Methods of Determining 

Permeability, Transmissibility 

and Drawdown

Compiled by RAY BENTALL

GROUND-WATER HYDRAULICS

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1S36-I

This compilation consists of reports by 

the following authors:

R. H. Brown, J. G. Ferris, C. E. Jacob,

D. B. Knowles, R. R. Meyer, H. E. Skibitzke,

and C. V. Theis

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1963
UNITED STATES DEPARTMENT OF THE INTERIOR 

JAMES G. WATT, Secretary

GEOLOGICAL SURVEY 

Dallas L. Peck, Director

First printing 1963 

Second printing 1964

Third printing 1967 

Fourth printing 1980

Fifth printing 1983

For sale by the Distribution Branch, U.S. Geological Survey, 

604 South Pickett Street, Alexandria, VA 22304
CONTENTS

Page 

Introduction..__________________________________________________ 243

Determining the permeability of water-table aquifers, by C. E. Jacob____ 245

Abstract__________________________________ 245

The theory of permeability__-_---_-_----___----__-_-_________ 245

Theis graphical solution using corrected drawdowns._______________ 247

Wenzel's limiting formula__________________________________ 254

The location of observation wells______________________________ 255

The gradient formula______________________________________ 263

Water-level corrections for wells tapping less than the full thickness 

of an aquifer..__________________________________________ 264

Correction of drawdowns caused by a pumped well tapping less than the 

full thickness of an aquifer, by C. E. Jacob.______________________ 272

Abstract______._______________________________ 272 

The formula for correcting drawdowns_______________--__-__---__ 272

Application of the formula____________________________________ 277

Determination of the drawdown at the point of stagnation________ 280

Drawdown distribution in the vicinity of a pumped well tapping less

than the full thickness of an aquifer_________________________ 281

The recovery method for determining the coefficient of transmissibility,

by C. E. Jacob__________________________________ 283

Abstract______________________________________ 283

The Theis recovery method ___ ___________________ 283

Wenzel's method of correcting water-level recovery data__________ 284

Constant, but different, coefficients of storage for the pumping and 

recovery periods_________________________-______-__------_- 285

The uniformly changing coefficient of storage_____-__-__-_____ 288

Evaluation of Wenzel's method of correcting recovery data.________ 289

Determination of the coefficient of transmissibility from measurements of 

residual drawdown in a bailed well, by Herbert E. Skibitzke. _________ 293

Abstract______________________________________ 293 

The bailed well as an instrument of hydrologic analysis___________ 293

Derivation of the formula for a vertical line source from the equation

for heat flow from a point source_-________-----------_-------- 293

Derivation of the formula for a vertical line source from the Theis 

recovery equation______________________________________--__- 296

Modified formula for the analysis of water-level recovery after repeated bailing_____________________________________________ 297

Use and limitations of the formulas________-__--__------_-------- 298

The slug-injection test for estimating the coefficient of transmissibility of

an aquifer, by John G. Ferris and Doyle B. Knowles_______________ 299

Abstract____________________.______________ 299 

Equation for drawdown in an instantaneous vertical line source or

line sink_____________________________________ 299 

Equipment and procedure for a slug-injection test.________-__----_ 300

A practical application of the slug-injection test___________--__---_ 303

m
IV CONTENTS

Page

Cyclic water-level fluctuations as a basis for determining aquifer transmissibility, by John G. Ferris___________________________________ 305

Abstract__ _ ________________________________________________ 305

Acknowledgments _____________________________________________ 305

Cyclic fluctuations of water levels_---_-_-_-____-___-__-_-___.___ 305

The stage-ratio method for determining the coefficient of transmissibility__________________________________________________ 306

The time-lag method for determining the coefficient of transmissibility. _________________________________________________ 309

Other related formulas __________________________________________ 309

Possible additional use and extension of the methods for analysis of

cyclic fluctuations___________________________________________ 310

A field problem involving cyclic fluctuations___-___-_-_-____-_-_-_ 311

Drawdowns resulting from cyclic rates of discharge by Charles V. Theis. _. _ 319 

Abstract___________________________________________________ 319

The drawdown effects of cyclic pumping________________________ 319

A formula for drawdowns caused by cyclic pumping_______________ 319

Drawdowns resulting from cyclic intervals of discharge, by Russell H.

Brown _________________________________________________________ 324

Abstract____________________________________________________ 324

Cyclic withdrawals of groundwater_____-__-__-______-_----_-__- 324

Formula for determination of drawdown in a cyclically pumped well. _ 324 

Estimating the transmissibility of aquifers from the specific capacity of

wells, by Charles V. Theis, Russell H. Brown, and Rex R. Meyer. _ _ _ _ _ 331

Abstract__--_-____-___-___-_____-_-____--____-________--__..- 331

The general relationship between transmissibility and specific capacity. 331 

Estimating the transmissibility of a water-table aquifier from the

specific capacity of a well, by Charles V. Theis.__-__-____-_____- 332

Estimating the transmissibility of an artesian aquifer from the

specific capacity of a well, by Russell H. Brown__________________ 336

A chart relating well diameter, specific capacity, and the coefficients of

transmissibility and storage, by Rex R. Meyer._________________ 338

References cited-________-_-___-_-_____---_-_-__-__---___-___--____ 341

ILLUSTKATIONS

Page 

FIGURE 72. Semilog graph of water-level drawdown during aquifer test

near Wichita, Kans------_------_--------_---_-------- 250

73. Log graph of water-level drawdown during aquifer test

near Wichita____________________--_____-___----___- 251

74. Map showing the location of wells used for aquifer test near

Grand Island, Nebr______-__--_-_____----___-------- 259

75. Log graph of water-level drawdown along lines A and C

during aquifer test near Grand Island.__________________ 260

76. Log graph of water-level drawdown along lines B and D

during aquifer test near Grand Island.-._____.___--___-- 261

77. Log graph of water-level drawdown along lines N, S, SW, and

W during aquifer test near Grand Island__________-_--_- 262

78. Data from aquifer test near Grand Island demonstrating the 

effects of a well tapping less than the full thickness of the 

aquifer_ __ _ ____________-______----__------------- 9RV
CONTENTS V

Page

79. Semilog graph of water-level drawdown during aquifer test

near Grand Island___________________________________ 268

80. Corrections for drawdowns along the top of an aquifer for 

differential fractional parts, A, tapped by the pumped 

wdl..... ...._____._____.._..__.__________________ 275

81. Corrections for drawdowns along the bottom of an aquifer 

for different fractional parts, A, tapped by the pumped 

welL_______________________________________________ 276

82. Graph illustrating the method of correcting drawdowns to

determine T______________________________________ 278

83. Graph showing the distribution of drawdowns near a pumped

well tapping 40 percent of the thickness of an aquifer. _ _ _ 282

84. Water-level recovery curves for different types of variation

of S with time.___________________________________ 284

85. Straight-line plots for cases 1 and 2 and Wenzel's empirical

correction applied to case 2-____.-___-_--______-_____- 286

86. Graph showing curves obtained by plotting s' against log

(tft1 ), aquifer test near Grand Island.__________________ 288

87. Straight-line plot for case 1, the uncorrected plot for case 3,

and Wenzel's empirical correction applied to case 3______ 290

88. Graph showing curves obtained by plotting s' against log

(tit1 ), aquifer test near Scottsbluff, Nebr_____________ 291

89. Equipment for making a slug-injection test _______________ 301

90. Graph showing decline of residual head in a 6-inch well at 

Speedway City, Ind., following the instantaneous injection 

of a 39-gallon slug of water__-_-_____-.___--__-_____-_ 304

91. Map of the Ashland well field of the municipally owned

water supply of Lincoln, Nebr-_-______-_______-___-___ 312

92. Generalized east-west geologic section through supply well 2

in the Ashland well field.._--_____-__--____.__--__-_-_ 313

93. Graphs showing the stage of the Platte River and the water

level in observation well 1 in the Ashland well field ______ 314

94. Semilog graph of the average ratio of rising and falling stages 

of the Platte River plotted against the distance of the observation wells from the riveredge.-__--_--__-.--_----- 315

95. Graph of the time lag between maximum and minimum 

stages of the Platte River and corresponding water levels 

in observation wells plotted against the distance of the 

wells from the riveredge---_--_-------------___---_-_- 317

96. Graph showing the drawdown in an observation well 2,000

feet from a well pumped at a cyclically varying rate____. 322

97. Hydrograph for a cyclically pumped well showing the symbolism used for time factors__.____-______-__-_____-_ 325

98. Semilog plot of the factor s-T/264<? against the number of

pumping cycles n for selected values of p___-_-_-__--__- 327

99. Diagram for estimating the transmissibility of an aquifer

from the specific capacity of a well______--_--___.-_-_ 334 

100. Graph showing relation of well diameter, specific capacity,

and coefficients of traosmissibility and storage_________ 339
VI CONTENTS

TABLES

Page 

TABLE 1. Data for aquifer test near Wichita, Kans., giving drawdowns

after 18 days of continuous pumping at 1,000 gpm_________ 252

2. Data for aquifer test near Grand Island, Nebr., giving drawdowns after 48 hours of continuous pumping at 540 gpm_ ... 256

3. Coefficients of transmissibility and storage determined from the 

corrected drawdowns after 48 hours of continuous pumping 

at 540 gpm during aquifer test near Grand Island, Nebr____ 258

4. Data used in determining the coefficient of permeability from 

radial components of the hydraulic gradient after 48 hours of 

continuous pumping at 540 gpm during aquifer test near 

Grand Island, Nebr--_-__--_-_______________ 264

5. Interruptions in the pumping from well 84 near Grand Island

I' _i" 

Nebr., and the corresponding values of the factor ^ 

{

used in correcting the 48-hour drawdowns______________ 270

6. Data for determination of drawdown at the point of stagnation 

beneath a well tapping less than the full thickness of an 

aquifer_____..____.-._-____________-___ 281

7. Data for a slug-injection test at Speedway City, Ind_________ 303

8. Ratio of the range in water-level fluctuation in observation 

wells 1, 2, and 3 to the corresponding range in stage of the 

Platte River at the U.S. Highway 6 bridge. _________ 311

9. Time lag between the minimum and maximum stages of the 

Platte River at the U.S. Highway 6 bridge and the corresponding minimum and maximum water levels in observation wells 1, 2, and 3__________________________-----_-_ 316

10. Summary of determinations of the coefficient of transmissibility by the stage-ratio and the time-lag methods.________ 318
GROUND-WATER HYDRAULICS

METHODS OF DETERMINING PERMEABILITY, 

TRANSMISSIBILITY, AND DRAWDOWN

Compiled by RAT BENTALL

INTRODUCTION

The development of the nonequilibrium formula by Theis (1935) 

was a major advance in the field of ground-water hydraulics, and 

Wenzel (1937, 1942) did much to make the formula a practical tool 

for the hydrologist. Subsequently, general modifications or adjustments that are applicable to the earlier methods were advocated by 

Theis and other workers and new formulas for the solution of special 

field problems also were developed. These papers include suggested 

corrections for drawdown measurements analyzed by the Theis graphical method; remarks pertaining to Wenzel's limiting formula, gradient 

formula, and the recovery method; a formula for corrections to be 

applied if wells used for aquifer tests tap less than the full thickness 

of the aquifer; formulas for the determination of aquifer constants 

from water level data obtained when a well is bailed or a slug of 

water is injected into a well; analyses of the effects of cyclic fluctuations of the water-level, the pumping rate, or the pumping interval; 

and formulas and a chart relating the specific capacity of a well to the 

coefficient of transmissibility of the aquifer tapped by the well.

In writing these papers, it has been assumed that the reader understands the basic definitions relating to ground-water hydraulics and is 

acquainted with the fundamental nonequilibrium and equilibrium 

formulas for determining the hydraulic constants of an aquifer. The 

symbols used, unless otherwise specified, are defined as follows: 

&=the height of the column of water in the pumped well or in an observation

well anywhere within the area of water-level drawdown, measured from

the bottom of the aquifer; 

m=the thickness of the aquifer; 

r=the radius of the pumped well or the distance from the pumped well to the

observation well or point at which the drawdown is desired; 

«=the water-level drawdown in the pumped well, in an observation well, or at

some point in the vicinity of the pumped well; 

f=the time since pumping began;

243
244 GROUND-WATER HYDRAULICS 

t'=the time since pumping stopped;

P=the coefficient of permeability of the aquifer ; 

p=the rate of discharge from the pumped well ; 

/Sf=the coefficient of storage of the aquifer ; and 

T=the coefficient of transmissibility of the aquifer.

Also, the terms "semilog" and "log" are used in reference to graphs to 

mean semilogarithmic and logarithmic, and the units gallons per day 

and gallons per minute are abbreviated to gpd and gpm, respectively. 

Furthermore, the aquifer, unless otherwise specified, is assumed to 

conform to the idealized conditions postulated by Theis (1935, p. 521) .
DETERMINING THE PERMEABILITY OF WATER-TABLE

AQUIFERS

By C. E. JACOB

ABSTRACT

If the Theis graphical method is used for determining the hydraulic constants 

of an aquifer under water-table conditions, the observed drawdowns should be 

corrected for the decrease in saturated thickness. This is especially true if the 

drawdown is a large fraction of the original saturated thickness, for then the 

computed coefficient of permeability is highly inaccurate if based on observed, 

rather than corrected, water levels.

Wenzel's limiting formula, a modification of the Theis graphical method, is 

useful where u=r*8/4Tt is less than about 0.01. However, a shorter procedure 

for determination of the coeflacient of transmissibility, as well as the coeflacient 

of storage, consists of plotting the values of the corrected drawdowns against 

the values of the logarithm of r.

Wenzel (1942) suggested that observation wells be situated on lines that extend upgradient and downgradient from the pumped well. However, a detailed 

analysis of aquifer-test results indicates that such a restriction is unnecessary.

The gradient method for determining permeability should yield the same 

results as the Thiem method. The former, when applied for a distance within 

the range of applicability of the latter, is merely a duplication of effort or, at 

best, a crude check. Because of the limitations of accuracy in plotting, the 

gradient method is much less satisfactory. That Wenzel (1942) obtained identical results from the two methods is regarded as a coincidence.

Failure to take into consideration the fact that the pumped well does not tap 

the full thickness of the aquifer leads to an apparent coeflacient of permeability 

that is much too low, especially if the aquifer consists of stratified sediments. 

The average coeflacient of permeability computed from uncorrected drawdowns 

may be only a little more than half of the true value.

THE THEORY OF PERMEABILITY

Formulas for the steady radial flow of water toward a well that taps 

the full thickness of an unconfined sand are based upon the premise, 

originally ^et forth by Dupuit (1863), that for low water-table gradients the average of the horizontal, or radial, velocity in a vertical 

section is proportional to the slope of the water table (6h/6r) 

that is

v=-k(dh/6r).

The horizontal component of velocity at the water table actually is 

equal to &(dA/dr)/[l + (dA/dr) 2 ] but, for slopes that are very small 

in comparison to unity, the (dh/6r) 2 in the denominator becomes insignificant. If the small vertical components are neglected, all flow 

lines in a given vertical plane through the well can be assumed to be

245
246 GROUND-WATER HYDRAULICS

both parallel and horizontal ; consequently, the distribution of vertical 

pressure is hydrostatic or, in other words, the head in a vertical section is uniform. Therefore, the horizontal component of the velocity 

in a vertical section is also uniform and equals the horizontal component at the free surface, or water table. The time rate of flow per 

unit width normal to the flow is then Jch (6h/dr) .

In the immediate vicinity of a pumped well that taps the full thickness of an unconfined aquifer, the slope of the water table is steep 

and the foregoing relations obviously do not pertain. At distances 

where the flow toward the well has not yet become steady, the water 

table is declining at radially differential rates that is, the slope of 

the water table is changing with time and again the above relations 

do not pertain. When applying the theory of Dupuit, these limiting distances should be approximated.

Between the two limits, Dupuit's assumption is valid. Inasmuch 

as the flow is steady, the inward flow of water through a cylindrical 

surface concentric with the well equals the discharge of the well, 

or

(1)

Separating the variables and integrating between TI and r2, which are 

both within the limiting distances,

Al-A?=(<2M)loge(r2/rO (2)

If one integration limit is considered to be fixed and the other moving, 

this equation defines, to a sufficient approximation, the lowered water 

table in the annular area, concentric with the well, over which Dupuit's assumption is valid. 

Solving equation 2 for k gives

K7._ ~

An equivalent expression was first used by Thiem about 1906 to determine the permeability of an aquifer from drawdowns in two observation wells near a pumped well (Wenzel, 1936). Principally 

through the work of Wenzel, this equation has had widespread application in this country. To minimize errors of observation as well as 

errors arising from inhomogeneities of structure, Wenzel has advocated using many observation wells spaced systematically on lines 

radiating from the pumped well, preferably in upgradient and downgradient directions; then from a modification of Thiem's equation 

known as the limiting formula, an effective average permeability is 

determined graphically from drawdowns observed at several points
PERMEABILITY, TRANSMISSIBILITY, AND DRAWDOWN 247

on the two opposing radii. The same result might be obtained more 

directly, however, by plotting values of h2 against Iog10r. If the 

equation is a valid engineering approximation, the graph should yield 

a straight line and the value of k can be determined from the slope 

of the straight line and Q.

Often the results of an aquifer test are desired in terms of the 

coefficient of transmissibility (T) of the water-bearing material. The 

coefficient of transmissibility is the product of &, which can be determined graphically from Thiem's relation or from Wenzel's limiting 

formula and the original saturated thickness, m, which is assumed 

to be uniform when the water table is in its undisturbed position. A 

graph of the values of the drawdown, corrected as indicated in the following pages, plotted against corresponding values of Iogi0r gives T 

directly, again by the straight-line method. A graph of this kind permits visualization of the distribution of drawdown and of the approximate limits of usefulness of the related linear mathematical 

expression. Moreover, it is useful in comparing methods involving 

steady-state drawdowns with those involving nonsteady-state drawdowns and in justifying application of the theory of nonsteady flow in 

a confined aquifer of uniform transmissibility to water-table aquifers 

wherein the thickness of saturated material diminishes appreciably. 

In fact, as will be seen in the following pages, only after such corrections have been made can the graphical procedure of Theis reasonably 

be applied to nonsteady-state drawdowns.

THEIS GRAPHICAL SOLUTION USING CORRECTED DRAWDOWNS 

From equation 3 above,

2.30Qlog10 (r2/r1 )

Substituting s=m h in this relation gives 

T= 2.30Qlog10 (r2/r1 )

or

T= 2.30Qlog10 (r2/r1 ) (4)

where s sP/Zm is the corrected drawdown.
248 GROUND-WATER HYDRAULICS

If the corrected drawdown is replaced by

s'=s-(s2/2m)=m-(h+s2/2m), (5) 

where

sf is the drawdown that would occur in an equivalent confined aquifer, 

then

Equation 6 is an expression in terms of the drawdown, s', for the coefficient of transmissibility of a confined aquifer of uniform thickness. 

To solve equation 6, and hence equation 4, graphically, plot values of 

sf against corresponding values of Iog10r- and find the slope of the 

straight-line plot. If As' =s( s'2 is taken as the change in drawdown 

over one log cycle, then \ogi0 (r2/r1 )=l. 

and

The nonsteady flow of water toward a well that taps the full saturated thickness and that discharges at a constant rate from an extensive aquifer of constant transmissibility obeys the relation

f

Jo

(8)

where S is in the coefficient of storage (Jacob, 1940, p. 579) . When the 

time rate of change of drawdown (6s/6t) becomes small in relation 

to its rate of change with distance, equation 8 reduces to equation 1, 

which applies to steady radial flow. The integration of equation 

8 yields

, 4H-T' 

where

-2^!+'"} (9)

u=

4Tt

For small values of u (that is, when r is small or t is large), equation 

9 can be approximated by
PERMEABILITY, TRANSMISSIBILITY, AND DRAWDOWN 249

When t is constant, this is the equation for the straight line (on semilogarithmic coordinates) in equation 6. After T is determined from 

the slope of the straight line, S can be determined from the intercept, 

rei on the r-axis (or the log r-axis). At that point s=0 ; 

hence

^" _ ., 0.5772_ A tiRO TjU~ e U.OOJ,

from which

5=4-0.562 ^.

Wenzel designated equation 9 as the nonequilibrium formula. It is a 

particular solution of the general second-order differential equation 

and is but one of a great many particular solutions for different limiting conditions. The given limiting conditions are that the discharge 

of the well is constant, that the initial drawdown (referred to the 

undisturbed piezometric surface) is everywhere zero, and that the flow 

across the upper and lower bounding planes of the aquifer is everywhere negligible.

Equation 9 is a valid engineering approximation of the actual flow 

only where T is virtually constant. This condition is satisfied in a 

confined homogeneous bed of approximately uniform thickness or in 

an unconfined homogeneous bed wherein the drawdowns are small 

compared to the initial thickness of saturated material. The nonequilibrium method, or graphical procedure of solving the exponentialintegral relation for T and S from observations of the variation of s 

with t or with r, was devised by Theis (Jacob, 1940, p. 582).

When the drawdown is a large fraction of the initial saturated thickness, the need for correcting the drawdown before applying the nonequilibrium method can be demonstrated by using data from an aquifer 

test conducted by S. W. Lohman near Wichita, Kans. (Wenzel, 1942, 

p. 142). Both the observed and corresponding corrected drawdowns 

in 6 wells after 18 days of continuous pumping at 1,000 gpm, or 

1,440,000 gpd, are given in table 1, and both are plotted against r in 

figure 72 and against r2 in figure 73. The average of the 18-day observed drawdowns in the corresponding observation wells along the 

north and south lines gives T 129,000 gpd per foot and £=0.47 by 

both the straight-line method (fig. 72) and the Theis graphical method 

(fig. 73), whereas the average of the 18-day corrected drawdowns 

in the same observation wells gives T=154,000 gpd per ft and £=0.35 

by the same two methods. The average thickness of saturated material at the test site at the beginning of the test was 26.8 feet and 

after the 18-day period of pumping was 22.3 feet.
250 GROUND-WATER HYDRAULICS

133d Nl '(,*) KMAOQAAVaa 031038800 CJNV (') NMOQAAVaCJ Q3Aa3SaO
PERMEABILITY, TRANSMISSIBILITT, AND DRAWDOWN 251

 a

133d Nl (,*) NMOQMWKl 031038800 QNV (0 N/IAOOMVMO 03AM3S8O
252 GROUND-WATER HYDRAULICS

TABLE 1. Data for aquifer test near Wichita, Kans., giving drawdowns after 18 days 

of continuous pumping at 1,000 gpm

Well

Distance 

from 

pumped 

well, r 

(ft)

r» 

(ft»)

Observed 

drawdown,

(ft)

«'/2m 

(ft)

Corrected 

drawdown,

(ft)

Ratio of 

observed to 

corrected 

drawdown

Line extending north from pumped well

1. ..................

2... ................

3. .._..._. _.._-...._

49.2

100.7

189.4

2,420

10, 140

35,900

5.91

4.58

3.42

0.65

.39

.22

5.26

4. 19

3.20

1. 12

1.09

1.07

Line extending south from pumped well

1. ..................

2...................

3... -_.--_-__-_.._..

49.0

100.4

190.0

2,400

10, 080

36, 100

5. 48

4.31

3. 19

0.56

.35

. 19

4.92

3.96

3.00

1. 11

1.09

1.06

That the two procedures (the straight-line method and the Theis 

graphical method) should give identical results for the test near 

Wichita is clear from figure 73. The approximation for w, upon which 

the straight-line plotting is based, does not differ by any significant 

amount from the type curve within the range of values of u that is 

involved. In this and similar instances, the nonequilibrium 

method becomes an equilibrium method and the two procedures should 

check each other within the limits of accuracy of plotting. Therefore, in the analysis of aquifer-test data, the straight-line method 

should be used to determine whether the flow is steady or nonsteady 

over the range of the distances involved. If the flow is found to be 

steady, the straight-line method suffices for determination of the hydraulic constants, but if the flow is found to be nonsteady, the Theis 

graphical method needs to be applied.

Dividing the value of T obtained from the corrected drawdowns by 

the initial thickness of saturated material, m=26.8 feet, gives k=5,750 

gpd per sq ft, which agrees reasonably with Wenzel's P=5,787 gpd per 

sq ft. The value $=0.47, which was determined from the uncorrected drawdowns, is believed to be about 0.18 too high because the 

value $=0.35, obtained from the corrected drawdowns, is only an 

approximation and becomes even smaller when corrected further for 

the reduction in saturated thickness. The corrected drawdowns used 

in determining $=0.35 were those that would have occurred in a confined aquifer having similar hydrologic properties and a thickness 

equal to the initial thickness of saturated material in the water-table 

aquifer. In order to determine the average coefficient of storage more 

closely, the above determined value may be multiplied by the average
PERMEABILITY, TRANSMISSIBILITY, AND DRAWDOWN 253

ratio of the final to the initial saturated thickness. The theoretical 

justification of this procedure follows.

The second-order differential equation governing the radial flow 

of water in an unconfined aquifer is

//-) (dA/dr)] =S(dfc/dO . (1 la) 

Substituting (m s) for h gives

(lib)

which can be expressed in terms of the corrected drawdown, s' rather 

than the actual drawdown, s, by determining the relationships between 

their respective differential coefficients. From

dsVdr=[(m-s)/m](ds/dr) (lie) 

and

dV/dr»= [(m-s)/m](52s/dr2 )- (1/m) (ds/dr)2. (1 1 d)

For low water-table gradient values of (ds/dr) 2 small in comparison 

with m(52s/^2 ) the last term of equation lid can be omitted and 

the equation becomes

d2«7dr2 =[(m-s)/m](d2s/dr2 ). (lie) 

The third relation required is

(llf)

Making the substitutions indicated by equations lie, lie, and llf in 

equation lib gives

which can be rewritten

(llh)

where

T=km

is the initial transmissibility and

S'=[m/(m-s)]S 

is the apparent coefficient of storage.

690-185 O 63 2
254 GROUND-WATER HYDRAULICS

If the variation of s is small in comparison with m, S' may be considered essentially constant, and the integration of equation llh gives 

equation 9, in which * is replaced by *' and S by S' as one solution. 

By application of the graphical method of Theis to the corrected 

drawdowns (*'), the values of T and S' can be determined; the 

approximate average coefficient of storage is then

S=[(m-s)/m~]JS'. (Hi)

In the test near Wichita, the initial saturated thickness was 26.8 

feet; the drawdown averaged over the logarithm of the distance 50 

to 200 feet that is, the drawdown at the geometric mean distance, 

100 feet was 4.5 feet; and hence S' was found to be 0.35. Therefore,

S= [(26.8-4.5)/26.8]0.35=0.3

instead of 0.47, as determined from the observed drawdowns. This is 

only an approximate spatial average (at a fixed time) of a coefficient 

of storage that varies not only with distance from the pumped well 

but also with time. Even if the coefficient of storage were invariable, 

its true value could not be determined precisely by this application, 

to an unconfined aquifer, of the theory of nonsteady flow in an aquifer 

of uniform transmissibility.

WENZEL'S LIMITING FORMULA

For the aquifer test near Wichita, Kans., Wenzel's limiting formula 

gives Pf = 5,805 gpd per sq ft, which does not differ significantly from 

the value obtained by the corrected drawdown methods in figures 72 

and 73. The steps involved in the application of Wenzel's limiting 

formula and in the straight-line method are described below.

Wenzel's limiting formula:

1. Tabulate well numbers, distances, and observed drawdowns.

2. Plot the water-level data- on graph paper having rectangular coordinates, 

and draw smooth curves through the points.

3. Determine values of drawdown from these curves for equal but opposite 

radii, preferably upgradient and downgradient from the well, and 

tabulate, for several different pairs of radii, values of B (half the 

difference in the averages of the upgradient and downgradient drawdowns).

4. Determine the average thickness of saturated material upgradient and 

downgradient between the same pairs of radii, and divide the logarithm of the corresponding ratios of outer to inner radii by these 

values. The resulting quotients are values of A..

5. Plot each value of A against the corresponding value of B, draw a 

straight line through the plotted points and the origin, and from the 

slope of that line determine Pt (or *).
PERMEABILITY, TRANSMISSIBILITY, AND DRAWDOWN 255

The straight-line method using corrected drawdowns:

1. Tabulate well numbers, distances, and observed drawdowns, and correct 

the drawdowns by subtracting (*V2m).

2. Plot the corrected drawdowns, *', against r on semilog graph paper, and 

draw a straight line through the plotted points; from the slope of 

that line and from its intercept, determine T and the apparent 8.

Although step 2 in Wenzel's method is not necessary if the observation wells are placed in pairs at equal distances upgrudient and downgradient from the pumped well, his method still entails two extra 

steps that involve considerable computation.

THE LOCATION OF OBSERVATION WELLS

As pointed out previously, Wenzel advocates that lines of observation wells be oriented upgradient and downgradient from the pumped 

well, their location being based upon a preliminary determination of 

the direction of the natural ground-water flow. Not only his limiting 

formula but also his applications of the nonequilibrium method make 

use of such orientation. However, it is questionable whether discrimination in the choice of direction is warranted.

The equations for unconfined flow are based upon Dupuit's premise, 

which holds for water-table gradients that, although considered low, 

are nevertheless steep compared to the usual undisturbed slopes in 

nature. Whenever an initial natural water-table gradient exists, the 

distribution of head, or potential, from different sources, natural and 

artificial, can be added directly by the principle of superposition. 

This principle is strictly applicable only when the transmissibility is 

independent of the head that is, when the aquifer is confined. The 

second-order partial differential equation representing the conditions 

of artesian flow is linear; hence, the validity of superposing or adding 

algebraically is verified. For unconfined aquifers, the differential 

equation is not linear but of the second degree; thus, the superposition, for example, of a theoretical radial drawdown distribution on a 

natural water table having a low gradient gives the approximate resultant distribution of head only at distances from the pumped well 

that are greater than a certain limiting distance.

If whatever occurs upgradient is offset by an opposite effect downgradient, averaging the slopes upgradient and downgradient would 

seem to counteract the nonlinearity for distances close to the well, 

where Dupuit's assumption also breaks down. This argument presupposes that the superposition of slopes is valid; then, from purely 

theoretical considerations, there would be no reason for preferring one 

direction to any other. From a practical viewpoint, both Dupuit's 

assumption and the principle of superposition seem uniformly valid 

hi all directions from the pumped well, especially for undisturbed 

gradients of the magnitude encountered in nature. Even if one
256 GROUND-WATER HYDRAULICS

direction were more significant than another, inhomogeneities and 

variations in the thickness of the aquifer are of greater significance. 

In fact the location of the observation wells should be based as much, 

or perhaps more, upon geologic considerations as upon the groundwater gradient. For instance, the configuration of the floor of the 

aquifer is as important as the configuration of the water table except 

in the immediate vicinity of the well. Often the geologic structure is 

not known in sufficient detail to aid in the location of observation 

wells. Even where the structure is known in detail, seldom does an 

aquifer meet the specifications regarding uniform initial transmissibility (that is, initial uniform thickness of saturated homogeneous 

material). Therefore, any advantage that might be postulated by 

placing the observation wells ungradient and downgradient is generally invalidated by the prevailing field conditions.

As an example of the above, consider an analysis of the water level 

data from an aquifer test near Grand Island, Nebr. (Wenzel, 1936, 

p. 26-57). As shown in figure 74, the observation wells for this test 

were drilled along lines radiating from the well which was to be 

pumped. During the test, pumping was continuous for 48 hours at an 

average rate of 540 gpm. The data needed for application of the 

graphical method to the corrected 48-hour drawdowns are given in 

table 2 and are plotted in figures 75, 76, and 77. Table 3 summarizes 

the determinations of the coefficients of transmissibility and storage 

for all lines except $, for which the water-level data were not sufficient 

for a separate analysis. (Actually, the fact that the pumped well did 

not tap the full thickness of the aquifer completely upsets the analysis 

and gives misleading results; this is discussed in a later section of 

this paper.)

TABLE 2. Data for aquifer test near Grand Island, Nebr., giving drawdowns after 

48 hours of continuous pumping at 540 gpm

Well

Distance from

pumped well,

T

(ft)

r*

(ft')

Observed

drawdown,

*

(ft)

g*!2m

(ft)

Corrected

drawdown. »'

(ft)

Line A

1. .-._-_.-.......

2-. .-.-.-..... ...

3... -------------

4---------------

5- 

6... -------------

7.... ......

8........ ......

9 ~ ..

10..-------.----.

11. - -------------

249

59.9

114 4

1642

229

354

429

479

604

755

904

620

3,590

13, 090

27,000

52,400

125, 300

184, 000

229, 000

365, 000

570, 000

817, 000

4. 01

2.79

2.03

1. 61

1. 14

.65

.52

.44

.26

. 16

. 11

0.08

.04

.02

.01

.01

.00

.00

.00

.00

.00

.00

3. 93

2. 75

2.01

1. 60

1. 13

.65

.52

. 44

.26

. 16

. 11
PERMEABILITY, TRANSMISSIBILITY, AND DRAWDOWN 257

TABLE 2. Data for aquifer test near Grand Island, Nebr., giving drawdowns after 

48 hours of continuous pumping at 640 gpm Continued

Well Distance from 

pumped well,

(ft)

r» 

(ft»)

Observed 

drawdown,

(ft)

(ft)

Corrected 

drawdown,

(ft)

LlneB

13.... ___________ 29.9 894

14.____________ 70.0 4,900

15-__-_--- 120.0 14,400

16.._. ___________ 184.9 34,200

17_______________ 255 65,000

18__............. 375 140,600

19.... ___________ 425 180,600

20-_____________ 500 250,000

21_______________ 650 423,000

22___.___________ 775 601,000

23...._._._...... 974 949,000

24_______________ 1,149 1,320,000

	Line W

25..-.----_-----. 49.7 2,470

26..._._--.....__ 170.0 28,900

27__________.____ 270 72,900

28............... 430 184,900

29__......_______ 625 391,000

SO---..---.------ 805 648,000

31_------------ 940 884,000

	LineD

32-_-_-_---__---_ 40.1 1,608

33______.___ ___ 95.1 9,040

34._. ____________ 144.7 20,900

35--------.------ 214 45,800

36.. _____________ 324 105,000

37-----------.--- 423 178,900

38..----_-..--._. 448 201,000

39-_----_----_-.- 573 328,000

40___----_-______ 723 523,000

41-.-------.----. 872 760,000

42___-____-______ 1,073 1,151,000

43....----.-.__-- 1,197 1,433,000

	LlneC

44-.-...-......- 39.3 1,544

45-----------.- 80.5 6,480

46--. ---------- 130.3 16,980

47--.--- ------ 195.6 38,300

48-.------------ 286 81,800

49_.._._._. 410 168,100

50____-------- 425 180,600

51------ ----.- 535 286,000

52__._._... 685 469,000

53________ 835 697,000

54...._____... 1,035 1,071,000

55...____--.. 1,175 1,381,000

3. 87

2.59

1. 86

1. 31

. 92

. 51

.40

.29

. 16

. 10

.05

.05

0.07 

.03 

.02 

.01 

.00 

.00 

.00 

.00 

.00 

.00 

.00 

.00

3. 80

2.56

1.84

1. 30

.92

.51

. 40

.29

. 16

. 10

.05

.05

2.98

1.44

. 84

.45

.21

. 11

.09

0.04 

.01 

.00 

.00 

.00 

.00 

.00

2.94

1. 43

. 84

.45

.21

. 11

.09

3. 15

2. 24

1.71

1.24

.77

.51

.46

.28

. 15

. 10

.06

.05

0.05 

.03 

.01 

.01 

.00 

.00 

.00 

.00 

.00 

.00 

.00 

.00

3. 10

2.21

1. 70

1.23

.77

. 51

. 46

. 28

. 15

. 10

.06

.05

3.23

2. 37

1.72

21

78

41

39

24

12

08

03

01

0.05 

.03 

.01 

.01 

.00 

.00 

.00 

.00 

.00 

.00 

.00 

.00

3. 18

2.34

1. 71

1. 20

.78

. 41

.39

. 24

. 12

.08

.03

.01
258 GROUND-WATER HYDRAULICS

TABLE 2. Data for aquifer test near Grand Island, Nebr., giving drawdowns after 

48 hours of continuous pumping at 540 gpm Continued

Well

Distance from

pumped well,

f

(ft)

r*

(ft*)

Observed

drawdown,

t

(ft)

»*frm

(ft)

Corrected

drawdown.

if

(ft)

UneSW

56_______________

57_______________

58... ._._..__.___

59-------__-_____

60____ .__._.___._

61____ _ .__. __

62.. _..._._._..__

63_.___ __________

64_________._.___

65- ___ ___-.-__

66_______ __ ____

67_._____________

68_______________

46.7

69.5

93.6

118.0

217

317

417

517

617

717

817

917

1,017

2,180

4,830

8,760

13,920

47,100

100,500

173,900

267,000

381,000

514,000

667,000

841,000

1, 034, 000

3. 12

2.58

2.18

1.86

1. 11

.66

.41

.28

.20

. 13

.09

.08

.07

0.05

.03

.02

.02

.01

.00

.00

.00

.00

.00

.00

.00

.00

3.07

2.55

2. 16

1 84

1. 10

.66

.41

.28

.20

. 13

.09

.08

.07

LlneS

73_______________

74_________._____

75 ___________ _

76_______________

130. 1

225

280

383

16, 930

50,600

78,400

146, 700

1.73

1.00

.76

.55

0.01

.01

.00

.00

1.72

.99

.76

.55

LineN

77

78

79

80

81

63.2

160.0

262

342

446

3,990

25,600

68,600

117, 000

198, 900

2.67

1.63

.96

.65

.42

0.04

.01

.00

.00

.00

2.63

1. 62

.96

.65

.42

TABLE 3. Coefficients of transmissibility and storage determined from the corrected 

drawdowns after 48 hours of continuous pumping at 540 gpm during aquifer 

test near Grand Island, Nebr.

Line

A.___________________

B_--.___________.____

W-_.______. _________

D..__________________

C _________________

SW.____. ____________

N __ ________________

Value of * for 

W(«)=4.04 

(ft)

2.45

2.78

2. 50

2.50

2.67

2.75

2. 50

Value of r* for 

«=0.01

(ft»)

5,800

3,700

4,700

5,300

3,800

3,600

4,700

Coefficient of 

transmissibility, 

T 

(gpdperft)

102, 000

90, 000

100, 000

100, 000

94, 000

91,000

100, 000

Coefficient of 

storage, S

0. 19

.26

.23

.20

.265

.27

.23
July 1931

FIGURE 74. Map showing the location of wells used for aquifer test near Ora
260 GROUND-WATER HYDRAULICS

bU *

u- <
PERMEABILITY, TRANSMISSIBILTTY, AND DRAWDOWN 261

ITIIIIII

T0'0= n

I I I I I I I

I I I I I I

z »

,7- a - s

133d Nl '(*) NAAOQAAVMQ
262 GROUND-WATER HYDRAULICS

iaaj NI '(*)
PERMEABILITY, TRANSMISSIBILrTT, AND DRAWDOWN 263

The average coefficient of transmissibility obtained from the data 

for all lines of wells is T= 97,000 gpd per ft and the average coefficient 

of storage is £=0.23. For the lines B and D, which extended upgradient and downgradient, respectively, from the pumped well, the 

averages are T= 95,000 gpd per ft and /S'=j.BC, and for lines A and 

C, which made a right angle with lines B and D, the averages are 

T= 98,000 gpd per ft and £=0.23. The results in table 3 show that, 

in general, where T is small S is large, a fact which probably indicates 

that the aquifer thickens somewhat toward the north or northwest. 

Any significance attached to the different average values of T from 

lines B and D and lines A and C is outweighed by anomalies arising 

very probably from differences in thickness.

THE GRADIENT FORMULA 

If equation 1 is solved for &,

and when (dh/dr) -and h vary with the angle 0, 

then

(13)

Wenzel's equation (Wenzel, 1942, p. 86, eq 82), which is the basis 

of the so-called gradient method, is obtained through approximation 

of the integral of equation 13 by averaging the gradient and the saturated thickness at the distance r upgradient with those at the distance 

r downgradient. This method should yield results that are equivalent 

to the results obtained by the Thiem method because both are based 

on equivalent relations and one is the integral of the other. If the 

former is applied at a distance within the range of applicability of the 

latter, the same data being used, the result is in effect a duplication of 

effort. The gradient method adds nothing to the Thiem method. 

Moreover, it is much less accurate because of the limitations of accuracy in plotting and drawing smooth curves.

The applicability of the gradient method is illustrated by computations from table 4, which is a modification of a corresponding table 

by Wenzel (1942, p. 124). Values of F/, computed by the gradient 

formula for r=115 feet and r=125 feet, are

^9,168 ft-min per dayX540 gpm 

'~ 115 (98.09+97.98) 0.21 ft3 

= 1,050 gpd per sq ft;
264 GROUND-WATER HYDRAULICS

and

p _9,168 ft-min per dayX 540 gpm 

' 125(98.19+98.08)0.21 ft3

=960 gpd per sq ft.

The 9 percent difference in the results of these computations is to be 

expected inasmuch as the differences in water-level altitude given in 

table 4 are accurate only to the nearest hundredth of a foot. To indicate that only 2 figures are significant, both results should be written 

as 1,000 gpd per sq ft. Moreover, because the slope (0.009) 115 feet 

downgradient from the well is smaller than the slope (0.010) 125 

feet downgradient from the well, the agreement between the value of 

Pf determined by the gradient method and the one determined by 

the limiting formula should be regarded as fortuitous.

TABLE 4. Data used in determining the coefficient of permeability from radial 

components of the hydraulic gradient after J^8 hours of continuous pumping at 540 

gpm during aquifer test near Grand Island, Nebr.

Direction from pumped well

Altitude of water table 

(ft)

110 ft from 

pumped 

well

1,808.44 

1,808.06

120 ft from 

pumped 

well

1, 808. 56 

1, 808. 15

Difference 

(ft)

0.12 

.09

.21

Altitude of water table 

(ft)

120 ft from 

pumped 

well

1,808.56 

1, 808. 15

130 ft from 

pumped 

well

1, 808. 67 

1, 808. 25

Difference

(ft)

0.11 

.10

.21

WATER-LEVEL CORRECTIONS FOR WELLS TAPPING LESS THAN 

THE FULL THICKNESS OF AN AQUIFER

If a pumped well does not tap the full thickness of the aquifer, the 

drawdowns measured in observation wells tapping only the uppermost 

part of the aquifer should be corrected (Muskat, 1937, p. 368). Because of the convergence of the flow lines in the vicinity of the well, 

the loss of head along the top of the aquifer is greater than the loss 

of head along the bottom and both differ from the head loss that 

would occur if the pumped well tapped the full thickness of the 

aquifer. This is especially true if, as is usually the case, the aquifer 

consists of stratified sediments. Even homogeneous beds of waterdeposited materials are invariably anisotropic. The inhomogeneities 

of stratification give a resultant permeability in the vertical direction 

that is many times smaller than the average permeability in the horizontal direction, producing in a sense an equivalent anisotropy. As a 

result, the effect of tapping only the upper part of the aquifer is accentuated; that is, the lateral extent of the disturbing influence is 

increased.
PERMEABILITY, TRANSMISSIBILITY, AND DRAWDOWN 265

The effects of a well tapping only the upper part of an aquifer are 

shown in figure 78, which is based on data from the aquifer test near 

Grand Island, Nebr. (Wenzel, 1936, 1942). Figure 784 is a section 

along line SW of observation wells. The pumped well, well 83, was 

only 40 feet deep and was perforated throughout its length. That 

the thickness of saturated materials at this site is about 100 feet is 

shown by the graphic log of well 84 (fig. 78#), which was 25 feet 

south of well 83. Thus, because the static water level in the well was 

about 5 feet below the land surface, well 83 effectively tapped only 

about 35 percent of the full thickness of saturated material.

Figure 78C' shows the streamlines and equipotential lines that would 

be obtained if the sediments were homogeneous and isotropic. The 

lines of equal potential, or in this case equal drawdown, were determined from Muskat's equation (Muskat, 1937, p. 268, eq. 9). The 

flow lines were sketched in to form an orthogonal system. In this 

hypothetical case, the tapping of only the upper part of the aquifer 

affects the distribution of head for a distance of about 150 feet out 

from the pumped well. However, because the sediments are stratified, 

the flow pattern actually is distorted as shown in figure 78A. Laboratory determinations of the permeability of bailer samples, plotted 

beside the log of well 84, range from 2 to 4,350 gpd per sq ft. Weighted 

according to thickness, for flow parallel to the plane of bedding, these 

permeabilities average 1,200 gpd per sq ft, but weighted according 

to reciprocal flow lengths, for flow across the plane of bedding, they 

average only about 150 gpd per sq ft. This suggests an equivalent 

anisotropy characterized by the permeability ratio (kr/ke ) =8. Actually, in order to make the observed drawdowns compatible with the 

computed drawdown at the point of stagnation underlying the well, 

this permeability ratio was found to be more nearly twice as large as 

indicated. In other words, the beds, on the average, are effectively 

about 16 times as permeable in the horizontal direction as in the 

vertical direction.

The flow pattern in figure 784 can be obtained by plotting the 

orthogonal net of figure 78/? on an elastic rectangular sheet which is 

to be stretched to four times its original length without reducing its 

width, or the stretching can be performed graphically.

The distribution of drawdown along the top and bottom of the aquifer is shown in figure 78Z?; a semilog plot of the identical data is shown 

in figure 79. In both diagrams, the points plotted as open circles represent the corrected drawdowns (see table 2) for wells 56 to 68 for the 

time when well 83 had been pumped continuously for 48 hours at 540 

gpm. Because none of the observation wells extended more than a few 

feet below the water table, the facts that the pumped well tapped only 

35 percent of the full thickness of the aquifer and that the effective
266 GROUND-WATER HYDRAULICS

12

i33j NI 'saruirw
<

Pum

we

1

3 =780.000 gpd. tt = l 

ped 

II 56 57 58 59

T = 180.000 gpd per ft. m = 

80

100 « 

61 42 63 64 

i i i i i i i i !

i*>H

1 1

1 1 

1 1

cJ fel fel hi

-J!i i i i i i i r 

i j i i

fel inl 4

_0| # C

1 1

 | | 

, 1 1

1

 fc

1

1,

1

IN N

 10_0

i

i 

i ,

i 

l~ 

jo 

! 

1 

1 

400 500 600 

RADIAL DISTANCE (r), IN FEET

D

Q =780,000 gpd. a = 0.70, kr/kz = 16, J- 180,000 gpd per ft,m = 100 ft 

Well 84 

' ^-i 56 57 58 59 60 61 62 

4000 2000 0 

k, GPO PER FT 2

400 500 600

RADIAL DISTANCE (r), IN FEET

E

FIOUHE 78. Data from aquifer test near Grand Island, Nebr., demonstrating the effects of a well tapping less than 

line 8 W showing idealized influence of well 83, which extends 35 feet into a supposedly homogeneous anisotropic w

of drawdown as observed along the top and as inferred along the bottom of a water-table aquifer 100 feet thick 

also the distribution of drawdown throughout a water-table aquifer 100 feet thick if the pumped well extends to th

potentials for a pumped well that extends 35 feet into an isotropic water-table aquifer 100 feet thick; also the appr

the well and along downward extension of well axis. D, Hypothetical flow pattern for purely radial flow induced 

artesian aquifer. E, Approximate pattern of flow in section along line SW produced by well 84, which has two leng

of permeability as determined in the laboratory and columnar section based on log of well 84.
(s') DRAWDOWN CORRECTED 

*i Ui T> 

_

-

-

\

K

\

\

>

\ o o * 

\ <T Q.IT 

X ?s y. <. 

\ ?A A f 

\ ;.< -2-S 

X go. £| 

01 o 3 A 

2.$ Q-S 

a <" (6"°- 

3 5 3 -» 

5 |f :l| 

\

\

\

\

_

iQ. 

O

o"

OQ

O 

13

2, V 

Vi

X

FEET IN PUMPING, OF HOURS 48 AFTER 

o _ r\> w 

L

^

"

V

f

tn

r »S. -zS. 

i IT z ° 5 "I? 

, > g-a SS <r n » = c t« 

_ Z Q) -,. QJ-O 

5ro N 

^ $. K) 

" S ^ 

. ., in A C A 

» 2. eg 

^ $ ? 

^ oo °w 

Q.CJ

_

1 90 1 

ho o. r\> 

* "2 K) 

"2 5 un 

So ^ 

_i_ '1 ^ 

§

71 ho p *» 

S* x 

^. 0 K) 

^" " 11 

P ° = -o 

(6 > QO x 

* oo 2.2 

Q. " * W _ 

*<

O (k> 

a. a. 11

f ? 

13 o 

C 3T 

||

oT2. \ 

\lf

a » \ 

\l

I

\

X vl V

Ny __ 

g

o

K)

X

^J

oo

§

FQ

3

a.

1

(Q

0>

<a + 

5 H 

-5 2- 

5'

O

1

3

ta

I

|

o

S

v v 

\

\

\

0

"o

K)

3,

3 

 )

+ 

1 ^ K' 

V \. 

^^

0^

"\\ u

N^

2^L

Aa»\

t'

*T

_

-

-

-

-

"

_

-

-

_

-

-

le -775'=r

~ v 

1" \ 

M2

1* ft 4mVk^=1600 
PERMEABILITY, TRANSMISSIBILITY, AND DRAWDOWN 269

permeability ratio of the aquifer was 16 were taken into consideration 

when the drawdowns along the bottom of the aquifer were computed.

As mentioned above, the drawdowns along the bottom of the aquifer, 

as determined from those observed along the top, must be compatible 

with the drawdown at the point of stagnation, which theoretically is 

independent of the degree of anisotropy. The drawdown at this point 

was determined by Muskat's method. The head at the stagnation 

point should be approximately 2.17 (Q/Zn-T) lower than the head at a 

distance equal to four times the thickness of an equivalent isotropic 

sand (or 16 times the thickness of the actual sand). The drawdowns 

for a well tapping the full thickness of the aquifer, indicated by a 

dashed straight line in figure 79, were also computed from Muskat's 

equation. The drawdown on this curve at r= 1,600 feet is 0.49 foot. 

Thus, the drawdown at the stagnation point probably is about 1.0 foot 

(see figs. 78Z? and 79).

The foregoing analysis of the effects of pumping from a well that 

taps only the upper part of an aquifer is based upon a theory that, 

strictly speaking, is applicable only to confined, or artesian, aquifers. 

However, it gives an approximation of the actual conditions in an 

unconfined aquifer that is close enough for the present discussion.

The uniform logarithmic drawdown distribution for steady flow 

toward a well that taps the full thickness of an equivalent artesian 

aquifer is shown in figure 78Z> by dashed lines representing traces of 

concentric cylindrical surfaces in the vertical plane. The distribution 

is approximately logarithmic out to a distance of about 400 feet. The 

nonsteady drawdowns at greater distances are represented by the solid 

vertical lines.

From the slope of the dashed straight line in figure 79, the coefficient 

of transmissibility is found to be about 184,000 gpd per foot. From 

corrected drawdowns, Wenzel (1942, p. 147) computed the permeability to be about 1,000 gpd per sq ft, which is equivalent to a transmissibility of 100,000 gpd per foot. Hence, for this aquifer test, failure to 

take into consideration the fact that the pumped well tapped only the 

upper part of the aquifer resulted in an apparent transmissibility that 

is only 55 percent of the probable true value.

The straight line in figure 79 intersects the zero-drawdown line at 

r=775 feet, the apparent external radius of the system after 48 hours 

of continuous pumping. The coefficient of storage determined from 

this is 0.18, whereas Wenzel obtained 0.217.

The above evaluation of the aquifer test made by pumping from well 

83 is confirmed by comparing that test with the test made by pumping 

from well 84. Well 84 was 105 feet deep, and because the top 24 feet 

and the bottom 48 feet of the casing were perforated it effectively

690-185 O 63 3
270 GROUND-WATER HYDRAULICS

tapped about 70 percent of the full thickness of the aquifer (see 

fig. ?8ff). This well was pumped intermittently for 2 days at about 

the same rate as well 83, or 540 gpm. Because of the interruptions in 

pumping and the uncertainties involved in correcting for them, computations for this test were not published, although drawdown curves 

for three observation wells were given by Wenzel (1936, pi. 5).

A reasonable correction for the interruptions can be made by assuming that the coefficient of storage remained constant. Then, if t' is the 

time interval back to a given shutdown and t" is the time back to the 

beginning of the pumping interval, the corresponding correction to be 

applied to the 48-hour drawdowns is

log, (t'lt") =

and the total correction is

log,

Corrections for the interruptions in pumping are given in table 5. 

Because successive interruptions of short duration have been lumped 

together, the computed correction is a liberal one :

778,000 gpdXO. 166 = ' A°f; -j j4xX 184,000 gpd per ft

. , . =0.056 foot.

TABLE 5. Interruptions in the pumping from well 84 near Grand Island, Nebr., 

and the corresponding values of the factor (t'-t")lt" used in correcting the 48- 

hour drawdowns

[Pumping began at 8:05 a.m. on Sept. 9,1931; average pumping rate was 540 gpm]

Period of interruption

Pumping

stopped

Septem

11:18 a.m.....

12:35 p.m __ .

2:00 . 

5:55 .. 

Septemb

4:26a.m.__-

8:57- 

9:36-.. .

9:39 . 

9:48 . 

11:17 . ..

11:49 . 

12:06 p.m ___

Pumping

started

er 9, 19S1

11:35 a.m.... .

12:37 p.m .....

3:38- - 

6:31...- 

er 10, 19S1

6:03 a.m __ .

9:32. 

9:38-.. 

9:40 

9:51. 

11:19... 

11:55. 

12:11 p.m.....

t'

Hours

44

43

42

38

27

23

22

22

22

20

20

19

Minutes

47

30

5

10

39

29

26

17

48

16

59

t"

Hours

44

43

40

37

26

OO

22

22

22

20

20

19

Minutes

30

28

27

34

2

33

27

OK

14

46

10

54

t'-t"

Hours

0

0

1

0

1

0

0

0

0

0

0

0

Minutes

17

2

38

36

37

35

2

1

3

2

6

5

t'-t"

t"

0.006

.001

.040

.016

.062

.026

.001

.001

.002

.002

.005

.004

2-0. 166
PERMEABILITY, TRANSMISSIBILITY, AND DRAWDOWN 271

The corrected drawdowns in the three wells are as follows:

Well 57._-----___.__.feet._ 2.16+0.06=2.22

59___.___________do_- 1.45+ .06=1.51

61_.-.__.-_______do-- .58+ .06= .64

These points are plotted in figures 78Z? and 79. Because the two 

perforated lengths of casing in well 84 are at the top and bottom of the 

sand, Muskat's theory, which assumes a single length of perforated 

casing at either the top or bottom, does not apply. However, the drawdowns observed when well 84 was pumped were much less than the 

drawdowns at corresponding distances when well 83 was pumped. 

Hence, the effect of tapping different fractional parts of the full thickness of an aquifer is clearly demonstrated.
CORRECTION OF DRAWDOWNS CAUSED BY A PUMPED 

WELL TAPPING LESS THAN THE FULL THICKNESS OF 

AN AQUIFER

By C. E. JACOB

ABSTRACT

If a pumped well taps less than the full thickness of a confined aquifer, it is less 

efficient than it would be if it were to tap the full thickness. Also, the distribution 

of head in its vicinity differs from that which would characterize a well having 

the same effective radius, discharging at the same rate, but tapping the full 

thickness of the aquifer at the same location. Because the formulas for determining the hydraulic constants of an aquifer are based on the assumption that the 

pumped well taps the full thickness, the observed water-level drawdowns caused 

by pumping from a well which taps less than the full thickness should be corrected 

before they are used in the formulas. Muskat (1932) and Kozeny (1933) described 

the problem in detail, and Wenzel (1942) gave it cursory treatment. This paper 

not only outlines the method of making the corrections, but also includes the 

necessary graphs. It also includes a table for determining the drawdown at the 

point of stagnation beneath a pumped well that taps less than the full thickness 

of an aquifer.

THE FORMULA FOB CORRECTING DRAWDOWNS

Whether or not a pumped well taps the full thickness of an aquifer 

is of practical significance. Not only is the productivity of the well 

affected but also the distribution of head in its vicinity. If a well 

that is pumped in making an aquifer test taps less than the full thickness, the water-level drawdowns observed during the test must be corrected before an accurate coefficient of transmissibility, or of permeability, can be computed. The corrections are necessary because the 

formulas for the determination of hydraulic constants from aquifertest data are based on the assumption that the pumped well taps the 

full thickness.

Muskat (1932, p. 329-364; 1937, p. 263-286) developed an equation 

for the discharge of a well that taps only a fraction of the full thickness of an aquifer in terms of the full thickness and permeability of 

the aquifer, the fractional part tapped by the well, and the total 

potential drop from the assumed effective external radius of the system 

to the known internal radius. Kozeny (1933, p. 101) summarized 

Muskat's analysis by a somewhat simpler empirical expression that is 

a sufficient approximation for many purposes. Kozeny's empirical 

formula can be written

2irTs v,a[l+7(rj2am)* cos Qra/2)] m 

V- loge (iv/r.) * ' 

or

Q=Qoa[l+7(rv,/2am)* cos ftra/2) ]=&«/#, (2) 

272
PERMEABILITY, TRANSMISSIBILITY, AND DRAWDOWN 273

where

l/Cr=l+7(rw/2am)» cos ( /2). (3) 

In these equations

Q=ihe rate at which water is discharged by a pumped well that

taps less than the full thickness of the aquifer, 

Q0 =ihe rate at which water would have been discharged if the

well had tapped the full thickness of the aquifer, 

T=the coefficient of transmissibility of the aquifer, 

m=the full thickness of the aquifer, 

a=the fractional part of m tapped by the pumped well, 

rw=the radius of the pumped well, 

Su,=the drawdown in the pumped well (that is, the drawdown at

distance rw). 

re=the external radius of the system (that is, the distance from

the pumped well to the locus of zero drawdown), and 

(7= the correction factor.

The apparent transmissibility, 7", referred to the total thickness of 

the aquifer and determined from observations of the drawdown in a 

pumped well, is given by

y/=Q log. WO. (4) 

2ir«u,

Similarly, the true transmissibility is

Hence,

f-K

which is the ratio of the true to the apparent transmissibility, or of 

the true to the apparent permeability. From equation 6,

(7)

which is the correction factor given by Wenzel (1942, p. 109), This 

correction is only for values of permeability or transmissibility determined from the drawdown inside the pumped well. Moreover, as 

$o is unknown, equation 7 does not lead to a solution of the problem ; 

however, Q0 can be determined from equation 2.
274 GROUND-WATER HYDRAULICS

Muskat (1937, p. 283) states that the flow of water toward a 

pumped well that taps only a fractional part of an aquifer becomes 

almost exactly radial at a distance from the well equal to twice the 

aquifer thickness. However, this is true of isotropic aquifers only, 

and most aquifers that consist of water-deposited sediments are 

stratified and, therefore, as a whole, are anisotropic. The flow toward 

a well that taps less than the full thickness of an anisotropic aquifer 

becomes radial at a distance from the well equal to twice the aquifer 

thickness multiplied by the square root of the ratio of the horizontal 

to the vertical permeability. For example, if an aquifer is 16 times 

as permeable in the horizontal as in the vertical direction, then purely 

radial flow occurs only beyond a distance equal to about 8 times the 

aquifer thickness. Drawdowns measured within the area in which 

flow toward the well is radial must be corrected if the coefficient of 

transmissibility computed from them is to be accurate.

In order to correct the drawdowns, the distribution of head throughout the aquifer must be known. However, because the foregoing 

equations relate merely to the difference in head between the inflow 

and outflow surfaces, they are inappropriate for use in determining 

the distribution of head in the vicinity of a pumped well. Instead, 

the distribution of head may be found from two equations derived 

by Muskat (1937, p. 268, eq. 8 and 9) one for small values of r and 

the other for large values of r. The latter contains one term for the 

logarithmic distribution of head for purely radial flow and a second 

term for the difference between the actual distribution and the logarithmic distribution. For convenience, consideration need be given 

only to the distribution of head as found from drawdowns measured 

along the top and along the bottom of the aquifer; such drawdown 

readings are found from one series of observation wells or piezometers 

that extend into the uppermost port of the aquifer only and from a 

second series that extend to and are open only in the bottommost part 

of the aquifer. The divergence of the head from a purely logarithmic 

distribution at a distance r from the well is given by

(2/«r) Zf I(± l)*Ko(n*r/m) sin (mm)]

 =8, (8)

n

where K0 stands for the modified Bessel function of the second kind 

of zero order, the plus sign is for the drawdown distribution along the 

top of the aquifer, the minus sign is for the distribution along the 

bottom of the aquifer, and 8 is the drawdown correction factor.

Figures 80 and 81 are based on data computed from equation 8 and 

give, for different fractions of aquifer tapped by the pumped well, the
PERMEABILITY, TRANSMISSIBILITY, AND DRAWDOWN 275

10- *

FIOUBB 80. Corrections for drawdowns along the top of an aquifer for different fractional parts, a, tapped

by the pumped well.

correction factor to be applied, respectively, to drawdowns measured 

along the top and the bottom of the aquifer. 

From equation 8

A»=8«?/3»r), (9)

where A* is the drawdown correction the difference between the 

observed drawdown and the drawdown that would have resulted if 

the pumped well, discharging water at the same rate, had tapped the 

full thickness of the aquifer.
276 GROUND-WATER HYDRAULICS

10

FIGUXI 81. Corrections for drawdowns along tne bottom of an aquifer for different fractional parts, a,

tapped by the pumped well.

If the well is assumed to tap only the upper part of the aquifer, the 

drawdown will be greatest along the top of the aquifer and least along 

the bottom because of the convergence of the streamlines upon the 

opening to the well. If the well is open to only the bottom part of 

the aquifer and no part of the aquifer is unwatered, the drawdown 

pattern is similar, though inverted, to that for the well open only to 

the top part of the aquifer. The curves in figures 80 and 81 do not 

apply if the screen is in some intermediate position, nor clo they apply, 

strictly speaking, to unconfined aquifers.
PERMEABILITY, TBANSMISSIBILITY, AND DRAWDOWN 277 

APPLICATION OF THE FORMULA

Application of the formula for correction of drawdowns caused by 

a pumped well that taps only the upper part or only the lower part 

of an aquifer and the procedure for determining the coefficient of 

transmissibility of an aquifer from the corrected drawdowns are best 

illustrated by an example. Assume that a well having -an effective 

radius of 1.0 foot and its screen set in the top 40 feet of an artesian 

aquifer 100 feet thick (hence, a =0.40) has been pumped at a constant 

rate of 840,000 gpd long enough to establish steady radial flow beyond 

the most distant of 3 observation wells. The drawdowns in the observation wells, which tap only the top of the bed and which are 50, 100, 

and 150 feet from the pumped well, are 5.02, 3.18, and 2.30 feet, 

respectively. The drawdown in the pumped well is 18.4 feet. If the 

screen loss is negligible and the aquifer is homogeneous and isotropic 

with respect to permeability, determine the coefficient of transmissibility of the aquifer from observed drawdowns.

First, plot the drawdowns against the logarithm of their respective 

distances from the pumped well (see fig. 82). Draw a straight line 

(line I) as closely as possible through the plotted points and make a 

preliminary determination of the coefficient of transmissibility from 

the formula

where As = the drawdown difference over one log cycle. 

From figure 79 and equation 10

 , 2.30X840,000 __ __ T= 2^(9.1-3.2) as58'00

and, hence,

_fi___Af__6.9_ 2xr~2.30~2.30~2'

Assume, as a first approximation, that the above value of T is correct 

and make the following computations and corrections from the given 

data and figure 80:

Wettl WeUt WeUS 

Distance, r,_ _ _ _ . __ _ ___ ._ ______ . _ __.feet__ 50 100 150

*r/m=7rr/100_ _...__. _ _____________ __ ________ _ _ 1.57 3.14 4.71

Observed «« ____ _ _______ _ _______ _ ____ feet__ 5.02 3.18 2.30

«, from figure 80___. ____ __ __ _______________ 0.31 0.045 0.008

As tov, from equation 9______ _ __ __ __ __ __ ___feet__ 0.80 0.12 0.02

Corrected drawdown =s As_____ ___ __ ____ _ _ do__ 4.22 3.06 -2.28
10 100 

DISTANCE (r), IN FEET

FIGURE 82. Graph illustrating the method of correcting drawdowns to 
PERMEABILITY, TRANSMISSIBILITY, AND DRAWDOWN 279

Plot the corrected drawdowns against their respective values of r, 

and draw a straight line through the points (line II) (see fig. 82). 

Make a second trial determination of T and Q/%irT:

 2.30X840,000 __ n _ _ , ^ 

2^(7.1-3.1) => gpd per ' 

and

P, _ _ -. == _ _ -.- 1. /4 It. 2-n-T 2.30 2.30

Further correct the observed drawdowns by repeating the above procedure using the corrected values of T and

Wett 1 Wett t Wett S

Observed stav,. ------------------------------- feet.. 5.02 3.18 2.30

«, from figure 80 _______ _ _______ _ ___________ _ _ 0.31 0.045 0.008

As, op, from equation 9-. _ ________________ ___.-feet-_ 0.54 0.08 0.01

Corrected drawdown = s As_---__-_-___------__do--- 4.48 3.10 2.29

Again, plot the new values of corrected drawdowns and draw a 

straight line through the points (line III) (see fig. 82). Make a third 

trial determination of T and Q/%irT:

and

Q As 4.5

Repeating the procedure a fourth time results in line IV (fig. 82), 

from which T = 69,000 gpd per ft and Q/%*T= 1.94 feet. These probably are close enough to the true value, and further computations 

are not necessary.

As a further check, however, the drawdown that would occur in a 

pumped well tapping the full thickness of the aquifer can be determined from equation 1. Solving this equation for the hypothetical 

drawdown, (Q/2irT)loge (re/rw), yields s^afC, where C is defined in 

equation 3. In the above example,

1 , . 7 cos (0.2ir) , . 7 (cos 0.628) _ _ . 

C=l+ V80 =1+ 8.94 =Lb34' 

and, therefore,

^£=18.4X0.40X1.634=12.0 feet,

which is identical to the s-ordinate of the point of intersection of line 

IV with the line r-rw=l foot in figure 82. This indicates that the 

corrected drawdowns represented by line IV are the drawdowns which 

would result if a well tapping the full thickness of the aquifer dis-
280 GROUND-WATER HYDRAULICS

charged 840,000 gpd and that the coefficient of transmissibility of the 

aquifer is T=69,000 gpd per ft.

The lower dashed line in figure 82 gives the probable distribution 

of drawdown as measured in wells that are open to only the top part 

of the aquifer. The drawdowns that would have been observed in 

wells 1,2, and 3 had they extended to and been open to only the bottom 

part of the aquifer can be determined by making a correction using 

figure 81:

Corrected drawdown, from line IV, figure 82_ _feet_ _ 

5, from figure 81 ______________________________

Asbottom, from equation 9____--___-_------_feet-_

Well 1 Well I Well 3

4. 42 3. 09 2. 28

-0. 28 -0. 044 -0. 008

-0.54 -0.09 -0.01

3. 88 3. 00 2. 27

The upper dashed line through the bottom points in figure 82 gives the 

probable distribution of drawdowns as measured along the bottom of 

the aquifer.

DETERMINATION OF THE DRAWDOWN AT THE POINT OP

STAGNATION

If the distribution of drawdown in the vicinity of a well that taps 

only the upper part of an aquifer is to be represented graphically, the 

drawdown at the point of stagnation beneath the well must be determined. In table 6 the difference between the drawdown, s0 , at the 

point of stagnation and the drawdown, s4m< at a distance of 4m, is given 

in units of (Q/2irT) for different values of a under two alternative 

assumptions regarding conditions at the well face. The data in the 

second column were computed from an equation derived by Muskat

TABLE 6. Data for determination of drawdown at the point of stagnation beneath a 

well tapping less than -the full thickness of an aquifer

Fractional part of aquifer tapped by pumped well

o.oo__. ________________________________________

0.05 _____________________________________ .-.

0.10_. __________________________________ _______

0.20_______ ____________ _____________________

0.25 ____________________________ ____________

0.30_._. _______________________________________

0.40._-_- _ .- _____

0.50... _________ _____

0.60__. ----__-..-_______.___.__.___.__.____-_._

0.70_-_-_._.___ _____

0.75 _______________ ______ __________

0.80__________._ _ _____

0.90.__________________ ___ _ _ _________

(*0-*4m)/(0/2irr)

For uniform 

velocity at well 

screen

1. 96

1. 971 

1.992

2.030 

2.087 

2. 170 

2. 287 

2. 459

2.733 

3. 264

For uniform 

drawdown at 

well screen

2.34 

2.25

2. 15

1 2. 25 

2.37

2.80

3.70

1 By interpolation.
PERMEABILITY, TRANSMISSIBILITY, AND DRAWDOWN 281

(1937, p. 268, eq 8) which is based upon the assumption that the flux 

through the lace of the well the screen velocity is uniform. The 

data in the third column were made available to the author through 

the courtesy of Dr. Muskat and the Gulf Research & Development 

Co. They were obtained by adjusting, largely by trial and error, 

the flux-density distribution so as to give a virtually uniform distribution of potential over the well face.

Assume, once again, that a well having an effective radius of 1.0 foot 

and ^its screen set in the top 40 feet of an artesian aquifer 100 feet 

thick has been pumped at a constant rate of 840,000 gpd long enough 

to establish steady radial flow beyond a distance of 4m, or 400 feet. 

At that distance, the drawdown, taken from line IV in figure 82, is 

0.40 foot. The drawdown at the piont of stagnation, if uniform screen 

velocity is asumed, is

s0=2.09 ;r%+0.40= (2.09X1. 94) +0.40=4.45 feet,

or, if uniform drawdown at the screen is assumed, is

s0=2.25 ^+0.40= (2.25X1. 94) +0.40=4.76 feet.

The latter assumption probably fits actual conditions more closely 

than the former and has been adopted for figure 82. However, if the 

pump intake is above the top of the screen, the drawdown will be 

greatest at the top of the screen and least at the bottom owing to the 

friction of the upward moving water inside the screen. Accordingly, 

the screen velocity actually may be more nearly uniform than it is 

when the drawdown is uniform. In other words, actual conditions 

probably lie somewhere between the two limits set by these assumptions. 

Theoretically, the drawdown at the point of stagnation in homogeneous beds is independent of the ratio of horizontal to vertical permeability, provided that ratio does not become infinite. At other 

points, radial distances are to be divided by the square root of that 

ratio before making corrections in the drawdown. (In the above 

problem the ratio is taken as unity.)

DRAWDOWN DISTRIBUTION IN THB VICINITY OF A PUMPED WELL 

TAPPING LESS THAN THE FULL THICKNESS OF AN AQUIFER

For aquifer tests in which the well to be pumped taps only a fractional part of the aquifer, the observation wells should be placed in 

pairs, one of each pair tapping the top part of the aquifer and the 

other the bottom part. Averaging the drawdowns measured along 

the top and bottom of the aquifer yields the approximate drawdown 

that would be caused by a pumped well that taps the full thickness of 

the aquifer. The data from the foregoing example verify this:
282 GROUND-WATER HYDRAULICS

WeUl 

5.02 

3.88 

4.42 

Average stov and s 4 o««om-------___--_-___-_____do_- 4. 45

Observed stov -- __---__-__--_--____-____--___.feetComputed Sbottom__ --___--_-_-_-__--______-_____do.

Final corrected drawdown_________________________do.

Wettt

3. 18

3.00

3.09

3.09

WellS 

2.30

2.27

2.28

2. 28

The difference between the final corrected drawdown and the average 

drawdown is within the error of plotting. For a =50 percent, the 

drawdowns average out exactly.

Figure 83 gives the approximate drawdown distribution for a=40 

percent, as measured along the top, the bottom, or at any point within 

the aquifer. The equipotentials (lines of equal drawdown) in this 

figure are to be regarded as diagrammatic, as they were sketched from 

known values of drawdowns along the top and bottom of the bed.

A remark about Slichter's formula for a well that taps only a fractional part of the full aquifer thickness may not be out of place here. 

(Reference is made to eq 123, Wenzel, 1942, p. 109.) This formula 

is unsound because it assumes radial flow superimposed upon hemispherical flow; the potential distribution along their common boundary cannot be corrected satisfactorily.

Pumped 

well 

Oin 

20

40

60

80

100

i

i 

i

i b

i -1

/ y

|

!

|OOJ

/

I/'/

i y

o-^i i i

i i

! i

III

1 /

/

/

1 '

i) Vy ' ' t\

/I 1

;_,

10

-

-

-

,' /Top of aquifer 

= 100 percent

100 200 

DISTANCE (r), IN FEET

300 400

FIOUBE 83. Oraph showing the distribution of drawdowns near a pumped well tapping 40 percent of the

thickness of an aquifer.
THE RECOVERY METHOD FOR DETERMINING THE 

COEFFICIENT OF TRANSMISSIBILITY

By C. B. JACOB

ABSTRACT

The Theis recovery method for determining the coefficient of transmissibility 

of an aquifer is applicable if the coefficient of storage remaics constant throughout both the pumping and recovery periods. However, if the coefficient of 

storage does not remain constant, the water-level drawdown data, when plotted 

on a graph, do not fall along a straight line that passes through the origin. 

Instead, most of the data fall along a smooth curve that does not pass through 

the origin, and the coefficient of transmissibility can be determined from the 

slope of this curve regardless of any variation in the coefficient of storage.

Wenzel (1942) suggested an empirical correction for water-level recovery 

data, but his suggested correction is thought to be invalid because it was based 

on a consideration of wells that tapped less than the full thickness of the aquifer.

THE THEIS RECOVERY METHOD

The recovery method developed by Theis (1935) has proved useful 

for determining the coefficient of transmissibility of an aquifer if the 

coefficient of storage remains virtually constant throughout both the 

period of pumping and the subsequent period of recovery. The 

residual drawdown during recovery is given by

where rw is the effective radius of the pumped well in which the 

water-level recovery is measured. If S (the coefficient of storage 

during the period of pumping) and S' (the coefficient of storage 

during the period of recovery) are constant and equal (case 1 in fig. 

84Z? ) , equation 1 can be simplified to

The graph of equation 2, where s' is plotted against log (£/£'), IS 

a straight line through the origin (case 1 in fig. 846"). If the value 

of Q is known, T can be determined from the slope of the line. A 

point moving on the recovery limb of the curve in figure 84J. follows the curve labeled case 1 in figure 84C"/ the latter curve is asymptotic to the straight line given by equation 2. As time becomes 

infinite, (t/f) approaches unity (that is, log t/t' approaches zero), 

the residual drawdown, s', approaches zero, and therefore the mov283
284 GROUND-WATER HYDRAULICS

Case 4

t

FIGUBE 84. Water-level recovery curves for different types of variation of Swlth time.

ing point, which represents the water surface in the well, approaches 

the static level. The position of the static level is determined by observations made prior to the beginning of pumping.

WENZEL'S METHOD OF CORRECTING WATER-LEVEL RECOVERY

DATA

For many tests, a graph of water-level recovery data is not a 

straight line passing through the origin. Wenzel (1942, p. 96) maintained that in such cases a correct value of T cannot be obtained without first applying an empirical adjustment to the data. The adjustment suggested by Wenzel is made by adding an arbitrary constant, 

c, to t so that each s' plotted against log (£+<?)/£' falls on a straight 

line through the origin; then T is determined from the slope of the 

new line.
PERMEABILITY, TRANSMISSIBILITY, AND DRAWDOWN 285

CONSTANT, BUT DIFFERENT, COEFFICIENTS OF STORAGE FOB 

THE PUMPING AND RECOVERY PERIODS

The failure of the water-level recovery data to produce a straight 

line through the origin is believed to be due to variability of S. The 

apparent average S in the vicinity of a pumped well often is greater 

during the drawdown period than during the subsequent recovery 

period. In- unconfined or semiconfined aquifers, this difference in S 

results from the hysteresis of the capillary fringe and from the envelopment of air bubbles by the rising water table. In confined 

aquifers, the difference results from the consolidation of deposits 

during pumping, especially in newly developed aquifers, and is less 

than in unconfined or semiconfined aquifers.

Case 2 in figure 84Z? illustrates what is probably the simplest form 

of variation of the storage coefficient with time. In case 2, the storage 

coefficient is idealized as a constant value S during the period of 

pumping and as a smaller constant value S' during the recovery 

period. The residual drawdown then is given by

This again is a straight-line equation; however, the line intersects 

the drawdown axis at value (Q/^irT) log(S/S') instead of passing through the origin. From the slope of the straight line, T is 

determined as by the Theis method; no adjustment is needed. The 

ratio ($/$') can be determined from the intercept on the log (£/£') 

axis.

The straight-line plot for case 1 (where $'=£), the straight-line 

plot for case 2 (where /S"=$/2), and three curves obtained by applying the correction factor suggested by Wenzel to the straight-line 

plot of case 2 are shown in figure 85. The correction factor is taken 

successively as 0.5, 0.8, 0.9, and 1.0 times the pumping interval ( t tr ) . None of the adjusted curves passes through the origin, 

nor would any curve obtained by using other correction factors. 

Secants or tangents to parts" of the curves for c= 0.8 (t £') and 

c= 0.9 (£ t') would pass through the origin, but the values of T 

obtained therefrom would be too small and would depend on the 

range of data available for consideration and on the reliance placed 

on different parts of those data.

To determine the value of c which would result for case 2 in a 

straight line that would pass through the origin, let s' equal zero; 

then

690-185 O 63 4
286 GROUND-WATER HYDRAULICS
PERMEABILITY, TRANSMISSIBILITY, AND DRAWDOWN 287

The value of c which would cause the straight line to pass through 

the origin now can be found by equating the logarithm of the corrected 

time ratio to that given by the quantity in brackets in equation 3 :

Therefore,

C == £o tQ t t

for all values of t or of t1 '. Hence,

i log =*>i

This is the equation of the drawdown axis. In other words, in case 2, 

the only correction that gives a straight line through the origin involves the subtraction of the constant (t-f) from t; such an adjustment results in a vertical line from which is obtained the absurdity 

T=Q.

Data from an aquifer test near Grand Island, Nebr., are used as an 

example of case 2. Wenzel's method of correcting the data from 

the Grand Island test is illustrated by figure 86, which is a reproduction of the data plots presented by Wenzel (1942, fig. 12). The 

minimum value of (t/f) in figure 86 is 2.79, and Wenzel's correction 

is 1,120 minutes, or

= 1,120=_

0.39.

t-t e 2,880

The curvature of a line through Wenzel's corrected data is similar, 

though less pronounced, to that of the corrected curves in figure 85. 

If the range of the data were greater at the lower end of the curve, 

the corrected curve would pass to the right of the origin, whereas if 

a larger c were subtracted from £, the corrected straight line could be 

made to pass through the origin simply by using the lower points on 

the graph.

The hypothesis that S has one constant value during the drawdown 

period and a smaller constant value during the recovery period is an 

adequate though provisional explanation for the position of the data 

plotted in figure 86. From the intercept log(t/t') =0.11, it is found 

that (S/S') =1.29, approximately. Consequently, the relative change 

in S is

which is not unlikely.
288 GROUND-WATER HYDRAULICS

1.4

1.2

5 1.0 

u.

z

n o

WDOWN c c

<

oc. 0.6

o

z>

o

Ul

oc.

0.2

n

L

I

/

i

' >

f t

>' /

oc '~ 1121

e r

ty

Cj 

/

C5

r

y/

o/

r /

/ 

/

Log

/

/ J ^ y4

/

x

f 

f

/

/

6

/

0.2 0.4 0.6 0.8 

Log -£

1.0 1.2 1.4

FIGURE 86. Graph showing corves obtained by plotting «' against log WO, aquifer test near 

Grand Island, Nebr.

If other factors could be excluded for the problem represented by 

figure 86, the value of T apparently could be determined from the 

original straight line without any adjustment. However, the well 

that was pumped in making that aquifer test extended only 40 feet 

into an aquifer that is 100 feet thick. In the first paper in this series, 

Jacob computed the probable true value of the coefficient of transmissibility for this aquifer to be about 180,000 gpd per ft instead of 

the 100,000 gpd per ft as determined by Wenzel by the Thiem and 

Theis methods. The value of T obtained from the unconnected recovery data is as close to the true value of T as that obtained by any 

of the other methods that do not make allowance for the fact that 

the pumped well tapped less than the full thickness of the aquifer.

THE UNIFOBMI/? CHANGING COEFFICIENT OF STORAGE 

Another hypothesis worthy of consideration is illustrated by case 3 

in figure 845. In case 3 the storage coefficient increases uniformly 

during the pumping period to twice its initial value and then, on
PERMEABILITY, TBANSMISSIBILTTY, AND DRAWDOWN 289

cessation of pumping, instantaneously drops back to its initial value 

and increases uniformly during the recovery period at the same rate 

as during the pumping period. The recovery curve for case 3 is 

shown schematically in figure 84(7.

The straight line for case 1, the uncorrected curve for case 3, and 

three curves obtained by applying Wenzel's correction factor to the 

uncorrected curve for case 3 are shown in figure 87. The three corrected curves were obtained by appling the correction factors 0.5, 

 0.8, and 0.9 times the pumping interval (t/tf). The unconnected 

curve for case 3 is asymptotic to the corresponding straight line of 

case 2 and is tangent to the logarithmic time-ratio axis at the origin. 

Although the corrected curves pass through the origin, they have 

points of inflection; and, if tangents or secants to the curves are drawn 

so that they pass through the origin, the value of T determined from 

them not only varies, depending upon the weight placed on different 

parts of the available data, but is also too small.

Data from an aquifer test made near Scottsbluff, Nebr., are used to 

illustrate case 3, although they also apply to case 2. Wenzel's method 

of correcting the data from this test is illustrated in figure 88, which 

is a reproduction of the data plots presented by Wenzel (1942, fig. 16). 

The minimum value of (t/f) plotted in figure 88 is 1.29. Although 

extraneous factors began to affect the recovery 2 or 3 days after pumping stopped, further corrected data probably would tend toward the 

origin. Wenzel's correction in figure 88 is 575 minutes, 

or

c -575

t t' 939

-=-0.61.

In drawing the straight line through the corrected data, Wenzel gave 

greater weight to those points in the higher range of (t/t') and 

virtually ignored the 17 lowest points. If a larger c had been subtracted and the lower points used in preference to the higher, an 

equally acceptable straight line could have been drawn through the 

origin but would have yielded an even smaller value of T.

EVALUATION OF WENZEL'S METHOD OF CORRECTING RECOVERY

DATA

The empirical correction for recovery data, suggested by Wenzel, 

is not justifiable and its use should be discontinued. According to 

Wenzel (1942, p. 96)", "It may be fortuitous that such a correction gives 

transmissibility values that check with those computed by the other 

formulas, and more tests must be made before such a correction can be 

applied with assurance." Actually, the data used by Wenzel not only 

fail to demonstrate the effectiveness of his correction factor but show,
FIGURE 87. Straight-line plot for case 1, the uncorrected plot for case 3, and Wenzel's emp
PERMEABILITY, TBANSMISSIBILTTT, AND DRAWDOWN 291

FE »-

Log f-575

y

0.2 0.4 0.6 0.8 1.0 1.2

FIOUBE 88. Oreph abowlng curves obtained by plotting «* against log W), 

aquifer test near Scottsbtaff, Nebr.

instead, that the values obtained from empirically corrected data are 

governed by that part of the data which is favored. Wenzel's results 

are inconclusive for the additional reason that no corrections were 

made to offset the fact that the pumped wells tapped less than the full 

thickness of the aquifer.

It is probable that hypothetical cases 1, 2, and 3 rarely, if ever, 

obtain in nature. Instead, during some tests, the variation in 8 probably more closely resembles case 4 of figure 84Z?, and, during others, the 

apparent 8 during the recovery period may be greater than during 

the pumping period. Apparently the coefficient of transmissibility is 

nearer the true value if it is determined from the slope of the curve 

drawn through plots of uncorrected, rather than corrected data.

On the basis of the hypothesis of either case 2 or case 3, the corrected data should not be expected to fall along a straight line. Instead, the transmissibility should be determined from the slope of the 

curve through the uncorrected data. If case 2 is, assumed to apply to
292 GROUND-WATER HYDRAULICS

figure 88, then from the intercept log(t/f) =0.14, it is found that 

S/S' = 1.38, approximately.

The water-bearing material at the site of the aquifer test near Scottsbluff, Nebr., is 123 feet thick, and the top 9 feet consists of silt and 

fine sand. As the pumped well was only 46 feet deep, the true coefficient of transmissibility probably is nearly twice that found by applying WenzePs correction factor. Although the value obtained from 

the unconnected data is more accurate, it is less than the true value 

because the pumped well tapped less than the full thickness of the 

aquifer.
DETERMINATION OF THE COEFFICIENT OF TRANSMISSIBILITY FROM MEASUREMENTS OF RESIDUAL DRAWDOWN IN A BAILED WELL

By HERBERT E. SKIBITZKE

ABSTRACT

The coefficient of transmissibility can be computed from measurements of the 

residual drawdown In a bailed well provided the aquifer tapped by the well 

has a low transmissibility. The vertical line-source formula, developed from 

both the point-source heat-flow equation and the Theis recovery equation, is used 

for the determination if the well has been bailed only once. A modified formula, 

which sums the effects attributed to each bailing, is used for the determination 

if the well has been bailed repeatedly. Although the modified formula requires 

much computation, it affords a means of calculating the coefficient of transmissibility when other methods of analysis are not feasible.

THE BAILED WELL AS AN INSTRUMENT OF HYDE OLD QIC ANALYSIS 

Drilling reveals the thickness and lithologic character of waterbearing materials but it does not provide sufficient information for 

accurate determination of the hydraulic constants. Therefore, if the 

water-bearing materials are not tapped by pump-equipped wells that 

are suitable for use in aquifer tests, or if the water-bearing materials 

lie at depths so great that the cost of installing a test pump would be 

prohibitive, consideration should be given to the possibility of using 

the drill rig to bail the well instead of installing a test pump. The 

hydraulic constants of the water-bearing materials can be computed 

from measurements of the water-level recovery subsequent to the bailing, provided the aquifer has not such a high transmissibility that the 

water level recovers before measurements can be begun.

The nonequilibrium formula developed by Theis (1935) generally 

is used in analyzing aquifer tests in which the drawdown varies as a 

function of time. One of the assumptions inherent in the derivation 

of that formula is that the discharge from the well is both steady and 

constant. The nonequilibrium formula is rarely applicable if a well 

is bailed because, obviously, Q is instantaneous if water is removed 

from a well by a single bailing and generally is discontinuous or intermittent if water is removed from a well by repeated bailing. Equations are derived in this paper for use in the analysis of residual drawdowns resulting from either a single bailing or repeated bailings.

DERIVATION OF THE FORMULA FOB A VERTICAL LINE SOURCE 

FROM THE EQUATION FOR HEAT FLOW FROM A POINT SOURCE

A fundamental differential equation of hydrodynamics, describing 

the nonsteady-state flow of an incompressible fluid in a compressible

293
294 GROUND-WATER HYDRAULICS

porous medium, has been given by Muskat (1937, p. 133) and Jacob 

(1950, p. 333) in the general form

&h &h ,&h=S^*>h m 

ax^dy^ds2 P d*' W 

where

h =a head, or potential, function (the sum of the gravity 

potential and the pressure potential), as represented 

by the piezometric surface of the incompressible fluid;

S' =the coefficient of storage of the water-bearing material 

(S), divided by the thickness of the water-bearing 

material (m); and

x, y, z= the linear distances from the origin of a three-dimensional coordinate system.

If Q' is the volume of water removed instantaneously at time £=0, 

then Q' '/S' is the strength of the point source. According to Carslaw 

and Jaeger (1947, p. 216-217), a particular solution of equation 1 for 

an instantaneous point source of strength Q'/S' at the coordinate 

origin, a?=y=s=0, is

h=

where h now is the change in head attributable to the instantaneous 

point source in a homogeneous and isotropic medium of infinite thickness and extent. A pumped or bailed well approximates a vertical line 

source. If the center of the well is considered to be the origin of the 

coordinate system and if the axis of the well is along the s-axis, equation 2, which is the solution for a point source, can be used to find the 

distribution of head in the vicinity of the bailed well by integrating 

or summing up all the effects of the point sources from oo to + oo 

along the s-axis. Over an infinitesimal length dz along the s-axis, 

the source strength is ( Q'/S' ) dz. Therefore,

"f

j <

(3)

Because 5, y, and t are not variables of integration, the relation

where

r=the distance from the center of the well, can be substituted in 

equation 3, which then can be written in the following form:

Q'/S'

MP/S')t

f"^

J
PERMEABILITY, TRANSMISSIBILTTY, AND DRAWDOWN 295

where z is the only variable of integration. The function to be 

integrated,

is symmetrical with respect to the xy plane because z occurs only as z2 ; 

therefore,

/( )-/(- ).

Consequently, the integration of f(z) between the limits oo and oo is 

double the result of integrating from 0 to oo . Use of this relation and 

a table of integrals (Peirce, 1929, p. 63, item 492) gives

f" e^lsl)tdz=2 f 

J-« Jo

__ ,

V*"/ <fe= 

o

Substitution of this result in equation 4 yields

8[ir(P/S')«]J 

or

MPisf)t

which represents the radial distribution of changes in head in the 

vicinity of an infinite vertical line source in terms of an instantaneous 

point source of strength Q'/S'.

From equation 5, the distribution of head is seen to be independent 

of z; therefore, the flow characteristics of the system would not be 

changed by confining a segment of the aquifer between two impermeable strata that are parallel to the xy plane. Also, the strength of a 

bailed well that taps the full thickness of the aquifer is more conveniently expressed in terms of the quantity of water, q, removed by the 

bailer in one bailer cycle. Thus, the terms $', £', and P in equation 5 

can be replaced by equivalent terms incorporating the thickness, m, 

and the conventionally defined hydraulic constants of the aquifer. By 

definition,

£'=-1; £'= ; and P= 

x m m m
296 GROUND-WATER HYDRAULICS

Substitution of these ratios for Q', /S", and P in equation 5 and adoption of the more commonly used symbol, $', (the residual drawdown 

of the piezometric surface), in place of h results in the equation

where

r=the distance from the center of the bailed well to the point at 

which the drawdown is observed.

In equation 6, as r becomes small and as t becomes large, the exponent

-T*S

rzS/±Tt approaches zero and the value of e iTt approaches unity. 

Thus, in and near the bailed well, when r is small in comparison with 

the extent of the aquifer and when t is large, equation 6 can be written 

in the simplified form

Although equation 6 could have been written directly from Carslaw's solution (Theis, 1935, p. 520), some of the steps in the integration procedures that are involved in the derivation from the more 

general or fundamental point-source solution were considered worthy 

of further study.

DERIVATION OF THE FORMULA FOR A VERTICAL LINE SOURCE 

FROM THE THEIS RECOVERY EQUATION

Most of the material presented in this section is credited to M. I. 

Rorabaugh, who independently derived equation 7 from the Theis 

recovery formula (Theis, 1935, p. 522, eq 7). In nondimensional form, 

the recovery formula is

" 4- (8)

where the times t and t' are large.

If, for a recovery test, a volume of water, #, is pumped from a well 

during' a short pumping period of length A£ and if tn represents the 

elapsed time from the midpoint of the A£ pumping interval, then

Q=-r-.' t=tn -\-^-> and t'=tn =--
PERMEABILITY, TRANSMISSIBILITT, AND DRAWDOWN 297

Substitution of these values in equation 8 yields

or

The log factor in equation 9 can be expanded into series form by 

referring to a comprehensive mathematical handbook, such as Hodgman (1952), which shows

In equation 9,

n=2tn/M; 

hence,

ptjAO + n ["At (At)* (AQ» n 

10g< L(2<«/AO-lJ~2 L2^+3(2^)3+5(2^+ J

For a well that is bailed, the pumping time, A£, per bailer cycle, is 

very small compared to the time, tn. Therefore, each term beyond the 

first in the series shown in equation 11 is so small that it can be neglected and the equivalent of the log term is then &t/tn. Substitution 

of this value in equation 9 gives

A*t'

or

/= g

which is identical to equation 7.

MODIFIED FORMULA FOR THE ANALYSIS OF WATER-LEVEL 

RECOVERY AFTER REPEATED BAILING

The residual drawdown in or near a bailed well for some time, £, 

after the removal of only one bailer of water is given by equations 7 

and 12; the residual drawdown after n bailer cycles is
298 GROUND-WATER HYDRAULICS

where the subscripts identify each bailer cycle in chronological order. 

Each time tn represents the interval between the occurrence of the indicated bailer cycle and the time at which s' is observed. The volume 

of water removed in each bailer cycle can be assumed to be constant. 

Therefore, equation 13 can be simplified to

or, in abbreviated notation,

T is in gallons per day per foot if s' is expressed in feet, q in gallons, 

and ti in days.

USE AND LIMITATIONS OF THE FORMULAS 

The development of equations 7, 12, 13, and 15 involves most of the 

assumptions inherent in the Theis recovery formula, particularly the 

stipulation that T be small and t large. As observed by Rorabaugh 

(written communication, Dec. 8, 1950) , the new equations will be most 

useful when the bailing is hit-or-miss and a reasonable average pumping rate cannot be ascertained. When bailing is at a fairly uniform 

rate, the cyclic effects might be dissipated before the large-tf requirement is satisfied. In this case, the Theis method is preferable because 

the recovery data are more easily studied graphically. The use of the 

analytical method described in this paper, particularly when the number of bailer cycles is large, requires much computation for each observed residual drawdown in the recovery period. However, if other 

methods of analysis are impractical and the artesian aquifer is of low 

transmissibility, the described analysis affords a means for computing 

the coefficient of transmissibility by a method that commonly is overlooked.
THE SLUG-INJECTION TEST FOB ESTIMATING THE 

COEFFICIENT OF TRANSMISSIBILITY OF AN AQUIFER

By JOHN O. FEBBIS and DOTLB B. KNOWLES

ABSTRACT

If a comparatively small volume, or slug, of water is instantaneously injected 

into or withdrawn from a well, the well is considered to be an instantaneous 

vertical line source or line sink. For artesian aquifers, the coefficient of transmissibility of the material in the immediate vicinity of the well can be estimated 

by an equation developed by Theis (1935) for a vertical instantaneous line sink. 

The equation can be solved graphically from a straight-line plot of the data. 

For a slug-injection test, the well should tap the full thickness of the aquifer 

and be fully developed, and the water-level measurements should be made in 

rapid succession after the injection or withdrawal of the slug of water into or 

from the well.

EQUATION FOB DRAWDOWN IN AN INSTANTANEOUS VERTICAL 

LINE SOURCE OR LINE SINK

In the development of the nonequilibrium formula for drawdowns 

caused by the pumping of ground water from storage at a constant 

rate, Theis (1935) presented the equation for drawdown in an instantaneous vertical line source or line sink and obtained, by integration, 

the equation for a vertical line source or line sink that is continuous 

at constant strength. Many field tests evidence the applicability of 

the nonequilibrium formula to the problem of the discharging well 

that taps an artesian aquifier of infinite extent; however, the importance of the nonequilibrium formula to quantitative ground-water 

hydrology so over shadowed the initial step of the Theis development 

that little attention has been accorded the equation for an instantaneous vertical line source or line sink. Although of limited application, the equation provides a useful method for estimating the 

coefficient of transmissibility in the immediate vicinity of a well, which 

is a physical approximation of the theoretical vertical line source or 

line sink.

The slug-injection test for estimating the coefficient of transmissibility involves the instantaneous injection (or withdrawal) of a slug 

of water into or from a well; hence, the term "slug-injection test" 

seems appropriate. Use of the slug-injection test should be limited 

to fully developed wells that are open to the full thickness of an 

artesian aquifer of small or moderate transmissibility less than 

50,000 gpd per ft The test probably cannot be used to determine the 

transmissibility of most water-table aquifers. Because the coefficient 

of transmissibility determined from this test generally applies only to 

the material close to the well, indiscriminate use of the results can

299
300 GROUND-WATER HYDRAULICS

lead to erroneous conclusions. Great care must be exercised in conducting the tests, in analyzing the data and, most particularly, in 

applying the results to the solution of field problems. Nevertheless, 

the simplicity of the test justifies its use provided the assumptions 

upon which the formula is based are essentially fulfilled and the 

limitations of the test are fully recognized.

The equation for residual head in an instantaneous vertical line sink 

is written

-T*S

(1)

where

*=the residual head after the injection of a slug of water, 

r=the distance from the injection well to an observation well, 

t=the time since the slug was injected, and 

2=the volume of the slug.

Ordinarily, only a small volume of water can be injected into a well 

as a slug. For this reason, the reaction to the injected slug usually is 

not measurable in the aquifer beyond the immediate vicinity of the 

well. Therefore, the water-level measurements are made only in the 

injection well; the distance is then the radius, r«,, of the well. For 

values of r as small as r«,, especially where S is small (as for artesian 

aquifers), the exponent of e in equation 1 approaches zero as t becomes 

large and the value of the exponential terms approaches unity. Then, 

if q is expressed in gallons, T in gallons per day per foot, t in minutes, 

and s in feet, equation 1 can be written in the form

where

£m=the time in minutes measured from the average of the times 

marking the beginning and cessation of the injection.

Equations 1 and 2 are equivalent to those derived by Skibitzke in the 

preceding paper for drawdowns near a vertical line source. The 

effects of instantaneously injecting a slug of water into a well are 

identical in the opposite direction to those of instantaneously withdrawing a slug of water of equal volume from the same well; that is, 

the rate of water-level decline following the injection of a slug equals 

the rate of water-level recovery following the withdrawal of a slug 

of equal volume.

EQUIPMENT AND PROCEDURE FOB A SLUG-INJECTION TEST 

A convenient apparatus for instantaneously injecting water into a 

well is shown in figure 89A. An oil drum has been adapted for use
PERMEABILITY, TRANSMISSIBILITY, AND DRAWDOWN 301

Oil drum of known volume

B5E

Block drum up above well casing to __j 

provide for escape of air ~^1

Land surface

Well casing

Rope

Cover plate with fixed eyebolt 

' Gasket attached to cover plate

Flange and nipple making watertight connection with bottom of 

drum

A. APPARATUS FOR INJECTING SLUG OF WATER INTO WELL

Pipe

Lead filler

Eyebolt for attaching steel tape

1"-

^"^ Shallow depression in the lead

B. PERCUSSION INSTRUMENT FOR RAPID MEASUREMENT OF WATER LEVELS 

FIQUBE 89. Equipment for making a slug-injection test.

060-185 O 63 6
302 GROUND-WATER HYDRAULICS

as a container to hold the slug of water to be injected. An opening 

has been made in the bottom of the drum, and a nipple has been attached by using a flange connection. (The nipple could have been 

welded directly to the container.) A cover plate to which a gasket 

is attached has been placed over the opening inside the drum. When 

the rope attached to the cover plate is pulled, the cover plate is raised, 

thus allowing the slug of water to be introduced into the well almost 

instantaneously. The drum should be supported by blocks so it can 

be removed quickly and easily once the injection is completed and also 

to provide a vent for the escape of air from the well. Water-level 

measurements must be made rapidly after the slug of water has been 

introduced into the well. Because the wetted-tape method generally 

is too slow, a percussion instrument attached to the end of a steel tape 

or an electric tape should be used. A percussion instrument, such as 

shown in figure 89Z?, can be constructed easily by filling a short section of steel pipe with lead, carving a shallow, oval depression in the 

lead at one end of the pipe, and attaching the other end of the pipe to 

a steel tape. The instrument usually is referred to as a popper because the depression in the lead causes a popping noise upon sharp 

contact with the water surface in the well. Water-level measurements 

made with the popper are as accurate as those made by the wettedtape method.

The residual head at any instant is the difference between the observed water level and the extrapolation of the antecedent water-level 

trend. In making a slug-injection test, the time at which the injection 

began and the time that it stopped should be recorded, and the average 

of these times should be used as the origin of time in analyzing the 

water-level measurements. The elapsed time during the injection must 

be small compared to the period of the water-level recession. 

A slug-injection test is made as follows:

1. Before injecting the slug of water into the well, define the existing waterlevel trend by making water-level measurements at frequent intervals.

2. By means of a suitable apparatus, instantaneously (or nearly so) inject a 

known volume, q, of water into the well.

3. Remove the injection apparatus from the well and resume water-level 

measurements.

4. Calculate the values of the residual head and l/tm.

5. Plot the corresponding values of 8 and l/tm on rectangular coordinate 

graph paper. The plotted values should define a straight line through 

the origin.

6. Calculate T from the coordinates of any point on the straight line.

If the observed data from a slug-injection test do not plot on a 

straight line, the well may be in need of additional development. The 

slug-injection test cannot be used to estimate transmissibilities that
PERMEABILITY, TRANSMISSIBILITT, AND DRAWDOWN 303

are much in excess of 50,000 gpd per ft, because the water-level buildup 

produced by introducing water into the well disappears so rapidly that 

the data curve cannot be defined accurately.

A PRACTICAL APPLICATION OF THE SLUG-INJECTION TEST 

A slug-injection test to estimate the coefficient of transmissibility 

of a water-bearing formation was made on a 6-inch test well at Speedway City, Ind. Measurements of the depth to water prior to and 

following the instantaneous injection of a 39-gallon slug of water, 

the residual head at the time of each water-level measurement, and 

the reciprocal of the time since the slug was injected, l/tm, are given 

in table 7. The residual head, *, was plotted against the reciprocal 

of the time since the slug was injected, l/£m, on rectangular coordinate graph paper, as shown in figure 90. A straight line drawn 

through points plotted from the observed data passes through the 

origin. Arbitrary selection of the point on the straight line where 

lAm=0.5 and *=0.165 foot and substitution of those values in equation 

2 gives

114.6X39X0.5 

0.165 =13,500 gpd per ft.

The well used for the slug-injection test was used later as a waterlevel observation well during an aquifer test made by pumping from 

a nearby well. The data from the second test were analyzed by the 

nonequilibrium formula and gave T=16,000 gpd per ft.

TABLE 7. Data for a slug-injection test at Speedway City, Ind. 

[39-gal slog of water injected into a well 6 in. in diam at time -0 min]

Time 

(min)

-20. ___ ... ____ ..

-15 _____ ... __ ...

-10... ..............

L 25........ ........

L33 ...............

1.50-....... .......

1.92........ .........

2.17....................

2.30 ...... ....

2.37.. ..............

2.42 ....................

2.67 ...............

2.72 

2.77 . 

2.92 ___________

3.00 __ ....... __ ...

3.22 ____ . ..........

3. 28.... .............

3.33. ..... ...........

3.40.... ...........

a 47 ............

a 55 .................

Depth to 

water 

below measuring point 

(ft)

42.39

42.40

42.40

42.14

42.15

42.20

42.23

42.24

42.25

42.26

42.26

42.28

42.28

42.28

42.29

42.29

42.30

42.30

42.30

42.31

42.31

42.31

Residual 

head 

(ft)

a 26

.25

.17

.16

.15

.14

.14

.12

.12

.12

.11

.11

.10

10

.10

.09

.09

1/f-

(1/min)

0.800

.750

.667

.521

.461

.436

.422

.413

.375

9AA

9A1

342

.333

.311

305

.300

.294

208

.282

Time 

(mln)

3.67

a TV... _.__ _ -- -

3.87 ____________

4.10 .... 

4.33. 

A JW>

4.58 __ ... .....

4.72

5.17

5.28 . ....

5.45 ....... ... ....

6.10 _______ .. .

6.40

6.83

7.17 .

7.75 __ ..........

8.58. . ____ ... _ .

9.37. ___ .........

10. 12.. _________ ....

11 00

12.5 ..................

13.0 ___ . ______ ...

Depth to 

water 

below measuring point 

(ft)

42.31

42.31

42.32

42.32

42.32

42.33

42.33

42.33

42.34

42.34

42.34

42.35

42.35

42.35

42.36

42.36

43.36

42.37

42.37

42.37

42.37

42.37

Residual 

bead 

(ft)

0.09

.09

.08

.08

.08

.07

.07

.07

.06

.06

.06

.05

.05

.05

.04

.04

.04

.03

.03

.03

.03

.03

1/t- 

(1/min)

0.272

.265

.258

.244

.231

.221

.218

.212

.193

.189

.183

.164

.156

.146

.139

.129

.117

.107

.099

.091

.080

.077
304 GROUND-WATER HYDRAULICS

133d Ml '(s) QV3H 1vnOIS3d
CYCLIC WATER-LEVEL, FLUCTUATIONS AS A BASIS FOR 

DETERMINING AQUIFER TRANSMISSIBILITY'

By JOHN G. FERBIS

ABSTRACT

In an aquifer bounded by a body of tidal water or by a regulated stream, 

the water level responds to changes In stage of the surface-water body. If the 

stage fluctuates as a simple harmonic motion, a train of sinusoidal waves is 

propagated through the ground-water body. Because the amplitude of each 

transmitted wave decreases as the distance from the boundary increases and the 

time lag of a given maximum or minimum increases as the distance from the 

boundary increases, the transmisslbility of the aquifer can be found by either 

the stage-ratio method or the time-lag method. The solution of the formula for 

each of these methods is facilitated by the use of a straight-line plot. For the 

stage-ratio method, the logarithm of the ratio of the ground-water stage in an 

observation well to the surface-water stage Is plotted against the distance of the 

observation well from the boundary. For the tune-lag method, the time between 

a change in surface-water stage and the corresponding maximum or minimum 

ground-water stage in an observation well is plotted against the distance of the 

observation well from the boundary. Both methods can be applied to fluctuations 

that are limited in duration to a single maximum or minimum.

ACKNOWLEDGMENTS

The author is greatly indebted to D. L. Erickson, city engineer and 

director of parks, public property, and improvements for Lincoln, 

Nebr., and to members of his staff for their hearty cooperation and 

effort in making available the data on which this paper is based. The 

basic data were assembled under the direct supervision of H. A. Waite, 

district geologist of the U.S. Geological Survey at Lincoln, Nebr.

CYCLIC FLUCTUATIONS OF WATER LEVELS

In coastal areas, many wells near bodies of tidal water exhibit 

sinusoidal water-level fluctuations in response to periodic changes in 

the tidewater stage. In many inland places, the regulation of a surface 

reservoir similarly produces correlative water-level changes in wells 

that are near either the reservoir or the stream which carries releases 

from the reservoir. As the surface-water stage rises, the head upon the 

subaqueous outcrop of the aquifer increases and thereby either increases the rate of flow into the aquifer or reduces the rate of flow from 

it. The increase in recharge or reduction in discharge results in a 

general rise of the water level in the aquifer. Conversely, a falling 

surface-water stage causes a corresponding decline of the water level 

in the aquifer. When the stage of the surface-water body fluctuates 

as a simple harmonic motion, a train of sinusoidal waves is propagated 

inland through the subaqueous outcrop of the aquifer. With increas-

> The analysis on which this paper is based was made independently and without prior knowledge of the 

similar study made by Werner and Noren (1951).

305
306 GROUND-WATER HYDRAULICS

ing distance from the subaqueous outcrop, the amplitude of each 

transmitted wave decreases and the time lag of a given maximum or 

minimum increases.

If the aquifer has no subaqueous outcrop but is confined by an 

extensive aquiclude, the rise and fall of the surface-water stage changes 

the total weight upon the aquifer. Resultant variations in compresive stress are borne in part by the skeletal mass of the aquifer and in 

part by its confined water. The relative compressibilities of the 

skeletal mass and the confined water determine the ratio of stress 

assignment and the net response of the piezometric surface to the 

surface force.

THE STAGE-RATIO METHOD FOB DETEBMINING THE COEFFICIENT 

OF TRANSMISSEBrLITY

The problem of potential distribution within a semi-infinite solid, 

with the face at *=0, normal to the infinite dimension and subjected 

to periodic variations of potential, was analyzed long ago and the 

solution was used by Angstrom (Carslaw, 1945, p. 41-44) to determine 

the thermal conductivity of various solids. Similar methods of analysis have been used by other investigators to determine the conductivity of the earth, the penetration of diurnal and annual temperature 

waves, and the flow of heat in the walls of a steam-engine cylinder. 

Because the physical nature of these problems is analogous to the problem of an aquifer that crops out under tidewater or a regulated surface 

stream, the solutions of these problems provide a ready pattern for 

evaluating their hydrologic counterparts.

Assume a homogeneous aquifer of uniform thickness and of great 

lateral extent inland from its subaqueous outcrop. Assume also that 

water is released immediately with a decline in pressure at a rate proportional to that decline. As a further simplification, assume that the 

flow is unidimensional and that the full thickness of the aquifer abuts 

the surface-water body that propagates the cyclic fluctuations. In 

those situations where less than the full thickness of the aquifer abuts 

the surface-water body or where the aquifer is under water-table (unconfined) conditions, the analysis will be satisfactory if (1) the observation well is far enough from the subaqueous outcrop that it is unaffected by vertical components of flow and (2) the range of the cyclic 

fluctuation in the observation well is only a small fraction of the saturated thickness of the formation.

The fundamental differential equation for the linear flow of water 

in an aquifer that is incised by a stream can be written as follows 

(Ferns, 1950, p. 286):

&***.?* 

do?
PERMEABILITY, TRANSMISSIBtLITT, AND DRAWDOWN 307

where, in terminology adapted to the given problem,

s=the net rise or decline of ground-water level with reference 

to the mean ground-water level over an observed period,

2= the distance from the subaqueous outcrop to an observation 

well, and

<=the time elapsed from a convenient reference node within 

any cycle.

Let 80 designate the amplitude, or half range, of the fluctuation in 

the surface-water stage. The problem resolves then to finding the particular solution of equation 1 that will satisfy the boundary condition

8=8Q sin ut (2)

at a5=0. The mathematical development leading to the particular 

solution of the differential equation is given in considerable detail by 

Ingersoll, Zobel, and Ingersoll (1948, p. 46-47). Only the final form 

is given below :

_ IuS

s=80e 2T sin

i jj^

(w< x-J^)- (3) 

\ if ^-*/

If, in accord with Jacob (1950, p. 365), the period of the uniform 

tide or stage is designated by £0, then < > can be expressed in radians 

per time unit as 2r/£0 and equation 3 becomes

8=8tle sin /2r* S I - z-i/T

\ tQ V t02

_

Equation 4 defines a wave motion where amplitude, s^e * UT) decreases rapidly with distance x. When the aquifer response is due to 

loading rather than to change in head, at the subaqueous outcrop, tiie 

amplitude factor should be reduced by the ratio a/ (a +00), where 

a is the vertical compressibility of the skeletal aquifer, ft is the compressibility of the water, and 0 is the porosity of the skeletal aquifer 

(Jacob, 1950, p. 356). Values of the compressibility of water can be 

obtained from published tables of physical data but information for 

estimation of the vertical compressibility of the skeletal aquifer is 

scanty. This factor may vary considerably, but its magnitude may 

be estimated from values which have been established by the aquifertest method for similar aquifers.

From equation 4, the range of the ground- water fluctuation at an 

observation well at distance x from the subaqueous outcrop is given by

(5)
308 GROUND-WATER HYDRAULICS

This equation indicates that the slower the fluctuation of the surface 

tide that is, the greater the value of t0 the greater the range of 

water-level fluctuation within the aquifer.

If the coefficient of transmissibility is measured in units of gallons 

per day per foot, equation 5 becomes

Sf=2s0e **', (6) 

from which

or

Therefore,

2.1 ^/^-log..(.,/2».). (8)

The logarithmic quantity (*r/2s0 ) is, in effect, the ratio of the range 

of the ground-water stage to the range of the surface-water stage. 

The form of equation 8 suggests the use of a semilogarithmic plot of 

the logarithm of that ratio against the distance a? for each observation 

well; the right-hand member of the equation represents the slope of 

this plot, and, if the change in the logarithm of the ratio of the two 

ranges is selected over one log cycle, the numerator of the expression 

for the slope reduces to unity. Thus, equation 8 becomes

A more convenient form of this equation, obtained by removing the 

radical, is

S

(Ax)1 

or
PERMEABILITY, TRANSMISSIBILTTY, AND DRAWDOWN 309

To evaluate the coefficient of transniissibility, 7*, from equation 10, 

the coefficient of storage, $, must be known. However, reasonable 

estimates of S can be made if the aquifer is known to be locally artesian or nonartesian; generally this can be determined from studies of 

well logs and water-level records.

THE TIME-LAG METHOD FOB DETERMINING THE COEFFICIENT 

OF TBANSMISSIBrLITY

Let £1 denote the lag in time of occurrence of a given maximum or 

minimum ground-water stage following the occurrence of a similar 

surface-water stage. Then, from Ingersoll, Zobel, and Ingersoll 

(1948, p. 48), the expression for tj. is

or

" l 4*T 

from which

(12)

If T is expressed in gallons per day per foot and to and ti are expressed 

in days, equation 12 becomes

which is solved readily from an arithmetic plot of corresponding 

values of x and t^.

OTHER BELATED FORMULAS

The apparent velocity of the transmission of a wave through an 

aquifer is

This equation indicates the apparent velocity of a given maximum 

or minimum and does not pertain either to the rate of pressure transmission (Muskat, 1937, p. 669) or the apparent rate of pressure transmission (Jacob, 1940, p. 585) within the aquifer. 

The wave length of the fluctuation is

(15)
310 GROUND-WATER HYDRAULICS

Physically, there is little opportunity in ground-water hydrology to 

obtain a snapshot view of a sinusoidal wave train, as would be required 

if equation 15 were to be used.

During half the cycle, water flows into the aquifer through its subaqueous outcrop; in the other half, it flows out again. The quantity 

of flow per half cycle is determined with the aid of Darcy's law, 

where

dz

' '-W^)l' (16)

as given by Ingersoll, Zobel, and Ingersoll (1948, p. 49), and Z=the 

length of the subaqueous outcrop through which the flow occurs.

It is convenient to set up the integral for the quantity of flow per 

unit length of subaqueous outcrop. Because the gradient 6s/6x is 

not in phase with #, the limits of integration are determined by noting 

from inspection of equation 16 that when a?=0 the gradient is zero at 

t = w/4o> = £0/8, reaches a minimum at t=»/4o>=£0/8, and returns to 

zero at £=3ir/4<i>=3£o/8. Therefore,

t^ dt, (17) 

, xOx/s.o

8

or

3w * 

4»

from which

POSSIBLE ADDITIONAL USE AND EXTENSION OF THE METHODS 

FOB ANALYSIS OF CYCLIC FLUCTUATIONS

Although greatest use of the described methods for analyzing cyclic 

fluctuations of ground-water levels will be in areas of tidal streams 

and seas or near regulated streams and lakes, Rambaut (1901) showed 

that these methods can also be applied with fair results to variations 

that resemble periodic motion but are limited in duration to a single 

maximum or minimum. Thus, the response of an aquifer to the passage of a flood crest in a stream to which the aquifer is hydraulically 

connected may lend itself to this analysis. A logical extension from
PERMEABILITY, TRANSMISSIBILITT, AND DRAWDOWN 311

this generalized problem of a simple sinusoidal motion would be to 

study the applicability of the unit functions or delta functions of 

electrical-network analysis to the response of aquifers to complex 

patterns of recharge from precipitation or to the response of a stream 

to various rainfall-runoff patterns.

A FIELD PROBLEM INVOLVING CYCLIC FLUCTUATIONS

To illustrate the applicability of these methods to field problems, 

data are presented for three riverside observation wells at the Ashland well field of the municipally owned water supply of Lincoln, 

Nebr. (see fig. 91). The three observation wells are equipped with 

water-level recording gages, as is also the gaging'Station on the Platte 

River at the crossing of U.S. Highway 6. An east-west geologic section through supply well 2 is shown in figure 92. Typical records 

from the autographic charts for the river-stage recorder and observation well 1 are reproduced in figure 93.

Observation wells 1,2, and 3 are 42,106, and 252 feet, respectively, 

from the riveredge when the river is at normal stage. Each well is 

screened and taps only the upper part of the aquifer. From records 

obtained during the period September 23-29, 1950, the ratio of 

ground-water fluctuation to change in river stage was computed for 

the rising and falling limb of each cycle. These ratios are listed in 

table 8. The length of the period of the river fluctuation, computed 

for both limbs of each cycle, ranged from 20.5 to 31.0 hours and 

averaged 24 hours, or 1 day.

TABLE 8. Ratio of the range in water-level fluctuation in observation wells 1, g, 

and S to the corresponding range in stage of the Platte River at the U.S. Highway 6 

bridge

ftagmmt of cnrvn of cyrHc fluctuations >

Rising stage 1-2... ..... ... ......... ...........

Falling stAgft 2-3

Rising stage 3-4__ ___ _ ____ _ __ _ ______ _ __ _

Falling stage 4-5.. ... ... ...... ... ....... ... ...

Rising stage 5-6__ ______ __ __

Falling stage 6-7 ____ _ __ _ __._ __ ___

Rising stage 7-8. ___ _ _ ___ ____ __ _ __

Falling stage 8-9 __ ______ __ ____ _

Rising stage 9-10. ... _ _ __ _ ___ ____ ___

Falling stage 10-11_,___ _ __ _ _ _ _ ____

Rising stage 11-12. __ _____ _ ______ __ __ ___

Average of rising stages. _____ _______

Average of falling stages -__.. _ . ._ __

Average of rising and falling stages ._ _ .. .__ _

Ol

1

0.73

.71

.77

.76

.74

.73

.69

.71

.72

.68

.71

0.73

.72

.72

Mffvatkmwi

2

0.52

.56

.54

.56

.59

.56

.47

.56

.52

.51

.53

0.53

.55

.54

afl

3

0.35

.46

.31

.29

.26

.29

.28

.20

.33

.37

. 14

0.28

.32

.30

1 Numbers correspond to those in figure 93.
R. 9 E. 

A 6

23

EXPLANATION

o

Supply well

 

Observation well

O

Test hole

Stream-gaging station

0_______2000______4000 FEET

14

FIOUBE 91. Map of the Ashland well field of the municipally owned water s
VERTICAL 

SCALE A 

FEET 

rSO

-60

 *>i.*«. ° » .'T

FIGURE 92. Generalized east-west geologic section through supply well 2 in the Ashland well field of the mun
314 GROUND-WATER HYDRAULICS

2.30

24 25 26 27 

SEPTEMBER 1950

28 29

FIGURE 03. Qraphs showing the stage of the Platte River and tbe water level in observation well 1 in the 

Ashland well field of the municipally owned water supply of Lincoln, Nebr.

The averages of the ratios for rising and falling stages (table 8) 

are plotted in figure 94 against distances from the riveredge. If, as 

indicated in figure 94 for one log cycle, Aa5=560, and if t0 1 day 

equation 10 becomes

g=1)400)000g.

Then, if values appropriate for a water-table aquifer are substituted 

for 5,

T= 140,000 if 5=0.10, 

r=210,000 if 5=0.15, 

T= 280,000 if 5=0.20, and 

r=350,000 if 5=0.25.
600

500

a AOO

300

200

o 

§100

-100

0.1 0.2

Well 3

Distance from riveredg

outcrop of a

0.3 0.4 0.5 

RANGE RATIO (,-'-)

FIGURE 94. Semllog graph of the average ratio of rising and falling stages of the Platte Rive

observation wells from the riveredge.
316 GROUND-WATER HYDRAULICS

At the subaqueous outcrop, where aj=0, the range of the water-level 

response in the aquifer, *r, is equal to the range in stage of the river, 

2*o; therefore, *r/2*0 approaches unity as a? approaches zero. Thus, in 

figure 94, the negative value of a? at the range ratio of unity represents 

the effective distance from the riveredge to the subaqueous outcrop.

Although the rate of withdrawal from the several nearby municipal-supply wells is relatively steady, minor changes in rate and distribution of pumping do occur. Such changes are more likely to 

affect the times the maxima or minima occur than the ratio of the 

changes in stage. Furthermore, the times of occurrence of maximum 

and minimum water levels cannot be determined as precisely as the 

ranges in stage because the time scale on the recorder charts is compressed to a greater degree than is the gage-height scale. Also, differences in the effective screen resistance of the observation wells would 

tend to distort observations of the timing of maximum and minimum 

water levels. The wide range in the observed lag of maxima and 

minima, as shown in table 9, may result from any one or a combination 

of several of the aforementioned causes.

TABLE 9. Time lag, in hours, between the minimum and maximum stages of the 

Platte River at the U.S. Highway 6 bridge and the corresponding minimum and 

maximum water levels in observation wells 1, 8, and 3

Point on corves of cyclic fluctuations '

Minimum !________. ._.._________ ______________

MaTimum 2

Minimum 3_-________________._______________-

Maximum4.... ...... .. .. ......_....___ ... ......

Minimum 5. __-_____..____________-___.._.____

Maximum 6_______ _____ _ __ _ ______ _ _ Minimum 7__ __ ____________ ___ ____________

Maximum 8__ ______ __ __ _________ __.__ ____

Minimum 9_. ___._...___._______..__._._______

Maximum 10______ ___ _ _ __ __ _____ ___

Minimum 11___________ _______ __ _______

Maximum 12____________. _________.____.._..._

Average of the minima

Average of the maxima____ ____________________

Observation well

1

1.25 

2.50 

2.00 

2.25 

1.75 

2.25 

1.50 

2.00 

2.50 

2.75 

2.25 

2.50

1.90 

2.40 

2. 1

2

3.75 

3. 50 

4.00 

2.75 

3.75 

3.25 

4.00 

2. 50 

4.00 

2. 25 

3. 75 

2.50

3.90 

2.80 

3.3

3

6.00 

7. 50 

6.75 

6.75 

5.75 

6.50 

5.50 

7.00 

6.75 

6.75 

5.50

6.70 

6.00 

6.3

i Numbers correspond to those in figure 03.

The average values of the time lag, £1, are plotted for each well in 

figure 95. The slope of the line through these plotted values is x/ti, 

which appears with a square exponent in equation 13. Substitution 

iri equation 13 of the coordinates of the slope of the line in figure 95 

gives

r0.60X(250) 2X11o ofinnm o T==l (5/24? J ^=860,000 s>
PERMEABILITY, TRANSMISSIBILITY, AND DRAWDOWN 317

from which

T= 86,000 if £=0.10, 

^=130,000 if #=0.15, 

T=170,000 if #=0.20, and 

^=220,000 if #=0.25.

At the subaqueous outcrop, where a?=0, the time lag, £i=0. Thus 

on figure 95, the negative value of x at the t^ Q axis is the effective 

distance from the riveredge to the subaqueous outcrop.

300

250

200

2 150

S

o

100

»

-50

Well 1

S

Well 2

g

 Distance from riveredge to subaqueous 

outcrop of aquifer

= 5 hours

We I 3

0123456 

TIME LAG (/j), IN HOURS

FIGURE 95. Graph of the time lag between maximum and minimum stages of the Platte River and corresponding water levels in observation wells plotted against the distance of the wells from the riveredge.

690-185 O -6$ 6
318 GROUND-WATER HYDRAULICS

The large difference between the coefficients of transmissibility 

obtained by the stage-ratio method and the time-lag method may be 

due to the influence of the nearby pumping on the gage height and 

on the time of each maximum or minimum. The values obtained by 

both methods and the resultant averages are summarized in table 10.

TABLE 10. Summary of determinations of the coefficient of transmissibility by the 

stage-ratio and the time-lag methods

Coefficient of storage, 5

0.10. _._..__..._.__....__._.__....

0.15. __.. _ ....___._...._.. _ ...

0.20. ___.__.___..___._.__._....__.

0.25. _-._._.__..._.__._..___._...

Coefficient of transmissibility, T (gpd per ft)

Method

Stage-ratio

140, 000 

210, 000 

280, 000 

350, 000

Time-lag

86,000 

130,000 

170, 000 

220,000

Average

110,000 

170, 000 

220, 000 

280, 000

If the time scale on the water-stage recorders were to be expanded 

through use of daily instead of weekly time gears, the computation 

of the coefficient of transmissibility could be refined appreciably. 

Also, if the computations were to be based on data obtained during 

a period when withdrawal from the nearby supply wells was held 

constant, the values of the coefficient of transmissibility would be 

more nearly consistent. A more adequate test of these methods might 

be possible if the observation wells were located along a line at right 

angles to the edge of the river and in an area remote from heavy 

pumping.

The average thickness of the water-bearing deposits in the Ashland 

well field area is about 70 feet. If this value for thickness and the 

average values for T in table 10 are used, the coefficient of permeability, in gallons per day per square foot, would be 1,600 for £=0.10, 

2,400 for £=0.15, 3^00 for £=0.20, and 4,000 for £=0.25. The coefficient of permeability for £=0.15 is almost the same as the value, 

2,200 gpd per square ft., which was computed in the City Engineer's 

office from gradient studies based on water-table contour maps.
DRAWDOWNS RESULTING FROM CYCLIC RATES OF

DISCHARGE

By CHARLES V. THEIS

ABSTRACT

Many large-supply wells are pumped at rates that change periodically according to a cyclic pattern. The drawdown at a given distance from such a well at 

the end of any period of cyclic pumping can be determined from a formula 

derived from the Theis nonequilibrium equation. A graph of the drawdowns 

at the end of successive periods of cyclic pumping is a series of hysteresis loops, 

the spacing of which indicates the prevailing relationship between the recharge 

to and the discharge from the aquifer.

THE DRAWDOWN EFFECTS OF CYCLIC PUMPING

Because the demand for water fluctuates with the season, the rate 

of ground-water withdrawals for municipal supply, irrigation, and 

cooling purposes generally follows a cyclic pattern. In reports on 

ground-water supplies in the vicinity of Atlantic City, N.J., Thompson (1928) and Barksdale, Sundstrom, and Brunstein (1936) describe 

the water-level response to changes in average daily pumpage. Their 

graphs showing the relation between depth to water and average daily 

pumpage are characterized by hysteresis loops that indicate a progressive net lowering of the water level each year. The study of 

such field problems should be facilitated by the following analysis of 

a theoretical problem involving cyclic pumping.

A FORMULA FOR DRAWDOWNS CAUSED BY CYCLIC PUMPING

Consider a well that taps an artesian aquifer of infinite areal extent. 

Assume that the coefficient of transmissibility, T7, of the aquifer is 

20,000 gpd per ft and that the coefficient of storage, 8, is 0.0003. 

Also assume that the annual schedule of discharge from the well is 

200 gpm during the first quarter, 400 gpm during the second quarter, 

600 gpm during the third quarter, and 400 gpm during the fourth 

quarter. Compute the drawdown in an observation well 2,000 feet 

from the discharging well.

The effects of the different rates of drawdown are the same as if 

on January 1 a well begins to discharge at a rate of 200 gpm, on 

April 1 a second well at the same place begins to discharge at the same 

rate, on July 1 a third well at the same place begins to discharge at the 

same rate, on October 1 a fourth well at the same place begins to recharge at a rate of 200 gpm, on January 1 of the next year a fifth well 

at the same place begins to recharge at the same rate, on April 1 a sixth 

wel1 at the same place begins to discharge at the rate of 200 gpm, and

319
320 GROUND-WATER HYDRAULICS

so on. The drawdown in the observation well at any time will be the 

sum of the effects of all these hypothetical wells.

The drawdown in the observation well due to any one of the hypothetical wells is given by Theis (1935) as

where

_1.87r*£ 

U~ Tt '

Consequently, if t is expressed in quarter years instead of days,

= 1.87X4,000,OOOX0.0003=0.00125 

U~~ 20,000X91* ~~ t

When u is small, as it is when t is greater than 1, the following approximate expression is very nearly correct :

W(u}=-0.577-2.3 log u=-2.3(0.251+log u). 

Therefore, for the observation well,

__ 114.6X200X2.3 F . /0.00125\"| 

20,000 L \ * / J 

= -2.64(0.251+0.0969-3-log t) 

=2.64(2.6521+log t),

which is the water-level drawdown or buildup caused by one of the 

hypothetical wells when t is expressed in quarter years since the real 

well began discharging.

At the end of the first quarter, inasmuch as only one well has been 

discharging, the total drawdown is

«i=2.64 (2.6521+log 1).

At the end of the second quarter, because one of the 200-gallon wells 

has discharged for two quarters and one for one quarter, the total 

drawdown is

*2 =2.64(2.6521+log 2+2.6521+log 1) 

=2.64(2X2.6521+log2+logl).

At the end of the third quarter

sa=2.64(3X2.6521+log 3+log 2+log 1), 

and at the end of the fourth quarter

S4=2.64(3 X 2.6521-2.6521+log 4+log 3+log 2-log 1).

The minus signs in the expression for drawdown at the end of the 

fourth quarter result from the fact that the well introduced at the 

beginning of that quarter was a recharging well. Similarly, at the 

end of the fifth quarter
PERMEABILITY, TRANSMISSIBILITY, AND DRAWDOWN 321

s6=2.64(3X2.6521-2X2.6521+log 5+log 4+log 3 log 2 log 1), 

and at the end of the sixth quarter

s6=2.64(4X 2.6521 -2X 2.6521 +log 6+log 5+log 4-log 3-log 2+log 1)

The quantity multiplied by 2.64 is seen to consist of a multiple of 

the constant 2.6521 and a log term. By inspection it is seen that the 

constant terms for the respective quarters of any year are: first 

quarter, 2.6521; second quarter, 5.3042; third quarter, 7.9563; and 

fourth quarter, 5.3042. If n is the number of a quarter, the log term 

corresponding to that quarter is

1 pa (n-1) (n-2) (n-5) (n-6) (n-9) .. 

g L («-3) (n-4) (n-7) (n-8) . . 

which is a series that continues until the last factor is 1. Thus the 

drawdowns for successive quarters are as follows:

st =2.64(2.65+log 1) = 7.0

st =2.64(5.30+log 2-1) = 14.8

83 =2.64(7.96+log 3-2-1) =23.1

st =2.64[5.30+log (4-3-2)/I]=17.6

8» =2.64[2.65+log (5-4-3)/(2-l)] = 10.9

st =2.64[5.30+log (6-5-4-1)/(3-2)]= 17.4

s7 =2.64[7.96+log (7-6-5-2-l)/(4-3)] = 25.1

s8 =2.6415.30+log (8-7.6.3.2)/(5-4.1)]=19.3

8, =2.64[2.65+log (9.8.7.4.3)/(6-5-2-l-)]=12.3

s,0 =2.64[5.30+log (10-9.8-5-4-l)/(7-6-3-2)]=18.6

s,,=2.64[7.96+log (ll-10-9-6-5.2.1)/(8-7.4-3-)]=26.2

s12 =2.64[5.30+log (12-ll-10-7.6-3-2)/(9-8.5-4.1)] = 20.2

Si3 =2.64(2.65+log 214) = 13.1

s,4 =2.64(5.30+log 113-5) = 19.4

sis=2.64(7.96+log 167) =26.9

s,,=2.64(5.30+log 413) =20.9

s,7=2.64(2.65+log 369) = 13.8

s,8=2.64(5.30+log 189) =20.0

Si,=2.64(7.96+log 269) =27.4

s»=2.64(5.30+log 650) =21.4

s«=2.64(2.65+log 566) = 14.3

sM =2.64(5.30+log 284) =20.5

s»=2.64(7.96+log 395) =27.9

sM =2.64(5.30+log 940) =21.8

»«=2.64(2.65+log 809) = 14.7

8M =2.64(5.30+log 399) =20.9

Sj7=2.64(7.96+log 545) =28.3

828=2.64(5.30+log 1,280) =22.2

s»=2.64(2.65+log 1,092) = 15.0

8ao=2.64(5.30+log 533) =21.2

83,=2.64(7.96+log 720) = 28.6

8a=2.64(5.30+log 1,675) =22.5
322 GROUND-WATER HYDRAULICS

If these drawdowns are plotted against the discharge rate, the line 

connecting successive points forms a series of hysteresis loops (see 

fig. 96). As may be seen from this graph, the successive points for

12

15

18

21

24

27

? §600-

UJ ^

iS0 0- 200-

il °J 

^

/t_4

2d quart*'

f 

_>

^ ls» q«»rtw -

Diagrammatic sketch

» 

T

^

of

Mquorfer

*h quarter 7 -i>

discharge cycle

Numbers beside points indicate 

total elapsed quarter years 

since pumping first began. 

Descending limbs of curves 

are full lines; ascending 

limbs are dashed

Diagram based on the assumptions:

Elastic artesian aquifer

of infinite area! extent

T= 20,000 gpd per ft

5=0.0003

Discharge cycle (see sketch)

1st quarter of each year, 2OO gpm 

2d " 4OO 

3d " 6OO 

4th " 4OO

200 400 600

PUMPING RATE DURING LAST QUARTER YEAR OF TOTAL PUMPING 

PERIOD INDICATED, IN GALLONS PER MINUTE

FIOUHS 96. Oraph showing tbe drawdown in an observation well 2,000 feet from a wen pumped at a

cyclically varying rate.
PERMEABILITY, TRANSMISSIBILITY, AND DRAWDOWN 323

the same rate of discharge fall closer and closer together. This is 

the expected situation if natural recharge continues at the same rate 

and natural discharge is allowed to continue unrestricted. However, 

if the natural discharge could be stopped or if the natural recharge 

could be increased, the loops for successive cycles eventually would 

nearly coincide. Conversely, if discharge from the aquifer were to 

be increased by artificial withdrawals and no recharge were to occur, 

the successive points for the same pumping rate would tend to become 

equally distant. Thus, a graphic representation of drawdowns resulting from cyclic pumping is indicative of the prevailing rechargedischarge regimen of the aquifer.
DRAWDOWNS RESTTI/TING FROM CTCLIC INTERVALS OF

DISCHARGE

By RUSSELL H. BBOWN

ABSTRACT

The water-level drawdown of a cyclically pumped well at the beginning of any 

period of pumping (or at the end of any period of recovery) can be determined 

from the formula developed in this paper. The drawdown can be determined 

arithmetically if the pumping period equals the period of recovery or if the 

number of complete cycles is less than 10; otherwise, the formula is more easily 

solved by the use of a straight-line semilogarithmic plot.

CYCLIC WITHDRAWALS OF GROUND WATER

In most areas where water for irrigation or air conditioning is obtained from wells, ground-water withdrawals for these purposes are 

cyclic that is, periods of pumping alternate with periods of nonpumping, or recovery. If the period of recovery in each cycle is 

sufficiently long, the water level in the well returns to its original 

position. Thus successive cycles of withdrawal and recovery will not 

result in a net lowering of the water level in the well. On the other 

hand, if the period of recovery in each cycle is not long enough for 

the water level in the well to return to its original position, successive 

cycles of withdrawal and recovery will result in a water level that is 

lower at the end of each cycle than that at the end of the previous 

cycle. Provided the cycles follow a regular pattern, the position of 

the water level at the beginning of any period of discharge (or at the 

end of any period of recovery) can be determined through use of the 

formula developed in this paper.

FORMULA FOR DETERMINATION OF DRAWDOWN IN A CYCLICALLY

PUMPED WELL

Consider a well that is pumped at rate Q in virtually definite cycles 

of a given pumping period followed by a given period of recovery. 

The hydrograph for such a well would resemble that shown in figure 

97. The numbers 1,2, and 3 in figure 97 identify the first, second, and 

third cycles, each of which consists of a period of withdrawal followed by a period of recovery. Points a, &, and c designate the beginning" of the first, second, and third periods of ground-water withdrawal and points d, e, and / designate the end of the corresponding 

periods of withdrawal (or the beginning of the recovery periods). 

The hydrograph is identical to that which would characterize a well 

that had discharged continuously at rate Q throughout the three-cycle 

period if, at precisely the same place and at the same rate Q, a second

324
PERMEABILITY, TRANSMISSIBILTTY, AND DRAWDOWN 325

Effect of first 

discharging well

FIGURE 97. Hydrograph for a cyclically pumped well showing the symbolism used for time factors.

well had begun recharging at time d, a third well had begun discharging at time 6, a fourth well had begun recharging at time e, a fifth well 

had begun discharging at time <?, and a sixth well had begun recharging at time /. Thus the net drawdown in the real pumped well at the 

end of the third cycle can be expressed as follows:

S «r «r, (1)

where the numbered ^-subscripts refer to the discharging wells during 

the first, second, and third cycles, respectively, and the numbered 

r-subscripts refer to the respective recharging wells. Inasmuch as 

the net drawdown is to be computed for the pumped well (that is, 

the radius r in the basic Theis equation is small), the approximate 

form of the Theis (1935, p. 522) equation can be used for values of 

time that are not too small. Also, if the component drawdown s^ 

paired with the component buildup, or negative drawdown, sri> the 

expression for their algebraic sum is identical with the Theis recovery 

formula. 

Thus,

264Q, *! S^ «rj= y-lOg p>

_ =264g 

. _264Q

tog 7^

lz

(2)

(3)

(4)
326 GROUND-WATER HYDRAULICS

Equation 1, which is the sum of equations 2, 3, and 4, can therefore 

be written as

"3 rrt "

The form of this equation suggests a simple expression for the drawdown after any number of cycles. Thus, the drawdown in the pumped 

well immediately after completion of the nth cycle and immediately 

before the beginning of the (n-f 1) pumping period is

From figure 97, each cycle is seen to span a time interval of length 

t. Furthermore, within each cycle, the recovery interval spans a 

time interval of length tr and pumping occurs over a time span />£, 

where p represents the pumping fraction of the cycle (that is, the 

ratio of pumping time to the period of the cycle, t) . Therefore, each 

time factor appearing in equation 6 can be replaced by an equivalent 

factor written in terms of p and t.

The time factors for the discharging wells can be replaced by multiples of t; figure 97 shows that, for the nth cycle, tn can be replaced 

by t- for the last two cycles, £n_i can be replaced by 2£; for the last 

three cycles, tn-z can be replaced by 3£; and so on through the first 

cycle, for which £t can be replaced by nt.

In similar fashion, substitutions can be made for the time factors 

related to the recharging wells. Thus, for the nth cycle, tn' can be 

replaced by (t pt) £ _/, for the last two cycles, can be replaced by 

(2£ pt) tn-z', for the last three cycles, can be replaced by (3£ pt) ; 

and so on through the first cycle for which £/ can be replaced by the 

factor (nt pt). Now equation 6 can be rewritten as follows:

Cancelling the t in each factor in the numerator with the t in the 

corresponding factor in the denominator and reversing the order of 

the two series gives the relation

-n "|

-p)]' -p) . . . (n

Therefore, the length of the cycle and the total length of time are 

seen to have no significance; the drawdown is determined by the 

number of cycles and not by whether those cycles are measured in 

minutes, days, years, or any other time unit.
PERMEABILITY, TRANSMISSIBILTTY, AND DRAWDOWN 

Equation 8 can also be written in the form

327

(9)

A semilog plot of values of snT/Z§±Q on the arithmetic scale against 

values of n on the logarithmic scale for various values of p (see fig. 

98) facilitates the solution of the equation. The change in the factor

2.8

2.4

2.0

1.6

1.2

0.8

0.4

0.0

10

p=0.25

20 40 60 80 100

FIGURE 98. Semilog plot of the factor *.T/264Q against the number of pumping cycles n for selected

values of p.
328 GROUND-WATER HYDRAULICS

sn7'/264<2 during the nth cycle is represented by the last term of the 

series expression,

n-p

On the semilog plot, the increment on the n-axis during the nth cycle 

(that is, between points n 1 and n) is log n log(n 1), or

Therefore, the limiting slope of this semilog plot the slope across 

the nth cycle is given by the change in the factor sn7V2640 divided 

by the change in log n. In mathematical terms, this limiting slope for 

the nth cycle is

2-3 =r~2 (^=r) +3 (^=i) " J

The series expansions for the two log terms can be found in most 

comprehensive handbooks of chemistry, mathematics, or physics. 

(See, for example, Hodgman, 1952, p. 274.) The expansions generally 

are given for the natural logarithm but are readily converted to the 

common logarithm by multiplying by 2.3.

If the two series expressions shown in equation 10 are multiplied 

by (n 1) , then the ratio becomes

P

n-l

As n approaches infinity, the value of this fraction approaches the 

quantity p. In other words, p is the limiting slope of the semilog plot 

previously described. Thus, a means for handily resolving equation 8 

begins to emerge.
PERMEABILITY, TRANSMISSIBILITY, AND DRAWDOWN 329

Note that equation 8 can be rewritten in the form

(11)

The right half of this equation can be evaluated for the end of any 

number of cycles, w, and for the value p. For example, if p=Q.f5 and 

n=100, then

^gL=log 100!-(log 0.25+log 1.25+log2.25+. . .+log 99.25)=2.060.

Computations of this nature have been made for selected values of 

n and p. The results of these computations are given below and are 

plotted in figure 98.

Number of 

cycles (n)

10-.----

20--_--_

30------

40----.-

50.-.-..

60.--__-

70-----.

80.-..--

90-__-._

100..--.

Valueof»n77264Q

p=0.25

0 342 

416 

459 

490 

514 

534 

550 

565 

577 

589

p=0.50

0.754 

.902 

. 989 

1.051 

1. 100 

1. 139 

1. 172 

1. 201 

1. 226 

1.249

p=0.75

1.313 

1.537 

1. 669 

1.762 

1.834 

1.894 

1.946 

1.987 

2.025 

2.060

Observe that within the limits of plotting accuracy each curve shown 

is a straight line over the log cycle from n= 10 to n=lQQ. Furthermore, the slope of each of these curves is only a few tenths of one percent less than the limiting slope p. General rules of procedure for 

the solution of practical field problems can now be stated.

When n is less than 10, solve equation 11 numerically, computing 

the factor s»77264$ in the manner indicated by the example. When n 

is more than 10 but less than 100, solve equation 11 graphically by use 

of figure 98. The family of curves shown in the figure can be expanded easily to include any other value of p merely by computing the 

factor snT/264Q for w=10 and n=100, plotting the two computed 

values, and joining them by a straight line.

When n is greater than 100, find the value of s100 7V264$ by computation or from figure 98. The value for the end of the nth cycle 

can then be computed by using the relation

SnT

264Q"

n

(12)
330 GROUND-WATER HYDRAULICS

Although it is an approximation, equation 12 gives values of 

that are only a few tenths of one percent too high. Because equation 

12 is based on the idea of extending the lines in figure 98 beyond the 

100th cycle, values of 8nT/^AQ for values of n greater than 100 can be 

determined without computation by extending the lines in figure 98 an 

appropriate distance beyond the n=100 ordinate.

When the pumping and recovery periods are of equal length that 

is, when p=0.5 equation 9 can be simplified for an easier solution 

as follows :

1.2-3.4-5- ...... -n "I

-l*-2*-3H*. - . -<n-i) J

, - .4-6-8- 

g . 2n)

In any of the above methods, after the value of the log term is 

determined, the drawdown, sn. of the pumped well at the end of the nth 

cycle is readily found by multiplying the value of the log term by 

264^/T7.
ESTIMATING THE TRANSMISSIBILITY OP AQUIFERS 

FROM THE SPECIFIC CAPACITY OF WELLS

By CHARLES V. THEIS, RUSSELL H. BROWN, and REX R. METER

ABSTRACT

The specific capacity of a well can be used as a basis for estimating the 

coefficient of transmissibility of the aquifer tapped by the well. From assumed values for the hydrolojric constants of the aquifer, separate formulas 

including a term for specific capacity are develoi>ed for the transmissibility 

of water-table and artesian aquifers. Prom a chart relating the well diameter, 

the specific capacity of the well, and the coefficients of transmissibility and 

storage, the transmissibility of the aquifer can be estimated from the known 

specific capacity of the well or the specific capacity of the well can be estimated 

from the known transmissibility of the aquifer. These methods are subject to 

limitations but are useful means of approximation.

THE GENERAL RELATIONSHIP BETWEEN TRANSMISSIBILITY AND

SPECIFIC CAPACITY

In many ground-water investigations, especially those of a reconnaissance type, the specific capacities of wells provide the only basis 

for estimating the transmissibility of the aquifers tapped by the 

wells. Generally speaking, high specific capacities indicate an aquifer 

having a high coefficient of transmissibility, T7, and low specific capacities indicate an aquifer having a low T. However, a precise correlation between the specific capacities of wells and the T values of 

the aquifers they tap has not yet been established.

The specific capacity of a well cannot be an exact criterion of T 

in the vicinity of the well because, obviously, the yield of the well per 

foot of drawdown is also a function of other factors such as the diameter of the well, the depth to which the well extends into the aquifer, 

the type and amount of perforation in the well casing, and the extent 

to which the well has been developed. However, estimates of T that 

are based on the specific capacities of wells should be reasonably reliable and could be made without the elaborate tests necessary for 

precise determinations. Therefore, if developed within the limits of 

idealized assumptions, a formula expressing the theoretically exact 

relationship between the specific capacity of a well and the transmissibility of the aquifer which the well taps would be highly useful 

n the making of reconnaissance ground-water studies provided the 

Jieoretical formula is empirically modified for prevailing field 

conditions.

331
332 GROUND-WATER HYDRAULICS

ESTIMATING THE TRANSMLSSIBILIT Y OF A WATER-TABLE AQUIFER 

FROM THE SPECIFIC CAPACITY OF A WELL

By CHARLES V. THEIS

The relation between the discharge of a well and the water-level 

drawdown a short distance from the well is given by an equation 

derived by Theis (1935). The value of u in that equation is small 

provided r is small, T and S are within the range of values for fairly 

productive aquifers, and t is at least several hours. For the purpose 

of this paper, the Theis formula can be written with negligible error 

as follows :

(i)

The computation can be made somewhat simpler by substituting 

values for S and T that are within the range of fairly productive 

water-table aquifers. However, if corrections for these values are 

included, the formula remains general. Thus, if T 1,000 gpd per 

ft, $=0.2, and t=l day, the formula for an average water-table 

aquifer corrected for variations of that aquifer from average is

(3 w 10-,) _logi

Therefore,

s s s

[-loglo (3.74f* 10- 6)-log10 5S+log10 t]. 

Let

then

< 10-6) _logio 5S+loglo t]

s s

=2 [-66-264 log,0 (3.74r2 - 10~6)-264 Iog10 5S+264 Iog10 *]. 

s
Let 

then

PERMEABILITY, TRANSMISSIBILITY, AND DRAWDOWN 333

#=-66-264 Iog10 (3.74T-2 - 10' 6) (3)

'= CK-264 log,0 5S+264 Iog10 0- (4)

s

Values of K, computed for selected values of r, are as follows:

r (ft)

0. 25

. 50

1.0

5. 0

10

K

1,684

1,524

1,367

996

838

r(ft)

20

30

40

50

K

680

588

521

469

The foregoing formulas indicate the importance of both the storage coefficient and the duration of pumping when the coefficient of 

transmissibility is estimated from a single measurement of drawdown 

in an observation well. If $=0.2, the influence of the S term is zero 

because the formula was derived on that basis. However, if $=0.1, 

the S term would equal 264 Iog10 5$= -264 Iog10 0.5=80, or about 

8 percent of the constant, K, for r=5 .feet, and if $=0.3, the S term 

would equal 45, or about 4.5 percent of the same value for K. 

Provided S is known, the correction can be made, but if S is unknown, 

the error for a water-table aquifer (for which S ranges from 0.1 to 

0.3) probably will be smaller than the errors inherent in the method. 

Although the correction for the duration of pumping also is comparatively small, it presumably should .be made if, as in many cases, 

the duration is known. For an artesian aquifer, S is very small and 

the S term correction will be large, making it inadvisable to apply 

the formula (in its present form) for artesian conditions; for if 

$=0.001, the S term would be about double K for r=5 feet.

The coefficient of transmissibility cannot be determined explicitly 

from the computed values of T'. However, from charts giving the 

values of Tf for various values of T and Q/s, the value of T can be 

ascertained from known values of T' and Q/s. Such a chart is shown 

in figure 99.

Thus, within the limits of the idealized assumptions, the coefficient 

of transmissibility of a water-table aquifer apparently can be computed without great error from a single measurement of drawdown 

in an observation well that is a short distance from a pumped well, 

even if the coefficient of storage is not known. However, the informa690-185 o es 7
334 GROUND-WATER HYDRAULICS

tion generally available concerns the specific capacity of the pumped 

well. In the foregoing formulas Q represents the discharge of the 

pumped well and * is the drawdown in a nearby observation well at 

a distance r from the pumped well. Obviously, the drawdown in the 

pumped well bears a relationship to the drawdown a short distance 

from the well. If this relationship can be ascertained approximately,

200

180

8

160

140

120

100

20

0 10 20 30 40 50 60 70

SPECIFIC CAPACITY, IN GALLONS PER MINUTE PER FOOT OF DRAWDOWN 

FIGURE 99. Diagram for estimating the transmissibility of an aquifer from the specific capacity of a well.
PERMEABILITY, TRANSMISSIBILITY, AND DRAWDOWN 335

the specific capacity of the pumped well can be substituted for the 

quantity Q/s for the appropriate distance from the well.

For small- diameter uncased wells that tap consolidated waterbearing rocks, or at least for wells that produced no sand or silt when 

developed, the distance r probably can be equated to the radius of the 

well. For instance, for a well 6 inches in diameter,

7"=C(l,684-264 log 10 5£+264 Iog10 *), 

in which

C= = the specific capacity of the pumped well.

o

In wells having perforated casing and for which no improvement 

in performance was noted upon development, some head is lost as the 

water moves through the perforations in the casing. The amount of 

head lost in this manner ranges widely according to whether or not 

the casing fits snugly against the wall of the hole. If it does, the 

drawdown within the aquifer at the wall of the hole presumably 

would be considerably less than within the well itself, and the specific 

capacity computed on the basis of the lesser drawdown would be considerably higher. An arbitrary increase, then, in the specific capacity 

probably would be justified for the computation of the coefficient of 

transmissibility. In consolidated formations in which the wall of a 

hole is rough and the casing does not fit tightly, the loss in head presumably is small and can be disregarded.

Many wells of large yield tap aquifers that consist of consolidated 

sand or gravel. Such wells yield readily to development and once they 

are developed the pumping level of the water both within and immediately outside the casing is generally higher than it would have been 

had they not been developed. It is difficult to estimate the extent to 

which the transmissibility of the materials in the immediate vicinity 

of a well has been increased by the development of the well. However, available data indicate that in many cases the effect is the same 

as if the well were 10 feet in diameter but had not been developed. 

Therefore, 996 (the factor for r=5 ft) would be a reasonable value to 

substitute for K in the equation for T' '.

Although many empirical data should be gathered as to the relation 

between the specific capacities of wells and the transmissibilities of 

the tapped aquifers before any final correlation is made, present 

knowledge seems to justify the following equation for wells that have 

a diameter of about 1 foot and that tap water-table aquifers consisting 

of unconsolidated sediments:

,300-264 log 

The factor (1±0.3) should be adjusted upward for wells having a
336 GROUND'WATER HYDRAULICS

small diameter, for wells that are poorly developed, and for wells with 

poorly perforated casing, and downward for larger and well-developed 

wells.

ESTIMATING THE TRAN8MIS8IBILITY OF AN ARTESIAN AQUIFER 

FROM THE SPECIFIC CAPACITY OF A WELL

By RUSSELL H. BROWN

The use of figure 99 can be demonstrated by the following example. 

Assume that examination of well logs and related data has led to an 

estimate of 0.15 as a likely coefficient of storage, $, for a given watertable aquifer, that a review of well records has revealed a number 

of completion (or acceptance) tests, and that data taken from the best 

controlled test show, for a 30-hour pumping period, the specific capacity of a 6-inch well to be 12 gpm per ft of drawdown. The order of 

magnitude of the coefficient of transmissibility is to be determined. 

From the preceding discussion by Theis,

r= (#-264 log105S+264 loglo*)

o

= 12(1,684-264 log100.75-|-264 Iog101.25)

= 12(1,684-1-33-1-26)

=20,900.

As shown by figure 99, the abscissa of Tf 20,900 gdp per ft intersects 

the ordinate of specific capacity equals 12 gpd per ft of drawdown 

about where 7*= 19,000 gpd per ft. If S should later prove to be 0.25 

instead of 0.15, the revised value of Tf would be 20,200 and, from the 

chart, T would be about 18,000 gpd per ft. Thus it is evident that 

even large differences in S do not materially affect the value of T and 

that exercising judgment in selecting a value for S will produce results 

of the correct order of magnitude.

As stated by Theis (p. 333), the formulas and related constants derived by him are not applicable to artesian conditions. The principal 

objection in attempting to extend their application from water-table 

conditions to artesian conditions is the large adjustment in the K 

factor that becomes necessary if, for example, £=2X10-*, which is 

one-thousandth the assumed £=0.2. However, a formula and set of 

constants for artesian conditions can be found by paralleling the Theis 

derivation and using an assumed coefficient of storage of 2 X 10~4 . If 

it is assumed again that T= 100,000 gpd per ft, Theis' diagram (fig. 

99) can be used without modification.
PERMEABILITY, TRANSMISSIBILITY, AND DRAWDOWN 337

If T= 100,000 gpd per ft and S=2X 10~4, then from equation 1 on 

page 332

114.6g r /1.87r» . 2 1Q-* S 100,000 IV] 1 r~ L~°-577~log<V Topoo 2 . io-*r ' t)\

(3.74r».10-«)(5S.l(

-Iog10 (58 10«)+logio (T 10-6)+log10 *].

Therefore,

Again let

Then

s s 

-Iog10 (3.74r* - 10-»)-log10 (58 103)+log10 *].

10_6)

=[-66-264 log,0 (3.74T-2 10-»)-264 log,0 (58

Let

= -66-264 log,o (3.73r2 lO"9).

Then

T'=Q [#-264 log,0 (58 10")+ 264log,0*]. 

s

Values of K, computed for selected values of r, are as follows:

logic*] 

(5)

(6)

r(ft)

0.25

.50

1.0

5.0

10

K

2,477

2,318

2,159

1,790

1,633

r(ft)

20

30

40

50

K

1,472

1,379

1,313

1,262
338 GROUND-WATER HYDRAULICS

If the value of S is as large as 2X10"3 (10 times the,assumed value) 

the effect will be to decrease K for r 5 feet by nearly 15 percent. For 

larger values of K this percentage obviously is lower, and for smaller 

values it is higher. Conversely, if 8 is as low as 2X 10~5 (one tenth 

the assumed value) the effect will be to increase K by nearly 15 percent.

The application of the equation derived for T' for artesian conditions can be demonstrated by an example. Assume that the best estimate of S for a given artesian aquifer is 4X10~5 . Furthermore, data 

collected during a 30-hour acceptance test of a 6-inch well show that 

the specific capacity of the well is 7.5 gpm per ft of drawdown. The 

coefficient of transmissibility may be computed by following the same 

procedure used in the previous example. 

Thus,

7" =2 [#-264 log,0 (5S 103 ) +264 Iog10 «] 

s

=7.5(2,477-264 log,0 0.2+264 log,0 1.25)

=7.5(2,477+184+26)

=20,200.

According to figure 99, T=18,000 gpd per ft (approx.) where the 

ordinate of 7.5 intersects the abcissa of 20,200. If it later develops 

that a value of 4X 10~4 is a better estimate of £, then T' would be 18,200 

and T would be about 16,000 gpd per ft.

A CHART RELATING WELL DIAMETER, SPECIFIC CAPACITY, AND 

THE COEFFICIENTS OF TRANSMISSIBILITY AND STORAGE

By REX R. METEB

The relationships of well diameter, specific capacity, and the coefficients of transmissibility, T7 , and storage, S, are shown graphically 

in figure 100. This graph was prepared by (1) computing, for various 

values of T and S, the theoretical drawdown in wells having diameters 

of 6, 12, and 24 inches, (2) computing the specific capacity of those 

wells (on the assumption that they are 100 percent efficient), and (3) 

plotting the specific capacity against S to form a family of curves 

which represent the different values of T. For the sake of clarity, 

the curves for a well 24 inches in diameter were not plotted in the 

upper part of the graph; they would be virtually parallel to the curves 

for a well 12 inches in diameter and lie above them at a distance equal 

to that between the curves for wells 6 inches and 12 inches in diameter. 

The specific capacity at the end of 1 day's pumping is shown on the 

left scale of the graph. The values of S, shown on the bottom scale, 

range from those for artesian conditions on the left to those for watertable conditions on the right. Each group of curves for a specific T is 

bracketed on the right margin.
PERMEABILITY, TRANSMISSIBILITY, AND DRAWDOWN 339

Figure 100 can be used to determine the approximate T of an aquifer 

if the specific capacities of wells are the only available data. It also 

can be used to determine the approximate specific capacity of a well 

which is to be drilled into an aquifer for -which T and S are known. 

The computed theoretical specific capacity is useful not only for planning purposes but also, when compared to the specific capacity determined from a field test, as a means of determining the approximate 

efficiency of a well. Although determinations made from figure 100 

may not be exact owing to unknown factors that must be estimated, the 

graph serves as a measure for approximation.

A cursory study of the graph reveals that it has certain limitations. 

One of the principal factors affecting the specific capacity of a well 

is the entrance loss of the water. The graph is based on the assumption that the wells are 100 percent efficient or, in other words, that 

when the wells are pumped the water level inside and immediately 

outside the casing or screen is the same. Because, in most wells, the 

water level immediately outside is higher than inside, the observed 

specific capacity is somewhat less than that of an ideal well. The 

specific capacity of a well is affected also by the diameter of the well. 

The well diameters shown on the graph 6, 12, and 24 inches are 

considered to be the effective diameters of the wells. If an aquifer is 

composed of consolidated rocks, the effective diameter probably is 

approximately the same as the diameter of the well. However, if 

the material in an aquifer consists of unconsolidated materials and if

200

} 200,000 " 

}150,000£

100,000^;

80,000 < 

60,000 °

}40,000 £

Well is assumed to have 100 percent efficiency

, , , i ,.,,1 , ,

0.00001 0.00005 0.0001 0.0005 0.001 0.005 0.01 0.05 0.1 

COEFFICIENT OF STORAGE

FIQUBE 100. Graph showing relation of well diameter, specific capacity, and coeflBclents of transmlsslblllty

and storage.
340 GROUND-WATER HYDRAULICS

the well has been highly developed, the effective diameter may be substantially larger than the diameter of the screen. On the other hand, 

a seemingly highly developed well may be very inefficient because of 

caving or faulty construction, and, accordingly, have an effective diameter less than the diameter of the screen. Other conditions being 

the same, a change in the effective diameter has the greatest effect on 

the specific capacity of wells in aquifers that have a low T and a 

high S.

The graph shows that large changes in /S correspond to relatively 

small changes in T and specific capacity; therefore, inaccuracy in 

estimating S generally is not a serious limiting factor. Moreover, 

from a general knowledge of the geology and hydrology, an aquifer 

usually can be classified as principally water table or artesian, and S 

can be estimated accordingly. However, the graph should not be 

used in an attempt to determine 8 even when accurate values of the 

specific capacity and T are available.

If the pumped well taps less than the full thickness of the aquifer 

thus introducing vertical components of flow or if it taps a thin 

water-table aquifer so that the water-level drawdown is a substantial 

fraction of the original saturated thickness, the graph obviously cannot be applied without serious error.

The time interval of 1 day used for computing the specific capacity 

scale on the graph was selected arbitrarily. An error will be introduced if the specific capacity determined in the field is based on a 

shorter or longer period of pumping. The amount of the error is 

small for high values of T and low values of /S but increases substantially for low values of T and high values of S.

The procedure for using the log graph to determine T from the 

specific capacity of a well is as follows:

1. Select the specific capacity on the left margin.

2. Move horizontally along the abcissa to the intersection of the ordinate 

through the estimated value of S.

3. From this intersection move along a curve or parallel to the family of 

curves, and find the value of T on the right margin.

Although the specific capacity at the end of 1 day's pumping can 

be computed for an ideal well tapping an aquifer having known 

values of T and $, it can be determined more easily and quickly from 

the graph. To determine the theoretical specific capacity of such 

a well, the procedure described above is reversed; move left along or 

parallel to the curve from the known value of T to the intersection 

of the ordinate through the known value of S ; thence move horizontally to the left margin and read the specific capacity.

If the graph is used with an understanding of its limitations, it 

should provide a useful tool in ground-water studies.
PERMEABILITY, TRANSMISSIBILITY, AND DRAWDOWN 341

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O