Time, Distance, and Drawdown

Relationships in a Pumped

Ground-Water Basin

GEOLOGICAL SURVEY
Time, Distance and Drawdown 

Relationships in a Pumped

*

Ground-Water Basin

By Fred Kunkel

GEOLOGICAL SURVEY CIRCULAR 433

Washington, 

7960
United States Department of the Interior

FRED A. SEXTON, SECRETARY

Geological Survey

THOMAS B. NOLAN, DIRECTOR

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

Abstract- _______________

Introduction. _ ___________

Page 

_______ __ ____ 1

______________ 1

Computation of drawdown______,

References__-__-____--_- __-_.

Page 

.---_.__ 1

. ___ __ 8

ILLUSTRATIONS

Figure 1. Graphs showing relation between drawdown and time at observation points 1,000 

and 5,000 feet from pumping well_-___-_--____-___-______--__--__-_______.

Graphs showing relation between drawdown and time at observation points 500 

and 5,000 feet from pumping well_________________________________________

Page

3. Graphs showing relation between drawdown and time at observation point 5,000 

feet from well pumping 4,000 gallons per minute __________________________

4. Graphs showing relation between drawdown and distance from pumping well for 

selected times since pumping began _____________________________________

TABLES

Table 1. Time and quantity of pumpage required to lower the water level 10 feet at 

observation points 1,000 and 5,000 feet from a pumping well___________

2. Time and quantity of pumpage required to lower the water level 10 feet at 

observation points 500 and 5,000 feet from a pumping well ____________

3. Time and quantity of pumpage required to lower the water level 10 feet at 

an observation point 5,000 feet from a well pumping 4,000 gpm ________

Page 

3 

3 

6

III
Time, Distance, and Drawdown Relationships 

in a Pumped Ground-Water Basin

By Fred Kunkel

ABSTRACT

Several reasonable values are assumed for coefficients of 

transmissibility and storage of lenticular alluvial deposits. 

These values when substituted in the Theis (1935) nonequilibrium formula as modified by Wenzel (1942) give curves 

from which time, distance, drawdown relationships are 

estimated.

recharge, if any, induced by the declining 

water levels; (d) the reduction in natural discharge, if any, and (e) boundaries of the 

ground-water basin. The lowering at arty 

point of the cone of depression is termed 

drawdown and is dependent upon the above 

variables and the distance from the pumping.

INTRODUCTION

In the development of a ground-water basin 

one of the principal problems commonly is to 

determine the effect of pumping on the water 

level in the basin. Where all the geologic and 

hydrologic factors are known, it is possible 

to determine these effects. In many groundwater basins most of the geologic and hydrologic conditions are reasonably well known 

except the coefficients of transmissibility 

and storage of the water-yielding deposits. 

For these basins it is possible to assume 

reasonable values for the coefficients of 

transmissibility and storage, and by use of 

standard formulas estimate probable waterlevel declines caused by pumping. In this 

paper a range of values is assumed for coefficients of transmissibility and storage that 

are considered by the author to be reasonable for the Quaternary alluvial deposits of 

many ground-water basins and are analyzed 

with reference to water-level declines for 

various distances and times.

Withdrawal of water from any permeable 

material causes the water level to decline in 

the vicinity of the withdrawal, and around a 

pumping well the shape of the water table or 

piezometric surface is somewhat like an inverted cone, its apex at the point of withdrawal. This cone is usually called a cone of 

depression. However, the overall size, shape, 

and rate of growth of this cone of depression 

are dependent upon (a) the rate and duration 

of pumping; (b) the coefficients of transmissibility and storage; (c) the increase in

COMPUTATION OF DRAWDOWNS

In the following analysis it is assumed that 

the aquifer is constant in thickness, infinite in 

areal extent, homogeneous, and isotropic. For 

these conditions the relationship among the 

hydraulic properties maybe expressed by the 

nonequilibrium formula developed by Theis 

(1935)

114.60

in which S is drawdown, in feet, at any point, 

in the vicinity of a well pumped at a uniform 

rate; Qis the discharge of a well in gallons 

per minute; Tis the coefficient of transmissibility of the aquifer in gallons per day per 

foot; r is the distance of the discharge well 

from the point of observation, in feet; S is the 

coefficient of storage, a decimal fraction; and 

t is the time the well has been pumped in 

days.

As the coefficient of transmissibility appears on both sides of the equation, the formula cannot be solved directly for T and S. 

However, T and s may be determined by a 

graphical method described by Wenzel (1942) 

from the following formulas.

T =

114.6Ql?(u)
TIME, DISTANCE, AND DRAWDOWN RELATIONSHIPS IN A PUMPED GROUND-WATER BASIN

O 4>

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il

Jt

il

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Hi

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COMPUTATION OF DRAWDOWNS

in which W(u) is the "well function of a " and 

the other terms are as previously defined, and

tTu

C -» __ _- __- 

1.87r2

Conversely, if these formulas are modified 

as follows:

1.87f 2S 

Tu

it is possible to estimate or assume reasonable values for T and Sand construct curves 

that show the time required for any given 

drawdown at any assumed distance from a 

pumping well. Accordingly, the accompanying 

figures graphically show these relationships.

Figure 1 shows two sets of curves for observation points in an aquifer having a coefficient of transmissibility of 100,000 and a 

coefficient of storage of 0.1. The first set of 

three curves are for points 1,000 feet from 

wells pumping 1,000, 2,000, and 4,000 gpm 

(gallons per minute). The second set of two

curves are for points 5,000 feet from wells 

pumping 1,000 and 4,000 gpm. (See table 1.)

The curves for the observation point 1,000 

feet from the pumped well show that it will require about 60 years of pumping at 1,000 gpm 

to lower the water level 10 feet at the observation point. However, if the pumping rate is 

increased to 2,000 gpm the 10 feet of drawdown will take about 270 days. If the pumping 

rate is increased to 4,000 gpm the 10 feet of 

drawdown will occur in about 28 days. Likewise, at an observation point 5,000 feet from 

the pumped well it will require about 1,300 

years of pumping at 1,000 gpm to lower the 

water level 10 feet at the observation point. 

However, if the pumping rate is increased to 

4,000 gpm the 10 feet of drawdown will occur 

in about 700 days or less than 2 years.

Figure 2 shows two sets of curves for observation points in an aquifer having a coefficient of transmissibility of 10,000 and a coefficient of storage of 0.1. The first set of 

curves are for points 500 feet from wells 

pumping 200 and 400 gpm. The second set of 

curves are for points 5,000 feet from wells 

pumping 400 and 800 gpm. (See table 2.)

Table 1. Time and quantity of pumpage required to lower the water level 10 feet at observation points 1,000 and 5,000 feet from a

pumping well

Pumping rate 

(gpm)

1,000 __ _ ____

2,000 __________

4,000 __ __ ___

Observation point 1,000 feet 

from pumping well

Time required

22,000 days 

(60 years) 

270 days 

(3/4 year) 

28 days

Pumpage required 

(acre-feet)

97,000 

2,400 

500

Observation point 5,000 feet 

from pumping well

Time required

470,000 days 

(1,300 years)

700 days 

(1-9/10 years)

Pumpage required 

(acre-feet)

2,100,000

12,000

Table 2. Time and quantity of pumpage required to lower the water level 10 feet at observation points 500 and 5,000 feet from a

pumping well

Pumping rate

Observation point 500 feet 

from pumping well

Observation point 5,000 feet 

from pumping well

Time required Pumpage required 

(acre -feet)

Time required PumPaSe required 

(acre-feet)

200___________ 610 days

(1-2/3 years) 

400___ _____ 70 days

540

120

800

7,000 days

(19-1/5 years,! 

1,950 da -s

12,000

6,900
TIME, DISTANCE, AND DRAWDOWN RELATIONSHIPS IN A PUMPED GROUND-WATER BASIN

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COMPUTATION OF DRAWDOWNS
TIME, DISTANCE, AND DRAWDOWN RELATIONSHIPS IN A PUMPED GROUND-WATER BASIN

The curvos for the observation point 500 

feet from the pumped well showthat it will require about 610 days of pumping at 200gpmto 

lower the water level 10 feet at the observation point. However, if the pumping rate is increased to 400 gpm the 10 feet of drawdown 

will occur in about 70 days. Likewise, at an 

observation point 5,000 feet from the pumped 

well it will require about 7,000 days of pumping 

at 400 gpm to lower the water level 10 feet at 

the observation point. However, if the pumping 

rate is increased to 800 gpm the 10 feet of 

drawdown will occur in about 1,950 days.

Figure 3 shows a set of five curves for observation points 5,000 feet from wells pumping 4,000 gpm in aquifers having a coefficient 

of transmissibility of 100,000 and coefficients 

of storage of 0.2, 0.1, 0.01, 0.001, and 0.0001. 

These curves show that .for an S of 0.2 it will 

require about 1,400 days (3-3/4 years) of 

pumping at 4,000 gpm to cause a drawdown of 

10 feet 5,000 feet from the pumping well. For 

an S of 0.1, 0.01, 0.001, and 0.0001 it will require 700, 70, 7, and 0.7 days, respectively, for 

a drawdown of 10 feet to occur at a distance 

of 5,000 feet. (See table 3.)

In addition to the time required for the 10 

feet of drawdown to occur, figures 1-3 show 

also the total quantity of water, in acre-feet, 

that would be pumped for the various combinations of conditions. For example, figure 

2 shows that if a well, under the conditions 

assumed, were pumped at 200 gpm for 610 

days it would pump approximately 540 acrefeet of water and would cause water-level 

decline of 10 feet 500 feet from the well. If

Table 3. Time and quantity of pumpage required to lower 

the water level 10 feet at an observation point 5,000 feet 

from a well pumping 4,000 gpm

(Selected values are assumed from the coefficient of storage)

Coefficient 

of 

storage

0.2__ __ ____

. 1__. ______

.01____ __ _

.001_______

.0001__ __ _

Time required

1,400 days

(3-3/4 years) 

700 days «/

70 days

7 days «/

0.7 day

(16-4/5 hours)

Pumpage 

required 

(acre-feet)

25,000

12,000

1,200

120

12

the pumping rate were doubled (increased to 

400 gpm) the 10 feet of water-level decline 5.00 

feet from the well would occur in 70 days and 

only 120 acre-feet of water would be pumped.

Figure 4 shows a set of seven curves after 

the method of Theis(1938) for selected times 

in an aquifer having a coefficient of transmissibility of 100,000 and a coefficient of S of 0.1. 

Drawdown is plotted against distance from a 

well pumping 1,000 gpm at the end of 0.1, 1, 

10, 100, 1,000, 10,000, and 40,000 days. These 

graphs show that, under the conditions assumed, a drawdown of li feet would occur in 

2 hours and 24 minutes at a point 100 feet from 

the well pumping 1,000 gpm. However, it would 

require about 2-3/4 years of pumping for the 

same drawdown to occur at 9,000 feet. Similar relationships can be shown for other times.

It justifiably may be questioned if a theoretical analysis as presented herein is applicable to ground-water basins as they occur in 

nature. For example, figure 1 shows that if a 

well, under the conditions assumed, were 

pumped at 1,000 gpm for 1,300 years it would 

pump about 2,100,000 acre-feet of water and 

would cause a decline of water level of only 

10 feet 5,000 feet from the pumped well. There 

probably is no ground-water basin in which 

this would occur even if the coefficients of 

transmissibility and storage were as assumed. Unless the aquifer or basin were 

truly of infinite areal extent, the cone of depression would have gone to the boundaries of 

the aquifer, and resulting image effects would 

have caused a drawdown much greater than 10 

feet in less than 1,300 years. In virtually all 

aquifers of limited areal extent, boundary effects commonly result in drawdowns greater 

than those calculated for aquifers of infinite 

extent. Also, the preceding analysis does not 

allow for either an increase in recharge or a 

decrease in discharge which eventually must 

occur in any pumped aquifer.

However, figures 1 and 2 graphically demonstrate that an increase of pumping in an aquifer accelerates the rate of drawdown of the 

cone of depression at a rate proportionally 

much greater than the increased rate of 

pumping. Figure 3 graphically demonstrates 

that the rate of growth of the cone of depression is inversely proportional to the coefficient of storage. This is an important consideration because artesian aquifers commonly 

have coefficients of storage of about 0.001 or
COMPUTATION OF DRAWDOWNS

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8 TIME, DISTANCE, AND DRAWDOWN RELATIONSHIPS IN A PUMPED GROUND-WATER BASIN

less, whereas nonartesian aquifers commonly 

have coefficients of storage of 0.1 or more. 

Hence, the cone of depression in an artesian 

aquifer grows 50 to 100 times as fast as it 

does in nonartesian aquifers.

Figure 4 demonstrates that within a few tens 

of years even for relatively low rates of 

pumping significant water-level declines occur at distances of several miles from the 

pumping. If these rates are increased as 

shown by figure 1 the water-level decline occurs at a greatly accelerated rate.

In summing up this discussion the following 

points, in large part after Theis (1938 and 

1940), should be emphasized.

1. All water discharged by wells is balanced 

by a loss of water somewhere.

2. This loss is always to some extent, and in 

many cases largely, from ground water 

in storage in the aquifer.

3. However, after sufficient time has elapsed 

for the cone of depression to reach the 

area of recharge, further discharge by 

wells will be made up in part by an increase in the recharge if previously there 

has been rejected recharge.

4. Again, after sufficient time has elapsed for 

the cone of depression to reach the areas 

of natural discharge, further discharge 

by wells will be made up in part by a 

diminution in the natural discharge.

5. In artesian aquifers, compared to nonartesian aquifers, relatively little water is 

taken from storage. In artesian aquifers, 

because the cones of depression spread 

with great rapidity, each well in a short 

time has its maximum effect on the whole 

aquifer and obtains most of its water by 

an increase of recharge or decrease of 

natural discharge. In large nonartesian 

aquifers, where pumping is done at great 

distances from the localities of intake or 

outlet, however, the effects of each well 

are for a considerable time confined to a 

rather small radius and the water is taken 

from storage in the vicinity of the well.

6. In localities developing water from nonartesian aquifers and remote from areas of

rejected recharge or natural discharge, 

the conditions connoted by the concept of 

perennial yield1 may not be reached for 

several centuries and virtually all the 

water used may be taken from storage. 

If pumping in such a locality is at a rate 

that will result during the course of 10 

years in a lowering of water level to a 

depth from which it is not feasible to 

pump, pumping at half this rate would not 

cause this lowering in 100 years. However, the converse also is true. If past 

pumping in an aquifer of limited extent 

has caused only a moderate decline of 

water levels during the period of pumping, 

doubling the rate of pumping will in the 

future cause a decline of water level 

many times greater than past pumping.

7. Finally, even if a well discharges only a 

short time and draws its water for the 

time entirely from ground water in storage in the aquifer, eventually the aquifer 

must either discharge just that much less 

water through natural outlets or take in 

that much more water in the intake area. 

Hence, the cone of depression can have 

no ultimate limits short of at least one of 

the hydrologic boundaries of the aquifer.

REFERENCES

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.

 1938, The significance and nature of the 

cone of depression in ground-water bodies: Econ. Geology, v.33, no. 8, p. 889-902. 

-1940, The source of water derived from

wells: Am. Soc. Civil Engineers, Civil 

Eng., p. 277-280.

Wenzel, L. K., 1942, Methods for determining 

permeability of water-bearing materials: 

U.S. Geol. Survey Water-Supply Paper 

887.

iPerennial yield of a ground-water basin may be defined as the 

rate at which ground water canbe withdrawn year after year without depleting the ground-water storage to such an extent that 

withdrawal at this rate is no longer feasible because of increased 

pumping costs or deterioration of 'water quality.