1

United States

Environmental Protection

Agency

Office of Solid Waste

and Emergency

Response

Office of

Research and

Development

Ground Water Issue

Fundamentals of Ground-Water Modeling

Robert S. Kerr Environmental

Research Laboratory

Ada, Oklahoma

Superfund Technology Support Center for

Ground Water

Jacob Beara, Milovan S. Beljinb, and Randall R. Rossc

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

Technology Innovation Office

Office of Solid Waste and Emergency

Response, US EPA, Washington, DC

Walter W. Kovalick, Jr., Ph.D.

Director

aTechnion - Israel Institute of Technology

bUniversity of Cincinnati

c U.S. EPA, Robert S. Kerr Environmental Research Laboratory

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 groundwater 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

EPA/540/S-92/005

April 1992
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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

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,
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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
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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 twodimensional 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 twodimensional 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 twodimensional model, because in most cases the hydraulic

approach is not justified . The contaminant may be transported 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:

S =  • ( K• φ ) o

ð φ

ð t , (1)

where So = specific storativity of porous medium

φ = piezometric head

Κ = hydraulic conductivity tensor

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  φ

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 φ ( x,y,z,t ).

Thus, equation (1) states that the excess of inflow over outflow

of water in a unit volume of porous medium, per unit time, at a

point, is equal to the rate at which water volume is being

stored, where storage is due to fluid and solid matrix

compressibilities.

Mass balance for 2-D saturated flow in a confined aquifer:

S = • (T• φ ) - P (x,y,t ) + R (x,y,t ) ð φ

ð t (2)

where S = aquifer storativity

φ = piezometric head

Τ = aquifer transmissivity tensor

P (x,y,t ) = rate of pumping (per unit area of aquifer)

R (x,y,t ) = rate of recharge (per unit area of aquifer)

The storativity, S, is defined as the volume of water added to
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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,(= -Τ • φ ),

expresses the excess of outflow over inflow per unit area, per

unit time. Note that here, the two operators are in the twodimensional horizontal coordinates, and the variable to be

solved is φ (x,y,t ).

Thus, equation (2) states that the excess of inflow over outflow

of water in a unit area of an aquifer, per unit time, at a point, is

equal to the rate at which water volume is being stored, where

storage is due to fluid and solid matrix compressibilities.

Mass balance for a solute in 3-D saturated flow:

The left side of equation (3) expresses the mass of the

conservative solute added to storage per unit volume of

porous medium per unit time (e.g., it does not adsorb, decay

or interact with the solid matrix).

 = -  • (c q + φJ* + φJ )+ Γ ð φ c

ð t (3)

where c = concentration of considered solute

φ = porosity of porous medium

q = specific discharge of water (= volume of

 water passing through a unit area of

porous medium per unit time)

J* = dispersive flux of solute (per unit area of

fluid)

J = diffusive flux of solute (per unit area of fluid)

Γ = strength of solute source (added quantity

per unit volume of porous medium per unit

time)

The first term on the right side of equation (3) (to be read as

“minus divergence of the total flux of the solute”), expresses

the excess of the solute’s inflow over outflow, per unit volume

of porous medium, per unit time. This total flux is made up of

an advective flux with the fluid, a dispersive flux and a

diffusive flux. The second term on the right side of equation

(3) expresses the added mass by various sources.

The dispersive and diffusive fluxes appearing in equation (3)

must be expressed in terms of the concentration, c, which

serves as the variable to be solved in this equation. For

example,

 J = - D *  c , J * = - D  c ,

where D = coefficient of dispersion

D * = coefficient of molecular diffusion

in a porous medium

Model Coefficients and Their Estimation

In passing from the microscopic level of describing transport to

the macroscopic level, various coefficients of transport and

storage are introduced. The permeability of a porous medium,

aquifer transmissivity, aquifer storativity, and porous medium

dispersivity, may serve as examples of such coefficients.

Permeability and dispersivity are examples of coefficients that

express the macroscopic effects of microscopic configuration

of the solid-fluid interfaces of a porous medium. They are

introduced in the passage from the microscopic level of

description to the macroscopic, continuum, level. The

coefficients of aquifer storativity and transmissivity are

introduced by the further averaging of the three-dimensional

macroscopic model over the thickness of an aquifer in order to

obtain a two-dimensional model. All these coefficients are

coefficients of the models, and therefore, in spite of the

similarity in their names in different models, their interpretation

and actual values may differ from one model to the next.

This point can be illustrated by the following example. To

obtain the drawdown in a pumping well and in its vicinity, one

employs a conceptual model that assumes radially converging

flow to an infinitesimally small well in a homogeneous,

isotropic aquifer of constant thickness and of infinite areal

extent. The same model is used to obtain the aquifer’s

storativity and transmissivity by conducting an aquifer

pumping test and solving the model’s equation for these

coefficients. It is common practice to refer to these

coefficients as the aquifer’s coefficients and not as coefficients

of the aquifer’s model. However, it is important to realize that

the coefficients thus derived actually correspond to that

particular model. These coefficients should not be employed in

a model that describes the flow in a finite heterogeneous

aquifer with variable thickness and with non-radial flow in the

vicinity of the well. Sometimes, however, there is no choice

because this is the only information available. Then, the

information must be used, keeping in mind that when

coefficients are derived by employing one model in another

model for a given domain, the magnitude of the error will

depend on the differences between the two models. In

principle, in order to employ a particular model, the values of

the coefficients appearing in it should be determined using

some parameter identification technique for that particular

model.

Obviously, no model can be employed in any specified domain

unless the numerical values of all the coefficients appearing in

it are known. Estimates of natural recharge and a priori

location and type of boundaries may be included in the list of

model coefficients and parameters to be identified. The

activity of identifying these model coefficients is often referred

to as the identification problem.

In principle, the only way to obtain the values of the

coefficients for a considered model is to start by investigating

the real-world aquifer system to find a period (or periods) in

the past for which information is available on (i ) initial

conditions of the system; (ii ) excitations of the system, as in

the form of pumping and artificial recharge (quality and

quantity), natural recharge introduction of contaminants, or

changes in boundary conditions, and (iii ) observations of the

response of the system, as in the form of temporal and spatial

distributions of water levels and solute concentrations. If such

a period (or periods) can be found, one can (i ) impose the

known initial conditions on the model, (ii ) excite the model by

the known excitations of the real system, and (iii ) derive the

response of the model to these excitations. Obviously, in order

to derive the model’s response, one has to assume some trial

values for the coefficients and compare the response
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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

Figure 1. Model Application Process.

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

Heijde et al., 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,

Formulation of Objectives

Review and Interpretation

of Available Data

Model Conceptualization

Code Selection

Field Data Collection

Input Data Preparation

Calibration and Sensitivity Analysis

Predictive Runs

Uncertainty Analysis

More Data

Needed

Improve

Conceptual

Model
7

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,
8

(Mercer and Faust, 1981):

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.

sometimes orders of magnitude apart.

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--there 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
9

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 informationgathering and dissemination business that performs

customized literature searches on various ground-waterrelated 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
10

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.

IIASA 11, Springer-Verlag, Berlin, West Germany.

Beljin, M.S. and P.K.M. van der Heijde. 1989. Testing,

Verification, and Validation of Two-Dimensional Solute

Transport Models. In (G. Jousma et al., eds.)

Groundwater Contamination: Use of Models in DecisionMaking. Kluwer Academic Publishers, Hingham,

Massachusetts.

Beljin, M.S. and P.K.M. van der Heijde. 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

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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

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

Anderson, M.P. 1979. Using Models to Simulate the

Movement of Contaminants through Ground Water Flow

Systems. Critical Reviews in Environmental Control, v.

9, no. 2, pp. 97-156.

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
11

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 GroundWater 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.

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Ground Water. USGS Techniques of Water-Resources

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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.

In T.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 ScaleDependent 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

Research. v. 4, no. 5, pp. 1069-1093.

Pinder, G.F. and W.G. Gray. 1977. Finite Element Simulation

in Surface and Subsurface Hydrology. Academic Press,

New York, 295 pp.

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

Evaluation. In Ven Te Chow (ed.), Advances in

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 LowLevel 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. FiniteDifference Model for Aquifer Simulation in TwoDimensions with Results of Numerical Experiments.

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Book 7, Chap. C1, 116 pp.

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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. et al. 1985. Groundwater Management:

The Use of Numerical Models. 2nd edition, AGU Water

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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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Modeling. Lewis Publishers, Chelsea, Michigan.

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Publishers, Chelsea, Michigan.

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Groundwater Modeling. W.H. Freeman and Co., San

Francisco, California.

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Procedures in Groundwater Hydrology: The Inverse

Problem. Water Resources Research, v. 22, no. 2, pp.

95-108.