Constant-Head Pumping Test 

of a Multiaquifer Well to 

Determine Characteristics 

of Individual Aquifers

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1536-G

Prepared in cooperation with the 

Pennsylvania Geological Survey, 

Department of Internal Affairs, 

Commonwealth of Pennsylvania
Constant-Head Pumping Test 

of a Multiaquifer Well to 

Determine Characteristics 

of Individual Aquifers

By GORDON D. BENNETT and EUGENE P. PATTEN, JR.

GROUND-WATER HYDRAULICS

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1S36-G

Prepared in cooperation with the 

Pennsylvania Geological Survey, . 

Department of Internal Affairs, 

Commonwealth of Pennsylvania

U. S. Geological Survey

[^J2I£J1\\//P i

[ DEC - 3 1964 j

jVlashville, Tennessee

L

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1962
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

Reprinted 

1964

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

Washington, D.C., 20402 - Price 15 cents (paper cover)
CONTENTS

Abstract________________________________________________________ 181

Introduction _________..__-_______-______-__-_--__---_______________ 181

Review of earlier work_____________________,_______________________ 182

Conditions in a multiaquifer well___________-__-___-_________________ 185

Behavior of the well during construction_______________________ 185

Behavior of completed well under condition of no withdrawal-______ 185

Behavior of the well when pumped__^ _______________ _________^__ 1,87

Application of step-drawdown analysis to the problem________________ 187

Period from completion of well to beginning of test ________________ 187

Test pumping of well at constant drawdown._____________________ 193

Substraction of original discharge______________________________,_ 198

Method of analysis of test data._______________________________ 196

Field procedures and quality of results._______-____--__--__________-_ 198

General evaluation of the test method______________________________ 199

Other possible methods of obtaining data on individual aquifers _________ 202

References _.______.___________-________-_-_____-_--_______-_-_--__ 203

ILLUSTKATIONS

Page 

FIGURE 50. Diagram showing conditions in a typical three-aquifer well

shortly after completion_______________________________ 186

51. Graphs showing fluctuations of artesian head in the upper 

aquifer of the three-aquifer well of figure 50, as the two 

lower aquifers are penetrated----_-_-_--__-_---_-___---- 189

52. Graphs showing fluctuations in artesian head in the middle 

aquifer of the three-aquifer well of figure 50, during penetration of middle and lower aquifers. __-_-____________.-__- 191

53. Graphs showing fluctuations in artesian head in the lower

aquifer of the three-aquifer well of figure 50________ ______ 192

54. Graphs showing fluctuations in artesian head in the threeaquifer well of figure 50______________________________ 194

55. Diagram showing entrance losses in a multiaquifer well__-_ 201

TAkLE

Page

TABLE 1. Values of G($) corresponding to values of in the range 10~4 to 10" 184

in
GROUND-WATER HYDRAULICS

CONSTANT-HEAD PUMPING TEST OF A MULTIAQUIFER

WELL TO DETERMINE CHARACTERISTICS OF

INDIVIDUAL AQUIFERS

By GORDON D. BENNETT and EUGENE P. PATTEN, JR.

ABSTRACT

This report describes the theory and field procedures for determining the transmissibility and storage coefficients and the original hydrostatic head of each 

aquifer penetrated by a multiaquifer well. The procedure involves pumping the 

well in such a manner that the drawdown of water level is constant while the discharges of the different aquifers are measured by means of borehole flowmeters. 

The theory is developed by analogy to the heat-flow problem solved by Smith. 

The internal discharge between aquifers after the well is completed is analyzed as 

the first step. Pumping at constant drawdown constitutes the second step. 

Transmissibility and storage coefficients are determined by a method described 

by Jacob and Lohman, after the original internal discharge to or from the aquifer 

has been compensated for in the calculations. The original hydrostatic head of 

each aquifer is then determined by resubstituting the transmissibility and storage 

coefficients into the first step of the analysis.

The method was tested on a well in Chester County, Pa., but the results were 

not entirely satisfactory, owing to the lack of sufficiently accurate methods of 

flow measurement arid, probably, to the effects of entrance losses in the well. 

The determinations of the transmissibility coefficient and static head can be 

accepted as having order-of-magnitude significance, but the determinations of the 

storage coefficient, which is highly sensitive to experimental error, must be rejected. It is felt that better results may be achieved in the future, as more reliable 

devices for metering the flow become available and as more is learned concerning 

the nature of entrance losses. If accurate data can be obtained, recently developed 

techniques of digital or analog computation may permit determination of the 

response of each aquifer in the well to any form of pumping.

INTRODUCTION

It is often desirable to know the coefficients of transmissibility (T) 

and storage (8) and the original hydrostatic head of each aquifer 

penetrated by a multiaquifer well. A conventional pumping-test 

(aquifer-test) analysis fails to give this information; the most that 

can be expected from a conventional test is the determination of 

composite values of transmissibility, storage, and hydrostatic head 

for all aquifers tapped by the well. Such results clearly have limited

181
182 GBOUND-WATER HYDRAULICS

significance. This paper describes an aquifer-test technique designed 

to determine these constants for each aquifer penetrated by a well. 

Both the testing procedure and the method of analysis are more 

complex than those used in, a conventional test, and the results are 

not yet satisfactory; however, refinements may be made in the testing equipment that will improve the accuracy of the test results and 

thus make ttxe test a very useful tool in ground-water hydrology.

The research described in this paper was carried out by the U.S. 

Geological Survey under a cooperative program with the Pennsylvania Topographic and Geologic Survey. It constitutes one phase 

of an investigation into the application of borehole geophysics to

hydrology,

REVIEW OF EARLIER WORK

The equation of ground-water flow, as given by Jacob (1950), for 

two-dimensional radial flow t0 a well at the origin is

where k represents head, or elevation of the piezometric surface above 

datum; $ is the storage coefficient of the aquifer; T is the transmissibility of the aquifer; r is the radial coordinate; and t is time.

An analogy exists between the theory of ground-water flow and 

the theory of beat conduction in solids. Recognition of this analogy 

was of great assistance in formulating the mathematical treatment 

of ground water flow. The mathematical analysis developed for a 

given problem in heat flow lias often been applied directly to the 

parallel problem in ground-water flow. The thermal analog of equation 1 may be obtained by substituting the inverse of thermal diffusivity for S/T, and temperature for h. The resulting equation deals 

with the problem of two-dimensional flow of heat in a slab of conducting material, with radial symmetry about the origin. The analogy 

may be further clarified by noting that transmissibility is mathematically equivalent to the product of thermal conductivity and slab 

thickness and that the storage coefficient is mathematically equivalent 

to the product of heat capacity per unit volume and slab thickness.

Smith (1937) presents & solution to the problem of heat flow in an 

infinite solid bounded internally by a very long cylinder, when the 

temperature at the cylinder wall is a step function of time. More 

precisely, Smith's solution deals with the problem in which (letting u 

represent the temperature at the cylinder wall)

and so forth.
CONSTANT-HEAD PUMPING TEST, MULTIAQUIFER WELL 183

It is assumed that prior to time to the temperature UQ prevails throughout the solid.

Smith's results may be stated in the notation of ground-water 

theory to obtain a quantitative analysis of the problem in which the 

water level in a well varies as some step function of the time. Let 

ho, hi, h2 , . . . hn represent the succession of water levels which 

make up the step function, so that

hv> =h2 , fi<<<<2 and so forth;

and let it be assumed that the well penetrates, at the origin of coordinates, an infinite artesian aquifer throughout which the head hQ 

prevails prior to time tQ . It can be shown from Smith's analysis that 

during the stepwise variation, at a time t, such that <(«_D<<<i«, the 

radial derivative of head at the cylindrical contact between borehole 

and aquifer is given by

where the function G is defined by

(2)

(3)

Tf JX = 0 {* «/oV

in which

J0 (^0=Bessel function of zero order of the first kind;

function of zero order of the second kind.

The volumetric rate of flow, Q(£), from the aquifer into the well at 

any time t is related to the derivative at that time by the equation
184 GROUND-WATER HYDRAULICS

Substitution of the expression from equation 2 for the derivative into 

equation 4 gives the following expression for Q(t):

(5)

Equation 5 provides a means of computing the discharge from an 

aquifer to a well, or vice versa, at any instant, as a result of any pattern 

of changes in head that can be approximated as a series of steps, provided that the head h0 prevails throughout the aquifer at time t0.

Jacob and Lohman (1952) first applied Smith's solution to a 

hydrologic problem. They used the first step in his analysis to obtain 

the discharge of a flowing well suddenly opened to atmospheric 

pressure, after being closed long enough to permit the aquifer head 

to reach a uniform level. Thus to was taken as the instant of opening 

of the well, and was set equal to zero. This led to the relation

for the time-varying well discharge.

{

Tt (6)

TABLE 1. Values of G (0) for values of & from 10~* to 1012 

[From Jacob and Lohman, 1952, p. 561]

No.

I., . .... .

2..... ............

Z.. ..................

4.. ................ ..

5... _ ..............

6. _ ...... .......

T. ...................

^.. ................. .

^. ...................

10....... ............

No.

\. ...................

^.. ..................

3. ...................

4.. ___ . ____ ....

5..... ............

Q. ...................

T. ...................

B.. ..................

9... ..............

10.. ..............

10-*

*tf Q

40 4

33.1

28.7

oe 7

23.5

21.8

20.4

1Q ^

18.3

10*

0 1QA4.

1SU1

.1777

.1733

.1701

.1675

.1654

.1621

IfiHft

10-3

1 Q O A

10 1 1

10 79

Q 41

ft 47

7.77

7.23

6

7Q

6.43

6.13

10«

0 1 ftftft

.1524

1479

1449

.1426

Idfift

.1380

1QCQ

IQfifl

10-2

6.13

3.74

a on

3 on

2 7ft

2 flft

2 dfi

O QE

2 OK

106

0. 1360

19QQ

.1266

.1244

1997

191^1

.1202

1 1Q9

.1184

.1177

10-1

2 04.0

1.716

1.477

1.333

1.234

1.160

1.103

1.057

1.018

10?

0. 1177

.1131

.1106

.1089

.1076

.1066

.1057

indQ

.1043

.1037

1

0.985

ono

71Q

.667

.630

fif»9

.580

.562

U7

.534

108

0. 1037

.1002

flQQO

.0968

.0958

.0950

.0943

.0937

.0932

.0927

10

0.534

461

d">7

.405

QQQ

.377

.367

.359

.352

.346

10»

0.0927

.0899

.0883

.0872 nftfid

.0857

.0851

.0846

.0842

.0838

102

0.346

.311

.294

.283

974

.268

.263

.258

.254

.251

10"

0.0838

.0814

.0801

.0792

.0785

.0779

.0774

.0770

.0767

.0764

103

0.251

.232

.222

.215

.210

.206

.203

.200

.198

.196

10"

0.0764

.0744

.0733

.0726

.0720

.0716

.0712

.0709

.0706

.0704
CONSTANT-HEAD PUMPING TEST, MULTIAQUIFER WELL 185

Jacob and Lohman present a table of values of the G function for 

a wide range of values of its argument. This table was obtained by numerical methods of approximation, as the 6? function cannot be integrated directly. Although used by Jacob and Lohman in a single-step 

analysis, the values of the G function given by them have general 

application, and are used in the present paper in an analysis of more 

than one step. The values of G(fi) for values of ft ranging from 10~4 

to 10 12 given by Jacob and Lohman are reproduced in table 1.

CONDITIONS IN A MULTIAQUIFER WELL 

BEHAVIOR OP THE WELL DURING CONSTRUCTION

During construction of a multiaquifer well several adjustments in 

water level normally occur. Consider, for example, a three-aquifer 

well (fig. 50), in which the undisturbed head of the upper aquifer is 

hu, that of the middle aquifer is hM, and that of the lower aquifer is 

hL . Let it be assumed that hu^>hL ^>hM. When the well has penetrated only the upper aquifer, the water level stands at hv. As the 

drill enters the middle aquifer, the water level begins to fall and flow 

commences between the aquifers. A cone of depression develops in 

the piezometric surface of the upper aquifer and a cone of elevation 

develops on that of the middle aquifer. The water level in the well 

normally stabilizes at some value between hv and hM within a relatively short time after the middle aquifer has been fully penetrated.

A further adjustment in water level occurs as the lowermost aquifer 

is entered. If the balance level between the heads of the upper and 

middle aquifers lies above the head of the lower aquifer, a drop in 

water level will occur, and flow into the lower aquifer will commence. 

If the balance between the upper and middle aquifers lies below the 

static head of the lower aquifer, a rise in water level will occur as flow 

from this aquifer enters the well. The latter situation is shown in 

figure 51. In either situation, however, the water level normally 

stabilizes at some new position within a relatively short time after 

the lower aquifer has been fully penetrated.

BEHAVIOR OF COMPLETED WELL UNDER CONDITION OP NO

WITHDRAWAL

After the well is completed, the internal flow between aquifers 

generally continues indefinitely at a rate that diminishes slowly with 

time. If the well is assumed, for simplicity, to be free from periodic 

changes in head due to recharge to or discharge from one or more of 

the aquifers elsewhere, then significant changes in head are not 

possible after the early part of the well's existence. For the period 

of undisturbed internal flow between aquifers, the time derivative

644518 0 62 2
186 GROUND-WATER HYDRAULICS

Stabilized water level after 

penetration of 3 aquifers

EXPLANATION

hv Static head of upper aquifer

hM Static head of middle aquifer

hL Static head of lower aquifer

Surface

Middle aquifer

Lower aquifer

FIGURE 50. Diagram showing conditions in a typical three-aquifer well shortly after

completion.
CONSTANT-HEAD PUMPING TEST, MULTIAQUIFER WELL 187

dA/di must approach zero as time increases, and the head must always 

remain between the limits hv and hM- Thus the assumption that, 

for the condition of no withdrawal, the head stabilizes at a constant 

level a short time after the well is completed is a close approximation 

to the actual conditions in almost all wells. This condition can be 

checked easily by making periodic water-level measurements after 

the well is completed.

BEHAVIOR OF THE WELL WHEN PUMPED

Pumping the well produces a new set of conditions which may be 

summarized as follows: An equal change in head is applied to each 

aquifer over a given period of time. The flows entering or leaving 

each aquifer are changed from their values prior to pumping in such 

a way that the algebraic sum of these changes in discharge is essentially equal, at any instant of time, to the discharge of the well at 

that instant. In adjusting to pumping conditions, aquifers that were 

discharging to the well prior to pumping will increase then- rate of 

discharge, while aquifers that were "thieving" from the well will 

decrease their rate of thieving and may even reverse flow completely 

and begin discharging.

The conditions outlined above for the discharge of a multiaquifer 

well are generally true regardless of the method of well discharge 

that is, whether the well is discharged at a constant rate, at a constant drawdown, or in some other manner. For a further discussion 

of the internal hydraulics of a multiaquifer well during discharge, the 

reader is referred to Bennett and Patten (1959).

APPLICATION OF STEP-DRAWDOWN ANALYSIS TO THE

PROBLEM

PERIOD FROM COMPLETION OF WELL TO BEGINNING OF TEST

Analysis of the multiaquifer problem by the methods proposed here 

is possible only insofar as the changes in head involved can be approximated by step functions. The analysis would become complex, 

moreover, if it were not possible to represent the interval from completion of the well to pumping as a single step. Therefore, the 

applicability of the step-function approximation must be considered.

Referring to the three-aquifer well shown in figure 50, let t0 denote 

the time at which the middle aquifer is penetrated during drilling of 

the well. Let ta denote a time after penetration of the middle aquifer 

but prior to penetration of the lower aquifer. Similarly, let fa represent the time at which the lower aquifer is penetrated by the drill, 

and let tc denote a time after penetration of the lowermost aquifer 

but prior to pumping of the well. Finally, let ti represent the time 

when the pumping test is begun, and let it be assumed that no pumping of the well occurs prior to ti. The various times, as defined here.
188 GROUND-WATER HYDRAULICS

are indicated on the abscissas of the graphs in figures 52-55. In order 

to facilitate the discussion of the fluctuations in head in each aquifer 

during drilling, the point (that is, the map location) at which the well 

penetrates each aquifer will be referred to as the origin of coordinates 

(or simply the origin) in that aquifer.

The changes in head in the upper aquifer at the origin during 

drilling are identical with the changes in the water level in the well. 

They are summarized in figure 51 A At time t0 the head begins to 

decline as penetration of the middle aquifer begins; by the time ta it 

has reached a nearly stable position. At time £&, when the lowermost aquifer is penetrated, the head begins to rise, as it is assumed in 

this well that the static head of the lower aquifer is greater than the 

equilibrium head between the upper and middle aquifers. By the 

time t c , the head has reached a new stable position. Figure 5lB 

shows a stepwise approximation to the changes in head shown in 

figure 51-4, and figure 5 1C" shows a single-step change from the original 

head of the upper aquifer to the pretest equilibrium level of the completed well. It is assume(J in this analysis that the time interval 

t0-ti is very much greater than the interval t0-t c, as it would be if the 

tests were not begun until several weeks after completion of the well.

The fluctuations in head shown in figure 51-4 and B presume a 

discharge rate from the upper aquifer that at time ti has reached a 

value that can be calculated by an application of equation 5, as 

follows:

(7)

The subscript £7 is used to indicate the properties of the upper aquifer. 

The time interval from t0 to ti is long relative to the time interval 

from to to th , and the quantity (£: £0 ) must therefore be relatively close 

in value to the quantity (ti tb ). Examination of table 1, moreover, 

indicates that at large values of ft the change in G(ff) for a given 

change in 0 is small. Thus, as the quantities (ti 10 ) and (ti th) are 

both large, and the difference between them is small, the quantities

u(t\ to)\ i ~ CTu(t\ tb)\ i . j j , i 

o 2 r and " X 5 o - > can be considered to be essen- S>» J \ Svr* J

tially equal. Equation (7) can therefore be rewritten as

(S)

which simplifies to

(9)
CONSTANT-HEAD PUMPING TEST, MULTIAQUIFER WELL 189

*,--

EXPLANATION

fej; Static head of upper aquifer

h' Stabilized water level after penetration of middle

	aquifer

hi Stabilized water level after completion of well

t 0 Time of penetration of middle aquifer

t a Time shortly after penetration of middle aquifer

t b Time of penetration of lower aquifer

t c Time shortly after penetration of lower aquifer

tt Time of start of test

h,--

FIGUEE 51. Graphs showing fluctuations of artesian head in the upper aquifer of the three-aquifer well of 

figure 50, as the two lower aquifers are penetrated: (A) water-level fluctuations, (B) stepwise approximation, (C) single-step change from original head to equilibrium.
190 GROUND-WATER HYDRAULICS

Equation 9 is exactly the expression that would be obtained from 

an application of equation 5 to the single-step drawdown illustrated 

in figure 51C. Thus it is established that the drawdown pattern of 

the upper aquifer, as illustrated in figure 51A, can be closely approximated by the single-step drawdown illustrated in figure 51(7, provided 

only that the interval from completion of the well to time of measurement is much greater than the time spent in drilling the well. In 

other words, minor changes in head occurring early in an interval of 

time may be neglected, if they are followed by a long period of stability.

The changes in head in the middle aquifer at the origin of coordinates 

(at the point of well penetration) are shown in figure 52A. It is 

assumed that penetration of the middle aquifer occurs instantaneously 

at time t0, and that prior to to the middle aquifer is isolated hydraulically from the borehole that is, that there is no gradually increasing 

leakage to the well through the material separating the middle aquifer 

from the bottom of the borehole as the well is drilled toward this 

aquifer. Under these assumptions, the head hm will prevail at the 

origin, as well as throughout the rest of the middle aquifer, prior to 

to. At the moment of penetration, however, the head in the well, 

resulting from the upper aquifer, will be instantaneously applied to 

the middle aquifer at the origin; this water level is shown as hu. Thus 

the graph of figure 52.4 shows an instantaneous rise from hm to hu 

occurring at to. Flow from the well into the middle aquifer begins 

immediately, however, and the water level begins to fall toward its 

new equilibrium level, hf . By the time ta the water level has essentially stabilized at this new level.

Once the well has penetrated the middle aquifer, the head of this 

aquifer at the origin is identical with the water level in the well; 

thus, the fluctuations in head after time t0 in figure 52A are simply 

the variations in water level in the well. At time fa the lower aquifer 

is penetrated, and flow begins from this aquifer into the well. The 

resulting rise in water level is shown in figure 52.4 as an increase in 

the head of the middle aquifer at the origin. By time tc this head 

has stabilized again to a new equilibrium at hi.

In discussing the fluctuations in head of the upper aquifer, it was 

demonstrated that minor variations in head occurring early in an 

interval can be neglected if they are followed by a long period of 

stability. Application of this principle to the fluctuations in head 

shown in figure 52.4 leads to the single-step approximation of figure 

52B, as the interval ti~t0 is assumed to be long relative to the interval 

te-t0.

Figure 53A shows the variations in head of the lowermost aquifer 

at the origin during drilling. Again it is assumed that the aquifer is 

hydraulically isolated from the well until the time of its penetration,
CONSTANT-HEAD PUMPING TEST, MULTIAQUIFER WELL 191

h'--

EXPLANATION

h v Static head of upper aquifer

hM Static head of middle aquifer

h' Stabilized water level after penetration of middle

	aquifer

h, Stabilized water level after completion of well

ta Time of penetration of middle aquifer

ta Time shortly after penetration of middle aquifer

tb Time of penetration of lower aquifer

te Time shortly after penetration of lower aquifer

t, Time of sta rt of test

h,--

FIGURE 52. Graphs showing fluctuations in artesian head in the middle aquifer of the three-aquifer well 

of figure 50 during penetration of middle and lower aquifers: (A) change in head during completion of 

well and (B) stepwise approximation of these changes.
192 GROUND-WATER HYDRAULICS

hih' --

EXPLANATION

hL Static head of lower aquifer

h' Stabilized water level after penetration of middle

	aquifer

h{ Stabilized water level after completion of well

t0 Time of penetration of middle aquifer

t b Time of penetration of lower aquifer

t c Time shortly after penetration of lower aquifer

t, Time of start of test

h'--

B

FIGURE 53. Graphs showing fluctuations in artesian head in the lower aquifer of the three-aquifer well of 

figure 50: (.4) Changes in head during completion and (B) stepwise approximations of these changes.
CONSTANT-HEAD PUMPING TEST, MULTIAQUIFER WELL 193

tb . Until this time, therefore, the head at the origin in the lower 

aquifer is hL, the static head of this aquifer. At the moment of 

penetration, the aquifer is instantaneously exposed to the head h' 

that is, to the water column in the well. This is indicated in figure 

53.4 by the sharp drop in head at th. The rise in head after tb reflects 

the rise in water level in the well, due to the flow that begins from 

the lower aquifer at the instant of penetration. Again, since tc t0 

is assumed to be short relative to t\ 10, the single-step approximation 

of figure 535 may be used to represent the fluctuations shown in 

figure 53A

The assumptions of instantaneous penetration and complete 

hydraulic separation until the moment of penetration do not represent conditions normally found in practice. However, the effects of 

gradual penetration and interconnection usually make the actual 

changes in head in the aquifers even more similar to the stepwise 

approximation of figures 525 and 535 than the graphs of figures 52.4 

and 53^4 would indicate.

TEST PUMPING OP WELL AT CONSTANT DRAWDOWN

The change in head in each aquifer at the well, following completion 

of the well, can be approximated, as indicated, as a single-step change. 

A second-step change of head can be imposed on each aquifer at the 

well by beginning a constant-drawdown pumping test at time t\. 

Figure 54^1 shows the drawdown created by such a test; figure 545 

is a composite graph showing the. relation of this drawdown to the 

changes of head in the various aquifers during drilling; and figure 

54£7 shows the corresponding composite stepwise approximation. 

Neglecting entrance and well-bore losses, the same change of head is 

applied to each aquifer. Recording the discharges of the various 

aquifers in response to this change of head should, therefore, permit 

computation of the coefficients of transmissibility and storage. The 

discharge from the upper aquifer at a time t during this constant 

drawdown test can be calculated by a two-step application of equation 

5, that is

) - (10)

SUBTRACTION OP ORIGINAL DISCHARGE

If the time of measurement, t, is chosen a few hours after the 

beginning of the pumping test, whereas the time since completion of 

the well is measured in days or weeks, the difference between the 

intervals (t to) and (t1 to) will be small relative to the total lengths

644518 O 62 3
194 GROUND-WATER HYDRAULICS

feyEXPLANATION

Static head of upper aquifer

Static head of middle aquifer

Static head of lower aquifer

Stabilized water level after penetration of middle aquifer

Stabilized water level after completion of well

Water level during constant-drawdown test

Time of penetration of middle aquifer

Time shortly after penetration of middle aquifer

Time of penetration of lower aquifer

Time shortly after penetration of lower aquifer

Time of start of test

B

*,-.-

Upper aquifer

Lower aquifer

Middle aquifer

All aquifers

FIGURE 54. Graphs showing fluctuations in artesian head in the three-aquifer well of figure 50: (A) 

drawdown pattern in a well during constant-head pumping test, (B) composite graph showing 

changes in head in all aquifers during completion of the well and constant-drawdown test, (C) 

final stepwise approximation to these changes.
CONSTANT-HEAD PUMPING TEST, MULTIAQUIFER WELL 195

{

T (I I )"\ ~^ci 5-^~ r

and 6 < 

rr«-mr/i ] 2 > will, accordingly, be negligible, and equation 10 

can be modified by substitution to

From equation 9, however, it can be seen that the first term on the 

right in equation 11 is equal to Qv (t\), the internal discharge from the 

upper aquifer immediately prior to the beginning of the test. Substituting and transposing then gives

The discharge from the upper aquifer at time t consists of two 

parts one due to the first-step change in head occurring at completion 

of the well and the other due to the second-step change created by the 

pumping test.

It can be assumed that at the time of measurement the component 

of discharge due to the first step will be virtually equal to the value 

at the beginning of the test. This is true because the time difference 

(t ti) is small relative to (t± to) and because the discharge due to the 

first-step change in head has been in progress for a considerable length 

of time and is, therefore, changing at a slow rate. .Thus subtraction 

of the discharge due to the first step isolates the component which is 

due to the second step, and this component can then be equated to the 

expression corresponding to the second step for purposes of test 

analysis.
196 GROUND-WATER HYDRAULICS

It should be noted that measurements must be taken at several 

times during the test, and the assumption that the discharge due to 

the first step has not changed appreciably since the start of the test 

must be true for the entire test interval. If this component of 

discharge is changing significantly with time at the beginning of the 

test, its trend may be established by taking successive measurements 

prior to pumping. Extrapolation will then give the value of this 

component corresponding to any later time, and subtraction will 

isolate the component corresponding to the second step of drawdown.

METHOD OF ANALYSIS OF TEST DATA

The test data will consist of records of aquifer discharge as a function of time for each aquifer penetrated by the well. Jacob and Lohman (1952) present two graphical methods of solution that may be 

applied to these records to compute the transmissibility and storage 

coefficients for the various aquifers.

The first method consists of plotting discharge (or, in this example, 

the net change in discharge obtained by subtracting the original internal discharge) against time on logarithmic paper. This plot is then 

overlain upon a plot of 6r(/3) versus (/3) on logarithmic paper of the 

same scale. The curves are adjusted until the best possible fit or 

match is obtained while keeping the coordinate axes of the two graphs 

parallel. With the graphs aligned in this manner, a constant difference between the values of [6r(/3)] and (A$) is indicated by the 

separation of the horizontal axes of the curves. Similarly a constant 

difference between the values of /3 and t is indicated by the separation 

of the vertical axes. The significance of these constant differences 

can be seen by using logarithms in the equations

(13) 

and

That is,

log AQ=log prr(A,-A,)]+log 0(0) , (15) 

and

log <0)=log (J^)+log (*-*,). (16)

The separation of the horizontal axes of the superposed logarithmic 

plots is thus proportional to log 2irT(hi h2 ), while the separation of 

the vertical axes is proportional to log (T/Srw2 ). When the curves
CONSTANT-HEAD PUMPING TEST, MULTIAQUIFER WELL 197

are properly alined, therefore, the coordinates of any pair of overlying points may be used in equations 13 and 14 to compute T and 

S for the aquifer.

The second method is based upon the fact that when (t t^ is large,

{

T 1 ~o~i2 (l ti) r is about equal to the function 

orw J

2

2.',

This approximate equivalence is demonstrated by Jacob and Lohman 

(1952) by comparison of the solution for constant drawdown with 

the solution corresponding to constant rate of discharge. Substitution of the expression indicated above for the G function in the 

equation for net discharge change gives:

2.;

Equation 17 can be rearranged to read

1 2.3 , 2.25T. 2.3 , ,. Iog (t~

Differentiating equation 18 gives

2.3

a log (*- ( k }'

The term on the right side of equation 19 is a constant. Thus if 

values of -r^ are plotted against values of (t ti) on semilogarithmic

paper, the slope of the plot should become constant once the abscissa, 

(t ti), enters the range of values for which equation 17 applies. 

The second graphical solution is obtained by the following procedure. Values of -r^ for the aquifer concerned are plotted against

values of (t t'} on semilogarithmic paper. When these values begin 

to define a straight line, the test has entered the region in which 

equation 17 applies. Measurement of the slope of this line then
198 GROUND-WATER HYDRAULICS

leads to a determination of the coefficient of transmissibility, 

because this slope may be substituted for

d(^

in equation 19. Once T is known, it can be substituted, together 

with the coordinates of any point on the semilogarithmic plot, into 

equation 17 for a determination of the storage coefficient ($).

When the transmissibility and storage coefficients of an aquifer 

have been determined, the original hydrostatic head of the aquifer 

may be calculated by an application of equation 9. The discharge 

value employed is the original internal discharge from the aquifer, as 

measured prior to pumping. The water level in the well at the start 

of pumping (hi), the time interval from completion of the well to the 

start of the test (t\ to), and the appropriate values of T, 8, and rw 

are also inserted to obtain a solution for the original head (hv in the 

notation of equation 9). This is a substitution of the newly determined 

values of T and 8 into the first step of the pattern of changes in head 

(sketched in figures 51-53) in order to find the height of the first 

step. The asymptotic expression

may be used in place of the Q function in equation 9, if this seems to 

be more convenient.

FIELD PROCEDURES AND QUALITY OF RESULTS

In order to carry out a test of the sort under discussion, the following field procedures are necessary: (1) lithologic or geophysical logging 

of the well, to locate the possible aquifers, (2) measurement of the 

water level in the well prior to pumping, (3) measurement of the internal discharge to or from each aquifer prior to pumping, (4) pumping 

of the well at a measured constant drawdown, and (5) measurement 

of the internal discharge to or from each aquifer, as a function of 

time, during the constant-drawdown test.

The method was tested by the authors on a well in Chester County, 

Pennsylvania. Electric logging of the well indicated two major 

sandstone aquifers separated by a considerable thickness of shale. 

The internal discharge between the aquifers before pumping was 

measured by releasing a slug of brine in the region between the 

aquifers and tracing its movement with a fluid-resistivity logging
CONSTANT-HEAD PUMPING TEST, MULTIAQUIFER WELL 199

device. The velocity of the brine slug multiplied by the borehole 

area (as determined from caliper logs) provided the measurement of 

internal discharge. The discharge was from the lower aquifer to 

the upper aquifer and was about 8 gpm.

The pump used in the test was a small centrifugal model powered 

by a gasoline engine. A 34nch globe valve in the discharge line 

provided a means for controlling the rate of pumping. A float-operated water-level recorder was placed on the well, and a constant 

drawdown was obtained by controlling the pump manually while 

observing the recorder. A drawdown of 8.08 feet was obtained during 

the first three minutes of the test. This drawdown was not allowed 

to vary more than 0.15 foot during the remainder of the 6-hour test.

Measurements of discharge during the test were made frequently, 

both at the surface and, by use of a Au borehole flowmeter, in the 

interval between the two aquifers. The Au-meter measurements provided a record of the discharge of the lower aquifer, a,nd subtraction 

of this discharge from the discharge of the well provided a record of the 

discharge of the upper aquifer. Algebraic subtraction of the original 

internal discharge provided a record of A$ as a function of time.

The data were analyzed by using the semilogarithmic-plot method 

of Jacob and Lohman (1952) to obtain values of S and T, and by 

using an equation of the form of equation 9 (but substituting the 

asymptotic expression for the 0 function) to calculate h. The results 

were poor and indicate that the method ia unsatisfactory at present 

for accurate determination of the aquifer constants.

The results indicate that determinations of transmissibility and 

static head, using the field procedures employed by the authors, may 

be* accepted as having order-of-magnitude significance. The determination of storage coefficient, which is highly sensitive to experimental error, cannot be accepted at all. The poor results are due to 

insufficient accuracy of measurement and to deficiencies in the theory. 

These topics are discussed in the following section.

GENERAL EVALUATION OF THE TEST METHOD

The method presented in this report leads to estimates of the 

transmissibilities and static heads of the various aquifers penetrated 

by a well; determination of storage coefficients may become feasible 

as the method is refined. Also it may sometimes be possible to 

recognize hydrologic boundaries in a given aquifer from the test 

results; a recharge boundary should be indicated by a levelling off 

in the aquifer's discharge, and an impermeable boundary should be 

indicated by an abnormal decrease in its discharge.

At present a number of difficulties detract from the utility of the 

method. The doubtful accuracy of the flow measurements is a
200 GROUND-WATER HYDRAULICS

serious problem because, even when the conditions assumed in the 

theory are fully met, reliable results cannot be achieved without an 

accurate means of measuring flow in the borehole and at the surface. 

Basically, however, this is a problem of developing better instruments, 

and it is to be expected that future development along these lines 

may minimize such sources of error.

The method is also subject to the limitations inherent in conventional methods for analyzing pumping tests that is, the theories on 

which the solutions are based assume that each aquifer is infinite, 

homogeneous, isotropic, and free from discharge or recharge boundaries 

of any sort. Also, the head of each aquifer is assumed to be uniform 

prior to well completion. Because all these conditions must be 

fulfilled for several aquifers, rather than for only one, the chances of 

finding ideal test conditions are correspondingly lessened for a 

multiaquifer well.

The theory neglects any interflow between aquifers, other than 

through the well itself. Thus if the aquifers are not separated by 

perfect aquicludes and if certain aquifers are recharged vertically by 

loss from other aquifers, the test results may be misleading. In the 

same way, the theory does not consider flow that originates in the 

confining beds themselves and that enters the aquifers by vertical 

percolation, except insofar as this effect is considered to contribute 

to the storage coefficient of the aquifer concerned.

The greatest weakness in the test theory, however, is probably 

connected with well-entrance losses. Entrance losses were not 

considered. A test will, therefore, give good results only when these 

losses are negligible. In testing a well penetrating a single aquifer 

it is possible to isolate entrance losses by pumping at a number* of 

different constant rates of discharge (Rorabaugh, 1953). In testing 

a multiaquifer well, however, there are two complicating factors. 

First, the entrance losses may differ from one aquifer to another, as is 

illustrated in figure 55, which shows the differences between the water 

level in the well and the piezometric surfaces of the various acquifers 

immediately outside the well. Second, the entrance losses of each 

aquifer, must vary with time in a constant-drawdown test, as they are 

proportional to some power of the aquifer's discharge. The complications that this may introduce into the analysis are obvious, as 

the constant total drawdown represents the sum of the formation 

loss and entrance loss, that is:

« =«/+«., (20)

where sw is the drawdown in the well, sf is the drawdown of the 

piezometric surface in the aquifer outside the well, and se is the 

entrance loss of the aquifer. Thus, if the entrance loss of an aquifer
 CONSTANT-HEAD PUMPING TEST, MULTIAQUIFER WELL 201

Stabilized water level after 

penetration of 3 aquifers

EXPLANATION

hv Static head of upper aqutfer 

hM Static head of middle aquifer 

h^ Static head of lower aquifer

Surface

Entrance losses, upper aquifer 

Entrance losses, middle aquifer,

1 Entrance losses, lower aquifer

. I Upper aquifer

Middle aquifer

-7 * , Lower aquifer

ky__

FIGUBE 55. Diagram showing entrance losses in a multiaquifer well.
202 GROUND-WATER HYDRAULICS

(se} varies, the piezometric surface of the aquifer immediately outside 

the well must also vary in such a way that the total drawdown in the 

well remains constant. In such a situation, the boundary condition 

of constant drawdown at the borehole wall is no longer satisfied in the 

aquifer, and the test results are erroneous in proportion to the variation in the piezometric surface.

If the range of variation in an aquifer's discharge during most of 

the test is small compared to the absolute value of its discharge at 

the end of the test, the problem of entrance losses varying with time 

will probably be negligible. The problem of determining the magnitude of the entrance losses for each individual aquifer may be resolved 

as further research is carried on into the nature of entrance losses. 

For the present, however, the only way to deal with the entrance-loss 

problem is to use a drawdown step that will produce relatively moderate discharges during most of the test, thus minimizing the size of 

the losses. For any well, the results of the test should be questioned 

whenever significant entrance losses are suspected.

The most serious shortcomings of the test method are insufficient 

accuracy of measurement and incomplete control of certain variables. 

These shortcomings, at present at least, tend to reduce the test 

results to order-of-magnitude significance. In many tests, however, 

even a general idea of the constants of the various aquifers is of 

considerable value. If the method can be refined to the point of 

yielding reasonable results, it should be possible to obtain useful 

information concerning multiple aquifers penetrated by a single well. 

Using the results in an equation of the form of equation 5, the performance of each aquifer in the well can be estimated for any drawdown pattern that can be approximated by a series of steps. Modern 

methods of digital computation have made analyses of this sort 

practical. The data also might be used to construct an electric 

analog of the well and the surrounding aquifers; this in turn could be 

used to determine the response of the well to virtually any set of 

hydrolpgic conditions.

OTHER POSSIBLE METHODS OF OBTAINING DATA ON 

INDIVIDUAL AQUIFERS

If accurate data on the transmissibility and storage coefficients of 

the various aquifers are needed, it is probably best to run a constantdischarge test on each aquifer hi turn. At present this involves 

isolating each aquifer tested from the other aquifers in the well. 

This might be accomplished by using packers in the tested well or 

by drilling new wells, each of which is open to only one of the aquifers.

It is possible that advances in instruments may lead to better 

methods of performing future tests on multiaquifer wells. For
CONSTANT-HEAD PUMPING TEST, MULTIAQUIFER WELL 203

example, a system might be developed in which the well pump is 

governed by a servomechanism that is controlled by flowmeters 

placed above and below the particular aquifer being tested. The 

flowmeters would not only measure the discharge from this aquifer 

but would maintain it at a constant rate decreasing the pump 

output to counteract increases in flow, and increasing the pump 

output to counteract decreases in flow. The drawdown pattern 

observed in the well would then correspond to the constant discharge 

measured from the aquifer in question, and the aquifer constants 

could be calculated by the conventional Theis analysis, provided that 

any internal discharge at the beginning of the test was compensated 

for in the calculations.

REFERENCES

Bennett, G. D., and Patten, E. P., Jr., 1959, Borehole geophysical methods for 

analyzing specific capacity of multiaquifer wells: U.S. Geol. Survey open-file 

report, 54 p.,-8 figs.

Jacob, C. E., 1950, Flow of ground water, in Rouse, Hunter, Engineering hydraulics: New York, John Wiley and Sons, p. 321-386.

Jacob, C. E., and Lohman, S. W., 1952, Nonsteady flow to a well of constant 

drawdown in an extensive aquifer: Am. Geophys. Union Trans., v. 33, no. 4, 

p. 559-569.

Rorabaugh, M. I., 1953, Graphical and theoretical analysis of step-drawdown test 

of artesian well: Am. Soc. Civil Engineers Proc., v. 79, Separate 362, 23 p.

Smith, L. P., 1937, Heat flow in an infinite solid bounded internally by a cylinder: 

Jour. Appl. Physics, v. 8, p. 441-448.

o