Borehole Geophysical 

Methods for Analyzing 

Specific Capacity of 

Multiaquifer Wells

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1536-A

Prepared in cooperation with the 

Pennsylvania Topographic and Geologic 

Survey, Department of Internal Affairs

Property of

U.S. GEOLOGICAL SIWVEY

WATER RESOURCE: DIVISION

CROUND WATER IRANCH

Trenton, New Jersey
Borehole Geophysical 

Methods for Analyzing 

Specific Capacity of 

Multiaquifer Wells

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

GROUND-WATER HYDRAULICS

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1536-A

Prepared in cooperation with the 

Pennsylvania Topographic and Geologic 

Survey, Department of Internal Affairs

Property of

U.S. GEOLOGICAL SURVEY

WATER RESOURCES DIVISION 

GROUND WATER IRANCH

Trenton, New Jersey

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1960
UNITED STATES DEPARTMENT OF THE INTERIOR 

FRED A. SEATON, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

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

Washington 25, D.G.
CONTENTS

Page

Abstract.-..----._________________________________________________ 1

Introduction._____________________________________________________ 1

Review of earlier work.____-__.____________________________________ 3

Conventional logging and interpretation.____-____-_-________-_--_ 3

Flow measurement_________-_.________________-_-__-___--_-_ 3

Quantitative hydrology______________________________________ 3

Theoretical considerations and definitions___________________________ 7

Specific capacity for varying discharge and drawdown._____________ 7

Modifications in the definition of specific capacity _________________ 9

Specific capacity of a multiaquifer well__---------_-__----_------- 11

Contribution of individual aquifers to specific capacity_---_.__---_- 11

Hydraulics of a multiaquifer well.__________________-_--_-----___ 13

Method of presentation of experimental data____________________ 17

Description of field tests and calculations____________-_-_-__._-------- 18

Equipment-_-_-___________________.____--_______-_----------- 18

Experimental procedure_______________-_________-_-___._------- 20

Presentation and evaluation of results_______-_____________------__- 23

Conclusions_ _-__________-_____-__-_____-_-_-_-_--_----_---_----_ 24

Literature cited___________________________________________________ 25

ILLUSTRATIONS

Page 

FIGTTHE 1. Conventional specific-capacity graph where water level is

plotted as drawdown, sw_i__________-___-_-_-_-_-_----- 6

2. Specific capacity at constant discharge and at constant drawdown _-___________-______-______-_____-_-_-_---_----- 7

3. Specific-capacity graph where water level is plotted as head

above datum_______________________________________ 10

4. Composite specific-capacity graph for test well for pumping

times of 1 hour_-________----_--_-___------------------ 14

5. Schematic diagram of conditions in test well not being pumped- 16

6. Physical and electric logs of test well_________.-_-__--_-__ 19

7. Schematic diagram of equipment at test well, showing brinetracing procedure.. ___________-_-__-_----_----------- 20

8. Fluctuations of water level due to pumping of test well __ 21

TABLES

TABLE 1. Summary of experimental data_________________-__----_- 22

2. Data obtained from specific-capacity graph of figure 4____-__- 24

m

GROUND-WATER HYDRAULICS

BOREHOLE GEOPHYSICAL METHODS FOR ANALYZING 

SPECIFIC CAPACITY OF MULTIAQUIFER WELLS

By GORDON D. BENNITT and EUGENE P. PATTEN, JB.

ABSTRACT

Conventional well-logging techniques, combined with measurements of flow 

velocity in the borehole, can provide information on the discharge-drawdown 

characteristics of the several aquifers penetrated by a well. The information 

is most conveniently presented in a graph showing aquifer discharges as functions 

of the water level in the well at a particular time.

To determine the discharge-drawdown characteristics, a well is pumped at a 

steady rate for a certain length of time. While the well is being pumped, meaiurements are made of drawdown and of the discharge rates of the individual 

aquifers within the well. Discharge rates and drawdowns are usually recorded 

as functions of time, and their values for any given time during the test are 

obtained by interpolation. The procedure is repeated for several different rates 

of total well discharge. The well may be allowed to recover after each step, 

or discharge may be changed from one rate to another, and changes in discharge 

and drawdown may be measured by extrapolation. The flow measurements 

within the well may be made by use of a subsurface flowmeter or by one of 

several techniques involving the injection of electrolytic or radioactive tracers.

The method was tested on a well in Mercer County, Pa., and provided much 

useful information on aquifer yields, "thieving," and hydrostatic heads of the 

individual zones.

INTRODUCTION

The U.S. Geological Survey, in cooperation with the Pennsylvania 

Topographic and Geologic Survey, has conducted an extensive welllogging program in Pennsylvania since 1956. Water wells throughout 

the State have been investigated with devices that measure selfpotential, point resistance, normal resistivity, temperature, fluid 

resistivity, borehole diameter, gamma radiation, and flow velocity.

The general objectives of this program were twofold: (a) the 

accumulation of geologic and hydrologic information in various parts 

of the State and (b) the development of new methods of interpreting 

the data of borehole geophysics in ground-water studies. Pursuit of 

this second objective was undertaken in order to find new or improved 

approaches to certain hydrologic problems. One such problem is that 

of the multiaquifer well the well yielding water from two or more

1
2 GROUND-WATER HYDRAULICS

aquifers. Jones and Skibitzke (1956, p. 292) suggest a method of 

log interpretation designed especially to deal with this problem. The 

theory underlying their method is described in detail in the following 

pages, and the results of some field tests are given.

The standard well-logging methods have all been in use for some 

time in hydrologic work, but the general practice has been to adopt 

(with little or no modification) the interpretive principles of the oil 

industry. The quality of the results thus obtained has not been 

consistent. When the assumptions implicit in the oil-well techniques 

have been met in the ground-water problem at hand, results have 

been excellent; when these assumptions have not been met, results 

have been correspondingly poor. In any event the parallels between 

oil-reservoir problems and hydrologic problems do not extend indefinitely. The future of well logging as a hydrologic tool depends, 

therefore, upon the development of new interpretive techniques 

suited especially to the problems of hydrology.

The methods of well logging and interpretation discussed in this 

paper have as their objective the determination of the dischargedrawdown relations of the individual aquifers supplying a multiaquifer well. The well is pumped at a given discharge rate for a 

certain length of time. While the well is being pumped, measurements 

are made of the drawdown in the well and of the discharge rates of 

the individual aquifers tapped by the well. Discharge rates and 

drawdowns are usually recorded as functions of time, and their 

values for any given time during the test are obtained by interpolation. 

The procedure is repeated for several different rates of well discharge. 

The well may be allowed to recover after each step, or discharge may 

be changed directly from one rate to another, and changes in discharge and drawdown may be measured by extrapolation. The flow 

measurements may be made by use of a subsurface flowmeter or by 

one of several techniques involving the injection of electrolytic or 

radioactive tracers. A normal-resistivity logging device was used to 

locate the possible aquifers, and a well caliper was used to obtain 

borehole area at points of velocity measurement.

The authors were fortunate in having at their disposal several 

methods of flow measurement, and the tests described in this paper 

were made by the methods that seemed best suited to the particular 

situation at hand. A detailed discussion of the relative merits of 

the various techniques of measurement is beyond the scope of the 

present paper.

A somewhat lengthy discussion of theory is necessary to clarify 

both the merits and the limitations of the methods under study. This 

discussion pertains in part to borehole geophysical methods and in
SPECIFIC CAPACITY OF MULTIAQUIFER WELLS 3

part to general ground-water hydrology. It involves a review of 

some earlier work and the introduction of certain new definitions.

The investigation was made under the immediate supervision of 

D. W. Greenman, formerly district geologist, U.S. Geological Survey, 

Harrisburg, Pa.

REVIEW OF EARLIER WORK

CONVENTIONAL LOGGING AND INTERPRETATION

Of the earlier papers on the application of borehole geophysics 

to ground water, only those of Bays and Folk (1944) and Jones and 

Skibitzke (1956) are of direct concern to the present problem. Bays 

and Folk (1944) suggested the use of brine as a flow tracer. Jones 

and Skibitzke (1956), in their discussion of flowmeter interpretation, 

suggested the methods forming the basis of the present study.

Various conventional logging techniques and methods of interpretation can be used to investigate geologic conditions in a well and 

to locate possible aquifers. The original publications describing 

these techniques are not reviewed here, for they are adequately summarized by Jones and Skibitzke (1956).

FLOW MEASUREMENT

Meinzer (1928) reported the use of a conventional Price current 

meter to detect leaking or "thieving" zones in wells in Hawaii. Fiedler (1928) described the use of the Au current meter, a mechanical 

device designed specifically for well-flow measurement. Skibitzke 

outlined the theory for a thermal well-velocity meter in a Government 

patent (1955) ; he has since then developed an instrument of this type.

Bird and Dempsey (1955), in a paper intended primarily for application in the oil industry, described a use of radioactive tracers that 

might be adapted for the measurement of flow in water wells. A 

method of discharge measurement described by Barbagelata (1928), 

may be modified for application in a well by using a fluid-conductivity 

logging device and a suitable brine injection apparatus.

QUANTITATIVE HYDROLOGY

The equation of ground-water flow as given by Jacob (1950, p. 333) 

for a single, infinite, homogeneous, and isotropic aquifer of uniform 

thickness, in which there is no vertical flow, is as follows:

where h represents the head, or elevation above datum of the piezometric surface ; $ is the storage coefficient, a dimensionless term defined
4 GROUND-WATER HYDRAULICS

as the volume of water released from or taken into storage per unit 

surface area of the aquifer per unit change in the component of head 

normal to that surface; and T is the transmissibility of the aquifer, 

or its permeability multiplied by its thickness, having the dimensions 

of length squared per unit of time.

The solution for equation 1, for radial flow to a well discharging at 

a constant rate Q and involving no recharge to the aquifer, is that 

given by Theis (1935), which may be written as

^rlS

where s is the drawdown at a distance r from the discharging well, 

at time t after the beginning of pumping. The drawdown is evaluated 

as the sum of an infinite number of elemental drawdown increments, 

each the result of an instantaneous discharge at some earlier time, 

t'; this discharge acts over a time interval dt' .

The integral in equation 2 is evaluated by means of an infinite 

series

__+ - (3) ^ ^ {J

After a short time of pumping has elapsed, the series terms after 

the first two become negligibly small, and equation 2 may then be 

written as

2.25Tt ,..

The drawdown at the discharging well after a sufficient lapse of 

time is given by equation 4, where rw , the radius of the well, is used in 

place of r:

2.30a 2.25* , K .

In addition to the drawdown predicted by equation 5, there will 

generally be a loss of head at the discharging well, representing the 

energy expended as the water enters the well itself and flows upward 

through the borehole. Rorabaugh (1953) has indicated that in a typi-
SPECIFIC CAPACITY OF MULTIAQUIFER WELLS "5

cal 12-inch well these losses will be approximately proportional to the 

first power of the discharge for discharges of less than 0.75 cfs (cubic 

feet per second) or 337 gpm (gallons per minute). Then the drawdown at the discharging well will be

where C' is the constant factor relating entrance head losses to discharge in this range of discharge values.

For higher values of discharge Rorabaugh indicates well entrance 

losses proportional to some higher power, n, of the discharge, where 

the drawdown at the discharging well becomes

2.25Tt

In equation 7, C is the well-loss constant in effect for the higher 

discharge range.

In the discharge range covered by equation 6, drawdown is readily 

observed to be a linear function of discharge, if pumping for equal 

intervals of time is understood. If a well is pumped at several rates 

of discharge, starting at zero drawdown and continuing for the same 

length of time at each pumping rate, the resultant drawdowns should 

plot as a straight line against the corresponding discharge rates. Such 

a plot is shown in figure 1 , with drawdown on the abscissa and discharge 

on the ordinate.

The slope of this graph, - } is known as the "specific capacity"

ASm

of the well. If the graph were extended into higher rates of discharge, it would assume some curvature, as can be demonstrated by 

solving equation 7 for the ratio Q/sw . An examination of equations 

6 and 7 will show also that specific capacity is a function of time. 

After a sufficient time has passed, however, the changes in specific 

capacity per unit change in time become quite small, so that considerable differences in the time of pumping at the various discharge 

rates can often be tolerated.

Jacob and Lohman (1952) have demonstrated that, after a reasonable time of pumping, discharge-drawdown points determined by pumping a well at a variable rate so as to maintain a constant drawdown 

value approach very closely those determined by discharging the well 

at a constant rate.

546681 O 6C
6 GROUND-WATER HYDRAULICS

The solution of equation 1 given by Jacob and Lohman for this 

case is

Q=2rrsw.(9(a), (8)

in which sw is the constant drawdown, Q is a function varying with 

time,

and Y0 (x) and Jg (x) are Bessel functions of zero order of the first and 

second kinds, respectively. The integral is approximated by numerical methods, -and a table of values of G(a) for various values of a 

appears in the reference.

Jacob and Lohman point out that, at large relative values of t, the 

value of 0(ot) approaches that of the function

0/001 2/2.3 log 2.257V

and that the specific-capacity ratio calculated from equation 8 will 

therefore approach the specific capacity calculated from equation 5.

0 DRAWDOWN (sw ), IN FEET BELOW STATIC WATER LEVEL 

FIGUBE 1. Conventional specific-capacity graph where water level is plotted as drawdown, « 
SPECIFIC CAPACITY OF MULTIAQUIFEE WELLS

Constant discharge^

Constant drawdown'

0 TIME - 

FIOUBE 2. Specific capacity at constant discharge and at constant drawdown.

Jacob and Lohman present a graph (see fig. 2) in which specific 

capacity is shown as a function of tune for both constant-discharge 

and constant-drawdown methods.

THEORETICAL CONSIDERATIONS AND DEFINITIONS 

SPECIFIC CAPACITY FOB VARYING DISCHARGE AND DRAWDOWN

Equations 5 and 8 form the basis of most of the discharging-wel 

techniques presently employed in quantitative hydrology, and the 

only published analytical expressions for the specific-capacity ratio 

are taken from these equations. They are particular solutions of 

equation 1, corresponding to two possible ways of discharging an 

aquifer. Their applicability to multiaquifer problems is not limited 

by the fact that they were derived in terms of a single aquifer, but 

rather by their basic assumptions, which demand either a constant 

rate of discharge or a constant drawdown. Equation 5 could be 

applied in turn to each individual aquifer in a multiaquifer well, if 

some means could be found to hold the discharge of each aquifer constant. Equation 8 likewise could be applied to each aquifer in turn 

if the drawdown of the well were held constant, head losses within 

the borehole between aquifers were negligible, and some means were 

used to measure the variations in discharge of each aquifer with time.

In general, however, neither the drawdown of the well nor the 

discharges of the various individual aquifers remain constant during 

the pumping of a multiaquifer well. It will be instructive, therefore,
8 GROUND-WATER HYDRAULICS

to supplement the theory reviewed above with a discussion of discharge 

under more general conditions, in which the rate of discharge and the 

drawdown are at liberty to vary.

The solution of equation 1 corresponding to these conditions is

This is a more general form of equation 2, in which Q, as a function 

of time, appears under the integral rather than outside it. As in 

equation 2, the total drawdown sw at time t is evaluated as the sum of 

an infinite number of elemental drawdown increments, each of which 

is the result of the action of the pump during some earlier time interval, 

of length dt' '. The solution is taken by analogy from that given for 

the equivalent heat-conduction problem by Carslaw and Jaeger (1959, 

p. 261).

It is not necessary for the purposes of this paper to find solution 

of equation 9 corresponding to various possible forms of the function 

Q,(t'}. The point of direct concern is whether a point (sw, Qw) reached 

on the specific-capacity graph by discharge under these general 

conditions will appreciably differ from that reached by discharge at a 

constant rate or a constant drawdown for an equal time interval. If 

the difference is negligible, specific-capacity points taken at variable 

discharges may be plotted on the same grajih as those taken at constant 

discharge, and the plot should remain approximately linear. The 

question in explicit terms is whether or not sw as evaluated by equation 

9 will be approximately equal to sw as evaluated by equation 2 if the 

constant Q of equation 2 is equal to Q(t) in equation 9. [Q(t) is here 

intenv, ] to mean Q(tf ) evaluated at t'=t.]

An examination of the two equations will show that the answer 

to this question is dependent upon the nature of the function Q(t'). 

If this function has a time derivative which approaches zero as t' 

increases, the discharge eventually appears nearly constant over 

long time intervals, and the significant variations in discharge are 

confined to the early periods of pumping. Unless these early variations in discharge are excessively large, the nature of the function

t_ t f&xp ( ±T(i"Lt'\ ) *8 8Ucn tnat at larSe values of t the effect of the early

periods of pumping, for which t is very much greater than t', is quite 

small. The drawdown is largely the result of the operation of the 

pump at an essentially constant rate during later intervals of pumping 

and, as such, will not differ greatly from the drawdown predicted by 

equation 2 for the same value of t. Consequently, at sufficient values 

of t the ratio QJsw calculated for a constant discharge, Qi, should be
SPECIFIC CAPACITY OF MULTIAQUIFER WELLS 9

approximately equal to the ratio Q(t)fsw calculated for a time-varying 

discharge in which

as

It will become clear during the discussion of the discharge of an 

individual aquifer to a multiaquifer well that in such discharge the 

condition of a continuously decreasing time derivative of Q(t') is 

nearly always satisfied, and the initial variations themselves are 

moderate. Thus it seems reasonable to expect that rpecific-capacity 

points taken for these aquifers under variable-discharge conditions 

will plot in approximately the same place as those taken at constant 

discharge, if the time of pumping is sufficient. (Equation 6 should 

then approximate the relation between well drawdown and aquifer 

discharge.) If doubt exists as to the length of a sufficient time of 

pumping, Q(t') may be recorded in the field as a function of t' . Occasionally it may be desirable to establish the trend of the function 

Q(t') in the field and extrapolate this trend to obtain the discharge 

corresponding to a longer time of pumping.

MODIFICATIONS IX THE DEFINITION OF SPECIFIC CAPACITY

It is sometimes preferable to speak of well "head level" above some 

datum rather than well drawdown, because the former term is better 

suited to problems in which the water level in the well may either 

rise above the nonpumping head of a particular aquifer or be drawn 

down below it. For such a problem the graph of figure 1 would be 

replotted as shown in figure 3. Values of Q lying above the horizontal axis represent discharge; those below represent recharge or input 

to the well. The specific-capacity graph crosses the horizontal axis 

at the point

Q=0

h=h0 ,

where h0 is the static (nonpumping) level of water in the well. In 

figure 3 the head increases to the right, indicating a higher water level. 

In figure 1 the converse is ^rue, inasmuch as drawdown increases to 

the right. Whereas specific capacity was defined as the slope of the 

graph of figure 1, it will be the negative of the slope of the graph of 

figure 3. The same statements and equations that apply to discharge 

can be made regarding well input; the only differences that need be 

introduced are the use of

fig flip

in place of sw, and the use of the negative sign to distinguish recharge.
10 GROUND-WATER HYDRAULICS

HEAD (/,), IN FEET ABOVE ARBITRARY DATUM

FIGURE 3. Specific-capacity graph where water level is plotted as head above datum.

In a well penetrating a single aquifer, the discharge of the aquifer 

is zero when the water level in the well is at its static level (zero 

drawdown). Here it is not necessary to use the notation A<2/Asw for 

specific capacity; we may simply use Qlsw . In work with multiaquifer wells, however, it is necessary to use the graph-slope definition, A#/Asw . Equations 5-9 do not demand that the initial values 

of Qw and sw be zero, because as long as the original amounts and trends 

of these quantities can be extrapolated and their effect removed from 

the data under consideration, the equations still apply.

If in equation 6 the symbol B is substituted for the quantity

2.3 , 2.25T* i, .. ,

-r-ffi log 2cr> the equation becomes

47T2 Tw 'O

sw=BQ+C'Q. (10)

The slope of a graph of Q versus sw then becomes A<2/Asw =l/ 

(B-\-Cr ), and the slope of a graph of Q versus head, hw, becomes

'B+C'' (ID
SPECIFIC CAPACITY OF MULTIAQUIFER WELLS 11

It was indicated in the preceding section that, within certain 

restrictions, points on the specific-capacity graph are largely independent of the manner in which they are approached. Thus equation 

11 may be considered a generally valid expression for specific capacity 

when discharge varies.

The quantitative theory developed heretofore has all been based 

on the assumption of an infinite, homogeneous, and isotropic aquifer. 

This is not to say, however, that the ratio AQ/As«, cannot be defined 

and have meaning unless these conditions are met. A solution zone 

in limestone, for example, or a fracture zone in crystalline rock may 

show a constant ratio of discharge to drawdown; the value of this 

ratio in itself constitutes useful information, even though an analytical 

expression such as that proceeding from equation 6 cannot be written 

for it.

SPECIFIC CAPACITY OF A MULTIAQUIFER WELL

The specific capacity of a multiaquifer well is defined in the same 

way as that of a single-aquifer well: it is the ratio of the change in 

well discharge to the accompanying change in drawdown of water 

level. There are two possible ways of writing an analytical expression for specific capacity. One is based upon equation 6 and involves 

the use of an equivalent transmissibility and storage coefficient the 

T and S of a single aquifer which would behave in the seme manner 

as the entire group of aquifers actually present. Obviously such an 

expression has only limited value and significance. As an alternative, 

an approximate expression may be built up involving the T and S 

of each aquifer present; the manner in which this may be done will 

become clear as the behavior of a multiaquifer well is analyzed.

CONTRIBUTION OF INDIVIDUAL AQUIFEBS TO SPECIFIC CAPACITY

Specific capacity is in general considered to be a property of a 

particular well, for it is partly dependent upon well radius. In order 

to facilitate the discussion of the hydraulics of a multiaquifer well, 

it is necessary to define the specific capacity as related to an individual 

aquifer tapped by such a well. This is in a sense a more restricted 

definition, meaning the property dependent on a single aquifer at 

a certain well, rather than a property of the well as a whole.

The specific capacity related to one aquifer at a multiaquifer well 

is defined as the ratio of the change in that aquifer's discharge to the 

accompanying change in water-level drawdown in the well. It is, 

therefore, the negative of the ratio of the change in aquifer discharge 

to the corresponding change in head in the well, measured above 

some arbitrary datum.

A graph such as that of figure 3 may be plotted for the discharge 

of a single aquifer into a multiaquifer well. The discharge from the
12 GROUND-WATER HYDRAULICS

aquifer at a given time of pumping is plotted as the ordinate against 

the corresponding head as the abscissa.

The slope of this plot, -TJ~> is the negative of the particular aquifer's

t^tlw

contribution to specific capacity. Under certain conditions to be 

discussed in another section the straight line drawn through the 

plotted points intercepts the horizontal axis of the graph at the undisturbed head of the particular aquifer being observed, whereas a 

similar plot for the entire well always intercepts the horizontal axis 

at the static water level of the well. In a single-aquifer well the 

specific capacity related to a particular aquifer, as defined here, and 

the specific capacity of the well are synonymous, and the static water 

level of the well is the undisturbed head of the aquifer. By contrast, 

the static level of a multiaquifer well need not be equal to the head of 

any of its individual aquifers; this also will be dicussed more fully 

in a later section.

The expression for the specific capacity related to a single aquifer 

may be broken down analytically in the same way as the expression 

for the specific capacity of a well as a whole. If the time derivative 

of the aquifer's discharge is a continually decreasing function, equation 

6 must eventually approximate the relation between the aquifer's 

discharge and the drawdown in the well. The formation-loss term 

in equation 6 should apply in a straightforward manner, where Tand S 

refer to the constants of the formation in question and rw refers to the 

radius of the borehole as it passes through that formation. The 

entrance-loss term cannot be applied unless some qualifying assumptions can be made.

If several aquifers are contributing to the discharge of a well, 

the losses due to pipe friction that occur in the well column above 

the uppermost aquifer will be a function of the total discharge; the 

losses occurring between the uppermost and the next lower aquifer 

will depend upon the discharges of all the aquifers except the uppermost; and so on. Identification of the pipe losses related to any 

one aquifer will therefore depend upon knowledge of the discharges 

of several aquifers, rather than upon the discharge of that aquifer 

alone, and the well-loss term in equation 6 becomes inadequate under 

these conditions. If, however, the losses occurring within the well 

bore as the water moves to the pump intake may be assumed to be 

negligible, the only significant well losses will be those occurring 

as the water leaves the formation, enters the well bore, and turns 

through 90°. In this case the entrance losses related to each aquifer 

are a function of the discharge of that aquifer alone, and an equation
SPECIFIC CAPACITY OF MULTIAQUIFER WELLS 13

in the form of equation 6 is justified. This assumption is usually 

permissible in wells of a reasonable diameter and moderate depth. 

Thus an equation in the form of equation 11 can be written for each 

aquifer in a multiaquif er well :

where AQW is the change in the discharge of the nth aquifer accompanying a change in water level of Ahw; Bn is a function of the transmissibility and storage coefficients of this aquifer; and C'n is the entranceloss coefficient for this aquifer.

Equation 12 indicates that, within the discharge range over which 

it applies (on the assumption that no time variations exist between 

the various measurements), the specific-capacity graph of an individual 

aquifer should plot as a straight line.

HYDRAULICS OF A MULTIAQUIFER WELL

When the discharge of a multiaquif er well is changed by a certain 

amount (AQW ), a corresponding change in the water level of the well 

(A^w) must occur over a given time interval. The discharges of the 

various aquifers supplying the well must also change in such a manner 

that the algebraic sum of their changes is equal to the change in well 

discharge :

 -, (13)

where the subscripts 1,2, etc., refer to the individual aquifers tapped 

by the well. Dividing by the change in water level, we obtain

Hence, the specific capacity of the well is the algebraic sum of the 

specific capacities of the individual aquifers. If the specific-capacity 

graphs for the well and for each aquifer are plotted on the same set 

of axes, as in figure 4, the slope of the well graph must be the sum of 

the slopes of the aquifer graphs. Figure 4 represents the plot of data 

from a two-aquifer well.

With reference to equations 12 and 14, an analytical expression 

for the specific capacity of the well of figure 4 may be written as 

follows :

23 2 25T t

where Bn=. 'T log 2ere > in which Tn and S* are the coefficients of

e rw ow
14 GROUND-WATER HYDRAULICS

600

500

400

300

200

100

100

\

AQUIFER 2

i i i i

.AQUIFER 1

i i ii

Static (nonpumping) 

head

I I I I I I II

15 20 25 30 

HEAD (h), IN FEET ABOVE ARBITRARY DATUM

35 40

FIGURE 4. Composite specific-capacity graph for test well for pumping times of 1 hour. The head values 

in this figure are referred to an arbitrary datum 50 feet below the top of the well casing.

transmissibility and storage of the aquifer in question, and where 

Cn' is the entrance-loss coefficient of the particular aquifer.

The graph of figure 4 is useful in illustrating in general the performance of a multiaquifer well. No significant time variations exist between the various points plotted in figure 4; in other words, for each 

plotted value of discharge the corresponding head value was observed 

when that rate of discharge had been in effect for 1 hour. Preparation 

of any graph similar to figure 4 should include notation of the time
SPECIFIC CAPACITY OF MULTIAQUIFER WELLS 15

period used for observing the head values to be plotted against the 

various discharge rates of the well and the individual aquifers.

For any head value in figure 4 the discharge of the well is the algebraic sum of the discharges of the individual aquifers. Over a certain 

range one of the aquifers acts as a thieving zone, and the well discharge 

is less than the total discharge of the yielding zone.

Figure 4 illustrates the fact that a multiaquifer well is in effect 

never truly static; the net discharge of the well is zero when hw h01 

but this zero discharge is the algebraic sum of several aquifer discharges. In figure 4 aquifer 1 is discharging an amount equal to the 

recharge to aquifer 2, when well discharge is zero. Figure 5 shows 

the flow directions and piezometric surfaces corresponding to this 

internal discharge; a cone of depression exists in the piezometric 

surface of aquifer 1, and a cone of impression (cone of elevation) in 

that of aquifer 2. No piezometric surface is shown for the lowermost 

sandstone, indicating that it is not a significant aquifer.

The magnitude and direction of the internal flow in a given well 

not being pumped depend upon the transmissibilities, storage coefficients, and hydrostatic heads of the aquifers involved. Upward 

flow is at least as common as the downward flow shown in figure 5.

For the well represented in figure 5 a system of equations describing 

the internal flow may be set up by applying equation 9 to each aquifer, 

by using on the left in each equation the difference between the current 

water level and the undisturbed head of the aquifer in question, and 

by adding the condition that at any time

Qi+Q2=0. (16)

The main difference between the internal discharge of a multiaquifer well when it is not pumped and its discharge during pumping 

is in the value of the right side of equation 16. Equation 9 still 

applies to each aquifer, but the sum of the discharges must be some 

nonzero constant or function of time. The sum usually will be 

essentially constant, and this is the basis for postulating a decreasing 

time derivative for the individual discharges. It is unlikely that 

rapid variations in the individual discharges will occur after the 

initial moments of pumping where the aquifers are adjusting to a 

constant total rate of discharge from the well.

The zero-discharge intercepts of the lines in figure 4 are points of 

special interest in that they approximate the static heads of the 

individual aquifers. Each line on the graph must be constructed 

through an initial point of reference, from which the changes in Qu 

and hw are measured. In the specific-capacity graph of a singleaquifer well (fig. 3) this reference point was taken at zero discharge
16 GROUND-WATER HYDRAULICS 

Land surface 

Piezometric surface aquifer I"

Piezometric surface aquifer 2

Aquifer 1

Aquifer 2

Lowermost sandstone

FIGURE 5. Schematic diagram of conditions in test well not being pumped.
SPECIFIC CAPACITY OF MULTIAQUIFER WELLS 17

and the static head of the aquifer. In the graph for an individual 

aquifer in a multiaquifer well the reference must be taken at the 

static (nonpumping) head in the well and at whatever discharge the 

aquifer shows for this head. This "static" point has generally been 

reached through a long period of internal discharge. In order to 

construct a line whose intercept on the horizontal axis of the graph 

represents the true static head of the aquifer under study, pumping 

intervals of the same duration as the internal discharge would have 

to be employed, and this is clearly impractical.

It has already been pointed out in the discussion of the time dependence of specific capacity that large differences in the time of 

pumping will cause relatively small differences in the ratio &Q/Ahw 

provided the times in question are both of reasonably large magnitude. 

The slope and, therefore, the zero-discharge intercept of the line 

drawn through data obtained in a test of relatively short duration will 

usually approach that of the line that would be drawn through data 

obtained by pumping for time intervals approaching the duration of 

internal flow. The zero-discharge intercept of the specific-capacity 

graph related to an individual aquifer is thus a good approximation 

of the static head of the aquifer, and the approximation is only slightly 

improved by extending the times of pumping.

METHOD OF PRESENTATION OF EXPERIMENTAL DATA

The specific-capacity data provided by the methods described in 

this paper can be presented most effectively in a graph such as that 

of figure 4. The practical value of such a graph is obvious: it provides 

the specific capacities and hydrostatic heads of the various aquifers 

and shows the drawdown ranges over which thieving occurs in the well. 

Such information can be of great use to the owner or consultant in 

planning the development of the well, or in planning the drilling and 

development of additional wells in the immediate vicinity.

In the review of theory it was indicated that several factors may 

influence the linearity of the specific-capacity graph; among these were 

mentioned time variation between points, excessive well entrance 

losses, or a general inapplicability of equation 6 where the effects of 

variable discharge are still appreciably different from those of constant 

discharge at the time of measurements. The graph checks itself to a 

certain extent against these errors; if the experimental points fall on 

a straight line, it may be assumed that these effects are negligible.

In some investigations considerations of time and cost may prohibit the construction of individual graphs for each aquifer; then 

graphs may be constructed for the zones of greatest interest, and an 

"equivalent aquifer" may be postulated for all the remaining zones 

of the well. An equivalent aquifer is defined as a single aquifer which
18 GROUND-WATER HYDRAULICS

would perform in exactly the same manner as the group of aquifers 

it represents, taken as a whole. The specific-capacity graph 

related to the equivalent aquifer bears the same relation to the graphs 

of the aquifers it represents as the graph for the well bears to those of 

the individual aquifers of the well.

DESCRIPTION OF FIELD TESTS AND CALCULATIONS

Field tests of the methods presented in this paper were made over 

a 3-day period in September 1958, by use of an abandoned oil well in 

Mercer County, Pa.

Electric logs of the well (fig. 6) were made prior to the actual testing; 

they included self-potential, single-point resistance, 16- and 64-inch 

normal-resistivity, temperature, fluid-resistivity, gamma-ray, and caliper logs. The logs showed the well to be typical of the abandoned 

oil wells in the area. A wooden plug had been placed in the borehole 

at a depth of 580 feet in an effort to prevent salt water from deeper 

sands from contaminating the fresh-water sands above. This effort 

was only partly successful, for the fluid-resistivity log shows salt water 

below about 500 feet. The resistivity logs also suggest that the lower 

sandstone contains salt water, but fortunately it did not yield this 

water to the well at the pumping rates employed in testing.

The upper and middle sandstones, referred to as the first and second 

aquifers respectively, belong to the Pottsville formation of Pennsylvanian age and are productive aquifers throughout a large area of 

western Pennsylvania. The two aquifers vary in thickness and locally 

may be represented entirely by shale that forms a barrier to the movement of ground water. The pumping data presented in this section 

do not reflect any such hydrologic boundaries, but it is felt that with 

longer pumping time such boundaries would be discovered and would 

affect the linearity of the experimental data.

EQUIPMENT

The arrangement of equipment at the test well is shown schematically in figure 7. The well was pumped with a centrifugal pump 

capable of discharging a maximum of 115 gpm against a 20-foot 

suction head. Discharge was regulated with a globe valve and 

measured by noting the time taken to fill a calibrated 55-gallon drum.

Water-level measurements to 0.01 foot were made frequently 

throughout the testing period, and drawdown data were obtained for 

all pumping rates. Figure 8 shows the prepumping levels, drawdown 

during pumping, and recovery trends after pumping.

Velocities of internal flow in the well between the aquifer 1 and 

aquifer 2 and between the second and third aquifers were measured 

by injecting brine into the flow at a selected depth through a K-i
SPECIFIC CAPACITY OF MULTIAQUIFEF WELLS 19

LU «i 

" I

QO

5 §

_w o 

??. o

>y« n

IlII

-ii

r -c ZLJL
20 GROUND-WATER HYDRAULICS

Valve Pump Brine Logging truck

Aquifer 2

Ohm-meters

FIGURE 7. Schematic diagram of equipment at test well, showing brine-tracing procedure.

plastic hose connected to a 15-gallon surface reservoir, and by tracing 

the movement of the brine with a fluid-resistivity electrode positioned 

from the logging truck. The fluid-resistivity apparatus contained in 

the truck included the measuring probe, a cable for lowering the probe 

into the well, a calibrated reel over which the cable passed so that 

the depth of the electrode was known at all times, and a device that 

continuously recorded the resistivity data against depth.

EXPERIMENTAL PROCEDURE

Information was obtained for several well-discharge rates relating 

the discharge of the individual aquifers to the drawdown of the well. 

As mentioned previously, the lower sandstone did not yield water to 

the well and therefore is not discussed.
SPECIFIC CAPACITY OF MULTIAQUIFER WELLS 21

The direction of internal flow was initially determined by injecting brine between aquifer 1 and aquifer 2 and tracing its movement with the fluid-resistivity probe. When the well was not being 

pumped, water was flowing from the first to the second aquifer. The 

brine "slug" essentially maintained its shape and concentration until 

it was opposite aquifer 2, where the concentration of the slug diminished rapidly. After the direction of flow was established, tests 

were conducted to determine the flow velocity. The procedure used 

in these tests involved injecting the slug of brine immediately below 

aquifer 1 and lowering the fluid resistivity probe through the slug. A 

stop watch was started when the recording device in the logging 

truck indicated that the probe had passed through the point of maximum concentration of the brine. The probe was then positioned 

just above aquifer 2 and held stationary until the leading edge of the 

brine slug caused a deflection on the recorder. The probe was then 

raised in the borehole, and the watch stopped when the point of maximum concentration was recorded on the log chart. To minimize 

velocity errors due to constrictions of the borehole, the brine was 

allowed to travel the longest practical distance between the two aquifers. Flow velocity was computed from the distance the brine peak

11 a.m. 12 m. 1 p.m. 2 p.m. 3 p.m. 

SEPTEMBER 17, 1958

4 p.m. 5 p.m.

32

2B

83 gpm

EXPLANATION

Measured trend

Extrapolated trend

Irregular discharge during 

setting of valves

105 gpm (A0=22gpm)

11 a.m. 12 m. 1 p.m. 2 p.m. 3 p.m. 

SEPTEMBER 18,1958

4 p.m. 5 p.m.

FIGURE 8. Fluctuations of water level due to pumping of test well.
22 GROUND-WATER HYDRAULICS

traveled and the time taken for the brine to migrate from peak to 

peak. Figure 7 shows the operation. Discharge in cubic feet per 

minute was computed as the product of the flow velocity and the 

average area of the borehole (obtained from the caliper log) and was 

converted, for convenience, to gallons per minute.

The same procedure was employed for pumping rates of 22, 44, 72, 

83, and 115 gpm. Measurements of water level were made prior to 

each period of pumping, and drawdowns were taken from the extrapolated trends of these measurements. During the 72-gpm discharge the pump stalled after approximately 20 minutes; however, a 

measurement of internal discharge had been completed by this tune, 

and drawdown data were extrapolated to 1 hour in order to obtain the 

points for figure 4 corresponding to this pumping rate. All other 

pumping rates were maintained for at least an hour, and the data presented were taken for 1 hour of discharge. All pumping rates except 

that of 22 gpm followed a period of no pumping and were directly 

determined by measuring the time taken to fill a calibrated 55-gallon 

drum. The 22-gpm discharge was measured by increasing the well 

discharge from 83 to 105 gpm, for a net increase j0f 22 gpm. Drawdown data corresponding to a discharge of 22 gpm were obtained by 

subtracting the extrapolated trend of the 83-gpm discharge from the 

measured water level during the 105-gpm pumping. Figure 8 summarizes the measured and extrapolated drawdown and recovery 

trends for the various pumping rates.

TABLE 1. Summary of experimental data

Well 

discharge 

Q» 

(gpm)

0 ....

22*

44. .

72.........

83.. 

115... ..

Depth traveled by 

brine slug 

(ft)

Max

57

63

60

59.

63

68

Min

55

62

58

56

62

66

Time 

interval 

(min, 

sec)

4:00

5:42

4:53

4:37

5:04

6:05

Internal-flow 

velocity aquifer 1 

to aquifer 2 

(fpm)

Max

14.3

11.1

12.3

12.9

12.4

11.2

Min

13.8

10.9

11.9

12.2

12.2

10.9

Mean 

recharge 

to aquifer 2' 

(gpm)

58.0

'51.9

49.5

51.8

50.8

45.7

Total discharge of 

aquifer 1 

(gpm

58.0

»74.6

97.5

121.8

133.8

160.7

Change 

in head 

Aft«, after 

Ihr 

pumping 

(ft)

0.00

»-.86

-1.50

-2.42

-2.72

-3.81

Head, A 

after 1 hr 

pump- »'

32.44

'31.58

30.94

30.02

29.72

28.63

i Computed by using borehole area of 0.55 square foot.

> Values of h w obtained by adding values of Aft* to head of well at zero pump discharge, taken as 32.44 feet 

above assumed datum which Is 50 feet below top of well casing.

»Values obtained by taking differences between data for pump discharges of 83 gpm and 105 gpm, and 

adding these differences to values for zero pump discharge.

Table 1 summarizes the discharge-drawdown data associated with 

1 hour's pumping and shows for each rate of pumping the distance 

between the brine peaks, the time of traverse of the brine, the flow
SPECIFIC CAPACITY OF MULTIAQUIFER WELLS 23

velocity, the discharge or recharge of each of the two aquifers, the 

observed change hi the head hi the well, and the resultant head which 

is referred to an arbitrary datum 50 feet below the top of the well 

casing.

The data in table 1 concerning well discharge, aquifer discharge, and 

head are plotted on the composite specific-capacity graph hi figure 4.

PRESENTATION AND EVALUATION OF RESULTS

The head values plotted on figure 4 were obtained by adding 

algebraically the changes in head determined in the test to the static 

head of the well. Likewise, the internal-flow values on figure 4 were 

obtained by adding algebraically the observed changes in internal 

flow and the flow when the pump was shut down. Thus figure 4 

illustrates the performance of the well after 1 hour of pumping, for 

situations in which the water level had recovered to about prepumping level. If pumping were to be started from a different 

initial point, as it may in a well showing seasonal fluctuations of 

water level, the slopes of the straight lines drawn in figure 4 would 

remain the same, but the lines would be displaced so as to pass through 

the new initial point.

The first aquifer gave the highest specific capacity, about 27 gpm 

per foot of drawdown. The second aquifer gave a specific capacity 

of about 3 gpm per foot of drawdown and acted as a thieving zone 

until relatively high pump discharges (about 500 gpm) were achieved. 

The zones below the second aquifer including the lower sandstone 

on the logs, and all zones beneath the plug exhibited no yield, either 

positive or negative, during the test, and consequently they were not 

included in the specific-capacity graph. When the water level in the 

well was essentially static, the upper aquifer discharged 58 gpm to the 

thieving aquifer.

Because of the extreme time difference between the 1-hour pumping 

interval and the internal-discharge span of several years, the zerodischarge intercepts of the lines for the two aquifers are only crude 

approximations of the static heads of these aquifers. These approximate static heads are summarized together with the specific-capacity 

data in table 2.

A somewhat better approximation of the static heads might be 

obtained by extrapolating the drawdown data to several years, on 

the assumption that the aquifer discharges would remain constant over 

that length of time, and by replotting the specific-capacity graph. 

This method gives a static head of roughly 20 feet below the arbitrary 

datum for the thieving aquifer and of 45 feet above datum for the 

upper aquifer. These results are open to serious question on the 

grounds that the measurements of internal flow were neither numerous
24 GROUND-WATER HYDRAULICS

enough nor accurate enough to determine whether the two discharges 

were constant or were following trends. The linearity of the specificcapacity graphs suggests that the points are approximately those that 

would be attained by constant discharge. In general, however, more 

conclusive data on the time variation of the discharges should be 

obtained before large extrapolations are attempted.

TABLE 2. Data obtained from specific-capacity graph of figure 4

Aquifer 1__ , . _____ _________ ._ _, _ , , ____

Aquifer 2 - , _ _ - ______._- -_ _ _ Well.. _____________ - - - .-_ - -

Static head (ft above datum)

34 5

16

32. 5

Specific capacity 

at 1 hr (gpm per 

ft)

27

3

30

The greatest shortcoming of the analysis was clearly the limited 

accuracy of the method of measuring internal flow. Table 1 shows 

that many of the flow measurements have ranges of uncertainty of 2 

or 3 gpm. (Each point of figure 4 represents the mean value of the 

range of uncertainty in that particular measurement.) The total 

variation in the internal flow over the entire test was about 12 gpm. 

A more accurate method of flow measurement is undoubtedly necessary for work involving differences of this order. On an expanded 

scale the points on the lower graph of figure 4 all deviate from the line 

to some degree. The deviations are quite random, however, and 

although the straight line of figure 4 must be regarded as an approximation, it is certainly the best fit that can be made to the experimental 

data.

The authors felt that, for the measurement of downward flow, 

the brine-tracing method was somewhat more accurate than any of the 

other means at theu^disposal. A substantial improvement in accuracy 

might have resulted had a wider range of internal discharge been 

covered in the test, but the small capacity of the pump made this 

impossible.

CONCLUSIONS

Information on specific capacities, thieving, and hydrostatic 

heads of the various aquifers penetrated by a well can be of considerable importance in the construction and operation of the well, 

or in the development of a well field. Although greater accuracy of 

flow measurement is desirable, the linear trend of the experimental 

data shows that the method is useful. As more refined techniques 

of flow measurement become available, it should be possible to achieve 

results of greater reliability with much less difficulty in fieldwork and 

calculation.
SPECIFIC CAPACITY OF MULTIAQUIFEE WELLS 25

LITERATURE CITED

Barbagelata, A,, 1928, Chemical-electric measurement of water: Am. Soc. Civil

Engineers Proc,, v. 54, p, 789-802.

Bays, C.A., and Folk, S. H,, 1944, Geophysical logging of water wells in northeastern Illinois: Western Soc. Engineers Jour., v. 49, no. 3, p. 248-266. 

Bird, J. M., and Dempsey, J. C., 1955, The use of radioactive tracer surveys in

water-injection wells: Kentucky Geol. Survey Spec. Pub. 8, p. 44-54. 

Carslaw, H. S., and Jaeger, J. C., 1959, Conduction of heat in solids: London,

Oxford Univ. Press, 2d ed. 

Fiedler, A. G., 1928, The Au deep-well current meter and its use in the Roswell

artesian basin, New Mexico: U.S. Geol. Survey Water-Supply Paper

596-A, A, 24-32.

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 Traris., v. 33,

no. 4, p. 559-569. 

Jones, P. H , and Skibitzke, H. E., 1956, Subsurface geophysical methods in

ground-water hydrology, in Advances in geophysics, v. 3, p. 241-300: New

York, Academic Press, Inc. 

Meinzer, 0. E., 1928, Methods of exploring and repairing leaky artesian wells:

U.S. Geol. Survey Water-Supply Paper 596-A, p 1-3, 

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. 

Skibitzke, H. E., 1955, Electronic flow meter: U.S. Patent No. 2,728,225. 

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., pt. 2, p. 519-524.

o

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