its credentials must be established.
Independently   of   its   developers,   through  systematic
testing  and  evaluation  of  the  model's  characteristics.
Code  testing is  generally  considered to encompass verifi-
cation  and  validation  of  the  model  181.   To  evaluate
groundwater  models in a  systematic  and consistent manner,
the  International Ground Water Modeling Center (IGWMC) has
developed  a  model  review,  verification,  and  validation
procedure   |5|.     Generally,   the   review   process  is
qualitative  in  nature,  while code testing  results  can be
evaluated  by quantitative performance standards.
Model  Review
model.   Such a  procedure  determines whether the concepts
of  a  model  adequately represent  the nature  of  the system
under  study,  and  identifies  the   processes and  actions
pertinent to the  model's  Intended  use.   The examination
also  determines  whether  the  equations  representing  the
various  processes  are  valid  within  the  range  of  the
model's  applicability,  whether  these  equations  conform
mathematically  to the  intended range of the model's use,
and  whether  the selected  solution  approach is  the  most
appropriate.    Finally,  model  examination determines  the
appropriateness   of   the  selected  initial   and   boundar>
conditions  and  establishes the  applicability range of the
model.
     For   complex   models,  detailed  examination   of   the
 Implemented  algorithms  is  required  to determine  whether
appropriate  numerical  schemes, in  the  form of a computer
code, have been adopted  to represent  the model  1101.  This
step should  disclose any  inherent  numerical problems such
as  non-uniqueness   of  the  numerical  solution,  inadequate
definition of  numerical  parameters.  Incorrect or nonopti-
mal  values  used  for  these parameters, numerical  disper-
sion, numerical Instability such  as oscillations or  diver-
gent  solution,  and  problems  regarding  conservation  of
mass.
      In  addition,  the  specific  rules  for proper  appli-
cation  of the model  should be analyzed from  the  perspec-
tive  of  Us  Intended  use.    These  rules   Include  data
assignment  according  to  node-centered or  block-centered
grid  structure  for  finite-difference  methods;  size  and
shape  of  elements   in   Integrated   finite-difference  and
finite-element  methods;  grid size variations;  treatment of
singularities  such  as  wells;  approach to  vertical  aver-
aging  1n two-dimensional   horizontal  models or  layered
three-dimensional  models;  inclusion  of  partial  solutions
 1n analytical  element methods:  and  treatment of  boundary
conditions.   Consideration  1s  also  given  to the  ease with
which the mathematical  equations, the solution procedures,
and the final  results can be physically Interpreted.
 Evaluation of Model Documentation
      Model  documentation  is  evaluated  through   visual
 Inspection,   comparison    with   existing    documentation
 standards  and  guidelines,  and  through*its  use as a guide
 1n   preparing   for   and  performing   verification  and
 validation runs.
     A  complete review  procedure  comprises examination of
model   concepts,   governing  equations,   and   algorithms
chosen,  as well as evaluation of documentation and general
ease-of-use,  and  examination of  the  computer  coding  15.
9|.   If  the model  has been verified  or  validated by  the
author,  the review  procedure should Include evaluation of
this process.
      To facilitate thorough  review of the model,  detailed
 documentation of  the  model  and  its developmental  history
 is required.   In  addition,  to ensure Independent  evalua-
 tion of  the  performed  verification and  validation,  the
 computer code  should  be  available or at  least  accessible
 for  implementation on the reviewer's computer  facilities,
 together with  a  file containing  the original  test  data
 used in the code's verification and validation.
      Review  should  be  performed  by experienced  modelers
 knowledgeable   In   theoretical   aspects  of   groundwater
 modeling.   Because review Is rather  subjective  1n nature.
 selection  of the  reviewers  is  a  sensitive and  critical
 process.
      Good  documentation  includes  a complete  treatment of
 the  equations  on which  the model   1s based,  of  the  under-
 lying  assumptions,  of the  boundary conditions that  can be
 Incorporated In  the  model,  of the  method used to solve the
 equations,  and of the  limiting conditions  resulting  from
 the  chosen method.   The documentation must  also include  a
 user's  manual  containing  Instructions  for  operating the
 code and preparing  data files, example  problems  complete
 with Input and output,  programmer's instructions,  computer
 operator's  Instructions, and a  report of  the initial  code
 verification.
  Evaluating  Ease of  Use


       The data  files  provided  by the  model  developer  are
  used to evaluate the  operation of the code and the user's
  guide through  a test-run process.   In this stage special
  attention   is   given   to  the   rules  and   restrictions
  (•tricks."   e.g..   to  overcome   restrictions   in  applic-
  ability) necessary  to operate  the code, and to the code's
  ease-of-use aspects 1111-
 Model Examination
      Model  examination determines whether  anything funda-
  ental was  omitted  1n the Initial conceptualization of the
 Computer Code Inspection


      Part of  the  model  review process is the inspection  of
 the computer  code.   In this  Inspection  attention  is  gwen
                                                                   ENVIRONMENTAL SOFTWARE. 1987, Vol 2, No. 1  21

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Groundwater Contamination: P.K.M. van der Heijde
 to the manner  in which modern programming principles have
 been applied with respect  to  code  structure, optimal use
 of the  programming  language,  and Internal  documentation.
 This  step  helps  reveal  undetected  programming  or  logic
 errors, hard to detect in verification runs.
                     MODEL  VERIFICATION

     The  objective  of  the  the  verification  process  is
 twofold:  (1)  to  check the  accuracy  of  the  computational
 algorithms  used  to solve  the governing  equations,  and (2)
 to assure th&t the computer code 1s fully operational.
     To  check  the  code  for correct coding  of  theoretical
 principles  and  for major  programming  errors ("bugs"),  the
 code  1s  run using  problems for which  an  analytical  solu-
 tion  exists.   This  stage  is  also used  to evaluate  the
 sensitivity  of  the code  to grid design,  to various  domi-
 nant  processes,  and  to  a  wide  selection of  parameter
 values |9.  12,  13,  14|.
      Although  testing  numerical  computer  codes  by  com-
 paring  results  for  simplified   situations  with  those  of
 analytical  models  does  not  guarantee  a  fully  debugged
 code, a  well-selected set  of  problems  ensures  that  the
 code's  main program and most of  Its subroutines,  Including
 all  of the  frequently  called ones, are-  being  used in the
 testing.    In  the three-level  test  procedure developed by
 the   International  Ground  Water Modeling  Center   (IGHMC),
 this type of  testing  is referred to as level I  115].
      Hypothetical  problems   are   used   to  test   special
 features  that  cannot  be  handled  by   simple  close-form
 solutions, as In testing  irregular boundary  conditions  and
 certain heterogeneous  and anlsotropic aquifer  properties;
 this is the IGWMC level II testing.
      For both  level  I  and  level  II testing,  sensitivity
 analysis is  applied to further evaluate  code  characteris-
 tics.
                       MODEL  VALIDATION

      Model  validation or  field  validation  1s defined  as
 the comparison  of model results  with  numerical  data Inde-
 pendently  derived  from laboratory  experiments  or  obser-
 vations  of  the environment  |10|.   Complete  model  valida-
 tion  requires  testing  over the  full  range  of  conditions
 for which  the model  Is designed.   Model  development Is an
 evolutionary  process responding  to new  research  results,
 developments  1n  technology, and  changes  in  user  require-
 ments.   Model  review and  validation  needs  to follow this
 dynamic  process and should be  applied each  time the model
  is modified.
       The  objective of model validation is to determine how
  well   U»e  model's  theoretical   foundation  describes  the
  actual  system behavior In terms of the "degree of correla-
  tion" between model  calculations  and actual measured data
  for  the   cause-and-effect   responses   of   the  system.
  Obviously,  a comparison with field data  is  required.  Such
  a  comparison  may  take either  of two  forms.    One form,
  calibration,  1s  sometimes  considered the  weaker  form of
  validation   Insofar  as  1t  tests  the ability  of the code
  (and  the  model) to fit the  field  data, with adjustments of
  the physical  parameters  1131.   Some  researchers prefer to
  classify  calibration as a  form of  verification rather than
  a form of validation.
       The other  for* of  validation is that  of  prediction.
  This  Is  a test  of  the  model's ability  to fit the  field
data with  no adjustments of  the  physical  parameters.    In
principle,  this  is  the  correct  approach  to  validation.
However, unavailability  and  inaccuracy of field data often
prevent such  a  rigid approach.   Typically,  a part of  the
field data  1s designated as calibration data, and  s cali-
brated  site-model  is  obtained  through reasonable  adjust-
ment of parameter  values.   Another part of  the field  data
1s  designated  as  validation  data;  the  calibrated   site
model  1s  used  in a predictive mode  to  simulate  similar
data for  comparison.   The  quality  of  such  a test  is there-
fore determined by  the  extent  to  which the  site  model  ^s
"stressed  beyond"  the  calibration  data  on  which it  is
based  113].   In the  IGWMC testing procedure, this approach
1s  referred  to  as  level  111  testing.
      For  many types of  groundwater  models,  a complete set
 of  test problems and adequate  data  sets  for the described
 testing  procedure  is  not  yet  available.    Therefore,
 testing of  such models  is generally  limited  to extended
 verification, using existing  analytical  solutions,  and to
 code 1ntercompar1son.
      Whether a model is valid for a particular application
 can be  assessed  by performance criteria, sometimes called
 validation  or  acceptance  criteria.    If various  uses  in
 planning  and  decision  making   are   foreseen,  different
 performance  criteria  might  be  defined.    The user should
 then  carefully check  the  validity  of the  model  for  the
 Intended use.
      Three levels of validity can be distinguished 110):

      (1)  Statistical    Validity:       using    statistical
           measures  to  check agreement  between  two  differ-
           ent  distributions,  the calculated  one and  the
           measured  one; validity  is  established by  using
           an    appropriate    performance   or    validity
           criterion.

      (2)  Devtative Validity:    if  not  enough  data  are
           available   for   statistical    validation,    a
           deviation  coefficient  0  can  be  established,
           e.g.,

                0  -  |(x-y)/x|10M

           where   x   =   predicted  value  and  y = measured
           value.     The  deviation  coefficient  might  be
           expressed  as  a  summation   of  relative  devia-
           tions.    If  ED is a deviative  validity criterion
           supplied   by  subjective  judgment,   a  model  can
           considered to be valid if   D  < ED.

       (3)  Qualitative Validity:  using  a  qualitative scale
           for  validity   levels  representing   subjective
            judgment:  e.g.,  excellent,   good,   fair,  poor,
            unacceptable.    Qualitative   validity  1s  often
            established through visual Inspection  of graphic
            representations  of  calculated and  measured data
            1161.

       The aforementioned  tests  apply  to single variables
  and determine local-or-single  variable validity;  if more
  than  one  variable  Is present  in  the  model,  the model
  should  also  be   checked   for   global   validity and   for
  validity  consistency  110].    For  a   model   with   several
  variables to  be  globally  valid,  all  the calculated  outputs
  should pass  validity  tests.   Validity  consistency  refers
  to  the  variation   of  validity  among   calculations  having
  different Input  or comparison  data  sets.   A  model  might.be
  judged  valid under one  data  set but  not under  another,
  even  within the range of  conditions  for which the model
  has been designed   or  1s  supposedly  applicable.   Validity
  consistency can be evaluated periodically  when models have
  seen repeated use.
      Often,  the  data used  for field  validation are  not
 collected  directly from the field but  are  processed  in an
 earlier  study.   Therefore, they  are  subject  to inaccur-
 acies,  loss  of  Information,   Interpretive  bias, loss  of
  22   ENVIRONMENTAL SOFTWARE,  1987, Vol 2, No. 1

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                                                                       Groundwater Contamination: P.K.M  van der Heijde
precision.   and   transmission   and   processing   errors,
resulting in a general degradation of the data.


     As  notec,  earlier,  for  many  types  of  groundwater
nooe'.s  no  field  data   sets   ere  available  to  execute  a
co»clete validtnon.  One  approach  sometimes  taken  is  that
of code  intercomparlsori, where * new1.y  developed model  1s
con^tred  wHr. existing  modeis designed to solve the  same
type  of  prct'ens  as  tht  ne»  node'.    K the  simulation
rfi_':j  from  ,h£  ntu  code  cc  not deviate  significantly
f"^r  those  cbtiirse!) wll* the  existing  cede,  a relative  or
coinperttU-c  validity is  estate Isneo.    H  ^s  obv'.ous  that
«s  soon «s  sdeouate data  sets  become  aveiiable,  all  the
Involved models  should  be  validated with those data.


      Further  development of databases for field validation
of solute  transport models 1s necessary.  This  1s  also the
case  for many other types  of groundwater models.   These
research  Databases   Should  represent  a  wide  variety  of
hydrogeclogical  situations and should  reflect the various
types  of  " flow,  transpo-t,  and  deformation  mechanisms
present  In the  field.   The  databases  should also contain
extensive  Information  on  hydrogeological, soil, geochemi-
cel.  and cUmatological  characteristics.   With the devel-
opment  of  such  databases  end  the  adoption  of   standard
model-testing and   validation  procedures, the  reliability
of  models  used  in  field  applications  can  be   improved
consioeratv,.
v6 i ids: ion Sc.-oer IQS


     Often,  various  approaches  tc  field  validation of  a
model  are  viable.    Therefore,  the  validation  process
Should start  with defining validation scenarios.   Planning
and   conducting   field   validation   should   include   the
following steps  117):

      (1)  Define  data needs  for  validation and  select  an
          available  data  set  or  arrange  for  a site  to
          study.

      (?)  Assess   the  data  quality   1n  terms  of accuracy
          (measurement      errors).      precision,     and,
          completeness.

      (3)  Define  model performance or acceptance  criteria.

      («}  Develop strategy for  senslti>1t>  analysis.

      (S)  Perform  validation   runs   and   compare  model
          performance    with     established     acceptance
          criteria.
 Sensitivity Analysis


      An   important   characteristic  of   a   model   1s  Us
 sensitivity  to  variations or  uncertainty   in  Input  para-
 meters.   "Sensitivity  analysis  defines  quantitatively  or
 semiquantitatlveiy  the dependence of a selected model per-
 formance  assessment measure  (or  an intermediate variable)
 on a  specific parameter or set of  parameters I IB).  Model
 sensitivity  can  be  expressed  as  the  relative   rate  of
 change  of selected output  caused by a  unit change in the
 Input.   If the  change m the  input  causes » large change
 in  the  output,  the  «odel   is  sensitive  to that  Input.
 Sensitivity  analysis  H used to  identify those parameters
 most  influential in deter»1ning the accuracy and precision
 of mode' predictions.   This information 1s of importance
 to  the  user,  as he  «ust establish  required accuracy tnd
 precision in  the model application as  a function of data
 quantity and  quality  1171.   In  this  context the use of  a
 sensitivity  index as described by  Hoffman  and Gardner  |19)
 is  of  interest.   It  should be  noted  that if  models  are
 cowled  as  in multimedia transport  of contaminants,  the
 propagation  of  errors and  the   Increase  in  uncertainty
 through  the   subsequent  simulations  must   be  analyzed  as
 part  of the sensitivity analysis.
                      MODEL  SELECTION

     Using  models  to  analyze  alternative  solutions  to
groundwater problems  requires a number  of  steps,  each of
which  should  be taken  conscientiously  and  reviewed care-
fully.   After the  decision to use  an  existing  model  has
been  made,  the selection process  1s  initiated.   AS model
credibility  is  a  major problem  IP  model  use,   specia1
attention  should  be  given  1n  the  selection process to
ensure  the use  of quellfled  models tntt  ns*e  undergone
adequate  review  and  testing.   Selactins  »r  appropriate
model  is crucial to the success of  e  modeling project.
     Model  selection  is  the  process  of  matching a  detailed
description   of   the   modeling   needs   with   well-defined,
quality-assured  characteristics  of  existing  models,  while
taxing  Into  account  the objectives of  the  study and  the
limitations  1n  the  personnel and material  resources  of  the
modeling  team.    In  selecting  an appropriate model,  both
the model  requirements and the  characteristics  of  existing
models  must  be  carefully  analyzed.    Hajor elements  in
evaluating modeling  needs  are:   (1)  formulation of  the
management problems to be solved and the level  of analysis
sought:  (2)  description of the  system under  study: and (3)
analysis   of  the  constraints   1n  human   and   material
resources  available  for  the  study.   Model selection  is
partly  quantitative  and  partly  qualitative.   Many subjec-
tive  decisions  must  be  made,  often  because  there  are
Insufficient data in  the selection stage of  the project to
establish  the importance of certain characteristics of the
system  to  be modeled.
                                                                        Definition  of  modeling needs  1s  based  on  the manage-
                                                                  ment  problem  at  hand,  questions  asked  by planners  and
                                                                  decision makers, and on  the  understanding of  the physical
                                                                  system.  Including the pertinent processes, boundary condi-
                                                                  tions,  and system stresses,   the  major criteria 1n selec-
                                                                  ting  a  model  are:  (1)  that  the  model 1s  suited  for the
                                                                  intended use;  (2) that the  model  1s thoroughly tested and
                                                                  validated for  the  intended  use;   and (3)  that  the model
                                                                  code  and documentation  are complete «nd user-friendly.
      Regardless  of  whether   problem-solving   performance
 standards are  set, management-oriented criteria need  to  be
 developed for  evaluating  and  accepting models.  Such  a  set
 of scientific  criteria should  include:

      • trade-offs  between  costs  of  running  a  model  and
        accuracy
      • profile  of  model  user  and  definition of  required
        user-friend 11 ness
      • accessibility   1n   terms  of   effort,  cost,   end
        restrictions
      • acceptable temporal  and spatial scale  and  level  of
        aggregation

      If  different  problems must  be solved, more  than  one
 model  might be  needed or  a  model  might  be used  in more
 than one  capacity.   In such  cases,  the  model  requirements
 for each  of the problems posed have  to  be clearly defined
 at  the outset of  the selection  process.   To a certain
 extent  this is also true for modeling the  same  system in
 different stages of  the  project.   Growing understanding of
 the system  and data availability might  lead  to  a  need for
 a  succession  of models of  increasing  complexity.   In such
 cases,  flexibility  of the  model  or  model  package  might
 become an Important  selection criterion.
      It should  be  realized  that  a perfect  match rarely
 exists  between  desired   characteristics   and  those  of
 available   models.    Many  of  the  selection  criteria are
 subjective or  weakly justified.    If  a ««tch  is  hard to
 obtain, reassessment of  these criteria and their  relative
 weight  in the  selection  process  is  necessary.   Hence.
 •odel selection is very much an iterative process.


      In   standardizing   model  selection,   three   major
 approaches are  employed  in  characterizing the  validation
 of  numerical  models.     In  one,   the  model  is  tested

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Groundwater Contamination: P.K.M. van tier Heijde
according  to  established  procedures;  when accepted,  the
model  U prescribed  in federal  or  state  regulations  for
use  in cases covered by those  regulations.   This  approach
does   not  leave  much  flexibility  for  incorporating  the
acvances  of  recent research and technological  development.
The  second  approach  includes  the establishment of  a list
of  oroundwster  simulation codes as  "standard" codes  for
various  generic ano site-specific management purposes.   To
be  listed,  a code should pass  s widely accepted review and
test  rfuceoure such  as  that  describsd in  a previous part
of  this  paper.    This  spproach  is  suggested   in  a recent
evaluation  of  the role of modeling  in tne U.S.  Environ-
mental  Protection  Agency   |6|.   It  should be  noted that
establishing 'standard" models will  not prevent discussion
of  the appropriateness Of  a selected model  for analysis of
a specific  problem  nor  of its proper  use   1n a particular
decision-making  process.      In  considering   these  two
approaches,  questions have been raised  such as  [6|:

  •  Are  there  legal   liabilities  for  setting  up  certain
    models as acceptable?   (for Instance, if an enforcement
    agency certifies  a model  for use, can that  agency no
    longer criticize an Industry's use of  that  model?)

  •  Does  certification  squelch  the  development   of  new.
    better models?

  •  What  balance  should there  be between   using the  newer.
    faster models  and  using mature models  already  subjected
    tr, peer review?

      A  third  approach  1s to  prescribe a revlew-and-test
 methodology  1r>  regulations  of  enforcement agencies,  and
 require  the  model development team to show that  the model
 cooe  satisfies  the  requirements.    This   approach  leaves
 room  to  update the  codes  as   long as  each version 1s  ade-
 quately  reviewed and  tested.    An  example 1s  the quality
 assurance  program  for  models  and  computer  codes  of  the
 U.S.  Nuclear Regulatory Commission 120).


       In  any case, a general  framework of  nondlscHmlnatory
  criteria  should  be  established |6|.   These criteria should
  include:

  •  publication  and  peer  review  of  the  conceptual   and
    mathematical frame-work

  •  full documentation and visibility of the assumptions

  •  testing  of  the  code  according  to prescribed methods;
    '.Ms  «hoi:la include verification (checking the accuracy
    of  tne  computational  algorithms  used to   solve  the
    governing   equations),   and  validation  (checking  the
    ability  of the  theoretical  foundation of the  code to
    describe  the  actual  system  behavior)

  • trade    secrets    (unique   algorithms    that    are   not
    described)  should not  be  permitted If   they might affect
    the  outcome of  the simulations; proprietary  codes are
    already  protected  by the copyright  law

       Finally,  as model  selection  Is  very closely  related
  to  system concep-tualIzat1on  and  problem  solving,  "expert
  systems"   integrating  system  conceptualization  and  model
  selection   on  a  problem-oriented  basis promise  to  be
  valuable tools.
       further Information on groundwater model  selection  1s
  presented 1n |21, 22. 23. 241.
                            SUMHARY

       During  the  1970s  a  rapidly  Increasing  awareness  of
  the threat posed to groundwater resources  by  human-induced
  chemical   and   biological   pollution  has  accelerated  the
  development  of   sophisticated  simulation  models.    These
  models  are   based  on  mathematical  descriptions  of  the
  physical, chemical,   and   biological  processes  that  take
  place  in  a  complex  hydrogeologtcal  environment.    The
extensive need  for  these models  in  assessing  current  and
potential  water  quality  problems   has   resulted  in  two
groups of modelers: (1)  model developers who are research-
oriented and who generally apply models only for* verifica-
tion  and  validation  purposes,  and   (2)  model  users  who
apply models  routinely  to actual generic or site-specific
groundwater problems.   The economic consequences of model
predictions  and  the  potential  liabilities   incurred   by
thefr   use   have  brought  quality  guarantees  and  coae
credibility  to  the  forefront  as major  issues  in ground-
water modeling.  Hence,  quality  assurance  (QA)  needs to  oe
defined  for  both model  development  and model  application.
There  Is a significant  difference  between these two:   the
first  is designed  to  result in  a  reliable code, and  the
second   to  Interpret  correctly  the  simulation  results.
Both  require  stringent  QA procedures to be established  anc!
enforced.   As  model  credibility has  become  a  m&jor con-
cern,  model  selection   should  focus on  those  codes  that
have  undergone  adequate review and  testing.    To  further
Increase the  applicability of  the models,  good documenta-
tion  and user-friendliness of the computer coding  involveo
should  receive  proper  attention.
                       ACKHOULEDGEtCKT

 The research described 1n this publication has been funded
 In  part  by  the  U.S.  Environmental  Protection  Agency
 through Cooperative Agreement  ICR-812603  with the Hoi comb
 Research  Institute.    It  has  not  been  subjected  to the
 Agency  peer  and   policy  review  and  therefore  does  not
 necessarily  reflect  the  views   of  the   Agency,   and  no
 official endorsement should be inferred.
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 (31    U.S. Office of Technology Assessment.   1982.  Use of
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 |4|   Javandel,  I.. C.  Doughty,  and  C.F.  Tsang.    1984.
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  17)   Taylor,  O.K.  1985.   What  is  quality  assurance?  in
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  |81   Adrlon,   W.R.,  M.A.  Branstad.  and  J.C.  Cherniasky.
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dation of  models for simulating solute transport  in
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comparison  of benchmark  techniques."   GWM]  84-13,
International Ground  Hater  Modeling  Center, Holcomb
Research  Institute,  Butler  University,  Indianapolis,
IN, 420 pp.

AS1K.    1985.   Standard  practices  for  evaluating
environmental fate models of cher.icals.  Annual book
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and Materials, Philadelphia, PA.

van  der   Heijde,  P.K.H.    1984.    Availability  and
applicability  of  numerical  models  for groundwater
resources  Management.   in Practical  Applications  of
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Columbus. OH. August 15-17.  1984.

Sykes.  J.F.,   S.B.   Pahwa,  D.S.  Ward,  and  R.B.
Lantz.   1983.   The validation  of SWENT, a geosphere
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(FE30GW)   flow  mode!:   Formulation,  computer source
list ings, ' and  user's   manual.    ONWI-548,   Battelle
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van  der  Heljoe,  P.K.M..  P.S.  Huyakorn,  and   J.W.
Mercer.      1985.      Testing  and  validation  of
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|20|  Silling,  S.A.    1983.   Final  technical  position  on
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|22)   Kincaid,  C.T.,   J.R.   Morrey,  and  J.E.  Rogers.
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                                                                  ENVIRONMENTAL SOFTWARE. 1987, Vol 2, No. 1  25

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                                   IGWMC Reprint
                         REPRESENTATION OF  INDIVIDUAL  HELLS

                      IN TWO DIMENSIONAL GROUND  WATER  MODELING
                                 Milovan S. Beljin
                     International  Ground Water Modeling  Center
                   Holcomb Research Institute, Butler University
                            Indianapolis, Indiana  46208
                                    presented at
                             The NWWA/IGWMC Conference
                    "Solving Ground-water Problems with Models"
                                February 10-12,  1987
                                  Denver,  Colorado
                                    GWMI - 87-06
                                  February  17,  1987
INTERNATIONAL   GROUND   WATER   MODELING   CENTER


                             Holcomb Research Institute
                                  Butler University
                            Indianapolis,  Indiana  46208

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                     REPRESENTATION OF INDIVIDUAL  WELLS

                   IN TWO-DIMENSIONAL GROUND WATER MODELING


                              Milovan S.  Beljin

                  International Ground Water Modeling Center
                Holcomb Research Institute,  Butler University
                         Indianapolis, Indiana  46208
Abstract
    Well-field simulation in ground water modeling  requires  grid  blocks with
dimensions  that  are  much  larger  than  the  well  diameter.    The  computed
hydraulic  head  in  the  well block  (i.e.,   in  the  block  containing a  well)
represents an average head  for  the  block and  is not the head in a particular
well.   The accurate computation  of hydraulic head  in  a well  is  needed for
both flow and transport modeling.

    The effect of an  individual  well  is  normally represented in ground water
modeling by an  imposed  discharge/recharge  rate  on the  well block.  The usual
assumption  is that the  flow near  the well reaches equilibrium  rapidly and
may  therefore  be  treated  as  steady-state flow.   Analytical methods  for
computing  the nodal correction use  the  equivalent well  block  radius,  which
is  defined as  the  radius at  which the  steady-state  hydraulic head  in the
aquifer  is  equal  to  the  numerically  calculated  head  of  the  well  block.
Numerical  methods  to improve  modeling of  near-well zones are based  on the
localized mesh refinement.

    The objectives  of this  paper  are to investigate the  effects of several
possible well situations on the nodal correction.   These include a well not
positioned  at the  center  of the well block,  a rectangular  well  block with
different  aspect  ratios,   an  anisotropic  medium,   and  more  than  one  well
within  the block.
 Introduction

      In  regional  ground  water modeling,  the dimensions of  a well  are too
 small  to  be  resolved by the computational grid.   The effects of the well are
 included  in  the modeling by an  imposed  discharge/recharge  rate on the block
 containing  the well  (the  well block).   The computed hydraulic head  in the
 well  block  represents an average head for the block and is not the head in a
 particular well;  but the accurate computation of hydraulic head in a well is

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important for both  flow  and  transport  modeling.   In  flow  modeling,  the  long-
term predictions  of drawdown in a well  field  are crucial to its design  and
determination  of  its  lifetime.    Simulation  of  solute  transport  in  ground
water  systems requires  the accurate  determination  of  the  velocity  field,
which  is  derived  from  the  computed  hydraulic heads.  Advection  and  disper-
sion,  two major  mechanisms  of solute transport,  are  both functions of  the
velocity.   Because  of the  radial  nature of  the flow near  a  well and  the
greatest  velocity  in  the vicinity  of the well,  solute  transport near  the
well  must  be modeled  with special  care.   Despite  the  importance  of  the
determination  of  the  velocity  near a well,  relatively  little  research  has
been  published  on   this  topic  in hydrological  literature   (Prickett  1967,
1971;  Trescott et  al.  1976; Prichett and  Garg  1980;  Bennett et  al.  1982);
most  of  the research  related  to  this  topic  comes from reservoir modeling in
the  petroleum industry  (e.g.,  van Poolen et al.  1970; Williamson  and Chap-
pelear 1981;  Peaceman  1978,  1983;  Abou-Kassem and  Aziz 1985).

     The  objectives  of  this  paper are  to identify  the  problems  in  repre-
senting   individual  wells  in  ground  water  modeling  and to  point  out  the
importance  of the  near-well  zone in modeling.   The  effects  of  several pos-
sible  well  situations are discussed.   These include a well positioned at the
center  of a  square  or a rectangular well block,  the  case of an anisotropic
medium,  a well not  positioned  at  the  center  of the  block, and more than one
well  within the  block.
Concept of Equivalent Well Block Radius

     With  horizontal ground  water  flow,  mass conservation
penetrating well can be expressed  in polar coordinates by
                                                              around  a  fully
     Q =
                                                                      (D
     where
Q
r
K
b
h
            is  the  discharge  rate  [L3/T];
            is  the  radius  from  well axis  [L];
            is  the  hydraulic  conductivity  [L/T];
            is  the  thickness  of the aquifer  [L];
            is  the  hydraulic  head  [L].
                                                 and
      By integrating between r = r  and some distance r, Equation (1) becomes
     where
            r   is  the  radius  of the  well  [L];
            h1"  is  the  head  at the  r   distance  [L]; and
            h*  is  the  head  at the  rw distance  [L].

-------
   Fig. La
                                                 H-2.J
  Fig.  1.b
  Fig. 1.c
                                                       ja
                                                       4
F  (h. .  . - h. .)
   i+l.J    _i3'

Z tn (Ax/rg)
                          JL
                          4

                                                                     ' "
Fig. 1. Flow from node  (i  + l,j)  to node (i,j).  (a) sectional view;  (b)

        equivalent radial  flow;  and (c) finite-difference representation.

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     From Equation  (2),  known  as the Thiem equation, it follows that h  will
increase indefinitely as  r  increases,  so that Equation (2) is valid only  in
the  close  proximity  of  a  well  as h  approaches  the  original piezometric
head ho.   The  Thiem  equation  is often used to extrapolate from the average
head   for   the   well  block   at  radius re to   the   head   at   the   well
radius rw (Prickett 1971;  Akbar  et al.  1974; Trescott  et  al.  1976).   For  a
specified head hjj at the distance  re  (Figure 1a),  and  with transmissivity  T
in place of the product Kb, Equation (2) can be  rewritten as

                 Q      r
     h . . - h  -. ^= in ( — } .                                        (3)
      i]    w   2nT    ^r '                                          VJ'
                         w

     The difference h^-  - hw represents the nodal  correction  to be  applied
to the  computed head  in  the interior well  block.   For  an unconfined aquifer,
the analogous equation is

                       P
     h? . - h' = 3_  m ( — 1  .                                         (4)
      ij    w   nK    lr  '                                           v  '
                        H

     Equations  (3)  and  (4)  assume that  (a)  the  aquifer  is homogeneous and
isotropic;  (b)  only  one  fully  penetrating  well  is in  the well block; (c)
flow is  laminar and in steady-state; and (d) well losses are negligible.

     If  h_j,- is  the average  hydraulic  head computed  for the well block, the
head  in the  well  may be  determined from Equation  (3)  as long as the  rad-
ius  re  can be  determined.   This equivalent well block  radius,  re,  is  defined
as  the  radius  of  a  hypothetical  well  for which  the  average value of hy-
draulic head  for  the well  block is applicable  (Trescott  et al.  1976).   From
Equation  (3)  it  follows  that  the  nodal correction  is  directly proportional
to  tne  equivalent  block  radius, re ,  and  to  the discharge/transmissivity
ratio.
The  Equivalent  Well  Block  Radius  Equations

(a )  Square  grid

     Herbert  and  Rushton  (1966) and  Prickett (1967,  1971)  were apparently
the  first  in the  hydrological literature to  point out  the  need for correc-
tions  in  modeling the true radius of a well.  The standard finite-difference
equation  applied  to  a square  mesh  of sides Ax (Figure  2a)  and  constant
transmissivity  T,  with a discharge well  at  the node  (i,j), has the form


     h .     . * h.     . + h .  .    +  h.  .    -  Hh. .  =    .                  (5)
      The Thiem equation for the flow  from an outer circle of radius Ax to an
 inner circle radius of re (Figure  1b)  is given  as
            tn
                  e

-------
                          V
T
Ay
                                                      /  . »
                                                     ->    *
       Fig. 2a. Square grid.
           Fig.  2b.  Rectangular grid.
     Assuming h« to  be  the  average  of hj_j  ,•,  h_^jtj,  hifj;-_j,  and hj (J>J ,
and  combining Equation  (5)  and  (6),  Prickeft (1971*) has'derived  the  fol-
lowing expression:
                                                                     (7)
    or
r  = exp I- :r
 e     K *•  2'
                      = 0.208AX
                                     (8)
The same  equation  can  be derived  by combining the Thiem equation and Darcy's
law (Figures  1b and  1c).

    In  the  petroleum literature,  among  the first to address  the  problem of
equivalent well block  radius  were van Poollen et al. (1970).  Their approach
assumed  that  the  computed  head of the  well  block equals the  areal  average
head, leading to the expression:
     r  = 0.342AX  .
                                     (9)
     Peaceman  (1978,  1983)  has shown  that Equation  (9)  is  incorrect,  but
that Equation  (8)  is a good approximation for the equivalent well radius.

-------
By solving the same problem numerically,  he derived the following equation:

     r  = 0.198Ax  .                                                 (10)
                                                             r
The constant  in  Equation (10) is the limit of the ratio  lim -—  , where  N  is
                                                          \l
the number of nodes.                                      N
-------
         Table  1.   Effects of aspect  ratio  (Ay/Ax) on equivalent
                   well  block radius.


M
AX
1
2
3
14
5
6
7
8
9
10


Eq. (11)
0.200
0.283
0.316
0.100
0.117
0.^90
0.529
0.566
0.600
0.632
r /Ax
e
Eq. (12)
0.208
0.312
0.416
0.520
0.621
0.728
0.832
0.936
1.010
1. 113


Eq. (13)
0.208
0.327
0.135
0.518
0.581
0.631
0.670
0.701
0.728
0.750


Eq. (11)
0.198
0.313
0.113
0.577
0.711
0.852
0.990
1.129
1.268
1.107
(c) Anisotropic medium

     Assuming  that  the  principal  axes  of  the  transmissi vity  tensor  are
parallel to  the  x  and y axes, Peaceman (1983) derived the  following  expres-
sion for the nodal correction in an anisotropic medium:
h.
          - h
                    r  T  :
                     xx yy
(15)
    where
     Txx is the transmissivity in x direction [L2/T];
      yy
         is the transmissivity in y direction [L2/T];  and
re = 0.28
                  xx
                      XX
                               yy
                                                                    (16)
For the  case Txx  = Tyy,  Equations  (15)  and (16) reduce to  Equations  (3)  and
(11), respectively.

(d) Some other cases

     Kuniansky  and Hillestad  (1980) derived  equations  for  the  equivalent
well block  radius for the cases in which the well block  is a  corner  or  edge
block and  in which the well is is not centered in a grid block.   Their  work
shows that  the  equivalent radius derived by Prickett  (1971)  for  an  interior

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well block,  Equation (8),  is  a  good  approximation  for both edge and  corner
well blocks.

    For  the case  in which  the  well  is not  positioned  at  the  center of  a
block,   two  different approaches  can  be used.   The  first predicts the  flow
rate with  an analytical  expression for the  well  at an  arbitrary  position;
this rate   is  used  to  compute the head  for  the  block.   To calculate  the
equivalent  well  block radius, the  flow rate and the computed head are  sub-
stituted  into  Equation  (3).   The  second  approach  generates the flow  rates
with a fine grid  so that  all the wells  are positioned  at the grid  block
centers.   These rates  are  used  to calculate the  equivalent  radii for  the
subsequent  simulations.   In both  approaches,  the calculated rate is assigned
to a centered well in the well block.

    Bennett  et  al.  (1982)  and McDonald (1985)  have  extended  the  equivalent
well block  radius theory  to  the multiaquifer  or  multilayered  wells.   Ac-
cording  to their  work,  the presence  of the wells changes  the nature  of the
equations  that  must be  solved  in  a  three-dimensional  ground water  flow
simulation.  The proposed method  allows for calculating  the head in the well
and individual aquifer discharges to  such a  well.

    All  previous discussions of  well  representation in ground water modeling
involve  finite-difference models.  Charbeneau and  Street (1979) presented a
finite-element  numerical method  for  obtaining  an  improved distribution of
head around a  well.   Instead  of  assigning all  the  discharge  of a well to a
particular  node, they place the well  within an  element,  and the discharge  is
distributed among  the  nodes  of that  element.    The numerical  results are
compared  with  the analytical  solutions for  confined and leaky aquifers, and
good agreements  were found.
 Example Problem

      To illustrate the importance of the correct representation of a well  in
 ground  water modeling, the two-dimensional finite-difference model (Trescott
 et  al.  1976) was applied to a  problem  of simulating  a  single pumping well  in
 an  infinite  aquifer.   This is  the classical  Theis problem  often  used  to
 verify  a  numerical  model.   The  aquifer  parameters  were chosen  from the
 benchmark  problem  described by  Ross  et al.  (1982).   The  aquifer  is  homo-
 geneous and  isotropic  with transmissivity  of 0.001 m /s and a storage  coef-
 ficient 0.001; the  discharge  rate is 0.003 m /s and the duration of pumping
 is  10 days.

     The constructed  numerical  model  has  a square  grid  design,  one pumping
 well  in the center  of  the model, and  no-flow  boundaries.  A number of runs
 have  been performed with  different  block  sizes.   To satisfy  the assumption
 of  an  infinite aquifer,  the no-flow boundaries  have  been placed  for  each run
 far  enough   from  the well  so  they  are not  affected  by  the  pumping.   The

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analytical and numerical  results are  in  good  agreement  for  the  nodes  outside
the well  block (Figure  3).   However,  tne computed  head  for  the  well block  is
significantly smaller  than  the head in  the well computed by  the  Theis  equa-
tion.   In order to  convert the average  well  block  head to the  head in  the
well,  the nodal corrections based on  Equation (3)  are applied.   Table  2
shows  the effects  of  the  well  radius and  the  block size on the nodal  cor-
rection for the given problem.
          Table 2.  Nodal corrections for the example problem
          (T = 0.001 m;/s,   S = 0.001 and Q = 0.003 mVs)

Ax [m]

10
100
200
300
500
1000
2000

r = 0.1m

1.^5
2.55
2.88
3.07
3.31
3.65
3.99
Nodal Correction
r = 0.25m

1.01
2.11
2.HH
2.64
2.88
3.21
3.54
, [ml
r = 1.00m
w
0.34
1.145
1.78
1.97
2.22
2.55
2.88
      It is  interesting  to note  that
 radius could also be determined graphi
 giver, example  problem  is  plotted in
 average drawdown for  the  well  block,
 m.   This drawdown is plotted on  the y-
 is  extended to intercept the anaJytica
 tion  represents  an  approximate  value
 For the given  example  it  is 40  m,  wh
 applying Equation (8).
the  approximate  equivalent well  block
cally.  The  analytical  solution  of  tne
 Figure 3-   The  numerically  computed
for  the  case Ax  =  Ay =  200 m, is  1.69
-axis and  a  line  parallel  to  the  x-axis
1 curve.  The  x  value of  the  intercep-
 of  the equivalent well  block radius.
ich  is  close to  the value obtained  by
     The  approximate nodal correction can  also  be  read  off the graph.  It is
the  distance on  the  graph between the  analytical  curve and the numerically
computed  average  drawdown  for  the  given  radius  of   the  well.    If,  for
example,  the radius of the  well  is assumed to be  1  m,  the nodal correction
from the graph is  1.8  m,  which is close  to the  value  given in Table 1.  It
is  important to note that for  the given example problem,  the values  obtained
from Equation  (3),  which represents a  steady-state situation,  are  in good
agreement with  the  values obtained with  the Theis  nonsteady-state equation.

     The  same problem  is  used  to  illustrate the effect  of anisotropy on the
value of the  nodal corrections.   The  Trescott model  uses  Equation (3) to
compute  the  nodal  correction  for a given Txx regardless of  the  degree of
anisotropy.   For the given example the  nodal correction  is  1.78  m.   However,
with Equations (15) and  (16),  the nodal correction is  1.29 m and 1.02 m for
TyV/Txx  equaling  2  and  3, respectively.

-------
    10 -g
o
TJ
   10
       -2.
        10
           -i
                  analytical

                  ax = 200 [m]

                     = 300 [m]
IIMI
\  i i nii| -- 1 — i i i mi)
     10
  distance
                              10
                              [m]
104
          Fig.  3. Analytical and numerical solution of  example problem.

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                                                                                  11
Discussion and Conclusions

     The  head  in the well block represents an  average  for the block and  is
not the  head  in the well itseslf.   To compute the correct value  of  the  head
in  the  well,  two approaches  are  possible.    The first  uses an  analytical
expression for   calculating  the  approximate nodal  correction,   the magnitude
of  which depends  on  the aquifer  parameters  and  on the  grid  design.    A
library of analytical  solutions  for  different  field  situations can easily  be
incorporated into a code.

    The  other  approach  to   improve  modeling  of  near-well  zones  develops  a
localized refined  mesh in the finite-difference grid.  The  radial nature  of
flow  near  the wells  makes  a  hybrid approach  most suitable.    The  entire
domain  is divided  into  a well region, represented with a  cylindrical  grid,
and a model  region represented with a rectangular  grid  (Pedrosa  and  Aziz
1986).

    Note  that  neither of the  two  methods for calculating the head  includes
any additional  drawdown  caused by well losses.

    Based on  study results  of  the  example  problem and  discussion of  the
different approaches,  it is  clear  that  the  near-well zone  needs  to be  mod-
eled  more carefully than is  usually  done at present.


                                  REFERENCES

Abou-Kassem,  J.H.  and  K. Aziz.  1985.  Analytical  Well  Models for  Reservoir
    Simulation.  Society  of Petroleum Engineers  Journal,  v.  25, no.
    4,  pp.  573-579.
Akbar,  A.M.,  M.D.  Arnold,   and  A.H.  Harvey.  1974.   Numerical  Simulation  of
    Individual  Wells  in a  Field Simulation  Model.  Society of Petroleum
      Engineers  Journal,  v. 1*4, no. 4,  pp. 315-320.
Bennett,  G.D.,  A.L.  Kontis, and S.P.  Larson.   1982. Representation of Mui-
    tiaquifer  Well  Effects  in  Three-Dimensional  Ground  Water  Flow Simula-
    tion. Ground Hater,  v. 20, no. 3,  pp. 334-3*41.
Charbeneau,  R.J. and  R.L.  Street.   1979.  Modeling Ground Water  Flow Fields
    Containing  Point  Singularities:  A  Technique  For  Singularity  Removal.
      Water Resources Research, v. 15,  no. 3, pp. 583-594.
Herbert,  R.  and K.R.  Rushton.  1966.  Ground  water Flow  Studies by Resistance
    Networks.  Geotechnique,  v.  16, pp. 53-75.
Kuniansky,  J.  and  J.G.  Hillestad.   1980. Reservoir  Simulation Using Bottom-
    hole  Pressure  Boundary  Conditions.   Society  of  PetroJeum  Engineers
    Journal, v.  20,  no.  6, pp. 473-486.
McDonald, M.G.   1985.  Development   of  a  Multi-Aquifer  Well  Option for  a
    Modular  Ground Water Flow Model. Proc.  Practical  Application  of Ground
    water Models,  pp.  786-796.
Peaceman, D.W.  1978.   Interpretation  of Well-Block  Pressures  in  Numerical
    Reservoir  Simulation.   Society  of  Petroleum Engineers  Journal,  v.  18,
    no.  3, pp.  183-194.
Peaceman, D.W.  1983-   Interpretation  of Well-Block  Pressures  in  Numerical
    Reservoir  Simulation with  Nonsquare Grid Blocks and  Anisotropic Perme-
    ability.  Society  of  Petroleum  Engineers  Journal,  v.   23(   no.  3, pp.

-------
    531-543.                                                                      12
Pedrosa,  O.A., and  K.  Aziz.  1986. Use of A Hybrid Grid in Reservoir Simula-
    tion. SPE Reservoir Engineering, v. 1,  no.  6,  pp.  611-621.
Prickett, T.A.  1967.  Designing  pumped well characteristics  into  electrical
    analog models. Ground Water, v. 5, no.  M,  pp.  38-46.
Prickett, T.A.  and  C.G. Lonnquist.  1971.  Selected  Digital  Computer Tech-
    niques  for Ground water  Resource  Evaluation.  Illinois  State  Water  Survey
    Bulletin 55, 62 pp.
Prichett, J.W.  and S.K.  Garg.   I960.  Determination  of Effective  Well Block
    Radii  for  Numerical Reservoir  Simulations.   Water  Resources  Research,
    v. 16,  no. 4, pp. 665-674.
Ross,  B.  et  al.    1982.   Benchmark  Problems  for Repository Siting  Models.
    U.S. Nuclear Regulatory  Comission.  NUREG/CR-3097.
Trescott, P.C.,  G.F.  Pinder, and  S.P.  Larson.  1976.  Finite-Difference Model
    for  Aquifer  Simulation  in  Two   Dimensions  with Results  of  Numerical
    Experiments. U.S. Geological  Survey, Chap. C1, Bk. 7.
van Poollen,  H.K.,  H.C. Bixel,  and J.R.  Jargon.  1970.  Individual Well Pres-
    sures in  Reservoir  Modeling.  Oil  and Gas Journal, pp.  78-80.
Williamson,  A.S.  and J.E. Chappelear.  1981.  Representing  Wells  in Numerical
    Reservoir  Simulation:  Part 1  -  Theory.  Society of  Petroleum Engineers
    Journal,  v.  21,  no.  3,  pp.  323-338.

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        IGWMC   GROUNDWATER   MODELING   REPRINT
                       REMEDIAL ACTIONS UNDER VARIABILITY OF
                               HYDRAULIC CONDUCTIVITY
                                         by

                                   Aly I. El-Kadi
                                    presented at

                              The NWWA/IGWMC  Conference
                     'Solving  Ground-water Problems with Models'
                                February 10-12, 1987
                                  Denver,  Colorado
                                    GWMI - 87-10
INTERNATIONAL   GROUND   WATER   MODELING   CENTER
                             Holcomb Research Institute
                                 Butler University
                            Indianapolis,  Indiana  46208

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                    REMEDIAL ACTIONS UNDER VARIABILITY  OF

                            HYDRAULIC CONDUCTIVITY


                               Aly  I. El-Kadi

                  International Ground Water Modeling Center
                         Hoi comb Research  Institute
                              Butler University
                            Indianapolis,  Indiana
Abstract
     Recent  research  has  demonstrated  the frequent  failure  of the  classic
dispersion equation in describing  the mass  transport  phenomena.   The  general
conclusion  of  that  research  has  been  that the  velocity field  should  be
described  in greater detail  by either a  deterministic  or a  stochastic  ap-
proach.   The stochastic  approach  is applied  here  to  evaluate  selected  reme-
dial actions involving  recovery wells.   A Monte-Carlo  technique  is  adopted
in the analysis of the two cases considered, an injection/recovery well,  and
plume  capture by- a  production well.   Variability causes  a  dispersion-like
phenomenon  which  affects  the  shape   of   plume   and  break-through  curves.
Variability  is highest  where  the plume  advances or  contracts.    Defining
effective dispersivities  representing the  average  stochastic  results  for  use
in  deterministic  analysis  fails   to   recognize   the  important  variability
features  described by stochastic analysis.   In  addition, stochastic analysis
allows   the   quantification   of  uncertainty   regarding  output   results.
Management decisions should be based on such uncertainty.


Introduction

     Several  physical,  chemical,  and  biological  techniques are  used  to
contain  spilled  or  leaked  contaminants  and  to  recover  and treat  ground
water.   (For details of  different techniques see, e.g., U.S.  EPA [1985]  and
Ehrenfeld  and  Bass  (1984)).    Containment  systems  such as  recovery  wells,
interceptor  trenches,  grout curtains,  and slurry walls,  interfere with  the
transport  process by  altering  the  flow field.   Air stripping,  a physical
process,  removes  volatile chemicals from  the soil by drawing or venting  air
through  the  soil  layer,  or by  passing contaminated water through a  packed
column  or tower with  counter-flowing  air and water.  To  increase the  effi-
ciency  of the removal process,  in situ air stripping is combined with  acti-
vated  carbon absorption.

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     Biological methods, performed above ground or  in  situ,  are  effective  as
remediation techniques.  Above-ground processes include fixed film  treatment
such  as  trickling  filters,  or  suspended-growth  systems  such  as  activated
sludge  (Jensen et al.  1986).    In  situ biodegradation includes  the use  of
existing soil microorganisms or  the  addition  of microorganisms  and  nutrients
to the contaminated aquifer.   The effectiveness of  in  situ biological  treat-
ment  depends  on a number of factors  such  as  type and concentration of  con-
taminants,  hydrogeology,  nutrient  availability,  dissolved oxygen,  pH,  tem-
perature, and salinity  (Engineering-Science 1986).

     Recovery wells are the  most commonly used remediation  techniques;  some
of their applications are  studied  here.   In  aquifer cleanup, polluted  water
is extracted  and  either reinjected after  treatment  or  released  to a surface-
water body  if that is  environmentally  and economically acceptable.   In  some
situations,  injection  wells   are  combined with  recovery  wells  to  enhance
recovery by altering  the hydraulic gradients.   The  recovery injection system
should  be  designed to intercept the  contaminant plume  so that no  further
degradation of the aquifer occurs.   Modeling is a  very  useful  tool  in the
design of such  systems  (Boutwell et al. 1985).

      Models to  study  contamination-related problems, including  the  design  of
remedial actions,  are based  on solution of the classic dispersion-convection
equation  (e.g., the  review  by Anderson (1984]).    However,  a number of re-
searchers,  including   Gelhar  et al.  (1979)   and  Matheron and  de  Marsily
(1980),  have  demonstrated that  this  equation fails to describe  contaminant
movement  near the pollutant  source or  over short time periods.   Application
of  the equation  has  resulted  in estimates  of dispersion coefficient  which
are  both time- and scale-dependent.   Although rese'archers  agree that  vari-
ability  of  hydrologic properties causes such  discrepancies, they disagree on
ways  of dealing  with  the situation.   Generally,  two approaches  have  been
adopted.   The first  includes  a  detailed  description of  the velocity,  either
deterministically  or  stochastically.   In  the  deterministic approach, hetero-
geneities,  e.g.,   stratification,  are  described  deterministically  (e.g.,
G*ven et al.  1984, Molz et al.  1983).   In the stochastic approach, a random
velocity field is employed  in  the  mass transport  model, based on a specified
spatial  and correlation structure  of the  hydraulic  conductivity field.   The
governing stochastic  equation  can  be solved  analytically, as reported in the
work  of Gelhar  et al. (1979),  Gelhar  and Axness  (1983),  Bresler  and  Dagan
(1981),  Dagan (1982),  and Tang  et al.  (1982).  The Monte-Carlo technique, a
contrast to closed-form analytical  solutions, was also reported by Smith and
Schwartz  (1980 and 1981a,b).

      The  second approach,  based on  a modification  of  the convection-disper-
sion  equation,  uses   time-dependent  dispersivities (e.g.,  Matheron and  de
Marsily  1980, Pickens and  Grisak 1981).  Gelhar  et al. (1979) proposed a new
one-dimensional equation in  a  perfectly stratified  aquifer.  At small times,
certain  restrictions  may  invalidate  this  equation.     Various  theoretical
studies  have  demonstrated  that the  classic convection-dispersion equation is
valid for  large  times  or  large distances  if dispersivities  are  estimated
from   various  statistical   characteristics   of   the   hydraulic  conductivity
distribution  (Anderson 1984).

      The  objective of the  study  is  to  analyze effects of uncertainty  in
hydraulic  conductivity on  the  design  of  remedial  actions.   Two  situations
that  involve  recovery wells  are studied:   a  case  of an  injection/recovery

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single well,  and plume  capture by  a  production well.   The study  examines
effects of  uncertainty  in  hydraulic  conductivity  on variability  of  concen-
trations  and  on  the  cleanup  time.   The  possibility  of characterizing  the
heterogeneous  aquifer   by   an  effective   dispersion   coefficient  is   also
examined.    A Monte-Carlo  simulation approach  is  employed  in  the analysis,
considering hydraulic  conductivity as  a stochastic  process.  The  stochastic
model is  based on the  USGS  two-dimensional mass-transport code  (HOC), devel-
oped originally by Konikow  and  Bredehoeft  (1978).


Application of MOC to Remedial  Actions

     The  Monte-Carlo  approach  is  applied  here  to  the  stochastic analysis of
the  transport problem.   MOC   is  solved  for a  different set  of  parameters
within each  Monte-Carlo run.   When  numerical  techniques are  employed in the
solution  of  the  problem,  care must be  taken  to minimize  numerical errors
that  may  contribute to variability of output  results.   MOC was  tested for
certain  situations occurring   in  remedial  actions  by  recovery wells.   The
three  cases   investigated   were  a  recharge/recovery   single  well,  a  re-
charge/recovery   doublet,  and  one or  two production  wells  for  plume  cap-
ture.   (For  details of  the  verification,  see El-Kadi |1987]).
                                                      ai_n,  MIUWII  aibu  as  LIIC
                                                       the dispersive proper-
                                                      tuation may also repre-
aquners  (e.g.,  fver
leanup process followi
i et ai.  iy»D).   me situ«
ng extended contamination.
      Some  hypothetical   experiments  were   simulated  and  the  results  were
 compared  to  the  analytical  solution  of  Gelhar  and  Collins  (1971) which
 solves  for  the  concentration  in the  well  during the  withdrawal  period.
 Sensitivity of  results  to  the  value  of dispersivity,  injection time,  and
 well  flux were examined.  Some  fluctuations were  noticed  in the breakthrough
 curves, yet  the overall behavior  of  the numerical results was good.  Some
 numerical dispersion  occurred due to  the  radial  flow situation;  its effect
 seems most severe for  larger  well  fluxes or longer injection periods.   Large
 relative  mass-balance  error  was  noticed,   with  a  maximum  value of  about
 -23%.   The  error value  appears  to be  irrelevant to the  accuracy  of  predic-
 tion.   The  mass error  may  be caused  by the method of removing  solute mass
 from the  aquifer at  sink nodes  (Konikow  and Bredehoeft 1978), rather  than by
 the radial flow situation.

      The  second  MOC   test  involved  application  to  a  recharge/recovery
 doublet.  The  solution of  the  purely  convective  transport  case was compared
 with the  semi-analytical  solution introduced by Javandel  et al. (1984).  The
 results  of  the simulation  showed  reasonable match of concentrations  in  the
 pumping  well for a short time  period  (less than 2.0 years).   The two  solu-
 tions  predicted  the  same value for the time at which the contaminant  reaches
 the  production  well.   For times larger than 2.0 years the  numerical solution
 for  the concentration  in  the pumping well  is not accurate  and  shows  large
 fluctuations  for which the  analytical  solution  represents  the  upper  enve-
 lope.   The  concentration at  the node  upstream of the production well showed
 much  less fluctuation  than at  the node  immediately to the  right or the  left
 of  the  production well.   The  relative mass  error  balance  was reasonably

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small,  approximately  -10%  to +  2*.    The  inaccuracy  of  model  results  is
caused  by  the  arrival  of contamination at  the  sink  node,  as was also  indi-
cated by Konikow and Bredehoeft (1978)

     The  last  test  case involved  plume  capture  by one  or two production
wells.   The technique prevents further degradation  of  the  aquifer by  using
one or  more wells.  The  simulation  results  were compared with the analytical
results  provided by  Javandel and  Tsang  (1986).   For  a  steady-state  flow
situation,  the numerical  model  was  run long enough  to  represent  the  steady-
state condition  for mass  transport.  An interactive  procedure,  sufficient  to
capture the plume, was needed to  estimate the  well flux.   HOC was capable  of
capturing  the  plume for  different values  of AH (the difference  in  hydraulic
head  value for the  upper and lower boundaries), which  ranged  between  10 and
90  ft.   However, the numerical model  predicted higher  values for  well  flux
over  the  entire range  of  AH.   The value  needed was  about  1.5 times  the
respective  analytical  value.   Sensitivity  of  results  to  the mesh size  was
not  studied;  as  the mesh size  decreases,  close agreement  is possible.   The
relative  concentration   in  the  well showed some  fluctuations,  yet the  be-
havior  of  the  curves  is generally acceptable.  The  case of  plume capture  by
two  wells  was  also simulated for &H = 20 ft.   HOC was  also  able to  capture
the  plume, yet  larger fluctuations  in the  pumping wells were  observed.   The
well  flux  was  also  larger than the  theoretical  value.   The simulation showed
that  HOC  is,  in general,  accurate in  simulating  plume capture  by  recovery
wells.   The  relative mass  error for all  cases  considered was  acceptable,
-2.7% to -6.4%.
 Stochastic  Analysis

     In  the  stochastic  analysis  of  mass  transport,  the  porous medium  is
 assumed  to be a  realization of a random field.   Hydraul.ic conductivity and
 other parameters  are  described  as stochastic processes and the flow equation
 is  then  solved to define  the  full  distribution of the velocity  field  or at
 least  its  first  few  moments  (i.e.,  mean  and  standard deviation or  covar-
 iance).    The spatial  correlation  structure of  hydraulic  conductivity,  an
 important  factor in  the  treatment, must  be defined 1n advance  on  the  basis
 of  field measurements, as  are  other  parameters in  the conductivity distri-
 bution.    The resulting  velocity  field,  a  random variable  in this  case,
 constitutes the  input to  the  transport  equation.    Finally,  the  transport
 equation  is solved to define the spatial  and temporal variability of  solute
 concentration.   Other parameters in the equation,  such as dispersivity, also
 may  be treated as stochastic processes.

      In  the Monte-Carlo technique,  a  deterministic problem is solved a large
 number  of times  with different sets  of  generated  parameters (called  reali-
 zations).   Each  realization is assumed to  be  an  equally probable represen-
 tation  of  the  actual set.  The  results  are then  analyzed statistically to
 define  the distribution of  output  variables.   Hydraulic  conductivity  reali-
 zations   were  generated   using   the  technique  developed   by  Mejia  and
 Rodriguez-Iturbe (1974)  (see also El-Kadi,  1986).   The approach  involves the
 addition of harmonics of random functions that are sampled from  the spectral
 density  function.    The  values of  Y,  the   logarithm  of  transmissivity, are
 generated from  a knowledge of the  mean Uy, the  standard deviation ov, and
 the  autocorrelation  coefficient «y.   The  autocorrelation structure  is re-
 presented by the relation

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     p(w) = e                                                      (1)

where c(w) is  the  autocorrelation  function,  and  w  is  lag.    The average
correlation  length, £,  may be  obtained explicitly  by  integrating  equation
(1) to yield

     ^-~                                                        (2)

Hydraulic  conductivity,  K,  assumed  as the  only  stochastic  variable, was
estimated  by  dividing  transmissivity  by  aquifer  depth  B,  a  deterministic
constant.  The  aquifer  was divided into a  number  of  conductivity  blocks and
the generated values  were inserted in  the  blocks,  assuming  a  constant  value
for each  block.   The  same finite-difference mesh was used as in the descret-
ization  of  the  block  system.   The  number  of Monte-Carlo  realizations was
taken  as  300 for  all  cases  considered.   The  computer  time  ranged between
three and eight hours on the VAX 11/780 minicomputer.

     HOC  was modified  by  adding  the  necessary  routines  that  included  a
generator and  a package  for  basic  statistical analysis.  New matrices  were
added  to  save  concentration  values  at selected times  at all  locations for
each Monte-Carlo  run.  These  data  are  analyzed statistically at  the end  of
the  run.   Also included   is  a simple  routine  to  check  the maximum concen-
tration  in  the aquifer during the remediation  process  and to  register  the
time to  reach  the  assigned (accepted) concentration.   Such  time  is  termed
the cleanup time;  its values  are  analyzed  at the end of the run.   To avoid a
potential storage  problem for output results, a new  "flag"  was  added  to the
program to  suppress the detailed output by  printing  only the  results  of the
statistical analysis.   Another flag  was added to  allow  the  use  of the  model
for both  stochastic and deterministic runs.
Results

    Based  on  the verification  of  HOC described in  a  previous  section,  only
the  cases  of  recharge/recovery  single  well  and  plume  capture  are  con-
sidered.   The  deterministic  parameters used for all  simulations are given in
Table 1.   Results of the analysis  follow.

Case 1;  A Recharge/recovery  Single Well

     Table 2  shows  the different  experiments,  run  in  a sensitivity analysis
framework.   The well  was  located at  node  (5,  6).   Results  of  Case  l.A are
given in  Figure 1.   The average concentration and the 10 and 90 percentiles,
as  obtained  by stochastic analysis,  are  compared  to the deterministic solu-
tion  for  uniform  soil  with  transmissivity  value  of 0.1 ft /s (i.e.,  K  =
0.005 ft/s and B =  20 ft.).   The average of  the  stochastic results agrees
with  the  deterministic solution at  small  and  large times.   Additional  dis-
persion  due  to  variability  of  conductivity  caused  the  deviation  at inter-
mediate times.   However,  when compared with Figure 2, the deviation from the
deterministic  solution  is most severe  in  the absence  of micro-dispersion
(i.e., a  = at  = a  = 0 as  in  Case  l.B).

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     A  number of  deterministic  runs  was  performed with  different  values
of a, in  an  attempt  to match  the deterministic  solution  and  the  average
concentrations obtained  from  Case  l.B.   The closest correspondence  is  shown
in Figure  3,  with  results obtained by using a = 50 ft.  The  solutions  match
only  at  intermediate  time but  deviate  at  small  and  large  times.   It  is
concluded  that no  single dispersivity  value can be defined as the  effective
value,  i.e.,   a  single  value to  reproduce the  average of  the  stochastic
solution.

     Figure 4 shows the  average concentration values  in  the aquifer  after
2.0 years  for Case l.B.   In  Figure 5  the 1% relative concentration  line  is
superimposed  on  the contours  representing  the  coefficient  of  variation.
Variability is smallest close to the aquifer boundaries due  to  their deter-
ministic  nature,  and close  to the well  where  the  injection/withdrawal  oc-
curs.   Variability  is maximum at the  plume boundaries  due  to the advancement
or contraction of the  plume.   Some of  the numerical  inaccuracies  at  lower
concentrations  may  have  contributed  also  to variability.   However,  vari-
ability  in this  zone   is  not really  important because  the  average  concen-
tration  is small.   The  coefficient  of  variation  is  about 8 at the 1% concen-
tration  contour.

     The single  well problem  is  of  special interest because the analytical
solution  obtained  by   Gelhar  and Collins  is independent  of the  value  of
hydraulic  conductivity.   Deterministic  runs  of HOC  showed  the  same  con-
clusion:   very close solutions for two  runs  involving  a change  in  K by one
order of magnitude (K was  taken as 0.005 and 0.05 ft/s).  Comparison between
the  results of Experiments  l.B  and  l.D, with an order  of  magnitude differ-
ence  in the  geometric  mean of  K,  showed practically identical  results for
both  the  average  concentration and  standard  deviation.   Hence the inter-
esting  conclusion  is that, although the expected value  of  the population of
K  does  not influence the  results of  the stochastic  analysis, variability of
K, represented by  the  standard deviation,  and  the autocorrelation structure,
have  profound effects  on  results  and  should be considered  in  the  analysis.
Including  variability in K values causes dispersion-like behavior and natur-
ally leads to variability  of concentration results.

     Sensitivity   of  results  to  the   degree  of  variability,  represented
by OY, and the degree  of  autocorrelation,  represented  by  *y, is illustrated
in Figures 6  and  7.   As expected, higher variability in concentration values
and more dispersion effects result as the  standard  deviation of K increases
(Figure  6.)   Figure  7  shows that  increasing  the  degree  of  correlation,
represented   by  a  decrease  in  »y or  an  increase  in a,  is  similar  to  an
increase  in   variability.      me   values  of *v used   are » («. = 0) and
0.0003  («.  =  3333  ft).    In  fact,  due  to  the finite   size  of   the  finite-
difference mesh,  the case  of  t  less  than half of the mesh increment length
cannot   be   simulated,   and   the   smallest  correlation   length   should
be £ = ax/2 (or Ay/2),  i.e.,  half of  the  mesh  increment  length  (450 ft).
The  reason is that  values of hydraulic conductivity are  taken  as constants
for each block.

     The dependency of i  can  be explained  as  follows  (see also  Smith and
Freeze  1979,  El-Kadi and  Brutsaert  1985).   If the  values  of hydraulic con-
ductivity  are highly correlated,  a  series of high or  low  values will  exist
in  the   block  system for  a given realization.    The  resulting concentration
will  then move further  away  from the  expected value.   Over the whole  series

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of Monte-Carlo  runs  the  standard deviation  in concentration  tends  to in-
crease,.    Howex'er,  the  effects  of  increased  correlation  on  the averaged
concentration ere  not large.   The reason for these effects can be explained
by the  influence  of  the  correlation  on the  sample  statistical   properties
(see  Lcucks  et  el.  1961,  c. I^c),   The  expected  value  of  the variance of a
conductivity realization  is smaller  than that for the population.  The bias
in that estimate decreases  as  the sample size  increases.  On  the  other  hand,
the variance  of this estimate  is inflated  by a  factor  that  depends  on the
correlation function  and  whose importance does not decrease  with  the  sample
size.   For the case  shown  in Figure  7,  the  variability in  the  variance  of
conductivity realization  apparently  caused  dispersion-like effects; yet the
effect was  not  too large  due  to the  decrease  in the expected value of the
variance  (the  sample  size  in  this  case  is 90; not  very large).   It can  be
concluded  that  increased correlation  will  cause  larger variability on con-
centration  values,  yet  the dispersion-like effects  are not  obvious.    For a
large number  of conductivity  blocks,  it is likely that increasing the cor-
relation  will   lead  to  similar  effects of  increasing the  variability  of
conductivity.

     The  results   regarding the  cleanup time  are shown  in_Table 3.   The
values  in  the  table are,  respectively, the  average time,  T, the standard
deviation, o-r, the  90  percentile,   T90,  the  maximum,  Tmax,  and  the time
needed to  reach an average concentration of  10%,  T.   Cleanup-time value was
defined  as  that needed  to  reach concentration value  of 10%  of  the  maximum
concentration (henceforth called  the cleanup  level).   In general, due to the
mixing effects  caused by variability  of  conductivity,  the  expected value  of
the  cleanup time  (T) is  smaller than  the  value  obtained  deterministically
(about 2.0 years  for all  cases).   Table 3 shows that  different management
decisions  could  be adopted based on different criteria.  A  decision  can  be
based  on the average cleanup  time  (uncertainty,   represented by  oj, must  be
considered);  the   time  needed  for   the  cleanup  process  to  satisfy  certain
probability  (e.g.,  T90 which represents  the time  needed  in  90%  of  the
cases);  the  time  to  reach  an average value  for  the  cleanup  level over  the
entire aquifer; and the maximum possible time for cleanup.   Decision  will  be
based  on a number  of factors including  available funds for  remediation  and
possible health effects caused by residual concentrations.

     Table  3 shows also that  results are sensitive  to the  variability  of
conductivity  and   its correlation  structure.    In general,  dispersive-like
behavior  results   for highly  variable  or  strongly  correlated  conductivity
fields.   The -value  of  the cleanup  time depends  then on  the  resulting  be-
havior  of the  curve  as well  as the  maximum  value accepted  for concentra-
tion.   For example,  for the deterministic solutions with  high and  low dis-
persivity,  the  relative cleanup  time  for the two cases should be different
if  the  cleanup level  was chosen  below or  above the  point where  the  two
curves intersect.

     Comparison between Cases l.B and l.E  shows  that  the  value of the popu-
lation mean has practically no effect on the estimates shown in Table 3.   As
might  be expected,  considering  the microscale dispersion  causes additional
dispersive  effects (compare between Cases  l.A  and l.B).  The highest varia-
bility  in the  estimate  of  the cleanup time has  been  found  for highly vari-
able or  strongly correlated conductivity fields.

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     It  is  concluded here  that  variability of  conductivity and  its  corre-
lation  structure  have  important  effects  on the  results  of the single  well
case; yet  the mean of  the  population  does  not influence the results  in any
way.  Variability  of conductivity leads to variability of  results regarding
concentration and dispersion-like effects.

Case 2:  Plume Capture by a Single Well

     Plume  capture  is a technique that  prevents further degradation  of the
aquifer through the  use  of  a  number  of production wells.   Recently, Javandel
and  Tsang   (1986)  introduced   a  technique, based on  the  complex  potential
theory,  to  analyze  the plume capture  by  using  different  aquifer  and  flow
parameters  including well flow, aquifer thickness, and Darcy's velocity.

     The well was  located  at  node (5,5) and a constant concentration  source
representing the landfill extended over three  nodes  from  (4,2)  to  6,2).   For
the  numerical  analysis  using HOC,  the  computer time needed  for   the plume
capture case  was  directly  proportional to Darcy's velocity and  consequently
to  the  well flux  (everything  else being  fixed).   Hence,  to reduce  the  amount
of  computer time a  value  of  AH  = 5  ft was used.   The analytical   value for
the  well flux,  Q,  as estimated from the equation of  Javandel and  Tsang, was
0.22  cfs.   The numerical  value  for  the uniform  case (as  obtained  by using
HOC)  was  about  0.35.   A number  of  tests  were performed for the  stochastic
analysis with  different values of well  flux.   The  conductivity values  were
generated using  uy  = -1.0, ov =  0.5,  and «y = «.  The results  were examined
for  a  few  realizations to test  the  ability  of the  well  to  capture  the
plume.  The model was run for 20 years of simulation time.

     Figure 8 compares  between  the  90 percentile of concentration for  Q  =
0.5  and  1.0 cfs.   The plume was  captured with  Q = 1.0 cfs,  indicating  that a
much  higher value  of Q  is  needed  to capture  the plume (more than 4.0 times
the  analytical  value and about three times the  value needed for the uniform
case).   The reason is the  introduction of  dispersion-like  effects caused by
the  variability in  the conductivity  field.    To illustrate variability of
results after 20 years  for the  case where Q  = 1.0  cfs, Figure 9 shows the
coefficient  of  variation  of concentrations   superimposed on  the  results
concerning  the  average  relative  concentration  of 1%.  The  figure  shows  that
variability is  highest  at  the boundaries  of  the plume  where  the  front is
advancing.    The coefficient  of  variation  of  concentration over   the plume
equals about 9.0 or  less.   Variability is  smallest at the source of contami-
nation  where  the  concentration  reached  its  maximum value.   As mentioned
before,  some  numerical  inaccuracies may also  contribute  to variability of
results in the low concentration zone.

     The time  change  of the  average  concentration  and  the coefficient of
variation of  concentration in  the well are  shown  in Figure 10.    Both are
practically  constant  after  about   10  years,   i.e.,  when   the  steady-state
situation is reached.   The  coefficient of  variation  is highest  at  small  time
(about 0.67) and declines to reach an asymptotic value of about 0.35.

     Some  deterministic  runs  were   performed  to   capture the   plume  with
different dispersivities,  in an  attempt  to match  the average plume  as ob-
tained  by  the stochastic approach.   The results concerning the IX relative
concentration are  presented  in  Figure  11  for  values of  a =  0   and  10 ft.
None  of the deterministic  solutions match closely  the  average of  the  sto-

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chastic  results.   The  plume was not  captured  for  cases with a higher than
10.   In  other words, 1t was  not  possible  to  match  closely the shape of the
plume and to capture the plume 1n the same time.

     It  is  concluded  that  although  plume  capture  can be  achieved under
variability  conditions,  a  higher  value of the well  flux is needed.   Disper-
sion-like  effects  result  from  the  variability  in  conductivity   values.
Variability  of  concentration  is  highest at plume boundaries where the front
is advancing.   It is  not  possible  to match  the  plume shape and to  capture
the plume by  including a high dispersivity value.   Sensitivity  of results  to
parameters  of  the  conductivity  distribution  (i.e.,  yy«  °v»  *v) was  not
studied.    However, additional dispersion-like  effects  are  expected by con-
sidering a highly  variable  or strongly correlated conductivity  field,  as was
the case for the single-well case.


Conclusions

     The  study  demonstrates  that  variability  of  conductivity  is  a very
important  factor  in  the  analysis  of  remedial  actions  by  recovery  wells.
Considering  variability  results  in  dispersion-like  effects caused  by the
variations in  the  velocity fields,  represented as  a  random  variable  in this
case.    In  addition,  uncertainty  in  concentration   results  1s quantified
through  the  stochastic  analysis.   Hence,  deterministic approaches fail  in
defining the  exact shape of  plumes  or the break-through curves and  also  in
describing variability of results.

     The analysis  performed  considered two situations that commonly  exist  in
remedial  actions  by recovery wells:   a  recharge/recovery  single well and
plume capture  by  a production well.   For the  single-well case,  variability
is maximum at  the  plume boundaries due to the  advancement or  contraction  of
the plume.   However, variability  in  this zone  is  not important due  to the
relatively low concentrations.   Variability  of  results  does not depend  on
the  expected  value  of  the  K  population, yet  the  standard  deviation and
correlation  coefficient  are  controlling  parameters.   The  cleanup time  is
influenced by  the dispersion-like  effects;  it also  is  affected by  varia-
bility and  the  correlation  structure as  well  as by the prescribed  cleanup
level  (defined  as the  maximum  residual  concentration after  remediation).
The study  indicates  that different management  decisions exist  in  choosing a
remediation  strategy  based on uncertainty in the cleanup time.  The deter-
ministic solution  for  the  concentration in the well  does  not  predict accur-
ately the average  of the stochastic solution.

     Although  plume capture -can  be  achieved under variability,  a higher flux
is needed  for the production  well.   For  the case  simulated,  this  value was
about three  times  the  respective  value for uniform aquifers.  This value  and
other  results  are influenced by  the distribution of conductivity  and  its
correlation  structure.   Again,  variability causes the development of disper-
sion-like  effects  as well  as variability in concentration values.   Varia-
bility of  concentrations is  highest  at the advancing  boundary  of  the plume
where the  concentration is  small.   It is not  possible  to capture  the plume
and  preserve  its  shape stochastically  through the  use of  a  deterministic
solution with  an altered dispersivity  value.

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     The  study  illustrates that  deterministic  solutions fail  to  predict a     10
number  of  important  features  provided  by  the  stochastic  results  and,
naturally, to quantify  uncertainty  in  output values.   However, as indicated
by a  number  of  studies  including  El-Kadi  (1984), parameters of the conduc-
tivity distribution  and the correlation structure  should  be based on field
studies.
References

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    transport-advection  and  dispersion.     In    Croundwater  Contamination,
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Boutwell, S.H.,  S.M. Brown, B.R.  Roberts,  and  Atwood  Anderson-Nichols  & Co.,
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    sources Research, v. 18,  no.  4, pp. 835-848.
Ehrenfeld,  J.  and J. Bass.   1984.  Evaluation of Remedial Action Unit  Opera-
    tions  at Hazardous  Waste Disposal Sites.   Pollution Technology  Review
    No.  10, Noyes Publications,  Park Ridge,  New  Jersey, 434 pp.
El-Kadi,  A.I.    1984.   Modeling  variability in  groundwater flow.   HRI Paper
    No.  75a,  Holcomb  Research  Institute,  Butler  University, Indianapolis,
     Indiana, 56  pp.
El-Kadi,  A.I.    1986.   A  computer  program for  generating   two-dimensional
    fields  of autocorrelated  parameters.    Ground  water, v.  24, no.  5, pp.
    663-667.
El-Kadi,  A.I.   1987.  Application of the USGS two-dimensional mass-transport
    model  (MOC)   to  remedial  action  by recovery wells.   Submitted  to  Ground
    Water.
El-Kadi,  A.I. and  W.   Brutsaert.  1985.   Applicability  of  effective para-
    meters  for  unsteady  flow in nonuniform aquifers.    water Resources Re-
    search, v. 21,  no.  2,  pp. 183-198.
Engineering-Science.   1986.   Cost model  for selected  technologies for re-
    moval   of  gasoline  components  from  groundwater.   Health  and  Environ.
    Sciences  Dept., API Pub.  No. 4422, American Petroleum Inst., Washington,
    D.C.
Gelhar,  L.W.  and C.L.  Axness.   1983.  Three-dimensional  stochastic analysis
    Of  macro-dispersion in aquifers.   Water Resources  Research, v. 19, no.
     1,  pp.  161-180.
Gelhar,  L.W.  and  M.A.  Collins.   1971.    General analysis  of  longitudinal
    dispersion   in  nonuniform flow.   Water  Resources Research,  V.  7,  no.  6,
    pp.  1511-1521.
Gelhar,  L.W., A.L.  Gutjahr,  and R.L.  Naff.  1979.   Stochastic analysis  of
    macrodispersion in  a stratified  aquifer,   water  Resources Research  v.
     15,  no. 6,  pp. 1387-1397.
G*ven,  0., R.W.  Felta, F.J.  Molz. and J.G. Melville.   1985.   Analysis and
     interpretation  of  single-well   tracer  tests  1n   stratified   aquifers.
     Water Resources Research, v. 21,  no.  5, pp.  676-684.

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G*ven,  0.,  F.J. Molz. and  J.G.  Melville.   1984.   An analysis of  dispersion     11
    in  o  Stratified aquifer.   Water  Resources  Research,  V.  20,  no. 10, pp.
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    Handbook  of  Mathematical  Models.    American Geophysical  Union,   Water
    Resources Monograph  10,  Washington, D.C., 228 pp.
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    groundw&ter  flow  in  6  bounded  domain^ 2.   Two-dimensional  simulations.
    Water Resources Research^  V. 15, no. 6, pp. 1543-1559.
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    analysis  Of macroscopic  dispersion.    Water  Resources Research,  V.  16,
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    tainty in prediction.  Water Resources Research,  v. 17, no.  2,  351-369.
Smith  L. and   F.W. Schwartz.   1981b.  Mass  transport,  3.   Role  of hydraulic
    Conductivity  data  in prediction.   Water Resources Research,  V.  17, no.
    5, pp. 1463-1479.
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    transport in  a random velocity field,   water Resources  Research,  V. 18,
    no. 2, pp.  231-244.
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Biographic Sketch                                                                 12

    Aly  I.  El-Kadi is  a research scientist  and  hydrologist  in Holcomb  Re-
search Institute's Water Science  Program, Butler  University.  He  received  his
B.S. and M.S. degrees  in Civil  Engineering  from  Ain Shams University,  Cairo,
Egypt, and  his  Ph.D.  degree  in Ground-Water Hydrology from  Cornell Univer-
sity  in  1982.   His current  research  includes modeling the effects of  para-
meter  variability  on  chemicals that penetrate  the soil  in conjunction with
water  or soil  moisture.  He  has  authored or coauthored papers  on  saturated
and unsaturated flow  in uniform and fractured porous  media,  and on stochas-
tic analysis  of flow  in heterogeneous porous media.   His publications  in-
clude  state-of-the-art  reports  on  modeling infiltration  and variability
studies  as they   apply  to ground-water  systems.   His  present address  is
Holcomb  Research   Institute,  Butler University,  4600  Sunset Avenue,  Indi-
anapolis,  Indiana  46208.

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                                FIGURE TITLES                                    13

Figure 1.  The  average  of the  stochastic  analysis,  and the  10  and 90 per-
           centile, compared to the deterministic solution  (Case  l.A).


Figure 2.  The  average  of the  stochastic  analysis,  and the  10  and 90 per-
           centile, compared to the deterministic solution  (Case  l.B).


Figure 3.  The  average  of the  stochastic  solution of  Case  l.B compared  to
           deterministic runs with a = 0 and 50 ft.


Figure 4.  The  average   concentration  in  the  aquifer  after  2.0  years
           (Case l.B).


Figure 5.  The  1  percent  average relative  concentration  superimposed on  a
           contour map  representing  the coefficient of variation of  concen-
           tration (Case l.B).


Figure 6.  Sensitivity of  results of  Case  l.B  to variation  in  the standard
           deviation of Y  (the logarithm of transmissivity).


Figure 7.  Sensitivity  of  results of Case  l.B to variation in the  corre-
           lation structure of Y.


Figure 8.  The  1 percent  relative concentration's  90 -percentile  with Q =  1.0
           and  0.5 cfs (plume capture  case).


Figure 9.  The  1  percent  average relative  concentration  superimposed on a
           contour map  representing  the coefficient of  variation  of concen-
           tration (plume  capture case).


Figure 10. The  average  and coefficient of  variation of  concentration in  the
           production well  (plume capture  case).


Figure 11. The   1   percent  relative  concentration  with a = 0 and  10  ft.
           (deterministic  solution)  compared to the  1  percent average rela-
           tive concentration for the stochastic solution.

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         IGWMC   GROUNDHATER   MODELING   REPRINT
                  A NEW ANNOTATION DATABASE FOR GROUNDUATER MODELS
                                         by

                      Paul van der Heljde and Stan A.  U1ll1ws
                                    presented at

                             The NWWA/IGWMC Conference
                     •Solving Ground-water Problems with Models*
                                February 10-12,  1987
                                  Denver,  Colorado
                                     GWMI  87-11
INTERNATIONAL   GROUND   HATER   MODELING   CENTER
                             Hoicomb Research Institute
                                 Butler University
                            Indianapolis,  Indiana  46208

-------
             A NEW ANNOTATION DATABASE FOR GROUND WATER MODELS
               Paul K.M. van der Heijde and Stan A. Williams

                 International  Ground Water Modeling  Center
               Holcomb Research Institute, Butler University
                             4600  Sunset Avenue
                         Indianapolis, Indiana 46208
Abstract
     The   International   Ground  Water  Modeling   Center   operates   as  a
clearinghouse for  Information  on ground water models.   In 1979, the Center
established  Us  first  annotated database of  Information on  models.   The
database  was  Initially  implemented  on a  UNIVAC 9030  mainframe  computer
using  COBOL software.    In 1982,  the database  was  transferred to  a VAX
11/780  minicomputer  and  implemented  with  a  database  management  system
called DATATRIEVE.  Since  installation of  the  database,  the  IGWMC staff has
continually maintained, updated,  and  used  the  annotation system  for storage
end  retrieval  of  information  that  1s pertinent to  specific  ground water
models.     However,   recent  developments  in  modeling  and  in  database
management  systems  have generated  an  awareness  of  the deficiencies in the
current   system.    Modeling  approaches  such  es  optimization  methods,
stochastic  techniques, hydrochemical  modeling, and  parameter identification
modeling  are not  adequately described  in the  current  annotation system.
Also,  developments  1n database  management  techniques  such as hierarchical
database  organization and  the  use of  expert systems  can  provide useful
tools  for Improving the organizational  structure and accessibility of the
database.

     In  response  to  new  methods  in  ground  water modeling  and  database
management,  the  International  Ground Water Modeling  Center has renovated
the  structure  of  Its annotation database for  ground water  models.   New
features  of the database  will  Include  sections  on  optimization models,
stochastic  methods,  parameter Identification models,  hydrochemical models,
and  data  processing programs.    The new database  will  also  incorporate  more
complete  descriptions of  the mechanics  of  models with  sections  on model
development,   solution  techniques,   boundary  conditions,   and  specific
hardware  and software requirements.   The  system 1s organized 1n a modular
format within  a  two-tiered  hierarchical  structure.   This structure, which
should  facilitate  accessibility  to  the  database,  will  be  amenable  to
additions  or modifications.   It 1s  anticipated  that  the new database may
eventually  be  Incorporated  Into a model  selection package by  coupling  it
with a complementary  expert system.

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Introduction

     In 1973, representatives  of  the  Robert S. Kerr Environmental Research
Laboratory of the U.S. EPA proposed that SCOPE  (the Scientific Committee on
Problems  of  the Environment)  initiate  an   investigation  into  the state of
the  art of  ground  water models.   SCOPE  requested that  Holcomb Research
Institute  perform  this   investigation,   since  the   Institute  had   just
completed  a  critical   evaluation  into the  role of mathematical  models in
environmental   decision-making  in  the  United  States.     The  Institute
coordinated  the organization  of  an  international  steering  committee to
pursue  SCOPE'S  original  recommended objective.   The  committee was chaired
by  John Bredehoeft of the U.S.  Geological  Survey.   Yehuda Bachmat,  then
Director  of  Research  of  the Hydrological  Services of  Israel, was recruited
as project director.   One of  the  principle  recommendations of the committee
was  for the  establishment  of a central "clearinghouse" which could provide
information  dissemination, technology transfer, and training  in groundwater
modeling  (van der Heijde 1982).

     The  first  project  report emanating  from  the investigations  of the
steering  committee was  entitled  "Utilization of  Numerical  Ground   Water
Models  for Water Resource Management," and  was produced by Holcomb Research
Institute  for  the  U.S.  EPA  and  SCOPE.    This report  was  a .comprehensive
review  of the  state-of-the-art of  ground water models.   It also presented
recommendations for further developments  in groundwater modeling (Bachmat
et al.  1980).   Four general problem areas were identified:

          - accessibility of models to potential users
          - communication between managers and technical staff
          - inadequacy  of data
          - inadequacy  of modeling effort

     The  report of  Bachmat  et al.  stated  that  the  first  of  these  four
problem  areas—accessibility  of   models—should  receive   the  highest
priority.    Accessibility  in  this context  includes  both  the  quality of
available  Information  on models  and  the  level of training of model  users.
This Initial  report emphasized the  need  for Improvements  1n  the quality and
availability  of  model  documentation  and  descriptive  information   about
models.   The recognition of  this need provided  the  early  impetus for the
establishment  of  the  MARS  database  of  annotated descriptions  of  ground
water models.

     The Model  Annotation and  Retrieval System (MARS)  was a  keystone  in the
development  of  the International Ground Water Modeling Center.  IGWMC was
established  in  early  1978,  and Us general  mandate was to serve as  a cen-
tralized  clearinghouse which  would bring  together Information on documen-
tation,  application,  verification, validation,  and availability of  ground
water models.   The Center  would  also serve  as a mechanism for  technology
transfer  by  offering  short courses, sponsoring conferences  on ground water
modeling, and generally  providing opportunities for Interaction  among model
users,  model  developers,   and those  Involved  1n wanagement  of projects
related  to  ground  water  modeling  (van  der  Heijde  1982).    Figure   1
Illustrates  the functions and  organization of  IGWMC.   Throughout the  early

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                    Figure  1.  The functions  and  organization  of  the
                                Modeling Center.
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years  of  IGWMC,  the MARS  database  became  an  Integral  component  of the
Center.   The  development of the database provided a means for pursuance of
many  of  the   established   objectives  of  IGWMC.    The  database  became   a
mechanism  for  improving the  accessibility  of  models,  and  it  indirectly
enhanced the quality and credibility of model applications.


Development of MARS

     The  development of  the MARS  database  was  initiated  in  1978  with  a
major  effort  at  collection of  the pertinent  information  on ground  water
models.   Requests for descriptive  information on available models appeared
in journals and  in  the  first newsletter distributed by the  Center.   Also  in
1978   IGWMC   distributed  a  questionnaire  on  model  use   to   users and
developers.   Results from these intial efforts were compiled  into a proto-
type  version  of MARS   that  was  presented  for  review  at   a  workshop  on
policies  and  operational  procedures  of IGWMC  1n April  1979.    After  this
initial  review.  Information collection efforts continued,  and the database
gradually  developed until   1982  by  which  time  a total of approximately 400
annotated  entries had  been collected and verified  (van der Heijde 1982).
In  1982, the emphasis  in  the  management  of  MARS was changed  from develop-
ment  to  review,  update,  maintenance,  and  use of the  database  contents.
Although  new  annotations  were  continually  entered  1n  response  to new
information,  the  Center  also  pursued  a strong Initiative  to  evaluate,
verify,  and update  stored  Information.   Many searches  were performed for
customers,  and   a  set  of  procedures  was  established  to  elucidate and
standardize  the use of  the database  (Srinivasan and  van der  Heijde 1985).
Also  during this period  the working  aspects of  MARS  were  studied,  with  a
view  toward potential  improvements.   The annotation  form  was  revised and
related  database programs  and  other  search mechanisms  were  Investigated.
Results  of searches for certain types of models were  published as documents
of  possible  interest  to  individuals  in  the  modeling  community (van der
Heijde  1982).   By 1984, the database  included  over  600 annotated entries  on
ground water models.


MARS—Organized Under DATATRIEVE

     In   1982.   IGWMC   transferred  the  MARS   database   from  a   UN IVAC
minicomputer,   using  in-house  COBOL-based   software,  to   a   VAX   11/780
minicomputer  and reorganized  the  database  under  DATATRIEVE,  a VAX-based
database  management system.   Within  the  DATATRIEVE  framework,  information
is  coded  in   a   binary  format  |(0,l)=(no,yes)).   This coded  information
includes  model  descriptors  on  aquifer   conditions,  boundary  conditions,
solution techniques,  processes  modeled,  and details  about  input/output
characteristics.   Also under DATATRIEVE,  several text fields were reserved
for information about model development and purpose  and for  references and
remarks  pertaining  to use  of the model  (van  der Heijde 1982).

     Several  advantages  are  inherent  in the organization  of  MARS under
DATATRIEVE.   This  system  avoids the  common  "key word"  type  of  search, and
is therefore  much more flexible than  other potential  systems.  The complete
11st  of descriptors  and textual  fields  1s  available during the searching
process.    In  this  system, no  remote  dial-up  1s  necessary.   IGWMC staff
members   perform  the   searches  according  to   customer   requests.    The

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Interaction  between  the  individual  requesting  the  search  and  the  IGWMC
staff  member  performing  the  search  provides  a  mechanism for  tailoring
results  to the  individual's  needs  and  for  assuring  the  quality of  the
Information sought.


MARS—Organized Under RBASE

     The current  renovation of the MARS database will be structured with a
format that  will  be compatible with the RBASE system V database management
system.   The  RBASE system  is compatible with  a PC  environment,  and its
inherent   flexibility  should  facilitate  the   new  MARS  organizational
structure.     Under   the  RBASE   system,   MARS  will   assume  a   two-tiered
hierarchical  structure,  and  descriptive  Information will  be  organized  in
modules.  Access  to the  new system will  be menu-driven.   The modular format
should ensure  greater flexibility than previous  systems to  make alterations
or additions  to  the system.   Figure  2 is a  representation of the  proposed
structure for  the new MARS database.

     As  Figure 2  indicates,  the  new system will  not only  be  structured
differently,  but  will   include  much more  descriptive  information   about
specific models  and  types   of  models.   Part  I of  the  structure  Includes
details   about   model   Identification,  model    development,    execution
requirements,  evaluation,   availability,  and  a general  description of the
physical system  that the model can address.   In part II of the  structure,
descriptive details  are organized in  modules  according to the type of  model
described.   For example, categories  for predictive models are fluid  flow,
solute transport, heat transport, and deformation.   Geochemical equilibrium
and   nonequilibrium  models  are   described   under  the   category  for
hydrochemistry.    The watershed  model  category  includes  conjunctive use
models,  which are usually  based  on methods for  integrating groundwater and
surface  components.   Management models usually combine governing equations
for  ground water flow or  transport with an optimization technique such  as
linear  or  quadratic  programing  (Gorelick  1983).   Data processing programs
include  pre-  and   postprocessors  for  models,  programs  for   statistical
analysis,  and database management systems.  Parameter identification  models
are  based  on efforts  to  solve  the  inverse  problem  of  defining  aquifer
parameters  through  analysis of  the  dependent  variables  of  the  system  of
interest.   In many  instances, a model may necessarily fall into two or more
of  these  modules.    For  example, a  management  model usually will  include
some  type  of predictive model.   Recently, some  predictive  models have been
coupled   with  hydrochemical  models.     IGWMC   staff  will  monitor  such
development  as they appear.   Information within each of  these  model-type
modules  is  based  on the   state-of-the-art   1n  model  development for the
respective category.  Details  included  within  each module describe general
characteristics  of  the  model,   processes and  phenomena  addressed in the
model,  boundary  conditions which may be simulated,  solution  methods, and
input/output   characteristics.    Other types of  detail  will  be addressed
according  to  the category  of the model.   The Appendix  provides specific
examples of  the  details  included.

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A New Database Structure for  IGWMC Model Information
                         I  PART I OF FORM |
                     CONTAINS INFORMATION ABOUT:

          (A) Model ID, development, code Information, evaluation, etc.

          (B) Physical system                      	
                          I PART II OF FORM
                          *         i
                I
       Fluid    Solute
       Flow   Transport
     I
Deformation
    I
Watershed
      I
Data Processing
                      Heat
                    Transport
          Hydro-
         chemistry
       Management
     r
 Saturated
        Unsaturated
                I
           Preprocessors
          Parameter
         Identification
              DBMS
                  Postprocessors
                    Statistics
                Figure 2.  Proposed structure of the new MARS database.

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Rationale for New Structure of MARS

     Several  justifications  underly  the  renovation  of  MARS.    The  new
structure  is  logical  in  both  the  vertical  and  horizontal  dimensions.
Vertically,  the   system   follows   a  hierarchical  structure  that  flows
logically  from  general descriptive  information  about  £ model  tc  detailed
inforraat'ion   about   the   solution  mechanics   of   a   specific   model.
Horizontally,  the modularity  of  the database  facilitates adaptability  to
dynamic developments  in the  field  of  ground  water  modeling.   Any  attempt  at
organization of  ground water models must be adaptable to  new findings  (van
der Heijde  and  Park 1986).   The new  database  will also provide  a  powerful
tool for  the improvement of quality control 1n model applications.  Recent
research  has shown that  quality of model  applications  is often  limited  by
the inaccessibility of Information about  available  models (van  der Heijde
end Park  1986).


     Another of  the principle reasons for the new organization of  the  MARS
database  is compatibility with expert  system technology.   Developments  in
artificial   intelligence  have  rendered   expert   systems   accessible   to
practical   applications   such  as  model  selection and  defining  potential
solutions  to hydrogeologic problems.  The conceptual basis of  the  database
is  a  function of the experience and expertise of  the IGWMC staff.   The use
of  this   expertise  1n   the   categorization  and  delineation  of  detailed
information on ground  water models may be  analogous  to  the logical  rules
and  knowledge base  that would be  used in  an  expert  system developed  for
hydrogeologic  problems or for ground water model selection.   In the future,
the MARS system  as  organized under RBASE  may be  Incorporated  into such  an
expert  system.    Also,  expert systems may eventually be used as  mechanisms
for determining  model reliability  and  for  the  Interpretation of  simulation
results  (van der  Heijde  and  Park 1986).
 Conclusion

      Since the inception of  the  International  Ground Water Modeling Center
 in   1978,   the  development   of  the  MARS  database  has  been  an  Integral
 component   in  the pursuit  of the  Center's objectives.    Its  most salient
 attribute  1s the contribution that  the  development of  MARS has made to the
 accessibility  of  detailed  Information  on ground  water  models.    In the
 future, the  database  may  also  be  incorporated  into  the  development  of
 expert  systems   that   may  provide   powerful   mechanisms   for   problem
 resolution.    The   current   modifications  of  MARS  will  ensure  that  the
 database  will  continue  to  be  an  Important  tool in ground  water modeling
 investigations.
 Appendix

 PART I:  General Information

 Model  Identification

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       Model name
       Model category
                Groundwater flow
                        saturated
                        unsaturated
                Solute transport
                Heat transport
                Subsurface deformation
                Hydrochemistry
                Watershed
                Management—Optimization
                Data-processing
                Parameter Identification
       Abstract (< 5 lines)
       Model history
                Date of first release
                Release dates of updated versions
                Current version
                        f
                        release date
                        IGWMC check date
                Built upon an existing model
                        which: 	  (see notes for more Info)
                Part of a program package
                        package name: 	  (see notes for more Info)
       Related preprocessing programs
                Name
                Purpose
       Related postprocessing programs
                Name
                Purpose
Model Development

Authors:       1
Name:
Address(l):
Telephone:
Address(2):
Telephone:
       Original Institution of Model Development
                Name
                Address
                Type of  institution
                         Federal/national government
                         State/provincial government
                         Municipal/county administration
                         Planning agency
                         Research institute
                         University
                         Consultant
                         Private industry
                         International organization
                         Other:

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       Code Custodian
                Name
                Address
                Type of institution
                        Federal/national government
                        State/provincial government
                        Municipal/county administration
                        Planning agency
                        Research institute
                        University
                        Consultant
                        Private Industry
                        International organization
                        Other:
       Contact person to obtain code
                Name                    (space for 2 contacts)
                Institution
                Address
                Telephone
       Contact person for model support
                Name                    (space for 2 contacts)
                Institution
                Address
                Telephone
                       Prograa Execution Requirements


Resident software requirements: 	  (e.g. IMSL.MPSX)

Hardware Requirements
       Computer of model residence
                type
                        Supercomputer
                        Mainframe
                        Minicomputer
                        Microcomputer
                Make and model
                Operating  system
       Computer—other  Implementations     (space for several)
                Type (indicate make and model)
                        Supercomputer: 	 (Cray 1, CDC Cyber 205)
                           Operating system: ^_^
                        Mainframe: 	  (CDC~~5500/7600)
                           Operating system:
                        Minicomputer: 	 IVAJTll/780, Prime)
                           Operating system:
                        Microcomputer: 	(IBM PC, AT; Compaq 386)
                           Operating system: 	
       Storage requirements
                Core: 	  (e.g.  640K)
                Mass: 	  (e.g.  10MB)
       Peripheral hardware                 Required        Optional
                Magnetic  tape unit            	              	

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                                                                          10
         Disc unit                     	
         Line printer                  	
         Plotter                       _
         Math coprocessor              	
         Graphic board
         type: 	 (e.g.  EGA,  HerculesT
         Other:
                          Evaluation
Documentation
         available (Y/N)
      Description
         Theory
         User's instructions
                 Data Input
                 Application rules
         Program description
                 Variable list
                 Flowchart
                 Structure
         Example input/output

         Code  listing   (Y/N)
         Documentation  reviewed?  (Y/N)
                 By  whom?  	
status*
             1. good
             2. sufficient
             3. incomplete
             4. poor
             5. under
                development
             6. not present
          Internal  documentation  (comment  statements)
                  Sparse
                  Moderate
                  Comprehensive

 Model  Testing
          Verification/validation
                  Analytical  solutions
                  Hypothetical  problems
                  Laboratory experiments
                  Field experiments
                  Code intercomparlson
          Review:   (Y/N)
                  By whom:  	 (e.g. IGWMC)
                  Level of testing    (e.g. 1,2,3; IGWMC levels)
                       Code Availability
 Terms of availability
          Available  (Y/N)
          Public domain
                  Unrestricted distribution
                  Restricted distribution (e.g.
          Proprietary
        sole source)

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                                                                          11
                 Lease, licensed use
                 Royalty-based use
                 Use as part of consulting service
Form of availability

         	Source code
         	Compiled code
         Available as:   Tape
                         Diskette
                         Paper listing
                         Telephone transmission
                   General Code Information
          Language  (level/version)
          Number  of statements
          Number  of subroutines

 General  type  of code
          Research  code
          Expert  code
          General use code
          Educational code
 Applications
          Research
          Field
                  Number of known applications 	
          Classroom
 Third-party users
 Support
          Can be used without support
          Support available
                  Author
                  Third party
          Level of support
                   Full support
                   Limited or conditional support
                   Support agreement available
  Additional software  capabilities
          Pre-processing
                   Data storage
                   Data inspection
                   Data formatting
                   Interactive data entry
                   Interactive data editing
                   Digitizing
                   Graphic display of  Input  data
           Postprocessing
                   Data Inspection
                   Data storage
                   Graphics

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                                                                                  12
                                Screen
                                Plotter
                                Printer
                                Color
Remarks:
                          PART I:  Physical Syste*

Subsurface system
        Saturated zone
                Aqulfer(s)
                        hydraulic
                                Single aquifer
                                Single aqulfer-aqultard
                                Multiple aquifers/aqultards
                        Hydrodynamic
                                Single layer
                                Multilayered
                        Confined
                        Semiconfined
                        Water table
                        Storage-confining  layer
                        Porous media
                        Discrete fractures
                        Dual porosity
                        Isotropic
                        Anlsotropic
                        Homogeneous
                        Heterogeneous
                        Aquifer deformation
                        Layering
                        Delayed yield from aquifer  storage
                        Changing aquifer conditions In  space/time
                                Saturated/unsaturated
                                Conf1ned/unconf1ned
                        Other:
                Aqultard(s)
                        Homogeneous  in depth
                        Heterogeneous in depth
                        Homogeneous  in area!  extent
                        Heterogeneous in areal extent
                        Surfldal
                        Interbedded  with aquifers
                        Aquitard compaction

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                                                                                 13
                        Storage in aqultard
                        Other:
        Unsaturated zone
                Isotropic
                Anisotropic
                Homogeneous
                Heterogeneous in depth
                Heterogeneous in area! extent
                Discrete fractures
                Macropores
                Dual porosity (including crusting)
                Perched water table
                Tension-saturated zone
                Other:

Surface system
        Land surface
                Polders
                Springs
                Overland flow
                Ponding
                Wetlands
                Hillslopes
                Drainage basins
        Surface water bodies
                Rivers, canals
                Lakes, reservoirs, ponds
                Oceans, seas
        Interface:  subsurface/surface/atmosphere
                Infiltration
                Evaporation
                Evapotranspiration
        Other:
                            PART II:  Fluid Flow


                        General  Model  Characteristics


        Temporal
                Steady  state
                Successive  steady  states
                Transient
        Spatial
                Subsurface
                         Saturated
                         Parameters:  	lumped:  	single cell  	multicell
                                         _d1str1buted
                                 ID   	horizontal  	vertical
                                 2D   	horizontal  	vertical  	radial
                                 3D   	fully  	layered  	spherical
                                      	ax1symmetric
                         Unsaturated

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                                                                         14
                Parameters:  	lumped:  	single cell   	multicell
                                _distributed
                        ID   	horizontal  	vertical
                        2D   	horizontal  	vertical   	radial
                        3D   	fully   	layered  	spherical
                            	axisymmetric
        Surface
                 Parameters:  	lumped:  	single cell   	multicell
                               _distributed
                        ID    single channel
                        2D    multichannel
                        3D    	reservoir
Grid design
        None
        Orientation
                Plan or horizontal
                Cross-section or  vertical
        Preparation
                Automatic
                Manual
                        Required
                        Possible
        Spacing
                Regular
                Variable
                Local refinement
        Movable
        Size
                Predetermined
                Variable
        Number of  nodes
                Fixed
                Variable
                Maximum?   no.	
        Cell  shape
                 Linear
                Triangular
                 Curved  triangular
                Square
                 Rectangular
                Quadrilateral
                 Curved  quadrilateral
                 Cylindrical
                 Spherical
                 Curvilinear
                 Polygon
                 Cubic
                 Hexahedral
                 Triangular prism
                 Tetrahedral

 Fluid conditions
         Physical
                 Homogeneous
                 Heterogeneous

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                                                                        15


               Density
                        Constant
                        Variable
                        Density-temperature relation
                        Density-concentration relation
               Viscosity
                        Constant
                        Variable
                        Viscosity-temperature relation
                        Viscosity-concentration relation
               Compressible
                Incompressible
               Multiple fluids
                        Misclble
                        Immiscible
                                011-water
                                Gas(vapor)-water
                                Saltwater-freshwater
                Multiphase
                        Water
                        Ice
                        Vapor
                        Steam
                Other:
Units
        _Metric _Engl1sh

Restart capability
    Updates possible:
        Parameters
        Perturbations
        Boundary conditions
        Simulation  parameters
                Matrix  solution
                Time sequence
                 Iteration criteria
                Stability criteria

Error  criteria
         Fluid balance over model
         Sum head/pressure change  over model  between iterations
         Maximum  head/pressure change at any  node
         Maximum  fluid flux change at any boundary node
         Maximum  head/pressure change over a  time increment
         Maximum  fluid flux change over a time increment
                        Model Dynamics
 Flow characteristics
         Laminar
         Turbulent
         Darcy flow
         Non-Darcy flow

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                                                                         16
Hydro!ogic processes
        Diffusion
        Infiltration
        Soil evaporation
        Evapotranspiration
        Interflow
        Condensation
        Condensation
        Precipitation
        Capillary uptake

Physical phenomena
        Crusting
        Freezing/thawing
        Change-of-phase
        Hysteresis
        Buoyancy
        Deformation
        Compaction
        Osmosis
        Skin effect
        Consolidation
        Expansion
                    Boundary Conditions
                                Spatial                 Temporal
                              variability             variability
   Stage
        Heads, pressures          	                      	
        Springs                   _                      _
        Surface water stage       	                      	
        Free surface              	                      	
   Flux
        Fluid flux                _                      _
        Head/pressure-
        Dependent flux            	                      	
        No flow                   _                      __
        Seepage face              	                      	
        Surface infiltration      	                      	
        Groundwater recharge      	                      	
        Well pumpage/injection    	                      	

   Other
        Moisture content          	                      	
        Tidal fluctuations
                      Solution Methods


Equations solved (e.g. convection-dispersion);

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                                                                              17
            1.
            2.
            3.
            4.
            5.

Type of solution
    Analytical
            Analytic elements
                    Method of  Images
                    Line sinks
                    Dipoles
                    Doublets
                    Vortices
                    Area! sources, sinks
            Inverse methods
                    Theis type-curve method
                    Cooper-Jacob semi logarithmic method
                Unconfined aquifers
                    Boulton type-curve method
                    Neuman type-curve method
                    Neuman semi logarithmic method
                    Neuman recovery method
    Numerical
            Space approximation
                    Finite difference
                            Block-centered
                            Node-centered
                    Integral finite difference
                    Finite element
                            Variational
                            Galerkin—method of weighted residuals
                            Collocation
                            Numerical integration  (e.g. Gauss
                                quadrature)
                    Boundary element
                    Lumped-cell approach
                    Upstream weighting

            Time approximation
                    Finite difference
                            Strongly implicit
                            Fully  implicit
                            Fully  explicit
                            Crank-Nicolson
                    Finite element
                    Automatic  time  increment selection
                    Upstream weighting
    Matrix-solving technique
            Iterative
                    Gauss-Seidel  (point-successive over-relaxation,
                        PSOR)
                    Line-successive over-relaxation (LSOR)
                    Block-successive over-relaxation (BSOR)
                    Alternating direction  (ADI)

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                                                                         18
               Iterative alternating direction (IADI)
               Predictor-corrector
       Direct
               Gauss elimination
               Cholesky square root
               Doolittle
               Thomas Algorithm
               Point Jacobi
       Minimum search technique
               Newton-Raphson
               Gauss-Newton
               Steepest descent
        Iteration  criteria
               Fluid balance over model
               Total head/pressure change over model  between
                   iterations
               Maximum head change at  any node between  iterations
               Maximum flux change at  any boundary  node
               Maximum head/pressure change over  time Increment
               Maximum flux change over  time  increment
        Other:

Spatial interpolation
        Lagrange  method
        Spline  functions
        Kriging
        Linear, bilinear,  trilinear
                Input/Output Characteristics


                            INPUT
Geometry
    Elevation
        Ground surface elevation
        Aquifer/aquitard top
        Aquifer/aquitard bottom
        Surface water bed
    Thickness
        Aquifer
        Aquitard
        Unsaturated zone
        Root zone
Parameters
        Hydraulic conductivity
        Transmissivity
        Intrinsic permeability
        Porosity
        Storativity
        Specific storage
        Specific yield
        Hydraulic diffuslvlty
        Aquitard leakance
        Resistance—confining layers

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                                                                                 19
                Resistance—surface water beds

                Hydraulic conductivity/moisture content  relation
                Pressure head/moisture content relation
                Hydraulic conductivity/potential relation
            Fluid
                Density
                Specific weight
                Viscosity
                Compressibility
                Temperature

Initial conditions
                Saturated thickness
                Head/pressure head/potential distribution
                Transmissivity
                Temperature
                Velocities
                Position of interface
                Soil moisture distribution

General characteristics
        Well Characteristics
                Maximum number of wells 	
                Fully penetrating
                Partially penetrating
                Well bore
                        Diameter
                        Depth
                        Storage
                Well screen
                        Diameter
                        Elevation/depth of top
                        Elevation/depth of bottom
                        Length
                Well characteristics  (??)
        Meteorological data
                Air:—temperature, wind speed,  humidity
                Precipitation
                Evaporation
                Evapotranspiration
        Land-use data
                Soil cover
                Impermeable surface area
Boundary conditions
            Stage
                Heads, pressures
                Surface water  stage
            Flux
                Head, pressure-dependent flux
                Specified  flux
                Well pumpage/injection
                Surface  Infiltration
                Artificial  recharge
                Groundwater recharge

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                                                                                  20
                Precipitation
            Other
                Moisture content
                Seepage face
                Free surface
Simulation specifications
        Grid
                Grid intervals
                Number of nodes or cells
                Node locations
                Number of layers
        Time
                Time step sequence
                Initial time step
                Number of time steps
        Matrix solution parameters
                Relaxation factor
                Stability criteria
                Error criteria
                                   OUTPUT
 Echo of input
 output
         Grid
         Initial  heads/pressures/potentials
         Initial  fluxes
    Parameters
         Hydraulic diffusivity
         Hydraulic conductivity
         Transmissivlty
         Storativity
         Specific storage
         Specific yield
         Moisture content
         Resistance-confining layer
         Resistance-beds
         Fluid density
         Fluid viscosity
     Simulation results
         Head/potential values
         Fluxes
                 Internal
                 Boundary
         Velocities
         Pathlines
         Traveltimes
         Isochrones
         Position of interface
         Stream function
         Water balance
         Precipitation
         Evapotranspiration
Tabulated
Graphic

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                                                                                 21
        Groundwater recharge
        Groundwater storage
                Total
                Change
        Surface water balance
        Other
                                 REFERENCES

Bachmat, Y.. B. Andrews,  D.  Holtz,  and  S.  Sebastian.  1980.  Utilization of
Numerical Groundwater  Models for Water  Resource Management.  Report/600/8-
78-012.     Environmental   Protection   Agency,  Office  of  Research  and
Development, Environmental Research Information Center,  Cincinnati, Ohio.

Gorelick, Steven  M.    1983.   A Review of  Distributed Parameter Groundwater
Management Modeling Methods.  Water Resources Research 19:2, pp.  305-319.

Srinivasan, P.,  and  P.K.M.  van  der Heijde.   1985.   IGWMC Model Annotation
Databases—User's  Manual.    IGWMC  Report  no.  GWMI 85-26.   International
Ground Water Modeling  Center, Holcomb Research Institute,  Butler University
Indianapolis,  Indiana.

van  der  Heijde, P.K.M..   1982.   Facilitation of General  Understanding and
Applications   of   Groundwater  Models.     IGWMC  Report   No.  GWMI   82-05.
International  Ground  Water  Modeling  Center,  Holcomb  Research Institute,
Butler University, Indianapolis,  Indiana.

van  der Heijde.  P.K.M.,  and Richard  Park.   1986.   U.S. EPA  Groundwater
Modeling Policy  Study  Group:  Report  of Findings and Discussion  of Selected
Groundwater  Modeling  issues.   International  Ground Water  Modeling  Center,
Holcomb  Research  Institute.  Butler University,  Indianapolis, Indiana.

van  der  Heijde,  P.K.M.    1986.   ITAC   and  Policy Board meeting  notes.
Internal  memorandum.   International  Ground Water  Modeling  Center,  Holcomb
Research Institute,  Butler University,  Indianapolis,  Indiana.

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        IGWMC   GROUNDWATER   MODELING   REPRINT
                   TECHNOLOGY TRANSFER IN GROUNDWATER MODELING:
            THE ROLE OF THE INTERNATIONAL GROUND WATER MODELING CENTER
                                        by

                              Paul K.M. van der Heijde
                                    presented  at

                             The NWWA/IGWMC Conference
                     'Solving Ground-water Probleas with Models'
                                February  10-12,  1987
                                  Denver,  Colorado
                                     GWMI  87-09
I N T E R M A T I 0 N A L   GROUND   HATER   MODELING   CENTER
                             Hoicomb Research Institute
                                 Butler University
                            Indianapolis,  Indiana  46208

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                TECHNOLOGY TRANSFER  IN GROUNDWATER MODELING:

         THE ROLE  OF THE INTERNATIONAL GROUND WATER MODELING CENTER


                           Paul  K.M.  van  der  Heijde

                  International  Ground Water  Modeling Center
                Holcomb Research Institute, Butler University
                         Indianapolis, Indiana  46208


Abstract

     The protection  of  ground  water  resources  has emerged in recent years as
a top  priority for  natural  resource management  around the world.   Antici-
pating  the  Increased  Importance of  protecting  ground water  resources,  the
International  Ground Water Modeling  Center  (IGWMC) was  established  in  1978
at  the  Holcomb Research Institute,  Indianapolis,  Indiana, USA, to  advance
the use  of  modeling methodologies  by regulatory  and  management  agencies in
the development  of  effective  ground water  management procedures.    To  meet
its goals,  the  IGWMC  has  developed an extensive technology  transfer,  re-
search,  and assistance  program which includes dissemination  of  Information
about the modeling process and the role of modeling in ground water resource
msnsgement; model  availability; development of selection and  testing proce-
dures;  promotion   of quality  assurance  in  the  application of  ground  water
modeling  software; acquisition  and  distribution  of models, supporting  soft-
ware,  and documentation;  education  of  model users, managers,  and  teachers;
snd publication of  the Ground  water  Modeling  Newsletter.   A second office
opened  in  Delft,  The  Netherlands,   in  1984,  further expands  the Center's
international  activities.


Technology  Transfer and Training In  Ground Water  Modeling

     Technology transfer means dissemination of  information on technological
advances  through  communication  and  education.   When applied to ground water
modeling,  technology  transfer  includes dissemination of  information  about
the role  of modeling in water  resource  management, model theory, model veri-
fication  and  validation,  modeling  methodology,  availability  and  applic-
ability  of models  and  related software, model  selection, modeling project
management, and quality assurance.

      In  a  report  on  the  use of  models  for   water  resources management,
planning,  and policy, the  Office  of  Technology Assessment  of  the   U.S.
Congress  (OTA 1982)   considers  specific  education  and training  of  model

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developers,  users,  and  managers  In   various  aspects  of  water  resources
nooe'i'ng;  such functions  are  critical  components  of  technology  transfer.
Otr
-------
    •  accessibility of models for potential  users

    •  communication between managers and technical  personnel

       inadequacies in data

       inadequacies in modeling

     The need  was stressed  for  a centralized modeling information dissemi-
nation and  software  distribution facility—a centralized  "clearinghouse"—to
include all forms  of  Information on models currently  available,  the problems
for which  models have been  tested  and successfully used, as well as  public
domain software  and  its  documentation.   Recommendations  made in the  report
included a detailed  outline  of  the Institutional  mechanisms and procedures
needed to  acquire and distribute models;  to provide  an effective and  widely
recognized  training  program  for  field specialists  and ground water project
managers;  to develop  a respected research and development program 1n support
of technology  transfer;  and  to  conmunicate  efficiently with  interested pro-
fessionals.

     Based  on the experience  obtained   in  its  previous  modeling  projects,
guided  by  the recommendations  cited above,  and supported by the U.S.  Envi-
ronmental  Protection Agency, HRI established  the  International  Ground Water
Modeling Center  1n 1978.   The  Center's main mission  1s  to  promote the  cor-
rect  and efficient use of  computer-based data analysis  and  prediction tech-
niques in  support  of  effective  ground  water  management.

     The Center  accomplishes its mission by:

     (1)  developing  and  promoting  a comprehensive  approach  to  the role  and
         quality-assured  use of modeling based  decision-support technology
         in ground water  management;

     (2)  assembling,   organizing,   analyzing,  and  disseminating information
         related to  the  development and qualified  use of models and related
         decision-support technology,  in response  to changing  demands from
         groundwater  management,   and benefiting  from  computer  technology
         development;

     (3)   improving model  accessibility  through model  acquisition  from uni-
         versities and from  federal and  other research agencies, with subse-
         quent  review,   testing,  documentation,  and  software  distribution;
         and

     (4)   providing   federal,  state,   and  local   ground   water  management
          agencies and the  private sector  with the tools  and training  to
          analyze  existing  problems  and to screen  management  options  in
          addressing these problems.


 Structure of  IGWMC

      To meet  its  objectives, the Center operates  through four  divisions and
 two offices:  one  1n Indianapolis,   Indiana,  U.S.A.,  supported by the  Holcomb

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Research  Institute  of  Butler University,  and one in  Delft,  The  Netherlands,
supported  by  the   Institute  of  Applied  Geoscience  of  the Dutch  research
organization TNO.   Two of the  four IGWMC divisions are  directly  involved  in
technology  transfer:    Clearinghouse,  and  Training  and  Education.   The
Research  and  Development  Division and  the Communication  Division  provide
basic support  to the Center's mission (Figure 1).

     The  Clearinghouse.   The IGWMC emphasizes  reducing  the  time-lag  between
innovations  in ground  water data processing  and  modeling  at  universities,
research  laboratories,  and  major research  agencies  (USGS,  EPA, NRC,  USDA^
NAS, U.S. Army. DOE) and  the availability of these  research results to state
and private  users.   To  this purpose  the Center's clearinghouse  makes model
and modeling-related  Information and  software  available  to  governmental and
private  users  by  using  extensive  referral-type databases,  and  by  distri-
buting  of a wide  collection of  management-oriented  ground water  software.
Extensive  contacts  with  researchers  and  model users, together  with the
experience of  the  Center's professional  staff, form the  basis of these fre-
quently accessed user-oriented  modeling  services.   The  clearinghouse  is the
framework on which  the Center's knowledge base  on ground water modeling has
been developed and  is maintained.

     Training  and  Education.   To  enhance the  use  of  ground water  models  by
qualified  personnel, the  Center  offers  a  comprehensive  annual  program  of
short  courses,  workshops,  and  seminars,  in  which  principles,  concepts,
theories,  and  applications  of  ground  water   models  are  featured.   New
approaches in  education and  training, based on blending  educational develop-
ments with  recent   advances  in  computer  technology,  are  being explored.   In
addition,  the  Center  provides  assistance to  governmental  agencies,  educa-
tional   Institutions,   and   private  groups  in  organizing   and   conducting
specially designed  training programs.

     Research  and   Development.   When  the  Center was  established   it was
realized  that  the  technology  transfer  in ground  water  modeling  could  be
successful only  1f supported by  a strong research and  development program.
Therefore, together with  the  technology transfer  activities, the Center's
staff  has developed  an  extensive research and development  program.  The
results  of  these   activities  are new  water  resource  management  decision-
support  information,  modeling and training methodologies, and  related soft-
ware.    The  following  activities  have  taken  place  through  the  Center's
research  and development  studies: the establishment of  guidelines  and meth-
odologies for  modeling-related  activities such  as quality assurance in model
development  and  application,  computer  program  selection,   code  implementa-
tion, model  evaluation  and testing,  model documentation, and  pre-  and post-
processing;  state-of-the-art reviews based  on  the content  of the Center's
information  bases;  and  extensive software development  for  its  educational
and distribution activities.

     Communication.  Communication is one basic  element  of  technology trans-
fer.  To  ensure  adequate communication with all the  groups  active  in ground
water management and research,  the Center has established  a seperate commu-
nication  division.   One  of  the Center's  major  channels  of  communication  is
the quarterly  Ground Mater Modeling Newsletter.

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                                  international ground water modeling center
                                      Policy Board
                                           I
                                       International Technical
                                         Advisory Cownittee
                                 International Coordinator
          Indianapolis Office
                serving
       North, Central, and  South
                America

         MOicomb  Research Institute
            Butler University
        Indianapolis, Indiana, U.S.A.
                                    Delft Office
                                        serving
                               Europe, Asia, Africa,  and
                                      Australia

                                   TNO-OGV institute o<
                                    Appiied Geoscience
                                  Oeitt.  The Netherlands
Clearinghouse
-knowledge base:
 collection,
 analysis, storage,
 and dissemination
 of model information

-software evalua-
 tion, acquisition,
 and distribution

-technical assis-
 tance and software
 support
Training and
 Education

-short courses
 and workshops

-Individual
 assistance

-computer-aided
 instruction
Research and
 Development

-mode 1i ng
 methodology

-quality
 assurance

-software
 performance
                          -educational
                          approaches in
                          modeling

                          -model needs and
                          status
Communications
-information
 services-

-publications
 distribution

-newsletter
 publication
                       -documentation
                        center
                                                                             -networking
                                                                              of modelers
                        Figure  1.  Organizational Structure of IGWMC

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     Through its extensive  contacts as an  intermediary  between model devel-
opers and models  users,  the Center is  in a unique position to contribute to
quantitative approaches  to ground water management.   Changes in management
focus by  the  states and the  private sector can be monitored closely, while
the Center's flexible  structure  allows  rapid response to needs for  analytic
and forecasting tools.


International  Cooperation through IGWMC

     In  August  1983  HRI  and  the  TNO  Institute  of  Applied  Geoscience
(DGV/TNO)  reached  an  agreement   regarding  the establishment  of  a  European
IGWMC office  in Delft,  The Netherlands.   The  agreement forms  the  base  for
expanded  access  to and  benefit  from other countries' experiences  in ground
water modeling.    Activities  under this  agreement  include efficient inter-
office  communication and  reporting  procedures, open exchange of  technical
information,  mutual technical   and  organizational  assistance,   Integrating
information collected  by  both offices  Into a  single IGWMC  knowledge  base,
and carrying out joint research and development projects.

     The  Center  is governed by a Policy  Board  representing the participating
institutes  (HRI  and DGV/TNO).    The  Policy Board  supervises  the  Center's
activities  and is  responsible for  setting policies  and  long-term  planning.
The  agreement  also provides for  an annual  meeting  of the International  Tech-
nical   Advisory   Committee  (ITAC)  with   the   Center's   Policy   Board,  and
addresses the role  of  the director of  the Center's  Indianapolis  office as
International  Coordinator  of  IGWMC.
 Future  Developments

      In the  past the efficient application of  ground  water models has often
 been  hampered  by limited access to the models and to user information during
 trie selection  process;  by poorly  written and documented software; and by in-
 sufficient  knowledge or awareness  of the modeling  process  and  a  lack  of
 training  on  the p&rt of  the users.   Through  the  establishment of the Inter-
 national  Ground  Water  Modeling  Center, many  of these  problems  have  been
 reduced.   IGWMC's first  six  years have emphasized upgrading  the quality of
 ground  water  modeling  through improved  access to  information and  by  pro-
 viding  extensive  training opportunities,  both  supported  by research  and
 development.   For the  near  future,  the Center will increase  its efforts in
 assuring  high  quality ground  water modeling through emphasis on internal and
 external  quality  assurance  approaches  and  programs  in  all  stages  of  the
 modeling   process,  from  model   development  (coding  and  documentation),
 analysis,  evaluation, testing, and selection, to  application and  integration
 in the  resource-management decision  process.

      In considering  its  role  in  ground water modeling  in the next three to
 five  years,  the Center  foresees its  functions as:


    •    continuing  to  advance the  quality of  ground  water modeling  studies
          through the development  of improved methodologies, procedures, and
          standards

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         developing  and   introducing  efficient,  integrated,  computer-aided
         decision-support methodologies  in  ground  water management

         promoting the use of quality-assured  computer-based technology

         expanding the IGWMC knowledge  base,  both vertically  (in-depth)  and
         horirontally (to  include  multimedia  transport  of pollutants, expo-
         sure and risk analysis,  integrated management  of surface water and
         ground water, and  nonsimulation  computer  techniques  applicable to
         ground  water   management,   such   as  graphics,  data  processing,
         kriging, stochastic analysis, optimization, and the use of informa-
         tion theory)

         continuing  to expand clearinghouse  services with updated informa-
         tion on modeling  methodology and  related software, and distribution
         of selected quality-assured and well-documented computer  programs

    •     continuing  to  provide high-quality  practical  training    opportuni-
         ties  for  ground  water  professionals  and  managers,   incorporating
         major  new  research  and  technology  developments,  and    regularly
         adjusting  to  user  needs  in   terms  of  topics  covered, audience
         addressed, facilities, and educational methodology

         continuing  to  support  the  clearinghouse   and  technology transfer
         functions with  research and development activities

         expanding  and  improving  ways and means  to  communicate with  various
         potential audiences in different  parts of the world


Acknowledgment

     The  research described in  this paper  has been funded  in  part  by the
U.S.  Environmental  Protection  Agency  through  Cooperative  Agreement  #CR-
812603  with The Holcomb Research Institute.   It has not  been  subjected  to
the Agency  peer and policy  review and therefore does not necessarily reflect
the views of the Agency, and no official endorsement  should be inferred.


References

Bachmat  Y., B. Andrews, D.  Holtz, and S. Sebastian.   1978.   Utilization of
    Numerical  Groundwater Models  for Water  Resource Management.    EPA-600/8-
    78-012,  U.S.  Environmental Protection  Agency, Ada,  Oklahoma.

HRI.   1976.   Environmental Modeling and  Decision  Making: The United States
    Experience.  New York: Praeger Publishers.

OTA.    1982.    Use  of Models  for  Water Resources  Management,  Planning, and
    "policy."    OTA-0-159,  Office  of  Technology  Assessment,  U.S Congress,
    Washington, D.C.

van der Heijde,  P.K.M..  and  R.A. Park.    1986.   U.S.  EPA   Ground-water

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    Modeling  Policy  Study  Group;  Report  of  Findings  and  Discussion  of
    Selected  Ground-water  Modeling  Issues.     International   Ground   Water
    Modeling   Center,   Holcomb   Research  Institute,   Butler   University,
    Indianapolis, Indiana.


Biographical Sketch

     Director of the  Water Science  Program, Holcomb Research  Institute,  and
Director,  International  Ground  Water  Modeling  Center,  van der Heijde  is
trained as  a geohydrologist.  A  native  of  The Netherlands, he  received  his
M.S. degree  in Civil  Engineering in 1977  from Delft Technical  University,
where  he  specialized  in  hydraulic engineering  and hydrology.   His  career
since  1977   has  focused  on  ground  water  and quantitative  analysis of  its
interaction with soil and bedrock systems.

     His  current  research concentrates  on improving  the  quality of  ground
water  modeling  and developing technology-transfer methods  and  facilities in
ground  water modeling.   Recently,  he chaired  the EPA  Groundwater  Modeling
Policy  Study Group, and  is  a member of  the  editorial board of  the journal
Ground

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IGWMC   PUBLICATIONS   IN   GROUND   WATER   MODELING
                          AVAILABLE  SOLUTE  TRANSPORT MODELS

                   FOR GROUNDWATER AND  SOIL  WATER  QUALITY MANAGEMENT
                                          by

                                Paul  K.M. van der Heijde

                                          and

                                   Milovan S. Beljin
                                       GWMI 86-08

                                      August  1986
  INTERNATIONAL   GROUND   WATER   MODELING   CENTER

                               Holcomb  Research  Institute
                                    Butler University
                           Indianapolis, Indiana 46208  U.S.A.
                                   Phone:   317/283-9458

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INTRODUCTION

    Models  are  useful  instrument  1n  understanding the mechanisms of  ground-
water systems and  the processes  which influence  their  quality.   Through their
predictive capabilities, models provide a means to analyze the consequences of
human  Intervention in  groundwater  systems.    In  managing water resources  to
meet  long-term  human  and  environmental  needs,  models  provide  necessary
analytic support.

    Three types of models are frequently used in groundwater quality  studies:

         Flow models  for  the  analysis of  flow patterns and for the determina-
         tion of streamlines, particle pathways, velocities, and traveltimes.

         Solute  transport models for  the prediction  of  movement,  concentra-
         tions,  and  mass  balances   of  soluable  constituents, and  for  the
         calculation  of radiological doses.

         Hydrochemical  models,  either equilibrium or  kinetic,  for the calcu-
         lation of chemical constituent concentrations.

    The  flow and  solute  transport  models  may  be  embedded  in a  management
model, describing  the system in terms of objective function(s) and  constraints
and solving  the  resulting equations through  an optimization technique  such as
linear programming (Gorelick 1983).

    Two  of  these model types can be  used to evaluate  the chemical  quality of
groundwater:    hydrochemical  models  and  solute   transport  models.     In  the
hydrochemical models, the chemistry is posed  independent of any mass transport
process.   These models,  which are general  1n nature  and are  used for both
ground  and  surface   water,  simulate   chemical  processes  which regulate  the
concentration  of dissolved constituents.   They  can be used to identify  the
effects   of  temperature,  speciation,   sorption,  and   solubility   on  the
concentrations of  dissolved constituents.   They  are described  in a  separate
publication  (Rice  1985).

Solute Transport Models

    Solute  or mass   transport  models  consider   quality  in  conjunction with
quantity.   In principle,  a mass  transport model  1s based on solving equations
for flow and solute  transport under  given  boundary  and  initial  conditions.
Under certain conditions  such as low concentrations of contaminants and negli-
gible  difference  1n   specific  weight  between contaminant  and the  resident
water, changes  in  concentrations do not  affect the  flow pattern (homogeneous
fluid  phase).    In such  cases a  mass transport  model  can  be  considered as
containing  two  submodels, a flow  submodel and a  quality  submodel.  The flow
model computes the piezometric heads.   The quality submodel then uses the head
data  to  generate  velocities  for  advective  displacement of  the contaminant,
allowing for additional  spreading through disperison  and for transformations
by chemical  and  microbial reactions.   The final  result is the computation of
concentrations and solute mass balances.   In cases of high contaminant concen-
trations in  waste  water or saline  water,  changes 1n concentrations affect the
flow patterns through changes  in density  and viscosity, which in turn affects
the movement and  spreading  of the contaminant  and hence  the  concentrations

-------
(heterogeneous fluid phase).   To solve  such  problems  through  modeling,  simul-
taneous solution of  flow and  solute transport  equations  or  iterative  solution
between the  flow  and  quality  submodels  is required  (van  der Heijde  et  al.
1985).   Mass  transport  models which  handle  only  convective transport  are
called immiscible transport models, whereas miscible transport models  handle
mixing  resulting  from  dispersive and  diffusive  processes.   Models  which
consider both  displacements  and  transformations  of  contaminants are  called
nonconservative.    Conservative models  retain  the mass  of   constituents  in
liquid form and only simulate convective and dispersive displacements.

    The transformations  in  nonconservative  models are  primarily adsorption,
radioactive decay, and biochemical  transformations.  Thus far, the simplified,
linear representation  of the  the adsorption  process has been  included princi-
pally in the nonconservative transport models.

    In general, current  solute  transport models  assume that the reaction rates
are limited and thus depend on  the residence time  for the contaminant, or that
the reactions proceed  instantaneously to equilibrium.

    Various  numerical   solution  techniques   are   used  in   solute  transport
models.  They  include the finite  difference method (FD),  the integral finite
difference  method  (IFDM),  the finite element  method  (FE),  the collocation
method,  particle  mass  tracking methods   (e.g.,  random walk IRWJ),   and  the
method of characteristics (MOC).

CURRENTLY AVAILABLE  SOLUTE TRANSPORT MODELS

    Although  the various processes playing a  role  in contaminant distribution
within  groundwater systems are  not completely understood, computer codes  have
been  developed  for  situations  which do  not  require  analysis of  complex
transport mechanisms or chemistry.  These  codes  range  from poorly documented
research  codes to  extensively documented and applied  program packages.  The
uses  of these  programs  are  generally restricted to  conceptual  analysis of
pollution  problems,  to  feasibility  studies  in  design and  remedial  action
strategies  and  to data acquisition guidance.

 IGWMC Model  Information Data Bases

     In the  following   pages,   numerical   and  semi-analytical  mass   transport
models which are documented to some degree, which have  undergone some form of
testing or  review,  and  for  which the code  is available, are   listed.   This
 listing is  obtained by  performing a computerized  search  in the  MARS  and PLUTO
model  information  data bases of  the  International  Ground Water  Modeling
 Center.

     The table is prepared in such a way that it could independently  be used  as
 a first step  in  learning about available mass  transport models.  Many of the
 listed models are  in  the public domain and  available at nominal  or  no cost  to
 the user.   Others  (marked by "*" in column  4) require special  agreements  on
 usage.  The columns of the table  are explained below.

 Column 1:      Serial number

-------
Column 2:


Column 3:
Column 4:
Column 5:
Column 6:

Column 7:
List  of  authors  at the  time of  model  development.
organization 1s given 1n some cases.
Name of
Address  at which  further  information  on  the  availability  of
model is known.  When no  one's  name  appears at  the top,  any one
of the  authors  can be contacted.  An asterisk  M*" mark  follows
the name to indicate that the contact address  is  different  from
the organization where the model was  developed.

Name with which the model is referred.   Year of latest update of
the  model  is given  1n  parentheses.    A "$" mark next  to  name
Indicates a special agreement 1s required for model  usage:   the
models marked "+" are also available  from the IGWMC

Here,  the  purposes  of  the model  are  such:    type of  model,
aquifer conditions, flow  conditions,  system-geometry, numerical
method, etc.

Processes considered in the model for mass transport.

It is  the  last  four digits of  a number, known  as IGWMC-key, by
which  annotation  of  each model  is stored and  retrieved  in the
IGWMC model Information data base.
Further  Information

    Complete  annotations describing  all  the model  characteristics  including
program  code  specifications and the  nature  of  availability (cost, agreement,
written  permission, etc.) are available at IGWMC at nominal costs.

    Documentations  of   the  models  listed   are  available  at  the  contact
address.  The IGWMC appreciates feedback from the users about their experience
in trying to  acquire  the documentation of the models listed, so that the most
recent information will  be  available to future users.

-------
REFERENCES

Van der Heijtie,  P.K.M.,  et  £l.  1985.   Groundwater Management:   The  Use  of
Numerical Models.   Water Resou**c.  Monogr.  5,  2nd edition.  Washington, D.C.:
Am.  Geoph, Union.

Rice,  R.  1985,    Listing of  Hydrocneaiicf."! Kode'is  which are  Documented  and
Available.  Internatioanl Ground  Water  Modeling  Center  Publication GWMI 85-15s
Holcomb Research  Institute,  Indianapolis,  Indiana.

Gorelick, S.  1983.   A Review of  Distrbuted Parameter Groundwater Management
Modeling Methods.   Water Resources Research 19(2):305-319.

-------
Ko.


i .
1
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I
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1
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Author (s)


S.M. Ahistrojn
H.P. Foote
R.j. Serne






R.G. Baca






H.C. Burkholder
M.O. Cloninger
M.V. Dernier
G. Jansen
P.J. Liddell
J.f . Weshburn

i. .^. Davis



1




L.A. Davis






L . A . Dav i s
G. Segol




O.l . Deangel is
G.T. Yen
D.D. Huff



Delft
Hydraul iCS
Laboratory










Contact Address


J.F. Mahburn
Batteiie Pacific NW Labs
P.O. Box 999
Rich land, MA 99352





Rockwel I Han ford
Operations
P.O. Box 250
Rich land, MA 99352



Natl. Energy Software
Center*
Argonne Natl. Laboratory
9700 S. Cass Avenue
Argonne, 11 60439
Tel: 312/972-7250

Mater, Maste, and
Land, Inc.
1311 S. Col lege Avenue
Fort Collins, CO 80524





Mater, Maste. and
Land, Inc.
1311 S. Col lege Avenue
Fort Col 1 ins, Co 80524



Mater, Maste, and
Land, Inc.
1311 S. Col lege Avenue
Fort Col I ins, CO 80524


G.T. Yeh
Oak Ridge National
Laboratory
Envionmental Sci. Div.
Oak Ridge, TN 37830

j.M. Hesse I ing
Delft Hydraulics Lab.
P.O. Bon 152
8300 AD Emmeloord
The Nether lands
Tel: (0)5274-2922







Model Na»e
(last update)


MHT-DPRM
(1976)







FECTRA
(1979)





GETOUT
(1979)





SEEPV
(1980)







GS2
(1985)





GS3
(1985)




FRACPORT
(1984)




GROMKMA
(1982)











Model
Description

*
A random-walk model to
predict transient.
three-dimensional move-
ment of racJio-nuc I ides
and other contaminants
in saturated/unsaturated
aquifer systems


A two-dimensioani , ver-
tical finite element
model to simulate steady
or unsteady transport
for a given velocity
field in saturated or
unsaturated porous media
To predict migration of
radionuci ides to bio-
sphere using a steady-
state, homogeneous, i so-
tropic, saturated model
of the gaosphere

A finite difference mod-
el to simulate transient
vertical seepage frona
tailings impoundment.
including saturated/un-
saturated modeling of
impoundment with liner.
and underlying aquifer

A two-dimensional hori-
zontal or vertical fi-
nite element model to
simulate flow and solute
transport in saturated/-
unsaturated porous media

A three-dimensional fi-
nite element mode! to
simulate flow and solute
transport in saturated/-
unsaturated porous media

An integrated
compartment a I model for
describing the transport
of solute in three-
dimensional fractured
porous medium
Transient finite element
simulations of two-
dimensional, horizontal
ground water movement of
nonconservat ive solute
transport in a multi-
layered, anisotropic.
hetero-geneous aqu i f er
system




Model
Processes
[

advect ion
dispers ion
•li f fusion
adsorpt ion
decay
chemica 1
reactions i
i
ion exchange I
advect ion
dispersion
di f fusion
adsorption
(chain) decay


advect ion
dispersion
d i f f us i on
adsorpt ion
ion exchange
(chain) decay

tdvect i on








advect ion
dispersion
dec By
adsorption



edvect ion
dispers ion


|

odvect ion
I dispersion
adsorption
decay


advect ion
dispersion
di f fusion
adsorpt ion
ion exchange
decay
chemica 1
react ions





IGWKCi
Key i
i

0760 '







'•
0790






2080


i



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2891 :





,
2891





3374
:
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298?
I





i
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-------
No.
Author(s)
            Contact Address
                          Model Name
                          (last  update)
              Model
              Description
                          Model
                          Processes
              Cey
9.
 10.
 l I .
 12.
 13.
 14.
 15.
R. T. D i I I on
R.M. Cranweli
R.B. Lantz
S.B. Pahwa
M.  Reeves
G.R.  Dutt
M.J.  Shaffer
W.J.  Moore
       O.R.  Friedrichs
       C.R.  Cole
       R.£.  Arnett
 S.P.  Garabedian
 L. F .  Kon i kow
 M.Th.  van
    Genuchten
 S.K.
 C.T.
 P.R.
 C.A.
 C.R.
Gupta
Kinkaid
Meyer
NewDiI I
Cole
       V. Guvanasen
             R.M.  CranwelI
             Sandia National  Labs.
             Albuquerque.  NM  87185
             Tel:  509/376-8451
                  (support only)
             Code  distributed by:
             Natl. Energy  Software
                Center*
             Argonne Nat I. Laboratory
             9700  S. Cass  Avenue
             Argonne, IL  60439
             Tel:  312/972-7250

             Bureau of Reclamation
             U.S.  Dept. of Interior
             715  S. Tyler. Suite 201
             AraariIlo, TX  79101
                   O.R. Friedrichs
                   Battelie Pacific NW Labs
                   P.O. Box 999
                   Rich I and, WA  99352
                   Tel: 509/376-8628/8451
             L.F. Kon i kow
             Water Resources Division
             U.S. Geological Survey
             431 National Center
             Reston, VA  22092
             M. Th. van Genuchten
             U.S. Salinity Laboratory
             U.S. Department of
                Agr icu!ture
             4500 Glenwood Drive
             Riverside. CA  92501
C.R. Cole
Battelie Pacific NW Labs
P.O. Box 999
Rich I and, WA  99352
Tel: 509/376-8451
                   T. Chan
                   Applied Geosci.  Branch
                   Whiteshell  Nuclear
                      Research
                   Atmic  Enercy  of  Canada
                   Pinawa Manitoba   ROE MO
                                                    SWIFT
                                                    Salt  Trans-
                                                    port  i n
                                                    Irrigated
                                                    Soi Is
                                                    (1976)
                                        PCP
                                        (1977)
                           FRONT-
                           TRACKING
                           MODEL
                           (1983)
                           SUMATRA-
                           (1978)
CFEST
(1985)
                                        MOTIF
                                         (1986)
                                         A three-dimensional  fi-
                                         nite-difference  model
                                         for simulation of  coup-
                                         led, transient,  density
                                         dependent flow and
                                         transport of  heat,
                                         brine, tracers or  ra-
                                         dionuclides  in aniso-
                                         tropic, heterogeneous
                                         saturated porous media
A finite difference mod-
el for transient one-
dimensional,  simulation
of vertical  solute
transport in  the unsa-
turated zone, coupled
with a chemistry model

A semi-analytleal, ad-
vectiv* transport model
which calculates travel
times and paths along  an
unconfined aquifer for
given potential surface

A f in i te di fference
model for simulation  of
convective transport  of
a conservative tracer
dissolved in groundwater
under steady or tran-
sient flow conditions.
The model calculates
heads, velocitites and
trancer particle posi-
tions.

To simulate the simul-
taneous movement of
water and solutes in a
one-dimensional  satu-
rated-unsaturated non-
homogeneous soi i pro-
f ile,  inc!uding  the
effects of linear ab-
sorption  and zero- and
f irst- order decay

A three-dimensional fi-
nite  element model to
simulate  coupled  t>-an-
sient flow,  solute- and
heat-transport in satu-
rated porous media

Finite-element model  for
one,  two  and three-
demensiona!  saturated/-
unsaturated  groundwater
 flow, heat transport,
and  solute transport  in
 fractured porous media,
 faciIitates  single-
 species  radionuclide
 transport and  solute
 diffusion from fracture
 to rock matrix
                                                                                             advection
                                                                                             dispersion
                                                                                             di ffusion
                                                                                             adsorpt ion
                                                                                             ion  exchange
                                                                                             decay
                                                                                             chemical
                                                                                               reactions
                                                                    advection
                                                                    ion  exchange
                                                                    reactions
                                                         3840
                                                         2960
                                         advect ion
                                         particle
                                         tracking
                                                                                                              0741
                                         convection
                                         di sperslon
                                         adsorption
                                         Ion exchange
                                         decay
                                                                                                              3430 '
advection
dispersion
di ffusion
                                                                    convection
                                                                    dispersion
                                                                    di ffusion
                                                                    adsorption
                                                                    decay
                                                                    advection
                                                                                                              2070
                                                                                                              0953

-------
t
No.
t
16.

:
i
1

17. ;
; j

1

t


S
*
j
18.









t
i 19.
;




j
i

70.



1


' 2.

-




22.








23.






Author(s)
S. Haji-Djafarl
T.C. wei is




P. Huyakorn










P. Huyakorn










P. Huyakorn


!




!
P. Huyakorn






INTERA
En vi room.
Consult., Inc.




INTERA
Environ*.
Consult., Inc.






F.E. Kas/eta
C.S. Simmons
C.R. Cole




Contact Address
D'Appo Ionia Haste
Mngmt. Services, Inc.
10 Duff Rd.
Pittsburgh. PA 15235
Tel: 412/243-3200

IGMMC*
4600 Sunset Avenue
Indianapolis, IN 46208
Tel: 317/283-9458







GeoTrans, Inc.
250 Exchange Place
Suite A
Herndon, VA 22070
Tel: 703/435-4400






GeoTrans, Inc.
250 Exchange Place
Suite A
Herndon. VA 22070
Te 1 : 703/435-4400




GeoTrans, Inc.
250 Exchange Place
Suite A
Herndon, VA 22070
Tel: 703/435-4400


K. Kipp*
U.S.. Geological Survey
Box 25046, mail Stop 411
Denver Federal Center
Lakewood, CO 80225


INTERA Envioronmental
Consultants, inc.
11999 Katy Freeway.
Suite. 610
Houston, TX 77079
Tel: 713/496-0993



Battelle Pacific NH Labs
P.O. Box 999
Rich land, MA 99352




Model Naae
(last update)
GEOFLOW*
(1982)




TRAFRAP*
(1966)









GREASE2*
(1982)









SATURN2*
(1982)







SEFTRANt
(1985)





SwiPR
(1979)





Hydro logic
Contaminant
Transport
Model
(HCTM)*
(1975)



MMT-1D
(1980)





Model
Description
A three-dimensional fi-
nite element model to
simulate coupled tran-
sient flow, solute- and
heat-treasport in satu-
rated porous media
A finite element model
to study transient, two-
dimensional, saturated
ground water flow and
chemical or radionucl ide
transport in fractured
and un fractured, aniso-
tropic, heterogeneous.
mu 1 t i - 1 ayered porous
media

A finite element model
to study transient.
multi-dimensional ,
saturated ground water
flow, solute and/or
energy transport in
fractured and unfrec-
tured, anisotropic,
heterogeneous, multi-
layered porous media

A finite element model
to study transient, two-
dimensional variably
saturated flow and so-
lute transport in
anisotropic, hetero-
genous porous media


A two-dimensional finite
element model for simu-
lation of transient flow
and transport of heat or
solutes in anisotropic.
heterogeneous porous
media
A finite difference
model to simulate
nonsteady, three-
dimensional ground water
flow as well as heat and
contaminant transport in
a heterogeneous aquifer
A three-dimensional mod-
el to simulate transient
flow and solute trans-
port in a seturated/-
unsaturated, anisotro-
pic, heterogeneous aqui-
fer system using finite
differences and method
of characteristics
To simulate transient.
one-dimensional movement
of radionucl ides and
Other contaminants in
saturated/unsaturated
aquifer systems

Model
Processes
advection
dispersion
di f fusion



advection
dispersion
d i f f us i on
Bdsorpt ion
decay
chem i ca 1
reactions




advection
conduct ion
dispersion
di f fusion
buoyancy
adsorption





advection
conduct ion
dispersion
diffusion
adsorpt ion
decay
chemical
reactions

advection
dispersion
di f fusion
adsorption
decay


advecT i on
conduct ion
dispersion
diffusion
sorpt i on


advection
dispersion
diffusion
sorpt ion
decay




advect ion
dispersion
d i f f us i on
sorpt ion
(chain) decay
chemical
reactions
IGUMC
Key
3220





0581










0582










0583








0588






0692






0693








0781







-------
 No.
Author(s)
Contact Address
Model Name
(last update)
Model
Description
Model
Processes
j 24.   K.L. Kipp
i 25. !  T.R. Knowles
 26.
 27.
 28.
 29.
  30.
  31.
L.F. Konikow
J.D. Bredehoeft
       N.M. Larson
       M. Reeves
E.  Ledoux
        I. Mi Iler
        J. Marlon-
          Lambert
 T.N.  Narasimhan
 A.E.  Reisenauer
 K.T.  Kay
 R.W.  Nelson
 R.W.  Nelson
                   AERE  HarwelI
                   B336.32
                   Didcot, Oxfordshire
                   United Kingdom 0X11 ORA
                   Texas  Dept.  of  Water Res
                   P.O. Box  13087
                   Capitol station
                   Austin, TX  78758
                   Tel: 512/475-3681
 L.F. Konikow*
 U.S. Geological Survey
 431  National Center
 Reston, VA  22092
 Tel: 703/648-5878
 Nat I. Energy Software
   Center*
 Argonne Natl. Laboratory
 9700 S. Cass Avenue
 Argonne, IL  60439
 Tel: 312/972-7250

 Ecole des Mines de Paris
 Centre d'Informatique
   Geologique
 35, rue Saint-Honore
 77305 - Fontaineble
 France
 Tel: (1)422.48.21

 Eileen Poeter
 Colder Associates
 2950 Northup Way
 Bellevue, WA  98004
 Tel: 206/827-0777
 C.R. Cole*
 Battelle pacific NW Labs
 Water & Land Res. Div.
 P.O. Box 999
 Rich I and, WA  99352
 Tel: 509/376-8451
 Battelle Pacific NW Labs
 P.O. Box 999
 Rich I and, WA  99352
 Tel: 509/376-8332
                            Column
                            Transport
                            with
                            Sorption
                            (1976)
                                                     GWSIM-II
                                                     (1981)
                                                     USGS-20-*
                                                     TRANSPORT/
                                                     HOC
                                                     (1986)
                                              OOMOO
                                              (1977)
 NEWSAAM*
 (1976)
 PATHS
 (1978)
                                              Colder
                                              Groundwater
                                              Computer
                                              Package*
                                              (1983)
 TRUST*
 (FLUX/
 MULTVL)
 (1981)
               A one-dimensional,
               steady-state model  to
               simulate vertical mass
               transport i n a soi I
               column and to solve the
               inverse problem

               A transient, two-dimen-
               sional, horizontal  model
               for prediction of  water
               levels and water quality  ,
               in an anisotropic,
               heterogeneous, confined
               or unconfined aduifer
               based on finite differ-
               ence method

               To simulate transient,
               two-dimonsional, hori-
               zontal ground water flow
               and solute transport in
               confined/semicon fined or
               water table aquifers
               using finite differences
               and method of character-
               istics

               Prediction of coupled,
               one-dimensional movement
               of water, and trace con-
               taminants through lay-
               ered, unsaturated soils
                                                                    A  finite difference mod-
                                                                    el  for  transient predic-
                                                                    tion of piezo-metric
                                                                    heads and salt transport
                                                                    in  a multi-layered
                                                                    aqui fer
 A transient  finite ele-
 ment  model to  simulate
 hydraulic  and  solute
 transport  cherscteris-
 tics  of  two-dimensional,
 horizontal or  axi-
 symmetric  ground water
 flow  in  layered aquifer
 systems

 A transient  integral
 finite difference model
 to compute steady and
 non-steady pressure head
 distributions  in multi-
 dimensional, heterogen-
 eous, variably satu-
 rated, deformable, por-
 ous media  with complex
 geometry

 To evaluate  contamina-
 tion  problems  in un-
 steady,  two-dimensional
 ground water flow sys-
 tems  using an  analytical
 solution for the flow
 equation and the Runge-
 Kutta method for the
 path Iine equation
                                                                                              advect ion
                                                                                              d i f fusion
                                                                                              sorption
                                                                                              decay
                                                                     advection
                                                                     di spersion
                                                                     di f fusion
                           advection
                           d i sparsi on
                           di ffusion
                           advection
                           adssorption
                                                                                       advection
                                                                                       adsorption
                                          advection
                                          dispersion
                                          d i f f us i on
                                          adsorption
                                          decay
                                          chemical
                                            reaction
                                                                                              advection
                                           1160
                                         -  0680
                 0740
                                            1450
                                                                                                               1010
                                                                                                               0120
 advection
 adsorption
 ion  exchange
                                                                                                               2120
                                                          8

-------
Ho.
: 32.
Author (s)
J. Noorishad
• ; M. Menran
! i
'•



i
i
: :


I
33.

i


i
I
|
i

34.



I .L. Nwaogazie









D.B. Oakes

>
: i

\
35.








36.

}
:



j 37.
i







38.







39.










J.F. Pickens
G.E. Grisek







G.F. Pinder






T.A. Prickett
T.G. Naymik
C.6. Lonnquist






A.E. Reisenauer
C.R. Cole






B. Ross
C.M. Kopl ik







Contact Address
Jahan Noorishad
Earth Sciences Division
Lawrence Berkeley Lab.
Univ. of Cal ifornia
Berkeley. CA 94720





i .L. Nwaogazie
Dept. of Civil Engnrg.
Univ. of Port Har court
PMB 5323
Port Harcourt, Nigeria





Water Research Centre
Medmenhanm Labs
Morlo-
Buck i nghaash i re SL7 2HD
U.K.

GTC Geologic Testing
Consultants. LTD.
785 Car I i ng Avenue ,
4th Floor
Ottawa. Ontario
Canada K1S 5H7



Pr i nceton Un i v .
Dept. of Civel Engn.
Princeton, NJ 08540
Tel: 609/452-4602



IL State Mater Survey
P.O. Box 505O. Sta. A
Champaign, IL 61820
Tel: 217/333-6775





Batteile Pacific NH Labs
Hater t Land res. Dlv.
P.O. Box 999
Rich land. NA 993S2
Tel: 509/376-8338/8451



Analytic Sciences Corp.
Energy & Environment Div
On« Jacob Way
Reading. MA 01867
Tel: 617/944-6850




Model Mae
dart update)
ROCMAS-HS
(1981)








SOTRAN
(1985)








NIMBUS
(1980)




SHALT t
(1980)







ISOOUAO 2
(1977)





Random Walk*
(1981)







vrr
(1979)






WASTE*
(1981)







Model
Description
A transient model to
solve for tuo-dimension-
al di spers ive-convective
transport of non-conser-
vative solutes in
saturated, fractured
porous media for a given
velocity field as gener-
ated by ROCMAS-M

A finite-element solute
transport model for two-
dimensional uncon fined
aquifer systems using
linear or quadratic
isoparametric quadri-
lateral elements and
adsorption, bi ode-
gradation and radio-
active decay.
A one-dimensional finite
difference model for
transient simulations of
vertical unsaturated
flow and transport of
nitrates in soi Is
A finite element model
for transient simulation
of 2-dimensional , den-
sity dependent coupled
flow and transport of
heat and solutes in
fractured variably
satureated porous media

A finite element model
to solve the transport
equation in non-steady.
confined, area!, two-
dimensional groundwater
flow

To simulate one-or two-
dimensional, steady/non-
steady flow and solute
transport problems in
heterogeneous aqu i f er
under water table and/or
confined or semi -con-
fined conditions using a
"radom-welk" technique
A transient finite dif-
ference model to calcu-
late hydraui ic head in
con f i ned/uncon f i ned ,
multi-layered aquifer
systems and generate
streamlines and travel -
times
To compute one- or two-
dimensional horizontal.
or one-dimensional ver-
tical, steady /unsteady
transport of radio-
nuc tides in confined or
Mmi-conf ined, an i so-
tropic, hetero-geneous
multi-aquifer systems
Model
Processes
convection
0 i spers i on
di f fusion
adsorpt ion
decay
react ions




dispersion
adsorption
decay
advect ion






advection
dispersion




advect ion
conduct ion
dispersion
diffusion
adsorption
ion -exchange
decay
chemical
reactions
advect ion
dispersion
diffusion




I6WMC
Key
3081









4320









1221





2034








0511





,
advection \ 2690 s
dispersion
d i f f us i on
adsorption
decay
chemical
reaction


advect i on







advection
dispersion
diffusion
adsorption
ion exchange
decay



j
f
|
?
I




2092







2610









-------
Ho.
40.






41.








42.









43.







44.





45.







46.






Author (s)
A.K. Runchal






B. Sagar








B. Sagar









R.D. Schmidt







G. Segol
G.F. Pinder
W.G. Gray



H.H. Set in
J.M. Davidson






B.J. Travis






Contact Address
A.K. Runchal
Analytic and Computa-
tional Research, Inc.
3106 Inglewood Blvd.
Los Angeles, CA 90066






B. Sagar
Analytic & Computational
Research, Inc. 3 106
Inglewood Blvd.
Los Angeles, CA 90066







B. Sagar
Analytic & Computational
Research, Inc.
3106 Inglewood Blvd.
Los Angeles, CA 90066





U.S. Dept. of the
Inter ior
Bureau of Mines
P.O. Box 1660
Twin Cities, MN 551 11



G. Segol*
Bechtel, Inc.
P.O. Box 3965
San Francisco, CA 94119
Tel: 415/768-7159

H.M. Selim*
Louisiana State Univ.
Louisiana Agricultural
Experimental Station
Agronomy Dept .
Baton Rouge, LA 70803
Tel: 504/388-2110

B.J. Travis
Los Almos national Lab.
Earth & Space Sci. Oiv.
Los Almos, MM 87545



Model Name
(last update)
PORFLOW
II & III
(1981)






FRACFLOW
(1981)








FLOTRA
(1982)








ISL-50
(1979)






INTRUSION
(1974)




NMOOEL
(1976)






TRACR3D
(1984)





Model
Description
Steady or transient, 2-D
horizontal, vertical or
radial and 3-0 simula-
tion of density depen-
dent flow heat and mass
transport in an i so-
tropic, hetero-geneous,
non-deformable saturated
prous media with time
dependent aquifer and
fluid properties
Steady and unsteady
state analysis of den-
sity-dependent flow,
heat and mass transport
in frctured confined
aquifers simulating tow-
dimensional 1 y the pro-
cesses i n the porous
•tedium and one-dimen-
sional ly in the frac-
tures, including time-
dependency of properties
steady or transient.
two-dimensional, areal.
cross-sectional or
radial simulation of
density-dependent flow,
heat and mass transport
in variable saturated,
anlsotropic, hetero-
geneous defornable
porous med i a
A three-dimensional
semi -analytical model to
describe transient flow
behavior of leechants
and ground water, in-
volving an arbitrary
pattern of injection and
recovery we 1 1 s
A two-dimensional, ver-
tical finite element
model to simulate tran-
sient, density dependent
f low in a coastal
aquifer
Steady or unsteady simu-
lation of one-dimension-
al, vertical water and
nitrogen transport and
nitrogen transformations
in saturated and unsatu-
rated, multi -layered,
homogeneous so i 1 s
A three-dimensional
finite-difference model
of transient two-phase
flow and mult {component
transport in deformable,
heterogeneous, reactive
porous/fractured media
Model
Processes
convection
conduction
dispersion
di f fusion
change of
phase
adsorption
decay
reactions


convect ion
conduction
dispersion
di f fusion
consol idation
adsorption
decay
reactions




convect ion
conduct ion
dispersion
di (fusion
consol idat ion
hysteresis
adsorption
decay
reactions

advect ion







advection
dispersion
diffusion



advect ion
di s pension
di f fusion





dispersion
di f fusion
adsorption
decay
advection


IGWMC
Key
3233






3232








3235









2560







0530
S
I



0290







4270






10

-------
No.
Author(s)
Contact Address
Model Naae
(last update)
Model
Description
Model
Processes
IGWHC
Key
 47.
C.i.  voss
 C.I.  Voss
 U.S.  Geological  Survey
 431  National  Center
 Reston,  VA  22092
 SUTRA*
 (1984)
48.
J.W. Warner
 Colorado State Univ.
 Civil Engineering Oept.
 Ft. Collins, CO  80523
 Tel:  303/491-5861
 RESTOR
 (1981)
       G.T. Teh
       D.D. Huff
                   Environmental  Sci. Div.
                   OBK  Ridge National Lab.
                   OaK  Ridge, TN   37830
                            FEMA
                            (1985)
 50.   G.T. Teh
     *  D.S. Hard
                   Oak Ridge nati.  Lab.
                   Environmental  Sciences
                     Division
                   Oak Ridge, TN  38730
                   Tel:  615/574-7285
                            FEHWASTE+
                            (1981)
A  finite element simula-
tion model for two-di-
mensional, transient or
unsteady-state, satu-
rated-unsaturated, fluid
density dependent ground
water  flo* with trans-
port of energy or chem-
 ically reactive single
species solute transport

A  finite element model
to calculate the dual
changes  in concentration
of  two reacting solutes
subject to binary action
exchange  in flowing
ground water by two-
 dimensional simulation
of area!  transient or
steady ground water  flow
 and transient coupled
 transport of two solutes
 in an  anisotropic,
 heterogeneous confined
 aqui fer

 A  two-dimensional finite
 element model to simu-
 late solute transport
 including radioactive
 decay, sorption, and
 biological and  chemical
 degradation.  This model
 solves only solute
 transport equation and
 velocity field  has to be
 generated by  a  flow
 model

 A two-dimensional cross
 sectional  finite element
 model  for transient
 simulation of  transport
 of dissolved  constitu-
 ents for a given  velo-
 city field  in a hetero-
 geneous, saturated or
 unsaturated  porous media
                                                                                            convect ion
                                                                                            dispersion
                                                                                            di ffus ion
                                                                                            adsorption
                                                                                            react ions
                3830
advection
dispersion
dif fus ion
ion-exchange
  3100
                                          dispersion
                                          diH usion
                                          adsorption
                                          decay
                                          advection
                                          advect ion
                                          dispersion
                                          d i f < us i on
                                          adsorption
                                          deccy
                                                                                                             337c \
                                                                                                             3371
                                                         11

-------
                The Fundamentals of Geochemical Equilibrium Models;

                                      with a

                          Listing of Hydrochemical Models
                         That Are Documented and Available
                                        by

                                  Richard E. Rice
                                    GWMI 86-04
                                   December 1986
INTERNATIONAL   GROUND   WATER   MODELING   CENTER

                            Hoi comb Research Institute
                                 Butler University
                           Indianapolis, Indiana  46208

-------
Equilibrium and Multiphase Systems

     Equilibrium—probably the  most fundamental concept  of  classical  thermo-
dynamics—is defined by Lewis and Randall (1961) as "a state of rest."   Rather
than implying  a  cessation of motion at the microscopic level, this definition
means either that  the  macroscopic properties of the  system  under a given set
of external constraints  remain  unchanged over the course of time, or that the
system returns to its original state after the external constraints are momen-
tarily altered.   Mahan (1963)  lists  the following criteria  as  necessary for
equilibrium:

     (1)  no unbalanced forces acting on or within the system,
     (2)  uniform  chemical  composition  in  each  phase  with no  net chemical
          reactions occurring, and
     (3)  uniform temperature equal to that of the surroundings.

The  general thermodynamic  requirement for this condition  is that the change in
the  appropriate  free energy  function be  zero.

     The  number  of external  constraints required  to  determine the state of a
system  is referred to as  the number  of degrees of freedom and depends on the
number  of  constituents  and  phases present.   Thus  for a  system  consisting
entirely  of gases,  only  one phase can  exist  at  equilibrium, since all  gases
are  infinitely soluble in  each other.  Completely  miscible liquids  also form a
single  phase,  but  immiscible  liquids  constitute separate  phases.  Solids,
which generally  have only  limited solubility in each  other,  can give rise to a
number of different phases at equilibrium.

     For  a single  phase   such as pure  water,  for example,  two  constraints—
usually pressure and temperature—are  required to  determine  its state, whereas
it  is necessary  to  specify only one constraint for two phases of  pure  water in
equilibrium with  each other.   For pure water existing in  all  three phases
simultaneously,  there are no degrees  of  freedom; i.e., the  so-called triple
point  of  water  occurs only  at  a  temperature of 273.16  K and saturation  vapor
pressure  of 6.11 millibars.

-------
     Mathematically stated, the phase rule  (Denbigh 1971) is

          - - C + 2 - P,                                             n
witrt r  represents  the degrees of  freedom,  C the number of components f and F
the  nu-Tibsr  of phases.   For a  system in  which  chemical reactions can occur,

          C = N - R,                                                  (2)

I.e.  , the number of constituents  that completely define  the  system is  the dif-
ference between the total  number  of chemical  species  N and the  number  of inde-
pendent  reactions  R  relating  the  various  chemical  species.   An  additional
r5Si.-ictioji  arises   in   the  case  of  electrolytes,  since  electroneutral i ty
requires  that  the  total  number of  cations  equal  the total  number  of  arn'ons,
and  thus

          C = N - R - 1.                                              (3)

The  restrictions  imposed  by  the phase rule  must be taken into accout in  any
mult; component  system,  including those  in which the  number of  phases changes
through dissolution/precipitation reactions.

     A  system  of  CaC03 ,  Ca(OH)2,  and water, for example s  can  undergo fHe  foi-
"lowiiij  reactions:

          CaC03 +->  Ca2+  + C032" ,
          Ca(OH)2 «-»•  CaOH+ +  OH* ,
          CaOH+ «-»•  Ca2+  + OH~,
          H20 *-»  H+ + OH",
          H   +  C032~ *-»
          H+  •»-  HCOa ^ H2C03.

As  lonq  as  the  system contains no gas  phase,  there are six independent reac-
tions,  though  not  necessarily this  particular  set.   The  second and  third
reactions above could be replaced by

-------
           Ca(OH)2  + H* «-> Ca2+ + H20,
           Ca2* + H20 «-»• CaOH* + OH~,

 without altering  the description of  the  system in terms  of  N,  C, R, and  F.
 Regardless of the  set  of  independent  reactions  chosen, the  total number
 of  different  species is ten,  and  thus C = 3  from  eq.  (3).  Since there are
 three  distinct  phases—the aqueous and  two solid  phases—eq.  (1) indicates
 that  the system has  only two degrees  of freedom;  i.e.,  it  is  completely char-
 acterized  by  any two  intensive  variables,  such as temperature, pressure, pH,
 or  the  concentrations of dissolved  species.

      In this  particular  system  the number  of  compounds originally chosen and
 the number of components are  the same, but  this  is  not  always true.   With the
 addition  of  CaO to  the above  system, the  parameters  N,  R,  and  P  are each
 increased  by  one,  and the additional  independent reaction can be written  as

           CaO * H20 «-» Ca(OH)2.

 Thus  the  number of  components  remains three, but  the number of  degrees  of
 freedom is reduced to  one  because  of  the  additional solid  phase present.   In
 general, the  phase  rule offers  a  useful  guide  in  organizing multicomponent
 systems and characterizing them  in the correct thermodynamic terms (Brinkley
 1946).

     Regardless  of the number of components or phases,  however, true  equilib-
 rium  can occur only  in a closed system,  i.e., one  that exchanges energy but
 not matter with  its  surroundings.  For  an open system, which can exchange both
 energy  and matter  with its surroundings, the  time-invariant condition is not
 equilibrium,  but the  steady  state.   The difference between these two condi-
 tions  is  that  the  former is  characterized  by  a  minimum  in  free  energy,
 while the  latter is not.

     All natural water systems are of  course open systems,  so the application
 of  thermodynamic equilibrium  models  to them should be undertaken with care.
 Even though the time available  for approaching equilibrium in typical ground
waters may be  on the  order  of  tens to hundreds of years  (Morgan 1967), some  of
 the dissolution/precipitation  or  oxidation/reduction reactions may still occur

-------
slowly  relative  to these  time scales.   In  a particular ground water at any
given time,  some  reactions may be very  near  equilibrium, while  others remain
quite  far from  it.   Although  thermodynamic  calculations  cannot provide any
information  about  the  speed with which the system is approaching equilibrium.
they do yield  a  description of the  state toward which the  system is tending.

     There are,  therefore,  a  number of  inherent  dangers in  uncritically ac-
cepting the  values calculated from any  of  the  numerous equilibrium computer
models  available.   The  problems  are  more  fundamental  than  simply computa-
tional, and  since  there  have been  recent  reviews  of the  different  models
(Nordstrom et  al.  1979, Jenne 1981, Kincaid  et al.  1984,  Nordstrom and Ball
1984),  this  report focuses on the varous aspects of the conceptual model  as
distinguished  from the  computer  code that  performs  the indicated operations
(Mercer et al.  1981).   Other  recent  reviews  include one  by  Plummer  et al.
(1983), which  distinguishes between static  and reaction-path models,  and  a
brief  one by  Potter  (1979),  which  does point  out some of the problems with
computer models and contains an extensive bibliography.
Gibbs Free Energy and  Equilibrium  Constants

     As a  criterion for the  spontaneity  of  any process, neither the enthalpy
nor  entropy  is  entirely satisfactory.   A  process may  be  characterized by a
large negative  value of enthalpy  and  still  not proceed  spontaneously, whereas
the associated  entropy change must include calculations  on  the surroundings as
well  as on  the  system  itself.   Because  of  this,  the American  chemist J.
Willard Gibbs  developed the  free-energy  function  in the late nineteenth cen-
tury  (Denbigh   1971,  Lewis and  Randall  1961,  Moore  1972).   For a process at
constant temperature and pressure, the change in  free energy G  is defined as

          AG =  AH - TAS,                                            (4)

where H represents  the system's  enthalpy, T  its absolute temperature, and S its
entropy.   The   advantage of  the free-energy function as a measure of sponta-
neity  is  that  it  depends  only  on the  system, not  on  the surroundings, and
incorporates  temperature  and pressure as  its natural  independent variables.

-------
     The change in Gibbs free energy can also be written in the form
where q  and  qrgv are the actual and the reversible heats, respectively,  asso-
ciated with a given process at constant temperature and pressure (Mahan 1963).
For  a  process occurring  under equilibrium conditions, q and  q    are equal,
and AG = 0.   If the process  proceeds  irreversibly,  however, q = q.   < q   ,
and  AG  < 0.   Therefore AG can  never  be greater than zero, and  for  a system
tending  irreversibly  toward   equilibrium,  the  free  energy  decreases  until
finally  reaching its minimum value at equilibrium.

     Thus the general chemical reaction

          aA + pB «— » yc + 6D,                                       (6)

where the Latin majiscules represent chemical species and the Greek miniscules
the  appropriate  stoichioroetric coefficients,  is defined  as  having  reached
equilibrium  when  the  total  Gibbs  free energy  of the  products  (the  final
state)  minus that  of  the reactants  (the initial state)  is zero,  i.e., when

          AG = yGc + 6% " «GA - ^GB = 0.                            (7)

Each of  the individual molar  free energies is related to the activity a- of the
particular species i by the expression

          G. = G? +  RT£n a. ,                                         (8)

where G? represents the free energy of  the species  in some  standard state and
R  is  the  gas  constant.    The  expression  for  AG  may  be  rewritten from
eqs. (7) and (8) as
                            r
          AG = AG° +  RT*n -^  .                                      (9)
                          aAaB
At equilibrium the ratio of activities  is  equal  to  the  equilibrium constant  K,

-------
so
          AG° = -RT£n K,                                             (10)

which  is  the  well-known  relationship  between  standard  free  energy and the
equilibrium constant.

     Each reaction  of  the set chosen for a  particular model  must  be  charac-
terized by an equilibrium  constant.   In any geological environment there is  an
extremely large number  of possible  reactions,  and this is  reflected  by the
data bases  of  many  of  the  models,  some of  which  consist  of several  hundred
reactions.   These  include not only  reactions  occurring  solely in the  aqueous
phase,  but  also heterogeneous reactions between  dissolved species and solid
phases,   such   as  precipitation/dissolution   and   ion  exchange,  as  well  as
oxidation/reduction   and  degradation  reactions that  may  be  catalyzed  by
microorganisms  in the soil.

     At   least  three  fundamental  problems  are associated  with  such tabu-
lations  of   thermodynamic  data.   A  particular species may  simply  be   omitted
from the  data  base,  so even though it  is present  in the  system being modeled,
it  will  obviously  not  appear in  the  final  speciation  results  nor will   its
effect on the  speciation  of other elements.   The  program  WATEQ3 (Ball et  al.
1981),  for  example,  is  an extension of WATEQ2  (Ball et  al.  1979)  through  the
addition  of  several  uranium species,  but  the expanded  data  base does  not
include vanadium,  which frequently occurs naturally with uranium, and  thus  the
influence of minerals  containing both  elements cannot be  taken  into  account.

      Even when the  data base does contain particular minerals,  thermochemical
data for them may  not be  known with  very great  precision.  This problem is
frequently   compounded   by   other  uncertainties   such  as  nonstoichiometry,
solution-dependent composition with  respect to replaceable cations, metastable
forms, and  variation  in  free energy and solubility with  the degree  of  crys-
tal! inity (Stumm and Morgan  1981).

      The tabulated thermodynamic data  is also usually not checked  for internal
consistency.   Because  the  data  for a particular  reaction  may  come from more
 than one source (and may  thus be determined by  different methods),  there is no
                                        6

-------
guarantee that all calculations were made with consistent values of the neces-
sary  auxiliary  quantities or that the data  satisfies  the appropriate thermo-
dynamic relationships.

Pressure Dependence of Equilibrium Constants

     Of all  the  available  computer models, only  SOLMNEQ (Kharaka and Barnes
1973)  contains  a pressure  correction for  the  equilibrium constants,  and at
moderate  temperatures and  pressures  this is usually  quite small.   From eq.
(10) and the thermodynamic  relationship

           36.
where  v\  is the partial molar volume of species i, the result is
                    -
            3P  T     RT
where AV represents the difference in partial molar  volumes between products
and reactants.

     The  term AV is ordinarily less than  30  cm3 unless the reaction is char-
acterized by a  net change in the number of covalent bonds; an increase in the
number of bonds  decreases AV and vice versa (Moore 1972).  Thus for reactions
that exhibit a  large change in AV and take place in deep ground-water systems
(where the  actual  pressure can indeed be very large), the pressure dependence
of K may no longer be insignificant.

Temperature Dependence of Equilibrium Constants

     The  effect  of  temperature  on the  equilibrium  constant is  much greater
than that of pressure,  and only a few of the computer models do not include a
subroutine  for  temperature  correction.    The temperature  dependence  of  the
Gibbs free energy at constant pressure P is related to the enthalpy change for
the  reaction  (AH°)  through the Gibbs-Helmholtz equation (Lewis  and Randall
1961):

-------
                                                                     (13)
The substitution of eq. (10) leads to the van't Hoff equation,

               K,  _ AH
or in integral form,
              2   -I    2 AMO
             K~ = B  J   TT-  dT  .                                    (15)
             Kl   R       '
     In  those cases  for which  the heat  capacity  (Cp) of  each  reactant and
product  is known  as  a  function of temperature, AH° can be  determined  as a
function of temperature,  since
               = ACp = a + bT + cT2  + •  '  *                           (16)
 Equation  (15)  may thus  be  integrated directly,  and the resulting  temperature-
 dependent  expression for the  equilibrium  constant is frequently given in the
 form

           log  KT = A + BT + C/T + D log T .                           (17)

 As  Table  I shows  for the WATEQ series, such  empirical  expressions  are avail-
 able  for  only a  small  fraction of the  reactions  included  in computer models.

            TABLE I.   NUMBER OF EMPIRICAL EXPRESSIONS FOR CALCULATING
                              log K IN WATEQ SERIES
Model
WATEQ
WATEQF
WATEQ2
WATEQ3
# Empirical
Expressions
9
15
17
22
Total #
Reactions
157
191
526
588
Reference
Truesdell and Jones (1974)
Plummer et al. (1976)
Ball et al. (1979)
Ball et al. (1981)
                                        8

-------
     For the  remaining reactions,  the  van't Hoff equation is used.  This  is
obtained  from eq.  (15) with  the  assumption  that AH°  is constant over  the
desired temperature  range.   The  reference temperature is usually 25°C,  so  the
integrated form is

                              ALJO
                                7Qft   1     1
          log KT = !og K298 - j-fj^ (i - ±).                      (18)
Whether  this  is a  good approximation or  not depends on  how  constant  AH° is
over  the particular  temperature range;  usually the  smaller the  range,  the
better  the  assumption.    But  a  good approximation or  not,  the  van't Hoff
equation  is  often  the only means for computing  equilibrium constants at tem-
peratures other than 25°C.

      As  a comparison  between log K values  calculated  from eq. (17) and those
from  eq.  (18),  Table  II  contains  these   values  over the  temperature range
0-100°C  for all the  reactions  in WATEQ (Truesdell  and Jones  1974) for which
there are  empirical  expressions.  The  values  from eq.   (17)  are  presumably
more  accurate than those  of eq.  (18),  since the  former expressions are de-
rived from  measurements  over  a range  of  temperatures,  though  the  WATEQ
program   does  not  indicate the  range  of  applicability  for any  of  these
expressions.

      Table  II provides a number of interesting comparisons.

Rx #25.        The  equilibrium  constant for the hydrolysis  of boric acid in-
               creases with  increasing temperature according to the van't Hoff
               equation,  but actually  decreases with  increasing  temperature
               according to  the empirical expression.

Rxs #35 & 68.  The  log  K's calculated  from the  empirical  expressions both
               exhibit  maxima,   but  the  van't Hoff  equation can  obviously
               predict     only    monotonic   changes     with   temperature.

Rx #72.        The  constants B  and  D are  both set equal  to  zero  in the em-
               pirical expression, which thus has exactly the  same  form  as the
               van't  Hoff  equation.   The agreement  between log K  values cal-

-------
TABLE II.   VALUES OF LOG K FROM WATEQ,  CALCULATED BY EQS.  (12) and (13) FOR COMPARISON
Reaction
/-ii\ u c-jn < 	 k H + w <;in~
vl-)J n^oiu^ • • n T ti3jiu^
/ 1 /i > u c»n . >. on i u c-"^ it nn < » n > u nn
V^3J rljuUj * ' n * n2DU3
^?C\ MU , , U i Kill
\£u) nn^ • • n T "nj
^ .
fjc\ u fn , ( u 4. upn
VJJy n2V»U3 ' • n T HUUg
fcBA urn" . i u + urn?"
\oo / nuug • » n T nuuj
f77^ K* + ^n2" 4 	 * K
-------
               culated  from  the two  equations  then appears  to  be  excellent,
               but is merely fortuitous.

Rx #89.        Despite  the  nearly twofold  increase in AH° over the tempera-
               ture  range  25-100°C,  the van't Hoff equation  still  provides  a
               better approximation to the  log  K values determined from solu-
               bilities (Leitske et al.  1961) than does the empirical expres-
               sion.

Rx #91.        The  relative  error  between  the log  K's  is  only 3.7% at 373 K,
               but it is 74% between the K's themselves.

     Each  computer  model  contains an extensive collection  of thermochemical
data   gathered   from   many   different   sources.   There   are   generally
no  criteria set  forth for  the particular choices,  although the  authors  of
WATEQ3  (Ball  et al. 1981)  state  that "log K values  derived from solubility
measurements are  considered to be more  meaningful in  studies of the natural
environment  than  those derived from  AH  and AS  values;  therefore,  the former
were selected."   They  do  not,  however, offer any  rationale  for claiming that
the former are more  "meaningful."

     It  is  also  interesting to note  that the empirical expression for Rx #25
is  actually derived from  the  parameters of the empirical  expression for the
log K of the reaction

          B(OH)3 + OH" *— B(OH);,                                   (19)

which was studied as a function not only of temperature, but of  KC1 concentra-
tion as  well  (Mesmer  et  al.  1972).   The  ionic-strength  dependence  is  not
particularly great, and  those terms are  simply   set  equal  to  zero  for the
expression  appearing  in   WATEQ (Truesdell  and  Jones  1974).   Nevertheless,
without examining each  of  the original references, users  of a  computer model
cannot  know the  particular details  concerning  the  calculated  or measured
thermochemical  data.  Because  of  the large amount of  tabulated data, this is
an unreasonable expectation in most cases, yet Plummer et al.  (1976) warn that
"the responsibility  for final  selection of constants used in WATEQ rests with
the user."
                                       11

-------
Electrolytes and Activity Coefficients

     Because equilibrium  constants  are defined in terms of activities,  it  is
necessary to  relate these quantities  to  experimentally  measurable  cnncentre-
r,ions.   The relationship (Moore 1972)
          ai =
where y.. is the activity coefficient and m,. the concentration of a component i
considered to  be  the solute, is based  on  a standard state that obeys Henry's
Law  (Fig.  1).   The  solution  becomes  ideal  (v. = 1)  at  low solute concentra-
                                              i
tions:
                a.
           lim  -1 = 1.                                              (21)
          m.-*0 mi

For  the case  of  more than  one  solute  in  solution,  all  the  solutes  must
simultaneously conform  to  the limit in eq. (21); such ideal behavior may also
be  stipulated  in the limit as  the mole fraction of water  goes  to unity.   If
the  expression  for  activity  in eq.  (20) is  substituted  into eq.  (8),  the
result may be written

          G. = G? + RT£n mi + RTAn y.,                               (22)

where the terms  G° + RT£n m.  represent the free energy of  component  i  in an
ideal solution,  i.e.,  one that follows Henry's Law over the  entire range of
concentrations.   Thus  the  term  involving  the  activity   coefficient  is  a
measure of the real solution's deviation from this ideality and represents the
extent of interaction between ions of the same kind.

     Since it is not possible to separate the effects of cations and anions in
an electrically  neutral solution,  the properties of individual ions cannot be
determined  experimentally.    It  is  necessary then  to  relate  the laboratory
value of the  mean activity coefficient of an electrolyte, which represents an
average over  both cations  and anions,  to  the calculated  values of single-ion
activity coefficients.   This requires  an assumption, often  the mean-salt or

                                       12

-------
Figure 1.  Activity vs. molality for HC1  solutions.
          Henry's Law behavior.
     -
The dotted line represents
 O
                                  MOLALITY  (m)
                                       13

-------
Maclnnes (1919) convention, which equates the single-ion activity  coefficients
of K  and  Cl   in KC1 solutions to each  other as well  as to  the mean  activity
coefficient of the salt.
     It  is  possible to  calculate  single-ion activity  coefficients  from  only
electrostatic considerations.   This was  first  done successfully  with  Debye-
Huckel  theory,  which  manages  to  provide  surprisingly good  results  despite
several  severe  contradictions  and  physically incorrect  assumptions  (Bockris
and Reddy 1970).   Essentially,  Debye-Huckel theory ignores short-range inter-
actions  between ions  of  the same  charge, and  thus   its  predictions  become
poorer  for  more  concentrated  solutions  in which  ions with the  same  charge
increasingly affect  each other  and those with opposite charge form ion pairs
through electrostatic attraction (Robinson  and Stokes 1959).

     Virtually  all  current  computer  models are  based  on  this  idea  of ion
pairing,  which  was  developed independently by  Bjerrum (1926)  and  Fuoss and
Kraus  (1933,  Fuoss  1958).   With  the  inclusion  of  these  short-range  ionic
interactions, the  so-called  "extended" Debye-Huckel equation for  species i is
                       A z*l
          log *  = -- *-i— r  ,                                    (23)
               1     1 + B a. I*
                          Y i

 in  which A   and  B  are the Debye-Huckel  constants  that depend on dielectric
           Y       Y
 constant and temperature, and  z.  is  the ionic  charge,  a^  an ion size para-
 meter,  b.  an ion-dependent  empirical  constant,  and I  the solution's ionic
 strength, defined  by the expression

          I  = h I  z*m..                                              (24)
 In  eq.  (23) the  numerator  accounts for  long-range coulombic  interactions,
 the denominator for short-range interactions that arise  from treating  the  ions
 as  hard,  finite-sized spheres.  As  a correction for short-range  ion-solvent
 interactions as  well  as  short-range  ion-ion interactions  that  are  not ac-
 counted for by  the hard-sphere model, a linear term is  often added  empirically
 to  eq.  (23) or  to some variation of it.   In place of the extended Debye-Huckel
                                        14

-------
 equation,  which  is valid only up to  about 0.1 M, the Davies equation (Davies
 1967),
                     A
           log y. = -  Y i ^ - o.2I,                                  (25)


 is  frequently used, since  it is supposedly applicable  to  solutions  of ionic
 strength up to 0.5 M (Stumm and Morgan 1981).

      A  theoretical  consideration in  the use  of  equations  for multi component
 systems is the thermodynamic requirement that

           3G    3G .
          an    8n.  '

 where n is  the number of moles  of the particular species i or j.  This leads
 to  the condition  (Lewis and Randall 1961)
                       Y-
           8n       dm.    '

which  the Davies  equation does  satisfy in  general,  but which  the extended
Debye-Huckel  equation satisfies only  in the  unusual  case that  the ion size
parameters for two different ions are equal.

     Thus  computer models that calculate activity  coefficients  by either (or
both)  of  these equations are  restricted  to fairly  dilute ground  waters.
Because  of increasing interest  in modeling  more  concentrated natural  waters
(brines,  e.g.),  Kerrisk  (1981)  compared experimental  solubilities  of  CaC03,
CaS04,  and BaS04  in  0-4 M NaCl  solutions with  solubilities calculated from
four  different  computer models:   WATEQF  (Plummer  et al. 1976),  REDEQL.EPA
(Ingle  et al.  1978),  GEOCHEM  (Sposito  and  Mattigod  1980),  and SENECA2,  a
modification of the earlier SENECA (Ma and Shipman 1972).

     Even  though  the  ionic strengths  far  exceeded the  limitations of  the
extended  Debye-Huckel  and Davies  equations,  the  study produced a  number of

                                       15

-------
interesting  results.    The  different  models  frequently  disagreed  with each
other even at low concentrations, a result most likely caused by the different
thermodynamic data  bases in  the different models.   Calculations on CaC03  by
GEOCHEM  differed markedly  from  experimental  observations  even  below 0.5  M
(Figure 2);  one  possible explanation for this  is  the inclusion of  an  equili-
brium  constant   of  about 4  for  the  formation  of  the  ion pair CaCl  .  This
particular  ion  pair  is  omitted from the other three computer  models,  and  in
fact Carrels and Christ (1965)  note  that  at  ordinary  temperatures  chloride
forms no  significant  ion pairs with any major cation of natural waters. This
very clearly points  to  some of  the  dangers inherent  in  the ion-pair method
used in these models.

     A different approach to the problem of ionic  interactions in  solution is
the  specific-interaction model  (Pitzer 1973), which  has  been applied  to sea-
water  (Whitfield 1975,  Millero  1982) and  hydrothermal  brines  (Weare 1981,
Barta  and  Bradley  1985).   It  has  been long  assumed  that  the  results  from
Debye-Huckel theory  could be extended by the addition of  power-series correc-
tions (Weare et  al. 1982):

                         HM
          log Y. = "log  ^  + I  B..(I)m  +11 Ci1-kmimk'              <28)
                •         i     •   ij     j   4 |^   'j*  j  *•

       DH
where Y-  is the Debye-Huckel  activity coefficient  and B..(I) and C...   are  the
       1                                                 I J         « J K-
second  and  third virial coefficients respectively (Lewis and Randall 1961),
the  latter  of  which  is  required only for solutions  of  ionic strength greater
than 3  M.   Pitzer (1973) has  succeeded  in modeling  the  second virial  coeffi-
cient  B.. as a function of  ionic strength  and  has also developed  a Debye-
Huckel term  of  the form
                       A
              9         U
= -  Y      +  2"'1 + bl >•                        (29)
           log  Y
               1       1 + bl

which  not only  has the correct  limiting  form and obeys the condition  in  eq.
(27)  (because b  is a  universal  empirical  parameter),  but also  fits  experi-
mental  data  better  than  the conventional  Debye-Huckel  term given by eq.  (23).
                                        16

-------
Figure 2.  CaC03 solubility  as  a function of NaCl concentration, experimental


          values and  those  calculated  by  four  different equilibrium models

          (Kerrisk 1981).
         =
         \f
         i
         at
         ~
         -
         o

         a?
                                                      LEGEND
                                                     REDEOL.EPRK

                                                     GEOCHEM

                                                     WflTEOrtD)

                                                     5ENECR2
             :
                 0.0
1.0        2.0        3.0       4.0       5.0

    SODIUM CHLORIDE CONTENT  IMOLRD
                                       17

-------
     Although the specific-interaction model is more complicated mathematical-
ly,  it  has the  distinct advantage of not  explicitly including  ion pairs  for
ions  that are only  weakly  associated,  such as  Ca2   and Cl .   Instead,  the
second  virial  coefficient  accounts for  these  weak associations through  its
dependence on the ionic  strength (Weare et  al.  1982).   Weare and his  coworkers
(Harvie and Weare 1980,  Eugster et  al. 1980, Harvie et al. 1982, Harvie et al.
1984) have begun applying this model  to  simple  electrolyte systems—the most
complicated thus far  is one containing  only  11 different ionic species—but
the preliminary  results appear to  be  a  significant improvement over  calcula-
tions based  on  ion  pairing.  Figure  3 compares experimental values  of CaS04
solubility with  those  calculated with  the ion-pair model  (Plummer et al. 1976,
Kharaka  and  Barnes  1973)   and  with  the  ion-interact!'on model (Harvie  and
Weare  1980).   There  is still considerable work to  be done before  the spe-
cific-interaction  model can  be  applied  to ground  waters  in general, but it
clearly has the  advantage of being able  to treat more concentrated solutions
than  ion-pair  theory.   Pitzer's equations  have  already been or are currently
being  incorporated  into at  least  three  geochemical  models:   EQ3NR (Wolery
1983), SOLMNEQ (Kharaka and  Barnes  1973), and  PHREEQE (Parkhurst et al. 1980).
Oxidation-Reduction  Reactions

     Of all the  reactions  included in any of the  computer models, only a small
fraction  consists  of  oxidation-reduction  reactions.   The  model  REDEQL-UMD
(Karri ss  et al.  1S84), for  example,  lists only twenty-two redox couples, and
its  authors caution  that the  kinetics of  many  oxidation-reduction reactions
may  be slow.

     The  emf  or Nernst  potential  (E)  for any  reaction  involving  electron
transfer  can be  determined from the expression (Moore 1972)
                                                                      (30)
where  n  is  the  number  of electrons  transferred,  F  is  the faraday,  and  the
chemical  notation refers  to  the general reaction  in  eq.  (1).   The term E° is

                                        18

-------
     Figure 3. CaSO.  • 2HJ)  solubility as a function of Na-SO. concentration at two
               different NaCl concentrations  (Weare et al. 1982).
.03
.03
I S 11
02


.01






H 0.5m NaCl -02
t_>
li • E
i
-\
\
»
\ WATEQF 8 SOLMNEQ
«^ ^S
"fc"*^^-'-»
"* ~~—

•
MIRABILITE -Ol

>


i
1 I.Om NoCI

\
\ V GYPSUM
I • •
" H
\ SOLMNEQ
j
Nc^*C*
l'"^**.
-------
the standard emf of the redox reaction and can be calculated from  the  standard
electrode potentials  of the  half reactions that sum to the  overall  reaction
(Latimer 1952).

     The fact that oxidation-reduction reactions can be characterized  electro-
chemical 1y  in  this manner  has  led  to  the idea that  a  ground-water  system's
"redox  state"  can  be  described  in  terms of  a  single  parameter, either  an
overall Nernst potential,  usually designated Eh (Freeze and  Cherry 1979),  or
the negative logarithm of the electron activity designated pe (Truesdell  1968)
in analogy  with  pH.   The idea that a single parameter like pe or Eh can  char-
acterize an  entire system is based  on  the assumption  that all the oxidation-
reduction reactions  occurring in the system are at equilibrium.  That this is
not. Lrue has been stated again and  again (Morris  and Stumm 1967, Jenne  1981,
Wolery  1983),  but apparently with  little  effect,  since suggestions  for some
particular  redox  couple as an overall  indicator of the  system continue  (Liss
et al. 1973, Cherry et al. 1979).

     Lindberg  and Runnel!s (1984) have quantitatively demonstrated the futil-
ity in  trying  to characterize an entire  ground-water system by a single redox
parameter.   The  field-measured  Eh  value  for  each  of  approximately 600  water
analyses was compared  with  the Nernst potential calculated  from the data on
ten different  redox  couples  by means of  the computer model WATEQFC  (Runnel Is
ana Lindberg 1981).   As these same  authors (Lindberg and  Runnels 1984) state:
"The  profound  lack  of  agreement   between the  data points  and  the dashed
line  [which represents equilibrium points]  shows that internal equilibrium is
not achieved.   Further,  the computed  Nernstian  Eh values do not agree with
each  other.  ...   If  any measured  Eh  is used as input  for equilibrium calcu-
lations,  the burden  rests with  the investigator  to demonstrate the  reversi-
bility of the  system."

     Because many of  the  important oxidation-reduction reactions  are  very slow
and some  are even irreversible,  it  is  virtually impossible that any natural-
water  system can reach equilibrium  with  respect to all  of its redox couples.
Improvements in  this area of computer  modeling will require the  inclusion of
experimental data  for each  of  the  major  redox  couples  in  the  water system
under  study.
                                        20

-------
Conclusion

     Combining concepts from  thermodynamics  and electrochemistry, equilibrium
models can be  a  valuable  tool in predicting  the  behavior of complex geochem-
ical systems.  They  contain  pitfalls,  however, and an understanding of—or at
least familiarity with—the underlying conceptual  model as well as the computa-
tional methods should aid in properly using a particular equilibrium model and
in making sound scientific judgements about its input and output.

     On the  other  hand,  an unsuspecting user can be lulled into a false sense
of  security.  These  models  calculate  concentrations to what appears  to be
extremely great  accuracy,  yet there are two major cautions: 1) equilibrium is
a  state that open  systems (i.e., real geochemical systems) can never achieve,
and  2)  the  results of any model cannot be any better than the assumptions and
raw  data  that  go into it.  In  spite of these limitations, equilibrium models
can  provide  a  useful frame of reference for the knowlegeable user.
                                       21

-------
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                                        25

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Parkhurst,  David  L. ,  Donald  C.  Thorstenson,  and  L.  Niel  Plummer.   1980.
     "PHREEQE—A  Computer  Program  for  Geochemical  Calculations."   Water-
     Resources   Investigations  80-96.    U.S.   Geological   Survey,   Reston,
     Virginia.

Pitzer,  Kenneth  S.   1973.  "Thermodynamics of electrolytes.  I.  Theoretical
     basis and general  equations." J. Phys. Chem.  77:268-277.

Pitzer,  Kenneth  S. ,  and Janice J.  Kim.    1974.   "Thermodynamics  of electro-
     lytes.   IV.   Activity and osmotic coefficients for mixed  electrolytes."
     J. Am. Chem. Soc.  96:5701-5707.

Plummer,  L.  Niel, Blair  F. Jones, and  Alfred  H.  Truesdell.   1976.   "WATEQF—A
     Fortran  IV  Version  of WATEQ, a Computer  Program For Calculating Chemical
     Equilibrium  of  Natural  Waters."   Water-Resources  Investigations  76-13.
     U.S. Geological Survey,  Reston, Virginia.

Plummer,  L.  Niel,  David  L.  Parkhurst,   and  Donald  C.  Thorstenson.    1983.
     "Development of  reaction models  for ground-water systems."  Geochim.
     Cosmochim. Acta 47:665—686.

Potter,  Robert W.,  II.   1979.   "Computer modeling in low temperature geochem-
     istry."   Kev. Geophys.  Space Phys. 17:850-860.

Robinson,  R.A.,  and  R.H.  Stokes.   1959.   Electrolyte  Solutions.    2nd  ed.
     London:   Butterworths.

Runnells, Donald D., and Ralph D. Lindberg.  1981.  "Hydrogeochemical explora-
     tion for uranium  ore deposits:  use  of  the computer model WATEQFC."  J.
     Geochem. Explor. 15:37-50.
                                        26

-------
Sposito, Garrison, and Shas V. Mattigod.  1980.  "GEOCHEM:  A Computer Program
     for  the Calculation  of  Chemical Equilibria  in  Soil Solutions and Other
     Natural Water Systems."  University of California,  Riverside.

Stumm,  Werner,  and James J. Morgan.   1981.   Aquatic  Chemistry.  2nd ed.  New
     York:   John Wiley.

Truesdell,  A.H.   1968.   "The  advantage of  using pe  rather  than Eh in redox
     equilibrium calculations."  J. Geol. Educ.  16:17-20.

Truesdell,  Alfred  H.,  and Blair F. Jones.   1974.   "WATEQ, a computer program
     for  calculating  chemical  equilibria of  natural waters."   J.  Ros. U.S.
     Geol.  Sunr. 2:233-248.

Weare,  John H.   1981.   "Geothermal-Brine  Modeling—Prediction  of Mineral
     Solubilities  in Natural Waters:   the Na-K-Mg-Ca-H-Cl-S04-OH-HC03-C03-C02-
     H20  System to  High  Ionic  Strengths  at  25°C."   Report DOE/SF/11563-T1.
     U.S. Department of  Energy.

Weare,  John H.,  Charles E. Harvie, and Nancy  Mrfller-Weare.  1982.   "Toward  an
     accurate  and efficient  chemical  model   for  hydrothermal  brines."  Soc.
     Pet. Eng. J. 22:699-708.

Whitfield,  M.   1975.   "Improved  specific interaction  model  for sea water  at
     25°C and 1 atmosphere total pressure."  Mar. Cham.  3:197-213.

Wolery, Thomas J.   1983.  "EQ3NR, A  Computer  Program for Geochemical Aqueous
     Speciation-Solubility  Calculations:    User's  Guide  and   Documentation."
     Report  UCRL-53414.    Lawrence  Liverraore  National  Laboratory,  Livermore,
     California.
                                       27

-------
 feje';  foarae
(List  Update)
                     Author (s)
[  D.L.  Parkhurst
•  L.N.  Plummer
i  D.C.  Thorstenson
    ; :;'A'6     ! T.J. Wolery

 t.t-  -.-.,-".. ion of j
 L(;3r:rn ;;:;. sited |


 F(r.5W;  ror-  the •

 Cre>'-' ,  .-a  we! I }
 es ;T"1 PJ i  '• 11 y *
 ro ';: '.  (jri < VAC, |
    c,i.t. -'AX!    i
     :-!•„,. iB     ; J.R. Morrey
     t !''>""(.;      ; D.W. Shannon
 ;;-.:•.':.-••' :<  •;',  tor J
  i'J';-. J-no VAX |
     Glcu-irw
                 I
     l ' V..';      [
  noi current Iy  i
 cvs i ! o£> i e , ne»  •
 version  expected !

 !rC':<'*:;-,-.' I V f or '
| G. Sposito
  S.V. Mattigod
J.C. WestalI
J.L. Zachary
F.M.M. Morel
 '•• i C:.; iiAlv i V I Or
 L»-:; VAC i 100 ana
       VAXi
                   A.R.
                   D.C.
                   E.A.
        Felmy
        Glrvln
        Jenne
                   D.L.
                   D.C.
                   L .N.
        Parkhurst
        Thorstenson
        PIumraer
                   G. Pcckrel I
                   D.D. Jackson
                         Contact Address
                   L.N. Plummer
                   U.S. Geological  Survey
                   Mater Resources  Division
                   12201 Sunrise  Valley Drive
                   Reston,  VA  22092
                      T.J. Wolery
                      Lawrence Livermore National
                        Laboratory
                      P.O. Box 808, L-204
                      Livermore, CA 94550
                    Vase I  W. Roberts
                    Electric Power Research
                      Institute
                    3412 Hi I(view Avenue
                    Palo Alto, CA  94304


                    G. Sposito
                    Department of SoiI and
                      Environmental Sciences
                    University of California
                    Riverside, CA  92521
                                       F.M.M. Morel
                                                         Model Description
                                             fedel
                                           hccersei
Using the chemical  compositions of
water samples from two points sionp e
flew path end e set c< muife!  phesei,
hypothesizes to be the rn«ciivf: con-
stituents in the systeir.,  the- pre-crow
calculates the i«6ss transfer necessary
to account for the observe^ changes  in
composition between the two Mater
samples.


EQ3NR is fi geochemical aqueous specie-
tion/solubiIity program that can  be
used alone or  in conjunction with  E06,
which performs reaction-patn calcula-   '
tions.  Accomodates up to 40 elements,  t
300 aqueous  species,  15 gases, and 275  [
minerals.                               i
                                                                                                            i^-aas
                                                                                          the resetionb


                                                                                         rsoox rer-"t i.-ivn
                                                                                          ntito not  D-= 3'
                                                   Models cheriiic&l equilibria  in 900-
                                                   thermal brines ai various eInvitee
                                                   temperatures.  Ccnteins 26 Glevants,
                                                   200 aqueous species, 7 oases, anu  166
                                                   minerals.
 A program for predicting the equilib-
 rium distribution of  chemicel  species
 in soil  solution and  other natural
 water systems.  Includes 45 elements,
 1853 aqueous species, 42 organic
 Iigands, 3 gases, and 250 miners is end
 sol ids.
                                                    A program  for  the  cslculotion  oi  chem-
miss b&isnct; vor
 eech spec!Si
redox reactions
cat icn t.Oscrp-
 tion 6»'.l
 o.cr.s-'uc.
                                       Dapt. of Civil Engineering  -icsl  equilibria in aqueous si
                                       Massachusetss Institute of
                                         Technology
                                       Cambridge, MA  02139
                    David Disney
                    ADP Section
                    Environmental Research Lab.
                    U.S. Environmental
                      Protection  Agency
                    College  Station  Road
                    Athens,  GA  30613
                   IL.N.  Plummer
                   (U.S.  Geological  Survey
                   I Water Resources  Division
                   S 122C1 Sunrise  Valley Drive
                     Reston,  VA 22092
                       D.D. Jackson
                       Laurence Livermore National
                         Laboratory
                       P.O. Box BOB, L-529
                       Livermore, CA  94550
 A program for celcul Gt ing goochunicfi!    r.i_^.. ..•- • :-;Ci  '. :>.
 equilibria., containing the IsATEOi date-:  vtic.fi co.;,ci>nn-
 base.  Includes 31 elements, 373        ; rcdox reeci ion;.
 aqueous species, 3 gases, and 323       | ion uxci-'cngt
 SOlidS.                                 , S l >: i;jr:ci.-v'> ccri-
 An equilibrium model that can calculate'
 mass transfer as a function of siepwise;
 temperature chenpe O1" dissolution.
 Inciuctes 59 el omen-fa, !?0 J^TJOO K-
 species, 3 gases, end 2! Riintrcis.
                                                  A coupled  k inel ic/eqji I icrium progren-  . \-.it,f  ;:  ^.j
                                                  for calculaiing  olssolution  react icfij.  :, ~.:v>-i •,
                                                  Of  inorganic  solids  in  equeous soUi-   j  -c.-jiric:
                                                  tion,  with specific  application to cor-J  -aiCaCMi;1;
                                                  rosion of  vitrified  nuclear  waste  by   j   si:sea
                                                  groundwater.   Incorporates equiliDrium |  -surface
                                                  routines fromMINEQL.                   j   co^cf ^gs

-------
Author(s)
    I S.t.  Ingle
     K.D.  SchUdt
    ' D,K.  Schults
     D :-. Hsrriss
     5 . E.  • ng i e
     O.K. Taylor
    i V.R. Magnuson
      Y.K.  Kharaka
      i.  Barnes
      E.w.  Goodwin
      J.«. BalI
      t.A. jenne
OR-   D.K. Nordstrom
      J.I:;. B»l !
      " . r.. Jenne
      ••.r.'. Contrel i
                            Contact Address
D.W. Schults
Hat fie I a Marine Sci.  Cntr.
U.S. Environmental
  Prot ec t i on Agency
Newport, OR  97365


V.R. Magnuson
Department of Chemistry
University of Minnesota
Duiuth, MN  55812
Y.K. Kharaka
U.S. Geological Survey,
  MS/427
345 Middiefieid Road
Men to Park, CA  94025
 B.W.  Goodwin
 Atomic  Energy of
  Canada Ltd.
 Hhitesheil  Nuclear Research
   Establishment
 Pinawa, Manitooa  ROE  1LO
 Canada


 J.W.  BalI
 U.S.  Geological Survey,
   MS/21
 345 Middiefieid Road
 Menlo Park, CA  94025
i J.W. Sal I
(U.S. Geological  Survey
\   MS/21
i 343 Middle* ieid Road
t Men I o Park , CA  94025
                                     Kodel Description
                                                                         Kodel
                                                                      Processes
                            A program to compute aqueous equilibria! mass  balance
                             or up to 20 metals end 30  I i genus in a  reoov >-o»cTi
                             ystem.  includes 46 elements, 94        cce-.z i e-- •*•' io°
                            aqueous species, 2 geses, and  i i
                            minerels/soi ids.
                                             A program to compute ecuii ibriurn dis-
                                             tributions of species concentrations In
                                             aqueous systems.  Standard version
                                             includes 53 elements, 109 aqueous
                                             species, 2 gases, and 27 mixed solids
                             A  program  for computing  the equilibrium
                             distribution of species  in aqueous
                             solution.   Includes  26 elements,  162
                             aqueous  species,  and 158 minerals.
                             An interactive chemical  speciat ion
                             program that  calculates  equilibrium
                             distributions for inorgenic  aqueous
                             species often round in groundwatcr,  a
                             FORTRAN version of  SCl.MNEQ.   Induces
                             28 elements,  239 equeous species,  and
                             IB) solids.
                                                                            Base-
                                                                            model
                                                                                      mass b£.i ance of
                                                                                       each e lenient
                                                                                      reoox
                                                                                              mass bslsr.ce of
                                                                                               C-
                                                                                               D !
                                                                                                    n i ur aficct.
                                                                                                    rei-jv ior,.*
                             A chemical  equilibrium model  for       I ness Diisr.ce
                             calculating aqueous speciat ion of  major) redox  reactions
                             and minor elements among naturally
                             occurring ligands.
                             The WATEQ2 n«odel with the addition 01
                             uraniua species.
      !_.<•>. Plureaer
      B.F". Jones
      A.C. Truesdeil
j L.N. Pluramer
(U.S. Geological Survey
 Water Resources Division
 12201 Sunrise Valley Drive
 Reston, VA  22092
                             A program to modei the thermocynamic
                             spec!at ion of inorganic  ions and com-
                             plex spec Ios in solution for a given
                             water analysis.  A FORTRAN version of
                             the original WATEO (1973)  in PL/1.

-------
IQA/V>C
international ground water modeling center
             Price List of Publications and

            Services Available from IGWMC


                         July 1987
       Holcomb Research Institute      •                TNO-DGV Institute
         Butler University                       ol Applied Qeoscience
        lnd>*napolu. Indiana 46208                      PO Bo. 285 260O AO 0*m
            USA                            Th* Netnwtand*

-------
                                   Contents


                                                                        Page

1.  General Ordering Information	1



2.  IGWMC Publi cat i ons	2



3.  IGWMC Groundwater Modeling Software

    3.1  FORTRAN Programs	6
    3.2  BASIC Programs	13
    3.3  Hewlett-Packard HP-41C Programs	18
4.   IGWMC's Groundwater Model  Information Retrieval System
       (MARS and  PLUTO  databases)	21
GMSBOOl

-------
1.    General Ordering  Information
     ORDERS  ARE FILLED  AS ITEMS  ARE AVAILABLE.   CUSTOMER WILL BE
INVOICED  AFTER   THE  ORDER  IS  COMPLETED.     REMITTANCE   IN   U.S.
DOLLARS SHOULD BE FORWARDED AFTER  RECEIPT OF  INVOICE.

     Postage  and  handling fees  are  not  included  in  prices.    Postage and
handling fees will be  based on  actual  costs.

     Shipment abroad will  be  by surface  rate  unless  otherwise requested.


All  listed prices are  in U.S. dollars.
Prices are subject to  change  without  notice.
Allow 2 to 3 weeks for preparation of software  shipments.

     IGWMC updates  this  price  list monthly.  Please contact us  for the latest
listing.

-------
      2.  IGHMC Publications

                                                                              Unit Price
1983                                                                          	

      GWMI 83-02/2    Walton, w.C.  Handbook of Analytical Ground Water
                      Model Codes for Radio Shack TRS-80 Pocket Computer
                      and Texas Instruments TI-59 Hand-held Programmable
                      Calculator.  Notes:  Short Course April 11-15, 1983.        10.00

      GWMI 83-09      El-Kadi, A.I.  Modeling Infiltration for Water
                      Systems.  HR1 Paper No. 21.                                15.00

      GWMI 83-10      El-Kadi, A.I. and P.K.M. van der Heijde.  A review
                      of Infiltration models:  identification and
                      evaluation.  Paper 83-2506, Winter Meeting Am.
                      Soc. of Agric. Eng., December 13-16, 1983,
                      Chicago, Illinois.  Reprint.                               free

      GWMI 83-11      van der Heijde, P.K.M. and P. Srinivasan.  Aspects
                      of the Use of Graphic Techniques in Ground Water
                      Modeling.  Proc. UCOWR Annual Meeting, July 24-27,
                      1983, Columbus, Ohio.  Reprint.                            free
1984

      GWMI 84-06      Walton, W.C.  Handbook of Analytical Ground Water
                      Models.  Notes:  Short Course April 9-13, 1984.            60.00

      GWM] 84-10      El-Kadi, A.I.  Modeling Variability in Groundwater
                      Flow.  HRI Paper No. 31, June 1984.                         8.50

      GWM] 84-12      El-Kadi, A.I.  Automated Estimation of the
                      Parameters of Soil Hydraulic Properties.                    3.50

      GWMI 84-13      Huyakorn, P.S. et al.  Testing and Validation of
                      Models for Simulating Solute Transport in Ground-
                      Water:  Development, Evaluation, and Comparison of
                      Benchmark Techniques.  The International Ground
                      Water Modeling Center.                                     25.00

      GWMI 84-14      van der Heijde, P.K.M.  Availability and Applica-
                      bility of Numerical Models for Ground Water Resources
                      Management.  Presented at the NWWA/IGWMC conference,
                      August 15-17. 1984, Columbus, Ohio.  Reprint.              free

      GWMI 84-15      Srinivasan, P.  PIG-A Graphic Interactive Pre-
                      processor for Ground Water Models.  Presented at
                      the NWWA/IGWMC conference, August 15-17, 1984,
                      Columbus, Ohio.  Reprint.                                  free

      GWMI 84-17      El-Kadi, A.I.  Stochastic Versus Deterministic
                      Modeling of Ground Water Flow.  Presented at
                      the NWWA/IGWMC conference. August 15-17, 1984.
                      Columbus, Ohio.  Reprint.                                  free

-------
1985
      IGVHC Publications (continued)
                                                                             Unit Price
      GUMI  85-01      Beljin, M.S.  Selected Bibliography on Solute
                      Transport Processes in Groundwater, January  1985.           7.50

      GWMI  85-06      van der Heijde, P.K.M.  Utilization of Numerical
                      Models in Groundwater.  Presented at the ASCE
                      Computer Applications in Water Resources Con-
                      ference in Buffalo, New York, June 10-12, 1985.             free

      GWMI  85-07      van der Heijde, P.K.M., P.S. Huyakorn and J.W.
                      Mercer.  Testing and Validation of Ground Water
                      Models.  Presented at the NWWA/IGWMC Conference,
                      "Practical Applications of Ground Water Models,"
                      August 19-20, 1985, Columbus, Ohio.                        free

      GWMI  85-08      van der Heijde, P.K.M.  Groundwater Contamination
                      Following a Nuclear Exchange.  Report of the
                      SCOPE/ENUWAR Workshop in Delft, The Netherlands,
                      October 3-5, 1984.                                         free

      GWMI  85-12      van der Heijde, P.K.M. and M.S. Beljin.  Listing  of
                      Heat Transport Models which are Documented and
                      Available.                                                 2.50

      GWMI  85-16      van der Heijde, P.K.M. and M.S. Beljin.  Listing  of
                      Models to Simulate Location and Movement of Fresh
                      Water-Salt Water Interfaces in Groundwater.
                      August 1984.                                               2.50

      GWMI  85-17      van der Heijde, P.K.M.  Review of DYNFLOW and DYNTRACK
                      Groundwater Simulation Computer Codes.  Report  findings
                      for the U.S. EPA, Washington, DC.                          free

      GWMI  85-28      van der Heijde, P.K.M.  Modeling Contaminant Transport
                      in Groundwater.  Presented at the 1985 Washington
                      Conference on Groundwater Protection and Cleanup,
                      November  12-13, 1985, Arlington, Virginia.                 free

      GWMI  85-29      van der Heijde, P.K.M.,  Spatial and Temporal Scales
                      in Groundwater Modeling.  Presented at the SCOPE/
                      INTECOL/ICSU workshop, "Spatial and Temporal Varia-
                      bility of Biospheric  and Geosheric Processes,"   October
                      27-November  1, 1985,  St. Petersburg, Florida.              free

      GWMI  85-30      Beljin, M.S.  Listing of Microcomputer Graphic  Software
                      for Groundwater Industry.  November 1985.                  free

      GWMI  85-31      Beljin, M.S.  Analytical Modeling of Solute Transport.
                      Presented at the NWWA/IGWMC Conference, "Practical
                      Applications of Ground Water Models," August 19-20,
                      1985,  Columbus, Ohio.                                      free

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1S36
19S7
      IGHHC Publications (continued)
                                                                              Unit Price

      GWMI 85-32      Huyakorn, P.S., P.P. Andersen, and H.O.  White,  Jr.,
                      and P.V.M. van der Heijde.  Testing and  Application
                      of a Finite-Element Groundwater Flow and Transport
                      Model.  Presented at the International  Syposium on
                      Management of Hazardous Chemical Waste  Sites,
                      Winston-Salem, October 9-10, 1985.                         free
      GWMI 86-04      Rice, R.E.  The Fundamentals of Geochemical Equilibrium
                      Models; with a Listing of Hydrochemical Models that are
                      Documented and Available.  December 1986.                   3.50

      GWMI 86-05      van der Heijde, P.K.M.  Listing of Review Publi-
                      cations and Textbooks in Groundwater Modeling.             free

      GWMI 86-06      Beljin, M.S.  Microcomputers in the Analysis of
                      Pump Test Data.  Presented at the Southeastern
                      Ground Water Symposium, October 30-31, 1986,
                      Orlando, Florida.                                          free

      GWMI 86-07      van der Heijde, P.K.M. and M.S. Beljin.  Selected
                      References on Programs for Hand-held Calculators.           2.50

      GWMI 86-08      van der Heijde, P.K.M. and M.S. Beljin.  Available
                      Solute Transport Models for Groundwater  and Soil
                      Water Quality Management.  August 1986.                     2.50

      GWMI 86-13      El-Kadi, A.I. and L. Smith.  Stochastic  and Geo-
                      statistical Analysis for Groundwatr Modeling:
                      Part II.  Notes:  Short Course 15-19,  1986.                35.00
      GWMI 87-01       van  der  Heijde,  P.K.M  and  A.I. El-Kadi.
                       Short  Course  Notes:  Basics of Groundwater Modeling.
                       March  18-20,  1987                                          75.00

      GWMI 87-02       Mercer,  J.W.,  P.F. Andersen,  and  L.  Konikow.
                       Applied  Groundwater Modeling.  Notes:  Short  Course
                       March  23-27.  1987.                                         75.00

      GWMI 87-03       van  der  Heijde,  P.K.M.  and P.  Srinivasan.  Summary
                       Listing  of  Groundwater Models for Mainframe  and
                       Minicomputers (MARS data base).                            25.00

      GWMI 87-04       van  der  Heijde,  P.K.M.  and P.  Srinivasan.  Selected
                       Summary  Listing  of Available  and  Documented  Ground-
                       water  Models  for Mainframe and Minicomputers
                       (MARS  data  base).                                          20.00

-------
IGfcfriC Publications (continued)
                                                                        Unit Price

GWMI 87-05      van der Heijde, P.K.M. and P. Srinivasan.  Listing
                of Available Groundwater Models for Microcomputers
                (PLUTO data base).                                         20.00

GWMI 87-06      Beljin, M.S.  Representation of Individual Wells in
                Two-dimensional Groundwater Modeling.  Presented at
                the NWWA/IGWMC Conference,  "Solving Groundwater Pro-
                blems with Models,"  February 10-12, 1987, Denver,
                Colorado.                                                  free

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3.    IGWMC Groundwater Modeling Software
     3.1  FORTRAN Coaputer Programs for Mainframe and MicrocoBputers

     The  IGWMC  distributes  a  rapidly  increasing  number  of FORTRAN  programs
     related to groundwater modeling.   All  FORTRAN  codes  are  implemented  on a
     DEC  VAX  11/780.   Unless specified  differently, the  programs are  also
     available for IBM PC/XT/AT microcomputers and compatibles.   Copies of the
     software are provided  on  a  magnetic tape in  user-specified  formats.   PC
     versions are provided  on  5.25" or 3.5" diskettes  (includes  source  code,
     executable  version  and sample data).    Unless indicated otherwise,  the
     programs are public domain.

     IGWMC  software  comes  complete with  documentation,  including  pertinent
     reports, user's instructions, program listing, and example  problems.

     Support Policy

     The  Center  provides  limited  support  for  the software it  distributes.
     Assistance  in  implementation  is  available  in  the  form  of written  or
     telephone  response  to user's  questions.   For some models,  IGWMC refers
     the  inquiring  party to model  developer(s)  for optional  code acquisition
     and/or support.  Software support provided by the Center pertains only to
     programs obtained directly from the Center.

     The  user of  IGWMC software accepts and uses the program material as it is
     at the user's own risk, relying solely upon his/her own inspection of the
     program  material   and without  reliance  upon  any  representation  or
     description  concerning  the  program  material.    Neither   HRI  nor  its
     individual  staff  members  make  any  expressed or implied warranty  of any
     kind with regard to the program material.  Therefore, neither HRI nor its
     individual  staff  members  shall be  liable for any damages  in connection
     with the furnishing, use, or performance of the program material.

     The  Center  maintains a list  of its software  purchasers.   Users receive
     notification of updates  regarding  distributed programs.   For users who
     obtained the  program from the Center, updated versions of  the software
     are available at cost.

     System  Requirements:  IBM PC  or compatibles with  640K,  Math Coprocessor
     8087/80287 and  printer.

     A  list of  currently available FORTRAN  programs is given on  the following
     pages.

-------
INTERNATIONAL
  GROUND  WATER
    MODELING  CENTER
 FORTRAN  MAINFRAME AND MICROCOMPUTER SOFTWARE  AVAILABLE  FROM IGHfcC
 HoicomD Research institute   Butler  University   Indianapolis, Indiana 46208  USA  Tel:3!7/283-9458
 TNO-DGv institute of  Applied Geoscience   P.O. Box 285, 2600 AG Dei ft  The Netherlands  TeI:15/569330
         AUTHORS
    NAME
(Version
       Date)
GWMC
KEY
PURPOSE
REMARKS
PRICE 1  ORDER
  s   !     j?
i  C. Su,
i  R.H. Brooks
   i. Javandei
i   C. Doughty,
j   C.F. Tsang
   M.S. Beij i n
     FP
 (1.0 11/85)
                                 AGU-10
                                               6170
                                  ITIRD
                               (1.0 02/85)
                 6310
                                  OOAST
                                (1.0 02/85)
                                  RES SO
                                (1.0 02/85)
                 6312
                 3940
                                    RT
                                (1.0  02/85)
                                               6313
      ART
  (1.0 10/86)
                                                6383
        To  determine  the  parameters  of
        the   retention   function   (the
        soil-water characteristic  func-
        tion) from experimental  data
         A  semi-ana Iyticai   solution  to
         radial   dispersion   in  porous
         media,  calculating   the  dimen-
         sioniess   concentration   of   8
         particular    solute,   injected
         into  an  aquifer, as a function
         of  dimensioniess time  for dif-
         ferent  values of dimensionless
         radi us.
         An  analytical  solution for one-
         dimensional   solute   transport
         including   convection,   disper-
         sion,  decay,  and adsorption   in
         porous media.


         A  semi-ana Iytical   model  for
         calculation  of  streamlines lo-
         cation   of   contaminant   fronts
         ana calculation  of  concentra-
         tion  of  Sinks  through  simula-
         tion  of  two-dimensiona<,  advec-
         tive  contaminant transport »;th
         adsorption   in  a  homogeneous,
         isotropic  confined  aquifer «itn
         steady-state  regional  fio».


         This   program  converts  a time
         series  of   concentration data
         from   one  or  more  oDservation
         •ells  into a spatial  concentra-
         tion  distribution  in  the aqui-
         fer at various times.   The mod-
         el  can  t>e  used  for cases when
         regional  flow can  be  neglected
         and  a  single   production well
         creates a  radial  flow field  in
         the aqui fer.
         A  pre-  ano  post-processor  for
         RT.    This  is  an  interactive
         program for  inputting  ne»  data
         and  storing  tnem   in  a  file,
         editing  enistmg   data.     ART
         also can be used to graphically
         display results  of  RT.
                                                           Single  pack-
                                                           age  wi th
                                                           6310,  6312,
                                                           3940,  6313.
                                                           and  6383
                                                                                                        70
                                                                          120
                                                                                                               FOSCl
                                                                                 FOS05
                                                           Preprocessor
                                                           and    Post-
                                                           processor
                                                            i nciuded
                                                            (see ART)
                       IBM PC
                       graphic
                       boaro (CGA)
                       requ i rea

-------
WTRAN MAINFRAME AND MICROCOMPUTER SOFTWARE  AVAILABLE  FROM IGkMC  (continued)
AUTHORS
J.V. Tracy





D.C. Kent,
L. LeMaster,
j. Wagner





L.F. Koniko«,
J.D. Bredehoe't














M.G. McDonaio,
A.M. Harbaugn





J.C. Parker,
M.Tn.van Genuchten



D.L . Parkhurs-t ,
L.N. Plummer,
O.C. Thorstenson

J.N. Plummer,
B.F. Jones,
A.M. Truesdei I
T.A. Prickett.
C.G. Lonnquist






NAME
(Version
Date)
MOCNRC
(1.0 03/85)




MOCNftCM
(1.0 12/86)






MOC
(2.3 04/87)














MOOfLOw
(1.1 05/86)





CXTFlT
(1.0 05/85)



BALANCE
(1.0 05/86)


WAllN/WATCOf
(1.0 08/86)

PLASM
(1.1 06/86)






IGWMC
KEY
074DM





_







0740















3980






3432




3400



3620


0322







PURPOSE
A modified version of the
original Kon i*O«/Bredehoe< 1 MOC
model to include radioactive
decay ana adsorption (Linear,
Langmuir arc Freundich
isotherm) .
This is a modi tied MOCNRC mode i
that can simulate water-table
aquifer conditions. in addi-
tion the model has option to
simulate only the head distri-
bution, ana to create an output
for use Kith SAS graphics pro-
grams.
A model to simulate transient.
two-dimensional, horizontal
groundwater flew and solute
transport in confined aquifers
using the method of character-
istics and including flow simu-
lating subroutines. The model
has option to simulate radio-
active decay and linear adsorp-
tion. The user can specify up
to 16 particles per ceil. Too
versions of the model ere in-
cluded m the package: one
using interetive ADI, and one
using SlP numerical solution
techn i que.
A Modular finite-difference
groundnater model to simulate
two-dimensional and quasi- or
f ull y-three-dimensionai , tran-
sient fio« in anisotropic, het-
erogeneous, layered aquifer
systems.
An inverse model to determine
values for tne one-dimensionai
analytical solute transport
parameters using a non linear
least-squares method.
A chemical equilibrium mode i
for calculation of the mass
transfer along a fiov path and
including redo« reactions.
A program for chemical equili-
brium calculations and speoa-
t ion including redo* reactions.
A finite difference model to
simulate t>o-dimensionat tran-
sient, saturated flow m an
anisotropic, heterogeneous sin-
gle- or nuiti- layered aquifer
system with water table and/or
confined or leaky confined con-
ditions.
REMARKS
Single
package «itr-
MOCNRC and
MOCNRC*


Preprocessor
i nc i uded
Part of
FOS06




Preprocessor
PREMOC
(3.1 04/67)
included




























Preprocessor
included

Preprocessor
avei labte
for
mainframe
version



PRICE
$
150













200















120






70




50



120


95







ORDER
*
rosoe













FOS07















FOS08





j
I
FOS09




FOS10



FOSH


FOS12








-------
FORTRAN MAINFRAME  AND  MICROCOMPUTER SOFTWARE AVAILABLE FROM IGWMC (continued)
AUTHORS
T. A. Pr i CKett ,
T.G. Naymik,
C.G. Lonnquist



i

P.C. TrescotT,
S.P. Larson,
L.J. Torak


P.S. Trescott,
G.F. Pmder,
S.P. Larson



M.Th. van Gefucnten





M.Tn, van Genucnten




M.Th. van GenuchTen,
w.J. Aives




M. Vauc 1 ' n
(modi f ied by
A.I. El -Kadi )






G.T. ren,
D.S. Ward







NAME
(Version
Date)
RANDOM WALK
( 1 .0 1 1/85)






USGS-3D-FLOW
(1.0 1982)



USGS-2D-FLOW
(1.0 1976)




SOHYP
( 1 .0 04/86)




UNSAT1
(1.0 07/85)



ONE-D
(1.0 07/85)




INF IL
( 1 .0 06/83)







FEMWASTE-1
(1.0 1981)







GWMC
KEY
2690







0770




0771





6226





3431




6220/
24




3570








3371








PURPOSE
A model TO Simulate one- or
tuo-d linens i ona i steady or un-
steady soiute transport pro-
blems m heterogeneous aquifers
under «ater table or confined
conditions based on the random
•alK method and including flow
simulating Subroutines.
A finite difference model to
simulate transient, quasi- and
fully three-dimensional, satu-
rated fio« in anisotropic, het-
erogeneous groundwater systems.
A finite difference model to
simulate transient, two-dimen-
sional horizontal or vertical
flow in an anisotropic and het-
erogeneous confined, leaky-con-
f i ned or water-table aquifer.
An analytical model for calcu-
lation of the unsaturated hy-
draulic conductivity function
using the soil moisture reten-
tion data via Muaiem or the
Burdine theories.
A fininte element model to sim-
ulate one-d imensionai satu-
rated-unsaturated flow in het-
erogeneous SOilS,

A package of 5 analytical solu-
tions to the one-dimensional
convect i ve-d i spers i ve transport
equation with linear adsorp-
tion, /ero-order production.
and first order decay.
To solve a one-dimensional in-
filtration into a deep homogen-
eous soil using finite differ-
ences; output includes water
content profile and amount and
rate of infiltration at differ-
ent simulation times. Soil
properties need to be expressed
in mathematical form.
A two-dimensional finite ele-
ment model for transient simu-
lation of areal or cross-sec-
tionai transport of dissolved
constituents for a given velo-
city field m an anisotropic.
heterogeneous porous medium.
The velocity field is generated
by the FEMWATER-l code.
REMARKS
reprocessor
vai i able
or
ma i nf rame
vers ion



Ma i n frame
version only



Ms i n frame
version only






























Ms i n f r ame
vers ion on 1 y







RICE
$
95







95




95





70





70




70





70








95








ORDER
1
FOS13



i



FOS15

1


FOS16





FOS17





FOS',8



t
FOS19





FOS20








FOS21









-------
FORTRAN MAINFRAME AND MICROCOMPUTER SOFTWARE AVAILABLE FROM IGWC (continued)
AUTHORS
G.T. Yen.
D.S. Ward




A.I. £ i-Kadi







A.I. El -Kadi








J.B. Kooi ,
J.C. Per kef.
M.Th.van Genuchten






C.i. voss









S . A . W i 1 1 i ams ,
A. 1 . El -kadi









M.Th. van Genuchten







NAME
(Version
Date)
FEMWATER-l
(1.0 1961 )




STAT1
(1.0 06/85)






ST2D
(1.0 1965)







ONE STEP
(1.1 10/65)







SUTRA
(1.0 1985)








COVAR
(1.0 01/86)









SUMATRA- 1
(1.0 02/86)






IGWMC
KEY
3370





6333







4160








3433








3630









6334










3430







PURPOSE
A two-dimens iona i finite ele-
ment model to simulate tran-
sient, cross-sect iona : flow m
saturated-unsaturstefl an i so-
tropic, heterogeneous porous
wedia.
A program for Das i c statistical
analysis of data. Various mo-
ments of the statistical dis-
tribution are estimated: the
mean, median, standard devia-
tion, coefficient of variation,
variance, standard error, maxi-
mum, minimum, and range.
This program is for the sto-
chastic analysis of gravity
drainage via the Monte-Car 10
technique. it consists of
three sections: a generator for
hydraulic conductivity realisa-
tions, a finite-element simu-
lation program, and a statis-
tical analysis routine.
This program will estimate
parameters in the van Genuctiten
soil hydraulic property model
from measurements of cumulative
out Ho* with time during one-
step experiments. The program
combines a nonlinear optimiza-
tion routine with a Gaierkin
finite element model.
A Two-dimens ionai model to sim-
ulate density dependent fluid
movement under saturated or
unsaturated conditions and
transport of either energy or
dissovied substances in a sub-
surface environment employing a
hybrid finite-element and
integrated-f in i te-di f ference
method.
A program for generating two-
dimensional fields of Butocor-
reiated parameters which are
log-normally distributed (e.g..
hydraulic conductivity). The
program uses a technique based
on matrix decomposition. the
generated parameter field rep-
resents a major requirement in
the stochastic analysis via
Monte-Carlo techniques.
A one-dimensional finite-
element model to simulate the
simultaneous movement of water
and solutes in saturated-unsa-
turated and non-homogeneous
soil. The effects of linear
adsorption and zero- and first-
order decay are included.
REMARKS
Me i n f r ame
vers ion on i y












Ma i n frame
version only
















Mai nf rame
version only



























PRICE
$
95





70







120








70








120









50










70







ORDER
#
i
FOS2? i





FOS24







FOS25








FOS26








FOS27


|

i




FOS29










FOS30


1




                                              10

-------
FORTRAN  MAINFRAME AND MICROCOMPUTER SOFTWARE AVAILABLE  FROM IGWMC (continued)
AUTHORS
P.R. ScProeaer, et ai.













NAME
(Version
Date)
HELP
(1.0 01/87)












IGWMC
KEY














PURPOSE
Hydroiogic Evaluation of
Landfill Performance program
for estimation of surface
runoff, subsurface drainage.
and leacnate that may be
expected from operation of a
variety of possible landfill
designs. The program models
the effects of precipitation,
surface storage, runoff.
infiltration, percolation.
evapotranspi rat ion , soil
moisture storage, and lateral
drai nage.
REMARKS














PRICE
$
120













ORDER
i
FOS32





I







   P.S. HuyaKorn, et ai
  TRAFRAP
(1.0 03/86)
                                                0589
    M.A.  Butt,
    C.D.  McE  *ee
    M.Tn.  van Genucnten
    VARQ
 (1.0 04/86)
                                                 6082
   CFIT IM
 (1.011/85)
    W.F. Sar.tord,
    L.F. KoniKCw
  MOCDENSE
 (1 .0 01/87)
                                                 6227
                                                 0742
A  two-dimensional   finite  ele-
ment code  for  simulating  fluid
flow  and   transport  of  radio-
nuclides  in  fractured and  un-
fractured  porous  media.    The
code can  be  used  to model  both
groundwater  flow   and  solute
transport,  or  either  process
separately.  The  TRAFRAP  model
accounts  for  1)  fluid interac-
tions between the fractures and
porous matrix blocks; 2)  advec-
11 ve-dispersive   transport    in
the  fractures  and diffusion  in
the  porous  matrix   blocks  and
fracture  skin; and  3)  chain re-
actions of  radionuciide compon-
ents.    A  major   advantage   of
TRAFRAP   is  the  capability   to
model  the fractured sustem us-
 ing  either  the  dual-porosity  or
the  discrete-fracture modeling
approach  or  a combination.


A program to calculate aquifer
parameters   by    automatically
 fitting   pump   test   data  and
Tneis  type curve.   The program
 allows  variable  discharge  rate
 during  the  test.


 A  program  for   estimation   of
 non-equi Iibrium   solute   trans-
 port  parameters  from  miscibie
 displacement experiments.    De-
 pending upon the e«act form  of
 the transport  model,  the  pro-
 gram allows up  to  five  differ-
 ent parameters  to be estimated.


 A  numerical  model   to simulate
 solute transport  and dispersion
 of  either   one  or   two  consti-
 tuents   in   groundwater   where
 there  is  two-dimensional,  den-
 sity-dependent flow.  The  mode!
  is  a   modified  version  of   the
 Konikow  and  Bredehoeft   Model
 HOC (1978),  which  uses  finite-
 difference   methods  and    the
 method  of  characteristics   to
 solve  the  flow  and  transport
 equations.
                                                           11
Ma i nf rame
vers ion on Iy
                                                                                                          250
                                                                                                           50
                                                                                 FOS33
                                                                                                                  FOS34
                                                                                                           70
                                                                                                           120
                                                                                                                   FOS35
                                                                                                                   FOS36

-------
ORTRAN MAINFRAME AND MICROCOMPUTER SOFTWARE AVAILABLE FROM IGtMC (continued)
AUTHORS
G.T. Ter











NAME
(Version
Date)
AT123D
(1 .0 6/B7)










I6WMC
KEY
6120











PURPOSE
An analytical solution 
-------
IGMMC Grcxindwater Modeling Software - (continued)


     3.2  BASIC Programs for Microconputers

     BASIC programs for IBM PC/XT/AT microcomputers and compatibles are avail-
     able from IGWMC.  The programs come with a documentation and include code
     listing and example oroblems.  A copy of the  BASIC program is  provided on
     5V  diskette   (includes  source  code,  executable  version,  and  document
     file).

     System Requirements:   IBM PC  or compatibles  with  128K  and printer.  Some
     programs require IBM compatible graphic board (or HERCULES graphic board)
     and HP7475A plotter.

     For more information contact  IGWMC.

     For ordering  information see  page 1.
                                       13

-------
INTERNATIONAL
  GROUND  WATER
     MODELING CENTER
BASIC MICROCOMPUTER  PROGRAMS AVAILABLE FROM IGUMC
•  Ho i COITID Research  institute  But ier uni vers i ty  Indianapolis,  Indiana 46206 USA  Tel :3l7/283-9458
-  TNO-DGV  institute of Applied Geoscience  P.O. Box ?85. 2600 AG Delft The Netherlands  Tel:15/569330

AUTHORS

i
j P.K.M. van der Heijde
i
i
i
(


i
i P.K.M. van der Heijde,
j P. Srinivasan
I



P.K.M. van der Heijde,
P. Sr i n i vassn



P.K.M. van der Heijde














P.K.M. van der Heijde




P.K.M. van der Heijde






A.I. Ei -Kadi





NAME
(Version
Date)

PLUME
(1.0 10/63)





PLASM
(4.0 01/66)



RANDOM WALK
(3.3 06/86)



THWELLS
(2.0 02/87)













THElSFIT
(1.0 09/83)



6WFLOW2
(2.0 04/87)





INF IL
(1.0 12/83)





IGWMC
KEY

6020






6010



6011




6022














60BO




6023






6335






PURPOSE


An analytical Model to calcu-
late three-dimensional concen-
tration distribution in a homo-
geneous aquifer with a contin-
uous solute injection in a one-
dimensional flow field.

A finite difference model for
simulating two-dimensional
transient saturated flow in
conf ined aquifers.

To simulate one- or two- dimen-
sional, steady or unsteady flow
and transport problems in homo-
geneous aquifers under confined
condi t ions.
An analytical model to calcu-
late drawdown or buildup in
non-steady groundwater flow in
an isotropic homogeneous non-
leaky confined aquifer with
multiple pumping and injection
wells. Boundary effects can be
included through use of image
•ells. Results are displayed
in tabular form, time-drawdown
curves, and contour plots. The
program has options to read
from and write to an external
file. Metric or English units
can be used.
To calculate aquifer parameters
by automatically fitting type
curve and pump test data from
pumping an isotropic homogen-
eous non leaky confined aquifer.
A menu-driven series of 7 rou-
tines each containing an analy-
tical solution to a groundwater
flow problem. Results are
displayed in tabular form or
time-drawdown curves. Metric
or English units can be used.
To calculate infiltration rate
and amount and water content
profile at different times us-
ing the Philip series solution
of a one-dimensional form of
the Richards equation.

REMARKS









A teaching
tool . For
rea 1 -wor I d
mode I i ng see
*FOS12
A teaching
tool . For
ree i -wor i d
mode i i ng see
JFOS13
IBM PC
graphic
board
required
















IBM PC
graphic
board
required



IBM PC
VAX 1 1 /780
MICRO VAX




PRICE
S

35






35



35




50














35




50






35






ORDER
#

BAS01






BAS02



BAS03




BAS04














BAS05




BAS06






BAS07





                                                    14

-------
3ASIC HICROCOMPUTER  PROGRAMS  AVAILABLE FROM IGkMC  (continued)
         AUTHORS
  P.K.M.  van  aer
  W.C.  Walton
  E.G.  McCa'I.Jr.,
  D.D.  Lane
  K.S.  Rathod,
  K.R.  RuShton
  L .A.  Abr!Oi a,
  G.F .  Pinoer
  A.  Verruijt
  A.I.  El-Kadi
    NAME
(Version
       Date)
                                 PLUME2D
                               (1.2  01/86)
                                35  Micro-
                                computer
                                programs
                               (1.1  03/85)
  PESTRUN
 (1.1 01/85)
  RADFLOW
 (1.0 09/84)
    TETRA
 (1.1 09/85)
 BASIC GWF
 (1.1 01/87)
                                  SOIL
                               (1.0  04/85)
IGWMC
 KEY
                6024
                6350
6280
6064
6430
6030
                6330
           PURPOSE
                         An  analytical  model   to  calcu-
                         late  the   tracer  concentration
                         distribution   in  a  homogeneous,
                         nonieaKy contained  aquifer  wirr,
                         uniform  regional   flow.     The
                         program uses  the  we I  I-function
                         for  solute  advection  and  dis-
                         persion in  a System  witn  con-
                         tinuously  injecting,  full,  pen-
                         etrating  wells,    it  includes
                         options   for   retardation   and
                         radioactive decay.
                         A series of analytical  and sim-
                         ple numerical  programs  to anal-
                         yze  flow  and  transport  of  so-
                         lutes  and  heat  in  confined,
                         leaKy confined,  and water table
                         aquifers with  simple geometry.
         A simple pesticide runoff  model
         to appro»imate runoff  values  to
         identify watersheds which  need
         attention  to  evaluate  effects
         of different conservation  prac-
         tices.
A  finite  difference model  for
transient  radiai  flow towards  a
•en   in a  homogeneous,  isotro-
pic aquifer.   The  model  allows
for switching  from  confined  to
unconfmed  conditions when  wa-
ter  levels   are  drawn  beneath
top of  aquifer.    Program  in-
cludes restart capabilities  for
varying pumping schedules.


A  simple  program  to  calculate
velocity  components  in   tn^ee
dimensions  from  hydraulic  head
measurements.   Groups  of  four
observation    points  are   con-
nected  to   form  tetrahedrons,
and a  linear  interpolation  is
used  to  calculate  head   gra-
dients  for  each  tetrahedron.
Application  of Darcy's  Law then
yields velocity components.


Analysis of   plane,  steady  or
unsteady groundwater flow  in  an
isotropic,  heterogeneous,  con-
fined  or unconfined  aquifer  by
the finite  element  method.


To estimate soil  hydraulic pro-
perties  using  a   non-linear
least-square   analysis;   major
input   to  the  code  includes
pairs  of measured water  content
and suction.
                                                        15
                                                                                          REMARKS
                                           IBM PC
                                           TRS-80/lIi
                                           APPLE Me
                                               PRICE
ORDER
                                                          35      BAS06
                                                 70
                                                 35
                                                          35
                                                        BAS09   i
                                                        BAS10
                                                                 BASI i
                                                          35      8AS12
                                                                                         Shareware:
                                                                                         price   does
                                                                                         not  i ncIude
                                                                                         contr i but ion
                                                                                         to author
                                                           IBM PC
                                                           VAX ]t/780
                                                           MICROVAX
                                                          10
                                                          35
                                                                 BASI 3
                                                                 BAS14

-------
BASIC  MICROCOMPUTER  PROGRAMS  AVAILABLE  FROM IGldKC  (continued)
i
i AUTHORS

M.S. Be ' j i r


NAME
(Version
Date)
SOLUTE
(1.0 01/85)

i
i




P.K.*. van der Heijde

based on FORTRAN program
by C.D. McElwee




* . i> . Be < j i n

















TSSLEAK
(1.2 09/85)






PUMPTEST
(1.0 06/86)


















K.R. B'-£.dfc.jrv, TGtlESS

IGWMC
KEY
6380




PURPOSE

A program pacKege of 8 analy-
tical mode i s tor solute tran-
sport in groundoBter, a metric-
to-English unit conversion pro-
j gram, ana e subroutine TO cei-
I cuiete trror functions. The
t'S-er-i r ienci y , menu-driven pro-
! grs(T!£, CG1K: with optional screen

6081







6382

















cr.s printer graphics.
This program fits the HentuSh
and Jacobs equation to experi-
ments! pump test data to obtain
the "best" values (or storage
cocf < 
-------
BASIC MICROCOMPUTER  PROGRAMS AVAILABLE FROM IGUMC  (continued)
         AUTHORS
  D.B. Thompson
   NAME
(Version
       Date)
  TlMELAG
 (1.0 05/87)
IGWMC
 KEY
                                            6580
PURPOSE
        A program  to estimate hydraulic
        conductivity    from    time-lag
        tests for  most well  configura-
        tions.    The  method  involves
        instantaneously   raising    and
        lowering the  water  level  in a
        we I  I  (Hvorsiev 1951).
                      ubi i shed
                      .round Wa<
                     237777 —
REMARKS
                                                                                             i n
                                                                                             er
PRICE
                                                                                                 10
ORDER
                                                                                                       BAS22
                                                      17

-------
ISMMC Sroundwater Modeling Software -  (continued)


3.3  Hewlett-Packard HP-41C

     Documented  programs  for  Hewlett-Packard  HP-41C   are  availatle  from
IGWMC.   The  programs  come  with  complete  documentation  and  include  code
listing.    Prerecorded magnetic cards  for each program  are  also available.
IGWMC differentiates between  four  product types.

                                                             Unit  Price

     IGWMC Standard Program Documentation including
     Program Listing (paper copy)                               $ 5.00


     IGWMC Standard Program Documentation and pre-
     recorded magnetic cards                                    $10.00


     Special Reports (Documentation and prerecorded
     magnetic cards), e.g., AQTST                              $25.00


     HP-41C Program Package (documentation and prerecorded
     magnetic cards of 11 selected programs)                   $55.00
For ordering information see page 1.
                                       18

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INTERNATIONAL
 GROUND WATER
   MODELING CENTER
          HEWLETT-PACKARD HP-41C PROGRAMS AVAILABLE FROM  IGWMC
 Hciccx-.D Research Institute Butler University  Indianapolis, Indiana  46208  USA  Tel: 317/263-9458
 'NO-DGv institute o' Applied Groscience P.O. Box 285, 2600 AG Delft  The Netherlands  Tel: 15/569330
         IGWHC-Key     TITLE

         HPS-01
Hantush "Well-Function"  in  the  Pocket Calculator
                                                      Program Name
         HPS-02


         HPS-03


         HPS-04


         HPS-05


         HPS-06


         HPS-07


         HPS-08


         HPS-09

         UPS-10


         HP5-11


         HPS-12
Theis Condition Well  Field                            NWELLS2£


One-Dimensional Non-Steady  Ground  Water Flow         EDELMAN'}:
Steady Radial Ground  Water  Flow in a Finite          ISLE1}:
  Leaky Aquifer

Streamlines and Traveltimes for Regional  Ground      FLOP-21}:
  Water Flow Affected by  Sources and Sinks

Advection and Dispersion  from a Stream with          STRDISP?*
  Regional Flow

Advection and Dispersion  from a Solute Injection     RADDISP2^
  Well

Analysis of Various  Flow  in a Single Aquifer         AQMODL"}:
  Including Leakance  Problems and Recharge

Evaluating Theis  Parameters from a Pumping Test

Inverse Solutions of  the  Theis Equation
  I - Single Parameter Calculation

Evaluation of Well Characteristics from Step-        FASTEP5*
  Drawdown Test Data

Economically Optimal  Well Discharge Rate             QOPTIM**
          •English  version by IGWMC
          -Original  program by IGWMC
          ^Modified  by IGWMC
          "Original  version
          ^Documented  In:  Helweg, O.J.  et  al.  (1983),  Improving Uell Pump  Efficiency.
          American  water  works  ASSOC.,   6666  West  Quincy  Ave.,  Denver,  CO  80235.
          Phone:    303/  794-7711.    ONLY  PRE-RECORDED  MAGNETIC  CARDS  AVAILABLE  FROM
          IGWMC.
          *Nonstandard pricing
          ^Included  in 11-program package
                                                19

-------
Hewlett-Packard HP-41C Programs (continued)

IGWMC-Key     TITLE	

HPS-13
HPS-14

HPS-15


HPS-21

HPS-22


HPS-23


HPS-24


HPS-25


HPS-26


HPS-27



HPS-28



HPS-29

HPS-30


HPS-31


HPA-32
Benefit-Cost Analysis for Replacement or
  Rehabilitation of Pump

Metric-English Units Conversions

Inverse Solutions of the Theis Equation
  II - Aquifer hydraulic constants

Aquifer Test Analysis with a Hand-held Calculator

Two-Dimensional Flow to a Horizontal Drain in a
  Confined Aquifer

Ground Water Unit Step and Rectangular Pulse
  Response

Parameter Analysis and Drawdown Calculation
  for Anisotropic Confined Aquifers

An  Idealized Ground Water Flow and Chemical
  Transport Model

Simulation of Well Pumping and Recovery in a
  Confined or Unconfined Aquifer

Calculating Drawdown for a Single Well in a
  Confined/Unconfined Aquifer Bounded by
  Two Parallel Impermeable Boundaries

Two-dimensional Pollution Plume in a Homogeneous
  Aquifer with a Uniform Horizontal Flow Field
  Including Dispersion and Retardation

List of Groundwater Reserves

Solute Transport of a Contaminant from
  Multiple Point Sources

A Program to Calculate Aquifer Transmissivity
  from Specific-Capacity Data

A Program to Calculate Mounding due to
Asymmetric Recharge
Program Name

PRA*


MECONV3
AQTST"*

DRAIN'


PULSE 3


ANSTPY3


S-PATHS"
PLUME 2D



LGWRES*

WMPLUME


SPCAP-



INVHAN"
'English version by  IGWMC              ^Original program by IGWMC
^Modified by IGWMC                     -Original version
^Documented in:   Helweg, O.J. et al.  (1983),  Improving  well  pump efficiency.
 Am. Water Works Assoc., only pre-recorded magnetic cards available from IGWMC
*2HPS 26 and 27 are  documented in a single standard priced note
*Nonstandard pricing
^Included in 11-program package
                                      20

-------
4.   IGWHC's Groundwater Model Information Retrieval System

     The IGWMC's data bases,  MARS  and  PLUTO,  are  designed  to  facilitate rapid
     accessibility  to  information  on  groundwater  models  for  mainframe  and
     microcomputers,  respectively.    Each  model  is  described  by  a  set  of
     annotations   of   its   operating  characteristics,   capabilities   and
     availability.   An  extensive  checklist,  the  model
     developed  to  describe  each   model  as  completely
     possible.    This  list  is  used   by   IGWMC  staff
     information in one of the databases.
                                                annotation  form,  is
                                               and  consistently  as
                                               to  enter  the  model
     For  retrieval  of specific model  information  from the databases  a  model
     annotation retrieval form is filled out by requestor or at the Center.  A
     computer search  is then executed by IGWMC staff, identifying those models
     which are suited for requestor's problem.  Information on these models is
     printed  either  1n  summary  form  or  as  a  listing  of  the  complete
     annotations.   Before sending it to the  requestor  the search results are
     evaluated by the Center's technical staff.

     For  the  rapidly expanding category of microcomputer  software,  IGWMC has
     recently developed  the PLUTO database.   PLUTO  and MARS  are based on the
     same concepts,  but  the presentation  of model  information differs in that
     PLUTO   has   more   emphasis   on  software  compatibility  and  hardware
     specifications.
     For both  databases  the  following  services are available.

     Search  and  Retrieval:
     MARS
      PLUTO
1.   Selected Summary  listing  (GWMI 87-04)

2.   Summary listing of all stored models  (GWMI 87-03)

3.   Complete annotation without search
        one annotation
        each additional annotation

4.   Executing  search
        each selected  complete annotation
        each selected  summary, per 20 annotations


"5.   Summary listing of available models  (GWMI 87-05)

6.   Executing  search
        each selected  annotation
Price


20.00

25.00
                                                                        5.00
                                                                        1.00

                                                                       15.00
                                                                        1.00
                                                                        1.00
                                                                       20.00

                                                                       15.00
                                                                         .50
      Special  selections from the databases are possible.   Contact  IGWMC.
 For ordering information see page 1.
                                       21

-------

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                                                                 November 1986


                U.S. EPA GROUND-WATER MODELING POLICY STUDY GROUP
                       Report  of  Findings  and Discussion of
                      Selected Ground-water Modeling Issues
                                        by

                             Paul  K.K.  van  der Heijde

                                       and

                                 Richard A. Park

                    International  Ground Water Modeling  Center
                            Holcomb Research Institute
                                Butler University
                           Indianapolis, Indiana  46208
                                Project Officers:

                                 Joseph F. Keely
                                  Clint W. Hall
                                  Scott R. Yates

                        Office of Research and Development,
                   R.S. Kerr Environmental Research Laboratory,
                               Ada,  Oklahoma  74820
                          This study was conducted under
                          Cooperative  Agreement  CR-812603
                  with the U.S. Environmental Protection Agency,
                   R.S. Kerr Environmental Research Laboratory,
                               Ada, Oklahoma  74820
INTERNATIONAL   GROUND   WATER   MODELING   CENTER

    Holcomb Research Institute, Butler University, Indianapolis, Indiana  46208

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                                   CONTEKTS

1.  Executive Summary	1

    Introduction	1
    Issues	1
    Code Selection and Acceptance	2
    Review and Procurement of Modeling Studies	2
    Research Needs	3
    Information Exchange	4
    Staff	5
         Recruitment and Retention	5
         Training	5
         Workl oad	5

2. The U.S. EPA Ground-water Modeling Policy Study Group	7

    Introduction	7
    Definition of Terms	7
    Responsibilities and Objectives	8
    Authority	8
    Reporting	8

3. Role  of  Ground-water Models  in U.S. EPA	10

    Mathematical  Ground-water Models	10
    Ground-water  Models  in  U.S.  EPA.....	11
    Site-specific Modeling	12
    Generic Modeling	13
    Development of  Regulations  and Policies	13
    Permitting	14
    Remedial  Action	15
    Ground-water  Modeling  in Program Offices	16
         Office of  Drinking Water	16
         Office of  Health  and Environmental  Assessment	16
         Office of  Pesticide Programs	16
         Office of  Policy,  Planning  and  Evaluation	18
         Office of  Toxic Substances	18
         Office of  Solid Waste	18
         Office of  Waste Programs  Enforcement	18

4.  Model ing Concerns	20

     Program Office  Concerns	20
          Office of  Drinking Water	20
          Office of  Pesticide Programs	20
          Office of  Policy, Planning and Evaluation	21
          Office of  Toxic Substances	21
          Office of  Waste Programs Enforcement	22
          Office of  Research and Development	23
               Office of Health and Environmental Assessment	23
               Environmental Research Laboratory, Athens, Georgia	23
     Concerns of Regional Offices	23
          Model Use in Regional Activities	24
          Regional Staff	25
          Quality Assurance	26


                                      iii

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         Procurement of Modeling  Studies	27
         Technology Transfer and  Training	27
         Facilities and Resources	28
         Legal Concerns	29
         Summary of Regional Concerns	29

5.  Discussion of Issues	30

    Adequacy of Modeling Theory and Data	30
    Ground-water Code Review and  Testing	33
         Model Evaluation	34
              Model Review	34
              Model Examination	34
              Evaluation of Documentation	35
              Evaluation Ease of  Use	35
              Computer Code Inspect i on	35
         Model Verification	36
         Model Val idation	36
              Validation Scenarios	38
              Sensitivity Analysis	38
    Proprietary Codes  versus
    Public Domain  Codes and Acceptance Criteria	39
         Banning the Use of Proprietary Codes	39
         Continuing the Use of Proprietary Codes	40
         Options for U.S. EPA	41
    Quality  and Usefulness  of Model  Studies	43
         Code Selection	43
    Quality  Assurance  in Ground-water Modeling Studies	46
         Definition and Role of Quality Assurance
         in  Ground-water Modeling	....46
         Current EPA Quality Assurance Policies	47
         EPA Quality Assurance Options for Ground-water Modeling	49
              Model Development	50
              Model Application	51
    Technology Transfer and Training to Sustain and
    Improve  Expertise  of Agency Personnel	52
         Technology Transfer and  Training 1n EPA	53
    Information Exchange on Ground-water Modeling	54
         Training	56
         Recruitment and Retention	57
References	59
Appendix:  Composition of Study Group	62
                                      iv

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                                   SECTION  1

                               EXECUTIVE  SUGARY
     Jn lute  1985,  the  Office of Environmental  Processes and Effects Research
of  the  U.S.  EPA invited  the International  Ground Water Modeling  Center  at
Hoicrmb Research  Institute  to coordinate and lead & Study Group, charged  with
examining issues related to U.S. EPA use of  ground-water models and associated
      p•;nts.   This  task has  been pursued  through meetings,  thorough fact-
      i".  documented  in  a  series   of   interim  reports,  end  a  final   report
    crising the  findings of the  Study Group and presenting the most prominent
     Vie  Study  Group included representatives  of  various FPA Program  Offices
£••-•:.    "erne!  ground-water  modeling  experts,  who  met  with  technical   and
!••;.••  -  "V! staff of  the  Program  Offices  end  selected  Regions.  Interim  reports
c. v.    vJ  findings  were circulated to  a  group of corresponding members  from
h-,-••;•,• M» and  Regional  Offices for their comment


     Ground-water   models  are   mathematical   tools   to  aid   in   organizing
information  pertinent   to  complex  ground-water  systems,  and  in  evaluating
*.lver;iative  options  for efficient mangement of ground-water resources.   It is
within such  a decision  support framework,  pertinent to EPA's mission,  that the
SJ.;K>y Group  meetings were held and modeling  related issues  explored.
       ::-.:  S'..uoy Group has  examined  the Agency':- UL-SS fif  proum-wete?  f'ov end
       ';.:nt   transport  models,  and  its  f.ssccisteri ne-^s  UK.  cap^m Vit i?s.
       ict'lly,  the use of  models in  regulatory decision  niL'or.t; I>..Q.,  Sinning),
      . v ing,  and enforcement actions was discussed  in the.  conie.A  of  pctennal
           s  and the subsequent need for Agency policies.
      Mathematical  models  are often  efficient means  for EPA  to develop  its
 ground-water protection programs.   Currently available models may  be  used to
 test   Hypotheses  about  site-specific   and   generic  problems  and  to  assist
 analysis  of  alternative  courses of action  in  solving  ground-water  protection
 p"Of-":tT",i.   Models are  also used  in research to develop a fuller understanding
 of  the physical,  chemical,  and biological  processes  that  affect ground-water
 Quolvy.    The  latter  use   is  aimed  at  developing  models which  accurately
 represent  complex situations  such  as those  involving  immiscible fluids, dense
 plumes, or fractured rock aquifers.


      The   role  of ground-water-flow  and  contaminant-transport  models  in  the
 development  of  policies and  regulations, and  in  permitting and  in planning
 monitoring  and  remedial  action,   is   continuing  to   grow  within  the   EPA.

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However, the  Study  Group found that this growth  does  not  seem to occur  in  a
structured  and  coordinated  manner.   At  present,  no criteria  or  policies
provide Agency-wide guidance in the use of models  for  regulatory  planning and
decision-making purposes.  EPA's ground-water modeling  needs currently seem to
outpace its actual use of models in virtually all  program areas.


     Areas  where  official  policy statements on aspects of  ground-water model
use may be advisable  include the  use of proprietary  models (as opposed to
public  domain software) for  enforcement  support  work, minimal  documentation
and quality control procedures for  the development and application of models,
and  possibly  the  promotion of  "standard"  models.    Such   policy  statements
should  be  consistent  with  the  policies  adopted  for  surface  water  and air
modeling.   Suggestions and recommendations  are organized under  the  following
five major  topics:   code selection and acceptance,  review  and procurement of
modeling studies, research needs. Information exchange, and staff.


CODE SELECTION AND ACCEPTANCE

     In  recent years both development and use of ground-water models in  studies
performed  by  or for  the  U.S.   EPA   have  become the  focus  of  professional
criticism,  public discussion, and even adversary legal procedures.  Therefore,
determining  code  reliability,  establishing  code  acceptance criteria, and
providing  guidance  in  model  selection have become increasingly important.   In
establishing  an Agency mechanism for such guidance,  proper  attention  should  be
given  to definition of study objectives,  determination of modeling scenarios,
system  conceptualization, and formulation of selection criteria.


     The  reliability  of codes  should be  established by  adopting  a widely
accepted  review  and testing  procedure. Agency acceptance of  a model  should  be
based  on  technical  and scientific soundness, user friendliness, and  legal and
administrative  considerations.   A  list  should  be  compiled  of  reviewed and
validated  computer  codes  acceptable  to   the  Agency,  and  the  Agency  should
advocate  the  use of such  codes.   Proprietary codes important to the Agency's
mission should, where  possible,  be  brought  into the  public  domain.


     The  Agency should  assess  the use of  "expert systems" for assistance  in
selection and use of  acceptable  models.   Such a  system should be oriented  to
solving problems  rather  than  identifying  systems  and  processes.    Options
include a  meeting  of  specialists  to detail  courses  of action,  and  a  pilot
study  to  explore  the  potential  of this new technology.


REVIEW AND PROCUREMENT OF  MODELING STUDIES

     One  of  the major  issues  emerging  from  the  Study Group concerned  the
quality of ground-water modeling  studies  carried out by or  for  the  U.S. EPA,
and the  usefulness  of the study  results  in  the Agency's  decision-making
process.

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     Agency  aec.Hior'c-   'houlc   be  based  on  the  use   of   technically   and
sclent 1 "v:.- ; :/    ic.j-:    cvtr   rol "lection,    information   processing,    and
iriterpre-;.-;,-;-  ,-.-.v:--,     t.v-v n   ;.;-.  overall  Agency  quality assurance  (QA)
prccrcn-.  •_;  ^-r,;^.-.--.:   i-.-;;  r^c:.;nrctional  framework  for  QA in  ground-water
mode line shc..V_ _•:-  -. •-  <-.- '•;:.-. M'."
     •'•'il  pT-ject;  ::'ict  Involve  modeling  should  have  an  adequate  QA  plan
defined.   Sucn c; pi»n  should  specify goals for the  quality of  resulting data
and  processed  Inform;:nor,  acceptable  to  the user;  should  contain  detailed
desc<-ipaions  o*"  xne  t^ns/jres  tc  be  taken  to  achieve  prescribed  quality
objectives;  -r^d sho^'d  cs^igr, responsibility  for achieving  the goals.   The
pi en  shoulcj  also cor.uin  procedures for documenting the activities  within a
project  in  order  to  er.t&r":ish  an  administrative  record  supporting  Agency
decision making,   EPA shoilc ri?.v= effective review  and auditing procedures in
place  ID  monitor QA  psrforminr.e  of  the modeling  project  teams.   More than in
the  pest,  attention  shojld  b£  given  to  applying  such quality  assessment
procedures  dur->.a  c.  pvojt.cz,  LHG  not just  at the end  of  it.   The Study Group
CDnsi^c-s  -.'.   ••.r:.i'>~:-\;:~'i   t.'i:;  :  iTK?re  direct  mechanism  for  reviewing  and
directing  the  t-r-rk  cf outsics  contractors be established.  Current procurement
proccrti^ss  lirs.'t  t-t  "f-'":.ci ivcness of EPA  project managers,  especially in the
Re q-ions.
RESEARCH  NEEDS

      Appropriate  models  do  not  yet  exist  for  all   types  of  ground-water
problems   because   many  of   the   assumptions   and   simplifications   common  to
existing  models do  not allow faithful simulations  in  unusual  situations, and
because  the natural  processes  that affect fluid and  contaminant movement are
not  yet  fully understood. This is especially  true  for chemical and  biological
procer, ",£•>.
      lrno'-oveni?r,-:..7  VT;  u^::;cf  concurrently,  in several  major areas.    Data
 acquisition m:.Ui;•:-•: ,nc  '•r.terpretvv'e  models  are needed that can examine  to  an
 unprecedented  ckgre:-   :he  physical,  chemical,   and  biological   processes
 controlling   the   transport   and   fate   of   ground-water   contaminants.
 Unfortunately, few of  the constants  and  coefficients  needed  to  incorporate
 chemical  and  biological  processes  into  contaminant transport evaluations are
 available presently.


      Development is need for  simulation of flow  and transport  in fractured and
 dual-porosity media and  in multimedia.   Further, representation of  stochastic
 processes  in predictive  modeling,  and  incorporation of  economic factors  in
 modeling to  Improve estimation of  clean-up  costs,  should be studied.  Models
 are  needed for management of ground-water  contamination plumes,  as well  as
 risk assessment  end risk  management.   Special attention  should  be given  to
 research  that  includes  volatilization,  multiphase  flow,   density-dependent
 flow, and  immiscible flow  in  ground-water models.

      Fundamental   research  supporting  ground-water  modeling   is  considered
 necessary  in  such  areas  as

-------
         transient   behavior   of   process  parameters   (e.g.,   retardation,
         hydraulic  conductivity)

         desorption for nonhydrophobic  chemicals

         multicomponent transport  and  chemical  interaction

         transport  of silt  with sorbed  chemicals  in  aquifers

         improved numerical accuracy,  stability,  and efficiency


INFORMATION EXCHANGE

     In recent years modeling for  ground-water  protection has  become a  rapidly
growing  area  of technology.   As  a result,  information on technological  and
scientific  advances   has   become   increasingly   available  for   ground-water
management.     Disseminating  this   information  through  communication   and
education  is  the   goal  of  technology  transfer.    In  its  broadest  sense,
technology   transfer   includes  the   distribution   of   modeling   codes   and
documentation, and providing training  and assistence in  model  use.


     The  Study Group  found  that  many  improvements in  information exchange,
training, and  software distribution can be made within the Agency.


     The  U.S.  EPA  should  establish  a systematic technology  transfer  program
with ground-water modeling as an integral component.  Such a program should be
based  on an active approach  in  providing information and should  be flexible
enough to disseminate  research results quickly.  Furthermore,  a program should
be developed to train, on  a continuous basis,  agency personnel in ground-water
modeling  as  an integral part  of  their involvement  with  ground-water  quality
issues.   Such  a training  program  should  incorporate  recent scientific  and
technological  advances and provide opportunity to share practical experience.


     Both  information exchange and  training  should reach each  staff member
involved  in  ground-water  projects.    For  ad  hoc  consultation on specific
problems, project managers should have access  to experts such as (1) in-house
Regional  experts,   (2)  experts  within ORD, perhaps located at  EPA labs,  (3)
contractors,  or (4) experts  from other agencies.   In  addition, a  networking
mechanism needs to  be  developed to promote increased communication  and sharing
of experiences  among staff of Regional Offices and  Program Offices.


     Results   from  relevant  research  projects   have  not  been disseminated
effectively  to Regions  and  some  Program Offices.   Reports  on ground-water
modeling  should be distributed  to a targeted  mailing  list that  is  updated
frequently.    Many  publications  are   in  the   open   literature,  and provision
should  be made for  distributing  these as reprints.  Each  division or branch
Involved  in  ground-water  modeling should  have  its own  working  library of
pertinent publications.

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STAFF

Recruitment and Retention

     The  difficulty  in  attracting  and  retaining  a   skilled  staff  with  a
background  in ground-water  model application,  and the  high turnover  rates
among  staff,   are  serious  problems   in  all  EPA  Offices.    The  Study  Group
suggests three steps to alleviate this situation:

         The  Office  of  Human Resources should establish the  position  of "hy-
         drogeologist."

    •    A  career  path for  hydrogeologists should  be  established  through the
         GS-15 level.

         The  Agency  should encourage staff to take short courses and graduate-
         level courses  in ground-water geology  and modeling  conditionally at
         the  Agency's expense.

Training

     Training of  Agency personnel in the  effective use of ground-water models
continues  to  be a  problem.   Many  staff members with  degrees in engineering and
environmental  science,  need  additional  basic   training  in  ground-water
geology.   Physical geology and ground-water geology should be offered through
the  EPA Training  Institute  and  other institutions,  and  should  be  required
prior  to taking a  ground-water modeling course.


     The training  of EPA staff in ground-water modeling should be aimed  mainly
at  providing  skills  needed  to evaluate the  effectiveness  of model codes and
modeling  work, since  only  a  few of  the  staff  will  be (or  should  be) in  a
position   to   become modeling  experts.    Training should  be  based  on the
realities   of   needs,    staff  backgrounds,   administrative   structures  and
constraints,  and  potential changes in staff.


      Management  should  be  sensitive  to  the financial  and time  requirements
necessary  for adequate training.  (One does  not become a  competent modeler  by
completing a one-week short course or training  program.)

Workload

      The workload problems  identified at  the  Study Group meetings  in Regions  I
 and  III   clearly  limit  the  scientifically sound  use  of   the   analytical,
 planning,  and design methods  provided  by  efficient modeling.   Furthermore,  to
 incorporate  efficient  modeling   into  projects,  project  managers should  be
 sensitized to the potential  and proper role of models in their work.   Staff
 with  training and experience in ground-water modeling  should be  available  in
 each Region.


      Because of  the time and  effort  required to  characterize  a  ground-water
 system before a suitable remedy  can be selected, management needs to recognize

-------
that it often  takes  a significant amount of  time  to  properly  perform studies
in which modeling is an integral part.


     The Study Group finds it extremely important,  especially under conditions
of  severe  time and  budget constraints, that models be  used  at an early stage
of the project in order to optimize data collection, data analysis, and design
of management  alternatives.

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                                   SECTION  2

            THE U.S. EPA GROUND-WATER MODELING POLICY STUDY GROUP


INTRODUCTION

    During 1984 and  1985,  a series of issues related to the selection and use
of ground-water  models within the  U.S.  Environmental  Protection Agency  (EPA)
was  brought  forward by  EPA staff  in  several  Program Offices  and  Regional
Offices.  These  issues included  assessment of the validity of computer codes,
criteria  for  selection  of appropriate models for  specific  applications,  and
review  procedures for  determining  the  applicability  and validity  of models
used by  third  parties.  Another  important issue brought forward was the need
for quality assurance  in modeling  projects carried out by or for the Agency.


     The  Office  of  Environmental  Processes  and Effects  Research  of EPA/ORD
(OEPER)  and  its Robert  S.  Kerr  Environmental Research Laboratory (RSKERL) in
Ada, Oklahoma,  held wide-ranging discussions on these issues to determine the
best way to  resolve them.   In early 1985, a  consensus  led to the formation of
a  Study  Group  charged  with examining issues  related  to  EPA use of ground-water
models  and associated  needs and constraints.   Specifically,  the Study Group
was  to  conduct a  thorough fact-finding, documented  in a  series  of working
papers  and  summarized  in  a report  intended to  inform EPA managers  of the
issues  and their  significance for various offices and programs.  The working
papers  might  later be  modified  and adopted as official  guidance  documents.


     Accordingly, the  Office  of Environmental Processes and Effects  Research
asked   the  International  Ground  Water  Modeling  Center  at  Holcomb  Research
 Institute,  Indianapolis,  to coordinate  the  activities of the Study  Group  and
 invite  other Program  Offices  and model  users in Regional Offices to partici-
pate in the Study Group activities.  Formal  requests for designated  participa-
tion went out  to the  Assistant  Administrators  for Solid Waste  and  Emergency
 Response, for  Water,  for  Pesticides and Toxic Substances,  and for  Policy,
 Planning and Evaluation.  Several offices responded and named representatives;
 the  persons  participating  in   the  Study   Group  activities  are  listed  in
 Appendix 1.


 DEFINITION OF TERMS

      The general  objectives of ground-water  management can be characterized as
 the  optimal  and  efficient utilization  of  ground-water  resources and  the
 protection of  those resources for  sustained  and future  utilization.   Because
 most of the modeling-related ground-water management  issues  important  to EPA
 pertain to specific  programs  as  administered  by the  various Offices,  the
 discussions  were  to  be  focused  on  those   elements  which  are or  should  be
 included in an Agency-wide  approach to  modeling.


      For  the  purposes  of  this  study,  ground-water  models  are  defined  as
 restricted  to the mathematical  framework describing a ground-water  system and

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its inherent processes and stresses, and the subsequent implementation of  that
mathematical framework in a computer code.   Physical  models  and  screening and
ranking models  (e.g.,  DRASTIC)  are  not  included.   Traditionally,  ground-water
was understood  to encompass the saturated  zone  of  the subsurface.   Because of
the  close   relationship  between transport  and fate  of  contaminants in  the
unsaturated and  saturated zones  of  the  subsurface,  and their equal  importance
in  addressing   ground-water  protection,  the  ground-water  modeling  issues
discussed in this report relate to both unsaturated and saturated zones.


RESPONSIBILITIES AND OBJECTIVES

The  Study  Group was  to review  problems  and issues  relating  to ground-water
modeling practices  in  EPA Program Offices and Regions, to evaluate model  needs
and uses in existing  programs,  and  to  present approaches and provide guidance
to  improve model   use and  solve   the  problems  identified.   The  Group was
responsible for conducting  thorough fact-finding  (including  site  visits to
Agency  Offices  in  Region I,  III,  and  X, and a meeting  with staff of Program
Offices  at  EPA Headquarters),  and  for  summarizing its findings in a  series of
 interim  reports and  a final  report containing selected working  or position
papers.


AUTHORITY

      Established by  OEPER/RSKERL,   the Study  Group  has operated  under the
auspices of the International Ground Water  Modeling Center at Holcomb Research
 Institute,  which   is  responsible   for   delivering  a  report  on  the Group's
activities  and  findings.   An  IGWMC staff member  (van  der  Heijde)   served as
Chairperson.    The  Study  Group's authority  to  produce working papers for the
EPA  was specifically  limited to fact-finding, reporting,  and suggesting new
policies  or changes in current Agency  policies,  and  in  providing guidance to
Agency  staff.


REPORTING

     The  Study  Group  met  five  times,  initially  at  the  Holcomb  Research
 Institute,  Indianapolis, Indiana,  and  subsequently in EPA  offices  in Boston,
Philadelphia,  Seattle, and Washington, D.C.  A report  of the  findings of each
meeting  was prepared  and  distributed to Study Group members.  The  present text
 is the final  report and  includes the major elements  of the individual meeting
 reports.

     The report is divided  into four  parts:   (1)  the  role of  ground-water
models in  U.S. EPA;    (2)  concerns of  various offices  and  individuals  within
 the EPA with  respect  to a variety  of modeling issues;  (3)  discussion of  major
modeling issues, including  adequacy of modeling  theory and data  reliability,
 and  acceptance of ground-water simulation  codes,  quality and usefulness  of
 model studies,  and technology  transfer  and training  of EPA staff, and   (4)
 various ways  to resolve  current  porblems and  improve  the qualified use  of
 models for decision-making procedures  within the Agency.    The third section
 contains the  four issue papers prepared  initially  by the Study Group.   The
 report starts with an executive summary.


                                       8

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    Before  submitting  the   final   report   to  the   EPA,   participating   and
corresponding members  were  asked  to  comment  on  a  draft  version.    Comments
received  have  led to  a rearrangement of  topics and an expansion of  polio-
related discussions.  All Study Group documents  and written conmunicat ions  are
filed at the Holcomb Research  Institute.

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                                  SECTION 3


                   ROLE OF GROUND-WATER MODELS IN U.S. EPA
MATHEMATICAL GROUND-WATER MODELS

     The analysis of ground-water flow and contaminant transport  cannot yet be
thought of  as  exact science.   Although the physical processes  involved  obey
known  mathematical  and physical  principles,  precise aquifer and  contaminant
character!2ation  is  hard to  obtain  and often  makes even plume definition  a
difficult  task.    However,  where these  characteristics  have been  reasonably
established, ground-water models may provide a viable, if not the only, method
to  predict  contaminant  transport,  locate  areas  of potential  environmental
risk,  and assess possible remediation/corrective actions.


     Mathematical  models  are used to  help  organize the essential  details of
complex  ground-water  management  problems   so  that reliable  solutions  are
obtained.    Applications  include  a  wide range of  technical,  economic,  and
sociopolitical  aspects of ground-water  supply and  quality  (Holcomb Research
Institute  1976;  Bachmat  et  al.  1978;  Mercer  and  Faust 1981; U.S.  Office of
Technology  Assessment  1982; Javandel et al. 1984; van der Heijde et al. 1985).


     Existing  models can be  categorized  by their technical  uses,  as follows
(Bachmat   et al.   1978;   van   der    Heijde   et al.  1985):   (1)   parameter
identification models, (2)  predictive models, (3) resource management models,
and  (4) data manipulation codes.


     Parameter  Identification  models  are  most  often  used to estimate the
aquifer coefficients for  fluid  flow  and contaminant  transport characteristics,
such as annual  recharge  (Puri 1984), coefficients of permeability and  storage
(Shelton  1982;  Khan  1986a.  1986b),   and  dispersivity  (Guven et  al.   1984;
Strecker  and Chu  1986).   Predictive models are the  most numerous because  they
are the primary tools  for testing hypotheses  (Andersen et al. 1984;  Mercer and
Faust  1981;  Krabbenhoft  and  Anderson 1986).


     Resource  management  models   are  combinations  of  predictive  models,
constraining  functions   (e.g.,  total  pumpage  allowed),   and  optimization
routines  for objective functions (e.g., optimization of well-field  operations
for minimum cost  or minimum drawdown/pumping lift).  Very few of these are  so
well  developed and  fully supported  that they  may  be considered practicable,
and  there does not  appear  to be an extensive effort to improve the situation
(van der  Heijde  1984a, 1984b; van der Heijde  et al.  1985).


     Data  manipulation   codes  also  have   received  little  attention  until
recently.    They  are now becoming  increasingly popular because they  simplify
Input  preparation  (as "preprocessors") for  Increasingly  complex models,  and
because    they   facilitate   the   production   of   graphic   displays    (as


                                       10

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"postprocessors") of  the model outputs  (van  der  Heijde  and Srinivasan  1983;
Srinivasan  1984;  Moses  and  Herman  1986).    Other  software  packages  are
available  for  routine  and  advanced  statistics,   specialized  graphics,  and
database management needs (Brown 1986).


GROUND-WATER MODELS IN U.S.  EPA

     A policy of resource protection based on monitoring  is by its very nature
always reactive, not preventive; however, model-based policies and regulations
can be  both  preventive and  reactive.   Because  adequate  on-site monitoring is
not always feasible  due to  costs,  available  manpower, or  site accessibility,
models can provide a viable  and effective alternative.  An optimal approach to
the  management  of  ground-water   resources   includes  the  integrated  use  of
modeling and monitoring  strategies.


     Mathematical  models  can  be  helpful  to  EPA  in managing  ground-water
protection  programs.    Currently  available models  may be used  to  test hypo-
theses  about  site-specific  and  generic problems,  and  to  develop a  fuller
understanding  of the physical, chemical, and biological  processes that  affect
ground-water  quality.   The  former  use  is  self-evident,  but the latter use is
also quite important  because many  improvements  are  necessary before models can
accurately  represent  complex  situations such  as  those  involving  immiscible
fluids,  dense  plumes,  or fractured  rock  aquifers.


     Few aspects of  the Agency's  ground-water protection programs can  function
efficiently  without the  use of mathematical models.   Any activity requiring
some  estimate  of  ground-water flow or  contaminant transport, including  data
gathering  and  interpretation,  can benefit  from the  judicious use of  ground-
water models.


     Some of the principal  areas where  mathematical models can  now be used to
assist  in the management of  EPA's ground-water  protection  programs  are:

          development of regulations and policies

          planning and  design  of  corrective  actions  and waste  storage  facil-
          ities

     •     problem conceptualization and analysis

          development of guidance documents

          design and evaluation of monitoring and data collection strategies

          enforcement

      Specifically, ground-water modeling plays or could play a role in:

          determining  or evaluating the need for  regulation  of specific waste
          disposal,  agricultural,  and  industrial practices


                                       11

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        analyzing  policy  impacts  such   as  evaluating  the consequences  of
        setting  regulatory  standards and  banning  rules,  and  of  delisting
        actions

        assessing  exposure,  hazard, damage, and health risks

        evaluating  reliability,  technical  feasibility  and  effectiveness,
        cost,  operation and maintenance, and other  aspects of waste-disposal
        facility  designs  and of alternative remedial actions

        providing  guidance  in siting  of  new facilities and  in  permit  issuance
        and petitioning

    •   detecting  pollutant sources

        developing aquifer  or well-head  protection zones

         assessing  liabilities  such  as  post-closure  liability  for  disposal
         sites

     These  activities  can be broadly  categorized  as either site-specific  or
generic modeling efforts, and these categories can  be further  subdivided  into
point-source  or  nonpoint-source  problems.    The  success  of  these  modeling
efforts  depends  on  the  accuracy  and  efficiency  with  which  the  natural
processes  controlling the  behavior  of  ground  water,  and the  chemical  and
biological species it transports, are  simulated.   The  accuracy and  efficiency
of  the simulations,  in  turn,  depend  heavily  on  the  applicability of  the
assumptions  and  simplifications  adopted  in the model(s),  and  on  subjective
judgments made by the modeler and management.


SITE-SPECIFIC MODELING

     Whether  for  permit  issuance,   investigation  of  potential  problems,  or
remediation  of  proven contamination,  site-specific models are  necessary  for
the  Agency  to  fulfill  its  mandate  under a  number  of major  environmental
statutes.   The National Environmental  Policy Act of 1970 stipulates a need to
show  the   impact  of  major   construction activities  in   Environmental  Impact
Statements;  potential  impacts are often projected   successfully  by  the use of
mathematical models.


     Some  of  the  most  difficult  site-specific  problems  facing  the Agency
involve  hazardous  waste  sites  falling  under  the  purviews  of  RCRA  and
CERCLA/Superfund.   Associated with  most of these  sites  is a  complex array of
chemical  wastes  and  the  potential   for  ground-water   contamination.    The
hydrogeologic  settings  of   such  sites  usually  appear  quite  Intricate  when
examined  at  scales  appropriate  for  technical  assessments   and  remediation
efforts  (e.g.,  hundreds  to thousands  of  feet).    In   all  phases  of  these
analyses, ground-water models are useful  Instruments.
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GENERIC MODE LIK3

     In  a nuii,,>:v  :    . •. ..3.;~ce.^ yhere  the Agency  has  limited  data or  other
constraints.  s"7.~-- •-'"'-'-  racelinc;  is  not feasible.    As  a  result,  many
decisions  s^e '-::i.    :    ,M  assistance of  generic  models.    Such  models  are
more  often c.r-c. •: •...:.   „-.•;%  numeric^,  in  contrast to  site-specific  models.
This  is  a  loc--cv  --".-,•?•.:.-e-ics of  the  simplified  mathematics  of  analytical
models,  the  tic -  ^ca-.:.-  greater data  requirements  of  numerical  models,  and
the higher cos:-,  r.r  nuT-'-rical  simulations.


     The  Agency hti T^nv  statutory  responsibilities that benefit from generic
modeling,  includino  '.r :•  estimation  of  potential  environmental  exposures  and
their   integrr tic*-,  •-^••'•.r\   dose-response  models  to  yield   health-based  risk
assessments.    There   assessments  are  necessary,   for  example,   in  issuing
compound-specific rj"Mr-jS on  products  subject to preregi strati on requirements
under  the Toxic  Subs-&n:es Control  Act  (TSCA)  and  the Federal Insecticide,
Fungicide,  and Rodenti--;d?  Act (FIFRA).   More  generalized policy  formulation
activities   ol?o  : •'        ~-i^  generic  modelinc;  examples  include  policy
decisions  ",;.-,          _  :";:.ial   "bdnninj,"   setting  Alternate  Concentration
Limits,  prepsT-.rc  '•-LC--•-!;>. 1 En'orcement Guidance  Documents  (i.e., for moni-
toring  netwen; ±'.-.'•-::''  c^d "d~ list ing" under RCRA.


DEVELOPMENT  OF REPUL-.l  ;CKS AKD POLICIES

      Evaluation  cf   A.he   impacts   (economic, health-risk,  and  otherwise)   of
regulations   or  policy  scenarios  requires  process-oriented,  generic models.
Some  specific uses  o'  such models  in  the evaluation of proposed and  existing
policies and  -eoulatir.ns vithin the  U.S.  EPA include:

          testinc;  t1,?  :"fi:cacy of  standards  such as  meeting 10-6  health  risk
                           .c!;•;'• Q-JT.;:! or  detection  limits)

                *••->.:••  .f c: nc-r"'cition tillage through the use of  linked surface
          wdi: , ;.-..•.-...  :  .-23-2L.ne,  and  saturated-zone ground-water  models

          developing  g^idr.nce  for  well-setback   with  pesticide  applications,
          using uncertainty  analysis

          evaluating  the  seriousness of  various  "failures"  of  injection  wells
          through  the use  of sensitivity analyses

      Generic models  are  often  used to  provide a  technical  rationale for policy
 development, as  illustrated by the following examples:

          An  analytical  contaminant  transport model  coupled with  Monte  Carlo
          analysis has been used  to provide  the technical justification  for
          restricting   the  land  disposal   of   hazardous   wastes,   under   the
          Hazardous  and  Solid  Waste Amendments  of  1984  (HSWA  1984)  of  RCRA.
          The  hazardous  disposal  ban  decisions   are  based  on the results  of
          model  simulations  for  a  wide  range  of  site-specific hydrogeologic
          characteristics.
                                        13

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        A model has  been used for the analysis of potential failure scenarios
        of   waste  injection   in  deep   wells  in  four  different  regional
        hydrogeologic  settings.  The results of the study will be used to set
        policy  and develop regulations  under  HSWA 1984  (Section  3004  f and
        g).

        Long-tern fate of hazardous waste  injected  into deep saline ground-
        water  environments has  been  studied by  means  of a  hydrogeochemical
        simulation model.   The  results  will be used to  aid in setting policy
        and  siting criteria for the petition  process  of the hazardous waste
        Injection well ban under HSWA 1984.

        A  computer  model   is  planned  for  use   in  evaluating  the  need and
        effectiveness  of ground-water monitoring  programs for hazardous waste
        injection wells.   The  results  will be used  to help develop regula-
        tions under HSWA  1984   (Section  3004  f  and g)  for  the ground-water
        monitoring of  hazardous  waste injection wells.

     Sometimes,  models are  used as  an  integral part  of  EPA  policies and
regulations.   Such models  are often published  in  the Federal  Register as  part
of the rule-making  process.   Examples  are the delisting model used to delist
wastes, and the banning model  in the land disposal restriction rule.


     Models are being  used increasingly  to  implement policies and  regulations
pertinent  to hazardous  waste facilities,  such as to prove or disprove  a CERCLA
endangerment, and to determine clean-up  levels.


PERMITTING

     In discussing  the role  of  modeling  in  the permitting  process,  the  Study
Group  differentiated  between  permit   applications  having  a   site-specific
character,  and EPA  product permit review procedures where  the models  are used
generically for  screening  purposes.  These  different  types  of usage  require
different  types of models and expertise.


     Reliable  data  on actual   transport  and  fate of  chemicals  are  often
lacking,   especially   in   the  case  of  nonpoint-source   releases,   as  for
pesticides.   Under such constraints,  generic models are used to  evaluate the
potential  for  pollution and  contaminant migration.   The lack of  data  often
prevents the  use  of complex numerical  models, thus  forcing the  permit  writer
to  make  rigorous assumptions  regarding the  system under  study.   However,
permit  writers  may not   have  the expertise  to evaluate  such  model  usage
adequately.


     In the  permitting process  for hazardous waste  facilities,  ground-water
models can  be used on  a site-specific basis by owners/operators  of hazardous
waste  facilities,  to  show compliance with the  permit  requirements, and  by
regulatory  agencies   to  validate  the   information provided for  permitting
purposes.   The permitting agency could  be  the   EPA or  a corresponding   state
agency.    These  models can  be  used  to  evaluate  site characteristics,  to


                                      14

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deternin? the optimal  location  of  monitoring  wells,  to  estimate  the  transport
and ffite of contaminants, and to assess corrective action plans.
REMEDIAL ACTION

     Gr~!jnd-wa*er mode"1!  are  used increasingly  in the  CERCLA response  process
for  retis-Matior.  of hazardous  substances  releases.    The  current  state  of
ground-water  r;ode-1ing  practices  for  remedial  response  analyses  is  highly
variable  from  site to  site.    A  typical model  application for  Superfund-
financed  or  enforcement-related remedial  response  actions  includes  site
investigation  to assist  in  problem  definition  and  system conceptualization
(thereby guia:ng data  collection and data  analysis),  to identify the  contam-
ination source,  and to predict future contamination and health risks.   Models
are  also used for  development  and evaluation  of remedial  alternatives during
the  remedial  investigation/feasibility study (RI/FS) stages, and for analysis
of design  specifications  for  the chosen remedial action alternative.   The use
of  ground-water models  is fairly standard  for the design  of pump  and treat
types  of  remedial  alternate VPS:  however,  they are  not  widely used  for other
types  of remedial  alternatives.  Further, models are sometimes used to assess
required  clean-up  levels, the  extent  of  reauired  source removal,   and  the
projected  performance  characteristics  of  remedial  action  designs,  as  well as
to formulate  postoperation and closure  requirements.


     Models   contribute  to   justifying  the basis   for  Agency  action (i.e.,
exposure  analysis  as  part of public health risk assessment procedures or as
part of an  enforcement  endangerment  assessment).   Some  examples  of source
identification  with models are contained in the literature for Superfund sites
as  are several  examples where ground-water models are  being  used to assist  in
the  interpretation  of  monitoring data after the implementation of the  remedial
alternative.


      Discretion as to when to use a computer  code and which  code to  use  in  a
remediation   project  is  often   left   to   the   EPA   contractors  and/or   the
responsible  parties who perform the  RI/FS.


      Some  impediments to model  applications   in remediation analysis result
from  the  segmented  nature  of   the  overall  remedial  response  process,   with
different activities  being  conducted  in  discrete steps,  at  different times,
 and often with different contractors.   Thus,  during  a  specific step ground-
water modeling may not  be  implemented  due to  time  or cost constraints,  or
models may  be  selected for  only a  few of the potential  uses rather  than for
 multiple  uses.   Few, if  any,  comprehensive  ground-water  model  applications
 exist from the  start  to the  finish of  a site-remedial response.
                                       15

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GROUND-WATER MODELING IN PROGRAM OFFICES

Office of Drinking Water

     The Underground  Injection  Control  (UIC)  Program,  which  originated  in the
Safe Drinking Water Act  (SDWA)  (1974)  and  is  now subject to provisions  of the
Resource  Conservation   and  Recovery   Act   (1984   Amendments),   requires  an
evaluation  of  the potential for  excessive pressure build-up  and  contaminant
movement  out of  the  injection zone.    Mathematical  models  are the  primary
mechanism  for  the  required  evaluation, due  in  part to  the difficulty  of
installing monitoring wells several thousand feet deep.


     Because  of  the  character  of  the  injected  waste  and  because  most
underground waste injection takes place  in deep sedimentary basins, the models
and assumptions required for the  UIC program (for example, saline aquifers at
5,000-foot   depth)   differ  from  those   common   to  most   other   Offices.
Particularly,  UIC uses  complex models such as  the three-dimensional  finite-
difference  density-dependent  flow  and  transport simulators SWIPR  and  SWIFT,
which were  developed for saline waste  injection problems.


     Because  of the  chemical characteristics  of  the waste, interaction of the
injected  waste with  the  resident ground  water is  often of major  concern.
Geochemical  equilibrium  models  (such  as  MINTEQ,   WATEQF,  and  EQ3/EQ6) are
currently  being  used  by the  Agency   and  its  consultants  to  represent the
chemical  processes  occurring in  the subsurface  when injected  waste interacts
with the resident ground water.


     The regulations  also call  for determinations of which aquifers serve, or
could serve,  as underground sources of  drinking  water  (USDW),  based on  a  lower
quality  limit of  10,000  ppm total  dissolved  solids.  Here, modeling has  been
found to be a useful  adjunct to  gathering and interpreting field data,  as in
the U.S. Geological  Survey's  efforts  to assist EPA  in determining USDW (e.g.,
the Regional  Aquifer System Analysis  (RASA)  program).  Another USDW program,
for the  designation of  Sole Source Aquifers(SSA),  has frequently used models
for establishing  and managing water quality goals.   Designation  of the  Spokane
Valley-Rathdrum Prairie SSA,  for instance, included an evaluation of nonpoint
nitrate  sources with  a  ground-water model  developed for  EPA  by the USGS.

Office of Health  and  Environmental  Assessment

     The  Office  of  Health and Environmental  Assessment (OHEA)  is developing
guidance for  exposure and health  risk  assessments.   This guidance  will  be used
to  support  the various provisions  of the RCRA Amendments  and  CERCLA.   Current
focus  is on selection  criteria for ground-water fate and transport models to
be  used  in  exposure  assessments.

Office  of  Pesticide  Programs

     The primary  ground-water-related concern  in  the Office  of  Pesticide
Programs (OPP) 1s the assessment  of  pesticide  leaching and  contamination  of
underlying  aquifers  resulting  from normal use  of  registered pesticides,  and


                                       16

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evaluation  of new  pesticides  for  registration.   In  conjunction  with  data
received  from registrants as required by the  Federal  Insecticide,  Fungicide,
and Rodenticide Act  (FIFRA),  OPP  uses  models  to assess  the  leaching potential
of pesticides.   Past and  present efforts have  focused on  predicting  whether
various pesticides  are  likely to leach to ground  water  following  normal  use,
rather  than  on their spreading potential within aquifers.   This focus  is due
to the  large  areal, nonpoint-source  loading  aspect of  the OPP problem,  as
opposed  to   localized  point   sources  often  of  concern  to  other  Program
Offices.  Also, the  current policy of OPP is to protect all  potable sources of
ground  water, not  only  those  which are currently  used for  drinking  water.
Therefore,  occurrence of  pesticide  residues  In potable  ground water  Is the
issue  of  concern,  rather than dilution  and  degradation prior to  arrival  of
residues  at  well  heads.   Initially, OPP used  the PESTAN model;   however,  it
has been  replaced  with  the more accurate and time-varying Pesticide Root Zone
Model  (PRZM) developed  by the EPA  Environmental  Research  Laboratory (AERL),
Athens, Georgia.


     Models  such  as  PRZM will  not be used as  the sole basis for  regulation,
but they  can  provide important information for the  regulatory process.   For
example,  predictions of  significant  concentrations  at a  point  deep  in the
unsaturated  zone  can Imply that  a  pesticide  has the potential to  contaminate
ground  water and  this can be  an important piece of evidence  for the regulatory
process.  Other  issues  that can be  addressed  using models include:   the effect
of rate and  timing  of applications  on  leaching, comparisons  between use sites,
and  relative  ranking of  pesticides.   For  example, using  PRZM,  it was  shown
that April  applications of aldicarb in Florida  reduced leaching in comparison
to  June applications.   In contrast, earlier application in  Long  Island  would
be subject  to heavy spring rains, and  application  in early  summer  would reduce
leaching.    Based on this analysis,  current  Florida  regulations  state  that
aldicarb  must be  applied prior to April  1.


     Models that  link an unsaturated zone portion  with  a saturated  portion  are
the  next step for modeling.   Such  linked  models  should give a more  accurate
estimate   of  ground-water  pesticide  concentrations   than  unsaturated   zone
models, which predict  only concentrations  above the water  table.   Prediction
of  migration  from  a use  site  1s of  less  concern because  of OPP's  policy  to
protect all potable  ground water.  Care must be taken when  using  linked models
to  avoid  the  unrealistic  belief  that predicted  concentrations  represent
reality.    Two primary reasons  for  this are  difficulties  in measuring  and/or
estimating  system  parameters,  and  lack  of  databases  to validate  the  models.
Currently, OPP is helping  to fund a project by the EPA R.S. Kerr Environmental
 Research Laboratory (RSKERL), Ada,  Oklahoma, and Oklahoma  State  University,
which will  link  an  unsaturated zone model,  most likely PRZM, with a saturated
 zone  model.


      One  unresolved  problem  in   pesticide  modeling  is  the  Influence  of
 "macropore  flow" on pesticide  leaching.  This  type of flow can be described as
 the initial  rapid downward  leaching of  water and solute through  preferential
 flow paths  (such as cracks or  empty root channels) in the soil at  the onset of
 a storm.    PRZM  and similar models assume the  classic water front flow, with
 solute  appropriately  retarded  due  to  adsorption.    With  this  assumption,


                                       17

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pesticide leaching can be underestimated.  The  dynamics  and  quantification of
macropore flow need to be studied and implemented in models.

Office of Policy, Planning and Evaluation

     For the  Office of  Policy,  Planning and Evaluation  (OPPE),  ground-water
models are  used  as part  of  the policy  analysis, together with  surface water
and air models.   Under the  new ground-water  protection policy,  regional staff
members  expect to  use  models  in  the  selection  and  management  of Class  2A
aquifers.

Office of Toxic Substances

     Currently, the role  of  ground  water as  pathway for exposure to hazardous
contaminants  is  being  studied by  the   office  of  Toxic Substances  under the
Toxic  Substance  Control  Act — TSCA,  and  various  scenarios  leading to exposure
via  ground  water  are  being  modeled.   Existing  ground-water  models  are
considered  adequate, especially for evaluating generic situations.

Office of Solid Waste

     The Office of  Solid  Waste  (OSW) must review ground-water models submitted
-.0  the Agency by hazardous  waste disposal facilities  seeking Part B permits.
As  an  example, OSW  recently  reviewed the modeling effort by SCA-Chemical Waste
Management  for the New York Model  City  facility, which resulted  in the design
of  an  acceptable  ground-water  monitoring   system.    Yet  because regulations
require  that  requestors  submit  only certain  information, evaluations of permit
requests  are  sometimes hampered by  lack of  data,  and this often makes Agency
       use  impractical for this  purpose.
 Office  of  Haste  Programs  Enforcement
 T'CR'  rnf or cement  Divisior
     The  Office of  Waste Programs  Enforcement  (OWPE) has  developed a  single
 cMc'iytic  framework for comparing risks from different ground-water  contamina-
 ;:un sources  occurring  in  a  wide  variety  of  climatic  and  hydrogeologic
 settings.   This framework is based on the Office of  Solid  Waste's  (OSW)  liner
 location model  that has  been developed over  the  last  few years.  OWPE modified
 this model  slightly  and  supplied six additional  source  terms in addition  to
 the  hazardous  waste  sources  developed  by  OSW.   OWPE currently  models the
 following source types:   sanitary  landfills, municipal, industrial  and  mining
 surface  impoundments, underground  storage  tanks,  septic  tanks, agricultural
 feedlots,   road   de-icing,   hazardous  waste  landfills,  and  hazardous   waste
 surface  impoundments.    Each  source  type   is  divided  into  three  to  five
 subcategories,  based on  such factors as  size and constituents.  Releases  from
 each source  type  are profiled  over time;  for Instance,  the  water balance
 method  is  used  for municipal  landfills.   Seventy-two environmental  settings
 are  used in the model, each composed of  a different  combination of  values for
 (1)  depth  to ground  water,  (2) net-recharge rate, (3) aquifer configuration,
 and  (4)  ground-water velocity.   Several  variables  such as  fraction  of organic
 carbon  in soil  are held  constant across  all  environments.
                                      18

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     The  subsurface  transport  portions  of   the  liner  location  model   are
composed of  an  algorithm which estimates the  amount  of  time  for  contaminants
to reach ground  water  (the  McWhorter-flelson wetting  front model),  and  a  satu-
rated zone  model which  estimates  the  time for contaminants to reach  a  well.
Next  steps  are  adding  pesticides  as  a  source,  and  improving  analysis  of
hydrogeologic variables.   Because  of  the  importance of resource  loss to  the
ground-water  protection strategy,  OWPE  is  also re-evaluating all  sources  in
terms of  impacts  on  resource  loss  (i.e.,  volume of aquifer  contaminated  by
each source  type).

CERCLA Enforcement  Division—
    As a result  of the following factors,  the frequency of  ground-water  model
applications  at  Superfund sites will  increase rapidly  under  the  reauthorized
Superfund  law:

         expansion  of  the  Superfund  program  (i.e.,  with  a  2.5-  to  5-fold
         increase  in funds)

         increased  emphasis  on  permanent  measures,  such as in situ treatment
         and ground-water restoration, which  will  require better understanding
         of  the  interaction of remedial  technology with  ground-water systems

         the need  to  address  contaminated  aquifer  "sites" with multiple
         sources,   such  as  the  San  Gabriel  Basin aquifer, and  other complex
         sites

         more sites will be undergoing actual design and implementation  of  the
         remedial  response  alternative

         the need to reduce uncertainty  in remedial  response  analyses

         the need  to quantify  the performance  (effectiveness)  of  remedial
         response  alternatives  rather than rely  only on field data and  best-
         engineering  judgment

      OWPE  is investigating the  use of ground-water  modeling  for  fund-financed
 CERCLA  actions, but  with  a  focus on  the  use of  simple, desk-top  fate  and
 transport   calculations to  predict the effects  that  leaching  from  residual
 soils at  Superfund sites could have on ground-water receptors.   For  example,
 the VHS model  of  Domenico  and Palciauskus  (developed by  the Office  of Solid
 Waste  for delisting  applications) was used at  a  site in  Maine  to  predict
 initial soil cleanup  targets for trichloroethylene  (TEC).


      An important  development is the  increase of model  use by Potentially Re-
 sponsible Parties  (PRPs),  usually  large companies with considerable amounts of
 money  and  liability  at stake.   Often they  employ models to contest  Agency
 decisions or to propose certain remedial  actions.
                                       19

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                                  SECTION 4

                              MODELING CONCERNS
PROGRAM OFFICE CONCERNS
     Two major  kinds  of modeling issues are  likely  to be of concern  to  EPA:
those most frequently  encountered  by the national Program Offices,  and  those
of  particular   interest  to  Regional Offices.    These  two  Study  Group  foci
provide structure to the following overview of the Issues.

Office cf Drinking Water

     Most of  the Office  of  Drinking Water's  (ODW)  ground-water modeling  is
related to the Underground Injection Control (UIC) Program.  The UIC Program's
use  of  ground-water models  1s unique  in  the Agency  because  of the  type  of
geology  involved, the physical  and chemical  characteristics of the injected
waste,  and  the  pressure  buildup during  injection.    Models and  assumptions
required to simulate this type of environment differ from those of  interest to
other  EPA Program  Offices.   Therefore, many  of the  solute  transport models
currently used at  EPA  are not suited for the UIC Program.   Instead,  special
models  such as  those developed  for saline  waste  injection problems  (e.g.,
SWIPR),  are  used.  It is Important,  therefore,  to  determine the adequacy and
adaptability of existing  transport  models in meeting  these  specific needs of
the  UIC  program,  and  to   establish  a  program for model  enhancement  and
development for UIC use.


     The Office of  Drinking  Water is also concerned about the applicability of
geochemical equilibrium model  codes (such as MINTEQ,  WATEQF,  and  EQ3/EQ6) to
adcress  the chemical  processes  that occur  in  the subsurface  when the waste
interacts with the  resident  ground  water.   These types of codes are currently
being  used  by   the  Agency  and  Its  consultants,   but   their validity  for
epplication  to most  of  the problems encountered in the  UIC  program  has not
been established.   Hence, there  is  an urgent need to evaluate these  models for
application to UIC  program conditions.

Office of Pesticide Programs

     The Office of  Pesticide Programs (OPP) uses  models to assess the  leaching
potential  of   pesticides.    Thusfar,  OPP  has   focused  on  the modeling  of
unsaturated  zone processes.   Currently, it  is  cofunding  the  development of
linkage  of  its pesticide transport model PRZM with a  saturated zone transport
model.    The   main  concern  of OPP  is  to  avoid  the  unrealistic  belief that
predicted  concentrations  represent reality.    Two  prime  reasons  for  the
uncertainty  occurring in predictions are the difficulties in measuring and/or
estimating  system  parameters,  and the  lack  of databases  to  validate  the
models.   This uncertainty is  further aggravated by the  unresolved  problem of
modeling  pesticide transport  in  the presence  of macropores (e.g.,  empty  root
channels, cracks in soil).
                                      20

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Office of Policy, Planning and Evaluation

     The main concern  of  the  Office of Policy,  Planning  and  Evaluation  (OPPE)
relates to  the  establishment of  a set of  ground-water models  as  "standards"
for various  policy purposes.   OPP considers this  an  important  Issue  because
use of a ground-water  model as part of a policy analysis  requires considerable
time  and  effort  in  demonstrating  to  reviewers  that the particular model  is
appropriate  and gives  accurate  results in  the  considered  case.   Typically,
OPPE  does  not  have this problem  with  surface-water dispersion  models  and air
models,  for which  there  are  well-established  Agency standards.   Given  the
increased  attention  ground-water  contamination  issues  will  demand  in  the near
future, they find it important  that performance standards and review  criteria
be established  for ground-water models.


      Regardless of whether  certain models are deemed acceptable,  or  whether
performance  standards  for  different models  are set  based on their planned use,
criteria will need to be developed for evaluating  the models.   In this regard
OPPE  finds it  useful  to  have  a ground-water modeling catalog at hand similar
to,  but  less extensive  than,  the  Agency's  wasteload  allocation handbook.   A
set  of administrative and  scientific criteria that  OPPE  finds particularly
important  includes:

     •   trade-offs  between costs of  running  a model and  accuracy

         profile of  model  user  and definition of  required user-friendliness

         accessibility in terms of effort,  cost,  and restrictions

         acceptable   temporal   and  spatial   scale  and  level  of   aggregation
         allowed or  required

         classification of types  of contaminants  (organics,  metals, etc.)  the
         model  can handle

         description  of the model input variables that  can  be  varied  (and  by
          how much) and the factors that are considered constant

          data   requirements  of  the model  in  the  context of  the cost for  data
          col lection

 Office of  Toxic Substances

      Although  most  regulations  are   not  yet based on  exposure to  hazardous
 contaminants via ground  water,  such  considerations are Increasingly used  in
 the evaluation of new policies.   If ground-water exposure is expected  to be an
 important  pathway,   various   scenarios  of  exposure  via  ground water  are
 modeled.   To  do so  efficiently,  the existing  database needs  to  be  expanded
 both  with  generic data  and  with regional or site-specific  data.  The  use  of
 models for  evaluation of new  chemicals  is currently  also  hampered by  lack  of
 product data,  often  because of their  proprietary nature.
                                       21

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Office of Waste Programs Enforcement

     Recent experiences  by the  Office  of Waste Programs  Enforcement  (OWPE),
particularly   within   the  CERCLA   (Comprehensive  Environmental   Response,
Compensation and Liability Act or Superfund) Enforcement Division, have led to
recognition of a critical gap in Agency procedures regarding the selection and
application  of ground-water  computer  models  used  for  simulating flow  and
contaminant transport.


     The primary modeling  concern  is  the  lack of any EPA policy on the use of
proprietary models by or for EPA.  This has been an especially thorny issue in
several  CERCLA enforcement  actions  and  is  likely to surface  in the  RCRA
Program as well.  To remedy this situation, two efforts are currently underway
in  OWPE.   The  first  entails   revising  the  modeling sections  of the  RCRA
Technical  Enforcement  Guidance  Document to promote  the use of nonproprietary
models.  The second is the  development of a policy memo concerning this issue
that  the Director of  OWPE will  distribute to  EPA  Regional  Offices.   It is
expected  that  this  memo  will   strongly  discourage  the  use  of  proprietary
models.     OWPE has  specifically requested  the Ground-water  Modeling  Study
Group  to  address  this  issue   and  to  make  recommendations  confirming  or
modifying  the  policy decisions being made  by OWPE.


      Furthermore,  it  is OWPE's   belief  that  it  should  not be the sole referee
or  arbiter  of ground-water  computer  codes  as they  are  encountered  in the
enforcement  process;  OWPE  has  neither the  resources   nor  the  broader Agency
responsibility to  establish unilateral  criteria by which a code may be judged
acceptable  to the Agency.  However,  OWPE feels  strongly  that such criteria
must  be  developed.


      Other modeling  issues that need to be considered  are  closely  tied to the
proprietary  model  issue  discussed above.   These include  definition  of  what
constitutes  "acceptance" of a model by the technical  community,  establishment
of  an adequate  level  of  "peer  review,"  and  establishing quality assurance
protocols  for  model   development,  selection,  and application.


      Because  most  of the  modeling projects  to be reviewed by the Agency are
site-specific  analyses where calibration data may or may not exist, a  well-
defined  set  of guidelines  for  model  calibration  and predictive  phases  is
necessary.


      There  is  general   agreement  that  many of  the  parameters  required for
evaluating contaminant  transport (especially spatially varying  parameters) are
lacking.     In the  absence  of  complete  information,  the  most  meaningful
simulation results are  those  expressed  in probabilistic terms.   Guidelines for
evaluation of  field  data and  simulation results are  necessary.
                                      22

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Office of Research and Development

Office of Health and Environmental Assessment—
     The Office  of Health  and  Environmental Assessment (OHEA)  is  developing
guidance;  for  health  and  exposure  assessments,  including  development   of
mathematical  selection  criteria  for  ground-water  fate and  transport mod-
eling.   This guidance will  be  used to support the various  provision;  of  the
RCRA Amandments  and  CERCLA.  The  exposure  assessment guidelines,  proposed  in
the Federal  Register on  November  23,  1984 and soon to  be  published  in final
format, outline the next phase of  guidance to be developed  by OHEA.


     As part of  this  effort,  the Exposure Assessment  Group  met with the model
users  in several  Program Offices  in early  1985 to discuss  the approach being
used  in developing selection criteria  for ground-water  transport modeling  and
the  needs  of  Program Offices.    Subsequently,  the Office  of  Waste  Programs
Enforcement  Initiated  several meetings  to bring issues  relating to the Agency
procedures  and  criteria  for model  selection  and  use  of computer codes to  the
attention of concerned parties.   A workgroup has been formed to review Agency
procedures   and   criteria  for  ground-water  model  selection,  particularly
directed toward  exposure  assessment.

Environmental Research Laboratory,  Athens, Georgia—
     Guidelines  to determine the adequacy of proprietary versus public domain
software, quality control  measures, and  liability are  integral  parts  of  any
modeling  assessment;   however,   correct   application   procedures  and  data
evaluation  for  regulatory  decision making  may  be  equally  important.   For
example, most modeling studies  utilize  lumped parameters that are a reflection
of the  mass  balance  approach for  advective-dispersive models.  Although these
types  of  approaches may  be appropriate  for  problems  in ground-water  systems
where  the contaminant is distributed over the entire ground-water basin, they
may not be  adequate  where dispersion is important.  Guidelines  relating to the
appropriate  methodology  used  should be made  available to managers.


CONCERNS OF  REGIONAL  OFFICES

     The  Regions  have  varied interests  in  model  development, selection,  and
use,  and  in  training  in   modeling.     In  discussions with  the  staff,  the
following major  issues surfaced:

         limited knowledge  of model availability

     •    the need for assistance in selecting and  using available  models for a
         specific site

         guidance in model  reliability  and interpretation of  simulations

     •    need   for  additional   models  for   multiphase  flow  and  contaminant
         behavior in the vadose zone

          improved interaction and  communication with technical staff  in other
         Regional Offices,  Headquarters,  and EPA  labs
                                      23

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         training in basic processes (geology, hydrology, fate and transport,
         etc.)  for the project officers  as  well  as modeling training for the
         technical experts in the Region.   (In  some Regions  it was  stated that
         EPA "prefers to  rely on  Agency  expertise  rather  than external consul-
         tants, because of the 'burden of  proof needs.")

    •    hiring  and  retaining   technical  staff  who  have  received special
         training in modeling

         need  for ground-water  modeling  policies consistent with those for
         surface water modeling

Model Use in Regional Activities

     Because the Study Group visited only three of EPA's  ten Regions, the fol-
lowing discussion 1s somewhat limited.   However,  the  Study Group  considers
these  findings Indicative of the  general  Regional situation in  ground-water
modeling.


     EPA Region  I is involved with a number of major Superfund  sites as well
as  other  hazardous waste disposal  sites.   The staff is actively  involved  in
site  investigations  (RI/FS)  and  regulatory/enforcement actions  involving  PRPs
(Potentially Responsible  Parties).   In the  Study Group  meeting  in  Boston  it
was  stated  that  PRPs,  usually   large companies with  considerable amounts  of
money  and  liability at stake, employ models to contest the cases or propose
certain remedial  actions.


     In  Region III  models  have  not been  used to  the  extent  that  they  have
become controversial;  there  are  as yet no cases  in which  they  are contested.
Staff  members  pointed out that  under the new  ground-water  protection policy,
they  expect  to  use models  in  the selection  and  management  of  class  2A
aquifers.    For  permit  requests  under  RCRA,  numerical  models  are not  yet
used.  This 1s partly due to  the  extensive  karstic limestone  aquifers  in  the
Region,  which  make  modeling  impractical,  although various analytical  models
are used in the  permit  review process.   Because regulations are rather  strict
with respect to the  type of  information to be submitted,  evaluations of  permit
requests are sometimes hampered  by  lack of data, and this too makes the  use of
models impractical.   Further guidelines  for  data  collection and use of  models
seem necessary.


     It appears  that in  Region X numerical  ground-water  models  have been used
for only one Superfund site  and  for no RCRA sites.


     At the beginning of what is  perhaps  a  three- to five-year project life,
it  is  difficult  to anticipate  staffing,  data,  and modeling needs.   Project
funding tends  to be  incremental,  and  therefore data collection and analysis
are  often  short-sighted.   The  approach sometimes  becomes ad  hoc.  when a
stepwise approach would be preferable.  (On the other hand, additional funding
nay  not  be  forthcoming,  discouraging  use  of  a  stepwise approach.)     If
litigation  is  anticipated,  the  project  needs to  be carefully  executed   and


                                     24

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documented.   Project  managers need to know what the  tools  are,  what  the  cost
will be, and how results can be used.


     Often  data  are  available  only  as  hard  copy;  significant  costs  are
involved in transferring  such  data into  machine processable form.   An example
is  an  ongoing  Superfund  demonstration  in Region  X  where  $400,000  has  been
spent to assemble and  digitize  existing data for a county.

Regional Staff

     The  rapid  expansion of  responsibilities  under Superfund has  forced the
Regions  in recent years to expand significantly  their  project staffs.   It is
apparent to the Study Group that  in many Regions, project managers (primarily
those  working  under  the  jurisdiction of  CERCLA)  are so involved  in project
work   itself  that  little  time   1s  left  to   do  anything  other  than  meet
administrative   requirements   and  deadlines.     In  general,  managers  are
instructed to perform RI/FS work  in less than  one year, with completion of the
Rl-phase  within four  to  six  months.  This  does not allow  time for adequate
data  collection, and much less  for extensive  modeling, which  is therefore
often  considered  an unaffordable  luxury.


     Because  of the  deadlines  and workload,  project managers,  who are  often
untrained  in ground-water  modeling, have no  time to keep abreast of devel-
opments  in modeling  and,  therefore, are often  not qualified  to  introduce
modeling   into  the  project  or  to evaluate modeling  done   by contractors or
PRPs.   Model  application and interpretation of results is  very  subjective and
may  be  the   core   of   an  expert's  testimony  in  court.    The  expert's
interpretation  of  results  represents the culmination of months of  technical
work.   EPA staff must be capable of  providing the  expert with both policy and
technical  oversight,  e.g., in  the quality assurance of the project.   If this
oversight   1s   lacking,  the  expert's  work may be  misdirected  or  poor in
quality.   That this  is a  significant problem is illustrated by the modeling
deficiencies  frequently displayed by EPA contractors.


      Regional  staff  in the  Hazardous Waste  divisions are  almost all  gener-
al ists with degrees in environmental science or environmental  engineering.  In
the three regions  none  of  the  project  managers  had  formal  training  as a
hydrogeologist  (nor  is  there  a  "hydrogeologist" position  in the  Agency).  A
broader multidisciplinary  team is viewed  as mandatory.     There  is a  tendency
to underestimate  staffing needs; and  even with breadth,   staff  tends  to be
 spread too thin.   Internal capabilities can  be provided by the Environmental
Services  Division,  present  in some of  the Regions, but are not  always  used
optimally.   If a project gets too complex, EPA staff is often  pulled  off the
project and the job  is given to  a  contractor.


      An additional problem facing the regions is that most of the good people
eventually go  to consulting  firms, once they  have  experience,  resulting in a
 high  turnover.   In  a  number  of  Regions,  the rapid  expansion  of  Regional
 project staff  and the high turnover rate have led to a situation where  many of
 the project managers  have  less than two years' experience on the job.


                                      25

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     There  is  also  a  significant   difference   between   Regions   in  their
management approach to optimal  use of the limited  human resources  available.
Various measures  have been implemented to accommodate the  new  tasks required
by recent  legislation.   For example. Region III  staff  has  established a pool
of in-house  experts (three  hydrogeolegists  and three toxicologists)  who  are
available to  help the  project officers.   In addition,  each site  has a formal
review  "team"  that  includes   a  hydrogeologist,  a  toxicologist,  and   an
administrator.  Regular in-house technical meetings provide an environment  for
good communication among staff concerned with ground water.


     A  strong interest was  expressed, expecially  by the   Regions,  in having
advice  and guidance available for mdel  selection and use,  given site-specific
conditions.    This can  be  achieved  through  the  establishment of  a blanket
agreement  with a nonprofit agency to  provide a  quick  means  of  bringing in
capable outside experts for  advice on specific cases.

Quality Assurance

      The  aforementioned conditions  force Regional  staff  to rely  heavily on
their  contractors for  modeling  in  Superfund  projects.    Contractors  are
 selected  through  competitive bidding  for large contracts.   Modeling, which is
often  only   a small  part  of  these  contracts,   is  sometimes  done  by  the
contractors  themselves,  but frequently  by  subcontractors who are not  chosen by
the  Region.   Thus, Regional  staff  has little control over  who  performs the
work and must use the  "national"  contractors  because of existing procurement
procedures.


      In addition,  there  has been  little  quality  assurance  (QA)  in the  past
over modeling work performed  by the  contractors, partly because of  the tight
 schedules under which the work must be carried out,  and partly because of the
shortage  of  experienced staff.  Once a project starts, often no  formal  review
takes place  until the project  reaches its final  stages.   The  project  officer
at EPA is the only person who might  review the  study while it  is  in progress,
and   in most  cases  the  final  QA  is conducted  within the  Region   itself  by
personnel  in divisions  that are involved administratively.   Sometimes Regional
staff forms   a technical  review team for such purposes.   In  some  Regions  part
of the modeling  work  is  reviewed  by a USGS  modeler available to  the  Agency
through an interagency  arrangement, but no formal review meetings are held.


      Under such conditions, many of the modeling studies in the Regions may  be
of limited  usefulness  due  to  incorrect  siting  or data collection;  incorrect
use  of available data;  inadequate modeling  of  the physical  system,  such  as
 flow in fractured bedrock;  or  invalid  boundary  conditions.  Major constraints
 in addressing  these  problems are procurement policies mandated  by  government
 units outside EPA, specifically  the  U.S.  Office  of Management  and Budget, and
 the  lack  of  in-house expertise.


      High turnover in project  managers,  together with the inability to review
 the  degree  of   success   or  failure  in  earlier  projects,   leaves  little
 institutional memory for learning from previous studies.


                                       26

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Procurement of Modeling Studies

     Regional problems  include a rapid expansion  of  the  EPA responsabilities
as well  as  high turnover of  personnel.   One  solution is  to  have contracts  to
meet   specific   needs,   as   opposed  to  large-scale  contracts  for  general
support.    For  example,  a  contractor  might  be  engaged  to provide  modeling
support  for all Regions.   (However,  at the present  time the  Agency  does  not
have enough technical expertise to  review its contract work.)


     Procurement  procedures  also  could  be  changed  so that  Regional  Offices
have more  control  over the selection  of  their  contractors.   Technical exper-
tise should be  specified and  given greater  weight in the selection of a con-
tractor.   At the  very  least, it should  be  recognized that  ground-water con-
cerns  tend  to drive hazardous waste  remediation (e.g.,  the  RI/FS process) in
terms  of time and  effort, both to characterize  the problem and  to clean up the
site.    This   should  be  more  emphatically  stressed  in   the selection  of
"national"   contractors  so   that   qualified   ground-water   contractors  are
chosen.     There  is  also  a  need  for  guidelines  in  selecting  modeling
contractors.


     The modeler or someone familiar  with the application (e.g., the modeler's
supervisor) should be available to serve as  an expert witness.  In addition,
the  quality  of the  presentation   of  the results  of a modeling  exercise to
management, and to  the  judge in  the courtroom, is  important for the  ultimate
success of  the  modeling  effort.    Because   model   use  in  enforcement  and
litigation  is  likely  to  continue  to grow  with  time,  continuing attention
should be  given to issues related  to such applications.


     Study  Group participants also considered  it important to  establish  a  more
direct  mechanism  for   reviewing  and   directing  the   work  of  outside
contractors.     This  would   involve   the  establishment  of   thorough  QA/QC
procedures  for modeling  studies,  and would include  a detailed  review process
to be  conducted  throughout  the  modeling process,  with  stop/go decisions at
each critical point.


      Postmortem analyses  of  selected  cases with  all staff  should  be encour-
aged,  so the lessons learned can  be communicated and applied  to  current  and
future site investigations.


      Managers   should   require   computer-processible data;   a  protocol   for
database management systems  and  better  data-processing techniques should be
 adopted.   This will  significantly  improve  the  efficiency of modeling-based
 data analyses  needed for the  resolution of many ground-water issues.

Technology Transfer and Training

      The Study Group  has  found only  a  limited  understanding  among  Regional
 staff  of  what  software  is available.   As  Regional   staff   anticipates an


                                       27

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increased use of models, there  is  a  need  for  improved  technology  transfer  and
training.


     Most of the  technical  staff  indicated strong interest in  (but  expressed
their concerns and frustrations over the lack  of opportunities for)  structured
training  or self-study  in  modeling.    However,  many of  those  in need  of
additional  training   should   first  be   trained  in   general   hydrogeology,
hydrochemistry, and data analysis, before focusing on modeling.   Because model
use is expected to  increase  in  the future, the development  of in-house  exper-
tise, by  whatever means,  appears to be  a  major priority.   Most  of  the tech-
nical and  managerial  personnel  recognize  that  they  need not become modeling
"experts"  and  only  want  sufficient  training  to be  knowledgeable  users  or
competent judges  of the appropriateness  of models used by  PRPs  and contracted
consultants.   A  strong interest was expresses,  expecially  by the Regions, in
having advice  and guidance available for  model  selection and  use, given site-
specific  conditions.    This  can  be  achieved  through  the  establishment of  a
blanket  agreement  with  a   nonprofit  agency  to  provide   a  quick  means  of
bgringing  1n capable outside experts for adivce on specific areas.


     A  network of  staff  concerned with ground  water  also is  needed so that
experience  can be  shared.   Technology  transfer is  ineffective  if  it  simply
consists  of reports sent to Regional libraries; a better environment is needed
in which  state-of-the-art technology is  distributed  and used.  Regional staff
indicated  the need for a better  institutional  relationship  between Regional
Offices  and EPA Laboratories.


     If  communication  is  facilitated,  the   synergisms  would  work   to  the
advantage  of the  Regions and would more than justify the expenditure of travel
money.

Facilities  and Resources

     Some  Regions  have a  rather extensive  collection of modeling software
available  internally,  either 1n their Program  Offices or  from their Environ-
mental Support Division.  Other Regions have some  Incidental  software,  but are
not aware  of  sources  of additional models.  Some Regions consider proprietary
models an  acceptable  alternative  if source code is available and if the model
is  well-tested   and   properly  documented;  others  use  only  public  domain
software.


     Computer  facilities  available in  the  Regions  for  ground-water assessments
include  microcomputers, mostly IBM PCs or  compatibles,  as  well  as terminal
access  to the Region's administrative minicomputer  facilities.  Some  Regions
have   MicroVAX super microcomputers.   A few  Regions  have Indirect  links  with
local,  external  computer  facilities  such as  a university campus  or  a  USGS
District  Office.   EPA  facilities  in Triangle  Park  are   not  used, however,
either  because the need has not  been  identified, or  because the use of these
facilities  is  relatively  cumbersome  1n terms of access time,  response time for
printed  output,  and  the like.    Procurement  regulations  sometimes  Inhibit
adequate  expansion  of existing  computer  systems  1n  terms  of  hardware  and


                                      28

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software  to   serve  efficient  data  acquisition  and   processing  as  well  as
    1^,; needs, thus limiting efficient software selection and model  use.
         - expertise,  which is readily available  in  some  Regions  (e.g.,  from
         .'uc,  universities  1r,  Region  I),   is  seldom  fully  exploited.    Short
         or  special  classes are sometimes  arranged,  but they are  often  con-
         relatively  unsuccessful  because  they  are  too  theoretical  or  too
condensed.

Legal Concerns

     During the  discussions at  Boston,  a  number  of  legal concerns were brought
up.   In  general, legal procedures become  important when technical regulations
are not  available  or do not cover  the issue under  consideration, and also when
existing  regulations are not  followed and need to  be  enforced.   In accord with
earlie-  findings of  the Study Group,  it  was mentioned  that  a model's use by
tne  Agency  enywbere in  the  country results  in de  facto  acceptance  of that
mode- •  'ir> litigation, and this presents  a  major  legal  problem.  There is also  a
lack  r.4'  information  exchange within  the  Agency  about which  models   are
avai laC'i£;  where  and  under  what  conditions  they  have been used;  what  the
results   from  the  models  provide  in  terms useful  to management;  and what
administrative,  technical, and legal problems  are  encountered.


      Related to the  legal  problems is the  Agency's  need  to  retain  its  expert
witnesses for  litigation.   The high  turnover  rate  within the Agency  causes
continuity  problems" in staffing (which are sometimes avoided by working with
consultants).    However,  consultants  may  get  certain  jobs  by  promising  the
assis.tancs   of   senior  modelers,  while  the actual  work  is done  by  junior
modelers.  This can  result in an  unsatisfactory modeling  effort,  and  in court
problems when  the senior  modeler,  who has not been directly involved  in  the
Jit^'ico work,  is presented as the expert witness.

Surety  2!  r'eciondl  Concerns

      In  sutTTiary,  many  issues raised with respect  to ground-water modeling  are
 identical to those  raised  by the  national  Program Offices.  This is especially
 true  in the areas  of  training needs, QA/QC  in modeling, and  legal  require-
ments.   Additional  attention  has been drawn to  administrative and technical
 constraints  in  the  Regions.    The management  of  technical  contracts is  a
 problem  because project managers  (on-site  coordinators)  do  not have  the time
 to remain  technically qualified, partly  due to lack of support from regional
 management  for  independent   study.    This problem  is  particularly  critical
 because  ground-water  modeling  is  sometimes   performed  by   unqualified  con-
 tractors and because model use will  continue to increase  in the future.
                                       29

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                                  SECTION 5

                             DISCUSSION OF  ISSUES


ADEQUACY OF MODELING THEORY AND DATA

     Appropriate models do not yet exist for all ground-water problems because
many of the  assumptions  and simplifications common to existing  models do not
allow  faithful  simulations,  and because  the  natural  processes  that  affect
fluid  and contaminant  movement  have yet  to  be  fully  understood.   This  is
especially  true  for  chemical   and  biological  processes.    Although  large
advances  have  been made  concerning the behavior of  individual  contaminants,
studies of the interactions  among  contaminants are  still  in  their  infancy.
These  studies  Include,   for  example,  the  ability  of  certain   solvents  to
increase  dramatically the nobility of ordinarily  slow-moving  pesticides,  of
polynuclear  aromatic  hydrocarbons,  and of others.


     Other areas where substantial  progress is needed lie in understanding the
immiscible  flow  and  transport  considerations  so  crucial  to   solving  the
problems   of  leaking  underground  storage  tanks,  and  the manner   in  which
contaminant  transport in  fractured  rocks differs from transport through porous
sediment.     Finally,  certain   we11-understood   phenomena  pose  unresolved
difficulties for  mathematical formulations, such as  the dynamic operation of
partially  penetrating wells in unconfined aquifers.


     Improvements  are needed,  concurrently,  in several major  areas.  Models
need to be  Improved  mathematically  so  that  errors  arising from computational
techniques (e.g.,  numerical  approximations)  are  minimized.  While continuing
research  in this area  has been  a  well-recognized  need, other  topics just as
important  have  received  less  than adequate attention.


     The  theories  on  which models  are based  need  to be developed further so
that proper representations  of  the  true  Influences of  various   natural pro-
cesses  can be incorporated into  model  applications.   Theories that  have been
used for   solving regional  water supply problems are generally applicable to
localized  water-quality  problems  such  as  hazardous waste sites.    However,
certain specialized  needs  are  peculiar to chemically complex  problems.  The
highly  variable distributions  of contaminants at  hazardous  waste sites, for
Instance,  create a  number of practical  difficulties.   These  result in part
from  Incomplete  understanding  of   chemical  Interactions  and  the  role that
microbiota may play 1n  the  transport  and  fate of  subsurface  contaminants.
Such  difficulties also  result from  limitations  of  available  data,  of  field
methods,  and of quantitative  tools.


     As a result,  data acquisition methods and Interpretive models are  needed
that  can  examine  to  an unprecedented  degree  the  physical,  chemical, and
biological  processes  controlling   the  transport  and  fate  of  ground-water
contaminants.   Unfortunately, few  of the constants and  coefficients  needed to
incorporate   chemical  and biological  processes  into  contaminant  transport


                                      30

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eve'i.,'t v--.;.  t-,-e  available presently.   This does not mean,  however,  that  some
lr'c;~c'"-~:'  r'7  tneir  contributions  cannot  be  estimated;  much of  the  existing
T.rr  r.i •:*."•  r.,-ri  bs  used  in  a semiquantitetive  manner  (i.e.,  sensitivity
er;f ;:M-   ;..";:' "'-.'orst-ccse"  scenarios).


     :. -rr-ts to  collect field data  and  to  estimate natural-process parameters
must   :c  expended  and  improved  so  that   model   applications  will  be  more
physicsily  b^sed  and  thereby  capable   of  more  accurate  predictions.    For
excJ-ip1.; s  so few  data  are available  concerning the  exact  location,  volume,
ccrr:,! : --on, and  timing of chemical  releases at existing sites that it is very
dlr^-ic: "'\  for  modelers  to  determine the appropriate configuration  of  what is
refer,-tc  to  os the  "source term" in modeling.  One prevailing misconception in
this  re:;c-d  is  the idea that  additional  chemical  sampling of monitoring wells
car, provide  definitive clues  to  the missing historical  data, but this is true
onV  ~u-?erficislly.   Although indication  of  the  source  term can be obtained
St"~>--'  •'•"   "Sterns  of  chemical  movement,  there is  no  guarantee  that  causal
'•"-  •    :'•"-:: cm be  discovered or  that the patterns will remain constant.


      ' c;rf,:io~,  misconception is  that all  field methods  and tools necessary for
obtain -c data to  run the models  are available,  if  not in optimal  form, at
leas"  in  u useful form.   In  fact,  however, direct measurements are unreliable
or cannot be  obtained  for  a number  of  parameters such  as  ground-water flow
velocity  and  direction,  rates  of  sorption  and   desorption  (retardation)  of
organic   chemicals,    and   the  potential   for   biotransformation.     This
misconception  parallels  the  mistaken idea  that   all  necessary  theories have
b££-i  worked out  and  that  further  refinements are needed  only  so that better
precision and  accuracy can be obtained.


           •Irr.e :,rf_-;.-ion   of   geologic,   hydro Tocic,   chemical,  and  biological
r-"''-'••-   '  ' -•'  ust.t^'it subsurface flow £.nd  t-ansport models is possible only if
'-"..  .'•:••:.  >-;v- concepts  invoked are  sound.   The data must be representative as
we  i  ••<.;, A; cit^is anc  precise.  The  degree  to  which the data  are representative
is  '::":.-:£  to  the   scile   of  the  problem  and  the  concepts  guiding  data
collect ic-i and  interpretative efforts.  Careful  definition of  these concepts
is  crucial:  special   attention  should  be   given  to  the  spatial  and temporal
variations involved.    The  use of newly developed  theories to help solve field
problems  is often a  frustrating exercise.   Most theoretical advances call for
some   data  not   yet   practically  obtainable  (e.g.,   chemical  interaction
coefficients and  relative permeabilities  of  immiscible  solvents and water).
In addition, the  incorporation of  theoretical relationships into mathematical
models  typically  is  made   possible  by   invoking  certain  assumptions  and
simplifications that  alter  the intended relationship.   Therefore, theoretical
advr.ncrS  in modeling ground-water problems  must be accompanied by  improvements
in cold  collection and mathematical representation efforts.


      The   Study  Group  has  identified a  variety  of  new models  and modeling
approaches as important to EPA's mission:

          simulation  of flow  and transport  in multimedia  (e.g.,  coupled models
          for surface water/ground-water  interaction)


                                       31

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   •    representation  of  stochastic processes  in predictive  modeling,  and
        improving the applicability of geostatistical models
        improved modeling of hydrochemical speciation
        simulation  of  flow  and  transport  in  fractured  and  dual-porosity
        media,  including diffusion  in dead-end pores
        simulation  of flow and transport  in soils containing macropores
        determination   of   effects  of   concentration-dependent  density  on
        ground-water flow and pollutant transport
    •    determination  of effects  of  alteration of geologic  media on hydro-
         logical and chemical characteristics (e.g., dehydration of clay when
         attacked by solvents,  change in sorptive  capacity of  material when
         heated)
    •     representation  of  the  three-dimensional effects  of  partially pene-
         trating wells on water  table  aquifers
         development of  models  for management  of ground-water contamination
         plumes
         development of expert  systems  (artificial  intelligence)  for such
         tasks  as selecting appropriate submodels or subroutines for  specific
         problems
         application of  parameter  identification models  to be used with site
         studies
         further development of  pre- and postsimulation  data processors
    •     continued development of risk assessment and management models
         modeling of volatilization, multiphase flow, and immiscible flow
    •     incorporation of economic  factors  to improve estimation  of  clean-up
         costs
    •     development of generic and site-specific parameter databases
     Fundamental  research  supporting  ground-water  modeling  is  considered
necessary in such areas as:
         transient  behavior   of   process   parameters   (e.g.,   retardation,
         hydraulic conductivity)
         desorption for nonhydrophobic chemicals
         multicomponent transport and chemical interaction
                                      32

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    •    enhanced  transport  mechanisms  (e.g.,  piggy-backing  on more  mobile
         chemicals)

         transport of silt with sorted chemicals in aquifers

         improved numerical accuracy, stability, and efficiency

     Although  an extensive  research and  development  program exists at  ORD,
more  attention  should  be brought  to  bear on  the  mechanisms by which  those
needs can be satisfactorily addressed.


GROUND-WATER CODE REVIEW  AND TESTING

     The last  few years have seen  a significant increase in the use of ground-
water models  in situations leading to  litigation,  congressional hearings, or
extensive public discussion.   In  such  cases,  both  the'theoretical  foundation
and  coding  of  £  model,  as well  as its application, may  be  contested.   This
situation is  typical  for  many  of the hazardous waste landfills, impoundments,
and  spill sites investigated by  EPA under  Suoerfund  legislation.  Ground-water
models are  used to determine the  extent  of  present ground-water  contamination,
to  identify the source of that contamination, to predict future contamination
and   health  risk,   and  to  assess  remedial  action  alternatives.    These
determinations often form  the basis for  the principal findings of  a site's
Remedial  Investigation  and  Feasibility  Study  (RI/FS) under  CERCLA.  Discretion
as  to when  to  use a  computer  code and  which specific computer code to use  is
often left  to  the EPA  contractors and/or  the responsible  parties who perform
the  RI/FS.


      Criticism of the modeling  aspect  of  the RI/FS can take  three forms:  (1)
use  of  an  erroneous  conceptual  model  of the  physical  system to  which  the
computer  code  is  applied;  (2)   inappropriate  aoplicfition  (or use)  of  the
computer  code;  and  (3)  errors  in the  code's  formulations,   assumptions,,  and
coding  which renuer it unreliable.  The first  two  points  may be apparent  from
a  critical  review  of the RI/FS  and  may be  sharply debated by  technical
experts.    The  last  point is not apparent from an evaluation  of the  RI/FS  and
can only be  determined  after a  detailed  assessment and extensive use of  the
computer code.  The resulting confusion, especially among  nonmodelers (judges,
attorneys,  regulatory agency  staff, and legislators),  has  led to doubts  about
the utility of the modeling methodology in general.


      Such  controversies  can be  avoided  by applying  adequate  quality  assurance
 (QA) to  all  stages  of  the modeling  project.   Selecting an  appropriate  and
well-tested model  is another  significant measure  that  can  be taken.   Model
testing  is an  Important aspect of QA  in model  development.  Evidence  of a
code's technical  soundness  can be established prior to any  legal  proceeding.
Where a computer code has not been peer-reviewed and independently  tested, the
criticism  of  the  code itself becomes a valid  way  to  attack  both  the computer
code  and  the RI/FS  which may  have  relied  on  1t.   However,  it  should  be
 realized that  code testing requires a large amount of time and effort.   Often,
time and resources are limited for post-simulation review efforts.
                                       33

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     Adequate model testing  and  validation should be an  integral  part  of all
research and development projects resulting in modeling software to be used in
support of the Agency's regulatory mission.

Model Evaluation

     Before a ground-water model is used as a planning and decison-making tool
by  EPA  or  a  cooperating  agency,  its  credentials should  be  established,
independently of  its  developers.   This can be  done  by systematically testing
and  evaluating  the characteristics of  the model.   Code  testing  is generally
considered  to  encompass  verification  and validation  of  the model  (Adrion
et al.  1982).   To evaluate  ground-water models in a systematic and consistent
manner,  some  institutions   have  developed  model  review,  verification,  and
validation  procedures  (van  der Heijde  et al.  1985; Moran and  Mezga  1982).
Sometimes,  independent  review and testing is sought.  Generally,  the review
process  is  qualitative  in nature,  while code  testing results can be evaluated
by quantitative performance  standards.

Model  Review—
     A   complete  review  procedure  comprises  examination  of model  concepts,
governing   equations,  and   algorithms   chosen,  as  well  as  evaluation  of
documentation  and general  ease-of-use,  and examination of the computer coding
 (Huyakorn  et al.   1984; van  der Heijde et al.  1985).   If  the  model  has been
verified or  validated  by  the author,  the  review  procedure  should include
evaluation of  this verification and validation  process.

     To facilitate thorough  review of the  model,  detailed documentation of the
model  and  its developmental  history  is  required.   In  addition,  to ensure
independent evaluation of  the  performed verification  and  validation, the
computer  code  should  be  available  for  implementation  on  the reviewer's
computer facilities,  together with  a file containing  the  original test  data
used in the code's verification and validation.

     Review should  be performed  by  experienced  modelers  knowledgable  in
theoretical  aspects  of  ground-water  modeling.     Because  review  is rather
subjective  in  nature,  selection of the reviewers is a sensitive  and critical
process.

Model  Examination—
     Model  examination  involves  determining  whether anything fundamental was
omitted in the initial  conceptualization of the model.   Such  a procedure will
determine  whether the  concepts of a  model adequately represent  the  nature  of
the  system under  study, and will identify the  processes  and  actions  pertinent
to  the model's intended  use.  The  examination is also intended  to  determine
whether the equations representing the various processes are valid within the
range   of   the   model's  applicability,   whether  these  equations   conform
mathematically to  the intended  range  of the model's use,  and whether the
selected  solution   approach   is  the   most  appropriate.     Finally,   model
examination should determine  the  appropriateness of the selected  Initial and
boundary conditions and establish  the applicability range of  the  model.
                                      34

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     For complex models, detailed examination of the implemented  algorithms  is
required to determine whether  appropriate  numerical  schemes,  in  the  form of a
computer code  (ASTM 1984),  have been adopted  to  represent  the  model.  This
step should disclose any inherent numerical problems such as  non-uniqueness  of
the  numerical   solution,   inadequate  definition   of   numerical  parameters,
incorrect   or   nonoptimal   values   used  for   these   parameters,   numerical
dispersion, numerical  instability  such as  oscillations  or divergent  solution,
and problems regarding conservation of mass.


     In  addition,  the  specific  rules  for  proper  application  of  the  model
should  be   analyzed  from the  perspective  of its  intended  use.    These  rules
include  data  assignment according  to  node-centered  or  block-centered  grid
structure   for  finite-difference  methods;  size  and   shape  of elements  in
integrated  finite-difference and finite-element  methods; grid size variations;
treatment  of  singularities  such  as  wells; approach to  vertical averaging  1n
two-dimensional  horizontal  models or  layered   three-dimensional  models;  and
treatment  of  boundary  conditions.   Consideration  is  also given  to the ease
with which  the mathematical equations, the solution procedures,  and the final
results  can be  physically  interpreted.

Evaluation  of  Documentation—
    The  documentation is evaluated through visual  inspection, comparison with
existing documentation  standards  and  guidelines, and  use  as  a  guide  in
preparing  for  and performing verification  and validation  runs.


     Good   documentation  includes  a  complete  treatment  of  the equations  on
which  the  model is based, underlying  assumptions,  boundary conditions that  can
be incorporated in the model,  method  used  to solve the  equations,  and limiting
conditions resulting  from  the  chosen  method.   The  documentation  must also
 Include a  user's manual  containing  Instructions   for  operating the code  and
preparing   data  files,   example  problems complete  with  input  and  output,
 programmer's instructions,  computer  operator's instructions, and a  report of
 the initial code verification.

 Evaluating Ease of Use—
     The data  files provided  by  the  model developer are used to  evaluate  the
 operation  of  the code  and the user's  guide through  a test-run process.   In
 this stage special attention  is given to  the rules and restrictions ("tricks",
 e.g.,   to   overcome  restrictions  in  applicability) necessary  to operate  the
 code,  and  to.the code's ease-of-use  aspects  (van der Heijde 1984).

 Computer Code  Inspection—
     Part  of  the model review  process  is  the inspection of  the  computer code.
 In  this   inspection   attention   is   given   to  the  manner  in  which  modern
 programming principles  have been  applied  to  code structure, optimal use of  the
 programming language,  and  internal documentation.
                                       35

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Model Verification

     The objective  of  the verification process  is  twofold: (1) to  check  the
accuracy  of  the   computational   algorithms   used   to   solve   the  governing
equations, and  (2)  to  assure  that  the  computer code is  fully operational.   To
check  the  code  for correct  coding  of theoretical  principles and  for  major
programming  errors  ("bugs"),  the code  is run  using problems  for which  an
anelytical  solution  exists.    This  stage  is   also  used  to  evaluate  the
sensitivity of  the  code  to grid design,  to various dominant processes, and to
a wide selection of parameter values (Huyakorn et al. 1984; Sykes et al.  1983;
Ward et al. 1984; Gupta et al.  1984).


     Although   testing  numerical  computer  codes  by  comparing   results  for
simplified  situations  with  those  of analytical  models  does  not  guarantee a
fully  debugged  code, a well-selected set  of  problems ensures  that the code's
main program and  most of  Its  subroutines,  Including  all of the frequently
called ones, are being used  in  the testing.   In the  three-level test procedure
developed  by the  International  Ground Water Modeling Center (IGWMC), this type
of  testing is referred to as  level I (van  der Heijde et al. 1985).


      Hypothetical  problems are  used to  test  special features that cannot be
 handled  by  simple  close-form  solutions,  as  in testing  irregular  boundary
 conditions and  certain heterogeneous and anisotropic aquifer properties;  this
 is  the IGWMC level II  testing.


      For both level I  and level II testing,  sensitivity  analysis is  applied to
 further  evaluate code  characteristics.

Model  Validation

      Model validation  is  defined as  the comparison  of  model  results  with
numerical   data   independently   derived   from   laboratory   experiments  or
observations of  the  environment (ASTM  1984).     Complete  model  validation
requires testing  over the  full  range of  conditions  for  which  the  model  is
designed.   Model  development  is  an evolutionary  process,  responding to  new
research  results,  developments   in     technology,   and  changes   in   user
requirements.    Model  review  and validation needs to  follow  this dynamic
process  and should be  applied each time the model is modified.


      The objective  of model validation is  to determine  how  well  the model's
 theoretical foundation  describes  the  actual  system behavior  in  terms of  the
 "degree of  correlation" between  model  calculations and  actual measured  data
 for the cause-and-effect  responses of the system.   Obviously, a comparison to
 field data is  required.   Such  a comparison may take either of two forms.   One
 form, calibration,  1s the weaker form  of validation insofar as  1t tests  the
 ability of the code (and the model) to fit the field data, with adjustments of
 the  physical   parameters (Ward  et  al.  1984).    Some   researchers  prefer  to
 classify  calibration  as  a  form  of verification rather   than  a  form  of
 validation.  The other  form of validation  1s that of prediction.   This  is  a
 test for the ability of the model to fit the field data with no adjustments of


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the  physical   parameters.    In  p-inciple,  this  is  the  correct  approach  to
validation.    However,   unaveinability   and  inaccuracy  cf   field  data  often
prevent  such   u  rig-id  uDpro&cr.   Typically,   a  part  of  the  field  data  is
designated  as  eel ibr?tion  oats,   ard   a  calibrated  sU^-mdel  is  obtained
through reasonable  adjurtme-it  of parameter  vsii"~s.   Another  pert  of the f^eld
data is designated  &s  validation data:  the  Cc.libr£teG site  isode";  1s used in a
predictive rcooe  to  v;mulgte  similar da:a fc." corr.pari~g".  The quality of such
a  test is  therefore  determined  by  the extent  to  which  tne  site model  is
"stressed  beyond"  the  calibration  data on which   it  is  based  'Ward  et al.
1984).  In the IG'wMC ter.tinn procedure, this approach H referred to as level
III testing.
     For  many types of  ground-water  models, a complete  set of test problems
and  adequate  data  sets  for  the  described  testing  procedure  are not  yet
available.    Therefore.,  testing  of  these  models  is  gener^y   limited  to
extended  verification,  using existing analytical  solutions,
     Whether  a  model  's valid  for a  particular apnlicaticn  car  be assessed
through   the  use  of  performance  criteria,  sometimes  called validation  or
acceptance  criteria.   If  various uses  in  pi .inning and decision  making are
foreseen,  different perfonr.anc?  criteria ^ight ***»  (jp.MnH.   The  user  should
then carefully ched: the validity of  the model  for the  intPnHed use.


     Three  levels of validity  can be  distinguished  (ASTM 1984):

     (1)   Statistical Validity—using statistical  measures  to check agreement
          between  two   different  distributions,  the  calculated  one  and  the
          measured  one;  validity  is  established   by  using   an   appropriate
          performance or validity criterion

     (2)   Deviative Validity—If  not enough  deta  are available  for  statistical
          validation, a deviation coefficient D can be established,  e.g.,

               D - [(x-y}/xMOG %

          where  x  = predicted  value and v  -   measured  value.   The  deviation
          coefficient   might  be   expressed   as  a  summation  of   relative
          deviations.   If  ED  is  a deviative  validity  criterion   supplied  by
          subjective judgment, a model  can considered to be  valid  if D < ED;

     (3)   Qualitative Validity—using  a qualitative  scale for  validity levels
          representing  subjective  judgment:  e.g., excellent, good,  fair, poor,
          unacceptable.    Qualitative"  validity  is  often  established  through
          visual   inspection  of   graphic  representations  of  calculated  and
          measured data  (Cleveland  and  McGill 1985).

      The aforementioned tests  apply to single variables and  determine local-
 or-single  variable  validity;   if  more than  one variable  is  present  in  the
 model, then   the  model  should  also be checked for  global validity  and  for
 validity consistency  (ASTM 1984).   For a model with  several  variables to be
 globally  valid,  all  the  calculated  outputs  should  pass  validity  tests.
 Validity  consistency refers  to the variation of validity  among  calculations
 having different  input  or  comparison data sets.  A  model might be judged valid
 under one data  set but not under  another,  even within  the  range of conditions
 for which  the model has  been designed or  is supposedly applicable.  Validity


                                        37

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consistency can be evaluated periodically  when models  have  seen  repeated  use.


     Often, the data used for field validation are not collected directly from
the field but are processed  in  an  earlier  study.   Therefore,  they are  subject
to  inaccuracies,  loss of  information,  interpretive  bias,  loss of  precision,
and transmission and  processing errors, resulting  in  a  general  degradation  of
the data.


     As  noted  earlier, for  many  types of  ground-water models no  field data
sets are  available  to execute a complete  validation.   One approach sometimes
taken  is  that  of  code  intercomparison,  where  a newly  developed model  is
compared  with  existing models designed to solve  the  same  type  of problems  as
the  new  model.    If  the simulation results from  the  new code  do not  deviate
significantly  from  those  obtained  with  the  existing code  a  relative  or
comparative  validity  1s established.   It  is  obvious  that  as  soon as adequate
data sets become  available, all the involved  models  should  be  validated with
those data.


     Further development of  databases for  field validation of solute transport
models  is necessary.  This  is  also  the case  for  many  other  types of  ground-
water  models.   These research databases  should  represent a wide  variety  of
hydrogeological  situations  and should  reflect  the  various  types of  flow,
transport,  and deformation  mechanisms  present in the  field.   The databases
should   also   contain  extensive   information   on   hydrogeological,  soil,
geochemical, and  climatological characteristics.   With the development of such
databases  and  the   adoption   of  standard   model-testing   and   validation
procedures,  the  reliability of  models  used  in field  applications  can  be
improved  considerably.

Validation Scenarios—
     Often.,   various  approaches  to  field   validation  of a model  are  viable.
Therefore^   the  validation  process  should  start  with  defining  validation
scenarios.    Planning  end  conducting field   validation  should  include  the
following  steps (Hern et  al.  1986):

     •  define  data  needs  for  validation  and  select  an available data  set or
       arrange for  a  site  to study

     •  assess  the  data quality  in terms of  accuracy (measurement  errors),
       precision, and completeness

     •  define model  performance or acceptance criteria

     •  develop  strategy for sensitivity analysis

     •  compare model  performance with  established acceptance  criteria

Sensitivity  Analysis—
     An important  characteristic of a model is its sensitivity to variations or
uncertainty  1n Input  parameters.  Sensitivity analysis defines quantitatively
or   semiquantltatively  the   dependence   of   a   selected  model  performance


                                      38

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assessment measure  (or an intermediate  variable)  on a specific parameter  or
set of  parameters  (Intera 1983).   Model sensitivity can be expressed as  the
relative  rate  of change  of  selected  output  caused by a  unit change in  the
input.   If  the change in the  input  causes  a large change in  the  output,  the
model  is  sensitive  to that  input.   Sensitivity analysis is used  to  identify
those parameters most  influential in determining the accuracy end precision of
model predictions.   This  information is  of  importance  to  the user, as he  must
establish  required  accuracy  and  precision  in the  model   application  as  a
function of data quantity and  quality  (Hern et al.  1986).   In this context the
use  of  a sensitivity  index  as described by Hoffman and Gardner  (1983)  is of
interest.    It  should be   noted  that  if   models  are  coupled,  such  as  in
multimedia  transport  of  contaminants,   the  propagation  of  errors  and  the
increase  in uncertainty through  the  subsequent  simulations must be analyzed as
part of the sensitivity analysis.


PROPRIETARY CODES VERSUS  PUBLIC  DOMAIN CODES AND ACCEPTANCE CRITERIA

      Is  the  use of  proprietary  codes  in solving ground-water  problems for or
by  U.S.  EPA  acceptable,  or  should  they  be  banned  in  favor  of  publicly
available  codes?   Deciding this  policy issue has become  imperative  to the
Agency  in recent  years,  as an  increasing  number  of  modeling-based  analyses
performed  by  consultants in regulatory  compliance cases  are contested in the
courtroom,  and  Agency decision-making  processes  in  general  are  subject to
increasing  public  scrutiny.


      A  proprietary  code  consists of  computer software  that  is  sold, leased, or
used  on  a  royalty  basis,  which generally  conditions its  use and limits  its
distribution.   Some proprietary codes are publicly accessible,  but restricted
in  transfer and use.  According  to this  definition, proprietary  codes used for
solving ground-water  problems  could include:    (1)  ground-water simulation
codes,     (2)   databases,   (3)  statistical  packages,  and    (4)   graphical
packages.   Public domain codes  consist  of software and documentation that can
be  used,  copied, transferred,  distributed,  modified, or  sold without  any  legal
restrictions  such as violation of copyrights.


      There are  various  reasons  why  the use of proprietary codes  is regarded
problematic  by  EPA:     lack  of peer  review  and  validation;  problems  with
 intercomparison and  reproducability of  results; administrative  complications;
and  lack  of   access  to   software and  theoretical   basis.    On  the other  hand,
owners of  proprietary code rights often propagate the use  of these  codes for
commercial, scientific,  and other reasons.


      In  this  section the concerns  of the  Agency  are reviewed  and  advantages
 and  disadvantages   of  the  use  of   proprietary  codes  for  Agency  purposes  are
 discussed.   Finally,  it lists  the  elements which should  form the basis  of  an
 Agency  policy  regarding the use   of models,   both  proprietary  and  publicly
 accessible.

 Banning the Use of  Proprietary  Codes
                                       39

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     The Office of Waste Program Enforcement  (OWPE) prefers  the  use  of public
codes if litigation  is  anticipated,  assuming such code is available,  even  if
the code is  less  efficient  than an alternative proprietary  code.  Banning  of
proprietary codes is expected to eliminate some of the problems encountered  in
court cases.   One of these  problems  is  related  to  the notion  that  the  code
itself  and  its theoretical  foundation  might become contested.   Unrestricted
access  to  the  computer  code and  documentation is considered  crucial  in  such
cases.   However,  if  adequate  model  selection guidelines existed,  including
requirements for  code  review,  validation, and documentation  and were applied
consistently, such problems might be less significant.


     The inaccessibility of some proprietary codes and documentation can cause
other  problems.    EPA   regulations   (40  CFR  124.11   and  124.12)  provide  a
mechanism  for  formal public hearings  during the RCRA permit  process.   All
aspects  of EPA's decision making,  Including the use  of  ground-water models,
are  subject to  public  scrutiny.   EPA  use  of models not  accessible  by the
public  may complicate the proceedings  and  result in  EPA  having to duplicate
the  modeling effort  with  a  publicly  accessible model.  EPA's continued use of
nonpublicly  accessible  models  increases   the   likelihood  of  Federal  Open
 Information  Act  litigation.   EPA policy  restricting the use  of nonpublicly
 accessible models may reduce this  likelihood.


      Another consideration  brought  to bear in  this  issue   is  that  Regional
 staff does  not  have enough time  and expertise  to  evaluate models  or to go
 through a proper model selection  process without support  from model  experts.
 This support can be  provided indirectly by establishing a  list of reviewed and
 validated  models acceptable  to  the  Agency,  and  through  various  forms  of
guidance such  as reports,  and  by expert systems.


      Not all proprietary codes are  publicly inaccessible. The  private sector
rn&y  control  the use  and dissemination of  its computer models  through  copyright
protection,   patent  protection,  trade   secret   protection,   or  through  a
contractual  or license agreement.  Most  of  the  issues discussed above result
from attempts  by  some  companies  to  maintain tight controls  over their models
through trade  secret protection.   The rationale  is that a model contains some
formulation  that makes  1t  superior to those  generally available, and that this
formulation  gives the company  an  advantage  over its  competitors.   Exercising
control through copyright protection and contractual  agreements might  be more
difficult  to enforce than  trade secret  protection.   However,  they  allow  for
greater and  quicker acceptance of the model  by the  technical community.

Continuing the Use of Proprietary Codes

      Several  reasons  have  been  brought forward  for  the   continued  use  of
 proprietary  ground-water simulation codes:

          Use of  proprietary codes encourages  code development  for  solving new
          problems.   If  proprietary codes are  banned, this  Incentive  will  be
          removed, greatly  inhibiting future  code  development  1n the private
          sector.   Because  it  is difficult for private  companies  to  obtain
          funding  for code  development,  the main justification for investing


                                      40

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       corporate  funds  in  code  development  is the  anticipation  that  some
       development  cosls will  be  recovered  through code sales or value-added
       use.   Lapiiel  c;ain  is  a major  incentive  for code development.

       User  of  p-oprietery  codes encourages  private companies  V- enhance
       CDue the code  in the best possible  manner.   *g£i;\
        this  oc-*  ,.ot.  i:lv:sys 6ccurt  as  the  smaller ground-water  node!1.no
        softws'-e crlstnbutors  often  have  'limited  resources for  support  and
        maintenance of their  software.   Often, consistent user support  is not
        available for public domain codes.

        The use of proprietary databases,  statistical packages, and graphical
         packages is  rather widely  accepted; the current regulatory questions
         focus  specifically  on  the  use  of  ground-water  simulation  codes.
         Policies   applied   to   ground-water   modeling   software  should  be
         consistent  with those  established  for  other software.

Options for U.S. EPA

     From  the Study  Group  discussions,  the U.S. EPA it  became apparent  that
needs  to  establish  a consistent  policy concerning  selection  and use  of  well
tested and validated ground-water models  in all projects  carried out by or for
the Agency    Such  a  policy should address the Issues of model acceptance and
use  of Dropriet&ry codes  and should  be  consistent with  policies regarding
surface SIter and  air »dels.  It should  focus  on the basic  needs  and concerns
of theSr^ra. Sf!'IceTll headquarters as well  as the   Regional  Offices, and


                                      41

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should be  realistic with  respect  to the  Agency's  capabilities to  implement
it.  With respect to proprietary codes opinions of the Study Group  varied from
out-right banning  to ensuring a proper  place  for them in Agency  policy.   In
general,   Study  Group  members  agreed  on  the  Importance  of  the  following
elements  for an Agency policy regarding model acceptance:

    •    Establish a formal Agency mechanism to review and validate models for
         use  for  and  by  EPA and  define  model  acceptance  criteria.    This
         approach  should  be  restricted to  publicly  available,  noncopyrighted
         codes  and should prevent  shifting the  burden  of  testing  and  peer
         reviewing proprietary codes from the contractor to the Agency.

         The  Agency should  compile  and  distribute  a list  of reviewed  and
         validated  computer  codes  acceptable  to   the   Agency,   and  require
         contractors  (and  urge  PRPs)  to  use  them.    (This  approach  is
         comparable  to  the  one  employed  by  the   Agency  with  respect  to
         simulation software for surface water and air systems.)

     •    EPA  should identify proprietary  codes  that  it  regards  important to
         the  Agency's  mission;  such  codes  should be  brought  into the  public
         domain, after passing the Agency review and  validation process.

         A  special list  should  be compiled of  those proprietary codes that
         have passed  a  comparable  review  and  validation  process;  however,
         their  use   for   Agency   purposes   should   be   restricted   to
         noncontroversial  issues.

     •    Wherever  possible  the  Agency  should advocate  the use  of  publicly
         accessible  ground-water modeling software.

     The  Study Group recommends  that a general framework of nondiscriminatory
criteria  should be established  by  the Agency to apply  to  both public domain
and  proprietary  codes.  These criteria should  include:

     •    publication  and  peer  review  of  the  conceptual   and   mathematical
         framework

         full  documentation  and  visibility  of  the assumptions

     •    testing of  the  code  according to prescribed  Agency  methods;  this
         should    include   verification   (checking    the   accuracy  of  the
         computational  algorithms  used to solve the  governing equations), and
         validation  (checking the ability of the  theoretical foundation of the
         code to describe the actual  system behavior)

         trade secrets (unique  algorithms  that  are  not described) should not
         be  permitted if  they  might  affect the outcome of the  simulations;
         proprietary codes are already protected  by  the copyright  law

     In  establishing  a  model use  policy the Agency might take  a hard  line
requiring  that  acceptable  public  domain computer codes, where available,  be
used  in  situations  where a contractor  or  PRP's nonpeer-reviewed  proprietary
code  1s   unavailable for  full   EPA review  or where  such  a  review  would  be
burdensome  to  the  Agency  (e.g.,   documentation is poor,  Agency  resources

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scarce, etc.)-  Establishing such a prophylactic policy that  would  require  all
computer  codes  to  have  passed  peer   review,  be  validated,  and  publicly
available before  EPA would rely  on their results,  allows all  parties  (PRPs,
EPA,  State,  Intervenors,  etc.)  access  to  the  same  code to  perform  similar
analyses.


     Another way  to deal  with  the problem is  to  develop test procedures  and
validation criteria  acceptable  to  the Agency,  but  leave the  actual review  and
validation  process to be  initiated by the contractor  or PRP.  In  that  case
precautions  need   to  be  taken  to  assure that  the contractor  and  PRP  cannot
influence  the outcome  of the  review and  validation.    This  approach  allows
unique  proprietary codes to be  used  when called for.   These particular codes
would have to  be  addressed on a  case-by-case basis.


      In  the  interim, before  any  Agency  policy  takes  shape. Regional  and
Headquarter  project  officers  need  to be  sensitized  to the potential problems
of   approving  the   use  of   proprietary,   undocumented,  nonpeer-reviewed,
nonvalidated computer codes.


      In  case  an Agency  policy   includes  acceptance  of  proprietary codes
provisions  should be made regarding distribution of  program  copies of licensed
software   and  documentation   to   the   Agency  for  purposes  of   regulatory
compliance,  as well  as  provisions for  reasonable  use  by  third  parties  (at
reasonable  cost)  of  code documentation and  an  executable  version  of  the
program code,  or, at a  minimum, access to the use of the code.   Unreasonable
cost  to  a group,  such as a public interest  group,  could  violate the  provisions
of  the  Freedom of Information  Act.


      For  proprietary codes, the Agency  might  also  require  from  a  contractor
proof of copyright, ownership,  or  license  to perform,  display  and use the
code.   In  case  the Agency intends to  use the software  internally,  a  license
should  be  obtained  to  perform,   display,  use,  and  reproduce the code  and
related documentation in all  parts of the Agency.


      Bids  to the  Agency  can be  written  so  that code is acceptable only if the
foregoing  criteria are met.  Other code users, such  as Potentially Responsible
Parties (PRPs),  are not restricted to these options, but they take the  risk  of
being challenged   if they do not adhere to them.  •


QUALITY AND USEFULNESS OF MODEL STUDIES

Code Selection

      Using  models to analyze  alternative  solutions  to  ground-water problems
requires a number of  steps, each  of which should  be taken conscientiously and
reviewed carefully.   After the decision  to  use  a model  has been made,  the
selection  process  is  initiated.   Selection  of  a  model  is the  process  of
matching a  detailed   description of  the  modeling  needs   with well-defined,


                                       43

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quality-assured  characteristics   of   existing  models.     In   selecting   an
appropriate  model,  both  the model  requirements and  the characteristics  of
existing models  must  be  carefully analyzed.    Major  elements   in  evaluating
modeling  needs  are:    (1)  formulation of  the  management  objective  to  be
addressed  and  the  level  of  analysis  sought;   (2)  description  of  the  system
under  study;  and   (3)  analysis  of  the  constraints  in  human  and  material
resources available for the study.  Model  selection is partly quantitative and
partly  qualitative.   Many  subjective decisions must  be made,  often  because
there are insufficient data in the selection stage of the project to establish
the importance of certain characteristics of the system to be modeled.


     Definition  of  modeling  needs  is  based  on the management problem at hand,
questions  asked  by  planners and decision makers, and  on the understanding of
the  physical system, including the pertinent  processes, boundary conditions,
and  system stresses.  The  major  criteria  in  selecting  a  model  are:  (1)  that
the  model  is  suited  for  the intended  use;  (2) that the  model  is thoroughly
tested  and  validated  for  the  intended use; and (3)  that the model  code and
documentation  are complete and user-friendly.


      If different problems must be solved, more  than one model might be needed
or a model might be used in more than one capacity.  In such cases, the model
requirements for each of  the problems posed have to be  clearly  defined at the
outset  of the selection  process.   To a certain extent  this is  also true for
modeling  the  same  system  in different  stages  of  the project.    Growing
understanding  of the system and data availibility  might lead  to a need for  a
succession of  models of  increasing complexity.  In such cases,  flexibility of
the  model  or model  package might become an important selection criterion.


      It should be realized that a perfect match rarely  exists between desired
characteristics  and those of available models.   Many of  the  selection criteria
are   subjective   or  weakly  justified.    If   a match   is  hard  to  obtain,
reassessment of  these criteria and  their  relative  weight  in  the selection
process is  necessary.   Hence,  model selection  is  very much  an iterative
process.


      Special attention should be  given to model selection  procedures because
there is a  strong  interest, especially in  the Regions, in  having advice  and
guidance  available   for   model   selection   and   use,  given   site-specific
conditions.    Selecting  an appropriate model  is crucial  to the success  of  a
modeling project.   Special  measures are necessary to  assure  a proper selection
process.


      A  major  problem in model  use  is model  credibility.   In  the selection
process special  attention  should be  given to  assure the use of qualified
models  that  have undergone adequate review and testing.


      The various elements of model selection  should be  addressed  in  an Agency
guidance document.   In  a related  activity  ORD/OHEA is currently  involved  in


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the development  of  model  selection guidance  focussed  on  exposure  assessment.
Further information on ground-water model selection is presented in Rao  et  al.
(1981), Kincaid  et  al.  (1984), Boutwell  et  al.  (1985),  and Simmons and  Cole
(1985).


     In standardizing model  selection,  three major approaches  are  employed in
characterizing  the  validation of  numerical  models.   In  one,  the model  is
tested  according  to  established  procedures;  when  accepted,  the  model  is
prescribed in regulations for  use  in cases covered by those regulations.  This
approach  dues  not  leave  much  flexibility  for incorporating  the  advances of
recent  research  and technological  development.   The second approach includes
the establishment of  a  list  of  ground-water simulation codes  as "standard"
codes for various generic and  site  specific  management puposes.  To be listed,
a  code should  pass a widely accepted  review  and test  procedure.   However,
establishing   "standard   models"  will   not   prevent   discussion  of   the
appropriateness  of  a selected model  for analysis of  a new policy nor  of its
use  in a particular  decision-making  process.    In  discussions  at  OPPE  the
following related issues emerged:

1.   Is it better to  establish standard models for the Agency, or should  cri-
teria  or  guidelines  be  developed  by  which EPA  analysts can  evaluate  use of
models.   In  considering  this issue, questions have been raised such  as:

          Are there  legal  liabilities for setting  up certain models  as  accept-
          able?    (For  instance, if the  Agency  certifies  a  model for use, can
          the Agency no longer criticize an industry's use of that model  in a
          Superfund  case?)

          Does  certification squelch the development  of new, better models?

          What  balance should  there be  between using the newer, faster models
          and using  mature models already subjected to peer  review?

2.   Different types of models are appropriate  for different uses.  Their  role
in Agency ground-water  concerns  seems  to  be divided between  the models  used
for   national,   priority-setting  purposes  (EPA  ground-water  strategy),  or
regulatory   purposes,  and   those  models   used  for  site-specific purposes
 (Superfund   sites,   leaking  underground storage  tanks).   Separate  sets  of
criteria, or different types  of  models, must be established for each of  these
classes.


      A third  approach  is  to  prescribe  a  review-and-test  methodology  in the
 regulations and require the model  development team to show that the model  code
 satisfies the requirements.   This approach leaves room to update  the codes  as
 long  as  each version is  adequately  reviewed and tested.   An example  of  this
 last  approach  is the quality assurance program for models and computer  codes
 of the U.S. Nuclear Regulatory Commission  (Silling 1983).


      The Agency should assess  the use of  "Expert  Systems"  for assistance  in
 selection  and  use  of  available  models  for problem oriented needs.   Such  an
 expert  system  might  contain a  database  of  "acceptable"  models,  and  might


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include  options  for  system  conceptualization,  code  selection  and  project
evaluation.  A select group of Agency experts and outside experts  (in modeling
and artificial  intelligence)  could  be  brought together  with  a selection  of
potential  users  for  such an  assessment  and  to recommend a course  of action.
In  addition,   &  demonstret ion  project  could  be  initiated  to  explore  the
potential of expert systems in providing the  required guidance.


QUALITY ASSURANCE IN GROUND-WATER MODELING STUDIES

Definition and Role of Quality Assurance in Ground-water Modeling

     One  of  the major  issues  emerging  from the  Study Group concerned  the
quality  of modeling  studies  carried out  by  or for  the  U.S  EPA,  and  the
usefulness  of the study results 1n  the  Agency's  decision-making  process.   As
the  Agency's   decisions  are  contested,  often  in litigation,   the  analytical
framework  on  which  these  decisions  are based  also  can  become  contested.
Hence,  Agency   decisions  should  be based  on  the  use of  technically  and
scientifically    sound    data   collection,    information   processing,   and
interpretation  methods.


      In  litigating a case where  the expert  witness  relies  on  a ground-water
model,  the model (and its results)  must be  "of a type reasonably relied upon
by experts in the particular  field  in forming opinions or inferences upon the
subject"  (Rule 703;  Federal   Rules of  Civil  Procedure).   Although no criteria
have  been  established  within the  technical community,  it  follows  logically
that  the  models  must  show some minimal  level  of credibility  and reliability
before   they   may be  relied  upon  by  experts.    Such credibility  can  be
established  by  documentation  of  the  program,  publication  of the  model's
underlying theory, review, and validation,  and  its widespread  use within the
technical  concnunlty.   EPA runs the  risk of  undermining  its csse if  the models
it uses  cannot  be shown  to be credible.  In addition, where EPA continues to
use models not generally accepteo or used by the technical community,  it will
continue  to be  feced with litigation on the formulations of such models.  If
on the other  hand EPA  uses models  that  are generally  accepted by the  technical
community, model  credibility  is  more  likely  to  be recognized  by  opposing
parties.   Furthermore,  concerns  raised in  the  Regions  and by OWPE  regarding
the often unsatisfactory use of models  in field studies, are  identicative of
the  need  for guidance and QA policies  in  model application.   Comprehensive
quality   assurance,  as   applied  to data  collection,  data   processing,   and
modeling,  and sound  model selection policies provide  the mechanisms  to ensure
that  decisions  are  based on the  best  available  data  synthesis  and  problem
analysis methods.


      Quality  assurance  (QA)  in  ground-water modeling  is  the procedural  and
operational framework put in place  by the organization managing the modeling
study,  to assure technically  and  scientifically  adequate execution of  all
project  tasks  included  in the study.   QA in ground-water modeling  should be
applied  to both model  development  and model  application  projects.   The  two
major elements of QA are quality control (QC) and quality assessment.  Quality
control   refers  to  the  procedures  that  ensure  the  quality  of  the  final
product.    These procedures  include   the   use  of  appropriate  methodology,


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adequate validation, and  proper  usage (Taylor 1S85).  To  monitor  the  quality
control procedures ana to evaluate  the  quality  of  the study  products,  quality
assessment is applied.  Quality  assessment  consists  of two elements:  auditing
and technical review.  Audits  are  procedures  intended to assess  the degree  of
compliance with QA requirements, commensurate with  the level  of  QA prescrioed
for the project.   Compliance  is  measured in terms  of traceability of records,
accountability   (approvals   from  responsible  staff),   and   fulfillment   of
commitments  in  the  QA  plan.   Technical  review  implies  independent evaluation
of the  technical  and  scientific bases  of a project  and  of  the  usefulness  of
its results.
     Various phases of quality assessment exist for both model development and
application.    First,  review  and  testing   is  performed by  the  author,  and
sometimes  by  other employees  not  Involved in  the project,  or by  invited
experts from outside  the organization.   Also  to  be considered is the quality
assessment  by the  organization  for which  the  project  has  been carried out.
Again, three  levels can  be distinguished: project or product review or testing
by  the  project  officer  or  project  monitor, by technical  experts within the
funding or  controlling organization, and  by  an external  peer review group.


     Although  the  U.S.   EPA has  a centrally  managed  QA  program  in   place,
pertinent  regulations  and guidance  are very much  oriented  to the quality of
measurements  and  monitoring  techniques.   Data   synthesis  methods such  as
modeling  are  not well covered  by the current policy.   At  present,  ad   hoc QA
requirements are being  developed  for  individual  modeling projects by various
project  officers  in  cooperation  with quality assurance officers,  and might
differ  from Office to  Office.   These  requirements  are  called for in part by
the   Administrative  Procedures   Act,   which   compels  EPA  to   maintain  an
administrative record  to support  its regulatory decision making.  If computer
models  are  used  by  EPA  in decision  making,  data  input   records  should be
maintained  as  part  of  the  administrative   record.   The Study  Group strongly
suggests  extending current  EPA QA  approaches  tc  data  synthesis and problem
analysis  methods,  especially modeling.  Such a  policy,   following the approach
taken  in the  QA  policy for environmental  measurements,  is outlined in  this
section.   The approach  presented  applies to the quality assurance and quality
assessment  of  both modeling research  and  development  projects as well as to
studies  in  which  existing  models are used to  assist management in decision
making.   Some related issues such as  model  testing and  the use of proprietary
codes  are discussed in  separate sections of this  report.

Current  EPA Quality Assurance Policies

      In  May  1979  EPA  initiated   an Agency-wide  quality assurance program to
ensure  that all environmental measurements  conducted by the regional offices,
program offices,  EPA  laboratories,  contractors,   grantees,  or  other  sources
would  result  in  scientifically  valid and  defensible  data  of  documented
precision,   accuracy, representativeness,  comparability  (standardization),  and
 completeness  (Stanley  and Verner  1985).   The Administrator delegated  to  the
 Office  of  Research and  Development  (ORD)  the  authority  and responsibility  for
 developing,   coordinating,  and   Implementing  the  mandatory  Agency-wide  QA
 program.
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     ORD established the Quality Assurance Management Staff  (QAMS)  to serve  as
the  central  management  authority  for  this  program.   The  QAMS  activities
involve:   (1)  the development  of  policies and procedures;   (2)  coordination
for  and  direction of the  implementation  of the Agency  QA  program; and   (3)
review,  evaluation,  and audit  of  program activities  involving  environmental
monitoring and other types of data generation.


     In  an  effort to ensure  consistency  of the QA  program with  the Agency's
mission  and objectives,  the  requirement  was  established  that  a  single  QA
Officer  be  designated for each  Agency  organization involved in  the program,
and  that adequate data documentation would  be prepared for  each  measurement
activity to ensure that the results were of known quality and defensible.


     In  April  1984,  EPA  Order  5360.1,  "Policy and Program Requirements  to
Implement  the  Mandatory Quality Assurance Program," was issued and,  for the
first  time, provided a  regulation  basis  for the Agency QA program.  The order
specifies  certain activities crucial to  the  implementation of a  QA program.
These  activities  include the  following:

     •   Development  and  regular updating  of a  QA project plan for all projects
        and  tasks  involving environmentally related measurements, in  accordance
        with approved guidelines

     •   Assuring  implementation of  QA  for all  contracts  and  financial  assis-
        tance  involving  environmentally related measurements,  as  specified in
        applicable EPA regulations,  including subcontracts and subagreements

     •   Conducting  audits  (technical  system,  performance  evaluations, data
        quality bench studies,  etc.) on  a scheduled  basis  of organizational
        units and  projects  involving environmentally  related measurements

     •   Developing  and  adopting   technical  guidelines  for  estimating data
        quality in   terms  of  precision,  accuracy,  representativeness, com-
        pleteness, and  comparability,  as  appropriate,  and  incorporating data
        quality requirements   in  all  projects and  tasks  involving  environ-
        mentally related measurements

     •   Establishing  achievable data  quality  limits  for  methods  cited  in
        regulations, based on results  of  method evaluations  arising from  the
        methods standardization process  (e.g.,  ASTM  Standard D2777-77)

     •   Implementing corrective actions, based  on audit  results,  and incorpor-
        ating  this process  into the management  accountability  system

     •   Providing  appropriate training  based on perceived needs,  for all  levels
        of QA  management,  to assure that  QA responsibilities and  requirements
        are understood at  every stage of  project Implementation

      Each Agency organization engaged  1n environmentally  related measurement
 is required to submit a QA Management  Plan (QAMP)  for approval  by the Agency's
QA Management  Staff.  The  QAMP sets forth  the management philosophy  of  the
organization with respect to quality assurance/quality control.   It identifies


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the key elements of the QA program, how they are to be implemented,  and who  is
responsible for the implementations.


     The QA  project  plan is  a  specific delineation of  an  offerer's  approach
for accomplishing  the QA  specification in  a  Statement  of Work.   When a  QA
project plan  is not  required  as a part of  the technical proposal,  the  Con-
tracting Officer  may  require the  QA  project plan as a  deliverable under  the
contract.   The plan should address the  following:

    •  A  statement  of  policy   concerning  the  organization's  commitment  to
       implement  a QA  program   to  assure  generation of  measurement  data  of
       adequate quality to meet  the requirements of the Statement of Work

    •  An  organizational  chart  showing the  position of  the  QA function  or
       person  within  the organization.   It  is highly desirable that the  QA
       function  or  person  be  independent  of  the  functional  groups  that
       generate measurement data

    •  A delineation  of  the  authority and  responsibilities of  the QA  function
       or  person  and  the   related   data  quality  responsibilities   of  other
       functional  groups of the  organization

    •  The  type  and  degree of  experience  in  developing and  applying  QA
       procedures  to  the proposed sampling  and measurement methods  needed for
       performance of  the Statement of  Work

    •  The  background and experience of the personnel  proposed to  accomplish
       the QA  specifications  in  the Statement of Work

    •  The  offerer's  general  approach for accomplishing the QA specifications
       in  the  Statement  of Work

EPA Quc'ity  Assurance  Options for Ground-water  Modeling

     The primary  goal  of an EPA ground-water modeling QA program should be to
ensure that  all  modeling-based  problem analysis supported by the Agency is of
a  known   and   scientifically  acceptable  quality,  and  is   verifiable  and
defensible.  Decisions by management  rest on the  quality of environmental data
and data analysis; therefore, program managers should be responsible for: (1)
specifying  the  quality  of   the data  required from  environmentally related
measurements;   (2)   indicating   the  level   of   problem-solving-oriented  data
analysis;  (3)  specifying the  quality  required  from the  tools used  in the
analysis  (e.g.,  models); and (4)  providing  sufficient  resources to assure an
adequate level  of  QA.


     All routine  or  planned projects or tasks  carried out for  the U.S.  EPA or
from which the results will form  the basis  of  Agency action and  which involve
environmentally  related  measurements,  information processing, and  modeling,
should  be  undertaken  with  an adequate  QA  plan.  Such  a plan should specify
goals  for  the quality  of  resulting data and processed  information acceptable
to  the user,  contain  detailed   description  of the  measures  to be  taken to
achieve prescribed quality  objectives,  and  assign responsibility  for  achieving


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the  goals  (QC).    It  should also  contain  procedures  for  documenting  the
activities within  a project  in  order to  provide evidence that  standards  of
quality have been met.  Different levels of QA can be distinguished, dependent
on use  of the modeling results; e.g.,  1f  results are expected to  be  used  in
litigation, a high level of QA (including quality assessment!)  1s required.


     EPA  should have  effective  quality  assessment  procedures  1n place  to
monitor  the  QA  performance  of  the  modeling project teams.    QA  should  be
applied  to all  stages  of  the  modeling project,  not just at  the end  as  is
currently  often  the  case;  QA should be  an  Integral  part of all  projects.   It
should not drive or manage the direction of a project nor is it intended to be
an after-the-fact filing of technical data.


     The  Agency's  quality  assessment  process should be conducted throughout
the modeling process, with stop/go decisions at each critical  point.


     A  paper  trail  for  QA  in model  development and application is required by
the  Administrative  Procedures  Act.    However,  at  present  there  are  no
guidelines for modeling projects,  nor  1s  there a central document  room for
material  collected  on a case.   As  an  example,  a report  on a  modeling project
should  give:

     •   assumptions

     •   parameter values and  sources

     •   boundary  and  initial  conditions

     •   nature  of grid and  grid design  justification

     •   changes  and verification  of changes  made  in code

     •   actual  input  used

     •   output  of model  runs  and  interpretation

     •   validation  (or at least calibration) of model

     In addition,  depending on  the  level  of  QA required,  the following files
may  be  retained  (in  hard-copy and,  at  higher  levels,  in digital  form):

     •   version of  the  source code used

     •   verification  input  and output

     •   validation  Input and output

     •   application  Input and output

     If any modifications  are made  to the  model  coding for a specific  problem,
the  code should be tested again; all  QA for  model  development  should  again be


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applied  including  accurate recordkeeping  and  reporting.    All  new input  and
output files must be saved for inspection and possible reuse.

Model Development—
    Ideally,  QA  should  be  applied  to  all  ongoing  and  yet-to-be-developed
codes,  and  should  include  such aspects  as  verification  of  the  mathematical
framework, field  validation,  benchmarking, and  code  comparison.   At  a minimum
the  theory  should  be peer-reviewed;  a  fully documented  version should  be
available for  testing.   Different types  of  QA  are required for numerical  and
analytical  models;  in  particular,   such a  procedure should  call  for  the
verification of  the assumptions underlying the use of an  analytical  model  and
for  validation  of  numerical  models.   A detailed  dicussion of  testing  and
validation of  ground-water models  is presented in the section on ground-water
model testing  and validation.


      QA   for   code  development   and  maintenance  should   include  complete
recordkeeping  of the model  development, of modifications made 1n the code, and
of the  code validation process.

Model  Application—
     A lack of consistency  in model  acceptance  and  use  prevails across  the
Offices.   This  is  of  particular concern because acceptance of a model in one
Office  confers  legal acceptability  for all  the Agency.  ORD's role should be
to   establish   criteria  for  evaluation  and  use of  models  and to provide
technical  expertise as  needed.


      QA  in  model  application   should  address  all  facets  of  the  model
application process:

     •  correct and clear formulation of problems to be solved

     •  project description and objectives

     •  modeling approach to the project

     •  is modeling the  best available  approach and  if  so, is  the  selected
        model  appropriate and cost-effective?

     •  conceptualization  of  system  and processes,   including  hydrogeologic
        framework,  boundary conditions,  stresses, and  controls

      •  explicit  description  of  assumptions and simplifications

      •  data acquisition and  interpretation

      •  model  selection  or justification for choosing  to develop a new model

      •  model  preparation  (parameter selection, data entry  or reformatting, and
        gridding)

      •  the validity of  the parameter values used in the model application
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   •  protocols  for parameter estimation  and model  calibration,  to provide
      guidance, especially for sensitive parameters

   •  level  of  information  in  computer  output  (variables  and  parameters
      displayed, formats, layout)

      identification  of calibration  goals and  evaluation  of how  well  they
      have been met

   •  sensitivity  analysis

    •  postsimulation  analysis  (including  verification  of reasonability  of
      results,  interpretation of results, uncertainty  analysis,  and the use
      of  manual  or automatic  data processing  techniques, as for contouring)

    •  establishment of appropriate performance  targets  (e.g.,  a 6-foot head
      error  should be compared with a 20-foot head gradient or drawdown, not
      with the 250-foot aquifer  thickness!); these targets should  recognize
       the limits of the data

    •   presentation and documentation  of  results

    •   evaluation  of  how closely the modeling  results  answer  the  questions
       raised by management

In addition,  a project QA plan should  contain:

    •   title page with provision  for approval  signatures

    •   table of contents

    •   project organization(s) and responsibilities

    •   quality  assurance objectives   for  modeling,  in  terms  of  validity,
       uncertainty, accuracy,  completeness, and comparability

    •   internal quality assessment checks and  frequency

    •   quality assurance performance audits and their frequency

    •   quality assurance reports to management

    •   specific procedures used  in  routinely  assessing  validity,  uncertainty,
       completeness, and comparability of modeling studies

    •   corrective action


TECHNOLOGY TRANSFER AND TRAINING TO SUSTAIN
AND IMPROVE EXPERTISE OF AGENCY PERSONNEL

    In  1982  the Office  of  Technology Assessment  of the U.S.  Congress  (OTA)
published  a   report on  the  use  of  models   for  water   resources  management,
planning, and policy.   As discussed in a previous section of this report, the


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Study  Group  found  that  many  of  OTA's  conclusions  and  recoimiendations  on
technology transfer  and training apply  to  ground-water  modeling at the  EPA.
Tnese include such OTA findings as the following:

    •  Levels of  communication between decision makers and modelers are  low,
       and little coordination  of  model development,  dissemination,  or  use
       occurs within individual federal agencies.

    •  Developing  and  using  models  is  a  complex  undertaking,  requiring
       personnel  with  highly developed  technical   capabilities,  as  well  as
       adequate  budgetary  support  for  computer  facilities,  collecting  and
       processing data, and  such support services as user assistance.

    Technology  transfer  in  general   means  dissemination  of  information  on
technological  advances through communication and education.   When applied to
ground-water   modeling,   technology   transfer   Includes  dissemination  of
information  about the  role of modeling  in water  resource management,  model
theory, the process  and management of  modeling,  availability and applicability
of  model   software,  information  on quality  assurance,  model  selection, and
model  testing  and   verification.   It also pertains  to the  distribution of
computer  codes  and documentation,  and  includes   assistance  in transferral,
implementation,  and  use of  codes.

     The  OTA report  considers  specific  education  and  training of  model de-
velopers,  users, and  managers in  aspects  of water resources  modeling,  as a
critical   component  of  technology   transfer.     Other  technology  transfer
mechanisms  include  the distribution  of  published,  printed, or  electronically
stored  materials, such as  reports,  newsletters, papers, computer codes,  data
files,  and other communications, and  discussions and  information exchanges  in
meetings,  workshops, and  conferences.


      Effective communication  forms the  basis   of  technology  transfer.    Com-
munication is often hampered because of insufficient communication channels.
 incompatible   language or  jargon use,  existence  of  different  concepts,  and
 administrative impediments.


      Despite recent examples  of  successful modeling use  in developing  ground-
 water protection policies  in the United States and abroad, managers  still  do
 not rely widely  on modeling for decision-making.   One  of  the major obstacles
 is   the   inability   of    modelers    and   program   managers   to   communicate
 effectively.    An  ill-posed problem  yields  answers  to  the  wrong  questions.
 Sometimes, this  is  the  result  of  managers and  modelers speaking different
 jargon.


      On  another  level of  communication,  managers  should  appreciate  how dif-
 ficult it  is  to explain  the results  of  complicated models to  nontechnical
 audiences such  as  in public meetings and  courts  of law.   One useful  means of
 overcoming these limitations  in  communication  Involves the  effective  use of
 audio-visual aids.
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Technology Transfer and Training in EPA

    Ground weter  and  ground-water modeling expertise is disjunct  within  EPA.
Program Offices display different  approaches to  the  role of  modeling  In their
activities end show different levels of sophistication in model use, either in
a generic sense during  the  development  of  policies  and  regulations, or in the
preparation of  site-specific  guidance.   Consequently, many inconsistencies in
ground-water modeling have resulted.

    The Study Group concluded  that among  the modeling  issues addressed during
its meetings, retaining and improving the expertise of EPA  personnel deserve a
high  priority.   Because  1t  1s expected that  model  use will  increase  in the
future, the  development of  in-house expertise,  by whatever  means, appears to
be a major priority.

    A  major Impediment  to  meeting the  modeling needs of  the Agency  is the
inadequacy  of  the  current  levels  of  model-related training  and  information
exchange.   If  models are to  be  used  effectively in water  resources analysis,
training  in basic concepts  of modeling and  in proper interpretation of model
result!:  must  be  offered to  decision  makers  at  all  levels  of  the  Agency.
Further,  there  is a  need  for  specific  training  in  the  use  of individual
models,  and  a need  for continuingly  informing of  and  educating  users and
managers  in research  developments, new  regulations and policies,  and field
experience.

    Most  EPA technical  and managerial personnel  do not need to become modeling
"experts,"  but  should  have  sufficient training to be knowledgeable users or at
least  competent  judges of  the  appropriateness of models  used  by  PRPs and
contracted  consultants.

    The  return on investments made in applying mathematical  models to ground-
water  problems depends a great  deal  on  the  training  and  experience  of the
technical  support staff involved in their use.  Managers should be aware  that
specialized  training   and  experience  are  necessary  to  develop  tna   apply
mathematical  node Is,  eno that  relatively few technical support  staff can be
expected  to  nave  such  skills.   This is  due in  part to the  need for familiarity
with  a number of  scientific  disciplines,  so that the model  may be structured
to faithfully  simulate real-world problems.  Managers should have  some working
knowledge  of  the  sciences  involved  so  that  they  might  put   appropriate
questions to specialists.   In practice this means that ground-water modelers,
technical  staff,  and their supervisors   should  become  involved in continuing
education efforts,  and managers should  expect  and encourage  this.   They  should
be  sensitive  to  the financial  and time  requirements  necessary  for  adequate
training.    (One  does  not become  a competent  modeler in  a  course of a  week;
that  takes  years  and  is a combination  of  training and experience.)

    Unfortunately,   because  of   the   staff's  professional  beckground  and
expertise and  because  of  the  nature  of  the  activities  in which  they are
involved, modeling often has a  lower  priority  than some  other ground-water-
related  training  needs, such  as  general hydrogeology,  data collection and
analysis, and the administrative and  legislative framework  in which  the  work
is  carried  out.    However,  knowledge  of  basic hydrogeology  is  a  necessary
preface  to effective  modeling.
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Information Exchange on Ground-water Modeling

    There  is  an  urgent  need  for  expanding  existing  and  developing  new
mechanisms to  disseminate and exchange technical  information.   Two  different
approaches exist to information exchange:  (1) the receiver actively  seeks  the
required  information  or  technoloyg;  (2)  the  receiver  has  a  passive  role
insofar   as   supervisors  or  internal   or  external   specialists  bring  the
information or technology  to the potential user.


     EPA  has  various  mechanisms  in place  to  facilitate  both  approaches.
Reports on  research and development  carried  out  with funding from the Agency
are  published and  distributed either through EPA's  Center for  Environmental
Research  Information  (CERI), in Cincinnati,  or through the National  Technical
Information  Services   (NTIS),  in  Springfield,  Virginia.     Furthermore,  an
extensive  technology  transfer  program  for  ground-water  modeling  has  been
developed by  the  International Ground Water Modeling  Center  (IGWMC) with major
funding  from EPA,  and  includes  information  exchange, software  distribution,
training  activities,   and  model  user assistance.   Finally, the  EPA-suoported
National  Ground  Water Information Center, operated by the National Water Well
Association  (NWWA),  provides  access to  a large  information  base on  ground-
water  literature.  Despite  these  efforts, information from research projects
often  is  not disseminated  effectively  to  potential  users  in the national
Program  Offices or  the Regional Offices, and  the staff is  not aware  of the
existence of  certain  EPA guidance  documents  and  software.   Technology transfer
is  ineffective if  it  simply consistes  of reports sent to  Regional or Program
Office  libraries.


     To  resolve  this  problem,  steps  should be  taken to  promote  increased
communication and sharing of experiences  among  staff  through:

     •   compiling a list of  all potential model  users and  "modeling experts"
        within the Agency; such  a  list  can be maintained by ORD and should  be
        accessible via  computer/phone networks  to all Agency  staff;  this  list
        should be  provided  to organizations  such as  the   IGWMC  and  NWWA  to
        enable them  to  reach  out  efficiently to  all EPA staff  interested  in
        ground-water modeling

     •   establishing an electronic bulletin board on ground-water modeling  that
        is open to all staff interested  in modeling

     •   organizing  regular  exchange  sessions  for  personnel  involved  with
        similar  modeling-related   activities   throughout   the  Agency;   these
        sessions would  permit staff  members  to  keep  track  of developments  in
        each  other's  projects  and  to  institutionalize  their experience  with
        applications of  particular models to particular sites

      Meetings of  Regional and ORD people should  be  held  to address questions
 such as:

     •   What support  is  available to  Regions?
                                       55

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    •  What  Is  the experience  with particular  problems  in  the  various  Re-
       gions?   (Turnover  of personnel may  be  partly  due  to the feeling  that
       they are alone, "out on a 11mb.")

     For assistance with  specific problems, project managers and other  staff
should  have   access   to   senior   hydrogeologists  with  extensive   modeling
experience,  such  as  (1)  in-house Regional  experts,  (2)  experts within  ORD,
perhaps  located  at  EPA  labs;   (3)  contractors, or  (4)  experts  from  other
agencies.   In  addition,  adequate use should  be made of federally  supported
organizations such as the  International Ground  Water  Modeling  Center, and the
National Center for Groundwater Research.


     When particular models are  applied at  particular sites,  the  Agency needs
to institutionalize the experience.  A central  clearinghouse should be created
for  keeping records  on models  used  1n  the Agency.   It  should  have readily
available  Information on:   (1)  where  and  under what conditions they have been
used;  (2) what  results were provided in terms of usefulness  to management; and
(3)  what administrative, technical, and legal problems were  encountered.  This
information  should  Include   contact   persons,   site   descriptions,   model
modifications,  and details on QA  procedures.


      Each  program office,  regional office, division,  or branch  involved in
ground-water modeling  should  have  its  own   working  library  of  pertinent
publications maintained  by a coordinator who will  ensure that the library is
up-to-date   and  that  each  staff  member   involved  in ground-water  projects
receives  important information  in a timely way.   Each  Regional  and Program
Office should have a  ground-water library and  subscribe to  at least  the major
ground-water journals.   Wide distribution of EPA manuals, guidance documents,
and  research  reports within the Agency  is crucial.  The "EPA  model  user"
mailing  list mentioned  above could be used  for the distribution.  Many publi-
cations  are in the open  literature,  and  provision  should be made for distri-
buting their reprints.

Training

     To address  effectively the  issue of improving Agency  expertise in ground-
water  modeling,  an  evaluation   should  be made  of  the  existing  modeling
expertise  of Agency  personnel  and  the  requirements  in  terms  of educational
background  and  level of  experience for  the   technical,  administrative, and
management  staff  Involved in modeling projects.  Based on such an analysis, a
training  program  for staff  of Regional  Offices  and Headquarters  should be
developed.    An Agency  training  program  should be based on  the realities of
needs,  staff  backgrounds,  administrative  structures  and  constraints,  and
changes  in staff.  Such training  should be mainly  to provide  skills  needed to
evaluate  adequately  the  effectiveness  of  model  codes and modeling  work, as
only a few of the staff will  be (or should  be)  in a position  to become experts
in modeling.

     There   are  many ways  to obtain  education on  such  advanced  ground-water
topics as  mathematical modeling.   Existing  and  potential  approaches  Identified
include:
                                      56

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    •   ongoing   training  workshops  at  selected  Regional  Offices  and  at
       Headquarters;  these should be expanded to include  all Offices

    •   training courses at the EPA research laboratories

    •   presentation of courses by EPA experts (in the EPA Institute)

    •   attendance  at  such  external  training programs  as professional  short
       courses   organized  by  universities,   the  International  Ground  Water
       Modeling  Center,   the  National  Water  Well Association,  and  the  U.S.
       Geological Survey

    There should be satisfactory  performance  on competency  tests  in  the  short
courses and an  achievement  of A or B in academic courses, or  the staff member
should pay back the Agency  for money spent.

Other possible  approaches include:

    •  stationing  Regional  or Headquarters staff at one of the Agency research
       laboratories or  the  USGS for a few months in order to become familiar
       with current research  1n general, and modeling in particular

    •  3-  to  6-month  positions  at  EPA  for  university  faculty  on sabbatical
       leave

    •  courses  offered  at EPA by  outside consultants

    •  lecture  tours  through  the  Regions

    •  structured  self-study

    •  seminars on particular case studies

    •  part-  or   full-time academic  study   conditional  to  long-term  public
       service  committment.

    -  the  use of television in  workshops presented  simultaneously  in all
       Regions  and Offices.

    Generally,  the more contact time with  instructors,  the greater the  benefit
 in terms   of   increased problem-solving  capabilities.    Unfortunately, the
 greater  the contact  time  with  instructors  in  formal education  settings, the
 greater  the  disruption  in terms of  increased absence from  the job.   EPA's
 current  ground-water training efforts tend  to  be  of  short contact time.   This
 is aggravated  by a lack  of  in-house   programs  to reinforce  training and
 education  received   in  formal  settings.   A promising  alternative  to  formal
 education  is self-study coupled  with  obtaining experience, under the  guidance
 of a  senior  modeling  specialist.    Review of  work in  progress by  someone
 capable  of acting  as  a  foil  for ideas and  approaches would  substantially
 improve   model  applications.     Given   the  shortage  of   senior   modeling
 specialists   1t  would  seem  advantageous  to maximize  the time  they  spend  on
 model applications,   and to  minimize the  time they spend  on  other  duties.
 Workshops  should  include case  studies of  model application to actual  problems
 encountered by Regional and  Program  Offices.   Manuals  should  be prepared  to


                                       57

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explain when and how models  should  be  used  and  when a geohydrological  modeler
should be consulted.

Recruitment and Retention

    Difficulty in attracting and retaining staff with suitable backgrounds for
applying ground-water models, along with high turnover rates among such staff,
are  serious  problems  in  all  EPA  Offices.    Losing  trained  personnel  to
consulting  firms seems  to be the result  of  the significantly higher salaries
offered  and   the   recognition  given   by   private  firms   to   the  special
qualifications  of  hydrogeologists.    Three  steps  should  be  taken  to  improve
this situation:

     •  The  Office of  Human  Resources  should  establish the position of "hydro-
       geologist."  OTA's Superfund Strategy report lists establishment of the
       position as  Us highest priority, and the Study Group concurs.

     •  A career path  for  hydrogeologists should be established through the GS-
       15  level.   The GS-15 slot  would be established for a  "National Expert"
       as   defined  by  Civil  Service  so  that  EPA  can  retain  senior-level
       experience  comparable to that found in consulting firms.

     •  The Agency  should encourage staff to take  short  courses and graduate-
        level  courses  in ground-water  geology  and  modeling at  the Agency's
       expense,  and   the  graduate  courses  should  be  allowed  for  degree
       credit.   As  in some  other federal  agencies,  EPA could  require a certain
       number of  years  of  service with  the  Agency for  a given  amount of
       additional  training  at the  Agency's  expense.   This  would control the
       rate at which  trained  staff are  lost,  and would  provide a continual
       supply of trained  younger staff.

     Another major  way to maintain  an  adequate  level  of in-house expertise  in
ground-water  modeling   is   by  assuring  that  properly  trained,  experienced
technical   staff can  continue  to perform in a  technical  role without  losing
opportunities for  career   advancement  (i.e.,  avoiding  promotions  of  good
technical   people  to administrative  positions  as the only available  career
path).
                                       58

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                                        61

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                                   APPENDIX

                          COMPOSITION OF  STUDY GROUP

     The Study  Group was composed  of three  EPA  modeling experts, three  EPA
model users, three non-Agency modeling experts, and the  Chairperson.   Members
were as follows:

Chairperson:

    Paul K.M. van der Heijde
     International Ground Water Modeling Center
    Hoicomb Research Institute
    Butler University
     Indianapolis, Indiana

EPA  experts:

     Douglas Ammon
     Hazardous Waste  Engineering Research Lab
     Cincinnati,  Ohio

     Robert  Carsel
     Environmental Research  Laboratory
     Athens, Georgia

     Joseph  F.  Keely  (until  July 1,  1986)
     Clinton W.  Hall  (after  July 1,  1986)
     R.S.  Kerr  Environmental  Research Laboratory
     Ada,  Oklahoma
 EPA users:

     Peter Ornstein
     Office  of Waste Programs  Enforcement
     Washington, D.C.

     David Morganwalp
     Office  of Drinking Water
     Washington, D.C.

     Zubair  Saleem
     Office  of Solid Waste
     Waster  Management and Economics Division
     Washington, D.C.
 Non-Agency experts:

     James W.  Mercer
     GeoTrans, Inc.
     Herndon,  Virginia
                                      62

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    Richard  A.  Park
    Holcomb  Research Institute
    Butler University
    Indianapolis, Indiana

    Suresh C. Rao
    Department of Soil Science
    University of Florida
    Gainsville, Florida

     In order to ensure the broadest possible participation  and representation
by  Program  Offices,  but  without increasing  the  size  of the  Study  Group to
unmanageable  proportions,  a  number of Agency  personnel  (listed  below)  were
designated  as  Corresponding  Members.   These individuals were copied  on all
correspondence  and were asked to  cownent  on rthe interim and final documents
produced  by  the Study Group.  Some  of  the Corresponding Members attended one
or more of the  Study Group meetings.

Study Group  Corresponding EPA Members:

    James Bachmaier
    Office of  Solid  Waste
    Waste Management  and Economics  Division
    Washington, D.C.

    Stuart  Cohen
    Office  of  Pesticide Programs
    Exposure Assessment Branch
    Arlington,  Virginia

    Norbert  Dee
    Office  of Ground Water  Protection
    Washington, D.C.

    Michael  Gruber
    Office  of Policy Analysis
    Washington, D.C.

    Stephen C. Hern
    Exposure Assessment Research Division
    Environmental Monitoring Systems Laboratory
     Las Vegas, Nevada

     Seong T. Hwang
     Office  of  Health and Environmental Assessment
     Washington, D.C.

     David Kyllonen
     Underground  Injection Control Section
     Region  IX
     San Francisco, California
                                       63

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    Matthew Lorber
    Office  of  Pesticide  Programs
    Exposure Assessment  Branch
    Arlington,  Virginia

    Tom Merski
    Office  of  Groundwater
    Region  III, Philadelphia,  Pennsylvania

    Lee Mulkey
    Environmental Research Laboratory
    Athens, Georgia

    William A. Mullen
    Office of Groundwater
    Region X, Seattle, Washington

    Annett Nold
    Office of Pesticides and Toxic Substances
    Washington, D.C.

    Hope Pillsbury
    Office of Policy Analysis
    Washington, D.C.

    Herbert Quinn
    Office of Research and Development
    Water  and Land Division
    Washington, D.C.

    Paul B. Schumann
    Office of Solid Waste and Emergency Response
    Hazardous Site Control Division
    Washington, D.C.

    Carol Wood
    Office of Ground Water
    Region I, Boston, Massachusetts

     EPA project officers  for  the  Study  Group  activities  were:   Joseph F.
Keely  (ORD/RSKERL, Ada,  OK)(until  July  1.  1986),  Clinton W.  Hall (ORD/RSKERL,
Ada, OK)(after  July  1,  1986); Coordinator  for ORD, Washington, D.C. was Steve
Cordle (ORD/Water and Land Div.)
                                      64

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vvEPA
           United States
           Environmental Protection
           Agency
              Robert S. Kerr Environmental
              Research Laboratory
              Ada OK 74820
           Research and Development
              EPA/600/8-87/003 Jan 1987
The  Use of
Models in
Managing
Ground-Water
Protection
Programs

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                            EPA 600 8-87 003
                                January 1987
THE USE OF MODELS IN MANAGING
   GROUND-WATER PROTECTION
               PROGRAMS
          Joseph F. Keely, Ph.D, P.Hg.
Robert S. Kerr Environmental Research Laboratory
      U.S. Environmental Protection Agency
            Ada, Oklahoma 74820
       Office of Research and Development
     U.S. Environmental Protection Agency
            Ada, Oklahoma 74820

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                     DISCLAIMER

   The information in this document has been funded wholly or in
part by the United States Environmental Protection Agency. It has
been subjected to the Agency's peer and administrative review, and
it has been approved for publication as an EPA document.
                                11

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                       FOREWORD

   The U.S. Environmental Protection Agency was established to
coordinate administration of the major Federal programs designed
to protect the quality of our environment.

   An important part of the Agency's effort involves the search for
information  about environmental problems, management
techniques and new technologies through which optimum use of the
Nation's land  and  water resources can be assured and the threat
pollution poses  to the  welfare of the American people  can be
minimized.

   EPA's Office of Research and Development conducts this search
through a nationwide network of research facilities.

   As one of the  facilities, the Robert S. Kerr Environmental
Research Laboratory is the Agency's center of expertise for
investigation of the soil and subsurface environment. Personnel at
the laboratory  are responsible for management of research
programs to: (a) determine the fate, transport and transformation
rates of pollutants in the soil, the unsaturated zone  and the
saturated zones of the suburface environment; (b)  define the
processes to  be  used in characterizing the soil and  subsurface
environment as  a receptor of pollutants; (c) develop techniques for
predicting the effect of pollutants  on  ground  water, soil and
indigenous organisms; and (d) define and  demonstrate the
applicability and limitations of using natural processes, indigenous
to the soil and subsurface environment, for the protection of this
resource.

   This report contributes to that knowledge which is essential in
order for EPA to establish and enforce pollution control standards
which  are  reasonable, cost effective and  provide adequate
environmental protection for the American public.
                             Clinton W. Hall
                             Director
                             Robert S. Kerr Environmental
                             Research Laboratory
                            in

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                      ABSTRACT

   Because ground-water quality protection is emerging as a
major National environmental problem of this decade, there is
increasing pressure on regulators and the regulated to identify,
assess or  even anticipate  situations involving  ground-water
contamination. Site-specific and generic mathematical models are
increasingly being  used by  EPA to fulfill its mandates under a
number of major environmental statutes which call for permit
issuance,  investigation of potential problems, remediation
activities, exposure assessment and  a myriad of other policy
decisions.

   Mathematical models can be  helpful tools to managers of
ground-water protection programs. They may be used for testing
hypotheses about conceptualizations and to gather a fuller
understanding of important physical, chemical and biological
processes which affect ground-water  resources. The possible
outcomes of complex problems can be addressed in great detail, if
adequate data are available. The success of these efforts depends
on the accuracy and efficiency with which the natural processes
controlling the behavior of  ground water, and the chemical and
biological species it transports, are simulated. Success also depends
heavily on the expertise of  the modeler and the communication
with management so that the appropriateness, underlying
assumptions, and limitations of specific models are appreciated.
                              IV

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                        CONTENTS

Foreword                                               iii

Abstract                                                iv

Figures                                                 vii

Tables                                                  ix

Acknowledgments                                        x

1.   The Utility of Models                                  1
    Introduction                                          1
    Management Applications                              3
    Modeling Contaminant Transport                       6
    Categories of Models                                  7
    Chapter Summary                                     8

2.   Assumptions, Limitations, and Quality Control           9
    Introduction                                          9
    Physical Processes                                     9
     Advection and Dispersion                             10
     Complicating Factors                                13
     Considerations for Predictive Modeling                14
    Chenisc--.1.; Processes                                   15
     Chemicai/Klectronic Alterations                      15
     Nuclear Alterations                                 16
     Chemical Associations                               16
     Surface Interactions                                 17
    Biological Processes                                  19
     Surface Water Modeling Analogy                      19
     Ground Water B ^transformations                    20
     A Ground Water Model                              20
    Analytical and Numerical Models                      22
    Quality Control                                       23
    Chapter Summary                                    25

3.   Applications in Practical Settings                      29
    Stereotypical Applications                             29
    Real-World Applications                              29
     Field Example No. 1                                 30
     Field Kxample No. 2                                 32
    Practical Concerns                                   44
    Chapter Summary                                    52

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4.   Liabilities, Costs, and Recommendations for Managers    55
    Introduction                                         55
    Potential Liabilities                                  55
    Economic Considerations                              56
    Managerial Considerations                            64
    Chapter Summary                                    66

References                                              69
                               VI

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                         FIGURES

Number                                             Page

1-1     Small'sand tank'physical aquifer model             2

1-2     Laboratory column housed in constant-temperature
        environmental chamber                           2

1-3     Electric analog aquifer model constructed by Illinois
        State Water Survey                               3
1-4     Typical ground-water contamination scenario and a
        possible contaminant transport model grid design
        for its simulation                                 4

2-1     The influence of natural processes on levels of
        contaminants downgradient from continuous and
        slug-release sources                             11

2-2     Examples of plots prepared with the Jacob's
        approximation of the Theis analytical solution to
        well hydraulics in an artesian aquifer              24

2-3     Mathematical validation of a numerical method of
        estimating drawdown, by comparison with an
        analytical  solution                               26

3-1      Location map for Lakewood Water District wells
        contaminated with volatile organic chemicals       31

3-2      Schematic illustrating the mechanism by which a
         downgradient source may contaminate a
         production well                                 33

3-3      Location map for Chem-Dyne Superfund Site       35
3-4      Chem-Dyne geologic cross-section along NNW-SSE
         axis                                           36
3-5      Chem-Dyne geologic cross-section along WSW-ENE
         axis
                                                         37
 3-6     Shallow well ground-water contour map for
         Chem-Dyne                                     38
                             VII

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Number                                             Page

3-7     Typical arrangement of clustered, vertically-
        separated wells installed adjacent to Chem-Dyne
        and the Great Miami River                        39

3-8     Estimates of transmissivity obtained from shallow
        and deep wells during Chem-Dyne pump test        41

3-9     Distribution of total volatile organic chemical
        contamination in shallow wells at Chem-Dyne
        during October 1983 sampling                     42

3-10    Distribution of tetrachloroethene in shallow wells
        at Chem-Dyne during October 1983 sampling       45

3-11    Distribution of trichloroethene in shallow wells at
        Chem-Dyne during October 1983 sampling          46

3-12    Distribution of trans-dichloroethene in shallow
        wells at Chem-Dyne during October 1983 sampling  47

3-13    Distribution of vinyl chloride in shallow wells at
        Chem-Dyne during October 1983 sampling          48

3-14    Distribution of benzene in shallow wells at Chem-
        Dyne during October 1983 sampling                49

3-15    Distribution of chloroform in shal low wells at Chem-
        Dyne during October 1983 sampling                50

3-16    General relationship between site characterization
        costs and clean-up costs as a function of the
        characterization approach                        54

4  1     Average price per category for ground-water
        models from the International Ground Water
        Model ing Center                                 57

4-2     Price ranges for IBM-PC ground-water models
        available from various sources                    59

4-3     Costs of sustaining ground-water modeling
        capabilities at two different computing levels, for
        a five-year period                                61
                               vm

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                          TABLES

Number                                             Page

2-1      N atural processes that affect subsurface
         contaminant transport                            10

3-1      Chem-Dyne pump test observation network         43

3-2      Conventional approach to site characterization
         efforts                                           51

3-3      State-of-the-art approach to site characterization
         efforts                                           52

3-4      State-of-the-science approach to site characterization
         efforts                                           53

4-1      Desired backgrounds and salary ranges advertised
         for positions requiring ground-water modeling      60

4-2      Screening-level questions to help ground-water
         managers focus mathematical modeling efforts      65

4-3      Conceptualization questions to help ground-water
         managers focus mathematical modeling efforts      66

4-4      Sociopolitical questions to help ground-water
         managers focus mathematical modeling efforts      67
                             IX

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                ACKNOWLEDGMENTS

   The author is indebted to the many fine scientists, engineers,
and support staff at the Robert S. Kerr Environmental Research
Laboratory for their assistance. In particular, Dr. Marvin D. Piwoni
and Dr. John T. Wilson made substantial contributions to Chapter 2
in the chemical and biological sections, respectively.  Ms. Carol
House and Ms. Renae Daniels typed many drafts of the document.
Ms. Kathy Clinton prepared most of the illustrations.

   The author is grateful to Drs. William P. McTernan and Douglas
C. Kent of Oklahoma State University for their technical reviews of
the manuscript. Thanks also go to Ir. Paul van der Heijde of the
International Ground Water Modeling  Center for his readings of
early drafts of many sections, and for the use of certain photographs.
Mr. Marion R. (Dick) Sea IPs guidance and encouragement as EPA
Project Officer on this project are deeply  appreciated. Comments and
suggestions from the readers are welcome; the author assumes all
responsibility for any errors, omissions, or misstatements.

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                        CHAPTERl
               THE UTILITY OF MODELS
INTRODUCTION

    Every time man attempts to simulate the effects of natural
phenomena, he is engaging in the scientific art of modeling. Models
are nothing more than simplified representations of reality, and
their creation and use involves a considerable degree of subjective
judgment, as well as an attempt to incorporate known scientific
facts. There are  many forms of models, each  having specific
advantages and disadvantages compared with the remainder.

    Physical models, such as sand-tanks used to simulate aquifers
(Figure 1-1) and laboratory columns used  to study the relative
motion of various contaminants flowing through aquifer materials
(Figure 1-2), provide an element of reality which is enlightening and
satisfying from an intuitive viewpoint. Their main disadvantage
relates  to the extreme efforts and  time  required to  generate a
meaningful  amount of data. Other difficulties relate to the care
required to obtain samples of subsurface material for construction of
these models, without significantly disturbing the natural condition
of the samples.

    Analog models are also physically based, but their operating
principle is one of similarity, not true-life representation. A typical
example is the electric analog model (Figure  1-3), in which
capacitors and resistors are able to closely replicate the effects of the
rate of release of water from storage in aquifers. The  clear
disadvantage is that "a camel is not a horse', even if both can carry a
load. As is the case  with  other physically based models,  data
generation is  slow and there is little flexibility for experimental
design changes.

    Mathematical models are non-physical,  relying instead on the
quantification of relationships between specific  parameters and
variables to simulate the effects of natural processes (Figure 1-4). As
such, mathematical models are abstract and provide little in the
way of a directly observable link to reality. Despite this lack of
intuitive grace,  mathematical  models can  generate powerful
insights into the functional dependencies between causes and effects
in the real world.  Large amounts of data can be generated quickly,
and experimental  modifications can be made with minimal effort, so

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Figure 1-1     Small "sand tank" physical aquifer model. Three
             Pumping wells (A,B,C) penetrate the homogeneous
             sand to study the effects of well hydraulics on plume
             movements. Vials  in foreground  contain various
             concentrations of water-active dyes
 Figure 1-2.    Laboratory column housed in constant-temperature
              environmental chamber Contaminated solutions are
              injected into column through inlet tubing in top, by
              action of hydraulic press in foreground. Samples of
              the advancing front are withdrawn  through ports
              visible on right-hand side and bottom of column.
                               2

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            MODEL  OF  GROUND  WATER  ffiSEBYWR
          (MOLTING  EAST  ST.  LOB1S  AREIt
                                v-L$-2-.-«•  J  ' JLJ
Figure 1-3.    Electric analog aquifer model constructed by Illinois
             State Water Survey. The regular array of resistors and
             the two electric "pumps" shown are hard-wired into
             a board covered with the appropriate geologic maps.


that many possible situations can be studied in great detail for a
given problem.

MANAGEMENT APPLICATIONS

    Mathematical models can and have been used  to help organize
the essential details  of  complex ground-water  management
problems so that reliable solutions are obtained (Holcomb Research
Institute, 1976; Bachmatand others, 1978; U.S. Office of Technology
Assessment, 1982; van der Heijde and others, 1985). Some principal
areas where mathematical models are now being used to assist in
the management of ground-water protection are:

    (1) appraising the physical extent, and chemical  and biological
       quality,  of ground-water reservoirs (e.g.,  for planning
       purposes),
    (2) assessing the potential impact of domestic, agricultural, and
       industrial practices (e.g., for permit issuance),
    (3) evaluating the  probable outcome of remedial actions at
       waste sites,  and aquifer restoration  techniques generally,
       and
    (4) providing health-effects exposure estimates.

    The success of  these  efforts depends on the accuracy and
efficiency with which the natural processes controlling the behavior
of ground water, and the chemical  and biological species it
transports, are simulated (Boonstra and de Ridder, 1976; Mercer

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Figure 1-4.    Typical ground-water contamination scenario and a
             possible contaminant transport model grid design for
             its simulation. Values for natural process parameters
             would  be specified at each  node  of the grid  in
             performing simulations. The grid density is greatest
             at the source and at potential impact locations.

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and Faust, 1981; Wang and Anderson, 1982). The accuracy and
efficiency of the simulations, in turn, are heavily dependent on
subjective judgements made by the modeler and management.

    In the current philosophy of ground-water protection programs,
the value of a ground-water resource  is bounded by the most
beneficial present and future uses to which it can be put (U.S. EPA,
1984). In most  instances, physical appraisals of ground-water
resources are conducted within a framework of technical and
economic classification schemes. Classification of entire  ground-
water basins by potential yield is a typical  first  step (Domenico,
1972). After initial identification and evaluation of a ground-water
resource, strategies for its rational development need to be devised.

    Development considerations  include  the  need to protect
vulnerable recharge areas and the possibility of conjunctive use
with available surface waters (Kazmann,  1972). Ground-water
rights must be fairly administered to assure adequate supplies for
domestic, agricultural, and industrial purposes. Because basinwide
or regional resource evaluations normally do not provide  sufficent
resolution for water  allocation purposes,  more detailed
characterizations of the properties and behavior of an aquifer, or of a
subdivision of an aquifer, are usually needed. Hence, subsequent
classifications may involve local estimation of net annual recharge,
rates of outflow, and the pumpage which can be substained without
undesirable effects.

    The  consequences of developments which might affect ground-
water quality may be estimated initially by employing generalized
classification schemes; for example, classifications based on regional
hydrogeologic settings have been presented (Heath, 1982; Aller and
others, 1985). Very  detailed databases,  however, must be created
and molded into useful formats before decisions can be made on how
best to protect and rehabilitate ground-water resources from site-
specific incidents of natural and manmade contamination.

    The latter are ordinary ground-water management functions
which benefit from the use of mathematical models. There  are other
uses, however, which ought to be considered by management. The
director of the International  Ground  Water Modeling Center
discussed the role of modeling in the development of ground-water
protection policies  recently, noting its success in  many policy
formulation efforts  in the Netherlands, the United States, and
Israel.  Nevertheless, he concluded that modeling was  not widely
relied upon for decision-making by managers; the primary obstacle
has been an inability of modelers and program managers  to
communicate effectively (van der Heijde, 1985). The top executives
of a leading high-tech ground-water contamination consulting firm
made the same point clearly, going on  to highlight  the  need for
qualified personnel appreciative of the appropriateness, underlying
assumptions, and limitations of specific models (Faust and others,

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1981). Because  these views are  widely  held by  technical
professionals, it  will be emphasized herein that mathematical
models are useful only within the context of the assumptions and
simplifications on which they are based. If managers are mindful of
these factors, however, mathematical models can be a tremendous
asset in the decision- making process.

MODELING CONTAMINANT TRANSPORT

    Associated with most hazardous waste sites is a complex array
of  chemical  wastes and the  potential  for ground-water
contamination. The hydrogeologic settings of these sites are usually
quite complicated when examined at the scale appropriate for
technical assessments and remediation efforts (e.g., 100's of feet). As
a result, data acquisition and interpretation  methods are needed
that can examine  to an  unprecedented degree the physical,
chemical, and biological processes that control  the transport and
fate of ground-water contaminants. The methods and tools that have
been in  use for large-scale characterizations  (e.g., regional water
quality studies) are applicable in concept to the specialized needs of
hazardous waste  site investigations; however, the transition to
local-scale studies is not  without scientific and  economic
consequences. In part, this stems from the highly variable nature of
contaminant distributions at hazardous waste sites; but it  also
results from the limitations of the methods, tools, and theories used.
Proper acknowledgement of the inherent limitations means that one
must project the consequences of their use within the framework of
the study at hand.

    Assessments of the potential for contaminant transport require
interdisciplinary analyses and interpretations.  Integration of
geologic, hydrologic, chemical, and biological approaches  into an
effective contaminant transport evaluation can  only be possible if
the data and concepts invoked are sound. The data must be accurate,
precise, and appropriate at the intended problem scale. Just because
a given parameter (e.g., hydraulic conductivity) has been measured
correctly at certain points with great reproducibility, is no
guarantee that those estimates represent the volumes of aquifer
material assigned to them by a modeler. The degree to which the
data are representative, therefore, is  not only relative to  the
physical  scale of the problem, it is relative to the conceptual model
to be used for  interpretation efforts. It is  crucial, then, to carefully
define and qualify the conceptual model. In  so doing,  special
attention should  be given to the possible spatial and temporal
variations of the data that will be collected.

    To circumvent the impossibly large numbers of measurements
and samples which  would be needed to eliminate all uncertainties
regarding the true  relationships of parameters (e.g., hydraulic
conductivity) and variables (e.g., contaminant concentrations and
rates of movement), more comprehensive theories are  constantly

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under development. The  use of newly developed theories to help
solve field problems, however, is often a frustrating exercise. Most
theoretical advances call for some data which are not yet practically
obtainable  (e.g.,  chemical interaction coefficients,  relative
permeabilities of immiscible solvents and water, etc.). The 'state-of-
the-art* in contaminant  transport assessments is necessarily a
compromise between the sophistication of "state-of-the-science*
theories, the current limitations regarding the acquisition of specific
data, and economics.  In addition, the best  attempts to obtain
credible data are limited by natural and anthropogenic variabilities;
and these lead to the need for considerable judgement on the part of
the professional.

    Despite  these technical limitations, how well the problem is
conceptualized  remains the most serious concern in  modeling
efforts. For example, researchers recently  produced dramatic
evidence to  show that, in spite of detailed  field measurements,
extrapolations of two-dimensional model results to a truly three-
dimensional problem lead to wildly inaccurate projections of the
actual behavior  of the system under study (Molz and others, 1983).
Therefore, it is incumbent on model users to recognize the difference
between an approximation and a misapplication. Models should
never be used strictly on the basis of familiarity or convenience; an
appropriate model should always be sought.

CATEGORIES OF MODELS

    The  foregoing is not meant to imply that appropriate models
exist for all ground-water problems, because a number of natural
processes have yet to be fully understood. This is especially true for
ground-water contaminant transport evaluations, where the
chemical and biological processes are still  poorly defined. For,
although great advances have been made concerning the behavior of
individual contaminants, studies of the interactions  between
contaminants  are still in  their infancy.  Even  the current
understanding of physical processes lags behind what is needed,
such as in the mechanics of multiphase flow and flow through
fractured rock aquifers. Moreover, certain  well-understood
phenomena pose  unresolved difficulties  for mathematical
formulations, such as the effects of partially penetrating wells in
unconfined (water-table) aquifers.

    The technical-use categories of models are varied, but they can
be grouped as follows (Bachmat and  others,  1978; van der Heijde
and others, 1985):
    (1) parameter identification models,
    (2) prediction models,
    (3) resource management models, and
    (4) data manipulation codes.

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    Parameter identification models are most often used to estimate
the aquifer coefficients determining fluid flow and contaminant
transport characteristics, like  annual  recharge (Puri,  1984),
coefficients of permeability and storage (Shelton, 1982; Khan, 1986a
and b), and dispersivity (Guven and others, 1984; Strecker and Chu,
1986). Prediction models are the most numerous kind of model, and
abound because they are the  primary tools for testing hypotheses
about the problem one wishes to solve (Andersen and others, 1984;
Mercer and Faust, 1981; Krabbenhoft and Anderson, 1986).

    Resource management models are combinations of predictive
models, constraining functions (e.g., total pumpage  allowed) and
optimization  routines for objective functions (e.g., optimization of
wellfield  operations  for  minimum  cost  or  minimum
drawdown/pumping lift). Very few of these are so well developed and
fully supported that they may be considered practically useful, and
there does not appear to be  a significant drive to improve  the
situation (van der Heijde, 1984a  and 1984b; van der  Heijde and
others, 1985). Data manipulation codes also have received little
attention until recently. They are  now becoming increasingly
popular, because they simplify data entry ('preprocessors') to other
kinds of models and facilitate the production of graphic displays
('postprocessors') of the data outputs of other models (van der Heijde
and Srinivasan, 1983; Srinivasan, 1984; Moses and Herman, 1986).
Other software packages are available for  routine and advanced
statistics, specialized graphics, and database management needs
(Brown, 1986).

CHAPTER SUMMARY

    Mathematical models can be helpful  tools to  managers of
ground-water protection programs. They may  be used for  testing
hypotheses about conceptualizations and  to  gather a fuller
understanding of important physical, chemical, and biological
processes which  affect ground-water resources. The possible
outcomes of complex problems can be addressed in great detail, if
adequate data arc available. Mathematical modeling is neither
simple nor impossible, but its successful application relies  heavily
on the expertise of the modeler and  the degree of communication
with management.

    The merits of any problem solving technique need to be judged
by many criteria, the most important of which may not relate to
mathematical sophistication. Qualitative judgements by prior
experience, 'back  of the envelope* calculations, analytical models,
and other non-numerical modeling methods should be considered for
a reason which deserves emphasis; the data available or obtainable
may not justify extensive numerical model analyses (Javendel and
others,  1984). After all, it is  neither pretty nor efficient to 'use a
stiver sledgehammer to drive a thumbtack*.

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                       CHAPTER2
 ASSUMPTIONS, LIMITATIONS, AND QUALITY
                        CONTROL
INTRODUCTION

   There are many natural processes that affect chemical transport
from point to point in the subsurface. These natural processes can be
arbitrarily divided into three categories: physical,  chemical, and
biological (Table 2-1). Conceptually, contaminant transport in the
subsurface is an undivided phenomenon composed of these processes
and their interactions (Figure 2-1). At this level the transport
process may be gestalt: the sum of its parts, measured separately,
may not equal the whole because of interactions between the parts.
In the  theoretical context, a collection of scientific laws and
empirically derived  relationships comprise the overall transport
process. The universally shared premise that underlies  theoretical
expressions is that there are no interactions, measureable or
otherwise.

    Significant errors may result from  the discrepancy between
conceptual and theoretical appproaches. Also the simplifications of
theoretical expressions used to solve practical  problems can cause
substantial errors in the  most careful analyses. Assumptions and
simplifications, however,  must often be made in order to obtain
mathematically tractable  solutions. Because of this, the magnitude
of errors that arise from each assumption and simplification must be
carefully  evaluated.  The phrase, "magnitude of errors", is
emphasized because highly accurate evaluations usually are not
possible. Even rough approximations are rarely trivial exercises
because they frequently demand estimates of some things which are
as yet ill-defined.

PHYSICAL PROCESSES

    Until  recently, ground-water scientists studied physical
processes to a greater degree than chemical or biological processes.
This bias resulted in large measure from the fact that, in the past,
ground-water practitioners dealt mostly with questions of adequate
water supplies. As quality considerations began to dominate
ground-water issues,  the need for studies of the chemical and
biological factors, as well as more detailed representations of the
physical factors, became apparent.

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Table2-1.    Natural  processes  that  affect  subsurface
             contaminant transport.
                 PHYSICAL PROCESSES
                  Advection (porous media velocity)
                  Hydrodynamic Dispersion
                  Molecular Diffusion
                  Density Stratification
                  Immiscible Phase Flow
                  Fractured Media Flow

                 CHEMICAL PROCESSES
                  Oxidation-Reduction Reactions
                  Radionuclide Decay
                  Ion-Exchange
                  Complexation
                  Co-Solvation
                  Immiscible Phase Partitioning
                  Sorption

                 BIOLOGICAL PROCESSES
                  Microbial Population Dynamics
                  Substrate Utilization
                  Biotransformation
                  Adaptation
                  Co-metabolism
    There are two  complimentary ways  to view the physical
 processes involved in subsurface contaminant transport:  the
 piezometric (pressure) viewpoint and the hydrodynamic viewpoint.
 Ground-water problems of yesterday could be addressed by the
 former, such as solving for the change in pressure head caused by
 pumping wells.  Contamination problems of today also require
 detailed analyses of wellfield operations, for example, pump-and-
 treat plume removals. However, solutions to such problems depend
 principally  on hydrodynamic evaluations, such  as computing
 ground-water velocity (advection) distributions and dispersion
 estimates for migrating plumes.

 Advection and Dispersion

    Ground-water velocity distributions can be approximated if the
 variations in hydraulic conductivity, porosity, and the strength and
 location of recharge and discharge sources can be estimated.

    While there  arc several field and laboratory methods for
estimating  hydraulic  conductivity, these are not directly
comparable because different volumes of aquifer material  are
                             10

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                                             (T) Adv*ctlon

                                             ($} Dl*p«r>lon

                                             (J) Sorpllon

                                             (t • Blotrinitormttlon
     DISTANCE  FROM CONTINUOUS CONTAMINANT SOURCE
     DISTANCE FROM SLUG-RELEASE CONTAMINANT SOURCE


Figure 2-1.   The  Influence of natural processes on levels  of
             contaminants downgradient from continuous and
             slug-release sources.
                              11

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   affected  by different tests.  Laboratory permeameter tests, for
   example, obtain measurements from small core samples and thus
   give point  value estimates. These tests are generally  reliable for
   consolidated rock samples, such as sandstone, but can be highly
   unreliable  for unconsolidated samples, such as sands, gravels, and
   clays. Pumping tests give estimates of hydraulic conductivity that
   are averages over the entire volume of aquifer  subject to the
   pressure changes induced by pumping.  These give  repeatable
   results, but they are often difficult to interpret. Tracer tests are also
   used to estimate hydraulic conductivity in the field, but  are difficult
   to conduct properly.

       Regardless of the estimation technique used, the best that can
.   be  expected  is order-of-magnitude estimates  for  hydraulic
IC-'conductivity at the field scale appropriate for site-specific work.
   Conversely, porosity estimates that are accurate to better than a
   factor of two can be obtained. Estimation of the strength of nonpoint
   sources of  recharge to an aquifer, such as infiltrating  rainfall and
   leakage from  other aquifers, is another order-of-magnitude effort.
   Similarly,  nonpoint sources of discharge, such as aquifer losses to
     a*nmS streams, are difficult to quantify. Estimation of the strength
    jf point sources of recharge or discharge (injection or pumping wells)
   can be highly accurate.

       Consequently, it is not possible to generalize the quality of
   velocity distributions. They may be accurate  to within a factor of
   two for very simple aquifers, but are more often  accurate to an
   order-of-magnitude only. This situation has changed little over the
   past 20  years because better field and  laboratory methods for
   characterizing velocity distributions have not been developed. This,
   however, is not the primary difficulty associated with defining the
   advective part of contaminant transport in the subsurface. The
   jjrimary difficulty  is that field tests for characterizing  the physical
)^-paramelers that control velocity distributions are not incorporated
   into contamination investigations on a routine basis.  The causes
   seem to be: a perception that mathematical models can "back-out* an
   approximation of the velocity distribution (presumably  eliminating
   the need for field tests); unfamiliarity with  field tests by many
   practitioners; and a perception that field tests arc too expensive. A
   more field oriented approach  is  preferable because the non-
   uniqueness of modeling results has been amply  demonstrated, and
   this leads to uncertain decisions regarding the design of  remedies.

       Dispersion estimates are  predicated on  velocity distribution
   estimates and their accuracy is therefore directly dependent on the
   accuracy of the estimated hydraulic conductivity distribution.
   Tracer tests have been  the primary  method used to determine
   dispersion coefficients until  recently. Presently   there are
   suggestions that any field method capable of generating a detailed
   understanding of the spatial variability of hydraulic conductivity,
   which in turn could give an accurate representation of the velocity
                                 12

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distribution,  may be used to derive estimates  of dispersion
coefficients. The manner in which data from field tests  should be
used to derive estimates of dispersion coefficients,  however, is a
controversial issue. There are both deterministic and stochastic
schools of thought, and neither has been conclusively demonstrated
in complex hydrogeological settings.

Complicating Factors

    Certain subtleties of the spatial variability of hydraulic
conductivity must be understood because of its key role in the
determination of velocity  distributions and dispersion coefficients.
Hydraulic conductivity  is also known as  the  coefficent  of
permeability because it is comprised of fluid factors as well as the
intrinsic permeability of the stratum in question. This means that a
stratum of uniform intrinsic permeability (which depends strictly on
the arrangement of its pores) may have a wide range of hydraulic
conductivity because of differences in  the density and viscosity of
fluids that are present. The result is a dramatic downward shift in
local flow directions near plumes that have as little as a \% increase
in density relative to uncontaminated water. Such density contrasts
frequently occur at landfills and waste impoundments.  It is often
necessary to correct misimpressions of the direction of a plume
because density considerations were not addressed.

    Many solvents and oils are highly insoluble in water, and may
be  released to the subsurface in amounts  sufficient  to form a
separate fluid phase. Because that fluid phase will probably have
viscosity and density different from freshwater, it will flow at a rate
and, possibly, in a direction different  from that of  the freshwater
with which it is in contact. If an immiscible phase has density
approximately the same or less  than  that of ground water, this
phase  will not move down past the capillary fringe of the ground
water. Instead, it will flow along the top of the capillary fringe in the
direction of the maximum  water-level elevation drop. If the density
of an immiscible phase is substantially greater than the ground
water, the immiscible phase will push its way into the ground water
as a relatively coherent blob. The primary direction of its flow will
then be down  the dip of the first impermeable stratum encountered.
There is a great  need for better means of characterizing such
behavior  for  site-specific applications. Currently, estimation
methods  are patterned after multiphase oil reservoir simulators.
One of the key extensions needed is the ability  to predict the
transfer of trace levels of contaminants from the immiscible fluid to
ground water, such as xylenes from gasoline.

    Anisotropy is a subtlety of hydraulic conductivity which relates
to structural trends of the rock or sediments of which an aquifer is
composed. Permeability  and  hydraulic  conductivity are
directionally dependent in anisotropic strata.  When molten
material from deep underground crystallizes to form granitic or
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basaltic rocks, for instance, it forms cleavage planes which  may
later  become the  preferred directions of permeability.  Marine
sediments accumulate to  form sandstone,  limestone, and shale
sequences that have much  less  vertical  than  horizontal
permeability.  The  seasonal  differences in sediments  that
accumulate on lakebeds, and  the stratification of grain  sizes
deposited by streams as they mature, give rise to similar vertical-to-
horizontal anisotropy. Streams also cause anisotropy within the
horizontal plane, by forming and reworking their sediments along a
principal axis of movement. These  structural  variations in
permeability would be of minimal concern except that ground water
does not flow at right angles to water-level elevation contours under
anisotropic conditions. Instead, flow proceeds along  oblique angles,
with  the degree  of deviation from a  right-angle pathway
proportional to the amount of anisotropy. This fact is all too  often
ignored and the causes again seem to be a reluctance to conduct the
proper field tests, combined with an over-reliance on mathematical
modeling.

    If the pathways created by cleavage planes and  fractures begin
to dominate fluid flow through a subsurface stratum, the directions
and rates of flow are no longer predictable by the equations used for
porous rock and sediments. There have been a number of attempts
to represent fractured flow as an equivalent porous medium, but
these tend to give poor predictions when major fractures are present
and when there are too few fractures to guarantee a minimum
degree of interconnectedness. Other representations that have been
studied are various dual porosity models, in which the bulk matrix
of the rock has one porosity and the fracture system  has  another.
Further development of the dual porosity approach is limited by the
difficulty in determining a transfer function  to  relate  the two
different porosity schemes. Research in this  area  needs to be
accelerated because there is a  great likelihood of fractured flow in
just those situations commonly believed to be the most suitable for
disposal of hazardous wastes,  such as building  landfills on
'impermeable' bedrock.

Considerations for Predictive Modeling

    Equations for  the combined advection-dispersion process are
used to estimate the time  during which a nonreactive contaminant
will travel a specific distance,  the pathway it  will travel, and its
concentration at any point. The accuracy of most predictions is only
fair for typical applications, because of the  complexity of the
problems and the  scarcity of site-specific hydrogeologic data. The
lack of such data can be improved on with much less effort than is
commonly presumed, especially when the cost of another  round of
chemical sampling is compared with the costs of additional borings,
core retrievals, geophysical logging, or permeability testing.
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    Equations that assume a nonreactive contaminant have limited
usefulness, because most contaminants react with other chemical
constituents in subsurface waters and with subsurface solids in a
manner that affects the rate at which they travel. Nevertheless,
nonreactive advection-dispersion equations are often used to
generate  'worst-case' scenarios, on  the presumption  that the
maximum transport velocity is obtained (equal to  that of pure
water). This may not be as useful as it first seems. Remedial action
designs require detailed  breakdowns of which contaminants will
arrive at  extraction wells and  when,  how long contaminants will VL/"
continue their slow release from subsurface solids, and whether the  '
contaminants will be transformed into other chemical species by
chemical or biological forces. To address these points,  special terms
must be added to the advection-dispersion  equations.

CHEMICAL PROCESSES

    As difficult as the foregoing complications may be, predicting
how chemical  contaminants move through the subsurface is a
relatively trivial matter when  the contaminants behave as ideal,
nonreactive substances. Unfortunately, such behavior  is limited to a
small group of chemicals. The actual situation is that most
contaminants  will, in a variety of ways,  interact with  their
environment through biological or chemical processes. This section
focuses on the dominant chemical processes that may ultimately
affect the transport behavior of a contaminant. As with the physical
processes previously discussed, some of the knowledge of chemical
processes has been  translated into  practical use in predictive
models. However, the science has, in many instances, advanced well
beyond  what  is  commonly practiced.  Furthermore,  there  is
considerable evidence that suggests that numerous undefined
processes affect chemical mobility. Most of the deviation from ideal
nonreactive behavior of contaminants relates to their  ability to
change physical form by energetic interactions with  other matter.
The physical-chemical interactions may be grouped into: alterations
in the chemical  or electronic configuration of an element  or
molecule, alterations in  nuclear composition, the establishment of
new associations with other chemical species, and interactions with
solid surfaces.

Chemical/Electronic Alterations

    The first of these possible changes  is typified by oxidation-
reduction or redox reactions. This class of reactions is  especially
important for  inorganic compounds and  metallic elements because
the reactions  often result in  changes in solubility, complexing
capacity, or sorptive behavior, which directly impact on the mobility
of the chemical. Redox reactions are reasonably well understood, but
there are practical obstacles to applying the known science because
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it is difficult to determine the redox state of the aquifer zone of
interest and to identify and quantify the redox-active reactants.

    Hydrolysis, elimination,  and substitution reactions that affect
certain contaminants also fit into this classification. The chemistry
of many  organic contaminants has been well defined in surface
water environments. The influence of unique aspects of the
subsurface, not the least of which is  long residence time, on such
transformations of important organic pollutants is currently under
investigation. There is also a need to investigate the feasibility of
promoting in-situ abiotic transformations that may enhance the
potential for biological mineralization of pollutants.

Nuclear Alterations

    Another chemical process interaction, which results in internal
rearrangement of the nuclear structure of an element, is well
understood. Radiodecay occurs by a variety of routes, but the rate at
which it occurs is always directly proportional to  the number of
radioactive atoms present. This fact seems to make mathematical
representation in  contaminant transport models quite
straightforward because  it allows characterization  of the process
with a unique, well defined decay constant for each radionuclide.

    A mistake that is often made when the decay constant is used in
models involves the physical form of the reactant. If the  decay
constant is applied to the fluid concentrations and no other chemical
interactions are allowed, then incorporation of the constant into the
subroutine which computes fluid concentrations  will  not cause
errors. If the situation being modeled involves chemical interactions
such as precipitation, ion-exchange, or sorption, which temporarily
remove the radionuclide from solution, then it is important to use a
second subroutine to account for the non-solution-phase decay of the
radionuclide.

Chemical Associations

    The establishment of new associations with other chemical
species is not as  well understood. This category includes ion-
exchange,  complexation, and co-solvation.  The lack of
understanding derives  from the nonspecific nature of these
interactions which are, in many instances, not characterized by the
definite proportion of reactants to products (stoichiometry) typical of
redox reactions.  While the  general  principles and driving
mechanisms  by which these interactions occur are known, the
complex subsurface matrix  in which they occur provides many
possible outcomes and renders predictions uncertain.

    Ion-exchange and complexation reactions heavily influence the
mobility  of metals and other ionic species in the subsurface in a
reasonably predictable fashion. Their influence  on organic
contaminant transport, however, is not well understood. Based on
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studies of pesticides and other complex organic molecules, natural
organic matter (such  as humic and fulvic materials) can complex
and  thereby enhance the apparent solubility  and mobility of
synthetic organic chemicals. Research is needed to define the
magnitude of such interactions, not only with naturally occurring
organic molecules but also with man-made  organics present in
contaminated environments. Research is also needed to determine if
these complexes are stable and liable  to transport through the
subsurface. Examination of the degree to which  synthetic organic
chemicals complex toxic metals is also necessary. There  is no
theoretical  objection to such interactions, and there is ample
evidence that metals  are moving through the  subsurface at  many
waste sites.

    Co-solvation occurs when another solvent is in the aqueous
phase  at concentrations that enhance the solubility  of a  given
contaminant. This occurs in agricultural uses, for example, where
highly insoluble pesticides and herbicides are mixed with organic
solvents  to  increase their solubility in  water prior to field
application. There is  every reason  to expect similar behavior at
hazardous waste sites,  where a variety of solvents  are  typically
available. At present, prediction of the extent of the  solubility
increases that might occur at disposal sites in the complex mixture
of water and organic solvents is essentially impossible. Researchers
have started examining co-solvation as an influence on  pollutant
transport, by working on  relatively simple mixed solvent systems.
This research will be extremely useful, even if the results are
limited to a qualitative appreciation for the magnitude of the effects.

    At the extreme, organic solvents in the subsurface may result in
a phase separate from the aqueous phase. In addition to movement
of this separate phase through the subsurface, contaminant mobility
that involves partitioning of organic contaminants between the
organic and aqueous phases must  also be  considered. The
contaminants will move with the organic phase and will, depending
on aqueous phase concentrations,  be  released  into the aqueous
phase to a degree roughly proportional to their octanol-water
partition  coefficients. An entire range of effects is possible, from
increasing to slowing the mobility of the chemical in the subsurface
relative to its migration rate in the absence of the organic phase.
The equilibrium partitioning process increases the total volume of
ground water affected by contaminants, by releasing a portion of the
organic  phase constitutents into adjacent waters. It may also
interfere with transformation processes  by affecting pollutant
availability for  reaction,  or by acting as a biocidal agent to the
native microflora.
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Surface Interactions

    Of those interactions that involve organic chemicals in the
environment, none has been as exhaustively  studied as sorption.
Sorption studies relate, in terms of a sorption isotherm, the amount
of contaminant in solution to the amount associated with the solids.

    Most often the sorption term in transport models is estimated for
simplicity from the assumption that the  response is linear. This
approximation can produce serious mass balance errors. Typically,
the contaminant mass in the solution phase is underestimated and
contaminant retardation is thereby overestimated. In practical
applications, this means that high contaminant levels can be
detected at a monitoring well long before they were predicted. To
resolve the discrepancy between predicted  and actual transport,
most  practitioners arbitrarily adjust  some  other  poorly-
characterized model parameter, for example, dispersion. This leads
to the creation of a model that does not present  various natural
process influences in proper perspective. The predictions from such
models are likely to be qualitatively, as well as quantitatively,
incorrect. More widespread consideration should be given to
accurate representation of non-linear sorption, particularly in
transport modeling at contaminated sites.

    The time dependency  of the  sorption process  is a related
phenomenon that has also been largely  ignored  in practical
applications of sorption theory. Most models assume that sorption is
instantaneous  and completely reversible. A growing body of
evidence argues to the contrary, not only for large organic molecules
in high-carbon soils and sediments, but also for solvent molecules in
low-carbon aquifer materials.  Additionally, there must be  some
subtle interplay between sorption  kinetics and ground-water flow
rates  which gains significance in pump-and-treat remediation
efforts, where flow rates  are  routinely  substantially increased.
Constant pumpage at moderate-to-high flow rates may not allow
contaminants that are sorbed to solids sufficient times of release to
increase solution concentrations to maximum  (equilibrium) levels
prior to their removal from the aquifer. Hence, treatment costs may
rise substantially due to the prolonged pumping required to remove
all of the contaminants  and due to the  lowered efficiency of
treatment of the less contaminated pumped waters.

    Evidence from Superfund sites and ongoing research activities
suggests that contaminant association with a solid surface does not
preclude mobility. In many instances, especially in glacial tills that
contain a wide distribution of particle sizes, fine aquifer materials
have accumulated in the bottom of monitoring wells. Iron-based
colloids have been identified in ground water downgradient from a
site contaminated with domestic wastewater. If contaminants can
associate  with these fine  particles, their mobility through the
subsurface  could be  markedly enhanced.  To  determine the
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