United States
Environmental Protection
Office of
Research and
Off ice of Sol id Waste
and Emergency
April 1992
&EPA      Ground   Water   Issue
                         Fundamentals of Ground-Water Modeling
                        Jacob Bear3, Milovan S. Beljinb, and Randall R. Rossc
Ground-water flow and contaminant transport modeling has
been used at many hazardous waste sites with varying
degrees of success. Models may be used throughout all
phases of the site investigation and remediation processes.
The ability to reliably predict the rate and direction of ground-
water flow and contaminant transport is critical in planning and
implementing ground-water remediations. This paper presents
an overview of the essential components of ground-water flow
and contaminant transport modeling in saturated porous
media. While fractured rocks and fractured porous rocks may
behave like porous media with respect to many flow and
contaminant transport phenomena, they require a separate
discussion and are not included in this paper. Similarly, the
special features of flow and contaminant transport in the
unsaturated zone are also not included. This paper was
prepared for an audience with some technical background and
a basic working knowledge of ground-water flow and
contaminant transport processes.  A suggested format for
ground-water modeling reports and a selected bibliography
are included as appendices A and B, respectively.

For further information, contact David Burden (FTS 700-743-
2294), Randall Ross (FTS 700-743-2355), or Joe Williams
(FTS 700-743-2312) at (405) 332-8800.

Modeling as a Management Tool

The management of any system means making decisions
aimed at achieving the system's goals, without violating
specified technical and nontechnical constraints imposed on it.
In a ground-water system, management decisions may be
related to rates and location of pumping and artificial recharge,
changes  in water quality, location and rates of pumping in
pump-and-treat operations, etc. Management's objective
function should be to evaluate the time and cost necessary to
achieve remediation goals.  Management decisions are aimed
at minimizing this cost while maximizing the benefits to be
derived from operating the system.

The value of management's objective function (e.g., minimize
cost and maximize effectiveness of remediation) usually
depends on both the values of the decision variables (e.g.,
areal and temporal distributions of pumpage)  and on the
response of the aquifer system to the implementation of these
decisions. Constraints are expressed in terms of future values
of state variables of the considered  ground-water system,
such as water table elevations and concentrations of specific
contaminants in the water. Typical constraints may be that the
concentration of a certain contaminant should not exceed a
specified value, or that the water level at a certain location
should not drop below specified levels.  Only by comparing
predicted values with specified constraints can decision
makers conclude whether or not a specific constraint has been

An essential part of a good decision-making process is that
the response of a system to the implementation of
contemplated decisions must be known before they are

In the management of a ground-water system in which
decisions must be made with respect to both water quality and
water quantity, a tool is needed to provide the decision maker
with information about the future response of  the system to the
effects of management decisions.  Depending on the nature of
"Technion - Israel Institute of Technology
b University of Cincinnati
0 U.S. EPA, Robert S. Kerr Environmental Research Laboratory
                        Superfund Technology Support Center for
                        Ground Water

                         Robert S. Kerr Environmental
                         Research Laboratory
                         Ada, Oklahoma
                     Technology Innovation Office
                     Office of Solid Waste and Emergency
                     Response, US EPA, Washington, DC

                     Waiter W. Kovalick, Jr., Ph.D.

the management problem, decision variables, objective
functions, and  constraints, the response may take the form of
future spatial distributions of contaminant concentrations,
water levels, etc. This tool is the model.

Examples of potential model applications include:

    • Design and/or evaluation of pump-and-treat systems
    • Design and/or evaluation of hydraulic containment
    • Evaluation of physical containment systems (e.g., slurry
    • Analysis  of "no action" alternatives
    • Evaluation of past migration patterns of contaminants
    • Assessment of attenuation/transformation processes
    • Evaluation of the impact of nonaqueous phase liquids
      (NAPL) on remediation activities (dissolution studies)

What Is a Ground-Water Model?

A model may be defined as a simplified version of a real-world
system (here, a ground-water system) that approximately
simulates the relevant excitation-response relations of the
real-world system.  Since real-world systems are very
complex, there is a need for simplification in making planning
and management decisions. The simplification is introduced
as a set of assumptions which expresses the nature of the
system and those features of its behavior that are relevant to
the problem under investigation. These assumptions will
relate, among other factors, to the geometry of the
investigated domain, the way various heterogeneities will be
smoothed out, the nature of the porous medium (e.g., its
homogeneity, isotropy), the properties of the fluid (or fluids)
involved, and the type of flow regime under investigation.
Because a model is a simplified version of a real-world
system, no model is unique to a given ground-water system.
Different sets of simplifying assumptions will result in different
models, each approximating the investigated ground-water
system in a different way.  The first step in the modeling
process is the construction of a  conceptual model consisting
of a set of assumptions that verbally describe the system's
composition, the transport processes that take place in it, the
mechanisms that govern them, and the relevant medium
properties. This is envisioned or approximated by the modeler
for the purpose of constructing a model intended to provide
information for a specific problem.

Content of a Conceptual Model

The assumptions that constitute a conceptual model should
relate to such items as:

    • the geometry of the boundaries of the  investigated
      aquifer domain;
    • the kind of solid matrix comprising the aquifer (with
      reference to  its homogeneity, isotropy, etc.);
    • the mode of  flow in the aquifer (e.g., one-dimensional,
      two-dimensional horizontal, or three-dimensional);
    • the flow regime (laminar or nonlaminar);
    • the properties of the water (with reference to its
      homogeneity, compressibility, effect of dissolved solids
      and/or temperature on density and viscosity, etc.);
    • the presence of assumed sharp fluid-fluid boundaries,
      such as a phreatic surface;
    • the relevant  state variables and the area, or volume,
      over which the averages of such variables are taken;
    • sources and sinks of water and of relevant
      contaminants, within the domain and on its boundaries
      (with reference to their approximation as point sinks
      and sources, or distributed sources);
    • initial conditions within the considered domain;  and
    • the conditions on the boundaries of the considered
      domain that  express the interactions with its
      surrounding  environment.

Selecting the  appropriate conceptual model for a given
problem is one of the most important steps in the modeling
process.  Oversimplification may lead to a model that  lacks the
required information, while undersimplification may result in a
costly model,  or in the lack of data required for model
calibration and parameter estimation, or both. It is, therefore,
important that all features relevant to a considered problem be
included in the conceptual model and that irrelevant ones be

The selection of an appropriate conceptual model and the
degree of simplification  in any particular case depends on:

    • the objectives of the management  problem;
    • the available resources;
    • the available field data;
    • the legal and regulatory framework applying to  the

The objectives dictate which features of the investigated
problem should be  represented in the model, and to what
degree of accuracy. In some cases averaged water levels
taken over large areas may be satisfactory, while in others
water levels at specified points may be necessary. Natural
recharge may be introduced as monthly, seasonal or annual
averages. Pumping may be assumed to be uniformly
distributed over large areas, or it may be represented  as point
sinks. Obviously, a more detailed model is more costly and
requires more skilled manpower, more sophisticated codes
and larger computers. It is important to select the appropriate
degree of simplification  in each case.

Selection of the appropriate conceptual model for a given
problem is not necessarily a conclusive activity at the  initial
stage of the investigations. Instead, modeling should  be
considered as a continuous activity in which assumptions are
reexamined, added, deleted and modified as the
investigations continue. It is important to emphasize that the
availability of field data required for model calibration and
parameter estimation dictates the type of conceptual model to
be selected and the degree of approximation involved.

The next step in the modeling process is to express the
(verbal) conceptual model in the form of a mathematical
model. The solution of the mathematical  model yields the
required predictions of the real-world system's behavior in
response to various sources and/or sinks.

Most models express nothing but a balance of the considered
extensive quantity (e.g., mass of water or mass of solute). In
the continuum approach, the balance equations are written  "at
a point within  the domain," and should be interpreted to mean
"per unit area, or volume, as the case may be, in the vicinity of
the point." Under such conditions, the balance takes the form
of a partial differential equation. Each term in that equation

expresses a quantity added per unit area or per unit volume,
and per unit time.  Often, a number of extensive quantities of
interest are transported simultaneously; for example, mass of
a number of fluid phases with each phase containing more
than one relevant species. The mathematical model will then
contain a balance equation for each extensive quantity.

Content of a Mathematical Model

The complete statement of a mathematical model consists of
the following items:

    • a definition of the geometry of the considered domain
      and its boundaries;
    • an equation (or equations) that expresses the balance
      of the considered extensive quantity (or quantities);
    • flux equations that relate the flux(es) of the considered
      extensive quantity(ies) to the relevant state variables of
      the problem;
    • constitutive equations that define the behavior of the
      fluids and solids involved;
    • an equation (or equations) that expresses initial
      conditions that describe the known  state of the
      considered system at some initial time; and
    • an equation (or equations) that defines boundary
      conditions that describe the interaction of the
      considered domain with its environment.

All the  equations must  be expressed in terms of the
dependent variables selected for the problem. The selection of
the appropriate variables to be used in a particular case
depends on the available data. The number of equations
included in the model must be equal to the number of
dependent variables. The boundary conditions should be such
that they enable a unique, stable solution.

The most general boundary condition for any extensive
quantity states that the difference in the normal component of
the total flux of that quantity, on both sides of the boundary, is
equal to the strength of the source of that  quantity. If a source
does not exist, the statement  reduces to an equality of the
normal component of the total flux on  both sides  of the
boundary.  In such equalities, the information related to the
external side must be known (Bear and Verruijt, 1987).  It is
obtained from field measurements or on the basis of past

The mathematical model contains the same information as the
conceptual one, but expressed as a set of equations which are
amenable to analytical  and numerical solutions.  Many
mathematical models have been proposed and published by
researchers and practitioners (see Appendix B). They cover
most cases of flow and contaminant transport in aquifers
encountered by hydrologists and water resource  managers.
Nevertheless, it is important to understand the procedure of
model development.

The following section introduces three fundamental
assumptions,  or items,  in conceptual models that are always
made when modeling ground-water flow and contaminant
transport and fate.
The Porous Medium as a Continuum

A porous medium is a continuum that replaces the real,
complex system of solids and voids, filled  with one or more
fluids, that comprise the aquifer.  Inability to model and solve
problems of water flow and contaminant transport within the
void space  is due to the lack of detailed data on its
configuration. Even if problems could be described and
solved at the microscopic level, measurements cannot be
taken at that level  (i.e., at a point within the void space), in
order to validate the model. To circumvent this difficulty, the
porous medium domain is visualized as a continuum with fluid
or solid matrix variables defined at every point.  Not only is the
porous medium domain as a whole visualized as a continuum,
but each of the phases and components within it is also
visualized as a continuum, with all continua overlapping each
other within the domain.

The passage from the microscopic description of transport
phenomena to a macroscopic one is achieved by introducing
the concept of a representative elementary volume (REV) of
the porous medium domain. The main characteristic of an
REV is that the averages of fluid  and solid properties taken
over it are independent of its size. To conform to this
definition, the REV should be much larger than the
microscopic scale  of heterogeneity associated with the
presence of solid and void spaces, and much smaller than the
size of the considered domain. With this concept of an REV in
mind, a porous medium domain can be defined as a portion of
space occupied by a number of phases: a solid phase (i.e.,
the solid matrix), and one or more fluid phases, for which an
REV can be found.

Thus, a  macroscopic value at a point within a porous medium
domain is interpreted as the average of that variable taken
over the REV centered at that point. By averaging a variable
over all points within a porous medium domain, a continuous
field of that variable is obtained.

By representing the actual porous medium as a continuum,
the need to know the detailed microscopic configuration of the
void space  is circumvented. However, at the macroscopic
level, the complex geometry of the void-solid interface is
replaced by various solid matrix coefficients, such as porosity,
permeability and dispersivity. Thus, a coefficient that appears
in a macroscopic model represents the relevant effect of the
microscopic void-space configuration.

In practice,  all models describing ground-water flow and
contaminant transport are written at the continuum, or
macroscopic level. They are obtained  by averaging the
corresponding microscopic models  over the REV. This means
that one must start by understanding phenomena that occur at
the microscopic level, (e.g., on the boundary between
adjacent phases) before deriving the macroscopic model.  For
most models of practical interest, this has already been done
and published.

Horizontal Two-Dimensional Modeling

A second fundamental approximation often employed in
dealing with regional problems of flow and contaminant
transport is that ground-water flow is essentially horizontal.
The term "regional" is used here  to indicate a relatively large
aquifer domain. Typically, the horizontal dimension may be
from tens to hundreds of kilometers with a thickness of tens to
hundreds of meters.

In principle, ground-water flow  and contaminant transport in a

porous medium domain are three-dimensional. However,
when considering regional problems, one should note that
because of the ratio of aquifer thickness to horizontal length,
the flow in the aquifer is practically horizontal. This
observation also remains valid when small changes exist in
the thickness of a confined aquifer, or in the saturated
thickness of an unconfined aquifer.  On the basis of this
observation, the assumption that ground-water flow is
essentially horizontal is often made and  included in the
conceptual model. This leads to an aquifer flow model written
in horizontal two dimensions only.  Formally,  the two-
dimensional horizontal flow model is obtained by integrating
the corresponding three-dimensional variable over the
aquifer's thickness.  This procedure is known as the hydraulic
approach. The two-dimensional horizontal flow model is
written in terms of variables which are averaged over the
vertical thickness of the aquifer and thus are  a function of the
horizontal coordinates  only.

In the process of transforming a three-dimensional problem
into a two-dimensional one, new aquifer transport and storage
coefficients (e.g., aquifer transmissivity and storativity) appear.
In addition to the advantage of having to solve a two-
dimensional rather than a three-dimensional  mathematical
model, fewer field observations may be required for the
determination of these coefficients.

Whenever justified on the basis of the geometry (i.e.,
thickness versus horizontal length) and the flow pattern, the
assumption of essentially horizontal flow greatly simplifies the
mathematical analysis  of the flow in an aquifer. The error
introduced by this assumption is small in most cases of
practical interest.

The assumption of horizontal flow fails in regions where the
flow has a large vertical component (e.g., in the vicinity of
springs, rivers or  partially penetrating wells).  However, even
in these cases, at some distance from the source or sink, the
assumption of horizontal flow is valid again. As a general rule,
one may assume horizontal flow at distances larger than 1.5
to 2 times the thickness of the aquifer in  that  vicinity (Bear,
1979). At smaller distances the flow is three-dimensional and
should be treated as such.

The assumption of horizontal flow is also applicable to leaky
aquifers. When the  hydraulic conductivity of  the aquifer is
much larger than that of the semipermeable layer,  and the
aquifer thickness is much larger than the thickness of the
aquitard, it follows from the law of refraction of streamlines
("tangent law") that the flow in the aquifer is essentially
horizontal, while it is practically  vertical in the aquitards (de
Marsily, 1986).

When considering contaminant transport in aquifers, the
model user must be cautious in attempting to utilize a two-
dimensional model,  because in  most cases the hydraulic
approach is not justified . The contaminant may be trans-
ported through only  a small fraction of the aquifer's thickness.
In addition, velocities in different strata may vary appreciably
in heterogeneous aquifers, resulting in a marked difference in
the rates of advance and spreading of a contaminant.
Momentum Balance

The third concept relates to the fluid's momentum balance.  In
the continuum approach, subject to certain simplifying
assumptions included in the conceptual model, the momentum
balance equation reduces to the linear motion equation known
as Darcy's law. This equation is used as a flux equation for
fluid flow in a porous medium domain. With certain
modifications, it is also applicable to multiphase flows (e.g.,
air-water flow in the unsaturated zone).

Major Balance Equations

The following examples of major balance equations constitute
the core of models that describe flow and contaminant
transport in porous medium domains. A number of simplifying
assumptions must be stated before any of these equations
can be written. Although these assumptions are not listed
here, they must be included in the conceptual model of the
respective cases.

Mass balance for 3-D saturated flow in a porous medium
where   S.   =  specific storativity of porous medium
             =  piezometric head
             =  hydraulic conductivity tensor
t   =
The specific storativity, So, is defined as the volume of water
added to storage in a unit volume of porous medium, per unit
rise of piezometric head. Hence, the left side of equation (1)
expresses the volume of water added to storage in the porous
medium domain per unit volume of porous medium  per unit
time. The divergence of a flux vector, q, written
mathematically as  •  q, expresses the excess of outflow
over inflow per unit volume of porous medium, per unit time.
The flux q is expressed by Darcy's law,

                q   =  -K •  (p

Note that in equation (1), the  operators   (scalar) (to be read
as "gradient of the scalar"), and  •    (vector)  (to be read as
the "divergence of the vector"), are in the three-dimensional
space. The variable to be solved is 
storage in a unit area of aquifer, per unit rise of piezometric
head.  Hence, the left side of equation (2) expresses the
volume of water added to storage in the aquifer, per unit area
per unit time. The divergence of a  flux vector,(= -T • 
observed in the real system with that predicted by the model.
The sought values of the coefficients are those that will make
the two sets of values of state variables identical. However,
because the model is only an approximation of the real
system, one should  never expect these two sets of values to
be identical. Instead, the "best fit" between them must be
sought according to some criterion. Various techniques exist
for determining the "best" or "optimal" values of these
coefficients (i.e., values that will make the predicted values
and the measured ones sufficiently close to each other).
Obviously, the values of the coefficients eventually accepted
as "best"  for the model depend on the criteria selected for
"goodness of fit" between the observed and predicted values
of the relevant state variables.  These,  in turn, depend on the
objective of the modeling.

Some techniques  use the basic trial-and-error method
described above, while others employ more sophisticated
optimization methods. In some  methods, a priori  estimates of
the coefficients, as well  as information about lower and upper
bounds, are introduced. In addition to the question of selecting
the appropriate criteria, there remains the question of the
conditions under which the identification problem, also called
the inverse problem, will result in a unique solution.

As stressed above, no model can be used for predicting the
behavior of a system unless the numerical values of its
parameters have been determined by some identification
procedure. This requires that data be obtained by field
measurements. However, even without such data, certain
important questions about the suitability of the model can be
studied. Sensitivity analysis enables the modeler to
investigate whether a certain percentage change in a
parameter has any real significance, that is whether it is a
dominant parameter or not. The major point to be established
from a sensitivity analysis is the relative sensitivity of the
model predictions to changes in the values of the model
parameters within the estimated range of such changes.

A successful model application  requires appropriate site
characterization and expert insight into the modeling process.
Figure 1 illustrates  a simple  diagram of a model application
process. Each phase of the  process may consist of various
steps; often, results from one step are used as feedback in
previous steps, resulting in an iterative procedure (van der
Heijdeetal., 1989).
Methods of Solution

Once a well-posed  model for a  given problem has been
constructed, including the numerical values of all the
coefficients that appear in the model, it must be  solved for any
given set of excitations (i.e., initial and boundary conditions,

More Data

Formulation of Objectives 1

Review and Interpretation 1
of Available Data 1
Model Conceptualiz

Code Selection 1
Field Data Collect

ion 1 *-*

Input Data Preparation 1
V '
Vibration and Sensitivit
A i - h

Predictive Runs 1


Uncertainty Analysis 1
Figure 1.  Model Application Process.

sources and sinks).  The preferable method of solution is the
analytical one, as once such a solution is derived, it can  be
used for a variety of cases (e.g., different values of
coefficients, different pumping rates, etc). However, for most
cases of practical interest, this method of solution is not
possible due to the irregularity of the domain's shape, the
heterogeneity of the domain with respect to various
coefficients, and various nonlinearities. Instead, numerical
models are employed.

Although a numerical model is derived from the mathematical
model, a numerical model of a given problem need not
necessarily be considered as the numerical method of
solution, but as a model of the problem in its own right. By
adding assumptions to the conceptual model of the given
problem (e.g., assumptions related to time and space
discretization) a new conceptual model is obtained which, in
turn, leads to the numerical model of the given problem.  Such
a model represents a different approximate version of the real

Even those who consider a numerical model as a model in its
own right very often verify it by comparing the model results
with those obtained by an analytical solution of the
corresponding mathematical model (for relatively simple cases
for which such solutions can be derived). One of the main
reasons for such a verification is the need to eliminate errors
resulting from the numerical approximations alone. Until  the
early 1970s, physical (e.g., sand box) and analog (e.g.,
electrical) laboratory models were used as practical tools for
solving the mathematical models that described ground-water
flow problems. With the  introduction of computers and their
application in the solution of numerical models,  physical  and
analog models have become cumbersome as tools for
simulating ground-water regimes. However, laboratory
experiments in soil columns are still needed when new
phenomena are being investigated and to validate new
models (i.e., to examine whether certain  assumptions that
underlie the model are valid).

Analytical Models

During the early phase of a ground-water contamination  study,
analytical models offer an inexpensive way to evaluate the
physical characteristics of a ground-water system.  Such
models enable investigators to conduct a rapid preliminary
analysis of ground-water contamination and to perform
sensitivity analysis. A number of simplifying assumptions
regarding the ground-water system are necessary to obtain an
analytical solution. Although these assumptions do not
necessarily dictate that analytical models cannot be used in
"real-life" situations, they do require sound professional
judgment and experience in their application to field situations.
Nonetheless, it is also true that in many field situations few
data are available; hence, complex numerical models are
often of limited use.  When sufficient data have been
collected, however, numerical models may be used for
predictive evaluation and decision assessment.  This can be
done during the later phase of the study. Analytical models
should be viewed as a useful complement to numerical

For more information on analytical solutions, the reader is
referred to Bear (1979), van Genuchten and Alves (1982), and
Walton (1989).
Numerical Models

Once the conceptual model is translated into a mathematical
model in the form of governing equations, with associated
boundary and initial conditions, a solution can be obtained  by
transforming it into a numerical model and writing a computer
program (code) for solving it using a digital computer.

Depending on the numerical technique(s) employed in solving
the mathematical model, there exist several types of numerical

    • finite-difference models
    • finite-element models
    • boundary-element models
    • particle tracking models
      - method-of-characteristics models
      - random walk models, and
    • integrated finite-difference models.

The main features of the various numerical models are:

   1.  The solution is sought for the numerical values of state
      variables only at specified points in the space and time
      domains defined for the problem (rather than their
      continuous variations in these domains).
   2.  The partial differential equations that represent
      balances of the considered extensive quantities are
      replaced by a set of algebraic equations (written in
      terms of the sought, discrete values of the state
      variables at the discrete points in space and time).
   3.  The solution is obtained for a specified set of numerical
      values of the various model coefficients (rather than as
      general relationships in terms of these coefficients).
   4.  Because of the large number of equations that must be
      solved simultaneously, a computer program is

In  recent years, codes have been developed for almost all
classes of problems encountered in  the management of
ground water. Some  codes are very comprehensive and can
handle a variety of specific problems as special cases, while
others are tailor-made for particular  problems. Many of them
are available in the public domain, or for a nominal fee.  More
recently, many codes have been developed or adapted for


Much uncertainty is associated with  the modeling of a given
problem.  Among them, uncertainties exist in

    • the transport mechanisms;
    • the various sink/source phenomena for the considered
      extensive quantity;
    • the values of model coefficients, and their spatial (and
      sometimes temporal) variation;
    • initial conditions;
    • the location of domain boundaries and the conditions
      prevailing on them;
    • the meaning of measured data employed in model
      calibration; and
    • the ability of the model to cope with a problem in which
      the solid matrix heterogeneity spans a range of scales,

      sometimes orders of magnitude apart.
(Mercer and Faust, 1981):
The degree of uncertainty is increased in most cases by the
lack of sufficient data for parameter estimation and model
validation. Errors in observed data used  for parameter
identification also contribute to uncertainty in the estimated
values of model parameters.

Various methods for introducing uncertainty into the models
and the modeling process have been proposed. For example,
one approach is to employ Monte Carlo methods in which the
various possibilities are represented in a large number of
simulated realizations. Another approach is to construct
stochastic models in which the various coefficients are
represented as probability distributions rather than
deterministic values.

Often the question is  raised as to whether, in view of all these
uncertainties, which always exist in any real-world problem,
models should still be regarded as reliable tools for providing
predictions of real-world behavior-f/iere is no alternative!
However, the kind of answers models should be expected  to
provide and the very objectives of employing models, should
be broadened beyond the simple requirement that they
provide the predicted response of the system to the planned
excitations. Stochastic models provide probabilistic predictions
rather than deterministic ones. Management must then make
use of such predictions in the decision-making process.
Methodologies for evaluating uncertainties will have to be
developed; especially methods for evaluating the worth of
data in reducing uncertainty. It then becomes a management
decision whether or not to invest more in data acquisition.

In view of the uncertainty involved in modeling, models should
be used for additional roles, beyond predicting or estimating
the deterministic or probabilistic responses to planned
excitations. Such roles include:

    • predicting trends and direction of changes;
    • providing information on the sensitivity of the system to
      variations in various parameters, so that more
      resources can  be allocated to reduce their uncertainty;
    • deepening our understanding of the system and of the
      phenomena of interest that take place within it; and
    • improving the design of observation networks.

Many researchers are currently engaged in developing
methods that incorporate the element of uncertainty in both
the forecasting and the inverse models (Freeze et al., 1989;
Gelhar, 1986; Yeh, 1986; Neuman et al., 1987, and others).

Model Misuse

As stated above, the most crucial step in ground-water
modeling is the development of the conceptual model.  If the
conceptual model is wrong (i.e., does not represent the
relevant flow and contaminant transport phenomena), the rest
of the modeling efforts —translating the conceptual model
into mathematical and numerical models, and solving for
cases of interest — are a waste of time and money. However,
mistakes and misuses may occur during any phase of the
modeling process (Mercer, 1991).

Following is a list of the more common misuses and mistakes
related to modeling.  They may be divided  into four categories
  1.   Improper conceptualization of the considered problem:

    • improper delineation of the model's domain;
    • wrong selection of model geometry: a 2-D
      horizontal model, or a 3-D model;
    • improper selection of boundary conditions;
    • wrong assumptions related to homogeneity and
      isotropy of aquifer material;
    • wrong assumptions related to the significant processes,
      especially in cases of contaminant transport.  These
      may include the type of sink/source phenomena,
      chemical and biological transformations, fluid-solid
      interactions, etc.; and
    • selecting a model that involves coefficients that vary in
      space, but for which there are insufficient data for
      model calibration and parameter estimation.

  2.   Selection of an inappropriate code for solving the

    • selecting a code much more powerful/versatile than
      necessary for the considered problem;
    • selection of a code that has not been verified and

  3.   Improper model application:

    • selection of improper values for model parameters and
      other input data;
    • misrepresentation of aquitards in a multi-layer system;
      mistakes related to the selection of grid size and time
    • making predictions with a model that has been
      calibrated under different conditions;
    • making mistakes in model calibration (history
      matching); and
    • improper selection of computational parameters
      (closure criterion, etc.).

  4.   Misinterpretation of model results:

    • wrong hydrological interpretation of model results;
    • mass balance is not achieved.

Sources of Information

In selecting a code, its applicability to a given problem and its
efficiency in solving the problem are important criteria. In
evaluating a code's applicability to a problem and its
efficiency, a good description of these characteristics should
be accessible.  For a large number of ground-water models,
such information is available from the  International Ground
Water Modeling Center (IGWMC,  Institute for Ground-Water
Research and Education, Colorado School of Mines, Golden,
Colorado 80401), which operates  a clearinghouse service for
information and software pertinent to ground-water modeling.
Information databases have been developed to efficiently
organize, update and access information on ground-water
models for mainframe and microcomputers. The model
annotation databases have been developed and maintained
over the years with major support from U.S. EPA's Robert S.
Kerr Environmental Research Laboratory (RSKERL), Ada,

The Center for Subsurface Modeling Support (CSMoS),
located at the RSKERL (P.O. Box 1198, Ada, OK 74820),
provides ground-water and vadose zone modeling software
and services to public agencies and private companies
throughout the nation. CSMoS primarily manages and
supports ground-water models and databases resulting from
research at RSKERL. CSMoS integrates the expertise of
individuals in all aspects of the environmental field in an effort
to apply models to better understand and resolve subsurface
problems.  CSMoS is supported internally by RSKERL
scientists and engineers, and externally by the IGWMC,
National Center for Ground Water Research and numerous
ground-water modeling consultants from academia and the
private consulting community.

The National Ground Water Information Center (NGWIC, 6375
Riverside Drive, Dublin, Ohio 43017) is an information-
gathering and dissemination business that performs
customized literature searches on various ground-water-
related topics, and locates and retrieves copies of available
documents. The center maintains its own on-line databases.
Appendix A: Suggested Format for a

Ground-Water Modeling Report

Following is a suggested standardized format for a report that
involves modeling and analysis of model results. The emphasis
is on the modeling efforts and related activities.  It is not an
attempt to propose a structure for a project report.  The Ground
Water Modeling Section (D-18.21.10) of the American Society of
Testing and Materials (ASTM) Subcommittee on Ground Water
and Vadose Zone Investigations is in the process of developing
standards on ground-water modeling.  Additionally, specific
infor-mation regarding the content of ground-water modeling
studies is addressed by Anderson and Woessner (1992,
Chapter 9).


The introduction may start with a description of the problem that
lead to the investigations. The description will include the
domain in which the phenomena of interest take place,  and
what decisions are contemplated in connection with these
phenomena. The description should also include information
about the geography, topography, geology,  hydrology, climate,
soils, and other relevant features (of the domain and the
considered transport phenomena).  Sources of information
should be given. The description of the problem  should  lead to
the kind of information that is required by management/decision
maker, which the investigations described in the report  are
supposed to provide.
This section should  continue to outline the methodology used
for obtaining the required information.  In most cases, a model
of the problem domain and the transport (i.e., flow and
contaminant) phenomena will be the tool for providing
management with the required information.  On the premise that
this section concludes that such a model is needed, the
objective of the report is to describe the construction of the
model, the model runs, and the results leading to the required

Previous Studies

This section should contain a description of earlier relevant
studies in the area, whether on the same problem or in
connection with other problems. The objective of this section is
to examine the data and conclusions in these investigations, as
far as they relate to the current study.

The Conceptual Model

Because the previous section concluded that a model is
required, the objective of this section is to construct the
conceptual model of the problem, including the problem domain
and the transport phenomena taking place within it.  The content
of a conceptual model has been outlined in the text.  However,
the  importance of the conceptual model cannot be
overemphasized.  It is possible that the existing data will
indicate  more than one alternative model, if the available data
(or lack of it) so dictates.

The Mathematical Model

The conceptual model should be translated into a complete,
well-posed mathematical one. At this stage, the  various terms
that appear in the mathematical model should be analyzed, with
the  objective of deleting non-dominant effects. Further
simplifying assumptions may be added to the original
conceptual model at this stage.

If more than one conceptual model has been visualized, a
corresponding mathematical model  should be presented for
each. This section should conclude  with a list of coefficients and
parameters that appear in the model. The modeler should then
indicate  for which coefficients values, or at least  initial ones, are
available (including the actual numerical value and the source of
information), and for which coefficients the required information
is missing. In addition, the kind of field work or exploration
required to obtain that information should be reported.  If
possible, an estimate should be given for the missing values,
their possible range, etc. At this stage, it is important to conduct
and report a sensitivity analysis in order to indicate the
significance of the missing information,  bearing in mind the kind
of information that the model is expected to provide.

Selection of Numerical Model  and Code

The selected numerical model and the reasons for preferring it
over other models (public domain or proprietary) should be
presented. Some of the questions that should be answered are:
Was the code used as is, or was it modified for the purpose of
the  project? What were the modifications? If so stated in the
contract, the modified code may have to be included  in the
appendix to the report.  The full details of the code (name,
version,  manual, author, etc.) should be supplied. This section
may include a description of the hardware used in running the
code, as well as any other software (pre- and post-processors).
More information about model selection can be found in
Simmons and Cole (1985), Beljin and van der Heijde (1991).

Model Calibration

Every model must be calibrated before it can be  used as a tool
for predicting the behavior of a considered system. During the

calibration phase, the initial estimates of model coefficients may
be modified. The sensitivity analysis may be postponed until a
numerical model and a code for its solution have been selected.

In this section objectives of the calibration or history matching,
the adjusted parameters/coefficients, the criterion of the
calibration (e.g., minimizing the difference between observed
and predicted water levels), the available data, the model
calibration runs, etc., should be described.

The conclusions should be the modified set of parameters and
coefficients to be used in the model.

Model Runs

Justification and reasoning for the various runs.

Model Results

This section includes all tables and  graphic output. Ranges and
uncertainties  in model results should be indicated. Results of
sensitivity analysis and the significance of various factors should
also be discussed.


The information required by the decision maker should be
clearly outlined.


Tables and graphs, figures, and maps not presented in the body
of the report,  along with a  list of symbols, references, codes,
etc., should be included.
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