Idf not destroy or throw away this publication. If you have no further use 

rrte to the Geological Survey at Washington and ask for a frank to return it

UNITED STATES DEPARTMENT OF THE INTERIOR 

Harold L. Ickes, Secretary

GEOLOGICAL SURVEY 

W. C. Mendenhall, Director

Water-Supply Paper 679 A

OS THIEM METHOD FOR DETERMINING 

[EABILITY OF WATER-BEARING MATERIALS

ITS APPLICATION TO THE DETERMINATION 

OF SPECIFIC YIELD

RESULTS OF INVESTIGATIONS IN 

THE PLATTE RIVER VALLEY, NEBRASKA

BY

LELAND K. WENZEL

Prepared in cooperation with the Conservation and Survey 

Division of the University of Nebraska

Contributions to the hydrology of the United States, 1935

(Pages 1-57)

Water Resources 

Geological Survey,

Box 3106, Car

UNITED STATES

GOVERNMENT PRINTING OFFICE 

WASHINGTON : 1936

Iby the Superintendent of Documents, Washington, D. C. Price 10 cents
CONTENTS

Abstract. ____-____----______________---_______----_-------------- 1

Introduction __ ____-________--__---__--__--_-__---_------------- 2

Investigation in the Platte Valley, Nebr______________---_____--_ 2

Acknowledgments. ____________--_--__-_-_----_---------------- 3

Hydrologic properties of water-bearing formations_______-__-____,--- 4

Outline of Thiem and pumping-test methods-_______-_---____-------- 8

Permeability. ______--_--_--________--___---_________--------- 8

Specific yield-______-____--___________-------_-_____---------- 9

Development of TMem 'a formula.__________-_-_-_----_-___--------- 10

Confirmation of Thiem's formula from other work done in the United

States..___________________________---___-_._.- ISFormula for determining the cone of depression-___.-______-------_---- 23

Graphic solution of Thiem's formula____________________-------_-__ 25

Pumping tests in Nebraska_________-________________--------_--_- 26

Results obtained from the pumping tests_______________--__--_-___- 32

Draw-down curves.____________-_----_________-----------____- 32

Recovery curves__________----_-___---______-_-_--------__l___ 34

Cones of depression______-___--_---_________-.-----------__---- 36

Computation of coefficients of permeability_____________________ 41

Differences between field and theoretical conditions _______________ 50

Determination of specific yield by the pumping method.___________ 53

ILLUSTKATIONS

Page 

PIRATE 1. Graph for computing coefficients of permeability.____________ 24

2. A, Line SW of 1-inch observation wells; B, Measuring the depth

to the watertable_______________________________________ 28

3. A, Weir for measuring the discharge of the pumped well;

B, Pumping arrangement in second test_________________ 29

4. Typical draw-down and recovery curves for test l____-____-__ 36

5. Typical draw-down and recovery curves for test 2____________ 36

6. Contours on the water table before pumping and at several times

after pumping began__________________________________ 36

FIGURE 1. Plan and section of ideal ground-water conditions assumed

by Thiem__________-___-_-_________________________ H

2. Plan and section showing assumed ground-water conditions for

the development of the formula from horizontal water table. 16

3. Plan and section showing assumed ground-water conditions 

for the development of the formula from horizontal artesian 

conditions____________________________________________ 17

4. Map showing location of wells used in pumping tests._______ 27

5. Computed recovery curve for well 5, test !_________________ 37

6. Profiles of the cone of depression at several times after pumping 

began and location of cylindrical sections used for computing specific yield-_______-_-__-_-----____________._____ 39

7. Relation of 2z>2 to P__________________________________ 48

in
TABLES

Page

1. Physical properties of samples of alluvium._________.--_____-___-__ 28

2. Log of well 84_-..----_-___._......___--__.__-_-._..___. 28

3. Location, diameter, depth, and altitude of wells______.......--___-_ 29

4. Record of pumping time_________________________________________ 31

5. Draw-down of the water table during test 1_.______________________ 36

6. Coefficients of permeability computed by Thiem's formula for all possible combinations of observation wells on line A_______-..--__--_ 42

7. Coefficients of permeability computed by Thiem's formula for several 

combinations of observation wells on line A and for.several periods 

of pumping.____._--__-______-_____-----_--___-_--__-------- 43

8. Coefficients of permeability computed by Thiem's formula, using a 

50 feet_---_____.-....-._._.____.....__-____--.___ 44

9. Draw-down of water table for several distances and directions from

the pumped well._----_-____-_--------- _.__----_---___---_ 45

10. Differences in average draw-down on lines A and C____------------ 45

11. Final computation of coefficients of permeability__________-- _--_- 47

12. Computation of v* for P=975......-..._-----_------------------ 48

13. Observation equations for t/2 _______________-_-_-_____-------_--_ 49

14. Volumes of water-bearing material unwatered around the pumped

well _________-_-__-_-_--__-___--___----__-_-________------- 54

15. Computed quantities of ground water that percolated through several

concentric cylindrical sections around the pumped well-______..._ 54

16. Quantities of ground water taken from storage___-_--_-__--_-_---- 55

17. Specific yield as computed for several concentric cylindrical sections

and for several periods of pumping-____-_____--___-_---____--- 55

IV
THE THIEM METHOD FOR DETERMINING PERMEABILITY 

OF WATER-BEARING MATERIALS AND ITS APPLICATION 

TO THE DETERMINATION OF SPECIFIC YIELD

By LELAND K. WENZEL

ABSTRACT

The Thiem method for determining permeability of water-bearing materials * 

consists of pumping a well, or, where the ground water is confined under pressure, 

allowing the well to flow and observing the decline of the water table or piezometric surface in nearby observation wells. The coefficient of permeability is 

computed by the formula

527.7 q logio -

where P is the coefficient of permeability; q is the rate of pumping, in gallons a 

minute; a and 01 are respective distances of two observation wells from the 

pumped well, in feet; m, for artesian conditions, is the vertical thickness of the 

water-bearing bed, in feet; m, for water-table conditions, is the average vertical 

thickness, at a\ and a, of the saturated part of the water-bearing bed, in feet; and 

s and si are the draw-downs at the two observation wells, in feet. This formula 

is mathematically developed by assuming ideal geologic and ground-water conditions, such as a uniform permeability, a uniform thickness of water-bearing bed, 

s horizontal water table or piezometric surface, and a cone of depression that has 

reached equilibrium in form. As these conditions are rarely approached, the 

applicability of the formula and hence of the method has been regarded as 

questionable.

Two rather elaborate pumping tests were made in 1931 near Grand Island, 

Nebr., to ascertain the accuracy of the Thiem method and to investigate the possibilities of determining specific yield by a pumping test. The behavior of the 

ground water was observed over a large area around the pumped wells by measuring the fluctuation of the water table in 81 observation wells during the period of 

pumping and after pumping was stopped. A study of the data obtained from 

these tests indicates that the Thiem method is applicable to conditions that are 

found in nature. However, to obtain consistent and accurate determinations of 

permeability it is necessary to employ an arbitrary procedure in computing the 

coefficient. The draw-down of the water table at any distance from the discharging well should be taken as the average of the draw-down at that distance upgradient and down-gradient from the well. In Thiem's formula only results for 

the draw-down of the water table that are obtained from the part of the cone of 

depression that has reached approximate equilibrium in form can be used. The 

part of the cone that has reached approximate equilibrium is determined by frei Thiem, Q., Hydrologische Methoden, Leipzig, 1906.
2 CONTRIBUTIONS TO HYDROLOGY OF UNITED STATES, 1935

quent measurements of the draw-down during the period of pumping. If the 

discharging well fails to penetrate through the water-bearing bed, the draw-down 

of the water table close to the well should not be used, because of irregularities in 

the cone of depression. Moreover, there are usually near the well some changes 

in the permeability of the water-bearing material resulting from the development 

of the well. In the first test described in this report the cone of depression reached 

approximate equilibrium in form out to about 200 feet from the pumped well 

after 48 hours of pumping and was affected by irregular conditions near the well 

as far as 40 feet from the well. Hence the draw-downs that were used for 

computations of permeability were selected from that part of the cone between 

40 and 200 feet from the pumped well. In the second test pumping was stopped 

several times, and the cone of depression did not reach approximate equilibrium 

in form.

Computations were made to determine the specific yield of the water-bearing 

materials from the data obtained in the pumping tests. The results show that 

the specific yield can be readily determined by this method. Samples of the 

material were analyzed in the laboratory for specific yield, and the results obtained 

compared favorably with those determined by the pumping method.

INTRODUCTION

INVESTIGATION IN THE PLATTE VALLEY, NEBR.

An investigation of the ground-water resources of Nebraska has for 

some time been in progress under the supervision of G. E. Condra, 

director of the Conservation and Survey Division of the University 

of Nebraska. At the request of Dr. Condra, a ground-water investigation of that part of the Platte River Valley lying between Chapman 

and Gothenburg, Nebr., was undertaken July 1,1930, as a cooperative 

project between the Conservation and Survey Division and the United 

States Geological Survey, under the general supervision of O. E. 

Meinzer, geologist in charge, division of ground water in the Geological 

Survey. The writer was assigned to this cooperative project and 

began work July 12,1930.

The investigation has for its purpose the determination of the 

source, quantity, and availability of the ground water, with a view to 

accomplishing maximum recovery and utilization. Field work has 

been carried on continuously since the project was begun, and comprehensive data have been collected concerning the occurrence and behavior of the ground water in that part of the valley. The area is 

one in which there is rather intensive irrigation by ground water, and 

it was found that determinations of permeability and specific yield 

of the water-bearing materials should be made in order to obtain a 

quantitative estimate of the ground-water supply. It was decided 

to use the Thiem method for the determination of permeability and 

the pumping method for the determination of specific yield.

Neither of these methods had been adequately verified by experiments for accuracy and practicability. Hence it was necessary to 

make rather elaborate tests that would determine the reliability of 

these methods and at the same time yield the actual figures for perme-
THIEM METHOD FOE DETERMINING PERMEABILITY 6

ability and specific yield. Accordingly, two tests of these methods 

were made in the summer of 1931 near Grand Island, Nebr.

The pumping method for determining specific yield was outlined by 

Meinzer 2 independent of the Thiem method for determining permea-' 

bility. However, the data necessary for the determination of specific 

yield and permeability by these methods can be obtained from one 

pumping test with a very small amount of additional effort. Where 

the ground water is confined under pressure the Thiem method for 

determining permeability may be used, but the pumping method for 

determining specific yield fails because there is usually no unwatering 

of the water-bearing materials. Hence the method for determining 

specific yield is strictly a pumping method, but the Thiem method 

applies also to areas where wells discharge water under artesian 

pressure. In the area where the tests described in this report were 

made the water-bearing materials consist of unconsolidated sand and 

gravel. As the upper surface of the zone of saturation lies several 

feet below the top of the water-bearing materials, the ground water is 

not confined under pressure, and both the Thiem method for determining permeability and the pumping method for determining specific 

yield could be used.

The behavior of the ground water near the pumped wells was observed in detail during and after the period of pumping, and these 

observations provided an opportunity to determine the effect of 

differences between theoretical and observed conditions on the 

computations for permeability and specific yield by these methods. 

The results obtained from the tests, a review of Thiem's development 

of the formula for permeability, and a theoretical review of the formula 

are incorporated in this report. Another report, now in preparation, 

will give the other data obtained in the Platte River investigation and 

the conclusions that were reached as to the ground-water conditions in 

that valley.

ACKNOWLEDGMENTS

The writer is indebted to G. E. Condra, A. L. Lugn, H. A. Waite, 

H. P. Burleigh, Howard Haworth, and Keith Miller, of the Conservation and Survey Division of the University of Nebraska, and to E. E. 

Brackett, chairman, and E. B. Lewis, research engineer, of the Agricultural Engineering Department of the University of Nebraska, for 

their assistance in making the tests, and to Fred Meyer for his cooperation in making his farm and pumping plant available for carrying out 

the tests. He is especially indebted to J. W. Shew, professor of 

mathematics, Junior College, Grand Rapids, Mich., and to C. E. Van 

Orstrand, geophysicist, United States Geological Survey, for their 

review of the mathematical sections of this report; to C. E. Slichter,

2 Meinzer, O. E., Outline of methods for estimating ground-water supplies: U. S. Geol. Survey WaterSupply Paper 638, p. 136, 1932.
4 CONTRIBUTIONS TO HYDROLOGY OF UNITED STATES, 1935

dean of the Graduate School, University of Wisconsin, and G. Thiem, 

consulting hydrologist, Leipzig, Germany, for their criticism of the 

manuscript; and to O. E. Meinzer, geologist in charge of the division 

of ground water of the Geological Survey, for his advice and suggestions throughout the investigation and for his review of the manuscript.

HYDROLOGIC PROPERTIES OF WATER-BEARING 

FORMATIONS

Most ground-water investigations are concerned with the quantity 

of water that is available for use by man. Perhaps the greatest 

difficulty in the determination of this quantity lies in the variability 

in the texture and hence in the hydrologic properties of the waterbearing materials. The hydrologic properties vary greatly, even with 

apparently slight differences in texture. Hence the ordinary geologic 

descriptions are quite inadequate for hydrologic investigations, and 

quantitative descriptions based on laboratory determinations have 

become essential.

The two hydrologic properties of greatest significance are permeability and specific yield. Mechanical analyses and determinations 

porosity and moisture equivalent are useful chiefly as indirect 

means of determining these two essential hydrologic properties.

About 1843 Poiseuille 3 discovered the law of flow through capillary 

tubes namely, that the rate of flow is .proportional to the hydraulic 

gradient. Later Darcy 4 verified this law and demonstrated its 

application to water percolating through the capillary interstices of 

sand and other porous media. He expressed this law by means of the

Kp'

formula v= j > in which v is. the velocity of the water through a fi

column of permeable material, pf the difference in head at the ends 

of the column, h the length of the column, and K a constant that 

depends upon the character of the material, especially on the size of 

the grains. Because it is usually more essential to determine the 

quantity of water flowing through a certain cross section of permeable 

material than to determine the velocity through the material, Darcy's 

formula is sometimes expressed as

in which Q is the quantity of water discharged in a unit of time, P the 

constant, which depends upon the texture of the material, / the 

hydraulic gradient, and A the cross-sectional area through which the 

water percolates. This formula serves as a basis for determining the 

quantities of ground water that percolate from areas of recharge to

* Poiseuille, J,, Recherches expfirimen tales sur le mouvement des liquides dans les tubes de trSs petits 

diamStres: Acad. sci. Paris M6m. sav. fitrang., vol. 9, p. 433, 1846. 

1 Darcy, H., Les fontaines publiques de la ville de Dijon, Paris, 1856.
THIEM METHOD FOR DETERMINING PERMEABILITY 5

areas of discharge, and consequently it is used for determining the 

safe yields of ground-water supplies.

The constant P in equation 1 is the most difficult factor to determine. The hydraulic gradient of an area can be obtained from contour 

maps of the water table or piezometric surface,5 and the cross-sectional 

area of the water-bearing material can be approximately determined 

from the logs of wells penetrating the material. The constant P has 

been designated by different names and has been expressed in various 

units. According to the present usage of the United States Geological 

Survey, it is called the "coefficient of permeability", defined as the 

rate of flow, in gallons a day, through a square foot of cross section, 

under a hydraulic gradient of 100 percent, at a temperature of 60° F.6 

In field terms the coefficient of permeability may be expressed as the 

number of gallons a day at 60° F. that is conducted laterally through 

each mile of the water-bearing bed under investigation (measured at 

right angles to the direction of flow), for each foot of thickness of bed 

and for each foot per mile of hydraulic gradient.6 Coefficients of 

permeability range widely. Fine sand is in general less permeable 

than coarse sand and therefore transmits less water through equal 

cross-sectional areas under the same hydraulic gradient. Clay may 

contain more water per unit volume than sand or gravel, but the 

permeability of a clayey material is generally low, and therefore the 

quantity of water transmitted through it is usually much less than 

is transmitted through sand and gravel. Coefficients of permeability 

ranging from 0.005 for clay to more than 20,000 for sand and gravel 

have been determined in the hydrologic laboratory of the United 

States Geological Survey.

The permeabilities of water-bearing materials may be determined 

by laboratory tests of samples of the materials or by determinations 

of ground-water velocities in the field. Hazen 7 and Slichter 8 have 

studied the rate of flow of water through sand and have developed 

formulas which essentially include the determination of the permeability of the sand. King 9 has reviewed the results of the investigators

»The upper surface of the zone of saturation in ordinary soil or rock is called the "water table." If a well 

Is sunk it remains empty until it enters a saturated permeable bed that is, until it enters the zone of 

saturation. Then water flows into the well. If the rock through which the well passes is all permeable the 

first water that is struck will stand in the well at about the level of the top of the zone of saturation that 

is, at about the level of the water table. If the rock overlying the bed in which water is struck is impermeable 

the water is generally under pressure that will raise it in the well to some point above the level at which it 

was struck. In such a place there is no water table, and the imaginary surface to which the water rises under 

its full head is called the "piezometric surface."

6 Stearns, N. D., Laboratory tests on physical properties of water-bearing materials: U. S. Qeol. Survey 

Water-Supply Paper 596, p. 148,1928.

' Hazen, Alien, Some physical properties of sands and gravels: Massachusetts State Board of Health 

24th Ann. Kept., p. 553, 1892.

s Slichter, C. S., The motions of .underground waters: U. S. Geol. Survey Water-Supply Paper 67, p. 26, 

1902.

6 King, F. H., Principles and conditions of the movements of ground water: U. S. Qeol. Survey 19th Ann. 

Kept., pt. 2, pp. 178-204, 1898.

18274 36 2
6 CONTRIBUTIONS TO HYDROLOGY OF DNITED STATES, 1935

on the flow of water through porous media and has described an 

apparatus for measuring the flow in the laboratory. 10 In the Geological Survey the permeability of water-bearing materials is now determined in the laboratory by means of apparatus devised by Meinzer. 11 

The coefficient of permeability of a water-bearing material is determined directly by measuring the rates of flow of water through a 

sample of the material with known cross section and thickness under 

observed differences of head. The laboratory methods are open to the 

criticism that the coefficients of permeability of the samples tested 

may differ widely from the average coefficient of the material as found 

in nature. The material that is tested in the laboratory must 

necessarily be removed from the ground, and as a result, especially 

with the more unconsolidated material, the soil particles do not 

remain in their original arrangement. Moreover, the coefficients of 

permeability determined in the laboratory necessarily apply only to 

very small samples, and unless a great number of samples are tested 

an average coefficient for a large area cannot be determined. These 

statements do not imply that laboratory determinations are not 

significant; they are intended merely to point out some of the inherent 

difficulties involved in such tests and to emphasize the importance of 

carefully and thoroughly investigating a method such as the Thiem 

method, which determines permeability in the field over a large area 

and without disturbing the water-bearing material.

A method for determining the natural velocities of ground waters, 

patterned after the method of the German hydrologist A. Thiem, was 

developed by Slichter. 12 Several small wells are driven into the waterbearing materials in such a manner that the water moves from one 

well toward one or more of the other wells. A salt is introduced into 

the up-gradient well and is allowed to move down-gradient with the 

ground water to the other wells, where its arrival is detected electrically. The rate of movement of the salt and hence the rate of movement of the ground water is computed from the elapsed time between 

the introduction of the salt in the central well and its detection in a 

well located down-gradient. The quantity of water flowing through 

a given cross-sectional area of the water-bearing material is computed 

by the formula

where Q= quantity of water;

p porosity of the water-bearing material; 

A= cross-sectional area; 

v= average velocity of the ground water.

" King, F. H., op. cit., p. 228. 

« Stearns, N. D., op. cit., p. 144. 

» Slichter, C. S., op. cit., p. 48.
THIEM METHOD FOR DETERMINING PERMEABILITY /

The coefficient of permeability of the water-bearing material is 

computed by equating equations 1 and 2:

=PZA and P-=.. ........ ......(3) J.^i 4

There are difficulties in the use of this method. The method is 

not satisfactorily adaptable to localities where the ground water has 

low velocity, because the salt solution, whose specific gravity is higher 

than that of the natural water, sinks rather rapidly and may not 

reach the down-gradient wells. In using this method in such a 

locality, the wells are located comparatively close to one another 

usually about 4 feet apart. Under these conditions errors in determining the velocity of the ground water are often introduced by 

failure to sink the wells exactly plumb, by the diffusion of the salt 

solution, and by increase in the hydraulic gradient caused by the 

rise of water in the up-gradient well at the time the salt is introduced.

The specific yield of a water-bearing formation is defined by 

Meinzer as the ratio of (1) the volume of water which, after being 

saturated, it will yield by gravity to (2) its own volume. 13 It is a 

measure of the quantity of water that a formation will yield when it is 

drained by lowering of the water table. Thus if 100 cubic feet of 

saturated water-bearing material when drained will supply 20 cubic 

feet of water, the specific yield of the material is said to be 20 percent.

The practical use of the specific yield is obvious. The quantity of 

water that a saturated material will furnish from storage depends upon 

its specific yield. To estimate the water supply obtainable from a 

material for each foot that the water table is lowered, or to estimate 

the available supply represented by each foot of rise in the water table 

during periods of recharge, it is necessary to determine the specific 

yield.

Meinzer 14 gives seven more or less distinct methods of determining 

specific yield namely, (1) saturating samples in the laboratory and 

allowing them to drain; (2) saturating in the field a considerable 

body of material situated above the water table and above the 

capillary fringe and allowing it to drain downward naturally; (3) 

collecting samples immediately above the capillary fringe after the 

water table has gone down an appreciable distance, as it commonly 

does in summer and autumn; (4) ascertaining the volume of sediments 

drained by heavy pumping, a record being kept of the quantity of 

water that is pumped; (5) ascertaining the volume of sediments 

saturated by a measured amount of seepage from one or more streams; 

(6) making indirect determinations in the laboratory with small

is Meinzer, O. E., Outline of ground-water hydrology: V. S. Geol. Survey Water-Supply Paper 494, 

p. 28, 1923.

n Meinzer, O. E., Methods for estimating ground^ water supplies: U. S. Geol. Survey Water-Supply 

Paper 638, p. 113, 1932.
8 CONTRIBUTIONS TO HYDROLOGY OF UNITED STATES, 1935

samples by the application of centrifugal force; and (7) making 

mechanical analyses and determinations of porosity and estimating 

therefrom the specific retention and the specific yield.

Much work has been done on the determination of specific yield by 

able investigators, but the methods just enumerated are still not 

thoroughly developed. The method of determining specific yield 

from pumping tests probably is the least developed of all.

OUTLINE OF THIEM AND PUMPING-TEST METHODS

PERMEABILITY

The Thiem method is very simple in principle. It consists of 

pumping a well that penetrates water-bearing material, the permeability of which is to be determined, and observing the decline of the 

water table or piezometric surface around the pumped well. Ground 

water obeys the law of fluids in that it always flows away from a 

point of high pressure toward one of low pressure. In other words, 

it flows in the direction of the hydraulic gradient. When a well is 

pumped some water inevitably is taken out of storage from the well 

and from the material surrounding it. This reduces the pressure, 

creates a hydraulic gradient toward the well, and causes ground water 

to flow into the well. If the water-bearing formation has a water 

table, considerable ground water may have to be removed from 

storage before a gradient will be developed that is steep enough to 

make the water flow toward the well at the rate that it is pumped and 

thus establish approximate equilibrium. If the formation is filled 

with water under pressure only a comparatively small amount of 

water has to be removed from storage in order to give the required 

gradient, and hence the draw-down will be more rapid and approximate equilibrium will be more quickly established.

When, with a constant rate of pumping, equilibrium is established, 

water is no longer removed from storage around the well but flows to 

the well as rapidly as it is withdrawn. If before pumping begins the 

water table or piezometric surface in a homogeneous formation is 

horizontal, water percolates toward the pumped well equally from all 

directions, and the same quantity of water percolates toward the 

pumped well through each of the indefinite series of concentric 

cylindrical sections around the pumped well. Because the areas of 

the large cylinders through which the water percolates are greater 

than the areas of the smaller cylinders, the velocity of the ground 

water passing through them is proportionally less and the hydraulic 

gradients are proportionally smaller.

According to equation 1 the discharge through any of the concentric 

cylindrical sections of water-bearing material, Qt is equal to PiA, and

the permeability of the material, P, equals - The symbol i is used in
THIEM METHOD FOR DETERMINING PERMEABILITY 9

this report to represent the hydraulic gradient at a point on the cone of 

depression around a well that is discharging water, and the symbol /is 

used to represent the normal hydraulic gradient that the water table 

or piezometric surface possesses when the well is idle. The two 

symbols are interchangeable in equation 1, their use depending upon 

whether the water table, or piezometric surface is cone-shaped or is 

approximately a plane. As previously explained, after approximate 

equilibrium has been reached the discharge through all concentric 

cylindrical sections of water-bearing material is the same, and the 

total discharge is equal to the quantity of water being pumped from 

the well. The hydraulic gradient at a given distance from the pumped 

well can be determined from the slope of the water table or piezometric 

surface. For artesian conditions the area of the cylindrical section 

through which the ground water percolates at that distance from the^ 

pumped well is equal to 2irxm, if x is the distance from the pumped 

well and m is the thickness of the water-bearing material. For 

water-table conditions the area is equal to 2irx(m-- s), where s is the 

draw-down at the distance x from the pumped well. Thus the permeability of the water-bearing material can be computed by substituting

these figures in the equation P = ^'

In 1906 G. Thiem, 15 son of the German hydrologist A. Thiem, published the results of his work in connection with the determination of 

additional water supply for the city of Prague and its suburbs. In 

this investigation he used what has since been known as the "Thiem 

method" for determining permeability and sunk 10 sets of wells, each 

set including 1 well that was pumped and 2 observation wells. The 

observation wells-were placed in line with the pumped well but in any 

convenient direction regardless of the natural hydraulic gradient. A 

formula was developed for computing the permeability from the data 

obtained from the pumped well and the two observation wells.

SPECIFIC YIELD

The determination of specific yield by the'pumping method is based 

on the withdrawal of water from storage during the period of pumping. 

Water is taken from storage until an approximate equilibrium is 

reached. Thus for a time the flow through successive concentric 

cylindrical sections around the pumped well will not be equal that is, 

the flow through a large cylinder will be less than the flow through a 

small cylinder, because a part of the ground water that percolates 

through the small cylinder is derived from storage between the two 

cylinders. The volume of material between any two cylinders that 

is unwatered in a given time can, of course, be computed from the

" Thiem, G., Hydrologische Methoden, Leipzig, 1906.
10 CONTRIBUTIONS TO HYDBOLOGY OF UNITED STATES, 1935

draw-down of the water table as shown by successive measurements 

of the depth to water in observation wells. The average hydraulic 

gradient that causes the water to percolate toward the pumped well 

can be determined from the same records of depth to water, provided 

the altitude of the tops of the observation wells is known. The total 

quantity of water that percolates through each of the two cylinders 

in the given time is computed by use of the formula Q=PiA. The 

difference between these quantities represents the volume of ground 

water taken from storage between the two cylinders. The specific 

yield of the water-bearing material is then determined by dividing 

this volume by the volume of material unwatered in the same time.

DEVELOPMENT OF THIEM'S FORMULA

Thiem's formula for computing the coefficient of permeability may 

be written in the convenient form

527.7g log ld-Jp- _________ ^_____________________(4) J. s \ \ /

in which P=the coefficient of permeability as defined on page 5; 

g=rate of pumping, in gallons a minute; 

a and ai= distances of two observation wells from the pumped

well, in feet; 

m (for artesian conditions) = vertical thickness of waterbearing bed, in feet;

m (for water-table conditions) = average vertical thickness 

(at <LI and a) of the saturated part of the waterbearing bed, in feet;

s and «i= draw-downs at the two observation wells, in feet. 

Thiem assumed a region where the water table or piezometric 

surface had an initial slope or hydraulic gradient before pumping 

began. His final formula did not contain a factor involving this 

slope, and he concluded that an initial slope of the water surface had 

no effect on the coefficient of permeability as computed by his formula. 

A review of the development of his formula indicates that during the 

development his original system of oblique coordinates was changed 

to a system of rectangular coordinates, which eliminated the factor 

involving the hydraulic gradient, and the resultant formula theoretically pertains only to regions where the water table or piezometric 

surface is horizontal. The following is Thiem's development, with 

some added interpretation, for water-table conditions:

A water-bearing bed of uniform permeability is assumed to rest on 

a relatively impervious formation, as indicated in figure 1. Water 

moves through the bed under a normal hydraulic gradient that is 

parallel to the slope of the underlying impervious bed. In this 

ground-water stream there is a well equipped with a pump extending
THIEM METHOD FOR DETERMINING PERMEABILITY 11

to the bottom of the water-bearing material, and two observation 

wells are placed in line with the pumped well. The pump is operated 

at a uniform rate during a period in which the water table declines 

and takes a form somewhat similar to an inverted cone around the 

pumped well. The draw-down (decline of the water surface) in each 

observation well, the distances of these wells from the pumped well, 

the rate of discharge of the pumped well, and the thickness of the 

water-bearing bed are measured. The coefficient of permeability is 

computed by substituting these measurements in Thiem's formula.

PLAN

 Direction of ground-water movement

Pumped well

s\

/ \

Observation well

SECTION

FIGTJKE 1. Plan and section of ideal ground-water conditions assumed by Thiem.

The following symbols, in addition to those previously given, are 

used in the development of the formula: 

I natural hydraulic gradient;

A=cross-sectional area in square feet that is, area of any 

designated cylindrical section through which the water 

percolates on its way to the pumped well; 

H= thickness, in feet, of the saturated part of the water-bearing

bed in the undisturbed condition of the water table; 

Q rate of pumping, in gallons a day;
12 CONTRIBUTIONS TO HYDROLOGY OF UNITED STATES, 1935

x and y= oblique coordinates of a point, J, on the cone of depression,

with reference to the point of intersection of the impermeable bottom of the formation with the axis of the well as

the origin;

i= hydraulic gradient, in feet per foot, at any point, J, on the

cone of depression;

x1 distance, in feet, of point Jfrom the pumped well; 

<£=the angle that a line through the pumped well and the two 

observation wells makes with the uninfluenced direction 

of ground-water movement;

6= the angle of inclination of the impermeable bottom; 

t distance, in feet, of the projection of point J from the 

pumped well measured along the uninfluenced direction 

of the movement of the ground water.

In the ground plan, figure 1, assume a small sector with an angle 

d$ whose apex is at the axis of the well and whose sides form an 

angle 0 with the uninfluenced direction of ground-water flow. At 

the point J, at a distance x from the well, the flow dQ passes through 

the sector d$. By using Darcy's equation, Q=PiA (1), it is possible 

to compute the flow dQ through the sector d<j> at the point «7.

The length of the arc at J=x cos Qd^>. The vertical thickness of 

the saturated water-bearing material is y. So the area through 

which the flow dQ passes is equal to xy cos Qd<l>.

The hydraulic gradient, i, at any point on the cone of depression 

is equal to the rate of change of the coordinates. These coordinates 

must be at riglit angles, so the horizontal coordinate is x1 and the 

vertical coordinate is (n4-y).

x1 =x cos 6_-____ -------------- _-_(5)

and n=x sin 6__ _____________________ (6)

tlms , * .d(n+y}_d(x  3? d(zcose) sin 6+y) ----------------(7)m

By substituting in Darcy's equation,

sin Q-\-y)xy cos Qd(f> 

 

, . d(x cos 9) ---- ------(8)

From the plan and section (fig. 1)

t=x cos 6 cos 0 ___________________ (9)

77 The natural hydraulic gradient is equal to , hence

,=»_- ._.___- _____._.

By equating (9) and (10),

n=xl cos 6 cos 0_ _________________ (11)

n=x sin 6 ________________________ (12)
THIEM METHOD FOR DETERMINING PERMEABILITY 13

Equating (11) and (12), we get

x sin 6=arJcos 6 cos 0 _________________ (13)

and sin 6= J cos 6 cos <f> _________________(14)

When both sides of equation 14 are squared,

sin2 6=72 cos2 6 cos2 0_ ______________ .(15)

1 sin2 6 may be substituted for cos2 6; thus equation 15 becomes

sin2 e=72 (l sin2 6) cos2 0 ______ ______.(16)

=P cos2 Q-P sin2 6 cos2 0_ ______ (17)

and 72 cos2 0=sin2 6 +P sin2 6 cos2 0___________ _ _(18)

=sin2 e(l+72 cos2 0) ____.____________(19)

Thus 72 cos2 0=l+72 cos2 0- ----- ----- -(20)

and, by taking the square root,

sin 6 1

Equation 14 may be written

sin 6

7cos0 T/l + P cos2 0 -- -

=cos Q _____ __________________ (22) 7 cos 0 

and by equating 21 and 22,

cos e=-7rr==L===_________________(23) VI +P cos2 <f>

1 cos2 6 may be substituted for sin2 6; thus equation 15 may be 

written

1 cos2 Q=P cos2 6 cos2 0________________(24)

l=cos2 e+/2 cos2 6 cos2 0___________.____(25)

l=cos2 6 (P cos2 0+l)__________________(26)

and 75 2 ... =cos2 6 ____---__________(27) P cos2 0+1 v '

By taking the square root,

\ ^TT=COS ©----- - -- ---(28) 

cos2 0+1

=cos 6 cos 07____________(29)

and by multiplying both sides of equation 28 by cos 0 I,

cos 07

V72 cos2 0+1 

Equation 14 may be substituted in equation 29, thus:

V72 cos2 0+1

18274 36 3
14 CONTRIBUTIONS TO HYDROLOGY OF UNITED STATES, 1935

Equations 28 and 30 may be substituted in equation 8, thus:

cos

Thiem states at this point that P is very small and therefore can 

be assumed to be zero, thus introducing a small error. If P Q, 

equation 31 becomes

cos <j>I+y)xyd<j> __ _ __ _ (32)

dx

Pxyd<}>(cos 0 ------ ---(33)

<^----- ----- -(34)

In order to integrate equation 34, Thiem changed from an oblique 

system of coordinates to a rectangular system of coordinates. Thus 

7=0, n Q, and 0=0, and Thiem's final equation will pertain only 

to horizontal water-table conditions. Equation 34 then becomes

By integrating with respect to $ and Q,

V (36)

and Q=27rP ------------ (37)

If the equation is now further integrated with respect to x and y,

- (38)

and +a.... .... (40)

This is Thiem's general equation. If equation 37 is integrated 

between limits x=a, x=a^ and y=h, y=hi (fig. 1, section) equation 

40 is developed into a more practical form. From equation 37

f1 *-?-? f

Ja X Q J h
THIEM METHOD FOR DETERMINING PERMEABILITY 15

r

loge#

L

i ,

loge di loge a= ~Qf) 9 I

H? L *Jh

ll2 h2

, ,

loge fli log. a=

and p=^(log -log. a)

A) -- -- -(46) 

and (hi h) is equal to the difference of draw-downs (s «i). Thus

V-A2=$i+fc)(s-Si) ----- (47) 

and equation 45 becomes

p=@ (loge fli loge a) _ (48) 

-------- ---

This is Thiem's final equation and applies to regions where theground water is not confined below an impermeable bed. Thiem 

developed a formula that applies to artesian conditions in the same 

manner. His artesian formula differs from equation 48 only in that 

(hi+h) is replaced by 2m. If m is used as defined in this paper 

(p. 10), the equation for both water-table and artesian conditions 

may be expressed

p^Q (loge QI loge a) ___ _____ __ ...(49),

2irm(s Si)

Equation 49 includes factors involving natural logarithms that is, 

logarithms with base e. It is developed to the more convenient form 

given on page 10 in the following manner:

From equation 49

~2Trm(s Si) 

logeX=2.30259 logiocc

2.30259 Qlogw ^

thus P= -= ? 

2irm(s Si)

If the rate of pumping is expressed in gallons a minute, the equation? 

becomes

2.30259(/l,440g Iog10-) 527.7g Iog10- 

p= V g /_ <

Si) m(s Si)

which is Thiem's formula in modified form, for convenient use in 

the United States.
16 CONTRIBUTIONS TO HYDROLOGY OF UNITED STATES, 1935

Thiem's formula may be developed more simply by starting with 

the assumption of a horizontal water table or piezometric surface 

(fig. 2). For water-table conditions the demonstration is as follows:

i, at any point on the cone of depression, is equal to the slope, or

fjfij

-/> and the increment area through which the flow dQ moves is 

equal to xyd<f>. Therefore

---- ................ (52)

PLAN

Pumped well

SECTION

_. . . , , Pumped well Static water table ^ \,

FIGURE 2. Plan and section showing assumed ground-water conditions for the development of the formula from horizontal water table.

With horizontal conditions, x and y are independent of $ ; hence $ 

may be integrated independently of x and y, and equation 52 becomes

/[*] * - -(53)

and .(54)
THIEM METHOD FOR DETERMINING PERMEABILITY

By integrating x and y,

17

Cdx 2irP C , ,.-. I -0- I ydy--.-- - _-__(55)

log«ic=27T.1

and « -*_*

PLAN

Observation we//\

J^ 

Observation we//-^

Pumped well.

Static piezometr/c surface r

SECTION

Pumping Observation 

/ we//s \i

Conftn/nghed'

/ / 7 / / /

FIGUKE 3. Plan and section showing assumed ground-water conditions for the development of the formula 

from horizontal artesian conditions.

Equation 57 is identical with equation 40 obtained by Thiem. The 

formula for computing the coefficient of permeability may be developed from equation 57 in the same manner as it was from equation 40. 

Thiem's formula for artesian conditions can be developed in a 

manner similar to the one that applies to water-table conditions (fig. 3).
18 CONTRIBUTIONS TO HYDROLOGY OF UNITED STATES, 1935

The demonstration is as follows, again beginning with

The hydraulic gradient at any point on the cone of depression is

dlJ equal to -£ The total area through which the flow, Q, passes is 

2m-xm, where the water bed is horizontal. Therefore

and

Cttl dx 2irmP Chl 

By integrating, = n <fy___ ___ _____________(60)

Ja <*' V Jh

, N ,_.,. i h)--- ___________ (61)

anda

Equation 62 is developed to equation 4 in the same way that equation 

4 is obtained from equation 50.

CONFIRMATION OF THIEM'S FORMULA FROM OTHER 

WORK DONE IN THE UNITED STATES

The theoretical work by Slichter and by Turneaure and Russell, 

done several years before Thiem's paper was published, is briefly 

described below, for the purpose of showing that Thiem's formula 

can be deduced from their results. Thus Thiem's formula is given 

essential confirmation by these eminent hydrologists in the United 

States, for the particular conditions, to which it applies namely, a 

homogeneous water-bearing material, an original horizontal water 

table or piezometric surface, and an original uniform thickness of 

the saturated part of the water-bearing formation. Thiem's formula 

has essentially been derived by the others, but because of differences 

in their use of symbols and in the final form of their formulas, these 

formulas have not generally been recognized as being different 

manners of expressing identical conclusions. Thiem's outstanding 

contribution was in the application that he made of his formula for 

determining permeability.

The following discussion outlines the development of Slichter's 

formula and points out its relation to Thiem's formula for artesian 

conditions. 16 Slichter starts with the assumption of a homogeneous 

water-bearing material overlain and underlain by impervious material, and an artesian well that completely penetrates the material.

16 Slichter, C. S., Theoretical investigation of the motion of ground waters: U. S. Geol. Survey 19th 

Ann. Kept., pt. 2, p. 359, 1899.
THIEM METHOD FOR DETERMINING PERMEABILITY 19

The following nomenclature and units are changed somewhat from 

that of Slichter to correspond to the nomenclature and units used 

previously in this paper:

ra, thickness of water-bearing material in feet (Slichter's a): 

K, a constant defined by Slichter as "the quantity of water that

would be transmitted in unit time through a cylinder of

stone of unit length and cross section, under unit difference

in head at the ends";

v, velocity of the ground water, in feet per day; 

r, radius of the well, in feet;

Q, rate of discharge of the pumped well, in gallons a day; 

h, amount of lowering of water in well by pumping, in feet; 

J, a point on the cone of depression; 

x, distance of point J from the axis of the well, in feet; 

Z, pressure at point J, in feet of water (Slichter's p) ; 

B, distance, in feet, from the wall of the well at which the pressure

may be assumed to be equal to its normal value (that is,

Z=0 when x=R+r}. 

The velocity at point J at distance x from the axis is given by

KdZ

The velocity varies inversely with the distance from the axis of 

the well, so

v=- 

x

in which c is a constant to be determined. After equating 63 and 

64,

__________ __ ...__. (65)

 C

From which c loge x=KZ+C1 . ...... _ ._____- (66)

When x=r, Z=h, and when Z=0, x=R+r-, thus

cbg.(B+r) = C'1 -(67)

and cloge r Kh=d __ ___ __ _ ____(68)

By equating 67 and 68,

cloge (fl+r)=c log. r-l£ft______ __-__-___ (69)

Therefore c= ^-x___.________________ (70)

and by substituting equation 70 in 64,

................(71) y
20 CONTRIBUTIONS TO HYDROLOGY OP UNITED STATES, 1935

The velocity at the wall of the well is found by placing x r. An 

expression for the total amount of water flowing into the well in 

unit time is obtained by multiplying the velocity at the wall of the 

well by 2irrm. Therefore

At this point Slichter solved equation 72 for Q, having determined 

K by means of a previously developed formula that depends upon the 

laboratory analysis of the water-bearing material for the effective 

size of the sand grains. However, equation 72 can be converted to 

Thiem's formula for permeability by proper substitution. Solving 

equation 72 for K, we get

K== ^ '___ ................ (73) 2-jrhm

K, as previously defined, is really a coefficient of permeability. Therefore the symbol P may be substituted for K, giving

Q P= 

(R+r) corresponds to the distance ax in Thiem's formula, and r 

corresponds to the distance a. Equation 74 can then be written

The term h is equal to the draw-down at the pumped well, at the distance r from the axis of the well. The draw-down at the distance 

(R-\-r) was assumed by Slichter to be zero. Therefore, h represents 

the difference in draw-downs between the two points on the cone of 

depression r and (B-\-r) and is equivalent to Thiem's term (s «i). 

Substituting in equation 75, we have

p_ $ lo&T =Qqogea1 -logea)

----- --------

Thiem's final formula for artesian conditions is identical with formula 

76.

Turneaure and Russell 17 published the development of *a formula 

which is similar to the simple development from horizontal watertable conditions given on page 16. Using Darcy's law as a basis 

for their development, they arrived at essentially the same equation 

as 57, with the exception that the factor p (porosity of the watera Turneaure, F. E., and Russell, H. L., Public water supplies, 1st ed., p. 269, John Wiley & Sons, Inc., 

1901.
THIEM METHOD FOR DETERMINING PERMEABILITY 21

bearing material) is included. As given by Turneaure and Russell, 

the equation is

_. __....___- _(77)

At this point c is evaluated by substituting x=r (radius of the well) 

and y=h (saturated thickness of water-bearing material at the wall 

.of the well). Then

c=Q\oger-TrKph2 .......... .....(78)

and by substituting in equation 77

Q logo

'+" -----"-<? >

If in equation 79 the value of x is taken to be R, or the distance from 

the axis of the well at which the change in water level is inappreciable, 

the corresponding value of y will be H, the original depth of water, and 

equation 79 will become

and

log.

The product Kp corresponds to Thiem's coefficient of permeability, 

P. Hence

In equation 82, H and h represent the thicknesses of the saturated 

part of the water-bearing bed at R and r, respectively. In Thiem's 

formula hi and h represent the thicknesses of the saturated part of 

the water-bearing bed at di and a, respectively. The characters used 

in Thiem's formula may be substituted in equation 82, and that equation then becomes

____ _ (83)

Cl 1 ^ V °Sea

This equation is Thiem's final formula 48, for computing the coefficient of permeability from water-table conditions.

As shown above, there is little difference between the formulas of 

Slichter, Turneaure and Russell, and Thiem. The principal variance 

occurs in that Thiem determined the coefficient of permeability,

18274 36 4
22 CONTRIBUTIONS TO HYDROLOGY OF UNITED STATES, 1935

whereas the others determined the quantity of water entering the well 

and obtained the coefficient of permeability from laboratory analyses 

of the water-bearing material. Thiem's formula includes the drawdown of the water level in observation wells at two definite and measurable points on the cone of depression, but the formulas of the others 

contain the draw-down at the more indefinite points r and R. The 

draw-down at the wall of the well at a distance r from the axis of the 

well has usually been taken to be the water level in the well while 

pumping was in progress. This sometimes introduces a large error, 

because a part of the draw-down in the pumped well is caused by the 

loss of head of the water as it enters the well. Moreover, the texture 

of water-bearing material, if it is sand or gravel, is likely to be disturbed for several feet around a pumped well by the development of the 

well, and therefore the effective diameter r may be considerably 

larger than the nominal diameter of the well.

The formulas of Slichter and of Turneaure and Russell include the 

determination of the radius of the cone of depression, R. Slichter 

assumed . this distance to be 600 feet, and Turneaure and Russell 

determined it with a formula derived by the following reasoning: 

"Assuming that all the water in the circle of influence flows into the 

well, the width of the strip of the ground-water stream tributary to 

the well will be 2R, and the original cross section of this portion of th& 

ground-water stream is 2RH." Then from formula 1, Q=PI(2RH)

and R o PJTJ' By substituting the value of Q, from equation 82 the 

formula, after reduction, becomes

_ ,.............. ................(85)

2IH log."

This formula involves the draw-down in the pumped well and the 

radius of the well, and therefore it is subject to the difficulties 

previously enumerated in ascertaining these items. It is certain 

that R is rather difficult to determine, and under some conditions it 

has been known to exceed 5,000 feet.

Recently the results of laboratory experiments on the flow of 

water through sand by Wyckoff, Botset, and Muskat 18 were published. They constructed a small apparatus in which the groundwater conditions around a pumped well were reproduced. They 

observed the draw-downs of the water table and piezometric surface 

at several distances from the well under various rates of flow. A 

formula was prepared from the data obtained from these experiments,, 

by which the flow into a well could be computed from a knowledge

is Wyckofl, R. D., Botset, H. Q., and Muskat, M., Flow of liquids through porous media under theaction of gravity: Physics, vol. 3, no. 2, pp. 90-113, August 1932.
THIEM METHOD FOR DETERMINING PERMEABILITY 23

of the permeability of the water-bearing material and the saturated 

thicknesses of the material at two points on the cone of depression. 

The formula with the nomenclature altered to correspond to usage 

in this report is

in which K is a coefficient of permeability, p is the density of the 

fluid, g is the acceleration of gravity, and h and AI are the fluid pressures at the respective distances a and ax from the pumped well. 

Equation 86 may be written

Q ^ (87) -" """ '" '" '" "

The density of water is essentially 1, and therefore p may be regarded 

as equal to 1. Kg in equation 87 is therefore equivalent to the 

coefficient of permeability, P, as contained in Thiem's formula. 

The fluid pressure is probably equivalent for most conditions found 

in nature to the saturated thickness of the water-bearing materials. 

Therefore

.... ...............(88)

which may then be reduced to equation 48 (p. 15).

FORMULA FOR DETERMINING THE CONE OF DEPRESSION

A formula can be developed for the cone of depression from Thiem's 

formula, provided the conditions are the same as those assumed in 

developing Thiem's formula for artesian conditions the waterbearing material is homogeneous and of uniform thickness, the ground 

water is confined between horizontal impermeable formations, and 

the piezometric surface is horizontal before pumping is started (fig. 3).

From Thiem's modified formula (equation 4, p. 15)

527. 7 Io 10

Let the quantity p^ be represented by B, a constant. Then

which is the equation for the cone of depression pertaining to artesian 

conditions. The draw-down, s, at any point on the cone of depression
24 CONTRIBUTIONS TO HYDROLOGY OF UNITED STATES, 1935

can be computed by substituting the corresponding figure for a, the 

distance from the pumped well at which the draw-down occurs.

The equation for the cone of depression pertaining to water-table 

conditions is developed as follows from equation 4:

527.72 Iog10~ 1055.4g log^1 

P= - = - 2 ---------(91)

1055.'_. 

or h2 *=hi2 p ________________(92)

Let the quantity 105p'4g be represented by F, a constant. Then

h? is equal to (fif s) 2 (fig. 2); therefore

^......__.........(94)

and s=flr- 12-^logio 1 -- ------- (95)

The draw-down, s, at any point on the cone of depression can be computed by substituting the corresponding figure for a, the distance from 

the pumped well at which the draw-down occurs.

The slope of the cone of depression at any point may also be computed from Thiem's formula. Starting with the formula as stated in 

equation 49 (p. 15),

s~Sl~

Let 'the quantity % p be represented by E, a constant. Then

............... .(97)

If di is a fixed point in any given pumping test, then log^ is a constant in that test. Therefore the quantity E log^x is also a constant 

and may be represented by L. Then

s Si=L E \ogea _____ __ ___ (98) 

and s=i+Si Elogtd _ _ __.___.__. (99)
tT. 8. GEOLOGICAL SURVEY WATER-SUPPLY PAPEK 67ft PLAffl i

oi «f

COEFFICIENT OF PERMEABILITY, P

I I ! ! I I I I I !

in

-90O

X

-SCO

i- -TOO

-20O

-IOO

0 .1 .2 .3 .4 .5 .6 .7 .3 .9 1.0 U 1.2 1.3 IA 1.6 1.6 1.7 1.6 1.9 2.O 3.1 22 2.3 2.4 2.6 26 2.7 2.8

i ! DIFFERENCE IN DRAW-DOWNS(s-St)IN\FEET \ \

400 , 500 i 60O ! 7OO ! 800 '. 900 , 1,000 1,100 1,200 1,3OO 1,400

'DISCHARGE OF PUMPED. WELL 7, //V GALLONS PER MINUTE \ 

0 IO 2O 3O 40 SO 6O 7O 80 30 100 110 12O 13O 14O I5O 160 17O 180 190 ZOO 2IO 220 23O 24O 25O 26O 27O 28O

THICKNESS OF SATURATED WATER-BEARING MATERIAL, m, IN FEET

GRAPH FOR COMPUTING COEFFICIENTS OF PERMEABILITY,

THIBM METHOD FOR DETERMINING PERMEABILITY 25

Subtracting both sides of equation 99 from h, we have

By substituting the general factors (h s}=y and a=x in equation 

100 (fig. 3), we get

Si _-- (101) 

and by differentiating with respect to y and x,

dy=^- -__-_-__. (102) 

The slope of the cone of depression is equal to dy/dx. Therefore,

and i=0 ____________ _________(104) 2irPmx ^ J

In a similar manner it can be shown that for water-table conditions 

the slope of the cone of depression at any distance, x, from the pumped 

well can be computed by the formula

---- --008)

This formula differs from equation 104 for artesian conditions only in 

that the thickness of the water-bearing formation, m, is replaced by 

the thickness of the saturated water-bearing material, y.

The slope of the cone of depression, i, at any point may be computed 

for both water table and artesian conditions by substituting for x, 

the distance of the point from the pumped well.

GRAPHIC SOLUTION OF THIEM'S FORMULA

By the use of the graph presented in plate 1 the coefficient of 

permeability may be determined without the usual computations, 

if the factors contained in Thiem's formula are known. This graph 

is particularly useful for determining the effect on the computed 

coefficient of permeability of changes in the factors in Thiem'sformula, and it provides a rapid method for calculating the permeability for several regions and comparing the ground-water conditions of those regions. The graphic solution for permeability is 

made in the following manner:
26 CONTRIBUTIONS TO HYDROLOGY OF UNITED STATES, 1935

Locate the point of intersection of the horizontal line corresponding to - and of the vertical line corresponding to (s s:} ; move inward

Cb

or outward along the radial line through this point to the intersection 

of the line with the vertical line representing the discharge of the well, 

q; move horizontally from this point to the right or left to the vertical 

line representing the saturated thickness of the formation; move 

radially outward from this point to either the upper or the right margin of the diagram, where the coefficient of permeability, P, is read. 

As an example, the graphic solution corresponding to the conditions

-=2, 8 s1 =1.5, 2=800, and m=100

df

is shown in plate 1 by a heavy dashed line. The coefficient of permeability so (determined is approximately 850.

PUMPING TESTS IN NEBRASKA

Two pumping tests were made near Grand Island, Nebr., during 

the summer of 1931 on the farm of Fred Meyer, about 4 miles east 

of Grand Island, in the NW% sec. 17, T. 11 N., R. 8 W. This location 

was selected after a thorough inventory of existing irrigation wells 

in the vicinity, as the one that most nearly approached the ideal 

conditions desired for the pumping tests. The irrigation well used 

for test 1 was in a pasture just west of a large field of corn (well 

83, fig. 4). The land near the well was rather flat, although the 

field of corn was slightly higher than the pasture. There was a dry 

slough about 800 feet west of the well, but as no drainage had entered 

it for some time preceding the pumping tests, it probably did not 

affect the normal level of the ground water. Throughout the area 

covered by figure 4 the water table ranged only from 2 to 10 feet 

below the land surface, and hence the sinking of observation wells 

was not difficult. It is probable that the water table was lowered 

sjomewhat during the period of tests by drafts made on the zone of 

saturation by plants, but the amount of lowering was small,, as indicated by the small decline of the water table in those wells located 

farthest from the pumped well. There were three irrigation wells 

within a mile of the test wells, but none of them were operated during 

the tests or for several days before the tests were begun.

Before the pumping tests were made a test hole was drilled near 

observation well 76 to determine the thickness of the water-bearing 

materials. Sand and gravel showing a great range in size and some 

clay were penetrated to a depth of about 110 feet, where bedrock 

was struck. The hole was continued into the bedrock to a depth of 

143 feet below the ground surface. Later a well was drilled about 

25 feet south of the existing irrigation well for the second pumping 

test, and samples of the water-bearing materials penetrated were sent
THIEM METHOD FOE DETERMINING PERMEABILITY 27

to the hydrologic laboratory of the United States Geological Survey 

for determinations of porosity, moisture equivalent, and permeability 

and a mechanical analysis (table 1). A log of the materials encountered in this well (84, fig. 4) is given in table 2. This well was 12 

inches in diameter and was drilled to a depth of 105 feet; the lower
28 CONTRIBUTIONS TO HYDROLOGY OF UNITED STATES, 1935

48 feet and the upper 24 feet of the casing were perforated. The 

existing irrigation well used for the first pumping test was 24 inches 

in diameter and 40 feet deep, and all the casing was perforated.

TABLE 1. Physical properties of samples of alluvium taken from well 84, near

Grand Island, Nebr.

[Determined in the hydrologic laboratory of the U. S. Geological Survey by V. C. Fishel]

Depth (feet)

Mechanical analysis (percent by weight)

Larger

than 2.0

mm

2.0-1.0

mm

1.00-0.50 

mm

0.50-0.25 

mm

0.25-0.125 

mm

0.125- 

0.062

mm

0.062- 

0.005 

mm

Less than 

0.005 

mm

6 to 10... 

10 to 16.. 

16 to 20.. 

20 to 25.. 

25 to 30.. 

30 to 39..

39 to 40..

40 to 42.. 

42 to 46.. 

46 to 51.. 

51 to 55.. 

55 to 61.. 

61 to 66.. 

66 to 71.. 

71 to 78. . 

78 to 86. . 

86 to 92.. 

92 to 99.. 

99 to 105.

29.7

14.1

16.8

18.6

7.5

36.4

3.4

15.9

15.4

17.3

39.6

27.4

20.6

18.1

179.3

14.3

36.2

15.1

25.8

16.9

17.9

15.2

18.8

17.2

20.8

3.6

11.0

15.2

10.7

12.8

14.9

19.6

18.0

3.5

11.9

10.3

10.4

13.3

18.9

31.2

25.8

21.3

25.0

21.4

1.8

20.1

20.2

13.1

9.5

16.3

19.7

17.7

3.9

18.2

14.6

22.8

13.7

17.1

30.4

29.4

24.8

30.0

15.0

4.7

33.4

19.5

29.4

15.7

22.4

19.1

23.7

6.3

25.1

17.1

31.1

21.9

15.4

5.5

10.5

13.8

16.0

4.7

26.0

15.4

16.4

24.4

13.5

11.8

9.7

14.0

4.0

18.7

11.6

13.9

14.3

1.3 

.3

1.6 

1.9 

3.4 

.8 

14.0 

2.6 

7.0 

3.2 

4.7 

3.7 

4.3 

3.3 

1.5 

6.5 

4.3 

3.0 

5.2

0.4 

.2

.5

.5 

31.5

.4 

4.5 

1.0 

2.5 

2.1 

4.4 

3.0 

1.0 

3.0 

3.4 

2.5 

3.5

0.2

.1 

.1 

.2

.a.1

13.6

.2 

1.5

.4 

1.0 

1.0

.9 

1.9

.3 

1.7 

2.3 

1.0 

2.0

Depth (feet)

Apparent 

specific 

gravity

Porosity 

(percent)

Moisture equivalent

Percent 

by weight

Percent 

by volume

Coefficient 

of permeability

6 to 10... 

10 to 16.. 

16 to 20.. 

20 to 25.. 

25 to 30-. 

30 to 39..

39 to 40-

40 to 42.. 

42 to 46.. 

46 to 51.. 

51 to 55.. 

55 to 61-. 

61 to 66-. 

66 to 71.. 

71 to 78.. 

78 to 86.- 

86 to 92.. 

92 to 99.- 

99 to 105-

90 27.1

30.9

32.3

28.5

31.0

30.6

40.3

31.2

26.3

30.2

26.2

25.6

25.0

26.3

22.8

41.8

21.5

29.9

27.6

1.4

1.5 

1.1 

1.4 

1.4 

1.0 

17.4

1.5

1.6

1.6

1.7 

1.6 

1.4 

1.6

2.6

2.7 

2.0 

2.6 

2.6 

1.9 

27.1 

2.7 

3.0 

3.0 

3.3 

3.0 

2.8 

3.0

1.6 

1.9 

1.2 

1.5

3.1 

3.9 

2.1 

2.9

480

1,685

1,460

1,095

1,095

4,350

2

925

150

350

780

730

2,095

1,050

2,185

220

495

430

285

J 76.0 percent larger than 5 mm.

TABLE 2. Log of well 84, drilled for second pumping test

Top soil ___________

Clay. - .. . ....

Thickness

Feet

38

10

5

Depth

Feet

39

40

50

55

Thickness

Feet 

16

7

8

6

13

Depth

Feet 

71

78

86

92

105
TT. S. GEOLOGICAL SURVEY

A. LINE SW OF 1-INCH OBSERVATION WELLS.

WATER-SUPPLY PAPER 679 PLATE 2

B. MEASURING THE DEPTH TO THE WATER TABLE.
U. S. GEOLOGICAL SURVEY WATER-SUPPLY PAPER 679 PLATE 3

A. WEIR FOR MEASURING THE DISCHARGE OF THE PUMPED WELL.

B. PUMPING ARRANGEMENT IN SECOND TEST.
THIEM METHOD FOR DETERMINING PERMEABILITY

A transit was set up over well 83, and six radiating lines of wells 

were laid out. Lines C and D were projections of lines A and B, 

and line W bisected the 90° angle formed by the intersecting lines 

A and B. lines N and S only approximately bisected the angles 

formed by the intersections of lines A and D and lines B and C, 

because the topographic features were such that actual bisections 

would have been difficult. Line SW was laid out from well 84 

(pl. 2, A).

More than 80 observation wells were sunk, most of them relatively 

close to the pumped wells, where the decline of the water table during 

pumping would be the greatest. Some of the observation wells were 

1 inch in diameter and were fitted with 18-inch screen drive points. 

These wells were driven into the saturated sand and gravel -to such 

depths that the water table during pumping would not drop below 

the bottoms of the wells. Several observation wells 3 inches in 

diameter were fitted with drilling bits at their lower ends and were 

jetted down with a drilling rig. Holes in the bits allowed water to 

enter the wells freely. The diameters and depths of the observation 

wells are recorded in table 3.

TABLE 3. Location, diameter, depth, and altitude of wells used in the pumping

Well no.

1 ___ ...

2 ____________ __ __ ...

3. __ ............. __ ..........

4. __ 

5 _________ - ___ .... _ ...

6 ...- _ .- . . .-. ...

7...... .......... .............

8 _ --_ . .--...--.._.-

9..--.. .. -.---......-..-.

11 

13 

14 

16--- 

16 

17 

18 

19 

20 

21, 

22 

23 

24 .

26 

26 

27 

28 

29 

30 

31 

32 

33 

34. 

35 .

Line

A

A

A

A

A

A

A

A

A

A

A

B

B

B

B

B

B

B

B

B

B

B

B

W

W

W

W

W

W

W

D

D

D

D

Diameter

Inches 

3

3

3

3

1

1

3

1

3

3

3

3

1

3

3

3

3

1

0

3

0

3

q

3

3

1

1

Depth 

of well 

below 

measuring point

Feet 

21.4

10.3

10.1

10.3

11.6

6.5

10.2

11.4

10.3

10.3

91 4

10.9

11.6

Q Q

9.9

in 9

1A 1

in 9

9.9

9 a

11.5

10.6

1A ^

10.1

10.7

21.5

16.5

16.5

16.5

Distance 

of measuring 

point 

above 

land 

surface

Feet 

0.7

.1

0

.1

1.4

.5

.2

1.2

.1

.4

.3

1.3

.2

.3

.5

.4

.3

.3

1.9

.3

.3

.3

.1

.3

.3

1.2

1.0

1.2

1.2

Altitude of 

measuring 

point

Feet 

1, 814. 34

1, 815. 66

1, 815. 26

1, 814. 63

1, 815. 83

1, 812. 35

1, 815. 39

1, 815. 52

1, 814. 97

1, 814. 73

1, 814. 05

1, 814. 84

1, 815. 17

1, 816. 10

1, 815. 67

1, 815. 46

1, 815. 08

1, 815. 86

1, 816. 32

1, 816. 39

1, 816. 95

1, 817. 12

1, 815. 39

1,814.78

1, 815. 33

1. 813. 46

1, 815. 68

1,815.39

1, 820. 42

1, 819. 17

1,818.99

1. 818. 93

D istance 

from 

pumped 

well 83

Feet 

24.9

59.9

114.4

164.2

229.0

354.1

429.3

478.9

604.0

754.6

903.8

29.9

70.0

120.0

184.9

254.7

375.3

499.7

649.7

775.3

974. 3

1, 149. 3

49.7

170.0

270.0

430.0

625.0

804.5

939.7

40.1

95.1

144.7

214.3

Distance 

from 

pumped 

well 84

Feet 

42.3

74.6

127.9

177.2

241.5

366.4

441.2

490.4

616.1

916.2

14 9

49.6

98.9

163.1

233.0

353.8

477.7

627.6

752.9

951.8

1, 127. 0

48.8

164.3

264.0

vIOQ A

618.0

797.6

932.5

63.1

117.7

166.9

236.7
30 CONTRIBUTIONS TO HYDROLOGY OF UNITED STATES, 1935

TABLE 3. Location, diameter, depth, and altitude of wells used in the pumping

tests Continued

Well no.

36 

37 

38 

39 

40

42 

43 

44 

45 

47 

48

49 __ 

60 

51 

52 

53 

54-.- __ 

55 

56. -

57 

58 _ -

59- __ _

60 _ _ .... ___ ......

61 ___ . ___ .. ....

62 

63 _ . ..................

64- __ . __ . _ ..

65- _____ ......

66 _ _ ..........

67 _ . ___ ....

68 ___ . _ ......

69- .

70-. , _ ....

71 . _ .... _ .

72 .........................

73 . ___________ . _ .

74-- . _ . _______ . .....

75-- ________ ......

76--- ___ ... _ ......

77 

78-- . _ ........

79 .......................

80- ____ ............

81 _ ___ ....... .....

82- . _____ .........

83.- _ . ............

84 _ ............. ...............

Line

D

D

D

D

D

D

D

D

C

C

C

C

C

C

C

C

C

C

C

C

sw

sw

sw

sw

sw

sw

sw

sw

sw

sw

sw

sw

sw

sw

sw

A

A

s

s

8

s

N

N

N

N

N

SW

Diameter

Inches 

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

24

12

Depth 

of well 

below 

measuring point

Feet 

16.5

16.5

16.5

16.5

16.5

12.7

12.7

23.0

17.2

11.1

12.7

12.6

11.4

12.3

12.4

12.7

12.6

12.7

12.6

11.0

10.5

10.6

11.0

11.0

10.6

10.9

10.7

10.8

10.9

12.5

10.9

11.1

6.1

11.4

12 0

12.6

13.0

6.1

12.8

 to n

12.3

11.8

97 ft

on K

102.0

Distance 

of measuring 

point 

above 

land 

surface

Feet 

0.9

.9

1.1

1.0

1.2

1.8

.5

1.0

.5

.5

.5

.8

.5

.7

.5

.6

9

.8

.7

.8

.9

.8

2.1

2.4

2.6

2.4

2 9

1.3

O A

.7

.5

9 3

1.7

.7

1 9

1 3

0

-.5

-3.0

Altitude of 

measuring 

point

Feet 

1, 818. 31

1, 818. 27

1, 818. 33

1, 818. 05

1, 818. 83

1, 818. 08

1, 817. 19

1, 816. 73

1, 820. 12

1, 818. 37

1, 818. 02

1, 817. 37

1, 817. 90

1,818.36

1, 818. 30

1, 818. 80

1, 819. 45

1, 817. 37

1 ST4. Q7

1, 815. 29

1, 816. 18

1, 815. 88

1, 816. 14

1, 816. 47

1 Q17 4.9

1, 816. 54

1, 815. 37

1

QTT QQ

1 B.19 RQ

1 Qi K QX

1 ci K nn

1, 816. 05

1, 817. 74

1, 813. 08

1, 815. 32

1,815.48

1, 816. 13

1, 816. 41

1, 812. 66

1 1 R14 On

Distance 

from 

pumped 

well 83

Feet 

323.8

423.2

448.2

572.9

722.7

872.2

1, 072. 5

1. 197. 0

39.3

80.5

130.3

195.6

285.6

410.2

425.2

535.4

685.3

1, 034. 7

1, 174. 9

46 7

69.5

93.6

118.0

216.9

316.6

516.5

716.5

816.6

1, 016. 6

1, 116. 9

1, 217. 1

2.6

12.3

m

-l

OOK O

382.7

160.0

342.0

445.8

04. Q

0

t)A Q

Distance 

from 

pumped 

well 84

Feet 

345.8

445.0

470.2

594.4

744.2

893. 6

1,094,2

1, 218. 3

35.0

71.9

120.2

185.0

274.7

398.7

413.9

524.2

673.9

822.8

1,022.9

1, 162. 8

OE Q

50.7

75.8

100.8

200.7

300.7

400.8

500.9

600 9

701.0

801 2

Qfll 9

I

ftm *?

1, 101. 5

1, 201. 8

26.0

DO O

105.3

9fifi Q

OKK 1

qee A

Q7 1

183.5

OCR 0

365.7

469 0

2 A

f)A Q

0

i First test. Altitude for second test 1,812.35 feet.

Each observation well was pumped with a pitcher pump until the 

water discharged was clear, indicating that the ground water had 

free access to the well and that the water level in the well showed the 

level of the water table outside the well. Definite points were 

established at each well from which measurements of depth to the 

water level could be made, and the distance of these measuring points 

above the land surface was recorded. To determine the altitude of 

the measuring points, instrumental levels were run to all the observation wells and to the two pumped wells (table 3).
THIEM METHOD FOR DETERMINING PERMEABILITY 31

Both pumping tests were started early in the morning. During the 

day preceding each of the tests several measurements were made of 

the depth to water in the observation wells, in order to determine 

the static level of the water table, and a few minutes before pumping 

began additional measurements were made as a check on the measurements of the day before. The measurements were made with a steel 

tape graduated in hundredths of a foot. The end of the tape was 

loaded with a swiveled weight, so that the tape would hang plumb, and 

the lower foot or so of the tape was coated with blue carpenter's chalk, 

so that the depth of immersion of the tape into the water could be 

plainly seen. The period of pumping in the first test was about 48 

hours, and the average rate of pumping 540 gallons a minute. During 

the second pumping test the pump was stopped several times because 

of trouble with the 50-horsepower gasoline engine that was used to 

drive it. In order to make the two tests as comparable as possible, 

pumping in the second test was continued a few hours longer than 

the 48-hour period, so that the total quantity of water pumped was 

about equal to the quantity pumped during the first test. Records 

of pumping time are given in table 4.

TABLE 4. Record of pumping time

Well 83 (test 1)

Started___ July 29, 1931, 6:05 a. mStopped.___ July 31, 1931, 6:04 a. m.

Well 84 (test 2)

Started___ Sept. 9, 1931, 8:05 a. m' 

Stopped__________-_____ 11:18 a. m'

Started_______________ 11:35 a. m.

Stopped________________ 12:35 p. m.

Started_________________ 12.37 p. m.

Stopped._______________ 2:00 p. m.

Started_______________ 3:38 p. m.

Stopped_____-_-________ 5:55 p. m.

Started_______________ 6:31 p. m.

Stopped__ Sept. 10, 1931, 4:26 a. m. 

Started___________ 6:03 a. m.

Stopped____ __________ 8:57 a. m.

Well 84 (test 2) Continued

Started___ Sept. 10, 1931, 9:32 a. m.

Stopped_____________ 9:36 a. m.

Started....___________ 9:38 a. m.

Stopped________________ 9:39 a.m.

Started_____________ 9:40 a. m.

Stopped_________-______ 9:48 a. m.

Started...______________ 9:51 a. m.

Stopped________________ 11:17 a. m.

Started,.____________ 11:19 a. m.

Stopped_____-__________ 11:49 a. m.

Started ------..-.---- 11:55 a. m.

Stopped...._____.-- 12:06 p. m.

Started____________ 12:11 p. m.

Stopped_ Sept. 11, 1931, 10:28 a. m.

Started___________ 10:34 a. m.

Stopped__-_______-_---- 2:05 p. m.

Twelve men were employed during the tests to make measurements of the depth to water in the observation wells, to measure the 

discharge of the pumped wells, and to operate the power unit. The 

men worked on alternate shifts of 6 hours during the first test and 

8 hours during the second test. During each shift three men made 

measurements of depth to water in the observation wells (pi. 2, B). 

These measurements were continued throughout the night with the 

aid of lanterns. The water from the pumped wells was discharged 

into a stilling basin about 25 feet east of the wells (pi. 3, J5), from
32 CONTRIBUTIONS TO HYDROLOGY OF UNITED STATES, 1935

which it flowed over a rectangular weir and through a small canal 

across the field of corn (pi. 3, A). The water was then used to 

irrigate the corn grown in the eastern part of the field.

After the completion of each pumping test measurements of depth 

to water in the observation wells were continued for at least 24 hours, 

so that the recovery of the water table could be determined. The 

measurements of depth to water made during pumping and after 

pumping had stopped, which number about 9,500, are on file and may 

be consulted in the office of the United States Geological Survey at 

Washington.

RESULTS OBTAINED FROM THE PUMPING TESTS

A considerable amount of study has been devoted to the records 

obtained in the pumping tests above described in an effort to determine the best procedure to be used in future tests for determiningpermeability by the pumping method. The present tests involved 

more time and expense than could ordinarily be spent on a field 

determination of the permeability of a water-bearing formation, and 

it was with the view of reducing the complexity of the tests that this 

study was made. The conclusions given below, under the heading 

"Computations of coefficients of permeability", show that satisfactory results can be obtained by less elaborate tests if certain facts 

developed in these tests are kept in mind.

The data collected in the tests are so adequate that a detailed study 

could be made of the behavior of the ground water in the vicinity of 

a pumped well, both during pumping and after pumping stopped. 

However, the writer has been able to make only a rather cursory 

examination of the whole mass of data, and hence the results here 

presented are not all that could be obtained if a more intensive study

were made.

DRAW-DOWN CURVES

To obtain the draw-down of the water table at any time it is only 

necessary to subtract the depth to the water level before pumping 

started from the depth at that particular time. The altitude of the 

water level at any time can be obtained by subtracting the draw-down 

at that time from the altitude of the normal water level. A continuous curve representing the decline of the water level in a well 

during the period of pumping is called a "draw-down curve."

Draw-down curves were plotted for many of the observation wells, 

chiefly for the first pumping test, because the curves for the second 

test show irregularities caused by interruptions in pumping. The ' 

draw-down curves for the first test are remarkably regular. A smooth 

curve could be drawn through most of the points, and the very few 

points that plotted far from the curve were obviously caused by errors
THIEM METHOD FOR DETERMINING PERMEABILITY 33

in making the measurements. The equation for the draw-down 

<;urve is not known, but the general form of the curve is shown by 

typical curves given in plate 4 for wells on line A. As that line was 

approximately at right angles to the natural direction of ground-water 

movement, the altitude of the static water level was nearly the same 

in all the wells.

The slope of the draw-down curve indicates the rate at which the 

water was withdrawn from storage in the sand and gravel. The form 

 of the draw-down curves of the wells close to the pumped well indicates 

that a large volume of water was withdrawn from storage immediately 

after pumping began, the amount withdrawn being in general 

inversely proportional to the distance from the pumped well. The 

draw-down curves of the wells comparatively far from the pumped 

well indicate that there was no withdrawal of water from storage at 

those distances from the pumped well for several hours after pumping 

began and that the maximum rate of withdrawal of water was not 

reached for some time after the first water was withdrawn. Obviously 

this lag was caused by the fact that all the water necessary to supply 

the pumped well was at first obtained from the sand and gravel 

nearby. The draw-down curves given in plate 4 show that the water 

levels in the observation wells were still declining after 48 hours of 

pumping. In other words, the cone of depression around the pumped 

well had not reached a condition of equilibrium.

The draw-down curves of observation wells near the pumped well 

reflected unavoidable changes in the rate of pumping. The pump 

was stopped several times during the second pumping test, and these 

ishut-downs caused the water levels in nearby observation wells to 

rise. Typical draw-down curves for the second test, given in plate 5, 

indicate that water continued to percolate toward the pumped well 

during the periods of interruption in pumping, and that this water 

began to refill the sand and gravel that had been unwatered during 

the periods of pumping. Of course, as soon as pumping was resumed 

the water table was again lowered. However, this lowering was 

resumed from a new level the level caused by the rise of the water 

table during the period in which there was no pumping and plate 5 

shows that sometimes it took several hours to lower the water level 

to the point where it stood before pumping was stopped. Only the 

wells comparatively close to the pumped well showed a rise of water 

level during the interruptions in pumping. The wells farther away 

showed a continuous decline of the water table, thus indicating that 

the water which caused the rise of the water levels in wells close to the 

pumped well came, at least in part, from storage in the area farther 

from the pumped well. There were several interruptions of pumping 

in the second test, and consequently considerable water was taken 

out of storage at some distance from the pumped well and stored
34 CONTRIBUTIONS TO HYDROLOGY OF UNITED STATES, 1935

temporarily in the sand and gravel close to the pumped well. This 

process tended to reduce the draw-down in the wells close to the 

pumped well, and therefore the coefficients of permeability computed 

from draw-downs observed in the second test are believed to be 

greater than the true permeability of the material. For this reason 

more study has been devoted to the first test, in which pumping was 

carried on at a nearly constant rate. It is probable that the 

inequalities in the cone of depression in the second test would have 

disappeared if pumping had been continued without interruption 

for several more days.

RECOVERY CURVES

A recovery curve is a continuous curve representing the movement of 

the water level in an observation well after pumping has stopped. 

Depending upon the location of the well, the movement after pumping 

has stopped may be either an immediate rise of the water level or a 

decline that is eventually followed by a rise. Measurements of the 

depth to the water level in the observation wells were continued in 

both tests after pumping had stopped in order to determine the rate 

and amount of recovery of the water table. Several typical recovery 

curves are shown in plates 4 and 5 as continuations of the draw-down 

curves. The recovery curves are usually smoother than the drawdown curves because of the absence of irregularities caused by interruptions in pumping or variations in the rate of pumping. The 

recovery curves show that in observation wells close to the pumped 

well the recovery was most rapid immediately * after pumping had 

stopped, whereas in observation wells comparatively far from the 

pumped well the recovery was most rapid several hours after pumping 

had stopped.

After pumping stopped, water continued to percolate toward the 

pumped well under the hydraulic gradient set up during the period of 

pumping, but instead of being discharged by the well it refilled the 

interstices in the sand and gravel that had been unwatered by the 

pumping. As the unwatered sand and gravel was gradually refilled 

the hydraulic gradient toward the well decreased, and the flow toward 

the well decreased proportionally. Thus the rate of recovery became 

progressively slower. The water level in wells comparatively far from 

the pumped well declined for several hours after pumping stopped. 

In these areas water continued to be taken from storage to supply the 

water that refilled the sediments around the pumped well. Of course, 

in time there was a general equalization of water levels over the entire 

region, and the water table assumed a form similar to that it had before 

pumping began. However, the ultimate level of the water table was a 

little lower than before pumping began, because water had been 

permanently removed from the zone of saturation during the period of 

pumping.
THIEM METHOD FOR DETERMINING PERMEABILITY 35

The rate of recovery of the water table is in general inversely proportional to the distance from the pumped well. However, this is 

true only for a short time after pumping ceases. Even though the 

water table close to the pumped well initially has a greater amount to 

recover, the rate of rise after a certain time is the same as the rate of 

rise of the water table at greater distances. After pumping is stopped 

the water table close to the well rises until the possible amount of 

recovery remaining that is, the remaining draw-down is equal to 

the remaining draw-down at distances farther from the pumped well. 

This is specifically shown in table 5. After 48 hours of pumping the 

decline of the water level in well 1, 24.9 feet from the pumped well, 

was 4.03 feet, and the decline in well 2, 59.9 feet from the pumped well, 

was 2.81 feet. After 2 hours of recovery the remaining draw-down in 

each well was about 1.68 feet, and the rates of recovery for the next 

22 hours were the same in the two wells. After 48 hours of pumping 

the draw-down in well 3, 114.4 feet from the pumped well, was 2.03 

feet about half of the draw-down of well 1. After 12 hours of recovery the remaining draw-down in all three wells was 0.77 foot, and 

the rates of recovery from that time on were the same. If the measurements of depth to water had been continued longer, the indicated rate 

of recovery in the wells farther from the pumped well would eventually 

have been nearly the same as the rate in wells 1, 2, and 3.

TABLE 5. Draw-down of the water table during test 1, at several times after pumping

stopped

Well no.

2. ... ...

3 _ 

4. 

5. 

6 .......

7 .-.. ...

8. ......

9 .......

10 

11 

Distance 

from 

pumped 

well 

(feet)

24.9 

59.9 

114.4 

164.2 

229.0 

354.1 

429.3 

478.9 

604.0 

754.6 

903.8

Draw-down (feet) at time indicated (hours) after pumping stopped

0

4.03 

2.81 

2.03 

1.62 

1.14 

.65 

.52 

.44 

.26 

.15 

.11

2

1.187 

1.69 

1.49 

1.30 

1.01 

.64 

.51 

.44 

.27 

.16 

.11

4

1.32 

1.33 

1.25 

1.11 

.91 

.60 

.49 

.44 

.28 

.16 

.11

6

1.12 

1.13 

1.17 

.98 

.83 

.56 

.47 

.43 

.28 

.17 

.12

8

0.98 

.99 

.94 

.88 

.76 

.53 

.45 

.42 

.28 

.17 

.12

10

0.86 

.87 

.85 

.80 

.70 

.50 

.43 

.41 

.28 

.17 

.12

12

0.77 

.77 

.77 

.72 

.64 

.48 

.41 

.39 

.28 

.17 

.12

14

0.70 

.70 

.70 

.67 

.59 

.45 

.40 

.38 

.27 

.17 

.13

16

0.64 

.64 

.64 

.60 

.55 

.43 

.38 

.36 

.27 

.18 

13

18

0.59 

.59 

.59 

.56 

.51 

.41 

.36 

.35 

.26 

.18 

.13

20

0.55 

.55 

.55 

.52 

.48 

.39 

.35 

.33 

.26 

.18 

.13

22

0.51 

.51 

.51 

.49 

.45 

.37 

.33 

.32 

.25 

.18 

.14

24

0.48 

.47 

.48 

.46 

.43 

.35 

.32 

.30 

.24 

.17 

.14

Table 5 also illustrates the decline of the water table after pumping 

ceases in wells comparatively far from the pumped well. Recovery 

started almost at once in wells 1 to 7, but in well 8 there was a lag of 

a few hours, and it was 6 hours before the water level reached a point 

0.01 foot above its level at the time when pumping stopped. In 

well 9 there was an actual decline of 0.02 foot during the first 4 hours 

of recovery, and it was not until 14 hours after pumping had stopped 

that there was any recovery from this low level. In well 10 there was
36 CONTRIBUTIONS TO HYDROLOGY OF UNITED STATES, 1935

a decline for 22 hours after pumping had stopped, and in well 11 the 

water level was apparently still declining after 24 hours. This lag 

is also shown by the recovery curves in plate 4.

The writer has devoted some time in an attempt to develop an equation for the recovery curves and their relation to the permeability of 

the water-bearing material. It would seem that the rate of recovery 

of the water level in an observation well is dependent on the quantity 

of water pumped, the draw-down of the water level at the time pumping stopped, the distance of the observation well from the pumped 

well, the initial hydraulic gradient, the thickness of the water-bearing 

formation, and the permeability of the formation. No equation was 

found that could be used for the draw-down curves of all the observation wells. However, the following general equation is suitable 

for many of the curves:

R= D

rpn l ! 

where R is the recovery of the water level, in feet; D is the draw-down 

from static water level at the time pumping stopped, in feet; K is a 

coefficient for each particular well; T is the elapsed period of recovery, 

in hours; and n is an exponent. K and n contain the varying distance 

factor as well as several other constants enumerated above. The 

formula is based on the assumption that when T equals 0, R equals 0; 

and when T equals infinity, R equals D. Of course, if pumping were 

carried on over an extended period during which there were no 

recharge, R would probably never equal D, because of the permanent 

withdrawal of water from storage. In figure 5 the recovery curve of 

well 5 is plotted as determined from the theoretical equation, and the 

actual field measurements are also indicated.

CONES OF DEPRESSION

Soon after pumping begins the water table around a pumped well 

assumes a form which is comparable to an inverted cone, although 

it is not a true cone. Where the water-bearing material is homogeneous, the so-called "cone of depression" will be circular if the initial 

water table is horizontal, but elliptical if the initial water table is 

sloping. The form of the cone of depression at any time can be shown 

by either profiles or contours on the water table. Profiles at different 

angles with the direction of initial slope of the water table may differ 

widely in form. The profiles in figure 6 are based on the draw-downs 

in wells on lines B and D, which have nearly the same direction as 

the initial slope of the water table. The development of the cone is 

shown by the several profiles, and it is interesting to note the rate at 

which the radius of the cone increased with the period of pumping. 

The profile of the cone after 2 hours of pumping shows that most of
tJ. 8. GEOLOGICAL SURVEY

_______________________________________- i i < 

4p.m. 6p.m. 8p.m. 10p.m. 12p.m. 2a.m. 4am. 6a.m 8a.m. fOam. 12m. 2pm. 4p.m. 6p.m. 8p.m. 10p.m. 12p.m. 2am. 4a.m. 6a.m. 8a.m. 10a

TYPICAL DRAW-DOWN AND RECOVERY CURVES FOR TEST 2.

U. S. GEOLOGICAL SURVEY 

&IQ.4O

W0.30

6a.m 8a.m. !0a.m. 12 m. 2p.m. 4p.m. 6p.m 8 p.m. 10p.m. 12p.m. 2a.m. 4a.m. 6a.m. 8a.m. 10a.m. 12m. 2p.m. -4p.m. 6p.m. 8p.m. 10p.m. 12p.m. 2a.m. 4

TYPICAL DRAW-DOWN AND RECOVERY CURVES FOR TEST 
C. B. GEOLOGICAL SUBVE* WATEB-SUPPLT PAFXR 878 PLAT* 6

X

Contour int«rv«l OJQ foot

CONTOURS ON THE WATER TABLE BEFORE PUMPING AND AT SEVERAL TIMES AFTER PUMPING BEGAN.

1, Before pumping began; 2, noood boor; 3, nidi hoar; 4, twelfth hour; 5, twenty-ftmrth hoar; 6, forty-eighth hour.

THIEM METHOD FOR DETERMINING PERMEABILITY 37
38 CONTRIBUTIONS TO HYDROLOGY OP UNITED STATES, 1935

the water pumped came from the unwatered sediments within a 

distance of 430 to 540 feet from the pumped well, because there was 

no draw-down of the water table beyond those distances. After 6 

hours of pumping the radius of the cone had increased to about 800 

feet, and after 12 hours of pumping there were small but measurable 

draw-downs at 1,200 feet from the pumped well. Further pumping 

undoubtedly increased the radius of the cone beyond the most distant 

observation wells.

It is probable that the rate at which the radius of the cone of depression develops depends somewhat upon the specific yield of the waterbearing sand and gravel. Most of the water that is discharged by the 

pumped well during the first hours of pumping is taken directly from 

the sediments around the pumped well. The amount of water taken 

from storage in this manner is indicated by the draw-down of the 

water table, which provides a method for determining the specific 

yield of the sediments (p. 9). If the water-bearing materials have 

a low specific yield, the radius of the cone of depression will probably 

develop more rapidly than if the materials have a high specific yield, 

for if the specific yield is low, there is less water available in the sediments, and hence the effect of pumping will be transmitted outward 

from the pumped well at a more rapid rate. The actual difference 

in the rate at which the radius of the cone of depression will develop 

in materials having different specific yields depends, of course, not 

alone on the specific yield but also on the relation between the specific 

yield and the permeability of the materials.

The slope of the cone of depression is steeper up-gradient from the 

pumped well than down-gradient (fig. 6). This indicates that if the 

permeability is the same, less water is percolating to the well from the 

down-gradient side. The slope of the cone down-gradient from the 

well becomes progressively less than the slope at the corresponding 

distance up-gradient, until at some distance down-gradient from the 

well the water table is horizontal. This point is called the groundwater divide. All water below this divide percolates away from the 

pumped well, and all water above this divide percolates toward the 

well. The ground-water divide moves down-gradient as the pumping 

period is increased and the cone of depression becomes larger. The 

ground-water divide in test 1, as indicated in figure 6, was about 280 

feet down-gradient from the pumped well after 2 hours of pumping. 

The divide gradually moved to about 360 feet below the pumped well 

after 6 hours of pumping, 440 feet after 12 hours, 500 feet after 24 

hours, 560 feet after 36 hours, and about 600 feet after 48 hours.

Contours on the water table before pumping began and at several 

times after pumping began are shown in plate 6. The limits of the 

area included in these maps are somewhat less than the distance of the 

farthest observation well from the pumped well, because there were
THIEM METHOD FOR DETERMINING PERMEABILITY 39
40 CONTRIBUTIONS TO HYDROLOGY OF UNITED STATES, 1935

too few observation wells to furnish data for contours over a 

larger area. The contours of the original water table were drawn 

readily by direct interpolation between the altitudes of water levels 

in the observation wells, but to draw the contours of the water table 

at given times after pumping began it was necessary to plot a profile 

of the cone of depression for each of the several lines and to determine 

the distance of each contour from the pumped well by inspection of 

the profile. A certain amount of judgment had to be used in tracing 

the contours between the several lines of wells.

The map showing contours on the water table before pumping 

began (map 1, pi. 6) indicates that the initial slope of the water table 

was nearly in the direction of the lines B and D and that this slope was 

very uniform, averaging about 6.9 feet to a mile. The contour 

interval selected was 0.1 foot for all the maps. These maps were 

drawn to illustrate the development of the cone of depression and 

especially the movement of the ground-water divide. The cone of 

depression is well denned close to the pumped well on map 2, showing 

contours on the water table after 2 hours of pumping, although there 

are some inequalities which are caused by the steep slope of the cone. 

The contours are nearly circular close to the well but gradually become 

elliptical at greater distances up to the ground-water divide. The 

divide as shown on these maps is a semielliptical line between the 

water which eventually enters the pumped well and that which percolates on down-gradient. A draw-down of the water table at some 

distance from the pumped well does not necessarily indicate that the 

water at that distance lies within the ground-water divide. It may 

indicate only that the draw-down of the water table created a slope 

of the water table that tended to move the water from its normal 

path, but this water may not reach the pumped well. The contours 

in map 2 show there was a draw-down of the water table for several 

hundred feet down-gradient from the ground-water divide. Also 

maps 2 and 3 show that there was a draw-down of the water table on 

lines A and C beyond the ground-water divide. The ground-water 

divide has been represented on maps 2 to 6 as the only line normal to 

the contours on the water table. Its position can be drawn in from 

the points of inflection of the contour lines. The steady decline of the 

ground-water divide down-gradient can be observed in maps 2 to 6, 

as well as by the profiles of the cone of depression in figure 6. The 

contour maps show in addition the lateral development of the divide 

and the increase in cross-sectional area through which water percolates 

to the pumped well.

The development of the cone of depression can probably best be 

seen by discussing one contour for example, the 1,809.5-foot contour on map 2 and observing the position of this line on the contour 

maps for subsequent times. After 2 hours of pumping this line is
THIEM METHOD FOR DETERMINING PERMEABILITY 41

somewhat elliptical, for it crosses line B at a point 85 feet up-gradient 

from the pumped well and line D at a point 120 feet down-gradient 

from the pumped well. After 6 hours of pumping this contour is still 

closed but includes a larger area. Its intersection with line B has 

advanced up-gradient to a point about 130 feet from the pumped well, 

and its intersection with line D has declined down-gradient to a point 

about 210 feet from the pumped well. After 12 hours of pumping this 

contour is no longer closed, and during the remaining 36 hours of pumping it diverges still more on the down-gradient side of the well, while 

its intersection with line B steadily moves up-gradient. The development of the cone close to the pumped well as the period of pumping 

lengthens can readily be seen by observing the increase in area included 

in the first closed contour, which is the same for maps 3 to 6.

After 2 hours of pumping the contours 300 to 400 feet up-gradient 

from the pumped well were affected but little by the pumping. As 

pumping continued the lines became closer together (indicating an 

increase in the slope of the cone of .depression), and after 48 hours 

they had a noticeable curvature.

The movement of the ground water was, ot course, always normal 

to the contours on the water table. Continuous lines drawn normal 

to the contours are called "lines of flow" of the ground water, and 

they trace the path of movement of a particle of water. The lines 

of flow on map 1, plate 6, are approximately parallel, but on maps 2 

to 6 the lines of flow included between the ground-water divide 

and the pumped well converge toward the well. The distances 

through which particles of water moved hi reaching the pumped well 

differed considerably. Thus a particle of water 400 feet from the 

pumped well on line B traveled to the pumped well through a much 

shorter path than a particle of water at the same distance from the 

pumped well on line A (map 6). Moreover, the particle of water on 

line B had to move with a greater velocity than the particle on line A, 

because the average hydraulic gradient along its path was greater. 

Consequently, the quantity of water that passed through a unit area 

on line B was much greater than the quantity that passed through a 

similar area on line A.

COMPUTATION OF COEFFICIENTS OF PERMEABILITY

Thiem's equation 4 was used for computing coefficients of permeability, and as the draw-downs at only two points on the cone of 

depression are required for the computation of a coefficient, the data 

obtained in the pumping tests are sufficient for a great many computations. Two general methods were used. In one method coefficients were computed by using the draw-downs that were measured 

in the observation wells, and in the other method coefficients were 

computed by using the interpolated draw-downs at selected distances
42 CONTRIBUTIONS TO HYDROLOGY OF UNITED STATES, 1935

from the pumped well, which were obtained from profiles of the cone 

of depression.

The coefficients of permeability computed by using the difference 

in draw-down at any two points on the cone of depression would be 

equal if the iorm of the observed cone of depression was the same as 

that of the theoretical cone of depression obtained by Thiem's formula. 

However, the cones of depression in both pumping tests were not 

identical with the theoretical cone, and the computed permeability 

ranged through wide limits. Computations of permeability were 

made by using the draw-downs in all possible combinations of observation wells on line A, after 48 hours of pumping, in test 1 (table 6). 

The coefficients of permeability thus computed ranged from 535 to 

5,630. The equation used was

527.7X540Xlog^

Tt Clr

in which the symbols are those given on page 10.

TABLE 6. Coefficients of permeability computed by Thiem's formula for all possible 

combinations of observation wells on line A

Well no.

72...... _ ..

1.. ...... ...... .

2...............

3_ -..-.. 

4....... ........

5_ -

6 -.-_ --.

7...... .........

8_. .......... ...

9...............

10 -- 

11...... _ .....

71

535

615

668

690

729

746

767

785

778

810

835

845

72

840

885

927

949

941

956

984

998

1,018

1,052

1,090

1

919

974

1,000

976

998

1,027

1,043

1,113

1,160

2

1,058

1,079

1,014

1,038

1,096

1,196

1,260

3

1,112

983

1,030

1,098

1,130

1,180

1,345

4

866

1,045

1,192

1,292

1,420

5

1,188

1,262

1,315

1,374

1,500

1,660

6

1,830

1,786

1,700

1,880

2,150

7

1,760

1,630

1,895

2,255

8

1,605

1,950

2,320

9

2,505

3,320

10

5,630

An inspection of table 6 shows that the computed coefficients were 

the smallest when both of the observation wells selected were close 

to the pumped well and the largest when both were far from the 

pumped well. This indicates that the difference in draw-down of 

wells close to the pumped well was great in comparison to the difference in draw-down of wells farther from the pumped well. In 

other words, the cone of depression had developed very little at 

distances far from the pumped well. These wide variations make the 

determination of the most nearly correct coefficient almost an impossibility from computations of this kind. The coefficients computed 

from combinations of observation wells on other lines gave similar 

results.

Coefficients could be computed, of course, from draw-downs at 

any time after pumping started. If the difference in draw-down 

changed during the period of pumping, the computed coefficient
THIEM METHOD FOR DETERMINING PERMEABILITY

would likewise change. In order to observe the effect of the length 

of the period of pumping on the computed permeability, coefficients 

were computed for several combinations of wells on line A and for 

several periods of time after pumping began (table 7). In general, 

the computed coefficients became smaller as the period of pumping 

increased, because the difference in draw-down became larger. However, the range in coefficients was smallest when the observation wells 

were selected close to the pumped well. At all times the coefficients became larger as the farthest observation well was selected 

farther from the pumped well, indicating that the form of the cone 

of depression was not equal to that of the theoretical cone.

TABLE 7. Coefficients of permeability computed by Thiem's formula for several 

combinations of observation wells on line A and for several periods of pumping

Well nos.

1 and 2 ____________________

1 and 3. ___________________

1 and 4. _ ...--.-......... -...... .. 

1 and 6- ___________________

1 and 7.. . .. ..... .......

Coefficients at different times (hours) after pumping began

2

960 

1,110 

1,180 

1,268 

1,413 

1,510 

1,550

6

868 

958 

1,008 

1,037 

1,132 

1,198 

1,225

12

879 

948 

974 

987 

1,040 

1,090 

1,107

24

935 

1,003 

1,025 

1,008 

1,040 

1,076 

1,095

36

918 

981 

1,003 

982 

1,010 

1,043 

1,060

48

919 

974 

1,000 

976 

998 

1,027 

1,043

Another computation was made by using the draw-downs obtained 

from profiles of the cone of depression. The draw-downs at several 

distances on lines A, B, C, and D were determined after 48 hours of 

pumping, and coefficients were computed by using a=50 feet and 

aj equal to several distances (table 8). The computed coefficients 

of permeability vary in about the same manner as those shown in 

table 6, but the variation is considerably less. The greatest variation occurred on line B, but this was only from 823 to 1,180. On lines 

C and D the coefficients computed for aj = 75 feet were larger than 

the coefficients obtained for ai = 200 feet, but on lines A and B the 

opposite was true. This fact suggests that the difference in the 

variation of the coefficients may be caused by the initial slope of 

the water table, because lines C and D are extensions of lines A and 

B, respectively. In general, this suggestion is confirmed by averaging the coefficients computed for several distances on line A with 

those computed for the same distances on line C, and similarly 

averaging the coefficients computed for lines B and D (table 8). 

The averages for lines A and C are very nearly the same as the averages at corresponding distances on lines B and D. The differences in 

the coefficients could be due to the fact that the wells penetrated 

materials of varying permeability, but it does not seem probable that 

the materials would be distributed areally in such a manner that all
44 CONTRIBUTIONS TO HYDROLOGY OF UNITED STATES, 1935

wells on one line penetrated a material of one permeability and all 

wells on another line penetrated a material of another permeability.

TABLE 8. Coefficients of permeability computed by Thiem's formula, using

a 60 feet

ai (feet)

75 

100 - ---------- ___ . _ .............

150 

200 

250 

300 ....... .... _

350 

400 - .

450 

500 

550 

600 ......

650 

700 

750 

800-.. .. ...... .......

850 

900 

Coefficients

Line A

957 

994 

1,055 

1,025 

1,000 

1,008 

1,022 

1, 050 

1,074 

1,096 

1,112 

1,130 

1,140 

1,156 

1,171 

1,192 

1,215 

1,232

Line B

823 

856 

888 

914 

929 

943 

958 

978 

994 

1,020 

1,040 

1,067 

1,086 

1,109 

1,130 

1,147 

1,169 

1,180

LineC

1,032 

980 

952 

970 

987 

1,002 

1,018 

1,030 

1,050 

1,075 

1,100 

1,122 

1,147 

1,170 

1,180 

1,210 

1,229 

1,224

Line D

1,200 

1,208 

1,114 

1,102 

1,087 

1,090 

1,100 

1,104 

1,115 

1,132 

1,152 

1,168 

1,182 

1,198 

1,216 

1,237 

1,260 

1,280

Average 

of lines 

A and C

995 

987 

1,004 

996 

994 

1,005 

1,020 

1,040 

1,062 

1,081 

1,106 

1,126 

1,144 

1,163 

1,176 

1,201 

1,222 

1,228

Average 

of lines 

BandD

1,012 

1,032 

1,001 

1,008 

1,008 

1,017 

1,029 

1,042 

1,055 

1,076 

1,096 

1,118 

1,134 

1,154 

1,173 

1,192 

1,215 

1,230

As the variations in the coefficients of permeability are caused by 

differences between observed and theoretical draw-downs it is well to 

examine the observed draw-downs and to determine the manner and 

amount of their deviation from theoretical draw-downs. The observed draw-downs of the water table as taken from profiles of the 

cone of depression after 48 hours of pumping in test 1 for several distances and directions from the pumped well are given in table 9. The 

draw-downs decrease regularly with the distance from the pumped 

well, but the draw-downs at equal distances from the pumped well are 

not equal. However, as would be expected from the previous averaging of the coefficients, the average of the draw-downs on lines A and C 

at equal distances from the pumped well are very nearly equal to the 

average draw-downs on lines B and D at the same distances from the 

pumped well.

The computations of coefficients of permeability have indicated that 

the computed coefficients become larger as the observation wells are 

selected at greater distances from the pumped well. As the differences 

in draw-down are substituted in the denominator of Thiem's formula, 

the reason for this increase in the coefficients is that the difference in 

draw-downs is relatively too small. This, in turn, indicates that the 

cone of depression has not reached a condition of equilibrium. The 

draw-downs at several distances on line A were averaged with the 

draw-downs at corresponding distances on line C, and the differences 

in draw-down (s sO were computed, s being taken as equal to the 

average of the draw-downs on lines A and C at 40 feet from the
THIEM METHOD FOR DETERMINING PERMEABILITY 45

TABLE 9. Draw-down of water table, in feet, after 48 hours of pumping during test 1, 

for several distances and directions from the pumped wett

Distance from pumped well (feet)

50.. ......... ......... ...... . ...........

75.. . loo.......................................

150............ ...........................

200... .................................

260.. . .................................

300. . . . ......... ... 

350.. ...... ...........................

400.. .......... .

450. ...................................

500.----.. .....

650 ..................................

600 

650-. _ .. ... .

700. .................. ...............

750...... ................... . .......

800... . ........... .............

850 

900 ............. ...............

950... .......................... ......

1,000.. . ..... .................

1,050 

 Draw-down

Line A

3.06 

2.52 

2.17 

1.74 

1.35 

1.03 

.82 

.66 

.57 

.49 

.41 

.34 

.28 

.22 

.18 

.15 

.125 

.12 

.105

LineB

3.14 

2.505 

2.11 

1.57 

1.22 

.95 

.74 

.575 

.455 

.36 

.29 

.23 

.195 

.16 

.14 

.115 

.085 

.075 

.065 

.06 

.05 

.05

LineC

2.98 

2.48 

2.08 

1.52 

1.17 

.92 

.73 

.57 

.44 

.35 

.28 

.23 

.195 

.16 

.14 

.12 

.10 

.075 

.06 

.05 

.04 

.025

Line D

2.93 

2.50 

2.20 

1.68 

1.33 

1.06 

.86 

.70 

.56 

.445 

.37 

.305 

.245 

.195 

.155 

.13 

.11 

.10 

.085 

.08 

.065 

.065

Average 

of lines 

A and C

3.02 

2.52 

2.13 

1.63 

1.26 

.96 

.78 

.62 

.51 

.42 

.35 

.29 

.24 

.19 

.16 

.14 

.11 

.10 

.08

Average 

of lines 

B and D

3.08 

2.50 

2.16 

1.63 

1.28 

1.01 

.80 

.64 

.52 

.41 

.33 

.27 

.23 

.18 

.15 

.13 

.10 

.09 

.08 

.07 

.06 

.06

pumped well. These differences in draw-down are given in table 10. 

They increase as the distance from the pumped well increases. The 

difference in draw-downs between 40 and 60 feet from the pumped 

well decreased as the period of pumping increased, and for distances up 

to 200 feet the difference was practically the same for the final 24 hours 

of pumping. For distances greater than 200 feet the difference in 

draw-downs increased throughout the period of pumping.

TABLE 10. Differences in average draw-down on lines A and C, in feet

Difference at different times (hours) after 

pumping began

40 and 60 ...

40 and 80. -_.-_ 

40 and 100---------....--- ----- ------------- - 

40 and 120_ 

40 and 140 

40andl60 

40andl80 

40and200. 

40and240.. - 

40 and 280... ---------

40and320._ - - 

40and380 

40aadt4ttU 

40and>;800. __ 

40 and 700-.. 

40 and 800. 

40 and 900. _ 

2

0.56

.88

1.08

1.24

1.37

1.46

1.53

1.59

1.67

1.72

1.76

1.79

1.81.

1.83

1.84

1.84

1.84

1.84

1.84

6

0.55

.95

1.21

1.41

1.57

1.70

1.81

1.91

2.04

2.15

2.23

2.29

2.33

2.39

2.42

 2.43

2.43

2.43

2.43

12

0.54

.94

1.23

1.45

1.64

1.78

1.92

2.04

2.23

2.36

2.47

2.55

2.61

2.72

2.78

2.81

2.82

2.83

2.84

24

0.53

.91

1.20

1.43

1.61

1.76

1.91

2.04

2.25

2.41

2.53

2.64

2,71

2.85

2.93

2.97

3.00

3.01

3.03

36

0.53

.92

1.20

1.42

1.61

1.77

1.92

2.05

2. 26

2.43

2.55

2.67

2.75

2.92

3.02

3.08

3.12

3.13

3.15

48

0.53

.91

1.19

1.44

1.63

1.79

1.93

2.07

2.29

2.46

2.60

3.71

2.80

2.99

3.11

3.19

3.23

3.25

3.28

>48+

0.53

.90

1.19

1.42

1.62

1.79

1.95

2.08

2.32

2.52

2.69

2.84

2.98

3.27

3.51

3.71

3.88

4.03

4.12

1 Theoretical difference in draw-down after the cone of depression reaches equilibrium.
46 CONTRIBUTIONS TO HYDROLOGY OF UNITED STATES, 1935

The data given in table 10 indicate that after 48 hours of pumping 

the cone of depression had reached approximate equilibrium in form 

for a distance of about 200 feet from the pumped well. The cone of 

depression is said to have reached "approximate" equilibrium, 

because as the water table was lowered the cross-sectional area through 

which the water percolated was decreased, and the form of the cone 

was changed slightly to compensate for this decrease. An inspection 

of table 10 shows that the cone had reached approximate equilibrium 

at 160 feet from the pumped well after only 12 hours of pumping. 

This means that at these distances from the pumped well the coefficients of permeability computed between 12 to 48 hours after pumping 

began would be approximately equal to the coefficients computed after 

a much longer period of pumping. With the equation developed for 

the cone of depression, the theoretical draw-downs at various distances 

from the pumped well were computed, and the differences in drawdowns were obtained. These are given in the last column in table 10 

and are the theoretical differences in draw-down that would occur 

after the whole cone of depression had reached a condition of equilibrium. The table shows that for distances up to about 200 feet from 

the pumped well the theoretical and observed draw-downs are practically equal, but that for greater distances the theoretical difference is 

larger than the observed difference. Hence the coefficients of permeability computed from the differences in draw-down up to 200 feet 

would be nearly equal, but the computed coefficients would increase 

beyond that distance.

It has been shown that the observed draw-down of the water table 

does not equal the theoretical draw-down at all points on the cone of 

depression; hence values for the coefficient of permeability may vary 

widely when computed directly from the difference in draw-down of 

two observation wells. However, the observed draw-downs on one 

side of the pumped well when averaged with the observed draw-downs 

at corresponding distances on the opposite side of the well approached 

or equaled the theoretical draw-downs at those distances. Coefficients 

of permeability computed from the draw-downs averaged in this manner were nearly equal lor that part of the cone of depression that had 

reached approximate equilibrium in form that is, for distances from 

about 40 to 200 feet from the pumped well. This indicates that the 

pumping method can be used in the field and that it will yield consistent results provided the observation wells are located within that 

part of the cone of depression that has reached approximate equilibrium and on a straight line through the pumped well preferably on 

a line along the natural hydraulic gradient and on both sides of the 

pumped well.

The coefficients of permeability computed from various combinations of the draw-downs of the water table between the limits of 40 and 

200 feet from the pumped well in test 1 differ but little and average
THIEM METHOD FOB DETERMINING PERMEABILITY 47

about 997 (table 11). In table 1 are given the coefficients of permeability of samples of the sand and gravel taken from well 84 during the 

process of drilling, as determined in the hydrologic laboratory of the 

United States Geological Survey. These coefficients reach a maximum 

of 4,350, but their average, weighted as to thickness, is about 1,200.

TABLE 11. Final computation of coefficients of permeability from the average 

draw-downs on lines A and C where the cone of depression had reached approximate equilibrium in form

0.1

(feet)

60

80

100

120

140

160

180

200

80

100

120

140

160

180

200

100

120

140

160

180

200

120

140

160

180

200

140

160

180

200

160

180

200

180

200

200

(feet)

40

40

40

40

40

40

40

40

60

60

60

60

60

60

60

80

80

80

80

80

80

100

100

100

100

100

120

120

120

120

140

140

140

160

160

180

, a\ loga~

0.176

.301

.398

.477

.544

.602

.653

.699

.125

.222

.301

.368

.426

.477

.523

.097

.176

.243

.301

.352

.398

.079

.146

.204

.255

.301

.067

.125

.176

.222

.058

.111

.155

.051

.097

.045

TO

(feet)

96.94

97.14

97.28

97.40

97.50

97.58

97.65

97.71

97.40

97.54

97.67

97.76

97.84

97.91

97.98

97.73

97.85

97.95

98.03

98.10

98.16

98.00

98.09

98.17

98.24

98-30

98.22

98.30

98.37

98.43

98.39

98.46

98.52

98.54

98.60

98.68

S Sl

(feet)

0.53

.91

1.19

1.44

1.63

1.79

1.93

2.07

.38

.66

.91

1.10

1.26

1.40

1.53

.28

.53

.72

.88

1.02

1.15

.25

.44

.60

.74

.87

.19

.35

.49

.62

.16

.30

.43

.14

.27

.13

P

976

970

980

969

975

982

987

985

962

983

965

975

985

992

994

1,010

967

982

994

1,002

1,005

919

964

987

1,000

1,003

1,023

1,035

1,040

1,043

1,050

1,071

1,043

1,053

1,038

1,000

Average, 997.

Two methods for determining the most probable coefficient of permeability from the observed data are suggested by C. E. Van Orstrand, geophysicist of the United States Geological Survey. These 

.methods as explained by Mr. Van Orstrand are as follows:

The value of P is to be determined from the equation

(40)

One method of obtaining a close approximation to the value of P 

consists in assuming at least three values of P} preferably at equal 

intervals and over a range so great that the true value of P falls 

between the extremes. From each observed value of y we thus obtain 

.a value of C as shown in table 12. The average value of C, 8,409.15,
48 CONTRIBUTIONS TO HYDROLOGY OP UNITED STATES, 1935

and the corresponding value of P, 975, are then substituted in formula 40, to obtain ye, the computed value of y.

TABLE 12. Computation of v2 for P=975 and Q = 777,600 gallons a day

(feet)

40 

60 

80 

100 

120 .. _ _ .................... _ _.

140 

160 

180 

200 

 ^log.x

913, 063. 35

1, 013, 421. 91

1, 084, 630. 29

1, 139, 861. 41

1,184,988.85

1, 223, 143. 71

1, 256, 194. 76

1, 285, 349. 88

1, 311, 428. 35

c

8, 410. 54

8, 410. 37

8, 411. 37

8, 409. 45

8, 412. 16

8, 410. 35

8, 407. 94

8, 405. 62

8, 404. 54

8, 409. 15

y°

96:68

97.21

97.59

97.87

98.12

98.31

98.47

98.61

98.74

S

y*

96.67

97.20

97.58

97.87

98.10

98.30

98.48

98.63

98.76

V. ......

»

+0.01

+.01

+.01

.00

+.02

+.01

-.01

-.02

-.02

.0017

2,500

775 875 975

VALUES OF P 

FIGUBE 7. Relation of S»2 to P.

1,075 1,175

The differences (r) between tile observed and computed values, the 

residuals, are tabulated in the last column, and we find 2y2=0.0017. 

Proceeding in a similar manner for the values P=775 and 1,175, 

we find for Sp2 the respective values 0.2835 and 0.0948. Plotting all 

the values as shown in figure 7, we see at once that the value of P
THIBM METHOD FOR DETERMINING PERMEABILITY 49

that makes 2r* a minimum is slightly less than 1,000. This, in accordance with the principle of least squares, is the most probable value 

of P.

Another method of procedure arrives at the correct result in a 

more direct manner.' Let us write down an observation equation 

(40) for each observed value of y as given in table 12.

TABLE 13. Observation equations for y2

Observation equations

C+ 91 3, 063 PI =9, 347

C+l, 013, 422 PI =9, 450

C+l, 084, 630 PI =9, 524

C+l, 139, 861 Pi=9, 579

C+l, 184, 989 PI =9, 628

C+l, 223, 144 Pi=9, 665

C+l, 256, 195 PI =9, 696

C+l, 285, 350 PI = 9, 724

C+l, 311, 428 Pi=9, 750

'4

v for y*

0

+.01

+.02

+.01

+.04

+.02

-.01

-.03

-.03

2»2

y»

96.68

97.21

97.59

97.87

98.12

98.31

98.47

98.61

98.74

»for y

0

.00

+.01

.00

+.02

+.01

+.00 -.02

-.02

.0014

-0.01

.00

+.01

.00

+.02

+.01

.00 -.01

-.01

.0009

Combining these equations by the method of least squares, we have 

the normal equations .

9(7+ 10,412,082P1 = 86,363 

10,412,082(7+12,186,687,602,840^=100,055,976,372

the solution of which gives P=980.083, £7=8,415.381. The residuals 

corresponding to these values of P and C are given in the next to the 

last column of table 13. These values are slightly erroneous, owing 

to the fact that y2 instead of y was used in evaluating P and (7,

To obtain the correct values, let us compute the partial differential 

coefficients in the equation

As dy is here the error in y that is, v we write the observation 

equations in the form

dC Q]Qg.xdp_j.

L\ ^\ y-vrt Ifrjl U/U V, 2y 2iryP*

Substituting the appropriate values from tables 12 and 13, we have

0.005172<&7-0.004916dP= 0.00 

0.005144rfCr-0.005426dP= 0.00 

0.005123<#7 0.005785rfP= +0.01 

0.005109d<7-0.006062dP= 0.00 

0.005096c?<7 0.006286dP= +0.02 

0.005086^(7 0.006476dP= +0.01 

0.005078^(7- 0.006640dP= 0.00 

0.005070<ft7-0.006785dP= -0.02 

 .005064<ft7- 0.0069 13dP= -0.02
50 CONTEIBUTIONS TO HYDROLOGY OF UNITED STATES, 1935

The solution of these equations gives dP 6.782, dC 8.162; hence

P= 980.083+6.782=986.87 

(7=8,415.381+8.162=8,423.54

The substitution of these constants in equation 40 gives the values 

in the last column of table 13. As shown in the table, the value of 

S#2 has been reduced from 0.0014 to 0.0009 by the last solution. 

Assuming that the work has been done correctly and that the numbers have been rounded off correctly, a further reduction hi the value 

of Z,v2 is impossible.

DIFFERENCES BETWEEN FIELD AND THEORETICAL CONDITIONS

The conditions found in the field rarely approach closely the theoretical considerations from which Thiem's formula was developed. 

The water-bearing formations usually differ from place to place in 

thickness and in permeability. Wells usually do not extend to the 

bottom of the formation, and the bottom is not always parallel to 

the water table or piezometric surface. Most water tables and 

piezometric surfaces are not horizontal, and few pumping tests are 

continued until the cone of depression reaches approximate equilibrium in form over a large area.

The errors introduced into Thiem's formula by the differences between field and theoretical conditions can be minimized. The thickness of the water-bearing material can be obtained from as many well 

logs as possible, and an average value used for m. Of course, the 

value used in Thiem's formula is not important if the computed 

coefficient of permeability is to be used to determine the quantity of 

water that percolates through some cross section of the material in 

which the factor of thickness appears, because in that case m cancels. 

The effect of variations in the permeability of the water-bearing material can be lessened by selecting an average value for the computed 

coefficient by one of the methods previously explained. Other 

differences such as pumped wells that do not extend through the 

formation, water tables that are not horizontal, and cones of depression that have not reached equilibrium in form cause part of 'the 

cone of depression to differ from the theoretical cone. The effect of 

these differences can be minimized by substituting in Thiem's formula 

only the draw-down of the water table from that part of the cone of 

depression which corresponds with the theoretical cone.

The reason why the observed cone of depression differs from the 

theoretical cone may be made clear by a review of the behavior of the 

water table during the first pumping test. As soon as the pump began 

discharging water from the well a hydraulic gradient from all directions was established toward the well, and the water table was lowered. 

The lowering of the water table unwatered a considerable volume of
THIEM METHOD FOR DETERMINING PERMEABILITY 51

sediments, the water in them gradually draining down to the new 

water table and eventually entering the well. Thus most of the water 

which the pumped well discharged was obtained from the unwatered 

sediments. The decline in the water table close to the pumped well 

was large at first, and then as a gradient adequate to discharge the 

540 gallons of water a minute required for the well was formed, the 

water table was lowered more slowly. Farther from the pumped 

well the water table lowered less rapidly, because (1) water was being 

supplied to the pumped well from sources closer to the well and (2) 

the hydraulic gradient necessary to force the required amount of 

water through a unit area of the sediments was smaller. Gradually 

the water table around the pumped well assumed a slope that was 

large enough to cause equal quantities of water to flow toward the 

pumped well without withdrawing large amounts of water from 

storage. Thus the slope of the cone of the depression for distances 

up to 200 feet from the well reached essential equilibrium in form 

after 48 hours of pumping. Beyond the distance of 200 feet, however, the slope of the water table was less than necessary to transmit 

540 gallons of water a minute through cylindrical sections of the 

sediments. If pumping had been continued the water table at 

distances greater than 200 feet from the pumped well would have 

continued to decline until the necessary hydraulic gradient had been 

developed. To develop the necessary hydraulic gradient at distances 

comparatively far from the pumped well, a very large quantity of 

water must be taken from storage. Of course, most of the water 

taken from storage must be discharged by the pumped well, and 

hence the time required for the cone of depression to reach equilibrium 

depends to some extent upon the specific yield of the water-bearingmaterials. The slope of the cone of depression necessary to force 

540 gallons a minute through cylindrical sections is developed at 

increasing distances from the pumped well only by the decline of the 

water table at those distances. Thus it is evident that in order for 

the cone to maintain its slope nearer the pumped well, the water 

table must continue to decline there, but at a diminishing rate. This 

decline in turn uriwaters more sediments, and the development of 

the cone proceeds at a still slower rate.

The fact that the cone of depression reaches essential equilibrium 

near the pumped well comparatively soon after pumping begins 

provides the opportunity to use Thiem's formula. Fortunately the 

difference in draw-down, s Si, is substituted in Thiem's equation, and 

so long as this difference is constant, the coefficients will be equal. 

Little error is introduced by the increase in absol ute draw-down as pumping is continued. Thus, though the cone of depression has not reached 

equilibrium it is still possible to use the formula. It is interesting to 

observe the difference between the theoretical and observed draw-
52 CONTRIBUTIONS TO HYDROLOGY OF UNITED STATES, 1935

downs shown in table 10. The differences are practically equal for 

distances up to 200 feet from the pumped well, but for distances 

farther from the pumped well the theoretical difference is larger. 

The theoretical difference in draw-downs of the water table at 40 

feet and 1,000 feet from the pumped well is 4.12 feet, and the observed 

difference after 48 hours of pumping was 3.28 feet. This indicates 

that with further pumping the water table at 40 feet and 1,000 feet 

from the pumped well would lower in such amounts that the difference 

between the declines would be increased by 0.84 foot. There is no 

way of ascertaining the net decline of the water table at either of 

these distances from the pumped well, as the net decline depends 

upon the length of the period of pumping.

The pumped well used in test 1 did not extend through the waterbearing sand and gravel, and it seems probable that this influenced 

the cone of depression close to the pumped well. As shown in table 6, 

the coefficients of permeability computed by using the draw-downs 

in wells 71 and 72, 2.6 feet and 12.3 feet, respectively, from the center 

of the pumped well, were comparatively small, indicating that the 

draw-downs near the pumped well were relatively great. It seems 

probable that these comparatively large draw-downs were an effect 

of the well's failing to penetrate the entire thickness of the formation, 

because the form of the cone of depression reached essential equilibrium from 40 feet to 200 feet from the pumped well, and the drawdowns were of such magnitude that the computed coefficients of 

permeability were practically equal, whereas the water table within 

40 feet of the pumped well reached essential equilibrium but did not 

correspond to the theoretical cone. A part of these large draw-downs 

may also have been caused by changes in the permeability of the 

formation due to the rearrangement of the sand and gravel during 

the development of the well.

It is unfortunate that there were interruptions during the period .of 

the second test, for these interruptions so changed the normal drawdowns in the observation wells that the computations of permeability 

from.the data obtained in this test are of doubtful value. Some 

computations were made, and the coefficients averaged about 1,300 

for wells 56 to 61 on line SW. Because there were interruptions 

in pumping it would be expected that the draw-downs would be 

smaller and consequently the coefficients larger. It is difficult to 

make an intensive study of this test, because only line SW extended 

through pumped well 84. The draw-down measurements made 

during these tests indicate the behavior of the water table at several 

distances from the pumped well when pumping is not continuous, and 

the measurements of depth to the water table made after the pumping 

stopped are valuable for determining the rate and amount of the 

recovery of the-water table.
THIEM METHOD FOE DETERMINING PERMEABILITY 53 

DETERMINATION OF SPECIFIC YIELD BY THE PUMPING METHOD

The data obtained during the first pumping test are adequate for 

detailed study of the use of the pumping method for determining the 

specific yield of water-bearing materials. This method was suggested 

by Meinzer 19 and essentially consists of determining the ratio of (1) 

the quantities of ground water that in a given time are taken from 

storage between concentric cylindrical sections around the pumped 

well to (2) the volumes of sediments between the cylinders that are 

unwatered in that time. A preliminary report by the writer on this 

method, with reference to the pumping tests described in this paper, 

has been published.20 As the quantities of water taken from storage 

are determined by ascertaining the difference in the quantities of 

water that percolate through the cylinders in a given time, the 

specific yield may be expressed by the equation

where y is the specific yield, YI is the quantity of ground water, in 

cubic feet, that percolates through the smaller cylinder, Y is the quantity of ground water, in cubic feet, that percolates through the larger 

cylinder, and V is the volume of water-bearing material, in cubic 

feet, that is unwatered between the cylinders.

The volumes of water-bearing material that were unwatered 

between concentric cylindrical sections around the pumped well 

during certain periods of pumping were computed by the formula

 -- (107)

where V is the volume of unwatered material, in cubic feet; a is the 

radius of the large cylinder, in feet; a,\ is the radius of the small 

cylinder, in feet; and s and Si are the draw-downs of the water table, 

in feet, at the distances a and a\, respectively, from the pumped well. 

The draw-down at any given distance from the pumped well was 

taken as the average draw-down at that distance on all lines. The 

volumes of material that were unwatered during 6, 12, 24, 36, and 48 

hours of pumping were determined by computing the volumes 

unwatered between 22 cylinders whose radii ranged from 10 feet to 

900 feet, but only the 'volumes of water-bearing material that were 

unwatered between several selected concentric cylindrical sections 

around the pumped well are given in this report (table 14).

19 Meinzer, O. E., Outline of methods for estimating ground-water supplies: U. S. Oeol. Survey WaterSupply Paper 638, p. 136, 1932.

" Wenzel, L. K., Specific yield determined from a Thiem's pumping test: Am. Oeophys. Union 

Trans., 1933, pp. 475-477.
54 CONTRIBUTIONS TO HYDROLOGY OF UNITED STATES, 1935

TABLE 14. Volumes of water-bearing material, in cubic feet, that were unwatered 

between concentric cylindrical sections around the pumped well for several periods 

of pumping

Radii of cylinders (feet)

50 and 280. _________________________

50 "arid 320 .- . .._ -.. .- 

50and»360 ..... .................... _ ............

50 and 400........ ...................................

50 and 500......... .... __ __._- __ ... _ . _ -.

Volume during different periods of pumping (hours)

0-6

160, 500 

179,800 

196,000 

209,400 

234,200 

248, 900 

259, 100 

263, 800 

263,800

0-12

227, 700 

260, 100 

289,200 

315, 000 

367, 300 

401,000 

421, 600 

437,000 

449,000

0-24

275,600 

319, 300 

359, 500 

395, 800 

474, 300 

532, 100 

575,000 

608,000 

634,800

0-36

308, 300 

361, 400 

411,400 

458,200 

561, 500 

640,000 

699,300 

745, 200 

785, 300

0-48

336, 400 

397,000 

455,200 

510, 600 

632, 400 

731,000 

806, 700 

865, 600 

916, 300

The quantities of water that were taken from storage between the 

concentric cylindrical sections were determined by computing the 

quantity of water that percolated through the smaller cylinder and 

subtracting from it the quantity of ground water that percolated 

through the larger cylinder. The quantities of water that percolated 

through each cylinder in a given time were computed by the formula

7.48X24-------------- (108)

where Y is the quantity of ground water, in cubic feet; P is the coefficient of permeability; i is the average hydraulic gradient, in feet per 

foot, at the distance a from the pumped well; h is the average thickness, in feet, of the saturated water-bearing material at the distance a; 

and T is the period of pumping, in hours. The quantities of water 

that percolated through several selected cylindrical sections in short 

periods were computed, and then a summation of these quantities 

was made (table 15). The computations were first made for short 

periods of time in order to decrease the error introduced by changes of 

the hydraulic gradient, especially in the first few hours of pumping. 

The periods of pumping given in table 15 were arbitrarily selected in 

order to show the change in specific yield with the period of draining.

TABLE 15. Computed quantities of ground water, in cubic feet, that percolated 

through several concentric cylindrical sections around the pumped well for several 

periods of pumping

Quantity during different periods of pumping 

(hours)

Jttaaius or cylinder (.leet;

SO........ ......................................... ..

280.... ...................................... ..... ...

320... ...............................................

360............................................. _ ..

400.... _. , -__-..-___ __ ...

500... ... ................... .... .... . - _ -

600..-............. ........... ...... ..... . 

700. . . . 

800..... .............................. ...-...-.-. .

900.... ... . ...-.-.-..-_-.__-.- _

0-6

23,800

8,000

7,200

5,700

4,700

2,000

1,400

1,200

700

0

0-12

51,400

22, 300

20, 300

16, 700

14,000

8,100

5,600

4,800

2,700

1,200

0-24

104, 000

57,900

53,000

45,200

39, 600

26, 300

19,500

12,800

8,800

5,800

0-36

155, 400

97,700

89,800

79, 300

70,400

51,300

36,400

23,600

18,100

12,800

0-48

206,800

139,600

128, 200

116,000

105, 200

80,800

60,200

38,800

29,400

20,300
THIEM METHOD FOR DETERMINING PERMEABILITY OO

The cylinder with a radius of 50 feet was chosen as the control 

cylinder, and the quantities of water that were taken from storage 

were computed by determining the differences in the quantities of 

water that percolated through that cylinder and the quantities that 

percolated through larger cylinders (table 16). Specific yield was 

then determined by dividing the quantities of ground water taken 

from storage in certain periods of time (table 16) by the volumes of 

water-bearing material that were unwatered in the corresponding 

periods (table 14). The results are given in table 17.

TABLE 16. Quantities of ground water, in cubic feet, taken from storage between 

several concentric cylindrical sections around the pumped well and for several 

periods of pum.ping

Radii of cylinders (feet)

 50 and 280.__.__ __ ................................

50 and 320..... ......................................

50 ana'-SQO-L.. ............... __ . __ . _ -...-.--_.

-50 and 400 .__.-_. --......_. __ ... _ ..........

50 and 600.......... ________ __________

50and700..... .-_..-.._.-.-.-_._._... 

SOandSOO....----- - _-.-- ..--. __ .-. 

50 and 900.... ...... ___ .... ____ .... _ ........

Quantity during different periods of pumping 

(hours)

0-6

15,800 

16,600 

18, 100 

19, 100 

21,800 

22,400 

22,600 

23,100 

23,800

0-12

29,100 

31,100 

34,700 

37,400 

43,300 

45,800 

46,600 

48,700 

50,200

0-24

46,100 

51,000 

58,800 

64,400 

77, 700 

84,500 

91,200 

95,200 

98,200

0-36

57, 700 

65,600 

76,100 

85,000 

104, 100 

119,000 

131. 800 

137, 300 

142, 600

0-48

67,200 

78,600 

90,800 

101,600 

126,000 

146,600 

168,000 

177, 400 

186, 500

TABLE 17. Specific yield as computed for several concentric cylindrical sections and 

for several periods of pumping

Radii of cylinders, (feet)

«) and 28CL.......... ........................................ .

SOsaA-330......... .......................... ..................

50 and 360-...-..--.. __ ............... ___ ..... __ .......

50 and 400 ________________________________

SOandSOO........ ___________________________

50 and 700.. ..... _ ........... _ . __ ... ____ .........

.50 and 800........ __ . .. _ . ______ . ___ -------

50 and 900-..-.- ____________________________ .

0-6

9.8

9.2

9.2

9.1

9.3

9.0

8.7

8.8

9.2

0-12

12.8

11.9

12.0

11.9

11.8

11.4

11.1

11.1

11.2

11.7

0-24

16.7

16.0

16.3

16.3

16.4

15.9

15.7

15.5

16.1

0-36

18.7

18.2

18.5

18.5

18.5

18.6

18.8

18.4

18.2

18.5

0-48

20.0

19.8

19.9

19.9

19.9

20.0

20.8

20.5

20.3

20.1

Specific yield during different periods of 

pumping (hours)

As an illustration, the specific yield will be computed for the volume 

of sediments unwatered during 48 hours of pumping between concentric cylindrical sections with radii of 50 and 280 feet. The average 

slope of the cone of depression during the last 12 hours of pumping 

was 2.59 percent at 50 feet from the pumped well and 0.369 percent 

at 280 feet from the well. The quantity of water that percolated 

through the 50-foot cylinder from 36 to 48 hours after pumping began 

was as follows:

2X3.1416X975X0.0259X50X96.92X12

7.48X24 =51,400 cubic feet
56 CONTRIBUTIONS TO HYDROLOGY OF UNITED STATES, 1935

This quantity added to the 155,400 cubic feet (determined in the same 

manner) that percolated through this cylinder in the preceding 36 

hours gives a total of 206,800 cubic feet ground water that percolated through the 50-foot cylinder in 48 hours. The quantity of 

water that percolated through the 280-foot cylinder in the last 12 

hours of pumping was as follows:

V 2X3.1416X975X0.00369X280X99.42X12 Qnn . . , 2= 7 4SV24 =41,900 cubic feet

This quantity added to the 97,700 cubic feet that percolated through 

this cylinder in the preceding 36 hours gives a total of 139,600 cubic 

feet of ground water that percolated through the 280-foot cylinder 

in 48 hours. The quantity of ground water taken from storage 

between the* 50-foot and 280-foot cylinders equals 206,800 minus 

139,600, or 67,200 cubic feet; the volume of material that was unwatered between these cylinders was 336,400 cubic feet (table 14), 

Hence the specific yield is computed as follows:

67,200X100 336,400 ~^U' U

The computed specific yield becomes larger as the pumping increases (table 17), the reason being that all the water in the material 

does not drain out of it immediately. Investigations have shown 

that a sample of material after being saturated will yield a very large 

percentage of its water within a few hours, though it may continue 

to yield small amounts for several years. When pumping first starts 

a comparatively large volume of material is unwatered, partly because 

only a small percentage of the water contained in the interstices of 

the sediments immediately drains down to the water table. As pumping 

is continued more water gradually drains out of the unwatered 

sediments, and hence the specific yield computed from the first few 

hours of pumping is relatively small. The specific yield computed 

from the volume of water-bearing material unwatered in the last few 

hours of pumping would be larger, because of the addition of water 

that percolated down from the material previously unwatered. The 

average values for specific yield (table 17) plotted against the periods 

of pumping fall on a smooth curve. By extending this curve the conclusion is reached that the true specific yield lies between 22 and 23.

A sample of the water-bearing material that was unwatered during 

this test was examined in the hydrologic laboratory of the United 

States Geological Survey. The porosity was found to be 27.1 and 

the moisture equivalent 2.6 (table 1). If the moisture equivalent is 

used roughly for specific retention, as is sometimes done, the specific 

yield of the sample is computed to be 24.5; if Piper's relation between
THIEM METHOD FOR DETERMINING PERMEABILITY 57

moisture equivalent and specific retention 21 is used, giving a specific 

retention of about 5 for materials with a moisture equivalent of 2.6, 

the specific yield is computed to be 22.1. Both of these values 

compare favorably with the value determined from this pumping test.

11 Piper, A. M., Notes on the relation between the moisture equivalent and the specific yield of waterbearing materials: Am. Geophys. Union Trails., 1933, pp. 481-487.

o