Status of Ground-Water Modeling 

in the U.S.Geological Survey

GEOLOGICAL SURVEY CIRCULAR 737
Status of Ground-Water Modeling 

in the U.S. Geological Survey

By Charles A. Appel and John D. Bredehoeft

GEOLOGICAL SURVEY CIRCULAR 737

1976
United States Department of the Interior

THOMAS S. KLEPPE, Secretary

Geological Survey

V. E. McKelvey, Director

Library of Congress Cataloging in Publication Data

Appel, Charles A.

Status of ground-water modeling in the U.S. Geological Survey.

(Geological Survey Circular 737)

Bibliography: p.

1. Water, Underground-Mathematical models. 2. Water, Underground-Data

processing. 3. Hydrology-Research-United States. 4. United States.

Geological Survey. I. Bredehoeft, John D., joint author. II. United States.

Geological Survey. III. Title. IV. United States. Geological

Survey. Circular 737. 

QB1001.72.M35A66 551.4'9'0184 76-608138

Free on application to Branch of Distribution, U.S. Geological Survey 

1200 South Eads Street, Arlington, Va. 22202
CONTENTS

Page

Abstract _____________________________________ 1

Introduction ___________________________________ - 1

Historical background ________________________ _ 1

Recent development _ __________________ _ 2

Status of modeling activity __________________--- 2

Recently published selected references _________________-_ 6

TABLE

Page

TABLE 1. Status of ground-water modeling, U.S. Geological Survey 4

in
Status of Ground-Water Modeling 

in the U.S. Geological Survey

BY CHARLES A. APPEL AND JOHN D. BREDEHOEFT

ABSTRACT

The U.S. Geological Survey is active in the development and use of models for the analysis of various types' 

of ground-water problems. Types of problems for which 

models have been, or are being, developed include: (1) 

ground-water flow in saturated or partially unsaturated 

materials, (2) land subsidence resulting from groundwater extraction, (3) flow in coupled ground waterstream systems, (4) coupling of rainfall-runoff basin, 

models with soil moisture accounting and aquifer flow 

models, (5) interaction of economic and hydrologic considerations, (6) predicting the transport of contaminants in an aquifer, and (7) estimating the effects of 

proposed development schemes for geothermal systems. 

The status of modeling activity for various models is reported as being in a developmental, verification, operational, or continued improvement phase. Recently published references that provide useful details on the 

characteristics of the models are identified.

INTRODUCTION

The use of models to aid in the analysis of 

ground-water problems has increased significantly in recent years. Both the applied researcher interested in establishing theoretical 

relationships of the dependence of a solution 

on problem variables and the field hydrologist 

interested in developing the capability to predict effects of stresses on an aquifer system 

find models useful.

HISTORICAL BACKGROUND

The U.S. Geological Survey (USGS) has 

contributed significantly to the development 

and use of models for the analysis of groundwater problems. The history of its contributions goes back at least as far as C. S. Slichter's 

classic analysis of the mathematics of steadystate flow through porous media, which was 

published in the U.S. Geological Survey's 19th

Annual Report, 1897-98. C. V. Theis' development of the nonequilibrium formula for transient ground-water flow toward a discharging 

well in a confined aquifer, published in 1935 *, 

is credited as being "one of the most important 

recent milestones in the development of 

ground-water hydrology." 2 This paper preceded a similar analysis of transient conditions in an infinite aquifer by petroleum reservoir engineers.3

Following Theis' 1935 paper, a number of 

analytical solutions were developed for differing geologic and pumping situations. These 

were closed-form analytical solutions. Although 

they often predicted the response of individual 

wells accurately, they were much less effective 

in the analysis of aquifer systems. The closedform analytical solutions generally require: 

(1) homogeneous and isotropic aquifer properties, and (2) simple aquifer geometry; but 

real aquifers commonly differ markedly from 

these idealizations.

In the early 1950's the pulsed resistor-capacitor (R-C) electronic network was developed 

as a direct analog for the equation of groundwater flow. A pulsed R-C network permitted 

solution of the general flow problem. Problems 

involving non-homogeneity, irregular boundary geometry, and pumping complexity could 

be analyzed. Petroleum reservoir engineers 

also adapted R-C networks for analysis of

1Tbeis, C. V., 1935, Relation between the lowering of the 

piezometric surface and the rate and duration of discharge of a 

well using ground-water storage: Am. Geophys. Union Tram., 

pt. 2, p. 519-524.

3 Ferris, John G., and Sayre, A. Nelson, 1955, The quantitative 

approach to ground-water investigations: Econ. Geol., Fiftieth 

Anniv. Volume, 1905-1955, pt. II, p. 714-747.

3 Muskat, M., 1937, The flow of homogeneous fluids through 

porous media: McGraw-Hill Book Co., Inc., 763 p.
flow problems. Research in the Geological Survey improved the methodology by (1) high 

speed pulsing of the networks, thus reducing 

the data acquisition time needed to complete 

a model analysis, and (2) demonstrating that 

low-cost, low-precision components gave results compatible with the accuracy of hydrologic data. With the development of the R-C 

network, most linear ground-water problems 

involving the flow of a single fluid could be 

readily solved in two or three space dimensions. The Geological Survey established an 

operational facility devoted exclusively to application of this methodology to aquifer analysis. About 100 hydrologic analog models have 

been developed and analyzed by Survey personnel in the past 15 years.

RECENT DEVELOPMENT

The R-C network is ideally suited for solving flow problems involving a single fluid of 

uniform density the usual ground-water supply problem. Most petroleum reservoir problems involve two or more fluids problems 

which are not particularly well suited to R-C 

analog solution. In the early 1950's reservoir 

engineers looked to numerical methods adapted 

for digital computers to solve large scale flow 

problems. Early numerical reservoir simulators were severely limited by the size and speed 

of the digital computers of the period. By the 

late 1960's digital computers could compete 

with the analog for the solution of transient 

problems involving a few thousand nodes in 

two space dimensions. With today's digital 

computers and powerful numerical techniques, 

three-dimensional problems involving up to 

10,000 nodes can be solved. Nearly all of the offices of the Geological Survey are now tied to 

a central computer through field terminals, 

and numerical ground-water models are used 

rather routinely for flow problems. Numerical 

models have been used for about 90 two-dimensional flow problems in the past 6 years. Due 

largely to the widespread availability of the 

digital computer, numerical methods have replaced the analog in most applications. However, the analog model remains a very useful 

tool for problems involving large multi-aquifer 

hydrologic systems which must be simulated 

using more than 10,000 nodes.

With the development of high-speed computers, it has become feasible to develop numerical models that consider more realistic 

representations of complex hydrologic systems 

than was possible earlier. Types of problems 

that have been analyzed include: (1) flow in 

water-table aquifers in which relatively large 

changes in saturated thickness take place, (2) 

flow in unsaturated or- partially unsaturated 

materials, (3) nonrecoverable compaction of 

fine-grained materials in response to pumpingrelated stresses, (4) flow in coupled ground 

water-stream systems, (5) coupling of rainfallrunoff basin models with soil moisture accounting and aquifer-flow models, (6) interaction of 

economic and hydrologic considerations, (7) 

predicting the transport of contaminants in 

an aquifer, and (8) estimating the effects of 

proposed development schemes for geothermal 

systems.

Many of these types of problems require 

that more than one equation be solved simultaneously. For example, general transport 

problems require the coupling and simultaneous solution of the partial differential equations that describe three (or more) components of a non-steady flow system. Typically, 

these would include, (1) an equation for pressure, (2) an equation for heat, in terms of temperature or enthalpy, and (3) an equation for 

the concentration of each chemical constituent 

of interest.

STATUS OF MODELING ACTIVITY

The Geological Survey continues to be active 

in modeling various types of ground-water 

problems. The status of our modeling activity 

is briefly summarized in the following table. 

With few exceptions the models handle transient conditions. Note that for many types of 

ground-water problems, models have been developed using more than a single solution 

technique. This is done so that the comparative 

strengths of different methods can be evaluated and used to advantage. Such evaluations 

provide a rational basis for determining the 

method likely to be most efficient for particular 

types of problems.

Some comments are in order on the classification of phases of modeling activity. To state 

that a model is in the "verification" phase is
intended, in most cases, to mean that development of a basic computer program has been 

completed and tests are being made to determine how well a model-derived solution 

represents the solution to the related differential equations used to describe the basic problem. For models of complex phenomena, the 

analog or numerical models often are tested 

by comparing the model results with analytical solutions. A difficulty here is that the most 

complex problem for which an analytical solution can be obtained usually is of trivial complexity compared to the degree of complexity 

of problems for which the model is developed 

to handle. A method used to check the usefulness of a model for complexity typical of realistic problems is to compare model-derived and 

observed responses to known stresses on the 

ground-water system for which the dependent 

variables and the system parameters are well 

known. In some cases, the required degree of 

certainty of these data in unavailable. In those 

cases one often relies on satisfaction of balance computations of mass, energy, and so 

forth to reflect the "correctness" of the computed values.

Models in the "operational" phase are being 

used to evaluate field problems. In some cases, 

experimental work is being done on operational models to develop useful improvements 

in model accuracy, flexibility, or convenience 

of application.

In many cases, investigators other than 

those given in table 1 are, or have been, involved in the listed modeling activity. The investigators listed can be contacted 4 for information on the subject models beyond that 

given in available publications.

The "recently published" references noted in 

table 1 provide useful details on the characteristics of the subject models and indicate the 

model capabilities. A more extensive, but not 

exhaustive, list of recently published references 

follows table 1.

4 The investigators can be located through the U.S. Geological 

Survey, Water Resources Division, National Center, Reston, Va. 

22092.
TABLE 1. Status of ground-water modeling, UJS. Geological Survey

Phase of activity

Con-

"

men- proetal tion al merit

Principal U.S. Geological 

investigators

FLOW

Saturated

Two-dimensional 

Analytical .

R-C Analog Networks __ _ . 

Numerical Finite difference _ ______

Finite element Galerkin

Finite element Variational 

Three-dimensional 

R-C Analog Networks __-__ .

Numerical (Finite difference)

Partly (or entirely) unsaturated 

One-dimensional 

Analytical 8 _______ _

Numerical Finite difference - -

Finite element Galerkin ____ ___ 

Two-dimensional

Numerical Finite difference ____________________

Cylindrical Region

 Numerical Finite element Galerkin _ __

LAND SUBSIDENCE Induced by ground water extraction

Analytical 

R-C Analog and Analytical ________

Numerical and Analytical ____ _ -

COUPLED GROUND WATER stream systems

Numerical Finite difference __ _ _ __ 

Numerical and Analytical 

COUPLED GROUND WATER RAINFALL-RUNOFF 

MODELS Numerical _______________ .

COUPLED GROUND WATER ECONOMIC SYSTEMS 

Numerical _ - _- -- -

COUPLED FLOW AND TRANSPORT OF CHEMICAL 

CONSTITUENTS

Saturated system

Conservative (or nonconservative trace constituents) 

Uniform density, inorganic - 

Two-dimensional

X

X

X

X

X X S. S. Papadopulos, R. L. 

Cooley.

X 

X 

X 

X 

X 

X

X 

X 

X

X 

X 

X

X

X

X S. M. Longwill _______

X P. 0. Trescott ___________

X G. F. Finder *, R. L. Cooley

__ R. T. Hurr ___________

X S. M. Longwill ___________

X P. C. Trescott _______

__ C. D. Ripple, J. Rubin, T. E .A

Van Hylckama.

X J. Rubin and C. D. Ripple _

__ __do _____________

X .do 

.do

__ F. S. Riley

__ D. G. "

X D. C. Helm

__ G. F. Finder 1 and S. P. Sauer 

__ A. F. Moench, V. B. Sauer, 

M. E. Jennings.

__ J. E. Reed and M. S. Bedinger

and John Terry. 

XT. Maddock, III and J. D.

Bredehoeft.
Analytical _ __________

Numerical Characteristics

X 

X

_ A. Ogata

Finite difference _____. 

Finite element Galerkin

X

Finite difference and finite element 

Galerkin _______________. 

Variable Density 

Two-dimensional 

Analytical 2 _____ _______________.

Analog and Analytical 2 ____________.

Numerical Characteristics __________.

Finite difference 2 __________. 

Finite element Galerkin ____.

X L. F. Konikow and J. D. 

Bredehoeft.

__ D. B. Grove ________. 

X D. B. Grove, J. Rubin, G. F. 

Finder.

__ D. B. Grove

Three dimensional

Numerical Finite element Galerkin _. 

Nonconservative, major constituents 

Uniform density

Inorganic constituents (one-dimensional) 

Numerical __ _____________.

X

X

_ A. Ogata _ .

__ G .D. Bennett .

__ G. F. Finder 1

" __do ____.

.do

Organic constituents (two-dimensional) 

Numerical ___ _ _ _____ .

CJl Unsaturated 

Nonconservative, Major constituents

Uniform density, inorganic constituents Numerical - 

COUPLED FLOW AND TRANSPORT OF HEAT 

Single phase (hot water) 

Two dimensional

Numerical Finite difference ___________.

Integrated Finite difference _____. 

Finite element Galerkin ________.

X

X

D. B. Grove, W. W. Wood, J. 

Rubin, and R. V. James.

J. B. Robertson and D. F. 

Goerlitz.

_ J. Rubin and R. V. James

Three dimensional

Numerical Finite element Galerkin 

Two Phase (steam-water) 

Two-dimensional

Numerical Finite difference _________

Finite element Galerkin .

X

C. R. Faust and J. W. Merce

M. L. Sorey ___ _ 

J. W. Mercer and G. F. 

Finder \

__ G. F. Finder 1

Three-dimensional

Numerical Finite element Galerkin _________ 

COUPLED FLOW AND TRANSPORT OF CONSERVATIVE 

(OR NONCONSERVATIVE TRACE) CONSTITUENTS AND 

HEAT (SINGLE PHASE) 

Two-dimensional 

Numerical Finite difference 28 ____________

T hr ee-dimensional

Numerical Finite difference 8 _______________

X

C. R. Faust and J. W. Merce

J. W. Mercer and C. R. Faus

__ G. F. Finder 1

__ D. B. Grove, S. P. Larson _

1 Part-time investigator with USGS.

2 Model limited to steady-state conditions.

3 Model prepared for USGS under contract.
RECENTLY PUBLISHED SELECTED 

REFERENCES

Ground-Water Flow Saturated

Bedinger, M. S., Reed, J. E., and Griffin, J. D., 1973, 

Digital-computer programs for analysis of groundwater flow: U.S. Geological Survey open-file report, 

'85 p.

Bennett, G. D., and Patten, E. P., Jr., 1962, Constanthead pumping test of a multiaquifer well to determine characteristics of individual aquifers: U.S. Geol. 

Survey Water-Supply Paper 1536-G, p. 181-203.

Bredehoeft, J. D., Cooper, H. H., Jr., and Papadopulos, 

I. S., 1966, Inertia! and storage effects in wellaquifer systems: An analog investigation: Water 

Resources Research, v. 2, no. 4, p. 697-707.

Bredehoeft, J. D., and Papadopulos, I. S., 1965, Rates 

of vertical ground-water movement estimated from 

the earth's thermal profile: Water Resources Research, v. 1, no. 2, p. 325-328.

Bredehoeft, J. D., and Pinder, G. F., 1970, Digital analysis of areal flow in multiaquifer ground water systems: A quasi three-dimensional model: Water Resources Research, v. 6, no. 3, p. 883-888.

Cooley, R. L., 1972, Numerical simulation of flow in an 

aquifer overlain by a water table aquitard: Water 

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Cooley, R. L., and Case, C. H., 1973, The effect of a 

water-table aquitard on drawdown in an underlying 

pumped aquifer: Water Resources Research, v. 9, 

no. 2, p. 434-447.

Cooper, Hilton H., Jr., 1966, The equation of ground 

water flow in fixed and deforming coordinates: 

Jour. Geophys. Research, v. 71, no. 20, p. 4785-4790.

Cooper, H. H., Jr., Bredehoeft, J. D., Papadopulos, I. 

S., and Bennett, R. R., 1965, The response of wellaquifer systems to seismic waves: Jour. Geophys. 

Research, v. 70, no. 16, p. 3915-3926.

Cooper, H. H., Jr., Bredehoeft, J. D., and Papadopulos, 

I. S., 1967, Response of a finite-diameter well to 

an instantaneous charge of water: Water Resources 

Research: v. 3, no. 1, p. 263-269.

Emery, Philip A., Patten, Eugene P., Jr., and Moore, 

John E., 1975, Analog model study of the hydrology 

of the San Luis Valley, South-Central Colorado: 

Colorado Water Resources Circ. 29, 21 p.

Frind, Emil 0., and Pinder, George F., 1973, Galerkin 

solution of the inverse problem for aquifer transmissivity: Water Resources Research, v. 9, no. 5, 

p. 1397-1410.

Hurr, R. T., 1972, Modeling ground-water flow by the 

finite-element method: in Proceedings of an International Conference on Variational Methods, held 

at University of Southampton, September 25, 1972, 

v. 1, p. 5/39-5/49.

Jenkins, C. T., 1968, Electric-analog and digital computer model analysis of stream depletion by wells: 

Ground Water, v. 6, no. 6, p. 27-34.

Jenkins, C. T., and Taylor, 0. James, 1974, A special 

planning technique for stream-aquifer systems: U.S. 

Geol. Survey open-file report 74-242, 16 p.

Maddock, T., Ill, 1970, A program to simulate an 

aquifer using alternating direction implicit-iterative procedure: U.S. Geol. Survey open-file report, 

73 p.

Moulder, E. A., and Jenkins, C. T., 1960, Analog-digital 

models of stream-aquifer systems: Ground Water, 

v. 7, no. 5, p. 19-24.

Papadopulos, I. S., 1966, Nonsteady flow to multiaquifer 

wells: Jour. Geophys. Research, v. 71, no. 20, p. 

4791-4797.

 1967a, Drawdown distribution around a largediameter well: Proc. Natl. Symposium on Ground 

Water Hydrology, San Francisco, Calif., Nov. 6-8, 

1967, p.157-168.

1967b, Nonsteady flow to a well in an infinite

anisotropic aquifer: Internat. Assoc. Sci. Hydrology 

Pub. 73, p. 21-31.

Papadopulos, I. S., and Cooper, Hilton H., Jr., 1967, 

Drawdown in a well of large diameter: Water Resources Research, v. 3, no. 1, p. 241-244.

Patten, E. P., Jr., 1965, Design, construction, and use 

of electric analog models, in Wood, L. A., and 

Gabrysch, R. K., Analog model study of ground 

water in the Houston district: Texas Water Comm. 

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Pinder, George F., and Bredehoeft, J. D., 1968, Application of the digital computer for aquifer evaluation: 

Water Resources Research, v. 4, no. 5, p. 1069- 

1093.

Pinder, George F., 1969, An iterative digital model for 

aquifer evaluation: U.S. Geol. Survey open-file report, 44 p.

Pinder, George F., and Frind, Emil 0., 1972, Application of Galerkins' procedure to aquifer analysis: 

Water Resources Research, v. 9, no. 1, p. 108-120.

Remson, Irwin, Appel, Charles A., and Webster, R. A., 

1965, Ground water models solved by digital computer: ASCE Jour. Hydraulics Div., v. 91, no. 

HY3, p. 133-147.

Skibitzke, H. E., and Robinson, G. M., 1954, The use 

of numerical and electrical methods in solution of 

ground-water flow problems: Unpublished manuscript.

Skibitzke, H. E., 1960, Electronic computers as an aid 

to the analysis of hydrologic problems: Pub. 52, 

Intern. Assoc. Sci. Hydrol., Com. Subterranean 

Waters, Gentbrugge, Belgium, p. 347-358.

Stallman, R. W., 1956, Numerical analysis of regional 

water levels to define aquifer hydrology: Trans. 

Am. Geophys. Union, v. 37, no. 4, p. 451-460.

 1963a, Electric analog of three-dimensional flow 

to wells and its application to unconfined aquifers: 

U.S. Geol. Survey Water-Supply Paper 1536-H, p. 

205-242.

1963b, Calculation of resistance and error in an

electric analog of steady flow through nonhomogeneous aquifers: U.S. Geol. Survey Water-Supply 

Paper 1544-G, p. G1-G20.

Trescott, Peter C., 1973, Iterative digital model for 

aquifer evaluation: U.S. Geol. Survey open-file report, 63 p.
 1975, Documentation of finite-difference model 

for simulation of three-dimensional ground-water 

flow: U.S. Geol. Survey open-file report 75-438, 

59 p.

Trescott, Peter C., Pinder, George F., and Larson, S. 

P., 1976, Finite-difference model for aquifer simulation in two-dimensions with results of numerical 

experiments: U.S. Geol. Survey Technique WaterResources Inv., book 7, chap. Cl. (In press.)

Weeks, John B., Leavesley, George H., Welder, Frank 

A., and Saulnier, George J., Jr., 1974, Simulated 

effects of oil-shale development on the hydrology of 

Piceance Basin, Colorado: U.S. Geol. Survey Prof. 

Paper 908, 84 p.

Ground-Water Flow Partly or Entirely Unsaturated

Cooley, R. L., 1971, A finite-difference method for unsteady flow in variably saturated porous media: Application to a single pumping well: Water Resources 

Research, v. 7, no. 6, p. 1607-1625.

Doherty, Patrick C., 1972, Unsaturated Darcian flow by 

the Galerkin method: U.S. Geol. Survey Computer 

Contribution C938, 70 p.

Ripple, C. D., Rubin, Jacob, and van Hylckama, T. E. 

A., 1972, Estimating steady state evaporation rates 

from bare soils under conditions of high water 

table: U.S. Geol. Survey Water-Supply Paper 2019- 

A, 39 p.

Rubin, Jacob, 1966, Theory of rainfall uptake by soils 

initially drier than their field capacity and its applications: Water Resources Research, v. 2, no. 4, 

p. 739-749.

 1967, Numerical method for analyzing hysteresisaffected, post-infiltration redistribution of soil 

moisture: Soil Sci. Soc. American Proc., v. 31, no. 

1, p. 13-20.

 1968a, Numerical analysis of ponded rainfall 

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1968b, Theoretical analysis of two-dimensional,

transient flow of water in Unsaturated and partly 

Unsaturated soils: Soil Sci. Soc. America Proc., v. 

32, no. 5, p. 607-615.

Stallman, R. W., and Reed, J. E., 1966, Steady flow 

in the zone of aeration: UNESCO Symp. Water 

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v. II, Internat. Assoc. Sci. Hydrology Pub. 83, p. 

565-575.

Land Subsidence

Helm, D. C., 1974, Evaluation of stress-dependent 

aquitard parameters by simulating observed compaction from known stress history: Ph.D. Thesis, 

Univ. of Calif., Berkeley, 175 p.

 1975, One-dimensional simulation of aquifer system compaction near Pixley, California: Water Resources Research, v. 11, no. 3, p. 465-478.

Jorgensen, Donald G., 1975, Analog-model studies of 

ground-water hydrology in the Houston District, 

Texas: Texas Water Devel. Board Rept. 190, 84 p.

Riley, F. S., 1969, Analysis of borehole extensometer 

data from central California, in Land Subsidence, 

v. 2, edited by L. J. Tison, Internat. Assoc. Sci. 

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Coupled Ground-Water Surface Water Systems

Luckey, Richard R., and Livingston, Russell K., 1975, 

Reservoir release routing model for the Upper 

Arkansas River Basin of Colorado: Colorado Water 

Conserv. Board, Water Resources Ser. Circ. 27, 

44 p.

Moench, Alien F., Sauer, Vernon B., and Jennings, 

Marshall E., 1974, Modification of routed streamflow by channel loss and base flow: Water Resources Research, v. 10, no. 5, p. 963-968.

Pinder, George F., and Sauer, Stanley, P., 1971, Numerical simulation of flood wave modification due 

to bank storage effects: Water Resources Research, 

v. 7, no. 1, p. 63-70.

Coupled Ground-Water Economic Systems

Bredehoeft, John D., and Young, Robert A., 1970, The

temporal allocation of ground-water A simulation

approach: Water Resources Research, v. 6, no. 1,

p. 3-21. 

Maddock, Thomas, III, 1972, Algebraic technological

function from a simulation model: Water Resources

Research, v. 8, no. 1, p. 129-134. 

 1973, Management model as a tool for studying

the worth of data: Water Resources Research, v.

9, no. 2, p. 270-282.

1975, A tax system for ground water management: Water Resources Research, v. 11, no. 1, p. 

7-14.

Young, Robert A., and Bredehoeft, John D., 1972, Digital computer simulation for solving management 

problems of conjunctive ground-water and surfacewater systems: Water Resources Research, v. 8, 

no. 3, p. 533-556.

Coupled Flow and Transport of Chemical Constituents

Bennett, G. D., Mundorff, M. J., and Hussain, S. Amjad, 

1968, Electric-analog studies of brine coning 

beneath fresh-water wells in the Punjab Region, 

West Pakistan: U.S. Geol. Survey Water-Supply 

Paper 1608-J, 31 p.

Bredehoeft, John D., and Pinder, George F., 1972, Application of transport equations to groundwateT 

systems: Proceedings of the Symp. on underground 

waste management and environmental implications, 

held December 6-9, 1971, in Houston, Texas, edited 

by T. D. Cook, AAPG Pub., Memoir 18, p. 191-199.

Bredehoeft, John D., and Pinder, George F., 1973, Mass 

transport in flowing groundwater: Water Resources 

Research, v. 9, no. 1, p. 194-210.

Grove, David B., 1970, A method to describe the flow of 

radioactive ions in ground water Final report, 

December 1, 1966 through June 30, 1968: Springfield, Va., Nat'1. Tech. Inf. Ser., 41 p.
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regional flow field: Water Resources Research, v. 

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Grove, D. B., and Beetem, W. AM 1971, Porosity and 

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Henry, Harold R., 1964, Effects of dispersion on salt 

encroachment in coastal aquifers, in Sea water in 

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Hughes, Jerry L., and Robson, S. G., 1973, Effects of 

waste percolation on groundwater in alluvium near 

Barstow, California: Proceedings of the Second 

International Symposium on Underground Waste 

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Louisiana, Sept. 26-30,1973, v. 1, p. 91-129.

Konikow, Leonard F., and Bredehoeft, John D., 1973, 

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preliminary report: Colorado Water Resour. Circ. 

17, 43 p.

Ogata, Akio, 1961, Transverse diffusion in saturated isotropic granular media: U.S. Geol. Survey Prof. 

Paper 411-B, 8 p.

 1963, Effect of the injection scheme on the 

spread of tracers in ground-water reservoirs: U.S. 

Geol. Survey Prof. Paper 475-B, p. B199-B202.

 1964, Mathematics of dispersion with linear 

adsorption isotherm: U.S. Geol. Survey Prof. Paper 

411-H, p. H1-H9.

 1964, The spread of dye stream in an isotropic 

granular medium: U.S. Geol. Survey Prof. Paper 

411-G, 11 p.

 1969, Two-dimensional dispersion in a granular 

medium-disk source emitting at constant rate: U.S. 

Geol. Survey Prof. Paper 650-D, p. 260-264.

 1970, Theory of dispersion in a granular 

medium: U.S. Geol. Survey Prof. Paper 411-1, p. 

11-134.

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8
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