Electric Analog of 

Three-Dimensional 

Flow to Wells and 

Its Application to 

Unconfined Aquifers

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1536-H
Electric Analog of 

Three-Dimensional 

Flow to Wells and 

Its Application to 

Unconfined Aquifers

By ROBERT W. STALLMAN

GROUND -WAT ER HYDRAULICS

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1S36-H

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1963
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

For sale by the Superintendent of Documents, U.S. Government Printing Office 

Washington 25, D.C.
CONTENTS

	Page 

Abstract__________________---__-_-____--___-----__----__- 205

Introduction- _____________--_.____-___--_-______-_-__-_____-__---_ 205

Equations of flow____________________________________________ 206

Finite-difference approximations of flow equations_____________________ 209

Resistance analog______________________ ___-_-____-_--_______-_- 211

Approach to pumping-test analysis._________________________________ 214

Design and construction of the model__-______-_-----________________ 215

Analog study of pumping-test data.___-_____--_-_______--__---_-_--- 219

Analysis of Grand Island pumping test--------------------------- 219

Analogic relations__________-__________--____--_----_-_-_----__ 219

Aquifer representation.______________-__-_____---_-------_-_-_- 231

Depth, penetration, and time lag of observation wells ______________ 234

Water-level control in pumped well._____________________________ 236

Model adapted for analysis of nonsteady flow______-___--___--_-_____- 236

Summary of results._______________________________________________ 241

References._______ ______________________________________________ 242

ILLUSTRATIONS

	Page 

FIGUBE 56. Instantaneous flow paths.______.---_-______--_--_-_-___- 207

57. Finite-difference grid.__________-_____________---__---___ 210

58. Grid subdivision of flow field_---_------__-__-----_---_-_- 211

59. Junction_____________________________-__--__--__ 212

60. Boundary elements------------------------------------- 213

61. Initial flow in unconfined aquifer._______-___---------__-_ 214

62. Grid reference__-_--_-__-_-___-___________________ 217

63. Grand Island drawdown conditions at i=0+_-______-___- 220

64. Relative drawdowns, Grand Island test________-__---____ 222

65. Initial vertical gradients.-------------------------------- 223

66. Selected hydrographs______---_____-_-_______________ 225

67. Drawdown data____-_--_----_____----__-___---_--__--__ 229

68. Relative drawdown for full penetration-________-___-__---- 232

69. Idealized response_____-_--_-_-----_-_--______----__--_- 235

70. Nonsteady flow, boundary analogic relations._______-_--___ 237

71. Nonsteady flow, interior analogic relations______---_---____ 239

in
IV CONTENTS

TABLES

Page 

TABLE]. Summary of R,r values for jR£ ' = 1.8X104 ohms, A2=0.05 m,

and A£=0.25_______.________________________ 218

2. Laboratory determinations of permeability of samples of alluvium from well 84, Grand Island, Nebr__________________ 221

3. Water-level data from selected wells and specific yield calculated

from model solution_______--__--___-____-----_-_-__-__ 226

5s 4. Estimates of ^- at the water table from drawdown data (r=50

ft) _ _-_ __ _ _ _- -- 231

5. Summary of values of permeability and specific yield calculated 

from model studies of Grand Island, Nebr., pumping-test 

data, well 35 ______-______________________________ 234
GROUND-WATER HYDRAULICS

ELECTRIC ANALOG OF THREE-DIMENSIONAL FLOW TO 

WELLS AND ITS APPLICATION TO UNCONFINED 

AQUIFERS ______

By ROBERT W. STALLMAN

ABSTRACT

Electric-analog design criteria are established from the differential equations 

of ground-water flow for analyzing pumping-test data. A convenient analog 

design was obtained by transforming the cylindrical equation of flow to a rectilinear form. The design criteria were applied in the construction of an electric 

analog, which was used for studying pumping-test data collected near Grand 

Island, Nebr.

Data analysis indicated (1) vertical flow components near pumping wells in 

un confined aquifers may be much more significant in the control of water-table 

decline than radial flow components for as much as a day of pumping; (2) the 

specific yield during the first few minutes of pumping appears to be a very small 

fraction of that observed after pumping for more than 1 day; and (3) estimates 

of specific yield made from model studies seem much more sensitive to variations 

in assumed flow conditions than are estimates of permeability.

Analysis of pumping-test data where vertical flow components are important 

requires that the degree of anisotropy be known. A procedure for computing 

anisotropy directly from drawdowns observed at five points was developed.

Results obtained in the analog study emphasize the futility of calculating unconfined aquifer properties from pumping tests of short duration by means of 

equations based on the assumptions that vertical flow components are negligible 

and specific yield is constant.

INTRODUCTION

Most analytical expressions used in analyzing discharging-well types 

of aquifer tests are based on the assumption that components of vertical flow are negligible. Usually artesian, or confined, flow to wells is 

adequately defined by this assumption. Methodology associated 

with such analysis has been described by Wenzel (1942), Ferris (1949), 

Jacob (1950), and Ferris and others (1962). Flow to wells in unconfined aquifers, on the other hand, is more often highly dependent on 

the velocity components parallel with the well axis. Boulton (1954) 

has developed an analytical solution of nonsteady flow to a fully 

penetrating well; and, more recently, Kirkham (1959) has presented

205
206 GROUND WATER HYDRAULICS

a means for analyzing steady flow to partially penetrating wells, 

both of which account for these vertical flow components.

In field testing, wells generally penetrate only a part of the aquifer 

thickness and are operated for testing purposes for such a short period 

of time that flow is nonsteady during the test. The investigations 

individually reported by Boulton and Kirkham are significant advances in test analysis, but a need exists for an analytical procedure 

that in effect combines their respective contributions. Specifically, 

therefore, a method of analyzing nonsteady flow to a partially penetrating well, treating both radial and vertical velocity components, 

is required.

The complexity of the analyses described by Boulton (1954) and 

Kirkham (1959) leads one to the belief that any expression obtained 

from a combination of the boundary systems used by each may be 

impractical for general use in hydrologic investigations. With this 

prospect in view, some technique other than the calculus appears 

to be desirable in developing methods of test analysis.

This report describes the design and use of an electrical model 

of the field of flow in the vicinity of a discharging partially penetrating 

well. The model, as devised, is a mechanism for solving the finitedifference approximations of the differential equations of flow. The 

interrelations among the aquifer and model characteristics are discussed and the model is applied to a particular analysis of selected 

aquifer-test data. Finally, the general utility of the analog for pumping-test analysis is outlined.

EQUATIONS OF FLOW

Confined flow and unconfined flow are illustrated in figure 56. 

Figure 56-4 represents the classic concept of nonsteady artesian flow 

to a well as defined by Theis (1935). If the aquifer can be considered 

homogeneous and isotropic, the drawdown, s, signifying the change in 

head due to a change in rate of pumping, at each point follows the 

relation:

52s , 1 ds S' ds /1N -f- == * (l/

where r is the distance from the centerline of the pumping well to a 

point at which s is observed; S' is the storage coefficient per unit 

volume of aquifer; P is the permeability of the aquifer material; and t 

is the elapsed tune since a change in rate of pumping occurred. In 

the flow system characterized by figure 56-4 and equation 1, the 

vertical flow components have been assumed to be negligible. This 

permits expression of the ratio S'/P in terms of the aquifer characteristics as S/ T, where S is the aquifer storage coefficient (approxi-
ELECTRIC ANALOG OF THREE-DIMENSIONAL FLOW 207

Pumping 

levelAQUIFER

Radius, r 

A. Confined aquifer

Watertable

Pumping 

level

Radius, 

. Unconfined aquifer

FIGURE 56. Schematic of instantaneous paths of flow in the vicinity of a pumped well. 

A, Confined aquifer; B, Unconfined aquifer.
208 GROUND WATER HYDRAULICS

mately equal to the aquifer thickness, m, times S'), and Tis the aquifer 

transmissibility (m times P), as defined by Wenzel (1942). Thus 

equation 1, rewritten for the total aquifer thickness, is

&s 1 ds_S ds . 

W+r Kr~T Dt ( '

Equation 2 follows also from the equation of continuity applied to the 

aquifer thickness and from the assumptions: (1) that the aquifer is 

homogeneous and isotropic; (2) that flow obeys Darcy's law; and (3) 

that discrete amounts of water are released from storage, because 

water expands and the aquifer contracts as pressure is reduced locally 

in the flow field.

In the flow system illustrated in figure 565, the expression for the 

drawdown at any point between the water table and lower confining 

bed may be written in the form

where PT is the permeability along the r axis and Pz is the permeability 

along the z axis. Equation 3 differs from equation 1 only in that the 

effects of vertical flow components are included through addition of d2s 

the term j ^ and that effects of stratification on directional permeability are recognized. The complete interrelations among s, r, z, and 

t depend on equation 3 and on the boundary conditions at the well 

screen and along the free surface. Boulton (1954, p. 568) described 

the boundary condition at the free surface, whence it can be shown 

that:

where Sv is the specific yield (Ferris and others, 1962) and it is assumed 

that no recharge or discharge crosses the water table. Equation 4 

applies along the free surface only, and Sy can be considered a variable 

in space or time.

In unconfined systems, the total effect on the relation between 

drawdown and time caused by the storage changes due to a change 

in the water-table position are generally of a much greater magnitude 

than those resulting from decompression of the water below the water 

table. Thus for the flow system shown on figure 565, drawdown may 

be defined in approximate form by
ELECTRIC ANALOG OF THREE-DIMENSIONAL FLOW 209

at all points between the water table and the confining bed for the 

boundary conditions that are considered.

ds The assumption S' ^7=0 appears to be a valid approximation for

the system analyzed later in this report. However, it must be recognized that under certain conditions including very low pumping rates, 

nonhomogeneity, low values of the ratio SyfPZ) or exceptionally thick 

aquifers the effects on water levels caused by release from internal 

storage may indeed be significant. Equation 5 seems to be a satisfactory approximation of the aquifer system analyzed in detail later 

in this report. Its use simplified the analytical solution developed 

by Boulton (1954) and will also simplify analog construction. However, use of the complete form, equation 3, offers no barrier to model 

design or analysis.

FINITE-DIFFERENCE APPROXIMATIONS OF FLOW

EQUATIONS

An electric analog is envisioned in which each component is the 

physical counterpart of a given block of aquifer material. The analog 

is to represent all the aquifer material between the water table and 

the confining bed, from the pumping-well radius rw out to some external 

radius re where the effects on water levels caused by pumping are 

negligible. In order to devise such an analog, the differential notation 

employed in stating the flow problem is expanded, so that the flow 

equation finally used as the basis for model design relates specifically 

to finite blocks of the flow field rather than to infinitesimal volumes. 

This might be accomplished by writing equation 5 in terms of finite 

differences directly, but the result would lead to an unnecessarily 

complicated model design. Consequently, equation 5 is transformed 

as suggested by Southwell (1946, p. 44).

Letting £=log r, equation 5 is rewritten

The finite-difference forms of the terms in equation 6 (Southwell, 1946, 

p. 16) used are

and

-~r \ fa. . 9.<?_4-.<?_,,v (8)

656086 62 2
210 GROUND WATER HYDRAULICS

The complete finite-difference form of equation 6 at any point (p, n) 

corresponding with (z, £) is obtained by substitution of equations 7 

and 8, from which

(9)

The finite lengths As and A£ and the subscript notation of equations 7, 

8, and 9 are identified on figure 57. Equation 9 must be satisfied at

Water table

n -1

Az

Az

-p-l

FIGURE 57. Finite-difference grid in a field of radial flow.

each point in the finite representation of the field of flow. In accordance with figure 57 and the basis for equation 9, these points are 

illustrated schematically in figure 58. The interval A£ is constant 

between all grid lines parallel with the z axis, and the interval A£ is 

constant between all grid lines parallel with the £ axis. By definition

(10)

'n-l

The accuracy with which the finite-difference grid represents the 

flow field increases with the number of grid intersections employed to 

represent a given region of the flow field. For a two-dimensional grid 

like that shown on figure 3, satisfactory accuracy might be achieved by
ELECTRIC ANALOG OF THREE-DIMENSIONAL FLOW 211

» t

> 

» i

A

1

-> A£

Lower confining beds

t Radius which drawdown at

is negligible

FIGTTBE 58. Finite-grid subdivision of the field of radial flow.

a minimum of about 100 grid intersections spread through the area of 

significant flow. In effect, such a grid would represent 100 equations 

like equation 9, containing 100 unknown values of s. Such a set of 

simultaneous equations could be written and solved directly to obtain 

the drawdown distribution for specified boundary conditions, but the 

electric analog appears to be a more convenient approach to solution.

[RESISTANCE ANALOG

A junction at which four resistors terminate, with the resistances 

oriented along the £' and z' axes, is shown on figure 59. Assuming there 

is a voltage distribution of some form arbitrarily established at the 

extreme ends of the resistances, the current to the junction (pf, n'} is 

given by Kirkoff's law as

where e is the voltage at the point indicated by the subscript; R? is 

the resistance of the elements between nf 1 and n' and between n' 

and w'+l; and R* is the resistance of the elements between p' l 

and p' and between p' and p'-}-l.

The analogy between the forms of equations 9 and 11 is obvious, 

and apparently the junction shown on figure 59 may be adopted as 

an analogy of the condition expressed by equation 9 if e is taken 

proportional to s, and if

(12)
212 GROUND WATER HYDRAULICS

/?'-!

'-I

FIGUEE 59. Junction of four resistance elements.

From the latter relations, a grid of resistance elements may be devised to fit between the grid intersections shown on figure 58. The 

resistor grid thus established is analogous to the finite-difference 

equations of the flow field internal to the boundaries.

Along the base of the aquifer the vertical flow component is zero. 

The spatial distribution of the grid intersections with respect to the 

lower boundary and the equivalent resistor grid are shown on figure 

60. The grid lines in the aquifer section have been located arbitrarily 

so that one line along the £ axis lies directly on the contact between 

the aquifer and the confining bed. Equation 9 applies at every grid 

intersection lying along the boundary, with the added condition that 

(sp-i Sp)/As=Q. The latter is derived easily from the requirement 

that the vertical gradient ds/ds=0 along the confining bed. Part of 

the resistor grid along the lower boundary representing the aquifer 

segment of figure 60.A is shown on figure 60 J5. Maintaining the electrical analogy assumed for the interior grid intersections requires that 

Bp'-i ep> Q. In the electric analog, this condition may be established by making all values of Ez> beneath the contact equal to infinity. 

This is accomplished by simply terminating the resistor grid in the 

z' axis directly beneath the analogous contact between the aquifer 

and confining bed.
^A£-> Grid intersections

*//////)*77T>

As

3 -,

V/////J

= 0

base

A. At base of aquifer

base 

/777T7

B. Resistance elements.along 

grid lines near base

FIGURE 60. Finite grid and resistance elements representing boundary regions. A, At base 

Eesistance values at base of aquifer; D, Resistance values 
214 GEOUND WATER HYDRAULICS

Comparison of figure 60^1 with 605 shows that R? along the contact 

represents an aquifer thickness of A0, which, in effect, extends below 

the contact position by an amount A0/2. Some adjustment in R? 

values along the boundaries must be made to account for the smaller 

flow segment they represent, as compared with the areas represented 

at interior grid intersections. According to the view held by Southwell (1946, p. 77), grid spacing along the boundary represents a 

prototype section with twice the resistance between comparable 

grid points in the interior of the flow field. Thus the boundary 

resistors should have twice the resistance of those in the interior at 

comparable £' values, as indicated on figure 60(7. If the water-table 

decline is small compared with A0, the resistor grid at points along 

the water table will have the same relative design, as indicated on 

figure 60Z>.

The effect of terminating the grid at large values of r will be discussed later. To this point, sufficient conditions have been established to identify the analogic resistance values as a function of £ 

and 0.

APPROACH TO PTJMPING-TEST ANALYSIS

The flow conditions in the vicinity of a partially penetrating pumped 

well in an unconfined aquifer, shortly after pumping begins, are shown 

schematically on figure 61. It is assumed that the pumping level is 

held constant as a function of time. Initially the water table is 

everywhere horizontal. Before water levels begin to decline along 

the surface of the saturated zone, a head gradient between the water

Initial water-table position

FIGURE 61. Initial flow conditions near a partially penetrating pumped well In an unconflned

aquifer.
ELECTRIC ANALOG OF THREE-DIMENSIONAL PLOW 215

table and well face is produced in response to the reduced water level 

in the pumped well. Along the water table, this gradient is initially 

directed downward and perpendicular to the water table. The 

downward motion at all points along the water table in response to 

this gradient as a function of the aquifer permeability, specific 

yield, and time is defined by equation 4. For a short period of 

time after pumping begins, the water table may be considered virtually horizontal over most of its area. Also, over most of the water 

table the term ds/ds will be much larger than the term 

Therefore, equation 4 may be simplified to

Both sides of equation 13 simply represent the velocity of the watertable decline parallel to the 3 axis.

All pumpage at the well is assumed to be derived from the downward movement of the water table and therefore the discharge rate, 

Q, at the well may be expressed as

Q=-Pe fT> (ds/ds)r27rn/r, (14)

JTw

where (ds/ds)r is the vertical gradient at the water table at radius r. 

The integral term of equation 14 may be evaluated easily by graphical 

analysis of an electric-model study, and Q is observed in the field 

test. Thus PS, may be obtained directly from use of equation 14. 

The rate of change of the water-table position is observed at a given 

distance from the pumped well, from which an estimate of ds/di is 

available; the value of (ds/ds)r may be estimated at that point from 

the model analysis, and Sv computed from these data may be put in 

equation 13. This, of course, presumes that the ratio Pr/P« is known 

from other sources of information and that the model was constructed 

and analyzed with full recognition of this ratio.

DESIGN AND CONSTRUCTION OF THE MODEL.

A particular model design was selected to illustrate the method of 

analyzing pumping tests described in this report. The model criteria 

discussed earlier and the shape of the flow system to be analyzed 

govern the selection of resistance elements to be assembled in the final 

model for any given group of problems. From equation 9 it is evident 

that all the interior resistance elements along any given value of p' 

(fig. 59) must be alike to attain greatest simplicity in the model for 

the form of flow equation adopted. This condition has already been 

assumed in the development of equation 11. Thus, in any one
216 GROUND WATER HYDRAULICS

particular model, all values of R$> are the same except along the 

water table and confining bed, where they are twice the internalelement resistance (2R^), as previously discussed. R$> will be used 

henceforth to identify the resistance of the internal elements along 

p' lines and as a reference for calculating Re ' by equation 12.

The interval A£ defines the degree of subdivision along the £ axis, 

and its selection partly determines the accuracy of the model results. 

Several calculated trial designs showed that A£=0.25 would be 

satisfactory for providing adequate detail along the £ axis.

Equations 9 and 12 and consideration of the difference between rw 

and a sufficiently large re ordinarily found under field test conditions 

indicate that values of Rz> must range over many orders of magnitude. 

Low-cost resistors can be obtained with values ranging from about 

2X107 to about 2X10" 1 ohms from commercial sources, and values 

of Rs> were restricted to this range for economy of model construction. 

The test to be analyzed later was conducted on a well with a screen 

radius, rw, of about 1 foot, in an aquifer about 100 feet thick. The 

value of As arbitrarily selected for the model is m/20; that is, the 

saturated thickness is divided by the chosen model grid into 20 equal 

increments along the z axis. At the well screen for such test-site

Az 5 dimensions and proposed model configuration, -^7=^ >, 2 -=20.

By referring to equation 12 and assuming an isotropic aquifer, it is 

seen that the ratio PT/PZ equals unity and along the ordinate rw 

represents the face of the well screen (Rg') Tjo=400R^, approximately. 

Unless the volume of the aquifer directly beneath the partially 

penetrating well is to be modeled, this latter relation provides an 

estimate of the largest value of Rz> required in the model for the test 

case to be studied herein. From the limits of the resistor values 

economically available, the maximum value allowable for R^ would

be about £^=1-^=^1^=5X104. A value of Ev less than the

maximum is desirable to allow space in the vicinity of rw in the model 

for control at smaller r values and treatment of anisotropic conditions. 

Thus a value JS> £< = 1.8X104 ohms was finally, and only somewhat 

arbitrarily, selected. The distribution of resistance elements, Rg> 

and R?, connecting all points (p' } ri) according to the reference 

system indicated on figure 62 is summarized in table 1.

The Rz> values are given in table 1 as a function of (Pr/P2) (r/m) to 

present the model-design data in as general a form as possible. A 

model constructed according to table 1 may be used for analyzing 

any pumping test in which the values of r/m lie within the range given
Pr (r//n,'=6.0xlO-3 7.8xlO~3 1.0xlO~2 1.3xlO~2 

/?£'=1.8xl07 1.2xl07 6.8xl06 4.7xl06 

*

z'

n' = l 23 4 

, . 

P !

2

3

t

Az

1

1

"^-kZ 

20

^

21

///////////////////////////////////

FIGURE 62. Grid reference for resistan
218 GROUND WATER HYDRAULICS

there because of the dimensional characteristics of equation 12. 

For example, n'=3, where (PT/Ps)(r/m) = 1.00XW~2 could be used 

to model a pumping well with a radius rw=0.1 foot in an iso tropic 

aquifer 10 feet thick, or a well with a radius of 10 feet in an isotropic 

aquifer 1,000 feet thick, or any other combination of rw and m for 

which (Pr/P9) (r/m) = 1.00X10- 2 .

According to equation 10, (?'n/m)/(rn_i/m)=eAJ. The latter relation

was used for calculating succeeding values of ( W" ) ( )> as given in 

table 1, by assuming that A£=0.25 and by beginning the grid computations arbitrarily at a value of (WO ( J=5X10~4. With the

successive values of ( j£ J ( j calculated in this way and the other

model-design assumptions that As = 0.05m, -Rj» = 1.8X104 ohms, 

equation 12 was used to calculate Re> for each value of n'. Because 

values of r/m for nonspecific test sites will seldom be predictable 

integers, the actual numerical value of r/m selected at any particular 

intersection along £ is immaterial, if the resistor grid is to be used for 

more than one test analysis. In most cases each pumping-test analysis 

will require a model solution for Pz by equation 14 for several assumed 

values of rw/m. The Pe values computed in this way may then be 

used for interpolation to find the exact value of Pz for the observed 

rw/m at the test site. This appears to be a more economical procedure 

than constructing a completely new model or resistor grid for each 

test analysis.

TABLE 1. Summary of values for #,£' = 1.8X10* ohms, Az 

(fig. 62)

0.05m, and

n'

1 ___ 

2 

3 ___ 

4 .

5

6 __

7 

8 .

9... ....

10 

11 ... 

12. __ 

13 .

14 __ .. 

15.. _ . 

16 

17 

18 

19.. __

Pr T

P,m

6. 03X10-"

7.82 i. ooxio-2

1.29

1.66

2.13

2.73

3.50

450

5.78 

7.42 

9.53 

1. 22X10-' 

1.57 

2.02 

2.59 

3.32 

4.27 

5.48

Calculated 

R,' (eq 12)

1.94X10' 

1.18 

7. 14X10« 

4.33

2 CO

1.59

9. 67X10*

5.86

3.56

2.15 

1.31 

7. 92XKH 

4.82 

2.92 

1.77 

1.07 

6. 51X103 

3.95 

2.39

R,' nominal 

resistance 

in ohms (±10 

percent)

1.8X10' 

1.2 

6. 8X10« 

4.7

o 7

1.5

1.0

5.6X10*

3.3

2.2 

1.2 

8. 2X10* 

4.7 

2.7 

1.8 

1.0 6.8X103 

3.9 

2.2

n'

20 

21 _ .. 

22 

23 __

24.. _ .

25. .....

26.. ....

97

28... 

29. ..... 

30 ......

31... ... 

32. ..... 

33 

34 ......

35 

36 

37 

Pr T

P,m

7.04 9.04X10-' 

1.16X10" 

1.49

1 Q1

2.46

3.16

4.05

5.20

6.68 

8.58 

1. 10X101 

1.41 

1.81 

2.33 

3.00 

3.84 

4.94

Calculated 

R»> (eq 12)

1.45 

8.80X102 

5.34 

3.24

1.96

1.19

7.22X10'

4.38

2.66

1.61 

9. 78X10" 

5.94 

3.60 

2.18 

1.32 

8.03X10-' 

4.88 

2.95

R,' nominal 

resistance 

in ohms (±10 

percent)

1.5 8.2X102 

5.6 

3.3

1.8

1.2

6.8X101

4.7

o 7

1.5 

1.0 5.6X10" 

3.9 

2.2 

1.2 8.2X10-1 

4.7 

2.7
ELECTRIC ANALOG OF THREE-DIMENSIONAL FLOW 219 

ANALOG STUDY OF PUMPING-TEST DATA 

ANALYSIS OF GRAND ISLAND PUMPING TEST

The resistance analog shown on figure 62 was used to study pumping-test data collected near Grand Island, Nebr., and was reported by 

Wenzel (1942, p. 117-122). At the test site, the aquifer is about 100 

feet thick and is composed of unsorted sand and gravel. Permeability 

coefficients determined by laboratory tests on disturbed samples are 

shown in table 2. The well pumped in the test was 24 inches in 

diameter and extended about 37.1 feet below the static water table. 

It was pumped continuously at a rate of about 540 gpm for 48 hours. 

During the period of pumping, drawdowns were observed in 80 nearby 

wells that were located at distances that ranged from 2.6 feet to 1,217.1 

feet from the pumped well.

Although it is evident from table 2 that the aquifer in profile is not 

homogeneous, it will first be assumed homogeneous and isotropic. 

Thus, Pr/Pz=l. (Other situations are discussed on p. 231-234.) In 

the resistor grid shown on figure 62, A£=0.25, A0=m/20=5 feet, 

and -Rj' = 1.8X104 ohms. Therefore, from equation 12, the value 

of RZ ', corresponding with the position of rw is

2

X1.8X10'=7.2X10'ohms.

From table 1, it is seen that the radius ru is very nearly represented by 

nf 3 on the resistance analog design given.

The drawdown in the pumped well was about 20 feet during the 

last part of the test period (Wenzel, 1942, p. 150). It was probably 

somewhat smaller during the first few minutes of pumping; and as 

indicated by Wenzel (1942, p. 150), the drawdown of the water table 

outside the well face was much less. But for illustration here, 20 

feet is assumed to be a satisfactory approximation of the drawdown 

in the pumped well. Because the pumped well was perforated along 

its full length (Wenzel, 1936, p. 28), the effective drawdown at any 

point along the well bore, but above the pumping level, numerically 

equals the depth of that point below the initial static water level. At 

all points below the pumping level in the pumped well, the head 

decline is very nearly 20 feet, assuming that flow between the screen 

and pump intake occurs with negligible head loss within the well. 

These relations are shown, in terms of s/sw, on figure 63 at the corresponding model grid intersections. The analogic electric controls 

established on the model for the first few minutes of pumping (£=0+) 

must reproduce the same head distribution as shown in figure 63. 

It has already been stated that voltage in the analog is proportional
220 GROUND WATER HYDRAULICS

n'=l

0.78 1.0 1.3 1.7 3000 3840 4940 feet 

2 3 4 5 35 36 37

1.00

S/S w ==

1.00

0

0.25 =

0.50

0.75

1.00

1.00

1.00

1.00

0 0

e/e w

19

20

21

0 0 0

u

10

HLJ 

LL.

Z

on LJ £.\j 1

GO

HCC

H-

£

O

30 

v/V/ 

d^~

C/J

1LJ

CD

X

40 £

LJ

o

90 

inn

FIGURE 63. Drawdown conditions at t= O+ for Grand Island pumping test in relation to model grid

intersections.

to drawdown in the aquifer. Therefore at all control points along 

the boundary the relation between voltage and drawdown may be 

expressed as

i^=^), (15)
ELECTRIC ANALOG OF THREE-DIMENSIONAL FLOW 221

where ew refers to the maximum voltage applied on the analogboundary points corresponding with points similarly located along 

the well screen.

TABLE 2. Laboratory determinations of permeability of samples of alluvium from 

well 84, Grand Island, Nebr.

[After Wenzel, 1942, p. 118]

Depth below water table 

(ft)

0-2.4

2. 4-8. 4

8. 4-12. 4

12. 4-17. 4

17. 4-22. 4

22. 4-31. 4

31. 4-32. 4

32. 4-34. 4

34. 4-38. 4

38. 4-43. 4

P (gpd per sq ft)

480

1,685

1,460

1,095

1,095

4,350

2

925

150

350

Depth below water table 

(ft)

43. 4-47. 4

47. 4-53. 4

53. 4-58. 4

58. 4-63. 4

63. 4-70. 4

70. 4-78. 4

78. 4-84 4

84. 4-91. 4

91. 4-97. 4

P (gpd per sq ft)

780

730

2,095

1,050

2, 185

220

495

430

285

To find the relative drawdowns occurring during the first few 

minutes of pumping, voltage was controlled at grid intersections 

(nodes) in accordance with figures 63 and equation 15. Readings 

were then made of the voltage at each node. The relative drawdown 

contours developed from these measurements are shown in figure 64. 

Instrumental limitations precluded definition of the drawdown configuration at the larger radii, but the contribution to flow from that 

region was found to be insignificant during the first few minutes of 

pumping and may be neglected. Values of s/sw observed along 

p'=2 on the model were used in obtaining the curve shown on figure 

65, which is a graph of r versus r(As/sw) per A.Z. Note that because 

s/sw=0 at all p'=l, the observed s/sw along p'=2 equals the difference 

between drawdown ratios through a depth kz. Because these 

measurements are made near the water-table position, they closely 

reflect the gradient found just beneath the water table. The latter 

curve is a convenient form for obtaining Pz from a finite-difference 

form of equation 14. In the relation

 = 2ir (16)

in which (AS/«W)AZI refers to the gradient observed just beneath the 

analog water-table position, between p'=l and p'=2 (fig. 62). The 

summation in equation 16 is equivalent to the integral of equation 

14, wherein constants have been removed from under the summation 

upon expression in terms of finite differences. The summation of
20-

40-

60-

80-

100-1

0.5

FIGURE 64. Initial drawdown configuration (<=0+) in terms of sls w assuming partial penetration 
ELECTRIC ANALOG OF THREE-DIMENSIONAL FLOW 223 

r (AS/S) PER Az, AT WATER TABLE w

10~6 (4) 10~5 (4) 10-4 (4) 10-3 (4) 10-2 (4) 1Q- 1 (4) 1.0

200

400

600

800

1000

FIGTJEE 65. Initial vertical gradients along the water table and their relation to radius from the pumped 

well, Grand Island pumping-test site.

equation 16 was evaluated for the study reported in figure 64 by 

graphically summing the area under the curve of figure 65 from 

which

rw =\

and

or

=536 gpd/ft2 (for s«=2° f t}

= 1,072 gpd/ft2 (for stt=10 ft).

The numerical result for sw=2Q feet is not in particularly close agreement with Wenzel's P= 1,000 gpd per sq ft obtained from analysis 

of later parts of the pumping-test data by other methods. The 

average permeability to horizontal flow should be about 1,200 gpd 

per sq ft, according to table 2. An assumed value of 10 feet for the 

drawdown in the pumped well may therefore be more nearly correct 

than the 20 feet assumed from the test for the model analysis of the 

flow occurring during the first few minutes of pumping.

However, other features of the Grand Island test and test site 

depart more fundamentally from the assumptions used in the model 

analysis. Study of these features is pursued in the following discussion not so much to obtain good numerical agreement between 

the model analysis and the more conventional forms of aquifer-test 

analyses as to illustrate those features of flow to wells that are signifi-
224 GROUND WATER HYDRAULICS

cantly related to the effects of vertical flow components, nonhomogeneity, and anisotropy.

Several interesting problems are associated with the computation 

of specific yield using equation 13 for the Grand Island test area. 

In obtaining the solution presented on figure 64, it was assumed that 

the drawdown along the water table was zero. The length of the 

time interval over which this condition is approximately satisfied 

in the field is critical for several reasons. Chiefly, it is important 

because it identifies the time interval over which the slopes of hydrographs for observation wells represent only the initial rate of downward movement of the water table. Boulton (1954, p. 572) has 

indicated that the flow pattern varies only slightly from its initial

Pt form in the interval 0<cv <0.05. ^Sym Thus the test-time interval

for valid analysis by equations 13 and 14 is from t=Q to Z=0.05 

Svm/P, approximately. By use of WenzePs results for Sv and P 

for the Grand Island test, it is seen that the critical time interval is 

evidently from t=Q to *=0.05X0.2X100X7.48/1,000=7.48X10~3 

day, or about 11 minutes. Hence, the hydrograph slopes observed 

in the field during about the first 10 minutes of pumping should 

adequately represent the flow regime resulting from the model boundary conditions assumed in developing figure 64, provided that the 

specific yield during the interval was approximately 0.2.

Fully defined hydrographs for the first few minutes of the Grand 

Island test cannot be constructed, because water-level measurements 

in the observation wells were made rather infrequently. A selected 

group of wells with the data used herein for computing Sy are listed 

in table 3 in order of distance from the pumped well. The selection 

from WenzePs data and the listing of wells was made on the basis that 

(1) the wells were measured to determine drawdown at least once 

during the first 10 minutes of pumping, (2) the well depth and diameter are known, (3) a measurable drawdown occurred in the well during 

the first 10 minutes of pumping, and (4) the well ends in the upper 

part of the zone of saturation. Selected hydrographs are given on 

figure 66. The distance of each listed well from the pumped well and 

the depth below the water table at which each is finished are shown 

on figure 64.

Factors related to data collection and test operation that might 

invalidate the estimate of Sv, using equation 13, will be discussed 

later. At this point it is of interest to proceed directly to the solution 

for Sy along the analytical lines already defined. Toward this end,
ELECTRIC ANALOG OF THREE-DIMENSIONAL FLOW 225

an estimated value of bsfot was obtained for each of the wells listed 

in table 3 by dividing the observed drawdown noted in the first measurement made by the time elapsed since pumping began. The time

10 20 30 40 50 60 70 80 90

/, IN MINUTES 

FIGURE 66. Hydrographs of selected observation wells, Grand Island pumping-test site.
226 GROUND "WATER HYDRAULICS

TABLE 3. Water-level data from selected wells and specific yield calculated from

model solution

Data from Wenzel (1942, p. 120, 151-171)

Well

1 . .... . ___ ....

13 

44 

32 

56 

25. ... .

57 

14 

45 

33 

46 

34 

26 

47... ...... 

35 

27 

{to

24.9 

29.9 

39.3 

40.1 

46.7 

49.7 

69.5 

70.0 

80.5 

95.1 

130.3 

144.7 

170 

195.6 

214.3 

270

Ati 

time first 

drawdown 

measured 

(minutes)

6 

6 

4 

2 

7 

4 

10 

9 

5 

4 

7 

6 

7 

9 

8 

9

(As) a 

for first 

time 

interval 

(ft)

0.96 

1.87 

.93 

.75 

.88 

.64 

.53 

.49 

.32 

.22 

.10 

.09 

.05 

.05 

.03 

.01

5*^(A*)a

dt Ati

( }

\day7

230 

448 

335 

540 

181 

231 

76.3 

78.3 

92.2 

79.2 

20.6 

21.6 

10.3 

8.0 

5.4 

1.5

Values obtained from model study

ds^(As)izi

dz A,

(nondimensional)

9.6X10-2 

7.0 

4.15 

3.9 

2.8 

2.4 

1.14 

1.14 

8X10-3 

5.3 

2.36 

1.88 

1.07 

6. 6X10-* 

4.67 

1.82

 536 A«4 2i/Az

* 7.48 (As) n/Ati 

?1 _ A*AU/AZ

(As) a/ Mi 

(nondimensional)

2.98X10-2 

1. 13X10-2 

8. 87X10-3 

5.17X10-3 

1.11X10-2 

7. 45X10-8 

1.07X10-2 

1.04X10-2 

6.22X10-3 

4.8X10-3 

8.2X10-3 

6. 23X10-3 

7.43X10-3 

5. 37X10-3 

6. 2X10-3 

8. 7X10-3

interval between commencement of pumping and the first drawdown 

measurement, A^, and the corresponding drawdown (As) H are given

ds in table 3. Estimates of > were obtained directly from the model

study as

for sw =20 feet, e p > = i=Q, and Az=5 feet. Given in the last column 

of table 3 are values of S« calculated from the relation

S,j=P, (17)

which is a finite-difference approximation of equation 13. The value 

of Pz used was 536/7.48 = 71.7 feet per day as obtained from equation 

16 and graphical analysis of figure 9.

The values of Sy, calculated from data collected during the first 

few minutes of pumping arrayed in table 3, are in disagreement with 

the value of 0.2 determined by Wenzel from the more commonly 

employed mathematical equations. In general, the results from the 

model study are, at least, an order of magnitude smaller than those 

obtained by Wenzel (1942, p. 125). Thus far, however, treatment 

of the test data has been rather superficial in that gross assumptions 

have been made regarding their meaning, and the aquifer itself has 

been modeled somewhat arbitrarily. Further investigation of the 

effects of these idealizing assumptions on the calculated results is
ELECTRIC ANALOG OF THREE-DIMENSIONAL FLOW 227

warranted. A summary of the numerical results obtained from the 

model study to this point suggests that calculated permeability is 

very nearly equal to, or is about one-half the value found by Wenzel 

(P= 1,000 gpd per sq ft), depending on the applicable value of sw. 

Specific yield calculated from the model study is about one order of 

magnitude smaller than that calculated by Wenzel.

These results indicate that possibly the values of (As)&ei/Az obtained 

from the model analysis are too small or that the assumptions attending data interpretation have resulted in values of (As) a/A£ that are 

too large. The present model configuration is not entirely a realistic 

analog of the Grand Island aquifer, and the hydrograph slopes appear 

to have been crudely estimated. Therefore, attention is focused on 

investigation of the validity of the model and the data used in obtaining values of the ratio

ANALOGIC RELATIONS

The resistance analog used for obtaining the drawdown distribution 

in the aquifer was constructed by assuming that the effects of internal

ds storage on flow are negligible. This was stated as S' jxT^O m developing equation 5 from equation 3. If internal storage changes are 

significant, they tend to dampen the response of the water-table 

fluctuations. Thus the computed values of (As) a/A^ would be relatively small and values of Sv calculated from the model analysis 

would be larger than the values calculated from the more commonly 

used mathematical relations. A very approximate appraisal of this 

effect might be made by comparing the volume pumped during the 

first few minutes of the test with the total change in water stored 

between the water table and the confining bed underlying the aquifer 

as a result of lowered heads in that region. The total change in volume 

of water stored beneath the water table may be expressed in finitedifference form as

AF=2mS"m Jb rAr(As)m, (18)

r»

where (As)m is the average drawdown along the z axis at radius r. The 

coefficient of bulk compressibility of water is about 5X10"5 per 

atmosphere, or about 1.5X10"8 per foot of water (Dorsey, 1940, p. 

245). According to Birch (1942, p. 61) the coefficient of bulk compressibility of the common rock types is about 4 X 10 ~6 per atmosphere, 

or about 1.3X10"7 per foot of water. Thus, for an assumed aquifer 

porosity of 0.3, the internal unit storage coefficient is (very) approximately £'=1.5X0.3X10-6+1.3X0.7X10-7=5.4X10-7. The summation in equation 18 for the data of figure 64 and sw =20 feet was found
228 GROUND WATER HYDRAULICS

by graphical analysis to equal 3,800. Substituting this sum, as well as 

appropriate values for the other terms in equation 18, yields

AF=27rX5.4XH)-7 XlX102 X3.8X103 =1.3 cubic feet.

Evidently, for the conditions postulated in the flow field, the initial 

drawdown distribution could be formed with little change in internal 

storage. The source of greatest error in the above estimate is probably 

the figure for 8'. However, an error of two orders of magnitude in S' 

would at most account for a volume of water pumped from the aquifer 

during only the first 2 minutes. The value of S' used here probably 

is not in error by this much. The effect of internal storage release on 

changes in the water-table position is, therefore, believed to be virtually negligible during the first 10 minutes of pumping.

Aside from consideration of the volumetric changes below the water 

table, the values of Sy summarized in table 3 also lend support to this 

belief. If release from internal storage were damping water-level 

changes appreciably, the calculated values of Sy should show an 

increase with increasing r. This is because (As) n /A^ would become 

progressively smaller than for the case of zero internal storage for 

increasing r. Data in table 3 indicate no such trend in the relation 

between Sy and r.

Use of equation 17 for calculating Sy presumes that (ds/dr)2 is 

very small compared with ds/ds and that ds/ds is small for the time 

interval used in analysis. These are the conditions at the water 

table prerequisite to pass from the conditions of equation 4 to those 

of equation 13 for an isotropic aquifer. The Grand Island data 

may be used directly as an independent check on the validity of 

these approximations. Both ds/d£ and ds/dr may be estimated 

from the test data by finite-difference or graphical methods. Use of 

Wenzel's values of Sy and P (0.2 and 1,000 gpd per sq ft, respectively) 

in a finite-difference form of equation 4, assuming PT=PZ , permits 

a solution for ds/ds. A comparison between the computed value

of ds/ds and -p ^-7 will indicate the importance of the radial-flow

component at the water table. According to Boulton's criteria for 

testing the validity of these approximations, time is the only significant variable. Thus it will suffice to perform the indicated check 

at only one value of r. A radius of 50 feet was selected arbitrarily 

and calculations were made for times of 30, 300, and 2,000 minutes 

after pumping began. An example of the data plots used is given 

on figure 67. Using the finite-difference approximations developed
ELECTRIC ANALOG OF THREE-DIMENSIONAL FLOW 229

20

r, IN FEET 

50

A. Drawdown profile for /=300 minutes

ds 

at

0.40

200 ft per min =2.88 ft per day

-A/=200 min

100 200 300 400 

MN MINUTES

500

B. Drawdown in well 25

600

FIOCTRE 67. Drawdown data used for estimating ds/dr and ds/di at r=50 feet, <=300 minutes. A, Drawdown profile for £=300 minutes; B, Drawdown in well 25.
230 GROUND WATER HYDRAULICS

on figure 67, taking SvfP= 0.2/140 (from Wenzel), and letting (ds/dz) =a 

in equation 4 yield

^X2.88= (0.029) 2+a2 -a

4.12X10-3 -8.4X10-4=3.28X10- 3=a2 -a,

from which a= 3.27 X10~3 or +1.00327. For the boundary conditions considered here (ds/d2)>0 is impossible, and the real solution 

is given by a= 3.27 X10~3 . A similar computation, using

5.36X10*

from the model study results hi the following: 

2.88 X 1.4 X 10~4= (0.029)2+a2 -<z 

4.0 X 10~4 -8.4 X 10~4= -4.4 X 10~4=a2 -a,

from which a=+4.4X10~* Or +0.9996, approximately. Neither 

of these values can be reconciled with the known features of the

Cf

flow field. Thus it is apparent that the -p ratio developed from

model analysis of the data collected during the first few minutes of 

pumping is probably incorrect, at least for £=300 minutes. Sv 

may have been considerably larger than 0.01 at Z=300 minutes. 

Using Wenzel's values for Sy and P, estimates of ds/ds were also 

made for r=50 feet, Z=30 minutes, and Z=2,000 minutes. The data 

employed and computed results are given in table 4. At those times 

for which the numerical values of a are nearly equal to (SvfP) 

AjS/A£, the water-table fluctuations are predominantly influenced by 

vertical-flow components. This can be recognized directly from the 

physical implications inherent in equation 4. To provide some numerical description of the relative effects of vertical or horizontal gradients 

on the rate of change of the water-table position, we may use, for 

example, the ratio (a2 a)/(/StfA is)/PAi. From table 4, for Z=30 

minutes, this ratio equals 0.967, which means that 96.7 percent of the 

rate of change of the water-table position was caused by the vertical 

component of gradient beneath the water table at Z=30 minutes. Or, 

using the complementary ratio, (ds/d/-) 2/('S'tfA is)/PAZ, one finds that 

only 3.3 percent of the rate of change of the water-table position was 

caused by the radial component of gradient beneath the water table. 

As time increases, the radial components of gradient become increasingly more significant. For the data of table 4, the vertical
ELECTRIC ANALOG OF THREE-DIMENSIONAL FLOW 231

component of gradient accounts for 79.4 percent at £=300 minutes 

and 14.6 percent at 2,000 minutes of the rate of change of the watertable position. Thus, even after 30 minutes of pumping, equations 

13 and 17 may be satisfactory approximations of equation 4 for some 

purposes, because radial components appear to have a comparatively

ds small effect on ^ at the water table. However, these comparisons

are highly dependent on the value of Sv and P. If it is assumed that 

SylP= 1.4X 10"*, obtained from analysis of data summarized on figures 

64 and 65, the rate of change of the water-table position appears to be 

controlled equally by radial- and vertical-flow components after pumping for 30 minutes. Information on the value of Sv developed from the 

model analysis is inconclusive. Nevertheless, it seems reasonable to 

conclude that the vertical-flow components were predominant in 

regulating the movement of the water table during at least the first 

10 minutes of pumping, if one restricts his analysis to the use of Sy 

values indicated by the model study. Evidently, for data collected 

during the early part of the Grand Island test, analytical methods 

based on the assumption that vertical-flow components are everywhere 

negligible yield spurious results at all times.

ds TABLE 4. Estimates of ^ at the water table from drawdown data (v=50 ft)

OZ

[All calculations based on <S, and P from Wenzel (1942, p. 122-127)]

t

(minutes)

30__ ________

300____---_-.

2,000_________

Observed 

A,s

A2 

(feet per day)

10.7

2.88

.452

Observed

ArS

Ar

2. 7X10-2

2. 9X10-2

2.35X10-2

Estimated 

ds

52

1.44X10-2

3. 27X10-3

9. 4X10-5

Observed

£BA*s

PA*

1.53X10-2

4. 12X10-8

6. 46X10-*

AQUIFER REPRESENTATION

Among the foregoing analyses, it has been assumed that the aquifer 

is homogeneous and isotropic and is effectively 100 feet thick, and 

the analytical results have been reviewed somewhat with the notion 

that Sy is independently constant in both time and space. Equation 

4, or an appropriate form derived from equation 4, might be utilized 

directly in conjunction with detailed field measurements of 

A^/Ar,Azs/As, and A 4s/Ai for calculating Sv as a function of time. 

The drawdown configuration from the Grand Island test observations is inadequate for this purpose. Thus a satisfactory estimate 

of Sv appears to be impossible to obtain from equation 4. Rather, 

one must rely on the model study for permeability and on a compara-
232 GROUND WATER HYDRAULICS

tively indirect estimate of Sy by utilizing observed values of

as in the development shown in table 3. But here one finds gross

uncertainties.

Obviously, from table 2, the aquifer is stratified to some degree. 

(See also table in Wenzel, 1942, p. 118). The assumed thickness of 

100 feet, heretofore accepted, may be effectively much less. Note 

the section of comparatively low permeability listed between depths 

of 31.4 and 32.4 feet below the water table. If this section is representative of an extensive lense, the pumped well might be considered 

as fully penetrating an aquifer 31.4 feet thick. Such a model representation may be more correct for the first few minutes of pumping 

than the one adopted in developing figure 64. Values of s/sw in 

the profile, obtained from a model by assuming that the pumped well 

fully penetrates an aquifer 31.4 feet thick and that the aquifer is 

isotropic, are shown on figure 68. The data of figure 68, graphically

Assumed screen rw = 1.1 ft 

/ r, IN FEET

IV 100 400

35-

FIGUBE 68. Initial drawdown configuration (<=0+) in terms of slsa, assuming full penetration to depth

of 31.4 feet, and P,=P,.

analyzed by means of equation 16, yield a permeability of about 

630 gpd per sq ft, if one assumes that sw =20 feet. The permeability 

calculated from the model, if one assumes a partial penetration of 

the aquifer 100 feet thick, was about 540 gpd per sq ft. A solution 

for Sv by use of equation 17, which uses data from WenzePs well 

35 at r-=214 feet and the model study of full penetration of an aquifer 

31.4 feet thick, yields $j,= lXlO~4. From table 3, the corresponding 

analysis for partial penetration of the aquifer 100 feet thick gave
ELECTRIC ANALOG OF THREE-DIMENSIONAL FLOW 233

$j,=6.2XlO~3 . Thus the assumption of m=31.4 feet yields a permeability coefficient more nearly in agreement, and Sy value in poorer 

agreement, with Wenzel's results than the modeling assumption of 

TO= 100 feet.

Unconsolidated rocks are almost invariably stratified to some extent, generally along nearly horizontal bedding planes. This produces 

a preferentially high horizontal permeability and a comparatively low 

vertical permeability. Data in table 2 indicate tha,t the aquifer is not 

homogeneous vertically, or in the direction of the z axis. If the entire 

aquifer thickness is considered, the gross permeability to one-dimensional flow along the z axis is only slightly more than 2 gpd per sq ft, 

and to one-dimensional horizontal flow, the gross aquifer permeability 

is about 1,200 gpd per sq ft. Thus results obtamed from the model 

design employed are probably in considerable error, and this error 

can be reduced adequately only by accounting for nonhomogeneities 

in the selection of Rs> values in the resistance network. Unless 

appropriately taken into account, anisotropy due to horizontal bedding of fine materials will also contribute to errors in calculated values 

of permeability and specific yield.

As an example, full penetration to a depth of 31.4 feet was also 

studied by assuming homogeneous anisotropy with Pr/Pz=26. On 

the model, the well-screen radius is represented for this ratio of 

PrIPz at n'=l, as can be found by comparing the computed value of 

Bj from equation 12 with the model-resistance data on table 1. 

From the model study and test data: Pr 1,300 gpd per sq ft, P^=49 

gpd per sq ft, and $j,=2.8X10~3 (from the data of well 35).

Drawdown data might be collected in the field specifically for 

finding the ratio Pr!Pz directly. Drawdowns could be measured in 

wells screened at an array of appropriate grid intersections, indicated 

schematically by closed circles on figure 58. Observations made 

simultaneously at an array of wells positioned as shown with A£, 

rn, and kz known would provide a direct solution for Pr/Pz from 

equation 9. With the ratio Pr/Pz found independently, the model grid 

location for rw is easily ob tamed directly through equation 12. In 

turn, this offers a sound approach to calculating Pz by model analysis 

through equation 16.

A summary of the analog test results obtained for this report is 

given in table 5. The numerical results are not significant hydrologically, but they do serve to emphasize the need for adequately defining 

the test site and analyzing the pumping-test data and model accordingly. Without such knowledge, pumping-test analysis is indeterminate if vertical-flow components are appreciable.
234 GROUND WATER HYDRAULICS

DEPTH, PENETRATION, AND TIMELAQ OP OBSERVATION WELLS

In the preceding analysis of the Grand Island pumping test, data 

from observation wells were assumed to yield correct values of 

(AS)A_I/AS and (As^/A^ without qualification. A comparison of the 

field conditions with the nature of the flow mod si reveals certain inconsistencies that affect the calculated permeability and specific 

yield. A review of these inconsistencies permits identification of 

improved field data-collection practices, designed to fit the electronicmodel analysis of pumping-test information.

TABLE 5. Summary of values of permeability and specific yield calculated from 

model studies of Grand Island, Nebr. pumping-test data, well 85

Assumptions

Well penetrates aquifer, 100 feet thick, to 

depth of 40 feet. Aquifer homogeneous 

and isotropic __ ____ _____

Well penetrates aquifer, 31.4 feet thick, 

to depth of 31.4 feet. Aquifer homogeneous and isotropic. __ ___ ___________

Well penetrates aquifer, 31.4 feet thick, 

to depth of 31.4 feet. Aquifer homogeneous and anisotropic_ ______ _________

Pr

540

630

1,300

p.

540

630

49

Sy

6. 2X10-3

1. OX 10-"

2. 8X10-3

Observation wells employed in the Grand Island test were either 3 

inches or 1 inch in diameter. The 1-inch wells were fitted with 18- 

inch long screened drive points, and the 3-inch wells were open to the 

aquifer only through holes in jetting points attached to the lower end 

of the well casing (Wenzel, 1936, p. 29). The 3-inch wells probably 

had a smaller efficiency for admitting flow from the aquifer to the 

well bore, and they certainly had a higher storage capacity per unit 

depth than the 1-inch wells. Thus it is evident that the 3-inch wells 

would respond more slowly to a sudden change in pressure in the 

aquifer than the 1-inch wells. On figure 66, the hydrograph of well 

25, a 3-inch well, may be compared with the hydrographs of wells 44, 

32, 57, and 45, all 1-inch wells. For these data, differences in timelag 

from an applied pressure change are not noticeable for times of the 

order of 10 minutes or more. Differences in the hydraulic conditions 

found at individual observation-well screens or holes may account 

for a wide range in response characteristics. This may be the reason 

for the great differences between hydrographs of wells 1 and 13 during 

the first 40 minutes of pumping. Both are 3 inches in diameter and 

penetrate the aquifer to a depth of 17 feet below the water table. 

Well 1 was 24.9 feet and well 13 was 29.9 feet from the pumped well. 

Accordingly, if the aquifer is reasonably homogeneous areally, both 

hydrographs should have very nearly the same shape.
ELECTRIC ANALOG OF THREE-DIMENSIONAL FLOW 235

In the model-test analysis, the effect of significant water-level 

timelag would be to greatly increase the calculated value of Sv, as 

can be seen from equation 17. Because the Su values calculated are 

all very much less than 0.2, it appears likely that possible errors due 

to timelag are relatively insignificant. In any event, the timelag 

for the observation wells used by Wenzel cannot be evaluated quantitatively from the data available.

According to the assumptions adopted for test analyses described, 

the water levels in the observation wells should have followed the 

idealized curve ABC of figure 69. The segment AB represents the

B

Decline of head between B and C at 

observation well due to dewatering at 

water table

Decline of head between A and B at 

observation well due to initial drawdown at pumped well assuming release from internal storage is zero

f

FIGUEE 69. Schematic of idealized observation well response.

initial head change due to creating an instantaneous drawdown at the 

pumped well. The length of AB is partly dependent on both the 

radius to, and depth of penetration, of the observation well. For an 

example, refer to figure 64. For well 13, s/s«,=5X10~2. Assuming 

that slf=20 feet, the length of segment AB in a hydrograph for well 

13 would be equal to 20X5X10~2=1.0 foot. The total drawdown 

observed in well 13 after 6 minutes of pumping was 1.86 feet. This 

latter observation was used without modification for estimating 

ds/d£ in calculation of Sy, as given in table 3. However, figure 69 

shows that the correct (As) n should be 1.86 1.00=0.86 and that a 

better estimate of (As) n/A*i would therefore be 0.86X1,440/6=206.4 

feet per day. This correction applied to the data of well 13 would 

increase the calculated value of Sy in table 3 by a factor of about 2. 

For wells located at a greater distance from the pumped well or at
236 GEOUND WATER HYDRAULICS

less depth below the water table, the AB segment of the hydrograph 

is a smaller proportion of the total drawdown to point C. For most 

of the wells listed in table 3, adjustment of the (As) 41 /A^ values for 

segment AB would alter the calculated Sy values by only about 20 

percent.

WATER-LEVEL CONTROL IN PUMPED WELL

In the Grand Island pumping test, the pumped well was controlled 

so as to maintain a virtually constant rate of pumping. Throughout 

the test analysis presented earlier in this report it was assumed that 

head was maintained constant in the pumped well throughout the 

period t^>0. To what extent screen efficiency and disturbance of the 

aquifer materials caused by drilling affected the drawdown near the 

pumped well eannot be surmised. The latter effects were assumed 

negligible in the test analysis. Published data on water levels in the 

pumped well are very fragmentary. Thus it is possible only to gain 

a vague definition of water-level control at the pumped well.

Wenzel (1942, p. 176) listed drawdown data from two observation 

wells, 71 and 72, both near the pumped well (at radii of 2.6 and 12.3 

ft, respectively). Neither their depths nor diameters were given. 

However, if it is assumed that both tapped the aquifer at such depths 

that the maximum initial drawdown would appear in each, it might 

be inferred from figures 64 and 68 that the effective drawdown at the 

pumped well was probably some value between 11 and 14 feet during 

the first 10 minutes of pumping. Also, the hydrographs from each 

well indicate that relatively stable heads were attained in the pumped 

well by the time pumping extended more than 4 minutes. This 

interpretation, however, does not preclude entirely the possibility 

that the effective drawdown was greater than 14 feet.

Although the discharge at the pumped well was controlled, the 

drawdown evidently became nearly constant just a few minutes after 

pumping began. The assumed initial conditions relating to drawdown control are therefore defined satisfactorily for the model analysis 

undertaken. However, the value of sw for calculating permeability 

is in question, and it cannot be resolved with accuracy from the data 

available. Permeability values calculated are directly proportional 

with the assumed sw, but values of Sv calculated by means of equation 

17 are unaffected by the assumed value of sw.

MODEL ADAPTED FOR ANALYSIS OF NONSTEADY FLOW

Pumping-test analysis described in the foregoing sections of this 

report has been based on the concept that, for a short period of time 

after pumping begins, the flow system may be treated as being in the 

steady state. The difficulty of adequately controlling either the draw-
ELECTRIC ANALOG OF THREE-DIMENSIONAL FLOW 237

down or the discharge rate at the pumped well and questionable 

response characteristics of observation wells during the first few 

minutes of pumping lead to gross uncertainties in the data and their 

application to aquifer analysis. Although there are advantages to 

the shorter testing period and the steady-state analysis, where control 

and response are inadequate, the total testing time must be extended 

from a few minutes to several hours. As was demonstrated earlier, 

even after several hours of pumping at the Grand Island site, verticalflow components were still controlling the downward movement of 

the water table. Also, Boulton's criteria suggest that changes in 

potential along the water table need to be considered in constructing 

the flow system as a function of time for test periods several hours 

in duration.

For as long a period as equation 13 remains an adequate approximation of the flow conditions at the water table, the flow system in 

nonsteady state form may be modeled easily, assuming that specific 

yield is constant with time. The nonsteady state model may be 

developed directly from the resistance grid already defined, adding 

electrical capacity to account analogically for the storage characteristics of the aquifer.

An array of nodes near the water table and its equivalent in the 

resistance grid are shown on figure 70. The capacitor, CWT, represents

n-1 n n+l n'- 1 ^ ^ n' n ^ /?'+1

-/=!

2/?f' | 2/?t

Az

ASA. Aquifer B. Analog

FIG-QBE 70. Analogic relations at water table for nonsteady flow. A, Aquifer; B, Analog.

the electrical storage equivalent of Sy . For electrical flow to the 

junction (p'=l } n'\ where radial-flow components are comparatively 

small, current flow to the junction may be expressed in the following 

form:

R*CWT "(V-I-^U (19)

where t' represents elapsed time on the model, and the subscript 

notation is defined by figure 705.
238 GROUND WATER HYDRAULICS

The flow conditions in the vicinity of the corresponding point 

(p=l, ri) in the aquifer can be derived from equation 13, from which:

kzS &s 

Pz d£ v ~ p ~

Comparison of equations 19 and 20 provides the relations between 

aquifer and electrical storage required for model scaling. It will be 

convenient to define two scaling constants for this purpose. One, the 

proportionality between voltage and drawdown has already been 

discussed. It may be defined as

s/e=Keg . (21)

The other, relating the aquifer prototype time, t, to time on the 

model, tf, may be stated as

t/t'=Kt . (22)

By differentiating equation 22, it is easily shown that

X,£-l. (23)

Substituting for s in equation 20, its equivalent expression as given 

by equation 21 yields

and multiplying the left side of equation 19 by equation 23 yields

KtRz,CWT de<y nt) = -(eP ' =l-epf=2)n'. (25) 

Equating 24 and 25, and solving for CWT, gives the relation

farads. (26)

Equation 26 is the final relation sought for determining the analogic 

storage characteristics at each point (p'=l, n') on the model. Within 

limitations of electronic instrumentation and measurement, the 

distribution of values of CWT may be selected inversely proportional 

to Rz> at each node of the resistance grid. The time constant, Kt, is 

then determined by the ratio kzSylPz, which can be found independently by test analysis.
ELECTRIC ANALOG OF THREE-DIMENSIONAL FLOW 239

Data from the Grand Island test site might be used to obtain a 

measure of a practical Kt value to be expected. For this aquifer 

system, use of £^=0.2 and Pz= 100 feet per day (the latter is a fictional 

value) and CW= 1,000 microfarads at .^=0.2 ohm, with A0=5 feet, 

results in

g= 5 X0.2 

1 1X102 X1X10- 3 X0.2

Thus each second on the analog represents 50 seconds of pumping 

time for the conditions assumed.

The release of internal storage might be modeled in similar fashion, 

where the effects from such storage on water-level changes are significant. The equation describing the drawdown interrelations at 

interior points of the flow field may be taken from equation 3 and 

written in finite-difference form as follows:

r A£

- 

*

- (27)

The array of grid intersections at which equation 27 applies is shown 

on figure 71 A

n-\ n+l

n'-l

p+1

Az

_ p

Az

i. P-I

A. Aquifer B. Analog

FIGURE 71. Analogic relations at Interior points for nonsteady flow. A, Aquifer; B, Analog.

The corresponding array on the analog is shown on figure 715, for 

which it can be shown that

~ (28)

Making use of equations 12, 21, 22, and 23 in equations 27 and 28, it 

can be shown that

'KtRz,P> farads. (29)
240 GROUND WATER HYDRAULICS

To provide a homogeneous model with respect to time scaling, the 

value of Kt in equation 29 must be the same as its value in equation 26. 

Thus at each value of £' in the model the relation between d and

(30)

as may be found by dividing equation 29 by equation 26. By again 

using the Grand Island test data for an example, where CWT= 1 X10~3 

farads, ^=0.2, &z=5 feet, and $'=5.4X10~7 , it is seen that

X1X1Q- 3=1.35X10~8=0.0135 microfarad.

According to equations 29 and 30, CWT and d are inversely proportional to Rz>. CWT was assigned a value of 1X 10~3 farads at ^=0.2. 

All other Rs, values are larger than 0.2, and therefore all capacitors in 

the model will be smaller than 1 X10~3 farads. Similarly, the internal 

capacitors would all be less than 0.0135 microfarad as each is inversely 

proportional to R2>. Thus at n=3, for example,

n 1X10- 3 X0.2 00 . . , , CWT= 7 -t^\x-in6 =28 micromicrolarads

i .14y\ 1U

and

i 3*vin-svn 2

-=3.8X10~16 farads

=3.8X10~4 micromicrofarads.

The latter value of d is probably much less than the capacity of the 

wire leads and resistance elements, and therefore it would be impossible 

to model internal storage accurately at the smaller radii without 

modification of the analog design from that already stated. The 

analog designer enjoys a certain amount of latitude for gaining the 

proper balance between component selection, test equipment, and 

prototype flow conditions. For this report, the fundamental equations that may be used as the basis for analog design are presented 

only in general terms. From them, analog grids may be designed to 

fit the particular flow system to be investigated.

Analogic treatment of nonsteady flow may require that the aquifer 

be modeled beyond the radius adopted in the resistance grid described 

earlier. This requirement did not arise in the example solutions,

T

because flow components from regions of large value were negligible;

lib

but it may arise for nonsteady flow analysis, because contributions
ELECTRIC ANALOG OF THREE-DIMENSIONAL FLOW 241

from storage at large values of r/m become increasingly significant to 

the drawdown distribution as time increases. The example resistance

PT r grid was terminated, as indicated on figure 62, at T =49.4 because of

limitations in low-cost resistor availability. In order to remain within 

these limitations, the model may be extended, if required, by changing 

the value of A£ to a smaller value at and beyond one of the larger 

radii. This would permit the use of higher-valued Rz> components of 

lower cost at the larger radii but would require more of them to simulate 

a given radial distance. For study of nonsteady flow such a design 

procedure would also reduce the CWr values and thereby decrease 

unit cost, but it would also require installation of more units. Extending the grid hi this way is similar to the use of "graded nets" 

as described by South well (1946) and is frequently employed in 

numerical analysis.

SUMMARY OF RESULTS

Design features of an electric analog of flow to wells in unconfined 

aquifers, including vertical-flow components, were derived from the 

basic differential equations of ground-water motion. The analog relations developed were applied to construct a resistance grid that was 

used for computing permeability and specific yield from the Grand 

Island pumping-test data of Wenzel (1936, 1942). The method of 

analysis used followed that of Boulton (1954), in which the form of 

drawdown distribution is assumed to be virtually constant for a short 

period of time after pumping begins.

Numerical results from the model studies show that permeability 

and specific-yield estimates are highly dependent on the flow conditions assumed to exist in the aquifer. Specific-yield estimates seem 

much more sensitive to variations in assumed-flow conditions, such 

as degree of penetration of the pumped well, nonhomogeneity, and 

anisotropy, than are the estimates of permeability to flow along 

horizontal planes as obtained by model studies. Adequate pumpingtest analysis by electrical models requires that the degree of anisotropy 

and nonhomogeneity be known independently of model analysis. 

A procedure for observing anisotropy relations directly in the field is 

suggested, for which a minimum of five observation points is required, 

all located in a prescribed pattern. Sufficient drawdown observations 

should be made near the water-table position to provide detailed 

hydrographs at observation wells and the pumped well for at least 

15 minutes after pumping begins. For the steady-state type of test 

analysis described, the pumped well should be operated at constant 

drawdown. For tests of short duration, the well array for measuring 

anisotropy and observation wells for the model analysis should
242 GROUND WATER HYDRAULICS

ordinarily be located at distances from the pumped well in the range 

of approximately 0.1 and 1.0 times the aquifer thickness.

Use of test data from pumping periods of only a few minutes in 

duration provides for good definition of the aquifer characteristics 

regulating vertical flow. However, as a prerequisite to electric-model 

analysis of such data, special pressure measuring devices with rapid 

response characteristics may be needed in place of comparatively 

slow-response observation wells.

Numerical investigations of the drawdown data collected by 

Wenzel (1936, 1942) near Grand Island, Nebr., show^that verticalflow components were predominantly responsible for regulating the 

rate of change of the water-table position several hours after pumping 

began and that specific yield was probably of the order of 0.01 during 

the first few minutes of pumping. This result serves to emphasize 

the futility of calculating permeability and specific yield from analytical expressions in which vertical flow components are assumed to 

be negligible and specific yield is assumed to be constant from 

pumping tests of short duration in unconfined systems.

REFERENCES

Birch, Francis, ed., 1942, Handbook of physical constants: Geol. Soc. America 

Spec. Paper 36.

Boulton, N. S., 1951, The flow pattern near a gravity well in a uniform waterbearing medium: Jour. Inst. Civil Engineers (British), p. 534-550.

 1954, The drawdown of the water table under non-steady conditions 

near a pumped well in an unconfined formation: Proc. Inst. Civil Engineers 

(British), p. 564-579.

Dorsey, N. E., 1940, Properties of ordinary water substance: New York, Reinhold Publishing Corp.

Ferris, J. G., 1949, Ground water, Chapter 7 in C. O. Wisler and E. F. Brater, 

Hydrology: New York, John Wiley and Sons.

Ferris, J. G., Knowles, D. B., Brown, R. H., and Stallman, R. W., 1962, Theory 

of aquifer tests: A summary of lectures: U.S. Geol. Survey Water-Supply 

Paper 1536-E, p. 69-174 (in press).

Jacob, C. E., 1950, Flow of ground water, Chapter 5 in Hunter Rouse, Engineering 

Hydraulics: New York, John Wiley and Sons.

Kirkham, Don, 1959, Exact theory of flow into a partially penetrating well: Jour. 

Geophys. Research, v. 64, p. 1317-1327.

Southwell, R. V., 1946, Relaxation methods in theoretical physics: London, 

Oxford Univ. Press.

Theis, C. V., 1935, The 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 Trans., p. 519-524.

Wenzel, L. K., 1936, The Thiem method for determining permeability of waterbearing materials: U.S. Geol. Survey Water-Supply Paper 679-A.

 1942, Methods for determining permeability of water-bearing materials: 

U.S. Geol. Survey Water-Supply Paper 887.

O