r

Methods for Determining the 

Proper Spacing of Wells in 

Artesian Aquifers

METHODS OF AQUIFER TESTS

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1545-B
Methods for Determining the 

Proper Spacing of Wells in 

Artesian Aquifers

By S. M. LANG

METHODS OF AQUIFER TESTS

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1545-B

The Theisnonequilibrium formula is 

modified to obtain equations useful 

in determining proper well spacing 

in artesian aquifers

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1961
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEW ART L. UDALL, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

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

Washington 25, D.C.
CONTENTS

Page

Abstract-____--_____--__-____--_______________---____-_---_------ B-1

Introduction.____________-_-_______-_______________-____---------- 1

Areally extensive aquifer___________________________________________ 2

Pumped wells near a geologic or hydrologic boundary________-_-_--__-_ 5

Pumped wells near an impermeable barrier._______--____---_--_-_ o

Wells arranged in a line parallel to the boundary.______--___-- 6

Wells arranged in a line at an angle to the boundary.__________ 9

Pumped wells near a recharge boundary-_______----_-_--_-------- 12

References cited ---_____-_________________________..--_-----.------ 16

ILLUSTRATIONS

Page

FIGURE 1. Locations of real and image wells near a straight-line boundary.. B-6 

2. Plot of the curve y=/(r2 )=r_4 + 2X10V23 +4X106r22 -26X1012 _ 6

m

METHODS OF AQUIFER TESTS

METHODS FOR DETERMINING THE PROPER SPACING OF 

WELLS IN ARTESIAN AQUIFERS

By S. M. LANG

ABSTRACT

The Theis nonequilibrium formula can be modified to provide for convenient 

solution of innumerable problems of limited scope in ground-water hydraulics. 

An example of the type of modification of the formula that can be made is presented herein.

Well spacing is a problem that cannot be solved directly by the application of 

the nonequilibrium formula or any of the available modifications. The formula 

is altered to obtain equations useful in determining the proper well spacing in 

artesian aquifers for the following situations: (a) pumped wells in an areally 

extensive aquifer, (b) pumped wells in an aquifer bounded on one side by a 

perennial stream, the wells being arranged in a line either parallel or at an angle 

to the boundary, and (c) pumped wells in an aquifer bounded by an impervious 

barrier, the wells being arranged in a line either parallel or at an angle to the 

boundary.

INTRODUCTION

A great advance in ground-water hydrology was made with the 

development of the nonequilibrium formula (Theis, 1935). This 

formula provides a means of determining the hydraulic characteristics of an aquifer; these characteristics define the ability of the 

aquifer to store and transmit water. The formula has also been used 

extensively in evaluating well performance and in predicting changes 

in water levels in aquifers of large areal extent and relatively uniform permeability.

The nonequilibrium formula may be used to obtain answers to 

many quantitative problems having to do with the recharge, movement, and discharge of ground water. However, problems remain 

that cannot be solved by the direct application of the formula, 

although it is possible to modify the nonequilibrium formula to provide shortcuts for solving innumerable problems of limited scope. 

In the past, modifications have been developed that have simplified 

methods of applying the formula (Cooper and Jacob, 1946) and have 

extended its application to situations where the pumped aquifer is 

limited \)y geologic or hydrologic boundaries (Stallman, 1952). 

This paper presents a further example of the sort of modification of the 

nonequilibrium formula that can be made.

Well spacing remains as one of the problems that cannot be solved 

directly by the application of the nonequilibrium formula or of any

B-l
B-2 METHODS OF AQUIFER TESTS

available modifications. Theis (1957) presented a method for determining well spacing in a thick and areally extensive aquifer; however, 

the prime consideration in this method was economics. According to 

the method, the farther apart the wells are the less their mutual 

interference caused by pumping but the greater the cost of pipeline 

and electrical installations. By equating the added cost of pumping 

due to mutual interference against the capitalized cost of these 

installations, Theis was able to derive an equation for optimum well 

spacing. In many situations, however, economics may be of secondary 

importance, or other considerations may be involved.

One or more of the following factors may be of importance in 

determining the proper spacing of pumped wells: (a) physical limitations of the aquifer, such as depth, thickness, piezometric head, or 

areal extent; (b) depths of existing wells; (c) limitations on available 

pumping equipment, such as suction lift, setting of bowls, or horsepower of motors; or (d) place and purpose of use of the water.

Presented herein is a modification of the nonequilibrium formula 

by means of which it is possible to determine the proper spacing of 

pumped wells in an artesian water-bearing formation, taking into 

account the physical limitations of the pumped formation.

AREALLY EXTENSIVE AQUIFER

The nonequilibrium formula is

114.6Q TI7, . /1N 

s= W(u) (1)

, . ,ON where u= ^ (2)

s= drawdown produced by the pumped well, in feet, 

Q= discharge rate of the pumped well, in gallons per minute, 

T=coefficient of transmissibility of the aquifer, in gallons per 

day per foot,

/jt2 nfi nA

__+_____+ . . .

(W(u) is known as the "well function of u")t

r= distance from the pumped well to the point of observation, 

in feet (where the pumped well is the point of observation, 

r is the effective radius of the well), 

S= coefficient of storage of the aquifer, dimensionless, 

2= time since pumping began, in days.
PROPER SPACING OF WELLS IN ARTESIAN AQUIFERS B~3

Most of the terms are self-explanatory. However, the coefficients 

of transmissibility and storage need further elaboration. The coefficient of transmissibility of an aquifer, T, is the rate of flow of water, 

under prevailing conditions including water temperature, in gallons 

per day, through a vertical strip of the aquifer 1 foot wide extending 

the full saturated height of the aquifer, under a hydraulic gradient 

of 100 percent. The coefficient of storage of an aquifer, S, is the 

volume of water it releases from or takes into storage per unit surface 

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

to that surface.

When two wells are pumped simultaneously, the total drawdown in 

each well is the sum of the effects due to its own pumping and to the 

interference of the other well. If their pumping rates are the same, 

then

(3)

where sp~ total drawdown in the first pumped well (well 1), in feet, 

Si~ drawdown in the first pumped well (well 1) due to its own

pumping, in feet, 

s2 == drawdown in the first pumped well produced by interference of the second pumped well (well 2), in feet, 

W(u)i= value of W(u) at well 1 resulting from its own pumping, and 

W(u)2== value of W(u) at well 1 resulting from the pumping of

well 2.

When t becomes sufficiently large that u is equal to or less than 0.02, 

the value of W(u) can be approximated by the first and second terms 

of the series expression, and equation 3 may be rewritten in the form

s,=^^ (-2(0.5772)-loge w^) (4)

log, ^=-+1.1544. (5) 

Converting to logarithms to the base 10,

(6)

By substituting the proper values for Tand Q and the desired value 

for sp in equation 6, the product of u^ can be computed directly. 

Then, from equation 2,

(^Tr1 r2)s (7)
B-4 METHODS OF AQUIFER TESTS

where

A T r__l.

?"!= the effective radius of wel] 1, 

and r2=the distance from well 2 to well 1. 

Therefore,

(8)

Within the limitation that u<0.02, equation 8 can be used to 

determine the well spacing in an areally extensive aquifer. The 

following example illustrates the use of the method.

Example 1. A water user requires about 500 gpm (gallons per 

minute) for 10 hours per day for 5 days per week. Ground-water 

investigations have identified an areally extensive shallow artesian 

aquifer (values of T and S are 50,000 gpd per foot and 1X10~4 , 

respectively). Test drilling has demonstrated that it is feasible to 

construct a 24-inch supply well capable of being pumped at a rate of 

500 gpm. However, another nearby water user already has a well 

tapping this particular aquifer operating at the same rate and schedule 

as the proposed well. Inasmuch as two 500-gpm wells will be pumping 

simultaneously, it is desired to determine the minimum spacing of the 

wells at which the artesian head of the aquifer will not be lowered 

excessively. It is determined that the permissible total drawdown in 

each well is 30 feet. If it is assumed that the total effect of all well 

losses (losses in head due to entrance of water into the well and travel 

of water in the well) and the residual drawdown due to the intermittent 

operation of the wells is 2 feet, the drawdown caused by the daily 

operation of the wells can be only 28 feet. It is assumed also that 

the effective radius of the proposed well is equal to the actual radius 

of the well.

Thus, sP=2S feet, T =50,000 gpd per foot, S=1X10~4 , <?=500 gpm, 

£=0.417 day, and ^ = 1 foot. 

From equation 6,

__/28X50,000 

~ V 264X500

log uiu*= 11.107 

=8.893-20.
PROPER SPACING OF WELLS IN ARTESIAN AQUIFERS B~5

Therefore,

p^l.sy^l.syXlXlO"4 p 

Tt 50,000X0.417

From equation 8,

ra= ~

8.97X10- 9 Xl 

= 312 feet.

PUMPED WELLS NEAR A GEOLOGIC OB HYDBOLOGIC

BOUND ABY

Stallman (1952) modified the nonequilibrium formula to broaden 

the scope of its use to situations where the pumped formation is 

adjacent to another formation of vastly different transmissibility, 

and the boundary between the formations approximates a straight 

line. The most common examples of this situation might be 

(a) permeable material of a buried valley abutting the relatively 

impermeable walls of the valley, and (b) an aquifer that is hj^draulically connected with a body of surface water. In either example, 

the solution is made by use of the image-well theory (Ferris, 1949). 

In the first example, the condition is imposed that no water flows 

across the boundary; in the second example, the condition is imposed 

that no drawdown occurs along the boundary. These conditions 

are simulated by assuming that the aquifer is infinite in extent and 

that an imaginary (image) well lies the same distance from the 

boundary as the pumping well but on the opposite side of the 

boundary. In the first example, the image well would operate at the 

same time and rate as the pumped well. In the second example, the 

image well would operate at the same time and rate but would be a 

recharging well instead of a discharging well. In this manner, the 

problem is converted from one involving boundary conditions to one 

of a multiple-well system in an infinite aquifer.

PUMPED WELLS NEAR AN IMPERMEABLE BARRIER

Each pumped well near a boundary has its own image well. Therefore, the total drawdown in each pumped well is the algebraic sum of 

the effects of all pumped and image wells.

602719 61-
B-6 METHODS OF AQUIFER TESTS

WELLS ARRANGED IN A LINE PARALLEL TO THE BOUNDARY

First, consider the case where two wells are arranged in a line 

parallel to the boundary (fig. 1A):

J. (9)

Image

well

a

Boundary

o 

o 

o

J[_ Pumped ^ 

well

Boundary

Pumped 

well 2

image well 

b

Pumped 

well 2

FIGURE 1. Location of real and image wells near a straight-line boundary. A, Wells parallel to 

the boundary; B. wells at an angle to the boundary.
PROPER SPACING OF WELLS IN ARTESIAN AQUIFERS B~7

When u is less than 0.02, the value of W(u) is given by W(u) 0.5772

 Iogei4 as discussed above.

Hence,

(-0.5772-loge ui-0.5772

-log, w2 -0.5772-loge Ua 0.5772 loge M>) (10)

«,-?^(-4 (0.5772)-loge WWW) (11)

log, «i«8wB«»=--+2.3088« (12)

Converting to logarithms to the base 10,

/ s T

log ttitt^tt»= +l ' (13)

The value of the second term on the right side of equation 13 is 

approximate; the actual value is 1.002.

By substituting the proper values of sp, T, and Q into equation 13, 

the product u^u^uau^ may be computed directly. From equation 2

1 070 

UiU2Uaub=K4tri2r2zra2r^) where K= 'j, (14)

It follows that

, , r i 7- UlUzUaUb /1Kx rW==Lt where L^-^-J-J- (15)

-tX /i / a

However, note from figure 1A that

?'B2 = ?'22 + ?'a2

and from equation 15

iYW+ra2)=i. (16) 

Thus,

r24+raV-Z=0. (17)

Again within the limitation that i*<0.02, equation 17 can be used 

to determine well spacing when the wells are to be arranged in a line 

parallel to an impermeable boundary. This equation is a quadratic 

of the classic form a»2 4-6x+c=0, where in this case aj=r»2. <z=l«
B-8 METHODS OF AQUIFER TESTS

b ra2, and c= L. The quadratic is solved by the formula

X~ 2a ' ^e ne^a^ve root °f this formula has no significance to the solution. The following example demonstrates the use 

of these equations.

Example 2. A new project is to have its own ground-water supply. 

It is estimated that it will require an average of 300 gpm to meet its 

expected demands. Although it is possible to obtain this amount 

from one well, it is decided to construct a second well in order to 

insure an adequate supply during periods of peak demand, as well as 

for a safeguard in case of well or pump failure. It is possible that 

the two wells will be operated simultaneously for periods as long as 

5 days.

Ground-water investigations have identified a relatively shallow 

artesian aquifer (T=3Q,QQQ gpd per foot, S=1X10-4 ) that is cut 

off by an impermeable boundary. Test drilling has demonstrated 

the feasibility of constructing 12-inch wells, each capable of being 

pumped continuously at 300 gpm, arranged in a line about 1,000 

feet from the boundary. It is desired to determine the well spacing 

if it is assumed that the well losses are negligible and that the permissible total drawdown is 40 feet in each well at the end of the 5 days.

The analysis evidently will require the use of two discharging 

image wells placed across the boundary opposite the supply wells 

and at a distance of 2,000 feet from the supply wells (fig. \A).

Therefore, sp=40 feet, ^=30,000 gpd per foot, £=1X10~4, #=300 

gpm, ra=2,000 feet, ^=0.5 foot, t=5 days. From equation 13,

= 740X30,000 \ 

\ 264X300 + /

= -16.2 

=3.8 20

Tt 30,000X5

6 6.3X10- 17

______ 

a* 2.41X10-36 X0.25X4X106

=2.6X1013 .
PROPER SPACING OF WELLS IN ARTESIAN AQUIFERS B~9

Solving equation 17 by the formula

x=-

2a

r24+4X10V-2.6X1013=0

-4X106 ±yi6X10l!4-4X2.6X1013

r2= 1,900 feet.

WELLS ARRANGED IN A LINE AT AN ANGLE TO THE BOUNDARY

A more complex problem would result from arranging the wells in 

a line at an angle to the boundary (fig. IB). 

Restating equation 15,

2 2 T U T

rfrf=Lt where L= .

'1 'a

Inasmuch as the image-well distances from the boundary are the 

same as for the real wells, except that they are on opposite sides 

of it, rb may be stated in terms of the known distance, ra) the angle, 

a, and the unknown well spacing, r2. Thus, applying the law of 

cosines, and recognizing that rb is also equal to the distance from 

image well a to pumped well 2,

r?=r£+r* 2rjrt cos (180° a) 

and

r62=r22 +ra2 -f 2rar2 cos a. (18)

Substituting equation 18 into equation 15,

r22 (r22+ra8+2rara cos «)=£' (19) 

and

cos a+rflV22 L'=Q. (20)

Equation 20 is a quartic equation that can be solved readily by 

graphics. By substituting various values of r2 into the modified 

equation 20 i

r24+2ra cos ca>f+r*rf-Lf (21)

corresponding values for y are obtained. The values for r2 and y 

then serve as coordinates for points plotted on rectangular coordi-
B-10 METHODS OF AQUIFER TESTS

nate paper. If the r2 values are taken along the horizontal axis, a 

curve fitted to the plotted points will yield the proper value for r2 

where it crosses the horizontal axis. The following example illustrates 

the use of these equations.

Example 3. Assume the same field situations as described in 

example 2 except that well 2 is located farther from the boundary 

and the line of wells makes an angle (a) of 60° with the impermeable 

boundary.

Therefore,

ra=2,000 feet, sp=40 feet, jT=30,000 gpd per foot,

^=0.5 foot, a=600, cos a=0.5, t 5 days. 

As previously determined, L'=26X1012 . Equation 20 is then

r24+2X103r23 +4X10V22 -26X1012 =0. 

Substituting values for r2 into the above,

r*

1X103

1.5X103

1.75X103

2X103

y=ra4 +2X10W+4X108r32 -26X1018

(l+2+4-26)1012=-19X1012

(5.1 + 6.8-f9.0-26)1012=-5.1X1012

(9.4+10.7+ 12.2-26)1012=6.3X1012

(16-H6+16-26)1012=22X1012

The computations shown above indicate that the wells should be 

spaced at a distance between 1,500 and 1,750 feet. The values for 

r2 and y are plotted (fig. 2) and the place where the curve fitted to the 

points crosses the horizontal axis represents the exact solution in 

this case about 1,600 feet.

An inspection of equation 20 indicates that the sum of the terms 

containing r2 must equal the value of the Lr term. Thus, a simple
PROPER SPACING OF WELLS IN ARTESIAN AQUIFERS B~ll

means of estimating the magnitude of r2 is to take a fourth of the 

power to which 10 is raised in the L' term. In the above example, 

10 is raised to the 12th power in the L' term. A fourth of 12 is 3. 

Therefore, all assumed values of r2 have a magnitude of 103, and all 

terms in equation 21 have a magnitude of 1012 .

y

-2

xio

30

20

10

12

XIO 3

FIGURE 2. Plot of the curve jr=/(r2)=r34-r-2X10W+4X106r22-26X1012. The curve indicates the solution 

of the problem where the pumped wells are arranged in a line at an angle to an impervious barrier.
B-12 METHODS OF AQUIFER TESTS

PUMPED WELLS NEAR A RECHARGE BOUNDARY

The total drawdown in each of two pumped wells near a recharge 

boundary is the algebraic sum of the individual effects of the pumped 

and image wells

,-W(u}a--W(u},}. (22)

When u is less than 0.02, W(u) is approximately

Therefore,

114 60 

sP ^r-^ ( 0.5772 loge ul 0.5772 log e«g

+0.5772+loge ufl +0.5772+loge u6) (23) 

and

i i A a f\ -. -.

(24)

Converting to logarithms to the base 10,

(25)

If the wells operate simultaneously, the only variable in the u is r2 and

sp~ log '"g 2 - (26)

Rearranging,

i ra2r62 sPT

and

=M, where M^antilog J. (28)

When the wells are placed parallel to the boundary (fig.

(29)

Hence,

2 2\z\

~M (30)

(32>

'2 'a
PROPER SPACING OF WELLS IN ARTESIAN AQUIFERS B~13 

^=^-1 (33)

and

r 2

r2 = f /"_== (35)

The determination of the proper well spacing when the wells are 

placed at an angle to the boundary (fig. IB) is made as follows: 

Repeating equation 28,

r Vft2 s T -~^=M. where M=antilog "

Applying the law of cosines, we repeat equation 18: 

r&2=r22 4-ra2 4-2^2 Cos a

ra2 r2 2+ra2 +2rar2 cos <*)

r22+ra2 +2rar2 cos a=^~ M (37)

-r22 (38)

a

ra2 +2rar2 cos a=r22 f ~ M- 1) (39)

\^a /

r22 C 3 M l)-2rar2 cos a ra2=0. (40)

\^a /

Equation 40 is a quadratic of the form ax2 -\-bx-\- c=Q, where

rizM 

a 2 1, b= 2ra cos a, and c= ra2. Thus, equations 35 and

T&

40 permit the determination of proper well spacing near a recharge 

boundary when the line of wells is parallel to and at an angle to the 

boundar}7 , respectively. The following example illustrates the use 

of these equations.

Example 4- It is desired to install a water-supply system. Groundwater investigations on a flood plain have identified a shallow 

artesian aquifer (T=50,000 gpd per foot, /S'=1X10~4 ) capped by 

about 35 feet of fine-grained alluvium. Test drilling has located a 

site about 1,000 feet from the stream for a 12-inch well capable of 

producing 600 gpm continuously. A site for a second well is to be 

located, and it is desired to determine the proper well spacing if the
B-14 METHODS OF AQUIFER TESTS

second well is to be arranged in a line either parallel to the stream 

or at an angle to the stream such that a=30°. Because of the small 

thickness of confining material, the total drawdown in each well 

should not exceed 30 feet. See figure 1 for an arrangement of wel]s. 

Evidently the analysis now requires the use of two recharging image 

wells placed across the recharge boundary.

Assuming that well losses are negligible, T=50,000 gpd per foot, 

£=1X10-4, r=0.5 foot, ra=2,000 feet, sp=30 feet, #=600 

a=30°, cos a=0.866

^^30X50,000 . ^= 

264X600 ^ " ^

From equation 35, we can determine the proper spacing for the case 

where the wells are arranged in a line parallel to the stream:

ra2 4X106 4X106

r2=-r-

-rfl2 V0.25X3.2X109 -4X106 2.8X104 

r2=140 feet if the wells are arranged in a line parallel to the stream. 

Equation 40 is solved by means of the quadratic formula

which gives the proper spacing for the case where the wells are arranged 

in a line at an angle to the stream.

rSM 0.25X3.2X109 , 10Q a 5 1= ~ 6 1 = 199 

ra2 4X106

b=~2ra cos o:=-2X2X103 XO.S66=-3.5X103 

c=_ra2=_4X106 .

3.5X103 ±Vl2.2X106+3.2X109 Therefore, rz=

398 

3.5X103 ±56.7X103

398

r2 =150 feet if the wells are arranged in a line at an angle 

of a=30° from the stream.

An inspection of equation 26 indicates that the total drawdown 

in a pumped well near a recharge boundary is dependent upon the
PROPER SPACING OF WELLS IN ARTESIAN AQUIFERS B~15

rate of discharge, the transmissibility of the aquifer, and the well 

distances. Time is not involved hi the equation; therefore, the drawdown or pumping level stabilizes and becomes a constant factor when 

u is less than 0.02, The time required for the drawdown to stabilize 

can be determined from the following analysis : The value of u varies 

with the square of the distance (r); if the time necessary for u to 

become as small as 0.02 is determined for the greatest distance in the 

well setup, u will be less than 0.02 for any other distance for that same 

time. In other words, the pumping level in the operating wells may 

be considered to have reached approximate stabilization at the time 

necessary for u at the maximum distance to equal 0.02.

When only one operating well is involved, the maximum distance 

is ra, the distance between the pumped well and its image well. With 

two operating wells, the maximum distance is rb, the distance between 

the first pumped well and the image well of the second pumped well 

(fig. 1). Thus, for the example given above, when the r2 values have 

been determined the maximum distances may be computed from equation 18 or 29 and then substituted in the following form of equation 2:

If the wells are arranged in a line parallel to the boundary, from 

equation 29

r62= (2,000) 2+ (140)2=4,020,000

rd=2,000 feet. 

Thus,

1.87X(2,000)2 X1X10-* ? 

50,000X0.02

If the wells are arranged in a line at an angle to the boundary, from 

equation 18

?V=r22 -|-?*a2 ~l~2rar2 cos «

r62=(150)2 +(2,000) 2+2X2,OOOX150X0.866=4,542,000 

r6 =2,130 feet. 

Thus, from equation 41

1.87X(2,130) 2 X1X1Q-* s 

50,000X0.02
B-16 METHODS OF AQUIFER TESTS

It must be remembered that the total permissible drawdown in the 

pumped wells must be specified before the equations presented in 

this paper can be applied. This may be done on the basis of (a) past 

experience with other wells tapping a particular formation; (b) 

physical limitations of the aquifer, such as depth, thickness, or areal 

extent; and (c) limitations imposed by pumping equipment, such as 

pumping by suction, setting of bowls, or horsepower of motors. 

The maximum permissible drawdown having been estimated, the 

accompanying equations then provide a convenient means of determining the proper spacing of pumped wells.

REFERENCES CITED

Cooper, H. H., Jr., and Jacob, C. E., 1946, A generalized graphical method for

evaluating formation constants and summarizing well-field history: Am.

Geophys. Union Trans., v. 27, no. 4, p. 526-534. 

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

Hydrology: New York, John Wiley and Sons, p. 198-272. 

Stallman, R. W., 1952, Nonequilibrium type curves modified for two-well systems:

U.S. Geol. Survey open-file report, 4 p., 1 fig. 

Theis, C. V., 1935, The relation between the lowering of the piezometric surface

and the rate and duration of discharge of a well using ground-water storage:

Am. Geophys. Union Trans., p. 519-524, August. 

 1957, The spacing of pumped wells: U.S. Geol. Survey open-file report,

4 p.

U.S. GOVERNMENT PRINTING OFFICE: 1951