United States
Environmental Protection
Agency
Office of
Research and
Development
Off ice of Sol id Waste
and Emergency
Response
EPA/540/S-92/005
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
violated.
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
implemented.
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.
Director
-------
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
systems
Evaluation of physical containment systems (e.g., slurry
walls)
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
excluded.
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
situation.
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
experience.
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
domain:
f
K-
(1)
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
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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
Needed
Formulation of Objectives 1
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Review and Interpretation 1
of Available Data 1
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Model Conceptualiz
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Code Selection 1
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Field Data Collect
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Input Data Preparation 1
V '
Vibration and Sensitivit
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Conceptual
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Figure 1. Model Application Process.
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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
system.
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
models.
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
models:
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
prepared.
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
microcomputers.
Uncertainty
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
model:
selecting a code much more powerful/versatile than
necessary for the considered problem;
selection of a code that has not been verified and
tested.
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
steps;
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,
Oklahoma.
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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).
Introduction
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
information.
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
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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.
Conclusions
The information required by the decision maker should be
clearly outlined.
Appendices
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.
Appendix B: Selected Bibliography
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Anderson, M.P. 1984. Movement of Contaminants in
Groundwater: Groundwater Transport-Advection and
Dispersion. In Groundwater Contamination, Studies in
Geophysics, National Academy Press, Washington, D.C.
Anderson, M.P. and W.W. Woessner. 1992. Applied
Groundwater Modeling. Academic Press, Inc. San
Diego, California.
Appel, C.A. and T.E. Reilly. 1988. Selected Reports That
Include Computer Programs Produced by the U.S.
Geological Survey for Simulation of Ground-Water Flow
and Quality. WRI 87-4271. U.S.G.S., Reston, Virginia.
Bear, J. 1972. Dynamics or Fluids in Porous Media.
American Elservier, New York, New York.
Bear, J. 1979. Hydraulics of Groundwater. McGraw-Hill, New
York.
Bear, J. and A. Verruijt. 1987. Modeling Groundwater Flow
and Pollution. Kluwer Academic Publishers, Hingham,
Massachusetts.
Bear, J. and Y. Bachmat. 1990. Introduction to Modeling of
Transport Phenomena in Porous Media. Kluwer
Academic Publishers, Hingham, Massachusetts.
Beck, M.B. 1985. Water Quality Management: A Review of the
Development and Application of Mathematical Models.
NASA 11, Springer-Verlag, Berlin, West Germany.
Beljin, M.S. and P.K.M. van derHeijde. 1989. Testing,
Verification, and Validation of Two-Dimensional Solute
Transport Models. In (G. Jousma et al., eds.)
Groundwater Contamination: Use of Models in Decision-
Making. Kluwer Academic Publishers, Hingham,
Massachusetts.
Beljin, M.S. and P.K.M. van derHeijde. 1991. Selection of
Groundwater Models for WHPA Delineation. Proc. the
AWWA Computer Conference, Houston, Texas.
Boonstra, J. and N.A. De Ridder. 1981. Numerical Modelling
of Groundwater Basins, International Institute for Land
Reclamation and Improvement, Wageningen, The
Netherlands.
Boutwell, S.H., S.M. Brown, B.R. Roberts, and D.F. Atwood.
1985. Modeling Remedial Actions of Uncontrolled
Hazardous Waste Sites. EPA 540/2-85/001, U.S.
Environmental Protection Agency, Cincinnati, Ohio.
de Marsily, G. 1986. Quantitative Hydrogeology. Academic
Press, Inc., Orlando, Florida.
Domenico, P.A. 1972. Concepts and Models in Groundwater
Hydrology. McGraw-Hill, New York, New York.
Freeze, R.A., G. DeMarsily, L. Smith, and J. Massmann. 1989.
Some Uncertainties About Uncertainty. In (B.E. Buxton,
ed.) Proceedings of the Conference Geostatistical,
Sensitivity, and Uncertainty Methods for Ground-Water
Flow and Radionuclide Transport Modeling, San
Francisco, California, CONF-870971, Battelle Press,
Columbus, Ohio.
Gelhar, L.W., A.L. Gutjahr, and R.L. Naff. 1979. Stochastic
Analysis of Macrodispersion in Aquifers. Water
Resources Research, v. 15, no. 6, pp. 1387-1397.
Gelhar, L.W. 1984. Stochastic Analysis of Flow in
Heterogeneous Porous Media. In (J.Bear and M.Y.
Corapcioglu, eds.) Fundamentals of Transport
Phenomena in Porous Media, Marinus Nijhoff Publishers,
Dordrecht, The Netherlands.
Gelhar, L.W. 1986. Stochastic Subsurface Hydrology from
Theory to Applications. Water Resources Research, v.
22, no. 9, pp. 135S-145S.
Gorelick, S.M. 1983. A Review of Distributed Parameter
Groundwater Management Modeling Methods. Water
Resources Research, v. 19, no. 2, pp. 305-319.
Grove, D.B. and K.G. Stollenwerk. 1987. Chemical Reactions
Simulated by Ground-Water Quality Models. Water
Resources Bulletin, v. 23, no. 4, pp. 601-615.
Herrling, B. and A. Heckele. 1986. Coupling of Finite Element
and Optimization Methods for the Management of
Groundwater Systems. Advances in Water Resources, v.
9, no. 4, pp. 190-195.
Hunt, B. 1983. Mathematical Analysis of Groundwater
Resources. Butterworths Publishers, Stoneham,
Massachusetts.
Huyakorn, P.S. and G.F. Pinder. 1983. Computational
Methods in Subsurface Flow. Academic Press,
New York.
Huyakorn, P.S., B.H. Lester, and C.R. Faust. 1983. Finite
Element Techniques for Modeling Ground Water Flow in
Fractured Aquifers. Water Resources Research, v. 19,
no. 4, pp. 1019-1035.
Istok, J. 1989. Groundwater Modeling by the Finite-Element
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Method. AGU Water Resources Monograph 13, American
Geophysical Union, Washington, D.C.
Javandel, I., C. Doughty, and C.F. Tsang. 1984. Groundwater
Transport: Handbook of Mathematical Models. AGU
Water Resources Monograph 10, American Geophysical
Union, Washington, D.C.
Keely, J.F. 1987. The Use of Models in Managing Ground-
Water Protection Programs. EPA/600/8-87/003.
Kinzelbach, W. 1986. Groundwater Modeling: An Introduction
with Sample Programs in BASIC. Elsevier Publication
Company, Amsterdam, The Netherlands.
Konikow, LF. and J.D. Bredehoeft. 1978. Computer Model of
Two-Dimensional Solute Transport and Dispersion in
Ground Water. USGS Techniques of Water-Resources
Investigations. Book 7, Chap. C2, 90 pp.
Ligget, J.A. and P.L-F. Liu. 1983. The Boundary Integral
Equation Method for Porous Media Flow. Allen and
Unwin, Inc., Winchester, Massachusetts.
Mercer, J.W. and C.R. Faust. 1981. Ground-Water Modeling.
National Water Well Association (NWWA), Worthington,
Ohio.
Mercer, J.W., S.D. Thomas, and B. Ross. 1982. Parameters
and Variables Appearing in Repository Siting Models.
Report NUREG/CR-3066, U.S. Regulatory Commission,
Washington, D.C.
Mercer, J.W. 1991. Common Mistakes in Model Applications.
Proc. ASCE Symposium on Ground Water, Nashville,
Tennessee, July 29-August 2, 1991.
Moore, J.E. 1979. Contribution of Ground-water Modeling to
Planning. Journal of Hydrology, v. 43 (October), pp.121-
128.
Narasimhan, T.N. and P.A. Witherspoon. 1976. An Integrated
Finite-Difference Method for Analyzing Fluid Flow in
Porous Media. Water Resources Research v. 12, no. 1,
pp. 57-64.
Narasimhan, T.N. 1982. Numerical Modeling in Hydrogeology.
/nT.N. Narasimhan (ed.), Recent Trends in
Hydrogeology, pp. 273-296, Special Paper 189
Geological Society of America, Boulder, Colorado.
National Research Council (NRC). 1990. Ground Water
Models: Scientific and Regulatory Application. National
Research Council, Water Science and Technology Board,
Washington, D.C.
Neuman, S.P., C.L. Winter, and C.M. Newman. 1987.
Stochastic Theory of Field-Scale Fickian Dispersion in
Anisotropic Porous Media. Water Resources Research, v.
23, no. 3, pp. 453-466.
Pickens, J.F. and G.E. Grisak. 1981. Modeling of Scale-
Dependent Dispersion in Hydrologic Systems. Water
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Pinder, G.F. and J.D. Bredehoeft. 1968. Application of the
Digital Computer for Aquifer Evaluation. Water Resources
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Pinder, G.F. and W.G. Gray. 1977. Finite Element Simulation
in Surface and Subsurface Hydrology. Academic Press,
New York, 295 pp.
Pinder, G.F. and L. Abriola. 1986. On the Simulation of
Nonaqueous Phase Organic Compounds in the
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1092-1192.
Prickett, T.A. 1975. Modeling Techniques for Groundwater
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Hydroscience, Vol. 10.
Prickett, T.A. and C.G. Lonnquist. 1982. A Random-Walk
Solute Transport Model for Selected Groundwater Quality
Evaluations. Bull. No. 65, Illinois State Water Survey,
Urbana, 105 pp.
Remson, I., G.M. Hornberger, and F.J. Molz. 1971. Numerical
Methods in Subsurface Hydrology. John Wiley and Sons,
New York, New York.
Rushton, K.R. and S.C. Redshaw. 1979. Seepage and
Groundwater Flow: Numerical Analysis by Analog and
Digital Methods. John Wiley and Sons, Chichester, U.K.
Schmelling, S.G. and R.R. Ross. 1989. Contaminant
Transport in Fractured Media: Models for Decision
Makers. USEPA Superfund Ground Water Issue Paper.
EPA/540/4-89/004.
Simmons, C.S. and C.R. Cole. 1985. Guidelines for Selecting
Codes for Ground-Water Transport Modeling of Low-
Level Waste Burial Sites. PNL-4980, Volume I and II,
Battelle Pacific Northwest Labs, Richland, Washington.
Strack, O.D.L. 1989. Groundwater Mechanics. Prentice Hall,
Englewood Cliffs, New Jersey.
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.G.S. Techniques of Water Resources Investigation,
Book 7, Chap. C1, 116 pp.
U.S. Office of Technology Assessment. 1982. Use of Models
for Water Resources Management, Planning, and Policy.
U.S. OTA for Congress of the United States, U.S.
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van der Heijde, P.K.M. etal. 1985. Groundwater Management:
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van der Heijde, P.K.M. and M.S. Beljin. 1988. Model
Assessment for Delineating Wellhead Protection Areas.
EPA/440/6-88-002. Office of Ground Water Protection,
U.S. Environmental Protection Agency, Washington, D.C.
van der Heijde, P.K.M., A.I. El-Kadi, and S.A. Williams. 1989.
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95-108.
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