A Formula for Computing 

Transmissibility Causing 

Maximum Possible Drawdown 

Due to Pumping

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1536-F
A Formula for Computing 

Transmissibility Causing 

Maximum Possible Drawdown 

Due to Pumping

By G. M. ROBINSON and H. E. SKIBITZKE

GROUND-WATER HYDRAULICS

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1536-F

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

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

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

Washington 25, D.C.
CONTENTS

Page

Abstract__ ______________________________________________________ 175

Introduction-_-___-_-_-__--_____-_-_---_____-_______-_____--______ 175

Effects of pumping-_______________________________________________ 176

Development of formula__________________________________________ 176

Illustration of formula-_________-____-___-----_-_---_-_-_-_---_-___ 178

Summary_ _ ________-_____-_-._______-_--____-----___-___-__._______ 179

References..-_____________________________________________________ 180

ILLUSTRATIONS

FIGURE 46. Profiles of drawdown cones near a pumped well-____________ 176

47. Graph showing point at which e~u =W(u) __________________ 178

48. Graph showing maximum possible drawdown for a given

radius and time-_____________________________________ 179

49. Graph of transmissibility at which drawdown is maximum for

a given radius and time-______________________________ 180

in
GROUND-WATER HYDRAULICS

A FORMULA FOR COMPUTING TRANSMISSIBILITY 

CAUSING MAXIMUM POSSIBLE DRAWDOWN DUE TO 

PUMPING

By G. M. ROBTNSON and H. E. SKIBITZKE

ABSTRACT

By modifying the Theis nonequilibrium formula a relation is found in which 

the maximum possible drawdown is expressed in terms of a unique value for 

the aquifer coefficient of transmissibility. The relation is valid for any specified 

period and rate of pumping, for a given aquifer coefficient of storage, and for 

any desired radial distance from the center of pumping.

INTRODUCTION

When planning ground-water investigations, it is often desirable 

to analyze the manner in which the release of ground water from 

storage affects the time required for changes in head to migrate from 

a center of pumping to the area of natural discharge. In many instances the effects of release would predominate over such a long 

period of time that the water from storage would become the controlling feature of the ground-water development in the region. 

Many ground-water reservoirs are of such large areal extent that 

pumping throughout any foreseeable economic or practicable pumping 

period would not cause drawdowns sufficient to recover water from 

the region of natural discharge. The purpose of this paper is to show 

the development of a formula describing for any desired pumping 

rate, pumping period, and distance from the pumped well the particular aquifer transmissibility at which the greatest possible drawdown occurs.

It is recognized that in studying any real hydrologic problem, a 

detailing of the effects of pumping and the release of water from storage requires knowledge of the variation in the coefficients of transmissibility and of storage throughout the region. If these data were 

known, it would be possible to describe the drawdown that would 

occur throughout the region because of any given pumping regimen. 

In the usual absence of such detailed data, however, highly significant 

perspectives of the hydrologic problem can be obtained through some 

idealization of the aquifer and use of the equation developed and 

described in this paper.

175
176 GROUND-WATER HYDRAULICS 

EFFECTS OF PUMPING

If a water-table aquifer were infinite in area! extent and transmissibility, pumping from the aquifer would cause no drawdown anywhere. If such an aquifer were infinite in transmissibility but not 

infinite in areal extent, the effect of pumping would be a uniform 

drawdown throughout the aquifer and the water table would remain 

a plane surface. If the aquifer were not infinite in transmissibility 

there would be a cone of depression that is somewhat steep sided near 

the well and that flattens out with distance from the well (Theis, 

1940); if the aquifer material were relatively impermeable, the drawdown cone near the well would be very steep sided. The relatively 

steep sides of the drawndown cone would flare out and intersect the 

horizontal or nearly horizontal water tables representative of the 

first two sets of conditions postulated in this paragraph.

DEVELOPMENT OF FORMULA

Obviously, for a specified rate and period of pumping, the drawdown near the pumped well increases as successively smaller aquifer

Re/otive/y impermeable

FIGURE 46. Profiles of drawdown cones near a pumped well.
FORMULA FOR TRANSMISSIBILITY CAUSING DRAWDOWN 177

transmissibilities are considered. Furthermore, as shown in figure 46, 

the intersection of the drawdown profiles with the plane of the water 

table occurs at distances that are progressively nearer to the pumped 

well as successively smaller transmissibilities are considered. The 

preceding statements suggest that for a specified steady and continuous 

rate of pumping, from an aquifer having a specified coefficient of 

storage, there is a unique combination of maximum possible drawdown 

and aquifer transmissibility for any given elapsed pumping time and 

distance from the pumped well. The expression for the maximum 

drawdown, in terms of the coefficient of transmissibility, can be found 

by differentiating the familiar Theis (1935) formula, which has the 

nondimensional form (Brown, 1953, p. 851)

Q 8= ~^fW (^

Differentiating

d* ___ Q_ ,,_Q_ ?>W(u) du "~ em 'dT

du_ r*S _ u 

T

ds The maximum or minimum may be found by setting 0, or

but

du u 

Hence

W(u)=e-»=-Ei(-u) (Wenzel, 1942).

The value of u where e~u = W(u) is obtained graphically, as shown 

in figure 47. The intersection of the two lines, representing plots 

of W(u) versus u and e~u versus w, occurs at values of W (u) 

0.64738 and u= 0.43482. From this value of u the transmissibility can 

be calculated for specified values of storage coefficient, elapsed pumping time, and distance from the pumped well. This value of transmissibility can be used to compute the maximum possible drawdown 

for a given rate of pumping. Thus the Theis formula can conven-
178 GROUND-WATER HYDRAULICS

iently be rewritten in nondimensional form to give an expression for 

the maximum drawdown in terms of the aquifer transmissibility:

where

= (0.647)^ (1)

0.6475

0.6471

0.4348 0.4350

FIGURE 4,7. Graph showing point at which e-u = W(u).

ILLUSTRATION OF FORMULA

The implications of equation 1 are illustrated in figure 48, where 

the maximum possible drawdown is plotted as a function of radius 

and time, and for the particular value of transmissibility determined 

for each selected combination of radius and time. The plot is for a 

pumping rate of 1,000 gpm (gallons per minute) and an aquifer
 FORMULA FOR TRANSMISSIBILITY CAUSING DRAWDOWN 179

1000

100 -

Storage coefficient, s=0.1 

Pumping rate, 0= 1000 gpm

1000 10,000 100,000 

RADIAL DISTANCE, r, IN FEET

FIGURE 48. Graph showing maximum possible drawdown for a given radius and time.

storage coefficient of 0.10. Figure 49 also illustrates the significance 

of equation 1 by showing the transmissibility associated with the 

maximum drawdown for a given radius and time. This plot is also 

computed for a pumping rate of 1,000 gpm and a storage coefficient 

of 0.10. Equation 1 can be used to plot graphs similar to figures 48 

and 49 for any other given values of storage coefficient and pumping 

rate.

SUMMARY

The relation shown as equation 1 can be used to determine quickly 

the maximum effects of proposed or predicted pumping in a region 

and to analyze the significance of these effects before proceeding with 

a hydrologic study. For example, if the maximum possible drawdowns at various points of interest in a very extensive aquifer indicate no drawdown in the region of recharge, it could be concluded
GROUND-WATER HYDRAULICS,

103 104 

RADIAL DISTANCE, r, IN FEET

105

49. Graph of transmissibility at which drawdown is at a maximum for a given 

radius and time.

that the problem always would be one of developing ground water 

from storage and would be independent of any relationships between 

recharge and the coefficient of transmissibility of the aquifer.

REFERENCES

Brown, B. H., 1953, Selected procedures for analyzing aquifer test data : Amer.

Water Works Assoc. Jour., v. 45, no. 8, p. 844-866. 

Theis, C. V., 1985, The relation between the lowering of the piezometric surface

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

Ain. Geophys. Union Trans., pt. 2, p. 519-524. 

 1940, The source of water derived from wells. essential factors controlling the response of an aquifer to development : Am. Soc. Civil Engineers,

Civil Engineering, v. 10, no. 5, p. 277-280. 

Wenzel, L. K., 1942, Methods for determining permeability of water-bearing

materials with special reference to discharging-well methods : U.S. Geological Survey Water-Supply Paper 887, p. 87-89.

U. S. GOVERNMENT PRINTING OFFICE : 1962 O - 655515