A Simple Method for 

Determining Specific 

Yield from Pumping Tests

By L. E. RAMSAHOYE and S. M. LANG

GROUND-WATER HYDRAULICS

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1S36-C

Prepared in cooperation with the

New Jersey Department of Conservation

and Economic Development

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON t 1961
U.S. DEPARTMENT OF THE INTERIOR 

BRUCE BABBITT, Secretary

U.S. GEOLOGICAL SURVEY 

Robert M. Hirsch, Acting Director

Any use of trade, product, or firm names in this publication is for

descriptive purposes only and does not imply endorsement by the

U.S. Government

First printing 1961 

Second printing 1965 

Third printing 1993

For sale by U.S. Geological Survey, Map Distribution

Box 25286, MS 306, Federal Center

Denver, CO 80225
CONTENTS

Page 

Abstract- __________________________________________________________ 41

Introduction_______________________________________________________ 41

Theory.._______________________________________________________ 41

Application to test data from Kearney, Nebr.._.___._____..______.-___ 45

Literature cited___________________________________________________ 46

ILLUSTRATIONS

FIGURE 11. Diagram showing drawdown of the water table in the vicinity

of a pumped well-.___.______________________________ 42

in
GROUND-WATER HYDRAULICS

A SIMPLE METHOD FOR DETERMINING SPECIFIC YIELD 

FROM PUMPING TESTS

By L. E. RAMSAHOYE and S. M. LANG

ABSTRACT

A simpler solution which greatly reduces the time necessary to compute the 

specific yield by the pumping-test method of Remson and Lang (1955) is presented. 

The method consists of computing the volume of dewatered material in the cone of 

depression and comparing it with the total volume of discharged water. The 

original method entails the use of a slowly converging series to compute the 

volume of dewatered material. The solution given herein is derived directly 

from Darcy's law.

INTRODUCTION

A pumping-test method to determine the specific yield of a watertable aquifer was presented by Remson and Lang (1955). The 

method involves the determination of the volume of dewatered 

material in the cone of depression during the course of a pumping test. 

The specific yield is then determined by comparing the volume of 

dewatered material with the total volume of discharged water.

The calculation of the volume of dewatered material requires the 

solution of an exponential series that converges very slowly and is, 

therefore, time consuming. The example presented by Remson and 

Lang needed 60 terms of the series and required more than ? hours of 

computation. This paper presents a more easily evaluated equation 

for rapidly computing the volume of dewatered material in the cone 

of depression.

The work was carried out under the supervision of Aller Sinnott, 

district geologist, as part of the investigation of the ground-water 

resources of New Jersey in cooperation with the New Jersey Department of Conservation and Economic Development. The senior 

author, a hydrologist-geophysicist with the British Guiana Geological 

Survey, participated in this study as a foreign trainee unde^ the program sponsored by the International Cooperation Administration.

THEORY

As pointed out by Remson and Lang, it may not be possible to apply 

the standard formulas to data from a pumping test in a shallow watertable aquifer because of the slow drainage of the aquifer material

41
42 GROUND-WATER HYDRAULICS

during the test and (or) because of a varying rate of discharge. However, the general equilibrium formula can be applied if a pumping rate 

Q is constant for a long enough period so that the cone of depression 

reaches approximate equilibrium form and is declining only very 

slowly. The condition of approximate equilibrium is described by 

Wenzel (1942, p. 98-99),

... as pumping continues, a hydraulic gradient that is essentially an equilibrium 

gradient will be established close to the pumped well, and water will be transmitted to the well through the water-bearing material in approximately the 

amount that is being pumped. The decline of the water table and the resulting 

unwatering of material in this area will then be much slower.

The assumptions used in the development of the general equilibrium 

formula and those used by Remson and Lang also apply here. The 

following is quoted from Remson and Lang (1955, p. 322). "Although 

the water table continues to decline slowly, the assumption that 

steady-state conditions have been reached involves only £, slight error 

no greater than that resulting from such a cause as fluctuation in pump 

discharge." The following paragraph, adapted from Wenzel (1942, 

p. 77), describes the requisite conditions of the test:

An isotropic and homogeneous water-bearing bed of infinite a.real extent is 

assumed to rest on a relatively impervious formation. The discharging well, 

equipped with a pump, is fully screened to the bottom of the water-bearing 

material. It is assumed that water movement from the outer radiu? of the screen

Pumped 

well Ground surface

FIGUBE 11. Diagram showing drawdown of the water table In the vicinity of a prmped well.
METHOD FOR DETERMINING SPECIFIC YIELD FROM PUMPING TESTS 43

to the pump intake occurs without loss of head or with a head loss that is regligible 

compared with the drawdown in the well. The water table before pumping, and 

the underlying impervious bed, are assumed to be horizontal. It is assumed also 

that there is no recharge to the aquifer during the test and that all the water 

pumped is removed from storage.

Figure 11 shows the shape of the cone of depression in the vicinity 

of a well pumping from a water-table aquifer.

The following symbols or nomenclature are used in the mathematical derivations in this report :

Q=the discharge rate of the pumped well in gallons per day 

P=the field coefficient of permeability of the aquifer in gallons per day per 

square foot under a unit hydraulic gradient and at the prevailing water 

temperature 

r = the horizontal distance from the axis of the pumped well to a point on the

cone of depression, in feet 

s = the drawdown at distance r, in feet

stt = the drawdown just outside the screen of the pumped well, in fe°t 

m=the thickness of the zone of saturation before pumping or the height of

the static water table above the aquifer bottom, in feet 

T Pm= the coefficient of transmissibility of the aquifer in gallons pe^ day per 

foot. It is the flow through a vertical strip of the aquifer 1 foot wide and 

extending the saturated height of the aquifer, at unit hydraulic gradient.

From Darcy's law:

Q=27rrP(-^)(m~S) (1) 

Therefore

Q 

where

dr , . T , x 7 /c^. (m s)ds= a(m s)ds (2)

Integrating,

as2 

Inr ams+- -=-- f InB (3)

z

where B is the constant of integration. 

Then

r^Be- '***2'* (4)

Equation 4 describes the cone of depression when it has reached 

virtually an equilibrium shape or position.

The volume of dewatered material in cubic feet, V, within the cone 

of depression is

(5)
44 GROUND-WATER HYDRAULICS

the limits of integration being chosen at zero drawdown (for example, 

the extent of the cone at equilibrium) and the drawdown outside the 

screen of the pumped well. The value for r in equation 4 may be 

substituted in equation 5 to give

'ds (6)

Observe in equation 6 that the exponent of e may be written in the 

equivalent form [ 2ams(l sj2m)].

Because s/2m is generally very small compared to unity, it may be 

neglected; therefore equation 6 becomes

2amt fa (7)

2am

For values found during field pumping tests

2amsw>l

Hence 2amtu is very small and can be neglected. Thu^ equation 9 

becomes

(10)'

From equation 4 B=ream°-a**i'i

Substituting the value for B in equation 10 there follow?

2am

If the exponent of e is again modified, in the manner shown in writing 

equation 7, it follows that equation 11 may be written in the form

2am

In equation lla the volume of dewatered material is expressed in 

terms of permeability, pumping rate, horizontal distance, drawdown,
METHOD FOR DETERMINING SPECIFIC YIELD FROM PUMPING TESTS 45

and aquifer thickness. In field practice, it is often necessary to make 

pumping tests, using wells that only partly penetrate the aquifer or 

for which incomplete data are available. Thus, it may not be possible 

to determine the coefficient of permeability, P, or the full e.quifer 

thickness, m. Under such circumstances, equation 10 canrot be 

used to determine the volume of dewatered material in the cone of 

depression. However, if the drawdown, s, at the point of observation 

is small compared to suspected thickness of the zone of saturation, 

the thickness may be assumed to remain uniform and transmissibility, 

T, may be used in place of the unknown permeability and rquifer 

thickness (T=Pm). Several standard ground-water formulas permit 

the direct determination of the coefficient of transmissibility. Therefore, equation lla may be further modified by substituting therein 

the equivalents 2ir PfQ for a, and T for the product Pm, which yields

Q.

(12) 4T 

Taking the logarithm of both sides of equation 12 produces

, T7 , Qrz . , log F=log fy+-7p log e

or

, T7 , Qr* , 5.45 Ts 

, 1QN (13)

The specific yield is the volume of water pumped during the test 

divided by the gross volume of dewatered material within the cone of 

depression.

where S= specific yield

<?=the average discharge rate of the pumped well, in gallons

per day

t=the time, in days, since pumping began 

V=the volume of dewatered material, in cu. ft., determined 

from either equation lla or 13.

APPLICATION TO TEST DATA FROM KEARNEY, NEBPASKA

The formulas derived herein were applied to field data from a 

pumping test near Kearney, Nebr. (Wenzel, 1942, p. 127-131). The
46 GROUND-WATER HYDRAULICS

same data were used by Remson and Lang (1955, p. 324). The data 

used are

m=48 ft

Q=l,584,000 gpd

s after 24 hr=3.97 ft for r=50 ft

s after 24 hr=2.99 ft for r=100 ft

P=4,100 gpd per square foot

T=Pm=1.96X105 gpd per foot

The specific yield computed for the drawdown at the 50-foot distance was 9 percent whereas its value for the drawdown at the 100-foot 

distance was 10 percent, which is the same value that was computed 

by Remson and Lang using the lengthy series.

The formulas derived herein may be used only with data from an 

equilibrium pumping test and the test should be long enough to 

permit the greatest possible dewatering in the cone of depression 

without it being affected by recharge.

LITERATURE CITED

Remson, Irwin, and Lang, S. M., 1955, A pumping-test method for the determination of specific yield: Am. Geophys. Union Trans., v. 36, p. 321-325.

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

materials with special reference to discharging-well methods: U.S. Geol. 

Survey Water-Supply Paper 887, 192 p.

a U.S. G.P.O. 1993-342-389:80018