Ground-Water Studies 

and Analog Models

GEOLOGICAL SURVEY \ 

CIRCULAR 468
Ground-Water Studies 

and Analog Models

By Charles J. Robinove

GEOLOGICAL SURVEY CIRCULAR 468

Washington 1962
United States Department of the Interior

STEWART 1. UDALL, SECRETARY

Geological Survey

THOMAS B. NOLAN, DIRECTOR

Free on application to the U.S. Geological Survey, Washington 25, D. C.
CONTENTS

Page Page

Abstract______________________________ 1 How does the hydrologist use the

Introduction____________________________ 1 basic information to describe the

The ground-water situation ___________ 1 ground-water system? _____________ 4

People are concerned-with water The electric analog model of a

problems _________________________ 2 ground-water system ______________ 6

How does the hydrologist collect Hpw are the results of ground-water

information needed to answer the studies presented to the public? ____ 12

questions? _______________________ 3 Ground-water studies and the future___ 12

ILLUSTRATIONS

Page

Figure 1. Block diagram of a ground-water basin i________________________-____--__---_ 1

2. Water is needed for many uses.. ___________________________--__----__---_ 2

3. Geologic and hydrologic data are collected in the field ________________________ 3

4. Water-level and well-discharge measurements are made in the field ___________ 4

5. Laboratory investigations and tests supplement field data _____________________ 4

6. Lines of equal transmissibility show the ability of the valley fill to allow

movement of water______________________________________________________ 5

7. Lines of equal storage coefficient are useful in describing the capacity of

the valley fill to store water ___________________--_-_--_----_--__-----_--- 5

8. The natural shape of the water table and the direction of flow of ground water ___ 5

9. The decline of water level in an area of pumping _____________________________ 6

10. Electric analog models of ground water systems _____________________________ 7

11. A technician transfers the lines of equal transmissibility to a sheet of pegboard _ 8

12. A technician is placing and soldering electrical resistors in the model _________ 8

13. The electrical capacitors are added to the model_______________________ ____ 8

14. The completed model is ready for analysis _______---___-__--___----__-.____- 9

15. An electrical stimulus being applied to the model ____________________________ 10

16. The effect of pumping being shown on an oscilloscope screen __________________ 10

17. The results of the analog model analysis are portrayed on graphs and maps_____ 11

HI
Ground-Water Studies and Analog Models

By Charles J. Robinove

ABSTRACT

Hydrologists make ground-water studies to aid managers 

and users of water resources in solving their problems in the 

development and management of ground water. Geologic and 

hydrologic information provides the basic knowledge for 

construction of electric analog models that portray the 

ground-water system in miniature. Analog models can be 

analyzed electrically, and the results of the analysis are 

presented in terms of the ground-water system so that the 

effects of alternative methods of water development can be 

assessed.

situation that is typical of many places in the. 

southwestern States. Our problem consists 

of determining how best to use the groundwater resource. Alternative methods of developing the ground water, such as spacing 

wells close together or far apart, and the 

possible effects of each alternative method 

can be determined from studies of the area. 

The water users then have a basis for deciding what action they should take in developing the ground water.

INTRODUCTION

All parts of the United States have water 

problems because expanding population and 

industry create ever-increasing needs for 

large supplies of good water. To cope with 

these problems, basic information is needed 

about water--its availability, quantity, quality, and use. This information is provided 

by hydrologists.

People feel an individual and a community 

responsibility to be aware of national and 

local water problems, but they cannot always 

find the information they want and need. This 

report explains how the United States Geological Survey studies ground-water systems 

and uses electric analog models. It is one of 

a series of reports being prepared by the 

U. S. Geological Survey to help people understand their water supply and how to manage 

it.

The electric analog is not new to science. 

However, its application to ground-water 

problems is new. It is one of the tools used 

to analyze ground water, which cannot readily be studied directly.

Although not all water problems are the 

same, all have certain basic features in 

common. Let us consider a ground-water

THE GROUND-WATER SITUATION

Our example is a desert basin that is underlain by sand, gravel, and other rock materials, 

and surrounded by mountains. (See fig. 1.) 

These mountains are composed of rock that

ferffcvei above 

the water table

Impermeable 

bedrock Permeable sand and 

gravel saturated with 

water

Figure l.__The ground-water basin contains water in die underlying sand and graved.
GROUND-WATER STUDIES AND ANALOG MODELS

is quite dense and contains little' water. Water moves readily through the underlying deposits of sand and gravel, however.

Rain and snow are the sources of water 

in the basin. Some water evaporates, some 

runs off in streams, and some seeps into 

the ground. Some of the water absorbed by 

the soil is taken up by the roots of plants. 

The rest of the water moves down deeper, 

pulled by gravity. At some distance below 

the floor of the basin, the spaces between the 

sand and gravel grains are saturated with 

water. This subsurface water is called ground 

water. The upper surface of the saturated 

zone is called the water table. The water table 

and the saturated zone are shown in figure 1.

Although some ground water is everywhere 

below the surface of the soil, it may not always be free to come into a well. Rocks 

through which water can move easily are 

said to be permeable. A layer of permeable 

rock that yields water in useful quantities is 

called an aquifer. The word "aquifer" comes 

from two Latin words: aqua or water, and 

ferre, to bring. The aquifer literally brings 

water (underground). An aquifer may be a 

layer of gravel and sand, sandstone, or 

cavernous limestone. In our example, the 

aquifer consists of deposits of sand and 

gravel, through which water moves readily. 

An aquifer may be only a few feet thick, or 

tens or hundreds of feet thick. It may be 

just below the surface, or hundreds of feet 

below. It may underlie a few acres or many 

square miles. In our example, the aquifer 

is about 500 feet thick and underlies about 

1,800 square miles and the water table lies 

20 to 100 feet below the surface.

The words "aquifer," "ground-water basin," and "ground-water reservoir" are used 

interchangeably here to mean the layers of 

rock in which ground water is stored by 

nature. Water in the ground-water basin is 

always moving. However, the rate of movement is so slow, compared with that of 

streams, that for all practical purposes, the 

ground-water basin is considered a storage 

reservoir.

Grou,nd water moves very slowly through 

the aquifer--sometimes only a few feet per 

year, whereas water in streams and rivers 

travels as much as several feet per second. 

Many problems of ground-water development 

and use arise from the fact that ground water moves very slowly from one place to 

another and more water is available in one 

place than another.

PEOPLE ARE CONCERNED WITH WATER 

PROBLEMS

Residents of the valley have drilled wells 

into the water-saturated sand and gravel. 

They have begun to pump water for irrigation, home supplies, industries, and public 

water supplies. As people realize that more 

water is being used and that more wells are 

needed, they become concerned about their 

future supply of water. The ground-water 

basin shown in figure 1 is an example of many 

areas where development of ground water 

is taking place or is contemplated. Problems 

concerning possible depletion of the groundwater resource, proper spacing of wells, 

increased pumping lifts, and many other 

problems confront users of water in such 

an area and they begin to ask searching 

questions about their particular uses of 

water and about the ground-water situation 

in general. (See fig. 2.) The hydrologistmust 

answer these questions, must aid water users 

to recognize the nature of their water problems, and must help them to choose between 

various alternative solutions" to their water 

problems.

Tfie suburban resident 

fhust have a perennial 

supply of water

\

ler needs 

constant irrigation 

upply throughout 

le growing se

The municipality must 

have enough water 

for growth and 

expansion

The industrialist needs 

large supplies of water

Figure 2,«.Water is needed for many
GROUND-WATER STUDIES AND ANALOG MODELS

HOW DOES THE HYDROLOGIST COLLECT

INFORMATION NEEDED TO ANSWER THE

QUESTIONS?

One basic requirement for study of a 

ground-water system is a geologic map. 

The geologist maps the extent, thickness, 

and types of deposits in the ground-water 

basin and the location of the impermeable 

rocks which bound it. The geologist has thus 

described for the hydrologist the shape, size, 

and physical boundaries of the ground-water 

system. The block diagram (fig. 1) shows 

the surface and cutaway sections of the basin 

and surrounding mountains. If we were to 

look directly down on the top of this block 

diagram, we would see the land surface as 

it would be portrayed on a geologic map.

Data on the ground water reservoir also 

are obtained in many other ways--from records and logs of water wells, records of 

rainfall and streamflow, descriptions of the 

rock units as seen on the surface of the 

ground and as sampled below the surface, 

measurements of the water-bearing properties of the rocks, and much other basic 

information. Data on the rocks and water 

are collected in the field (fig. 3 and 4). 

Measurements of depth to water in wells 

while the wells are idle or pumping are made 

to determine the decline of water level caused 

by pumping.

The cuttings brought to the surface by the 

drill from test holes are sampled and some 

of the properties of the material are measured (fig. 5). Such information includes data 

on the size of the individual sand and gravel 

particles; the amount of space between these 

particles that can be filled with water; the 

rate at which water moves through the material; and the composition, layering, and 

other physical and chemical properties of 

the material.

All these data help the hydrologist to 

determine three major characteristics of the 

ground-water basin: first, its shape and 

size; second, the amount of water that can 

be stored; and third, the rate at which the 

earth materials allow water to move through 

them.

We use the term "storage coefficient" to 

describe the ability of the earth materials 

to release water that is stored in them and 

the term "coefficient of transmissibility" to 

describe the rate of movement of water 

through the material.

Samples of material at the ground surface are collected and 

tested for porosity and permeability.

Power augers and well-drilling rigs are used to collect samples 

of earth material at depth and to drill holes for further testing.

The electric logger records the electrical properties of rocks in a 

test well. Much valuable geologic and hydrologic data can 

be interpreted from the electric logtf.

Figure 3. Geologic and hydrologic data are collected in 

field.

the
GROUND-WATER STUDIES AND ANALOG MODELS

The discharge of wells is measured with a current meter inserted 

in the end of the discharge pipe.

saiuraieu 

with water

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Samples of sand and gravel are analyzed with sieves and a mechanical shaker to determine the size and sorting of the grains.

Water levels are measured by lowering a steel tape into the well.

Figure 4. Water-level and -well-discharge measurements are 

made in the field.

Samples of earth material enclosed in the large tubes are tested

for their capacity to transmit water.

Figure 5. Laboratory investigations and tests supplement field 

data.

HOW DOES THE HYDROLOGIST USE THE BASIC

INFORMATION TO DESCRIBE THE GROUND-WATER

SYSTEM?

The ability of the ground-water reservoir, 

or aquifer, to transmit water is shown in figure 6. Each line connects points of equal 

transmissibility of the aquifer. The number 

represents the number of gallons of water a 

day that can pass through a 1-mile width of 

the water-bearing sand and gravel across that 

line for each foot-per-mile slope of the water
HOW DOES THE HYDROLOGIST USE THE BASIC .INFORMATION TO DESCRIBE THE GROUND-WATER SYSTEM?

160,000 gallons of water 

pass a 1-mile width 

of this line per day

Figure 6. Lines of equal transmissibility show the ability of the 

valley fill to allow movement of water.

table. Thus, if the water table has a slope of 

10 feet per mile in the vicinity of the 160,000 

line, the, flow through a 1-mile width of aquifer 

would be 1,600,000 gallons of water per day.

The water-storage capacity of the valley 

fill is described by the storage coefficient 

and is shown in figure 7). As mentioned before, storage coefficient represents the 

amount of water that is released from 

storage in a given unit of the aquifer ma^ 

terial when the water level is lowered a 

unit distance. For example, 1 cubic foot of 

sand and gravel in the valley fill may, when 

filled with water, contain 0.2 cubic feet of 

water. If the water table is lowered 1 foot 

by pumping and this material is drained, only 

about 0.15 cubic feet of water will be released. The rest remains in the rock as water 

that clings tightly to the individual particles 

and will not move toward the well. The storage 

coefficient is then considered to be 0.15. A 

water-table aquifer, 1 square mile in extent 

and 100 feet thick, with a storage coefficient 

of 0.15, can store more than 3 billion gallons 

of water. The storage coefficient is important 

because it indicates the maximum amount of 

water that can be withdrawn for use from, an 

aquifer at a given time.

The shape of the water table, or upper 

surface of the saturated zone, before pumping begins is shown in figure 8. Note that, 

the water table slopes toward the lower end 

of the basin. The ground water moves in the 

direction shown by the arrows.

16 percent of the total 

volume of sediments 

and water along this 

line is water that is 

available to wells

Figure 7. Lines of equal storage coefficient are useful in describing the capacity of the valley fill to store water.

Altitude of 

water table

Line showing 

direction of 

ground-water 

flow

Figure 8. The natural shape of the water table and the direction 

of flow of ground water.

When a well is pumped, as in figure 9, the 

shape of the water table is altered drastically 

and it assumes the shape of a funnel or cone 

around the pumped well. Within the cone the 

water flows radially toward the well. The 

longer the well is pumped, the larger and 

deeper the cone becomes. Eventually the water table is lowered throughout a broad area. 

Additional wells that are drilled within this 

area of influence will reach the water table 

at a depth lower than that occurring naturally and consequently pumping lifts will be
GROUND-WATER STUDIES AND ANALOG MODELS

Water table // . i / 

before pumping

Water table 

after pumping

Funnel-shaped volume 

drained of water by 

. r, - ,._ - pumping

Figure 9. The decline of water level in an area of pumping.

greater. The changes of water level caused 

by pumping water commonly are known only 

in the areas where water development has 

taken place. The effects of withdrawal of 

water outside of the immediate area of development may be largely unknown.

Predictions of water-level changes are 

needed not only in the developed area but 

also in adjacent areas of potential development. One of the major problems in management of ground water is the prediction 

of future changes in water levels in the 

aquifer as water continues to be pumped. 

Various patterns of pumping can be analyzed to determine answers to the following 

questions.

1. Which areas in the basin are capable of 

producing enough water for use over a period 

of many years without the water levels declining to depths from which it is uneconomical to pump?

2. What is the maximum amount of water 

that can be pumped from an individual well 

or well field over long periods of time?

3. If wells are closely spaced, will their 

pumping interfere with each other to the 

extent that water levels will be lowered to 

depths too great for economical pumping?

4. How long will the water last if sustained large-scale pumping is undertaken?

THE ELECTRIC ANALOG MODEL OF A GROUNDWATER SYSTEM

We want to find out what the effect of pumping wells on the flow of water in the aquifer 

will be. The-decline of the water table caused 

by pumping water from the aquifer is one 

of the major factors we wish to analyze. A 

large-scale approach would be to pump the 

actual wells and then measure and analyze 

the effects of the pumping. But if we want to 

find out what the effects will be 50 years 

from now, it is hardly practical to pump the 

wells for 50 years and then measure the 

effects. The answer would come too late to 

be of use.

Many ground-water systems can be analyzed by mathematical methods. Mathematical formulas can take into account the shape, 

size, and hydrologic properties of the system; the amount and rate of water development, and the effect of water development 

on the system. The more complex the system is, however, the more complex the 

mathematical techniques must be, the more 

tedious and time-consuming is the analysis, 

and the less clearly the mathematics describe the system.

The hydrologist must look for some way to 

scale down the problem that he is studying-- 

to look at it in miniature and with a reduced 

time scale, and to apply certain stresses, 

such as pumping, in a manner that will show 

effects on the ground-water system.

If the ground-water system being studied 

departs greatly from an ideal situation in 

which the flow of ground water is regular 

and the aquifer is uniform, the mathematical 

model may be replaced by constructing an 

electrical model of the ground-water system 

and analyzing it by electrical means.

The ground-water hydrologist can construct and analyze a model of the groundwater system in much the same way that 

the aeronautical engineer tests model planes 

in a wind tunnel, and the ship builder tests 

scale models of ships in tanks of water. An 

electrical model of the ground-water system 

is a useful analogy because an electrical 

circuit reacts to changes in the flow of electricity in the same way that the ground-water 

system reacts to changes in the flow of 

ground water. The electrical terms used to 

describe this relation and a description of
GROUND WATER SYSTEM ! ELECTRICAL SYSTEM

TRANSMISSIBILITY - £ -ELECTRICAL RESISTANCt 

STORAGE COEFFICIENT HI - EL EC TRIG AL CAPACITANl

Figure 10. Electric analog models of groond-water systems.
8 GROUND-WATER STUDIES AND ANALOG MODELS

the analogy between ground-water systems 

and analog models are shown in figure 10.

Readers who are unfamiliar with electricity 

and electrical terms may find that the terms 

used here are not very clear. However, it is 

not really necessary to understand the terminology of electricity to appreciate how the 

analog model is used. The point is that the 

electric analog model is illuminating for the 

hydrologist. For him, using the analog model 

is akin to putting the whole aquifer under a 

microscope.

The electrical model is constructed in a 

shape similar to that of the actual aquifer 

system; that is, it is built upon a small-scale 

map of the aquifer. The thickness of the 

aquifer and the properties of water storage 

and transmittal are representedby electrical 

properties. For example, the water table, 

which rises and falls as water is added to or 

removed from the aquifer, is represented by 

a changing electrical voltage in the model.

You will see by looking at figure 10 that a 

small piece of the aquifer has been removed 

to show how its properties are modeled. A 

certain amount of water is stored in this 

unit of the aquifer. Our model has an electrical capacitor that stores a certain amount 

of electricity. Water flows through the aquifer at a certain rate; in the model an electrical resistor limits the rate of flow of 

electricity to that amount that is analogous 

to the flow of water through the ground-water 

system.

At any particular point in.the ground-water 

system there is a combination of these factors. In the electrical model, we call this a 

junction--a joining of several resistors and a

capacitor that represent a rate of flow and 

an amount of storage at that particular point. 

The electrical model is constructed by connecting together enough of these junctions 

to simulate the entire ground-water system.

Figure 11 shows how the lines of equal 

transmissibility" of the water-bearing rocks 

are transferred to a sheet of pegboard. The 

resistors are placed on the pegboard and 

are soldered together (fig. 12); they then are 

removed from the pegboard and hung in a 

frame and the capacitors are added (fig. 13). 

The model is now complete and is ready for 

analysis (fig. 14).

The model is a physical entity representing the shape, size, and characteristics 

of the aquifer. The analyzer computes the 

electrical changes that take place in the 

model and records the effect of the change^ 

on the model.

Figure 12. A technician is placing and soldering electrical resistors in the model to represent the ability of the sediments to 

transmit water.

Figure 11. A technician transfers the lines of equal transmissibility to a sheet of pegboard on which the electrical model will 

be constructed.

Figure 13. The electrical capacitors that represent the capacity 

of the sediments to store -water are added to the model...
THE ELECTRIC ANALOG MODEL OF A GROUND-WATER SYSTEM

Figure 14. The completed model is ready for analysis.
10 GROUND-WATER STUDIES AND ANALOG MODELS

We wish to simulate the pumping of water from the aquifer, so we will "pump" or 

withdraw electricity from the model. We 

know that pumping water causes a decline 

of the water table which is analogous to 

changes in voltages in the model, and so 

we will measure the decline in voltage 

in the model as a measure of the decline of 

the water table. Figure 15 shows the electronic apparatus used to produce the stimulus 

in the model that is analogous to pumping of 

water from the ground-water system. This 

stimulus is fed into the model and is shown 

as a graph on the oscilloscope.

The effects of the stimulus are measured 

electrically and demonstrated on another 

oscilloscope screen. Figure 16 shows a graph 

of the decline of the water level for one point 

in the model. The graph shows the waterlevel drawdown at a well if a given amount of 

water were pumped in a given period of time. 

The scales of the graph on the oscilloscope 

screen can be read directly in feet of change 

of the water level and in days, months, years, 

or any unit of time that is desired. Various 

patterns and rates of pumping are tried and 

their effects are measured.

With this model it is possible to analyze 

quickly the long-term effects of pumping a

 * 

 ***** «

 * «

Figure 15. An electrical stimulus, representing pumping of 

ground water, is applied to the model by electrical function 

generators. The graph on the oscilloscope screen represents a 

well pumping a certain amount of water for a. given period of 

time.

Figure 16. The effect of pumping water from the system is pictured on the oscilloscope screen as a graph showing the decline 

of water level in the aquifer at some distance from the pumping well during the period of pumping.

large ground-water system. A few weeks or 

months are required to construct and analyze the model; most of the time is spent 

in calculating the design of the model and 

in the construction of the model itself. Actual analysis of the model may take only a 

few days.

The information obtained by using the analog model is presented in graphs and maps, 

as shown in figure 17. The first graph (A) 

shows the decline of water levels at different 

intervals of pumping. Two sets of graphs (B) 

illustrate the water-level decline at a given 

time in closely spaced and widely spaced 

wells. Finally, two sets of contour maps (C) 

show water-level declines at the end of 

several periods of time, one with wells placed 

close together, the other with wells spaced 

far apart. With this information available, 

hydrologists can recommend the best pattern 

of water development for this particular 

aquifer. The water users then have a basis 

for deciding what pattern and rate of water 

development will be most efficient and beneficial.
THE ELECTRIC ANALOG MODEL OF A GROUND-WATER SYSTEM 11

Example of graph representing decline 

of water levels with time as taken 

from the oscilloscope screen

Wells closely spaced

te at point A

at point B

at point C

at point D

at point E

[ at point F

10 20 30 

TIME, IN MONTHS

40 50

Water level decline at point A

at point B

The water-level changes at the various 

points are used to construct maps 

showing, by contours, the water-level 

declines throughout the aquifer at 

various times since pumping began

Wells far apart

Water level declines at the 

end of 10 months

20 months

30 months

'40 months

50 months

Wells closely spaced

Water level declines at the 

end of 10 months

20 months

30 months

40 months

50 months

Figure 17. The results of the analog model analysis are portrayed on maps and graphs.
12 GROUND-WATER STUDIES AND ANALOG MODELS

HOW ARE THE RESULTS OF GROUND-WATER 

STUDIES PRESENTED TO THE PUBLIC?

Each year the Geological Survey publishes 

many reports describing the results of 

ground-water studies throughout the Nation. 

These reports are published as water-supply 

papers, hydrologic investigations atlases, 

professional papers, circulars, and bulletins. 

Some are also printed in technical journals of 

scientific societies. Reports also are published by the State agencies with which the 

Geological Survey cooperates in hydrologic 

studies.

The maps and graphs resulting from the 

electric analog model analysis are published 

in reports along with the maps of the groundwater system, the basic data on the geology 

and water resources, and the interpretations 

of the data made by the hydrologist.

All these reports are available to the public 

and are announced in monthly lists or other 

releases. Geological Survey Water-Supply 

Paper 1492 "Bibliography of Publications 

Relating to Ground Water Prepared by the 

Geological Survey and Cooperating Agencies, 

1946-55," by R. C. Vorhis, 1957, is recommended for further information on ground 

water. Water-Supply Paper 1536-D "The 

Ground-Water Flow System in the Snake 

River Plain, Idaho-An Idealized Analysis," 

by H. E. Skibitzke and J. A. daCosta, 1962, 

describes the use of an electric analog

model in predicting the drawdown that could 

be caused by pumping in the Snake River 

Plain, Idaho. These two papers may be obtained from the Superintendent of Documents, 

Washington, D.C.

Two articles outline the methodology and 

use of analog model studies: "FromGeologic 

Data to Aquifer Analog Models" by R. W. 

Stallman,. Geo-Times Magazine, v. 5, no. 7, 

April 1961, and "Electronic Computers as an 

Aid to the Analysis of Hydrologic Problems" 

by H. E. Skibitzke, publication 52 of the 

International Association of Scientific Hydrology, 1960.

GROUND-WATER STUDIES AND THE FUTURE

The use of electric-analog-models in the 

solution of hydrologic problems provides the 

scientist and engineer with an effective new 

method of evaluating our water resources. 

They have been used in the science of groundwater hydrology for only a few years but already many improvements have been made. 

Let us make clear, however, that the analog 

model is not a final answer to our water-resource problems. It is an excellent tool which, 

in the hands of capable hydrologist, can be of 

great benefit and help. Research for new 

techniques and improved methods of solving 

water problems is, and will remain, the key 

to the proper management and conservation 

of our water resources.