Cover. See figures 2 and 3, page 6.
Guidelines for Evaluating Ground-Water Flow 

Models

By Thomas E. Reilly and Arlen W. Harbaugh

Scientific Investigations Report 2004-5038

U.S. Department of the Interior

U.S. Geological Survey
U.S. Department of the Interior

Gale A. Norton, Secretary

U.S. Geological Survey

Charles G. Groat, Director

U.S. Geological Survey, Reston, Virginia: 2004

For sale by U.S. Geological Survey, Information Services

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Although this report is in the public domain, permission must be secured from the individual copyright owners to reproduce any copyrighted materials contained within this report.

Suggested citation:

Reilly, T.E., and Harbaugh, A.W., 2004, Guidelines for evaluating ground-water flow models: U.S. Geological Survey Scientific Investigations Report 2004-5038, 30 p.
 iii

CONTENTS

Abstract. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Appropriateness of the computer model program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Different modeling approaches to address a problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

A calibrated model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

A hypothetical model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Superposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Particle tracking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Spatial and temporal approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Discretization and representation of the hydrogeologic framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Cell size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Specifying properties of cells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Time steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Representation of boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Representation of initial conditions in transient simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Accuracy of the matrix solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Adequacy of calibration for intended use of model results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Model input data, output listing, and report consistency check. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Model reporting and archiving. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Appendix – Office of Ground Water Technical Memorandum No. 96.04 – Policy on 

documenting the use of ground-water simulation in project reports. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Figures

1. Maps showing extent of the south-shore confining unit on Long Island, New York,

as represented in a regional ground-water flow model grid, and

as represented in a sub-regional ground-water flow model grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5

2. Maps showing representation of stream and shoreline boundaries on Long Island, 

New York, as represented in a regional ground-water flow model grid, and 

as represented in a sub-regional ground-water flow model grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6

3. Simulated drawdown from two wells using different grid spacings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6

4. Large finite-difference cells may be inadequate to represent some important 

features of a ground-water system. Map of the distribution of horizontal 

hydraulic conductivity showing a channel of high hydraulic conductivity. 

Finite-difference cells representing the high hydraulic-conductivity 

channel are no longer continuous, because there is no direct

connection between diagonal cells in the finite-difference method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7

5. Graph showing a cross section of drawdown showing the effect of grid spacing . . . . . . . . . . . . . . . . . . .7
iv 

6. Schemes of vertical discretization for aquifer cross section, aquifer cross 

section with rectilinear or uniform grid superimposed, and aquifer cross 

section with deformed grid superimposed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8

7. Horizontal and vertical discretization using uniform layers for the model simulating 

ground-water flow on Cape Cod, Massachusetts. 

A. Horizontal grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9 

B. Hydrogeologic cross section near column 85. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

C. Vertical grid using uniform layers along column 85. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

8. Horizontal and vertical discretization using deformed layers for the model simulating 

ground-water flow on Long Island, New York: horizontal grid, hydrogeologic 

cross section, and vertical grid using deformed layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

9. Maps showing comparison of areas contributing recharge to seven public-supply 

wells as determined by two different numerical models, Cape Cod, Massachusetts:

results from a two-dimensional single-layer model, and results from a 

three-dimensional eight-layer model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

10. Examples of interpolating data for cells from measured data. Cell value is 

the nearest measured value, cell value is the inverse-distance-squared 

weighted average of measured values, and cell value is the 

distance-weighted average of the two adjacent contours. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

11. Graph showing drawdown versus time for different numbers of time steps . . . . . . . . . . . . . . . . . . . . . . . .15

12. Flow chart of the ground-water flow modeling process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

13. Extent and model grid of the finite-difference model used to illustrate 

initial conditions: areal grid, and cross-sectional grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

14. Graph showing head distribution along a model row from the divide to the constanthead node for three different initial conditions used for a transient simulation. . . . . . . . . . . . . . . . . . . . . . 20

15. Graphs showing head in a cell through time in response to a well 

discharging at a rate of 100,000 ft3/d: the head in layer 1 at the divide, 

and the head in the cell with the discharging well in layer 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

16. Graph showing head in the well for the first 5 years after the start of 

pumping for the cases using the initial conditions of the steady-state head 

distribution and the linearly varying head distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

17. Cross-sectional view of a finite-difference representation simulating a variable thickness 

ground-water system with flow to a specified head due to areal recharge: starting 

head set at 20 meters, and starting head set at 100 meters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

18. Maps showing hydraulic-conductivity zones identified for automatic parameter estimation 

in a ground-water flow model of the Albuquerque Basin, New Mexico: zones in 

model layer 1, and zones in model layer 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Tables

1. Types of problems that may initiate a hydrologic study involving 

a ground-water flow model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3

2. Heads calculated for the same system with areal recharge 

and two different initial heads. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
 v

Conversion Factors

English units are used in all original work presented in this report. Figures and results from published studies are also presented throughout this report. The system of units that were originally 

used in these previously published studies are retained in this report in order not to introduce 

any errors and to show the level of approximation used in the investigator’s estimates.

Multiply By To obtain

foot (ft) 0.3048 meter (m)

mile (mi) 1.609 kilometer (km)

square mile (mi2) 2.590 square kilometer (km2)

cubic foot per second (ft3/s) 0.02832 cubic meter per second (m3/s)
Guidelines for Evaluating Ground-Water Flow Models

By Thomas E. Reilly and Arlen W. Harbaugh

Abstract

Ground-water flow modeling is an important tool frequently used in studies of ground-water systems. Reviewers and 

users of these studies have a need to evaluate the accuracy or 

reasonableness of the ground-water flow model. This report 

provides some guidelines and discussion on how to evaluate 

complex ground-water flow models used in the investigation of 

ground-water systems. A consistent thread throughout these 

guidelines is that the objectives of the study must be specified 

to allow the adequacy of the model to be evaluated. 

Introduction

The simulation of ground-water flow systems using computer models is standard practice in the field of hydrology. 

Models are used for a variety of purposes that include education, hydrologic investigation, water management, and legal 

determination of responsibility. In the most general terms, a 

model is a simplified representation of the appearance or operation of a real object or system. Ground-water flow models represent the operation of a real ground-water system with mathematical equations solved by a computer program. A difficulty 

that faces all individuals attempting to use the results of a model 

is the development of an understanding of the strengths and limitations of a model analysis without having to reproduce the 

entire analysis.

The primary purpose of this report is to help users of 

reports that document ground-water flow models evaluate the 

adequacy or appropriateness of a model. A secondary purpose 

for this report is to provide for model developers a guide to the 

information that should be included in model documentation. 

The information in this report is mainly qualitative. It reflects 

the views developed by the authors on the basis of over 50 years 

combined experience with ground-water modeling. The authors 

have used models, reviewed modeling studies and reports, provided modeling advice, taught modeling courses, and developed computer model programs.

It is important to distinguish among three terms we use to 

discuss the modeling process: conceptual model, computer 

model program, and model. A “conceptual model” is the 

hydrologist’s concept of a ground-water system. A “computer 

model program” is a computer program that solves groundwater equations. Computer model programs are general purpose in that they can be used to simulate a variety of specific 

systems by varying input data. A “model” is the application of 

a computer model program to simulate a specific system. Thus, 

a model incorporates the model program and all of the input 

data required to represent a ground-water system. The modeler 

attempts to incorporate what he or she believes to be the most 

important aspects of the conceptual model into a model so that 

the model will provide useful information about the system.

The information provided in this report is generally relevant to all types of ground-water flow model programs; however, the examples cited throughout the report use the model 

program MODFLOW (Harbaugh and others, 2000).

This report reviews the important aspects of simulating a 

ground-water flow system using a computer model program 

and explains the ramifications of various design decisions. An 

important part of the information necessary for evaluating a 

model is the intended use of a model, because it is impossible to 

develop a model that will fulfill all purposes. Further, the 

intended use must be specific as opposed to general. For example, saying that a model will be used to evaluate watermanagement alternatives is inadequate. Specific information 

about the alternatives to be considered also would be necessary. 

Thus, a consistent thread throughout this report is the need to 

consider the purpose of a model when evaluating the appropriateness of the model.

Appropriateness of the Computer Model 

Program

Many computer model programs are available for simulating ground-water systems. Each computer model program can 

be characterized by the mathematical method used to represent 

ground-water equations (Konikow and Reilly, 1999), assumptions, and the range of simulation capabilities. For example, the 

mathematical method in MODFLOW is finite difference in 

space and time, with backward difference for time. Major 
2 Guidelines for Evaluating Ground-Water Flow Models

assumptions are (1) confined three-dimensional flow with 

water-table approximations, and (2) principal directions of 

hydraulic conductivity are aligned with the coordinate axes. A 

variety of hydrologic capabilities are included, for example, the 

simulation of wells, rivers, recharge, and ground-water evapotranspiration. There also are simple analytical models that 

assume homogeneous conditions for one or two dimensions that 

can be used to solve some problems. The tool or computer 

model program used can be as simple or as complex as required 

for the problem, but the method, assumptions, and capabilities 

must be evaluated to assure that the tool is appropriate and can 

provide scientifically defensible results.

Questions to be answered in the evaluation of the appropriateness of the modeling program are:

1. Are the objectives of the study clearly stated?

2. Is the mathematical method used in the computer model 

program appropriate to address the problem?

3. Does the numerical or analytical model selected for use 

simulate the important physical processes needed to 

adequately represent the system? 

Different Modeling Approaches to Address 

a Problem

A general-purpose computer model program such as 

MODFLOW can be used in many ways to address a problem as 

illustrated in table 1. Approaches to a problem that are commonly used are: calibrated model, hypothetical system model, 

sensitivity analysis, superposition, and particle tracking. Frequently, several approaches are combined to address a problem.

A Calibrated Model

A model that is “calibrated” is required to address many 

hydrologic problems. Model calibration in its most limited 

meaning is the modification of model input data for the purpose 

of making the model more closely match observed heads and 

flows. Adjustment of parameters can be done manually or automatically by using nonlinear regression statistical techniques. 

In the broader meaning of model calibration, parameter adjustment is only one aspect of model calibration. Key aspects of the 

model, such as the conceptualization of the flow system, that 

influence the capability of the model to meet the problem objectives also are evaluated and adjusted as needed during calibration. For example, it may be noticed that some of the parameters 

that result in the best match to observations are not reasonable 

based on other knowledge of their values. This may indicate 

that there is a conceptualization problem with the model. Thus, 

the closeness of fit between the simulated and observed conditions, and the extent to which important aspects of the simulation are incorporated in the model are both important in evaluating how well a model is calibrated. In practice, calibration is 

conducted differently by each investigator; some examples that 

discuss calibrated models are Luckey and others (1986), Buxton 

and Smolensky (1999), and Anderson and Woessner (1992, 

section 8.3 and 8.4). 

The amount of effort that is required in calibrating a 

ground-water flow model is dependent upon the intended use of 

the model (that is, the objective of the investigation). Most models of specific ground-water systems that are used to estimate 

aquifer properties, understand the past, understand the present, 

or to forecast the future are calibrated by matching observed 

heads and flows. Determining if the calibration is sufficient for 

the intended use of the model is very important in evaluating 

whether the model has been constructed appropriately. (See 

later section for more on evaluating the adequacy of model 

calibration.)

A Hypothetical Model

A hypothetical model is a model of an idealized or representative system as opposed to a model of a specific system. In 

an attempt to understand the basic operation of a ground-water 

system, the determination of whether to develop a model of a 

hypothetical idealized system or a model of an actual system 

greatly affects the amount of data needed to construct the 

model. Hypothetical models are not calibrated, but input data 

are frequently adjusted during model development to make the 

model fit the idealized system or to test how the model 

responds. The utility of hypothetical models is that the system 

can be defined exactly and the cause and effect processes under 

investigation can be clearly identified with minimal cost. The 

input data needed to define the hypothetical system can be as 

simple or as complex as required to investigate the processes of 

interest. No effort is required to collect and interpret data from 

an actual ground-water system and no uncertainty exists in the 

ability of the model to represent the system, which results in 

substantial cost savings compared to making a model of a specific system. Hypothetical models have been used to examine 

various processes that affect or are affected by ground-water 

flow, for example: boundary conditions (Franke and Reilly, 

1987), contributing areas to wells (Morrissey, 1989; Reilly and 

Pollock, 1993), and model calibration (Hill and others, 1998).

Sensitivity Analysis

Sensitivity analysis is the evaluation of model input 

parameters to see how much they affect model outputs, which 

are heads and flows. The relative effect of the parameters helps 

to provide fundamental understanding of the simulated system. 

Sensitivity analysis also is inherently part of model calibration. 

The most sensitive parameters will be the most important 

parameters for causing the model to match observed values. For 

example, an area in which the model is insensitive to hydraulic 

conductivity generally indicates an area where there is relatively little water flowing. If the model is being calibrated, then 

changing the value of hydraulic conductivity in this area will 
Different Modeling Approaches to Address a Problem 3

not help much in causing the model to match observations. The 

calibration will not provide much certainty about the value of 

the parameter, but the uncertainty will not matter provided the 

model is not used in situations where large amounts of water 

will flow in that area. Such a model, however, would probably 

not be suitable for evaluation of recharge or withdrawal in this 

area because the amount of flow in the area would be much 

greater than it was when the model was calibrated, and the 

uncertainty from the calibration would be unacceptable. Anderson and Woessner (1992, p. 246-257) provide some examples 

of sensitivity analyses.

Sensitivity analysis can be conducted manually or automatically. In the manual approach, multiple model simulations 

are made in which ideally a single parameter is adjusted by an 

arbitrary amount. The changes to the model output for all of the 

parameter changes may be displayed in tables or graphs for 

evaluation. The automatic approach directly computes parameter sensitivity, which is the change in head or flow divided by 

the change in a parameter. Automatic sensitivity analysis is 

inherently part of automatic parameter adjustment for model 

calibration. The automatic parameter adjustment algorithm uses 

parameter sensitivity to compute the parameter values that 

cause the model to best match observed heads and flows.

Table 1. Types of problems that may initiate a hydrologic study involving a ground-water flow model.

Problem Type Reason for Undertaking Study Approach to Model the Problem

Basic Understanding of GroundWater System

Investigation of hydrologic processes

• Hypothetical system model

• Superposition

• Particle Tracking

Determination of effective data collection 

network

• Calibrated model

• Hypothetical system model

• Superposition

• Sensitivity analysis

Preliminary model to determine current 

level of understanding

• Calibrated model

• Hypothetical system model

• Superposition

• Sensitivity analysis

Estimation of Aquifer Properties

Aquifer test analysis • Calibrated model

• Superposition

Determination of aquifer properties • Calibrated model

Understanding the Past

Understanding historical development of an 

aquifer system • Calibrated model

Estimation of predevelopment conditions • Calibrated model

Understanding the Present

Determination of the effect of ground-water 

pumpage on surface-water bodies

• Calibrated model

• Superposition

• Particle Tracking

Determination of sources of water to wells • Calibrated model

• Particle Tracking

Determination of responsible parties causing 

impacts on the system

• Calibrated model

• Particle Tracking

Forecasting the Future Management of a system

• Calibrated model

• Superposition

• Particle Tracking
4 Guidelines for Evaluating Ground-Water Flow Models

Superposition

Superposition (Reilly and others, 1987) is a modeling 

approach that is useful in saving time and effort and eliminating 

uncertainty in some model evaluations. Models that are 

designed to use superposition evaluate only changes in stress 

and changes in responses. Most aquifer tests that analyze drawdown use superposition. Only the change in heads (the drawdown) and change in flows are analyzed, which assumes the 

response of the system is only due to the stress imposed and is 

not due to other processes in the system. The absolute value of 

the head and a quantification of the actual regional flows are not 

needed. In the past, superposition was frequently used with analog model analysis of ground-water systems because electrical 

simulation of areal stresses and boundary conditions was 

extremely difficult. As modern numerical computer models 

made simulation of all stress conditions easier, superposition 

was used less frequently in areal models. If the problem to be 

solved involves only the evaluation of a change due to some 

change in stress, however, the application of superposition can 

greatly simplify the data needs for model development. Superposition is strictly applicable to linear problems only, that is, 

constant saturated thickness and linear boundary conditions. If 

the system is relatively linear, however, for example the saturated thickness does not change by a significant portion (no 

absolute guidance can be given, but some investigators have 

used a 10 percent change in thickness as a rule of thumb), superposition can still provide reasonably accurate answers. Currently, superposition is used primarily in the simulation of aquifer tests, in that only changes due to the imposed change in 

stress (that is, the well discharge) are simulated and zero drawdowns are specified as the initial and boundary conditions; 

example simulations are presented in Prince and Schneider 

(1989) and McAda (2001).

Particle Tracking

Particle tracking (Pollock, 1989) is the determination of 

the path a particle will take through a three-dimensional 

ground-water flow system. The determination of the paths of 

water in the flow system aids in conceptualizing and quantifying the sources of water in a modeled system. For example, 

Buxton and others (1991) used particle-tracking analysis to 

determine recharge areas on Long Island, New York, and Modica and others (1997) made use of particle tracking in the context of a ground-water flow model to understand the patterns 

and age distribution of ground-water flow to streams of the 

Atlantic Coastal Plain. Although particle tracking is useful in 

determining advective transport, this report does not address the 

use of models to determine transport of chemicals, but rather 

refers to the approach of using particle tracking to understand 

the flow system. 

Spatial and Temporal Approaches

In addition to the overall modeling approaches discussed 

above, many model programs can be used in one, two, or three 

dimensions, and they can be applied as transient or steady state. 

The simplification of the model domain to one or two dimensions, either in plan view or cross section, is used to minimize 

the cost of constructing a model. The simplification of the system to one or two dimensions, however, must be consistent with 

the flow field under investigation and consistent with the objectives of the study. Consistent with the flow field, means that 

there is no or negligible flow orthogonal to the line or plane of 

the one- or two-dimensional system being simulated. 

Steady-state models are used widely, although true steadystate conditions do not exist in natural systems. All natural systems fluctuate in response to climatic variations that can be seasonal, annual, decadal or longer. In steady-state models, an 

assumption is made that a system can be represented by a state 

of dynamic equilibrium or an approximate equilibrium condition. If the objectives of the investigation do not require information on the time it takes for a system to respond to new 

stresses or the response of the system between periods of relative equilibrium, then simulation of the system as a steady-state 

system may be a reasonable approach. However, if the system 

is not at a period of equilibrium or approximate equilibrium during the periods of interest, then a transient analysis is required. 

Questions to be answered in the evaluation of the appropriateness of the modeling approach to analyze the problem are:

1. Is the overall approach (calibrated model, hypothetical 

system model, sensitivity analysis, superposition, and 

particle tracking) for using simulation in addressing the 

objectives clearly stated and appropriate?

2. If the analysis is not three dimensional, is the 

representation of the system using one or two dimensions 

appropriate to meet the objectives of the study and 

justified in the report?

3. If the model is steady state, is adequate information 

provided to justify that the system is reasonably close to 

a steady-state condition?

Models of ground-water systems may be very different in 

their level of complexity. Whether the model design and 

approach are appropriate for the problem being investigated 

must be evaluated. This evaluation requires a clear statement of 

the problem to be investigated and the modeling approach. A 

further requirement is an understanding of the model design. 

The remainder of this report focuses on specific aspects of 

model design that should be examined in determining the worth 

of a particular model. These aspects are: discretization and representation of the hydrogeologic framework, boundary conditions, initial conditions, accuracy of the numerical solution, and 

accuracy of calibration for the intended use of the model.
Discretization and Representation of the Hydrogeologic Framework 5

Discretization and Representation of the 

Hydrogeologic Framework

A fundamental aspect of numerical models is the representation of the real world by discrete volumes of material. The 

volumes are called cells in the finite-difference method, and the 

volumes are called elements in the finite-element method. The 

accuracy of the model is limited by the size of the discrete volumes. Further, for transient models, time is represented by discrete increments of time called time steps in most model programs. The size of the time steps also has an impact on the 

accuracy of a model. The issue of the size of the discrete volumes and time steps is discussed for the finite-difference 

method.

Cell Size

The size of cells determines the extent to which hydraulic 

properties and stresses can vary throughout the modeled region. 

Hydraulic properties and stresses are specified for each cell, so 

the more cells in a model, the greater the ability to vary hydraulic properties and stresses. If the cell size is too large, important 

features of the framework may be left out or poorly represented. 

Accordingly, it is important to evaluate the known (or assumed) 

variation of hydraulic properties and stresses of the system 

being simulated compared to the size of the cells. For example, 

the differences in the representation of a confining unit in a 

regional ground-water flow model and a sub-regional model of 

Long Island, New York (Buxton and Reilly, 1987) are substantial (fig. 1), and the locations where the clay is absent is much 

better represented at the finer scale. In a parallel sense, the representation of the streams and shoreline are different depending 

on the scale (fig. 2). The intended use of the model and the 

importance of the features being discretized affect both the 

evaluation of whether the model is discretized appropriately 

and whether important features are missing that would cause a 

systematic error or bias in the simulation results.

Figure 3 shows the difference in simulated drawdown 

when different cell sizes are used to simulate pumping from two 

wells in a one-layer model. The 3,300 ft by 3,300 ft system is 

confined with a uniform transmissivity of 10,000 ft2/d. No-flow 

boundaries surround all sides except the northern boundary, 

which has a specified head of 0 ft. The wells are 200 ft apart, 

and each is pumped at a constant rate of 100,000 ft3/d. 

Figure 3A shows drawdown with a grid spacing of 300 ft. With 

this grid spacing, the two wells are located in a single cell, so 

the model “sees” the two wells as a single well pumping at 

200,000 ft3/d. Figure 3B shows the same system using a 100-ft 

grid spacing; this spacing allows each well to be represented 

separately. Both grids result in nearly identical drawdown for 

distances greater than 500 ft from the wells, but the drawdown 

is quite different close to the well.

Continuity of geologic deposits can be disrupted when 

cells are too large; for example, isolated cells, unintended holes 

in confining units, and breaks in channels with high conductivity can occur. An example of this is shown in figure 4 where a 

high hydraulic-conductivity channel becomes discontinuous 

when discretized with finite-difference cells that are too large to 

accurately define the important feature of the framework. The 

effect of the high hydraulic-conductivity channel is not adequately represented in a model with this discretization because 

it is not represented as a channel but rather as a set of discontinuous pockets of high hydraulic conductivity.
6 Guidelines for Evaluating Ground-Water Flow Models

Further, selecting a cell size that is just adequate to represent the variation of hydraulic properties and stresses generally 

is inadequate. A change in a property or stress in a system has 

an effect on the computed head some distance away. A complex 

distribution of hydraulic properties and stresses results in a 

complex head distribution. Many cells are needed to simulate a 

complex head distribution because the finite-difference method 

computes a single value of head for each cell. Many single values are required to approximate a complex distribution. Thus, it 

is important to incorporate a sufficient number of cells to allow 

the complexity of head distribution to be simulated. A simple 

example is shown in figure 5. A system is simulated with two 

different grid spacings, as described for figure 3, except that a 

single well pumping 200,000 ft3/d is being simulated. The figure shows a cross section of head along the row containing the 

well. The head distribution is most complex near the well, and 
Discretization and Representation of the Hydrogeologic Framework 7

accordingly, there is noticeable difference in drawdown for the 

two grid spacings near the well. If accuracy of head near the 

well is not important to the problem, then the coarse grid is 

probably acceptable. But, if accuracy is needed near the well, 

then the finer grid would be necessary.

Some of the examples in this report have used uniform 

horizontal grid spacing; however, finite-difference models generally allow the widths of rows and columns to vary, which is 

called variable grid spacing. The use of variable grid spacing 

allows some flexibility to make cells smaller in some areas and 

coarser in other areas. Another approach to allowing cell sizes 

to vary, called telescopic refinement, is to couple a finer grid 

model to a subregion of a coarser grid model. This approach can 

avoid having the elongated cells, which are characteristic of 

using variable grid spacing. An approach for implementing 

telescopic refinement with MODFLOW is documented in 

Leake and Claar (1999).

In the vertical direction, two approaches commonly are 

used to represent the hydrogeologic framework in the 

model—uniform model layers (a rectilinear grid) and deformed 

model layers (fig. 6). Deformed model layers allow horizontal 
8 Guidelines for Evaluating Ground-Water Flow Models

continuity to be maintained with fewer cells at the expense of 

introducing some error in the finite-difference method. As 

examples, the discretization of the geologic framework into uniform model layers was used in the simulation of ground-water 

flow on Cape Cod, Massachusetts as shown in figure 7 (modified from Masterson and others, 1997), and the discretization of 

the geologic framework by deformed or hydrogeologic model 

layers was used in the simulation of ground-water flow on Long 

Island, New York as shown in figure 8 (modified from Buxton 

and others, 1999).

A two-dimensional (single-layer) model and a threedimensional (eight-layer) model of Cape Cod, Massachusetts, 

provide an example of the effect of vertical discretization on 

model results. The number of layers used to discretize the aquifer affects the resultant flow field and estimation of the area 

contributing recharge to pumping wells. The ground-water flow 

system in the example consists of a thick (250–500 ft) multilayered sequence of unconsolidated deposits or materials that 

range in grain size from gravel and sand to silt and clay and 

includes numerous overlying ponds and streams and variable 
Discretization and Representation of the Hydrogeologic Framework 9
10 Guidelines for Evaluating Ground-Water Flow Models

recharge rates from precipitation. More than 30 public-supply 

wells, screened at various depths, withdraw water from the system at widely differing rates. The three-dimensional model was 

developed first and then simplified into a two-dimensional 

model that was calibrated independently; consequently, the 

total transmissivities of the two models are not identical. The 

contributing recharge areas for the two-dimensional model and 

three-dimensional model (fig. 9) are different, however, even 

though both models represent the flow field on Cape Cod, Massachusetts. In the two-dimensional model (fig. 9A), the contributing areas are fairly typical of the simple ellipsoidal shapes that 

are delineated by two-dimensional analytical and numerical 

modeling techniques. In comparison, however, the shapes of the 

contributing recharge areas using the multilayer threedimensional model (fig. 9B) are more complex (Barlow, 1994; 

Franke and others, 1998).

In evaluating a ground-water flow simulation, the proper 

or sufficient discretization is not straightforward to determine. 

Enough detail is required to represent the hydraulic properties, 

stresses, and complexities of the flow field for the objectives of 

the study; yet, the cost will be less if the model is kept as simple 

as possible so that data entry, computer resources, and analysis 

of model output are as minimal as possible. Thus, the determination of the proper discretization is always a compromise. Ideally, the modeler would test the effect of grid spacing on a 

model to help determine the optimal grid spacing; however, the 

authors have not seen this done with any frequency. The model 

documentation should justify the discretization that is used.

Specifying Properties of Cells

A second aspect of representing the hydrogeologic framework is the choice of the hydraulic properties assigned to the 

cells. When simulating an actual system (as opposed to a hypothetical system), the properties of a system are generally not 

known at every cell in the grid; therefore, interpolation from 

limited real-world data must be done. Given the uncertainty of 

knowledge of the distribution of hydraulic properties, groups of 

cells are sometimes given a uniform value rather than attempting to define an individual value for every cell. Interpolation 

schemes, such as distance weighting and various geostatistical 
Discretization and Representation of the Hydrogeologic Framework 11

methods, also are used. The user of a model should evaluate the 

appropriateness of the interpolation scheme. To make such 

evaluation possible, the model documentation should specify 

the interpolation method used and include the rationale for 

using that interpolation method.

Three examples of interpolated hydraulic conductivity 

data for a hypothetical system are shown in figure 10. All three 

examples are based upon the assumption that values are known 

(presumably from aquifer tests) at four points. Figure 10A 

shows the use of the nearest-neighbor method. For every cell, 

the data point that is closest to the center of a cell is used as the 

cell value. An even simpler approach would be to use a single 

value for all the cells that is the average of the four known values. This simpler approach could be justified if the known values are not considered to be accurate. Figure 10B shows grid 

values determined by using a weighted average of the four 

known values based on the inverse distance squared from the 

center of a cell to the four points. Finally, figure 10C shows grid 

values determined from the hydraulic conductivity of the two 

adjacent contours. The value for a cell is the distance-weighted 

average of the two contour values. Contours were drawn based 

on the four known points plus additional geologic information 

about the types of sediments throughout the area (which was 

made up for this example). The three distributions shown in figure 10 differ significantly even though they are all based on the 

same four data points. There are many other methods available 

for interpolation that would each produce different parameter 

distributions.

The authors are aware of only one general guideline to help 

determine the best interpolation method to use in a particular 

situation. This guideline states that it is best to use the simplest 

interpolation method that is consistent with the known data. The 

rationale for this guideline is that unwarranted complexity in the 

discretized values builds a bias into a model that affects all 

future use. Ideally the model developer would evaluate the 

importance of the interpolation method by testing different 

methods and comparing the effect on model results. Such testing is not always practical depending on the resources available 

for model development.

The chosen interpolation method is often implemented by 

a computer program. The model documentation should reference the program that is used. Some model programs incorporate interpolation capabilities. For example, the HydrogeologicUnit Flow (HUF) Package (Anderman and Hill, 2000) in MODFLOW vertically averages hydraulic properties for cells based 

on real-world geometry of hydrogeologic units.

The discretization of the storage properties of the groundwater system has some intricacies of its own. The two main 

types of aquifer storativity are confined storage (specific storage) and unconfined storage (specific yield). Unconfined storage is related to the release of water as the water table lowers 

(dewatering of the aquifer material); thus, it occurs only along 

the top boundary of the saturated flow system. Confined storage 

is related to the release of water as the head drops because of 

expansion of the water itself as the pressure changes and 

changes in the solid framework of the aquifer (no dewatering 
12 Guidelines for Evaluating Ground-Water Flow Models
Discretization and Representation of the Hydrogeologic Framework 13

occurs). In simulating the changes in storage for transient systems, it is important that the unconfined storage occurs only at 

the top boundary (or top active layer), even if the water-table 

aquifer is divided into many layers. Some model programs, 

such as MODFLOW, control which storage coefficient is used 

based on the layer geometries and heads, thus ensuring that the 

proper (either the specific storage or the specific yield) coefficient is used. Other model programs require the user to specify 

the coefficient for each cell. Some investigators have erroneously specified specific yield for all layers in an unconfined 

aquifer, when it should be specified only for the uppermost 

active layer, causing incorrect quantities of water to be simulated from storage. Thus, care must be taken in determining if 

the proper storativity is simulated in a model.

Models that simulate a water table also can have a uniqueness problem related to the representation of the hydrogeologic 

framework by discrete volumes. Ground-water model programs 

such as MODFLOW allow cells representing the water table to 

go dry (desaturate) so that ground-water flow is not simulated 

in those cells. Cells also can convert from dry to wet in some situations. Cell wetting and drying depends on a variety of factors 

such as initial conditions, the iterative solution process, and 
14 Guidelines for Evaluating Ground-Water Flow Models
Discretization and Representation of the Hydrogeologic Framework 15

user-specified options to control wetting and drying. By varying 

these factors, it is possible to change the number of dry cells, 

and thus the head will vary. Careful evaluation is required to 

detect the potential for nonuniqueness and reject solutions that 

are unreasonable.

To avoid solver convergence problems that sometimes 

occur when cells can convert between wet and dry, some investigators have resorted to specifying cells representing the water 

table as having a constant saturated thickness. It is important to 

evaluate the extent to which this has been done and the degree 

to which the thickness represented by the simulated heads varies from the assumed specified thickness. For steady-state models, the following process can be repeated until the simulated 

saturated thickness is reasonably close to the specified saturated 

thickness: 

1. Run the model.

2. Compare the simulated saturated thickness (head minus 

bottom elevation) to the specified saturated thickness.

3. Adjust the specified saturated thickness to match the 

simulated thickness.

For transient models, the changes in saturated thickness 

throughout the simulation can be compared to the specified saturated thickness to insure that the change is small compared to 

the total saturated thickness.

Time Steps

Transient models simulate the impact of stresses over time. 

In MODFLOW, time is divided into time steps, and head is 

computed at the end of each time step. Many time steps are 

required to simulate a complex distribution of head over 

time. This is similar to the need for many cells to represent 

the spatial distribution of head. It is important to incorporate 

enough time steps to allow the temporal complexity of head 

distribution to be simulated.

Figure 11 shows the effect of using different numbers 

of time steps to simulate the drawdown of a well. The system is the same as that used for the fine-grid simulation in 

figure 3, with a dimensionless storage coefficient of 0.01 

and a well located in the cell at row 17 and column 17. The 

hydrographs are for the cell at row 17, column 13, which is 

the 4th cell directly to the left of the pumping cell. At the 

start of the simulation, the well is turned on with a pumping 

rate of 100,000 ft3/d. Each time step is 1.5 times longer than 

the previous time step, which results in more time steps in 

early time when head is changing most rapidly. Use of six 

or more time steps in this model produces nearly the same 

results, but four or less time steps produces much different 

results, especially in early time.

MODFLOW also makes use of stress periods to facilitate specification of stress data. A stress period is a group of 

one or more time steps in which stress input data are constant. In many situations, it is appropriate to maintain the 

same stresses for multiple time steps, so combining time 

steps into a stress period for the purposes of data input minimizes the data preparation effort. A new stress period must start 

whenever it becomes necessary to change stress input data. If 

stress periods are too long, important dynamics of the stresses 

may be left out or poorly represented. For example, the Well 

Package of MODFLOW (Harbaugh and others, 2000) allows 

pumping rates for wells to change every stress period, and 

within a stress period the pumping is constant. If the simulation 

is broken into stress periods of one year, for example, but the 

actual pumping rate changes more frequently, then stress periods may need to be shorter.

The intended use of the model is also an important factor 

in evaluating whether the size of stress periods and time steps is 

appropriate. Considering again the simulation of wells, if a 

model is used to analyze the average response of a system over 

many years, then pumping might be represented as yearly averages using yearly stress periods. There would likely be multiple 

time steps in each yearly stress period, but the stress would 

remain constant for each year. Thus, hourly, daily, and seasonal 

variations in pumping would be ignored. But, if a model is used 

to simulate seasonal system response, then pumping should be 

represented with shorter stress periods – perhaps monthly.

Questions to be answered in evaluating the appropriateness of the discretization and the representation of the hydrogeologic framework in the simulation of the ground-water system are:

1. Does the horizontal discretization represent the important 

features of the hydrogeologic framework to meet the 

objectives of the study? 
16 Guidelines for Evaluating Ground-Water Flow Models

2. Are the physical boundaries represented appropriately in 

space by the discretized representation?

3. Is the horizontal discretization appropriate to represent 

the degree of complexity in the aquifer properties and 

head distribution (flow system)?

4. Does the vertical discretization adequately represent the 

vertical connectivity and transmitting properties of the 

hydrogeologic framework to meet the objectives of the 

study? Does the method of vertical discretization, either a 

rectilinear grid or deformed grid, introduce any bias into 

the representation of the hydrogeologic framework?

5. Is the method of assigning parameter values to individual 

cells explicitly explained? Is the method appropriate for 

the objectives of the study and the geologic environment?

6. If the ground-water system is transient, then is the 

specification of storage coefficients appropriate?

7. If the ground-water system is unconfined in some areas, 

then is the treatment of changes in saturated thickness 

and the potential for cells to go dry explained and 

appropriate? If cells have gone dry, does the resultant 

solution seem appropriate?

8. Is the time discretization fine enough to represent the 

degree of complexity in stresses and head distribution 

over time?

The evaluation of the proper or sufficient discretization of 

the hydrogeologic framework of a ground-water flow simulation is not straightforward to determine. The continuity of 

deposits and the reasonableness of the specification of values 

for each cell in light of the depositional environment of the 

hydrogeologic framework must be considered. As always, the 

objectives of the study also determine which features must be 

represented in the model and the level of detail required to adequately represent their effect on the flow system.

Representation of Boundary Conditions

Boundary conditions are a key component of the conceptualization of a ground-water system. The topic of boundary 

conditions in the simulation of ground-water flow systems has 

been discussed in Franke and others (1987) and Reilly (2001). 

As discussed in Reilly (2001), computer simulations of 

ground-water flow systems numerically evaluate the mathematical equation governing the flow of fluids through porous 

media. This equation is a second-order partial differential equation with head as the dependent variable. In order to determine 

a unique solution of such a mathematical problem, it is necessary to specify boundary conditions around the flow domain for 

head (the dependent variable) or its derivatives (Collins, 1961). 

These mathematical problems are referred to as boundary-value 

problems. Thus, a requirement for the solution of the mathematical equation that describes ground-water flow is that boundary 

conditions must be prescribed over the boundary of the domain. 

Boundary conditions also represent any flow or head constraints within the flow domain. For example, recharge from 

percolation of precipitation, river interaction, and pumping 

from wells are simulated as boundary conditions. Three types of 

boundary conditions—specified head, specified flow, and headdependent flow—are commonly specified in mathematical 

analyses of ground-water flow systems. The values of head (the 

dependent function) in the flow domain must satisfy the preassigned boundary conditions to be a valid solution.

In solving a ground-water flow problem, however, the 

boundary conditions are not simply mathematical constraints; 

they generally represent the sources and sinks of water within 

the system. Furthermore, their selection is critical to the development of an accurate model (Franke and others, 1987). Not 

only is the location of the boundaries important, but also their 

numerical or mathematical representation in the model. This is 

because many physical features that are hydrologic boundaries 

can be mathematically represented in more than one way. The 

determination of an appropriate mathematical representation of 

a boundary condition is dependent upon the objectives of the 

study. For example, if the objective of a model study is to understand the present and no estimate of future conditions is 

planned, then local surface-water bodies may be simulated as 

known constant-head boundaries; however, if the model is 

intended to forecast the response of the system to additional 

withdrawals that may affect the stage of the surface-water bodies, then a constant head is not appropriate and a more complex 

boundary is required. A model of a particular area developed for 

one study with a particular set of objectives may not necessarily 

be appropriate for another study in the same area, but with different objectives. All of these aspects of boundary conditions 

must be considered in evaluating the strengths and weaknesses 

of a ground-water flow model.

In the ground-water flow modeling process (fig. 12), 

boundary conditions have an important influence on the areal 

extent of the model. Ideally in developing a conceptual model, 

the extent of the model is expanded outward from the area of 

concern both vertically and horizontally so that the physical 

extent coincides with physical features of the ground-water system that can be represented as boundaries. The effect of these 

boundaries on heads and flows must then be conceptualized, 

and the best or most appropriate mathematical representation of 

this effect is selected for use in the model. 

When physical hydrologic features that can be used as 

boundary conditions are far from the area of interest, artificial 

boundaries are sometimes used. The use of an artificial boundary should be evaluated carefully to determine whether its use 

would cause unacceptable errors in the model. For example, a 

no-flow boundary might be specified along an approximated 

flow line at the edge of a modeled area even though the aquifer 

extends beyond the modeled area. The rationale might be that 

the artificial boundary is positioned far enough from the area of 

interest that whatever is simulated in the area of interest would 

not cause significant flow across that area of the system. The 

rationale for artificial boundaries can generally be tested using 

the model. In the example of an artificial no-flow boundary, the 
Representation of Initial Conditions in Transient Simulations 17

appropriateness can be tested by looking at how much the head 

changes near the boundary when the model is used for its 

intended purpose. Substantial change in heads near the boundary is an indication that significant flow across the region would 

occur if the artificial boundary were not imposed. 

Another example of an artificial boundary is a specifiedhead boundary at a location where there is no source of water to 

maintain the head at its specified value. The appropriateness of 

this boundary can be tested by evaluating the flow from the 

boundary and the change in flow due to changes in parameter 

values or stresses within the model. If a stress causes a large 

change in flow from the boundary, then the head would probably change at the boundary if it were not artificially fixed. Artificial boundaries, if applied improperly and not evaluated, can 

overly constrain the response of the system and bias the results 

of an analysis. A frequently observed example is when the area 

of interest for a study is artificially bounded by specified heads, 

without regard to the flow being simulated from this boundary 

into the study area. In this case, the model may not be sensitive 

to parameter values and stresses because the specified heads 

artificially keep the simulated heads from deviating much. For 

further discussion of this topic, see Franke and Reilly (1987).

The objective of the modeling analysis and the magnitude 

of the stresses to be simulated also influence the selection of the 

appropriate approach to simulate the physical features that 

bound the ground-water system. When ground-water systems 

are heavily stressed, the physical features that bound the system 

can change in response to the stress. Any representation of these 

features must account for these potential changes, either by 

understanding the limitations of the simulation or by representing the physical feature as realistically as possible.

In evaluating the appropriateness of a ground-water flow 

model, the boundary conditions are key because they determine 

where the water enters and leaves the system. If the boundaries 

are inappropriate, the model will be a poor representation of the 

actual ground-water flow system. Questions to be used in 

evaluating the boundary conditions of a ground-water flow 

model are: 

1. Are all the external boundaries of the model associated 

with a definable physical feature?

If no –

A. Why not?

B. Is sufficient justification provided to warrant the use 

of artificial boundaries?

C. Are the effects of the “artificial” boundaries tested in 

the calibration of the model and documented in the 

report? Does the documentation of their use and their 

testing make a convincing argument for their reasonableness?

If yes –

A. Is the mathematical representation of the physical 

feature appropriate?

B. Are there conditions under which the representation 

of the boundary used in the model would become 

invalid? Are these conditions discussed?

2. Do the boundary conditions of the model overly constrain 

the model results so that the calibration is insensitive and 

the predictions are not realistic?

Representation of Initial Conditions in 

Transient Simulations

Initial conditions represent the heads at the beginning of a 

transient simulation. Thus, initial conditions serve as a boundary condition in time for the transient head response of a 

ground-water model solution. Initial conditions are used only in 

transient simulations, and are different from starting heads (or 
18 Guidelines for Evaluating Ground-Water Flow Models

the initial guess) in steady state solutions. In steady-state solutions, the starting heads can and do affect the efficiency of the 

matrix solution, but the final correct solution should not be 

affected by different starting heads. In transient solutions, however, the initial conditions are the heads from which the model 

calculates changes in the system due to the stresses applied. 

Thus, the response of the system is directly related to the initial 

conditions used in the simulation.

The changes in head that occur in the transient model due 

to any applied stress will be a combination of the effect of the 

change in stress on the system and any adjustments in heads as 

a result of errors in the initial head configuration (the initial conditions). Adjustments in heads resulting from errors in the initial head configuration do not reflect changes that would occur 

in the actual system, but rather occur because the heads specified as the initial condition are not a valid solution to the numerical model. Because errors in the initial head conditions cause 

changes in head over time during the simulation, it is best to 

begin all transient simulations with a head distribution that is a 

valid solution for the model. This ensures that there are no discrepancies (or errors) between the specified initial conditions 

and a valid head solution for the model.

For simulations that start from a period when the aquifer 

system was in a steady-state equilibrium, the development of 

appropriate initial conditions is straightforward. A simulation 

of the steady-state period should be made. The results of this 

simulation should then be used as the initial conditions for the 

transient simulation.

Sometimes, however, it is not possible to start a simulation 

from a point in time where the aquifer was in steady-state equilibrium. This condition could occur if the simulation is intended 

to simulate seasonal or other cyclic conditions where the system 

is never at steady state, or in instances where there is a period of 

unknown stress that cannot be reproduced accurately, or when 

it is not feasible to simulate the entire period of record from a 

time of steady state because of time and money constraints. 

Under these conditions, it is important that the initial conditions 

used do not bias the results for the period of interest. Some rules 

of thumb for the evaluation of the appropriateness of the initial 

conditions in these non-ideal situations are to evaluate the time 

constant of the system under investigation and to test the effect 

of different initial conditions on the results of the model.

The time constant for a ground-water system is derived 

from a dimensionless form of the ground-water flow equation 

and is defined as (Domenico and Schwartz, 1998, p. 73):

,

where T is the time constant (T), Ss is the specific storage of a 

confined aquifer (L-1), L is a characteristic length of the system 

(L), and K is the hydraulic conductivity (LT-1). The effect of any 

transient condition will not be observable if the time after the 

condition occurs is significantly larger than the time constant 

for the aquifer (T) (Domenico and Schwartz, 1998). Thus, the 

effect of a poor or erroneous initial condition (assuming the rest 

of the model including boundary conditions is correct) should 

not be observable in model results that are for periods of time 

significantly larger than the time constant for the aquifer. The 

time constant is developed from the ground-water flow equation for a confined system with homogeneous hydraulic conductivity. Thus, its application in actual systems is not always 

exact. The appropriate characteristic length (L) of the system is 

usually chosen to represent the distance between major boundaries. The specific storage (Ss) represents the compressible storage characteristics of the system; however, an equivalent 

storativity for unconfined aquifers could be calculated as the 

specific yield (Sy) divided by the thickness (b) of the unconfined aquifer. For unconfined aquifers, an approximate time 

constant would be:

.

The determination of the importance and duration of 

effects of erroneous or imperfect initial conditions can also be 

accomplished by testing the effect of different initial conditions 

on the model under study. This test is accomplished by simulating the same system with the stresses and different initial conditions. When the simulations for all the different initial conditions produce the same result, then one can assume the 

influence of the inaccurate initial conditions is negligible at all 

following time periods.

A simulation of a simple transient ground-water system 

can illustrate some of these points. In the illustrative simulation, 

the simple transient ground-water system is 20,000 ft long and 

20,000 ft wide with two aquifers separated by a confining unit, 

and bounded by no-flow boundaries with a stream along one 

edge. The aquifer has uniform areal recharge of 0.003 ft/d. The 

upper aquifer is unconfined and both aquifers have a horizontal 

hydraulic conductivity of 50 ft/d and a vertical hydraulic conductivity of 5 ft/d. The confining bed is 10-ft thick with a vertical hydraulic conductivity of 0.001 ft/d. The system is discretized as shown in figure 13, and simulated using the finitedifference model MODFLOW. The areal grid size is 1,000 ft by 

1,000 ft, and the two aquifers are each represented by two layers; the bottom aquifer is represented by a lower layer (layer 4) 

50-ft thick overlain by a 40-ft thick layer (layer 3), and the 

unconfined aquifer is represented by a 50-ft thick layer (layer 2) 

overlain by a layer (layer 1) with a uniform bottom at –50 ft, 

which allows changes in thickness as a function of the head. The 

stream is represented as a constant head of 0 ft along the righthand boundary in the top layer. The specific yield for the top 

layer is 0.2 and the specific storage for the entire model domain 

is 1.0 x 10-6 1/ft. 

The steady-state head distribution for the simple system in 

layer 1 is symmetric perpendicular to the stream and varies from 

67.94 ft at the ground-water divide to 0.0 ft at the stream 

(fig. 14). A transient simulation is run from the initial steady 

state to examine the effect of a well discharging 100,000 ft3/d 

from layer 3 in cell 10, 10 (9,500 ft from the divide). The correct 

simulation has as the initial condition the steady-state head 

T

SsL2

K = ----------

T

SyL2

bK = ----------
Representation of Initial Conditions in Transient Simulations 19

distribution before the well began discharging; the response of 

the system through time is shown at the divide in layer 1 

(fig. 15A) and at the cell containing the well in layer 3 

(fig. 15B). The effect of inaccurate initial conditions can be 

observed in the response of the aquifer at these same locations. 

Two different initial conditions, as shown on figure 14, are used 

to test the response of the system to inaccurate initial conditions. These two other conditions are a uniform head of 100 ft 

everywhere (all layers), except at the stream, and a linearly 

changing initial head ranging from 95 ft to 0 ft at the stream. 

The response of the system over time in response to the pumping well compared to the correct response that used the steadystate head distribution is shown in figure 15 for a cell in layer 1 

at the divide and for the cell containing the well in layer 3. The 

time constant can also be calculated for this system, although 

some approximations must be made to estimate a saturated 

thickness. If the saturated thickness of the unconfined aquifer is 

assumed to be 100 ft (the thickness at the stream), then the time 

constant is calculated as:

.

As shown in figure 15, the curves for the two 

inaccurate initial conditions do not approach 

the correct transient response until about 20 

to 40 years after the start of pumping. Thus, 

inaccurate initial conditions can cause errors 

for a significant time period in transient simulations. 

Examination of the simulated response 

through time from 0-5 years in the finitedifference cell containing the well illustrates 

some interesting points. The correct 

response of the system is simulated for the 

case with the steady-state heads as the initial 

conditions (fig. 16); the initial value for the 

head is 50.09 ft in the cell containing the 

well. The case with the linearly varying 

heads as initial conditions has the initial 

value for the cell containing the well equal to 

50.0 ft, which is almost the same as the correct steady-state value. Even though the initial conditions in the individual cell are 

almost the same, the response is different, 

because the initial conditions over the entire 

model domain affect the head response. The 

response of the system with the linearly 

varying initial conditions is obviously in 

error because the response of the system 

shows an increase in head after the first time 

step in response to pumping, which is not 

physically reasonable.

Questions to be used in evaluating the 

initial conditions of a ground-water flow 

model are:

1. Does the transient model simulation start from a steadystate condition?

If yes –

A. Were the initial conditions generated from a steadystate simulation of the period of equilibrium, which 

is the preferred method?

B. If the initial conditions were not generated from a 

steady-state simulation of the period of equilibrium, 

then is there a compelling reason why they were not 

generated, or are the initial conditions invalid?

If no –

A. Was it possible to select a period of equilibrium to 

start the simulation and make the determination of 

initial conditions more straightforward? If it is possible, then the model should have simulated the transient period from the period of equilibrium.

B. If it was not possible to select a period of equilibrium 

to start the simulation, then what was the justification for selecting the starting time and the initial conditions for the simulation? How was it shown that the 

initial conditions used did not bias the result of the 

simulation?

T 0.2 20 000ft ( ) , 2

100.ft 50 ft/d ( ) ------------------------------------- 1.6 104 = = × days = 44 years
20 Guidelines for Evaluating Ground-Water Flow Models

Accuracy of the Matrix Solution

Discrete numerical models involve the solution of 

large sets of simultaneous algebraic equations (Harbaugh and others, 2000). This solution of large sets of 

algebraic equations usually involves the use of sophisticated matrix solution techniques. Most of the solution 

techniques are iterative in nature whereby the solution is 

obtained through successive approximation, which is 

stopped when it is determined that a “good” solution has 

been obtained (Bennett, 1976). The criterion used in 

most iterative solution techniques is called the “head 

change criterion.” When the maximum absolute value 

of head change from all nodes during an iteration is less 

than or equal to the selected head change criterion, then 

iteration stops.

When evaluating a ground-water flow model, even 

if the computer model has output results, one must 

check to determine if indeed a solution has been 

obtained by the matrix solution technique. The first 

check is to evaluate the head change criterion. Was the 

head change criterion set small enough to obtain a 

model solution with minimal error? One means of evaluating the head change criterion is to examine the global 

mass balance for the model. If the error in the mass balance (for example, total inflow minus total outflow 

divided by one half the sum of the inflow and outflow) 

over the entire model domain is small, usually less than 
Accuracy of the Matrix Solution 21

0.5 percent, then the head change criterion is assumed to have 

been sufficient. If the error in the mass balance calculations is 

significant, then the matrix solution was not good and the model 

should be corrected by improving the matrix solution. The 

matrix solution can be improved by lowering the head change 

criterion, adjusting iteration parameters (if the solution techniques use iteration parameters), using different starting heads 

for steady-state simulations, or using a different solution technique.

Even if the head change criterion is met and the global 

mass balance error is small, the model solution may not be 

appropriate for the system under investigation. Two potential 

reasons are that some models can either be mathematically nonunique or very nonlinear. The mathematically nonunique problem usually is a poorly posed problem where a model has only 

specified-flow boundary conditions and no other boundary condition that specifies a head or datum (such as, constant head, 

river stage, general head boundary, etc.). In this type of problem, there is a family of solutions all with the same gradients but 

different absolute heads. The matrix solution technique may not 

converge or it may converge to one of the infinite number of 

possible solutions.

In nonlinear problems, the solution affects the coefficients 

of the matrix being solved; thus, the solution affects the problem being solved. As a result, the manner in which the iterative 

solution technique approaches a solution can affect the final 

solution. An example from Reilly (2001) illustrates this point. 

Consider a one-dimensional water-table system with a sloping 

impermeable bottom that contains a specified head and extends 

5,000 m, with an areal recharge rate of 0.5 m/yr. The starting head for the equation solution is specified at 20 m, 

which is above all the bottom elevations of the cells but yet 

close to the magnitude of the expected results. Figure 17A 

is a cross-sectional view of a finite-difference representation of the steady-state solution. The cell farthest from the 

specified head is simulated as being dry. The total recharge 

flowing to the specified head cell for a 500-m width is 

2,740 m3/d. The convergence criterion of the model was 

met and the mass balance was excellent (showing 0.00 percent budget discrepancy). Now consider figure 17B, which 

is the result of a simulation of the same problem, except the 

starting head for the matrix solution was set at 100 m. As 

is shown in figure 17 and table 2, three cells are now simulated as being dry. The result is that less recharge is simulated as entering the model and the heads and water budgets are reduced accordingly, with only 2,055 m3

/d being 

represented as recharge entering the system for a 500-m 

width. Although both solutions converged and had excellent mass balances, at least one of them is incorrect. 

Because it is a nonlinear problem, it is not easy to determine which solution is correct. The rate of convergence 

and the method of making cells inactive must be considered and evaluated. After evaluating these aspects, and 

noting that the head in cell 7 (table 2 and fig. 17) of the second model is above the bottom elevation of cell 8, which 

was converted to dry during the iterative process, it seems 

that the first model most likely is correct. In the second model, 

the iterative solution, in attempting to converge, apparently 

overshot the bottom of some of the cells, which prematurely or 

erroneously truncated the area from the active model domain, 

Table 2. Heads calculated for the same system with areal recharge 

and two different intitial heads.

[m, meters]

Cell 

number

Bottom 

elevation of cell

Head 

calculated 

with the initial 

head at 20 m

Head 

calculated 

with the initial 

head at 100 m

1 -30.0 0.00 0.00

2 -25.0 1.93 1.46

3 -20.0 3.83 2.86

4 -15.0 5.68 4.17

5 -10.0 7.49 5.38

6 -5.0 9.24 6.42

7 0.0 10.90 7.20

8 5.0 12.45 Dry

9 10.0 13.81 Dry

10 15.0 Dry Dry
22 Guidelines for Evaluating Ground-Water Flow Models

and resulted in the wrong problem being solved. The model 

developer or user must carefully evaluate nonlinear problems 

and monitor the rate of convergence to ensure that cells that 

should be part of the active problem domain are not removed.

The accuracy of the matrix solution usually is not an issue 

with ground-water models that meet the head change criterion 

and have small mass balance errors. It is important when using 

models and especially nonlinear models, however, to keep in 

mind that the accuracy of the solution is not assured, which is 

another aspect for continued evaluation. Some models do not 

converge smoothly, and investigators use non-standard methods (tricks) to obtain a model solution. For example, some nonstandard methods that have been used include: the saving of 

intermediate solutions that have not yet converged and changing matrix solution parameters when restarting the model; making a nonlinear water-table simulation linear by fixing the saturated thickness of the model; and obtaining a steady-state 

solution by using storage to slow convergence and damp the 

approach to the solution through simulating a long transient 

time period. As long as the non-standard method does not violate any important hydrologic process, they are usually transparent to the final solution and are appropriate. However, these 
Adequacy of Calibration for Intended Use of Model Results 23

non-standard techniques should be evaluated to determine 

whether they cause potential errors to be introduced to the 

model solution.

Questions to be addressed when evaluating the adequacy 

of the matrix solution in the simulation of a ground-water system are:

1. Is the ground-water system and set of matrix equations 

linear or nonlinear?

If linear –

A. Was the head change criterion met and was it sufficiently small to obtain an acceptable (that is, less 

than 0.05 percent error) global mass balance?

If nonlinear –

A. Was a nonlinear matrix solution technique used?

B. Was the head change criterion met and was it sufficiently small to obtain an acceptable (that is, less 

than 0.05 percent error) global mass balance?

C. Did the nonlinear terms, such as cells going dry or 

drains turning off, behave smoothly during the iteration process? Or were there large oscillations that 

would indicate a potential for convergence to an 

incorrect solution?

D. Were any “tricks” used to smooth convergence, such 

as setting saturated thickness as a constant in watertable simulations, and are the assumptions used in 

defining these artificially constrained features reasonable for the solution obtained?

2. Does the solution seem reasonable for the problem posed? 

If it is not and there are no input data errors, then another 

matrix solution technique should be tried to determine 

whether it is a matrix-solution issue or some other 

problem.

Adequacy of Calibration for Intended Use of 

Model Results

As discussed previously, not all objectives of using a 

ground-water model require calibration. For models that require 

calibration, however, an evaluation of the adequacy of the calibration is another difficult task. There are different quantitative 

measures that investigators use to show the accuracy of the calibration of a ground-water flow model. Some of these are: the 

mean error, the mean absolute error, and the root mean squared 

error (Anderson and Woessner, 1992). The areal distribution of 

residuals (differences between measured and simulated values) 

also is important to determine whether some areas of the model 

are biased either too high or too low. The difficulty that arises, 

however, is how to determine what is good enough. 

As stated previously, key aspects of the model, such as the 

conceptualization of the flow system, that influence the appropriateness of the model to address the problem objectives, are 

often not considered during calibration by many investigators; 

their focus is on the quantitative measures of goodness of fit. 

However, the appropriateness of the conceptualization of the 

ground-water system and processes should always be evaluated 

during calibration. Thus, the method of calibration, the closeness of fit between the simulated and observed conditions, and 

the extent to which important aspects of the simulation were 

considered during the calibration process are all important in 

evaluating the appropriateness of the model to address the problem objectives.

Freyberg (1988) reported on a class exercise where different models were calibrated by students using the same model 

and identical sets of data. Freyberg’s observations of the exercise showed that “success in prediction was unrelated to success 

in matching observed heads under premodification conditions.” 

He concluded, “good calibration did not lead to good prediction.” This is not to imply that matching heads is unimportant, 

only that there are other factors that need to be considered in 

determining the “goodness” of a model. Put in terms of logic, a 

good match between calculated and observed heads and flow is 

a necessary condition for a reasonable model, but it is not sufficient. The conceptual model and the mathematical representation of all the important processes must also be appropriate for 

the model to accurately represent the system under investigation. Thus, a model that matches heads and flows well must also 

be evaluated to determine if it is a reasonable representation of 

the system under study. As stated by Bredehoeft (2003), “A 

wrong conceptual model invariably leads to poor predictions, 

no matter how well the model is fit to the data.”

Thus, the evaluation of the adequacy of the calibration of 

a model should be based more on the insight of the investigators 

and the appropriateness of the conceptual model rather than the 

exact value of the various measures of goodness of fit. For 

example, it would be possible to specify every cell in a model 

that had an observation associated with it as a specified head 

cell in the model. This would produce a perfect match between 

simulated and observed heads, however, it is conceptually 

unreasonable to simulate random cells as specified heads that 

could serve as sources and sinks of water. Thus, although the 

measures of calibration might make it appear to be a wellcalibrated model, in effect the violation of a reasonable conceptual model makes it a poor model. A model developed according to a well-argued conceptual model with minor adjustments, 

in our opinion, is generally superior to a model that has a 

smaller discrepancy between simulated and observed heads 

because of unjustified manipulation of the parameter values. A 

reasonable representation of the conceptual model and sources 

of water is more important than blindly minimizing the discrepancy between simulated and observed heads.

Models can be calibrated by trial and error or by automatic 

parameter estimation techniques, such as nonlinear regression 

to minimize some measure of goodness of fit between the simulated and observed values. A key concept in automatic parameter estimation methods is that a limited set of parameters used 

in the model is designated to be automatically adjusted. These 

parameters usually are identified for specific regions (or zones) 
24 Guidelines for Evaluating Ground-Water Flow Models

of the model that are determined before the calibration process 

(a priori). An example of parameter zones for hydraulic conductivity is shown in figure 18 for the top two layers of a model of 

the Albuquerque Basin, New Mexico (Tiedeman and others, 

1998). In this example, the zones represent different hydrogeologic units. The areal extent of these units remains fixed during 

automatic calibration, and the conceptualization of the location 

and extent of these zones is part of the information specified 

before the automatic calibration process. The parameters and 

boundary conditions that are not identified for automatic calibration either remain fixed at their initial values or must be calibrated by trial and error. In addition, most automatic calibration methods weight observations according to the investigators 

insight into the reliability of the observations. Obviously, if the 

model is conceptualized incorrectly, the parameter zones are 

not representative of the actual parameter distribution, the fixed 

parameters and boundary conditions are poorly chosen, or the 

weighting functions are not appropriate, then the resultant estimates of the parameter values will be inaccurate even if the 

residual between observed and simulated conditions is automatically minimized.

If there are errors in the model conceptualization, the 

parameter zones selected, and the weighting functions defined 

for observed values, then the parameter estimation methods will 

provide the best parameters for the poorly defined model. This 

does not mean that the model will be an accurate representation 

of the system or will produce reasonable predictions. Perhaps 

the best use of the formal parameter estimation methods is to 

test different model, zone, and weighting function conceptualizations and determine which conceptualizations are most reasonable. In testing alternative models, Hill (1998) states that 

better models will have “three attributes: better fit, weighted 

residuals that are more randomly distributed, and more realistic 

optimal parameter values.” This approach was used by Yager 

(1996) to test three different model conceptualizations for the 

Niagara Falls area in New York and by Tiedeman and others 

(1998) to test six different system conceptualizations of the 

Albuquerque Basin system. This use of parameter estimation 

provides a quantitative means (although some subjectivity 

comes into determining which model is good enough) to test 

different conceptualizations. 

In trial and error calibration, investigators have the ability 

to continuously change their conceptualization of the system 

and parameter distributions in order to improve the calibration 

fit, although the benefits of these changes are frequently difficult to quantify. It is the insight and skill of the investigator during a trial and error calibration that will control how well a 

model represents the ground-water system under investigation. 

In evaluating the adequacy of a model calibration, the conceptual model and the insight of the investigators generally are 

more important than just an evaluation of quantitative measures 

of goodness of fit.

Questions to be addressed in evaluating the adequacy of 

calibration of a model using either trial and error or automatic 

methods are:

1. Is the conceptual model of the system under investigation 

reasonable?

2. Are the mathematical representations of the boundary 

conditions reasonable for the objectives of the study?

3. Does the simulated head and flow distribution mimic the 

important aspects of the flow system, such as magnitude 

and direction of the head contours?

4. Does some quantitative measure of head and flow 

differences between the simulated and observed values 

seem reasonable for the objectives of the investigation?

5. Does the distribution of areas where simulated heads are 

too high and areas where simulated heads are too low 

seem randomly distributed? If they are not randomly 

distributed, then is there a hydrogeologic justification to 

change the model and make the residuals more random 

areally?

Just because a model is constructed and calibrated, does 

not ensure that it is an accurate representation of the system. 

The appropriateness of the boundaries and the system conceptualization is frequently more important than achieving the 

smallest differences between simulated and observed heads and 

flows.

Model Input Data, Output Listing, and Report 

Consistency Check

In evaluating the adequacy of a model, the input data, output listing, and report ideally should be compared with each 

other to ensure that they all represent the same analysis. 

Depending on the level of evaluation being undertaken, this 

comparison can vary greatly in its thoroughness. Many times 

the output listing and input data sets are not available to the person evaluating the model, so there is nothing that can be 

checked.

If the listing file is available, then it is useful as a minimum 

to compare some of the model output to information in the 

report. The simulated water budget in the output listing can be 

compared to budget values determined from the system conceptualization and real-world measurements provided in the report. 

For example, if the areal recharge rate is specified in the report, 

the total recharge over the modeled area can be calculated and 

compared to the reported recharge in the model budget. Heads 

or drawdowns in the model output listing can be compared to 

values in the report.

If a more thorough evaluation is required, then the input 

data can also be checked. Although it is impossible to ensure 

that all the preprocessor steps and manual data entry were 

undertaken correctly, data checking can increase confidence 

that the model is consistent with the description in the report. 

Whether the model data files were constructed by manually 

entering information into files or by using a graphical user interface, there is the possibility that the data files contain errors. 
Model Input Data, Output Listing, and Report Consistency Check 25
26 Guidelines for Evaluating Ground-Water Flow Models

Examples of possible errors are: numbers scaled improperly, 

inconsistent data, data entered into incorrect fields, data 

assigned to incorrect cells, typographical errors, and many others. An example of inconsistent data is the use of inconsistent 

time or space units for different parts of the data. For example, 

pumping might be entered in cubic feet per second (ft3/s) and 

hydraulic conductivity in feet per day (ft/d). An example of data 

assigned to incorrect cells is the specification of stress data, for 

example pumping wells located in inactive cells.

The extent to which the input data can be checked depends 

on the size of the model, available resources, and how the data 

were entered. Typical models vary in size from several thousand cells to over a hundred thousand cells. There are multiple 

data values per cell, so it is impractical to check every input 

value in even the smaller models. Thus, data scanning is a better 

term to describe the data-checking process. If data files are 

available, then they can be checked or scanned directly. If the 

output listing is available and if this listing contains an echo of 

the input data, then usually it is easier to examine the output listing than the input files. Also, seeing the data in the output listing 

provides added confirmation that the data files have been properly read by the model program. 

Some checks that can be considered are:

1. Do the model water-budget quantities seem appropriate 

for the values described for the actual system in the 

report?

2. Are the input data the same as those described in the 

report? 

3. Are data values consistent and assigned to appropriate 

cells?

Checking the information that is read directly by the model 

increases confidence that the simulation is indeed a solution to 

the problem described. The level of evaluation required determines the thoroughness of the consistency check that should be 

undertaken.

Model Reporting and Archiving

Because models are embodiments of scientific hypotheses, 

a clear and complete documentation of the model development 

is required for individuals to understand the hypotheses, to 

understand the methods used to represent the actual system with 

a mathematical counterpart, and to determine if the model is 

sufficiently accurate for the objectives of the investigation. As 

stated in U.S. Geological Survey Office of Ground Water Technical Memorandum 96.04 (see appendix), there is no rigid 

checklist or recipe for reporting on the use of simulation in a 

ground-water study. The appropriate level of documentation 

will vary depending on the study objectives and the complexity 

of the simulations. A valuable result of the ground-water modeling effort is the insight gained by the investigator during the 

modeling process about the functioning of the flow system. This 

understanding of the flow system gained during the modeling 

process can be an important product of the study and should be 

appropriately discussed and documented in the modeling 

report.

The general structure of a well-constructed report describing simulation is much the same as that for any investigative 

study. It should present (1) the objectives of the study, (2) a 

description of the work that was done, (3) logical arguments to 

convince the reader that the methods and analyses used in the 

study are valid, and (4) results and conclusions. 

Ten specific topics that should be addressed in reports that 

describe studies in which simulation is used are listed and 

explained in U.S. Geological Survey Office of Ground Water 

Technical Memorandum 96.04 to aid individuals in documenting their model studies. These 10 topics are:

1. Describe the purpose of the study and the role that simulation plays in addressing that purpose.

2. Describe the hydrologic system under investigation. 

3. Describe the mathematical methods used and their 

appropriateness to the problem being solved.

4. Describe the hydrogeologic character of the boundary 

conditions used in the simulation of the system.

5. If the method of simulation involves discretizing the 

system (finite-difference and finite-element methods for 

example), describe and justify the discretized network 

used.

6. Describe the aquifer system properties that are modeled.

7. Describe all the stresses modeled such as pumpage, 

evapotranspiration from ground water, recharge from 

infiltration, river stage changes, leakage from other 

aquifers, and source concentrations in transport models. 

8. For transient models, describe the initial conditions that 

are used in the simulations. 

9. If a model is calibrated, present the calibration criteria, 

procedure, and results. 

10. Discuss the limitations of the model’s representation of 

the actual system and the impact those limitations have 

on the results and conclusions presented in the report. 

Once the study is finished, it is always useful to organize 

and archive the model files. The purpose of the archive is to 

ensure that the results are reproducible in the future either by the 

model developer or other interested parties. Thus, the archive 

should reference any published reports on the model and provide enough explanation in a text “readme” file for the model to 

be used by others. The archival of the model provides good scientific practice and reproducibility of results.
Summary 27

Summary

Ground-water models are designed and built to meet specific objectives. Models must be critically evaluated to ensure 

that there are no data input errors and that the conceptual model 

does indeed accurately represent the actual ground-water system sufficiently to meet the objectives of the study. The items 

to be evaluated are: the appropriateness of the model program, 

the discretization and representation of the geologic framework, 

the representation of the boundary conditions, the representation of the initial conditions, and the accuracy of the matrix 

solution.

Ground-water flow models attempt to reproduce, or simulate, the operation of a real ground-water system using a mathematical counterpart (a mathematical model). Thus, the evaluation of the model is intended to ensure that the model program 

and numerical representation of the important aspects of the 

system are sufficient to meet the objectives of the study. The 

guidelines presented in this report raise some of the important 

aspects of model evaluation.

Acknowledgments

The technical review and improvement of the report by 

William Alley, James Bartolino, William Cunningham, and 

Keith Prince are gratefully acknowledged. Sandra Cooper provided editorial review and assistance. The final report and illustrations were prepared by E.G. Baker and K.A. Waltenbaugh, 

and J.E. Banton, respectively.

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Prince, K.R., and Schneider, B.J., 1989, Estimation of hydraulic 

characteristics of the Upper Glacial and Magothy aquifers at 

East Meadow, New York, by use of aquifer tests: U.S. Geological Survey Water-Resources Investigations Report 87-

4211, 43 p.

Reilly, T.E., 2001, System and boundary conceptualization in 

ground-water flow simulation: U.S. Geological Survey Techniques of Water-Resources Investigations, Book 3, 

Chapter B8, 26 p.

Reilly, T.E., Franke, O.L., and Bennett, G.D., 1987, The principle of superposition and its application in ground-water 

hydraulics: U.S. Geological Survey Techniques of WaterResources Investigations 3-B6, 28 p.

Reilly, T.E., and Pollock, D.W., 1993, Factors affecting areas 

contributing recharge to wells in shallow aquifers: U.S. Geological Survey Water-Supply Paper 2412, 21 p.

Tiedeman, C.R., Kernodle, J.M., and McAda, D.P., 1998, 

Application of nonlinear-regression methods to a groundwater flow model of the Albuquerque Basin, New Mexico: 

U.S. Geological Survey Water-Resources Investigations 

Report 98-4172, 90 p.

Yager, R.M., 1996, Simulated three-dimensional ground-water 

flow in the Lockport Group, a fractured-dolomite aquifer 

near Niagara Falls, New York: U.S. Geological Survey 

Water-Supply Paper 2487, 42 p.
Appendix 29

Appendix

April 24, 1996

OFFICE OF GROUND WATER TECHNICAL MEMORANDUM NO. 96.04

Subject: PUBLICATIONS—Policy on documenting the use of 

ground-water simulation in project reports

It has been more than two decades since Ground Water 

Branch Technical Memorandum No. 75.11 was released on the 

subject of documenting the use of ground-water simulation in 

project reports. Because of the time lapse, changes in modeling 

techniques, and the frequency of problems found when reports 

are reviewed, a revisit to policy on this subject is appropriate.

There is no rigid checklist or recipe for reporting on the use 

of simulation in a ground-water study. The appropriate level of 

documentation will vary depending on the project objectives 

and the complexity of the simulations. The general structure of 

a well-constructed report describing simulation is much the 

same as that for any investigative study. It should present (1) the 

objectives of the study, (2) a description of the work that was 

done, (3) logical arguments to convince the reader that the 

methods and analyses used in the study are valid, and (4) results 

and conclusions.

Specific topics that should be addressed in reports that 

describe studies in which simulation is used include the following.

1. Describe the purpose of the study and the role that simulation plays in addressing that purpose. 

The objective of the simulation must be clearly stated. 

The model should be represented as a tool to help solve 

specific problems or answer specific questions rather than 

as an end product.

2. Describe the hydrologic system under investigation. 

The extent, nature of boundaries, transmitting properties, 

storage properties, sources of water, discharge 

mechanisms and other relevant components of the 

ground-water system should be described as known or 

conceptualized. Usually this can be accomplished in part 

by referencing previous works, but major relevant system 

characteristics should be summarized in the report that 

describes the simulation.

3. Describe the mathematical methods used and their 

appropriateness to the problem being solved. 

In most cases, a reference to a readily available 

publication will be sufficient to document mathematical 

details; however, it will usually be desirable to briefly 

summarize the methods that are used. For a welldocumented computer program, this will often require 

only a paragraph or two. If a documented computer 

program is modified such that computed values are 

affected, the modifications should be documented and 

evidence that the modifications are correct should be 

supplied.

4. Describe the hydrogeologic character of the boundary 

conditions used in the simulation of the system. 

In many cases, the model boundaries are placed where 

the aquifer terminates against relatively impermeable 

rocks or is intersected by a perennial stream whose head 

variation in time and space is known. In other cases, the 

aquifer may be so extensive relative to the area of interest 

that the modeled area may need to extend beyond the 

project area to accurately simulate the natural boundaries 

of the aquifer system. If the modeled area is arbitrarily 

truncated at some distance from the area of interest, it 

should be shown that the selection of the arbitrary 

boundary condition does not materially affect the ability 

of the model to simulate the system for the purposes of 

the study. Internal boundaries such as streams, lakes, and 

pinchouts of important hydrogeologic zones should be 

identified and their representation in the model should be 

described in the report. A clear, convincing argument of 

the appropriateness of the boundary conditions used in 

the model to represent the actual system should be made 

for the entire bounding surface of the modeled volume or 

cross section, as well as for any internal boundaries.

5. If the method of simulation involves discretizing the 

system (finite-difference and finite-element methods for 

example), describe and justify the discretized network 

used. 

The spacing and distribution of the blocks, elements, or 

subregions should reflect, in part, the spatial variability 

of the hydraulic parameters and the location of 

boundaries (for example streams, lakes, bed pinchouts), 

human-made features (for example wells and dams), and 

stresses. In most cases, a map showing the discretized 

network superimposed on the study area is required. 

Vertical discretization should be described and/or shown 

on illustrations. The manner in which time is discretized 

for transient models also should be described. If a steadystate model is used to simulate an average or approximate 

steady-state condition, discuss the errors that could be 

introduced in the study results as a consequence of using 

a steady-state model.

6. Describe the aquifer system properties that are modeled.

Explain whatever inferences are made from field data 

and previous studies as to the spatial variation of 

hydraulic properties of aquifers and confining beds and 

how discretized values are computed throughout the 

simulated area. During model calibration (see item 9), 

modeled values are often changed; the final aquifer 
30 Guidelines for Evaluating Ground-Water Flow Models

system properties that are modeled should be described 

in the report. This can be through maps or descriptions in 

the text. Lists of model arrays do not generally provide 

much understanding of the model and accordingly should 

not be included in the report unless it is expected that 

readers will want to repeat the simulations. If lists of 

arrays are included, they should usually be provided on 

electronic media. Note that Office of Ground Water 

Technical Memorandum No. 93.01 describes the separate 

requirement for archiving the complete model data sets 

used in ground-water projects.

7. Describe all the stresses modeled such as pumpage, 

evapotranspiration from ground water, recharge from 

infiltration, river stage changes, leakage from other 

aquifers, and source concentrations in transport models. 

The relations between observed and modeled stresses 

should be described. For example, it usually is desirable 

to provide a representative sample of actual pumping 

histories and the corresponding modeled pumping 

histories, although such information would not 

necessarily be provided for every pumped well. The 

manner in which stresses are averaged within the 

discretized time and space scheme should also be 

described. If a steady-state model is used to simulate an 

average or approximate steady-state condition, describe 

how the average stresses representing this system are 

calculated.

8. For transient models, describe the initial conditions that 

are used in the simulations. 

Ideally, a transient simulation will start from a steadystate condition, and the steady-state initial conditions will 

be generated by a steady-state simulation using the same 

model. In this case, the steady-state simulation must use 

the same hydraulic and stress parameters that are used in 

the transient simulation, except that the transient stresses 

are removed. In situations where it is not possible to start 

a transient model from a simulated steady-state 

condition, it is necessary to describe how the initial 

conditions were derived. It is also important to estimate 

the error in the derived values and the possible impact on 

the model results. 

9. If a model is calibrated, present the calibration criteria, 

procedure, and results. 

Describe the source of the observed data to which model 

results are compared. Explain the appropriateness of 

using these data for model comparisons and the rationale 

for any adjustments made to actual observations when 

making the comparisons. For example, when steady-state 

models are used to simulate an approximate steady-state 

condition, it is important to explain to what extent the 

observations that have been made at specific points in 

time correspond to the approximate steady-state 

condition being simulated. Give a representative sample 

of the actual comparisons used for calibration, and show 

the locations of the observation points on maps. When 

the number of observations is extensive, locations of 

representative points can be shown. It is important to 

report and use as many types of data as possible for 

calibration. For example, in a flow model, both head and 

flow observations are desirable for use in calibration.

10. Discuss the limitations of the model’s representation of 

the actual system and the impact those limitations have 

on the results and conclusions presented in the report. 

Evaluating the sensitivity of the computed model 

responses to changes in parameter values that reflect 

plausible parameter uncertainty helps to assess the model 

reliability. If the model is to be used to make specific 

projections, it is useful to estimate the impacts of the 

uncertainty of parameter values on the projections. In 

calibrated models, a concern is nonuniqueness, which is 

the extent to which other combinations of parameter 

values or configurations may result in an equally good fit 

to the observed data. Discuss the extent to which 

nonuniqueness may affect the use of the model in the 

study.

In summary, a report describing a study in which simulation is used should address the above topics; however, there is 

considerable flexibility in the form of such a report. The report 

should describe the purpose of the simulation and convince the 

reader that the use of simulation is credible. The report should 

further describe the system being simulated, the methods of 

simulation, and the data that are used.

William M. Alley

Chief, Office of Ground Water

Distribution: A, B, S, FO, PO

This memorandum supersedes Ground Water Branch Technical 

Memorandum No. 75.11