Theory of Aquifer Tests


By J. G. FERRIS, D. B. KNOWLES, R. H. BROWN, and R. W. STALLMAN 

GROUND -WATER HYDRAULICS 

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1536-E 

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

MANUEL LUJAN, JR., Secretary 

GEOLOGICAL SURVEY 

Dallas L. Peck, Director 

Any use of trade, product, or firm names in this publication is for descriptive purposes only and does not imply endorsement by the U.S. Government 

First Printing 1962 

Second printing 1963 

Third printng 1989 

For sale by the Books and Open-File Reports Section, U .S . Geological Survey, 

Federal Center, Box 25425, Denver, CO 80225 
CONTENTS 

Page 

Abstract--------------------------------------------------------- 69 

Introduction------------------------------------------------------ 69 

Aquifer tests-the problem----------------------------------------- 70 

Darcy'slaw--------------------------------------------------- 71 

Coefficients of permeability and transmissibility ------------------- 72 

Coefficient of storage------------------------------------------- 74 

The artesian case------------------------------------------ 75 

The water-table case--------------------------------------- 76 

Elasticity of artesian aquifers--------------------------------------- 78 

Internal forces------------------------------------------------ 78 

Transmission of forces between aquifers-------------------------- 80 

Effects of changes in loading-----------------------___________-- 80 

Excavations---------------------------------------------- 80 

Moving railroad trains------------------------------------- 81 

Changes in atmospheric pressure---------------------------- 83 

Tidal fluctuations----------------------------------------- 85 

Ocean, lake, or stream tides-------------------------____ 85 

Earth tides------------------------------------------- 86 

Earthquakes---------------------------------------------- 87 

Coefficient of storage and its relation to elasticity------------------ 88 

Aquifer tests--basic theory----------------------------------------- 91 

Well methods-point sink or point source------------------------ 91 

Constant discharge or recharge without vertical leakage-------- 91 

Equilibrium formula ----------------------------------- 91 

Nonequilibrium formula-------------------------------- 92 

Modified nonequilibrium formula------------------------ 98 

Theis recovery formula--------------------------------- 100 

Applicability of methods to artesian and water-table aquifers------------------------------------------------ 102 

Instantaneous discharge or recharge-------------------------- 103 

"Bailer" method-------------------------------------- 103 

"Slug" method---------------------------------------- 104 

Constant head without vertical leakage----------------------- 106 

Constant discharge with vertical leakage--------------------- 110 

Leaky aquifer formula--------------------------------- 110 

Variable discharge without vertical leakage, by R. W. Stallman- 118 

Continuously varying discharge------------------------- 118 

Intermittent or cyclic discharge------------------------- 122 

Channel methods-line sink or line source------------------------ 122 

Constant discharge---------------------------------------- 122 

Nonsteady state, no recharge--------------------------- 122 

Constant head-------------------------------------------- 126 

Nonsteady state, no recharge, by R. W. Stallman--------- 126 

Steady state, uniform recharge -----------------------_ -- 131 

Sinusoidal head fluctuations :------------------------------- 132 
IV CONTENTS 

Page

Aquifer tests-basic theory-Continued 

Areal methods ------------------------------------------------ 135 

Numerical analysis, by R. W. Stallman---------------------- 135 

Flow-net analysis, by R. R. Bennett------------------------- 139 

Theory of images and hydrologic boundary analysis ---------------- 144 

Perennial stream-line source at constant head-_ --- - - - _ - - _ _ _ - - 146 

Impermeable barrier --------------------------------------- 147 

Two impermeable barriers intersecting at right angles---------- 151 

Impermeable barrier and perennial stream intersecting at right 

angles -------------------------------------------------- 152 

Two impermeable barriers intersecting at an angle of 45 degrees- 154 

Impermeable barrier parallel to a perennial stream------------- 156 

Two parallel impermeable barriers intersected at right angles 

by a third impermeable barrier____________________________ 156 

Rectangular aquifer bounded by two intersecting impermeable 

barriers parallel to perennial streams----------------------- 159 

Applicability of image theory involving infinite systems of image 

wells --------------------------------------------------- 159 

Corollary equations for application of image theory------------ 161 

Applicability of analytical equations ----------------------------- 166 

References________________________________________________________ 168 

Index____________________________________________________________ 173 

ILLUSTRATIONS 

Page 

FIGIIRD 17. Diagram for coefficients of permeability and transmissibility-- 72 

18. Diagrams for coefficient of storage ------------------------ 77 

19. Diagrams for elastic phenomena in artesian aquifers-------- 79 

20. Effect of excavating in confining material overlying artesian 

aquifer ---------------------------------------------- 81 

21. Effects of changes in loading on artesian aquifer ------------ 82 

22. Effect of atmospheric pressure loading on aquifers---------- 83 

23. Logarithmic graph of well function W(u)-constant discharge- 95 

24. Relation of W(u) and u to s and r2/t---------------------- 98 

25. Equipment for a "slug" test_____________________________ 105 

26. Logarithmic graph of well function G(a)-constant drawdown- 108 

27. Logarithmic graph of modified Bessel function Ko (x) ------- 114 

28. Nomenclature for continuously varying discharge ----------- 119 

29 . Family of curves for continuously varying discharge-------- 121 

30. Logarithmic graph of drain function D (u) Qconstant discharge---------------------------------------------- 124 

31. Logarithmic graph of drain function D(u)w,-constant head_- 128 

32. Steady-state flow in aquifer uniformly recharged----------- 130 

33 . Finite grid and nomenclature for numerical analysis-------- 137 

34. Flow net for discharging well in infinite-strip aquifer-------- 142 

35. Image-well analysis for discharging well near perennial stream- 146 

36. Flow net for discharging well near perennial stream --------- 148 

37. Image-well analysis for discharging well near impermeable 

boundary -------------------------------------------- 149 
CONTENTS V 

Page 

FIGURE 38. Flow net for discharging well near impermeable boundary--_ 150 

39. Image-well system for discharging well in quarter-infinite 

aquifer-two impermeable boundaries--_ _ - _ _ _ _ _ _ -------- 151 

40. Image-well system for discharging well in quarter-infinite 

aquifer-one impermeable boundary -------------------- 153 

41. Image-well system for discharging well in 45° wedge aquifer - 155 

42. Image-well system for discharging well in infinite-strip aquifer- 157 

43. Image-well system for discharging well in semi infinite-strip 

aquifer---------------------------------------------- 158 

44. Image-well system for discharging well in rectangular aquifer-- 160 

45. Geometry for locating a point on a hydrologic boundary _ - _ - 165 

TABLES 

Page 

TABLE 1 . Approximate average velocities and paths taken by different 

types of shock waves caused by earthquakes --------------- 88 

2. Values of W(u) for values of u between 10- ' S and 9.9 ---------- 96 

3 . Values of G(a) for values of a between 10-4 and 10 12 --------- - 109

4. Values of Ko (x), the modified Bessel function of the second kind 

of zero order, for values of x between 10-z and 9 .9 ---------- 115 

5 . Values of D(u) c, u, and u z for channel method-constant discharge formula------------------------------------------ 126 

6. Values of D(u) h u, and u 2 for channel method-constant head 

formula ----------------------------------------------- 127 
SYMBOLS 

(The following symbols are listed in alphabetical order. Each symbol indicates the basic term usually 

represented, with no attempt to show the many and unavoidable duplicate uses . In the text, various 

subscripts are used in conjunction with these symbols to denote specific applications of the basic terms . 

A few of the more important combinations of this type are given; others are defined where they appear in 

the text. 

A Area of cross section through which flow occurs . 

a Distance from stream or drain to ground-water divide . 

e Base of natural (Napierian) logarithms, numerically equal to 2 .7182818 . 

g Local acceleration due to gravity. 

h Head of water with respect to some reference datum,, 

I Hydraulic gradient. 

L Length (width) of cross section through which flow occurs . 

m Saturated thickness of an aquifer . 

m.' Saturated thickness of relatively impermeable bed confining an aquifer . 

P Coefficient of permeability of the material comprising an aquifer . 

P' Coefficient of vertical permeability of the material comprising a relatively 

impermeable bed that confines an aquifer . 

Q Rate of discharge, or recharge. 

r Radial distance from discharge or recharge well to point of observation. 

r  Effective radius of discharge or recharge well . 

S Coefficient of storage of an aquifer . 

s Change in head of water, usually expressed as drawdown, or recovery or 

buildup . 

s' Residual change in head of water, usually reserved for use in conjunction 

with the term, "drawdown". 

T Coefficient of transmissibility of an aquifer . 

t Elapsed time with respect to an initial reference . 

t' Elapsed time with respect to a second reference . 

V Volume. 

TV Rate of accretion or recharge to an aquifer . 

w Spacing of grid lines used to subdivide a region into finite squares . 

x Distance from stream or drain to point of observation . 

BE Barometric efficiency of an aquifer. 

TE Tidal efficiency of an aquifer. 

D(u) n Drain function of u, constant head situation . 

D(u) c Drain function of u, constant discharge situation . 

G(a) Well function of a, constant head situation . 

W(u) Well function of u, constant discharge situation . 

Jo (x) Bessel function of first kind, zero order. 

I0(x) Modified Bessel function of first kind, zero order. 

Yo(x) Bessel function of second kind, zero order. 

Ko (x) Modified Bessel function of second kind, zero order. 

a Bulk modulus of compression or vertical compressibility (reciprocal of the 

bulk modulus of elasticity) of the aquifer skeleton. 

v1 
CONTENTS VII 

Bulk modulus of compression, or compressibility of water ; approximate 

value for average ground-water temperature is 3.3 X 10-a in 2/lb . 

y Specific weight of a substance . 

70 Specific weight of water at a stated reference temperature ; numerically 

equal to 62 .41b per cubic foot at 4°C or 39°F. 

0 Porosity of an aquifer. 

Density of a substance . 
GROUND-WATER HYDRAULICS 

THEORY OF AQUIFER TESTS 

By J. G. FERRIS, D. B. KNOWLES, R. H. BROWN, an,, R . W. STALLMAN 

ABSTRACT 

The development of water supplies from wells was placed on a rational basis 

with Darcy's development of the law governing the movement of fluids through sands and with Dupuit's application of that law to the problem of radial flow 

toward a pumped well . As field experience increased, confidence in the applicability of quantitative methods was gained and interest in developing solutions 

for more complex hydrologic problems was stimulated. An important milestone was Theis' development in 1935 of a solution for the nonsteady flow of 

ground water, which enabled hydrologists for the first time to predict future 

changes in ground-water levels resulting from pumping or recharging of wells . In the quarter century since, quantitative ground-water hydrology has been 

enlarging so rapidly as to discourage the preparation of comprehensive textbooks . This report surveys developments in fluid mechanics that apply to ground­ water hydrology . It emphasizes concepts and principles, and the delineation 

of limits of applicability of mathematical models for analysis of flow systems in the field . It stresses the importance of the geologic variable and its role in 

governing the flow regimen . The report discusses the origin, occurrence, and motion of underground water 

in relation to the development of terminology and analytic expressions for selected 

flow systems . It describes the underlying assumptions necessary for mathematical treatment of these flow systems, with particular reference to the way in 

which the assumptions limit the validity of the treatment . 

INTRODUCTION 

Lectures on ground-water hydraulics by John G. Ferris provide 

most of the source material for this paper, which was organized by 

Doyle B . Knowles. Subsequent refinements of concepts and standardization of nomenclature and method of presentation were accomplished by Russell H. Brown and Robert W. Stallman, with the 

important collaboration of Edwin W. Reed. Appropriate individual 

authorship is recognized for several sections of the text. 

The material presented herewith concerns the theory supporting 

many hydraulic concepts . Applications of the theory to field problems are to be shown in another report. 

69 
70 GROUND-WATER HYDRAULICS 

AQUIFER TESTS-THE PROBLEM 

The basic objective of ground-water studies of the U.S . Geological Survey is to evaluate the occurrence, availability, and quality of 

ground water. The science of ground-water hydrology is applied toward attaining that goal. Although many ground-water investigations are of a qualitative nature, quantitative studies are necessarily 

an integral component of the complete evaluation of occurrence and 

availability . The worth of an aquifer as a fully developed source of 

water depends largely on two inherent characteristics : its ability to 

store and its ability to transmit water. Furthermore, quantitative knowledge of these characteristics facilitates measurement of hydro­ logic entities such as recharge, leakage, and evapotranspiration . It is recognized that these two characteristics, referred to as the coefficients of storage and transmissibility, generally provide the very foundation on which quantitative studies are constructed . Within the 

science of ground-water hydrology, ground-water hydraulics methods 

are applied to determine these constants from field data. Ground-water hydraulics, as now defined by common practice, can 

be described as the process of combining observed field data on water 

levels, water-level fluctuations, natural or artificial discharges, etc., with suitable equations or computing methods to find the hydraulic characteristics of the aquifer; it includes the logical extension of these 

data and computing methods to the prediction of water levels, to the 

design of well fields, the determination of optimum well yields, and 

other hydraulic uses-all under stated conditions . The selection of 

equations or computing procedures to be used for analysis is governed largely by the physical conditions of the aquifer studies, insofar as 

they establish the hydraulic boundaries of the system . The extraordinary variability in the coefficients of storage and transmissibility,

combined with the irregularities in the shape of flow systems encountered in many ground-water studies, precludes uninhibited support of 

calculated coefficients based on vague or meager data. One quantitative test does not satisfy the demand for a quantitative study of an 

aquifer. It is merely a guidepost, indicator, or segment of knowledge, which must be supported by additional tests. Often the initially calculated results may require revision on the basis of the discoveries 

resulting from additional testing as the field investigation proceeds. 

Obviously the results from ground-water hydraulics must be completely in accord with the geologic characteristics of the aquifer or 

of the area under investigation . Circumstance frequently demands 

that tests be conducted without prior knowledge of the geology in the 

vicinity of the test site . To varying degrees, lack of knowledge of 

the geology in most cases reduces the reliability of the test results 

to a semiquantitative category until more adequate support is found. 
THEORY OF AQUIFER TESTS 71 

The principal method of ground-water hydraulics analysis is the 

application of equations derived for particular boundary conditions. 

The number of equations available has grown rapidly and steadily 

during the past few years. These are described in a wide assortment 

of publications, some of which are not conveniently available to many

engaged in studies of ground-water hydraulics . The essence of each 

of many concepts of hydraulics is presented and briefly discussed, 

but frequent recourse should be made to the more exhaustive treatment given in the cited reference. 

Where the definition of a hydraulics or ground-water term is considered necessary, it is stated where the term first appears, and the 

symbol and units in which the term is ordinarily expressed are given . 

Much of the terminology is assembled under "Symbols," following

the table of contents . 

DARCY'S Law 

Hagen (1839) and Poiseuille (1846) were the first to study the law 

of flow of water through capillary tubes. They found that the rate of 

flow is proportional to the hydraulic gradient . Later Darcy (1856) 

verified this observation and demonstrated its applicability to the 

laminar (viscous, streamline) flow of water through porous material 

while he was investigating the flow of water through horizontal 

filter beds discharging at atmospheric pressure . He observed that, 

at low rates of flow, the velocity varied directly with the loss of head 

per unit length of sand column through which the flow occurred and 

expressed this law as 

__P_h 

v l 

in which z is velocity of the water through a column of permeable

material, h is the difference in head at the ends of the column, 

l is the length of the column, and P is a constant that depends on the 

character of the material, especially the size and arrangement of the 

grains . 

The velocity component in laminar flow is proportional to the 

first power of the hydraulic gradient . It can be seen, therefore, that 

Darcy's law is valid only for laminar flow. The flow is probably 

turbulent or in a transitional stage from laminar to turbulent flow 

near the screens of many large-capacity wells. Jacob (1950) agrees 

with Meinzer and Fishel (1934) that because water behaves as a 

viscous fluid at extremely low hydraulic gradients, it will obey Darcy's 

law at gradients much smaller than can be measured in the laboratory. 

He points out, however, that Darcy's law may not be valid for the 

flow of water in sands that are not completely saturated, or in extremely fine grained materials . 
72 GROUND-WATER HYDRAULICS 

COEFFICIENTS OF PERMEABILITY AND TRANSMISSIBILITY 

The coefficient of permeability, P, of material comprising a formation is a measure of the material's capacity to transmit water. 

The coefficient of permeability was expressed by Meinzer (Stearns, 

N.D., 1928) as the rate of flow of water in gallons per day through a 

cross-sectional area of 1 square foot under a hydraulic gradient of 1 

foot per foot at a temperature of 60°F. In figure 17, then, it would 

be the flow of water through opening A, which is 1 foot square. In 

field practice the ajdustment to the standard temperature of 60°F. 

is commonly ignored and permeability is then understood to be a 

field coefficient at the prevailing water temperature. Theis (1935)

introduced the term coefficient of transmissibility, T, which is expressed 

-, " 

. , _-/ . %ice ; ; 

Opening A, I foot square 

FIGURE 17 .-Diagram for ooeffleients of permeability and transmissibility. 
THEORY OF AQUIFER TESTS 73 

as the rate of flow of water, at the prevailing water temperature, in 

gallons per day, through a vertical strip of the aquifer 1 foot wide 

extending the full saturated height of the aquifer under a hydraulic

gradient of 100 percent. In figure 17 it would be the flow through

opening B, which has a width of 1 foot and a height equal to the 

thickness, m, of the aquifer. A hydraulic gradient of 100 percent 

means a 1-foot drop in head in 1 foot of flow distance as shown schematically by the pair of observation wells in figure 17 . It is seldom 

necessary to adjust the coefficient of transmissibility to an equivalent 

value for the standard temperature of 60°F., because the temperature 

range (and, hence, range in viscosity) in most aquifers is not large. 

The relation between the coefficient of transmissibility and the 

field coefficient of permeability, as they apply to flow in an aquifer, 

can be seen in figure 17. 

A useful form of Darcy's law, which is often applied in studies of 

ground-water hydraulics problems, is given by the expression 

Qd=PIA 

in which Qd is the discharge, in gallons per day; P is the coefficient of 

permeability, in gallons per day per square foot ; I is the hydraulic 

gradient, in feet per foot ; and A is the cross-sectional area, in square

feet, through which the discharge occurs . For most ground-water 

problems, this expression can be more conveniently written as 

Qd=TIL 

in which Qd and I are defined as above, T is the coefficient of transmissibility in gallons per day per foot, and L is the width, in feet, of 

the cross section through which the discharge occurs . In many 

field problems it may be more practical to express I in feet per mile 

and L in miles. The units for T and Qd will remain as already stated . 

The coefficient of transmissibility may be determined by means of 

field observations of the effects of wells or surface-water systems on 

ground-water levels . It is then possible to determine the field coefficient of permeability from the formula P=T/m. Physically, however, 

P has limited significance under these conditions . It merely represents 

the overall average permeability of an ideal aquifer that behaves 

hydraulically like the aquifer tested . 

In general, laboratory measurements of permeability should be 

applied with extreme caution. The packing arrangement of a poorly 

sorted sediment is a critical factor in determining the permeability, and 

large variations in permeability may be introduced by repacking a 

disturbed sample . Furthermore, a laboratory measurement of permeability on one sample is representative of only a minute part of the 

water-bearing formation . Obviously, therefore, if quantitative data 
74 GROUND-WATER HYDRAULICS 

are to be developed by laboratory methods, it is desirable to collect 

samples of the water-bearing material at close intervals of depth and 

at as many locations within the aquifer as is feasible. 

COEFFICIENT OF STORAGE 

The coefficient of storage, S, of an aquifer is defined as the volume of 

water it releases from or takes into storage per unit surface area of the 

aquifer per unit change in the component of head normal to that 

surface. 

A simple way of visualizing this concept is to imagine an artesian 

aquifer which is elastic and uniform in thickness, and which is assumed, 

for convenience, to be horizontal . If the head of water in that aquifer

is decreased, there will be released from storage some finite volume of 

water that is proportional to the change in head. Because the aquifer is horizontal, the full observed head change is evidently effective 

perpendicular to the aquifer surface. Imagine further a representative

prism extending vertically from the top to the bottom of this aquifer,

and extending laterally so that its cross-sectional area is coextensive 

with the aquifer-surface area over which the head change occurs . 

The volume of water released from storage in that prism, divided by

the product of the prism's cross-sectional area and the change in head, 

results in a dimensionless number which is the coefficient of storage. If this example were revised slightly, it could be used to demonstrate 

the same concept of coefficient of storage for a horizontal water-table 

aquifer or for a situation in which the head of water in the aquifer is 

increased. 

As with almost any concise definition of a basic concept it is necessary to develop its full significance, its limitations, and its practical 

use and application through elaborative discussion. The coefficient 

of storage is no exception in this respect, and the following discussion 

will serve to bring out a few ideas that are important in applying the 

concept to artesian and water-table aquifers in horizontal or inclined 

attitudes. 

Observe that the statement of the storage-coefficient concept first 

focuses attention on the volume of water that the aquifer releases 

from or takes into storage . Identification and measurement of this 

volume poses no particular problem, but it should be recognized that 

it is measured outside the aquifer under the natural local conditions 

of temperature and atmospheric pressure ; it is not the volume that 

the same amount of water would occupy if viewed in place in the 

aquifer. 

Although the example used to depict the concept of the storage 

coefficient was arbitrarily developed around a horizontally disposed

artesian aquifer, the concept applies equally well to water-table aqui­
THEORY OF AQUIFER TESTS 75 

fers and is not compromised by the attitude of the aquifer. This 

flexibility of application relies importantly, however, on relating the 

storage-coefficient concept to the surface area of the aquifer and to 

the component of head change that is normal to that surface. In 

turn this relation presupposes that the particular aquifer prism involved in the movement of water into or out of storage is that prism 

whose length equals the saturated thickness of the aquifer, measured 

normal to the aquifer surface, and whose cross-sectional area equals 

the area of the aquifer surface over which the head change occurs . 

Furthermore, water moves into or out of storage in this prism in 

direct proportion only to that part of the head change that acts to 

compress or distend the length of the prism . In other words, the 

component of the head change to be considered in the release or storage 

of water is that which acts normal to the aquifer surface. The mathematical models devised for analyzing ground-water flow usually re 

quire uniform thickness of aquifer. However, the storage coefficient 

concept, as defined here, applies equally well to aquifers that thicken 

or thin substantially, if the "surface area" is measured in the plane 

that divides the aquifer into upper and lower halves that are symmetrical with respect to flow. The imaginary prism would then be 

taken perpendicular to this mean plane of flow . 

TM AMT101" G"11 

Consider an artesian aquifer, in any given attitude, in which the 

head of water is changed, but which remains saturated before, during, 

and after the change . It is assumed that the beds of impermeable 

material confining the aquifer are fluid in the sense that they have no 

inherent ability to absorb or dissipate changes in forces external to 

or within the aquifer. Inasmuch as no dewatering or filling of the 

aquifer is involved, the water released from or taken into storage can 

be attributed only to the compressibility of the aquifer material and 

of the water. By definition the term "head of water" and any changes 

therein connote measurements in a vertical direction with reference to 

some datum. In a practical field problem the change in head very 

likely would be observed as a change in water-level elevaiion in a well . 

The change in head is an indication of the change in pressure in the 

aquifer prism, and the total change in force tending to compress the 

prism is equal to the product of the change in pressure multiplied by 

the end area of the prism. Obviously this change in force is not affected by the inclination of the aquifer, inasmuch as a confined pressure 

system is involved and the component of force due to pressure always 

acts normal to the confining surface. Thus any conventional method 

of observing head change will correctly identify the change in pressure 

normal to the aquifer surface,and may be considered as a component 

of head acting normal to that surface. 
76 GROUND-WATER HYDRAULICS 

Examine figure 18A, which depicts, in schematic fashion, a horizontal artesian aquifer. Shown within the aquifer is a prism of unit 

cross-sectional area and of height, m, equal to the aquifer thickness. 

If the piezo.-netric surface is lowered a unit distance, x, as shown, a 

certain amount of water will be released from the aquifer prism . This 

occurs in response to a slight expansion of the water itself and a slight 

decrease in porosity due to distortion of the grains of material composing the aquifer skeleton . 

Summary statement .-For an artesian aquifer, regardless of its attitude, the water released from or taken into storage, in response to a 

change in head, is attributed solely to compressibility of the aquifer 

material and of the water. The volume of water (measured outside 

the aquifer) thus released or stored, divided by the product of the head 

change and the area of aquifer surface over which it is effective, correctly determines the storage coefficient of the aquifer. Although 

rigid limits cannot be established, the storage coefficients of artesian 

aquifers may range from about 0 .00001 to 0 .001 . 

THE WATER-TABLE CASE 

Application of the storage coefficient concept to water-table aquifers 

is more complex, though reasoning similar to that developed in the 

preceding paragraphs can be applied to the saturated zone of an 

inclined water-table aquifer. Consider a water-table aquifer, in any 

given attitude, in which the head of water is changed. Obviously 

there will now be dewatering or refilling of the aquifer, inasmuch as it 

is an open gravity system with no confinement of its upper surface. 

Thus the volume of water released from or taken into storage must 

now be attributed not only to the compressibility of the aquifer material and of the water, in the saturated zone of the aquifer, but also to 

gravity drainage or refilling in the zone through which the water 

table moves. The volume of water involved in the gravity drainage 

or refilling, divided by the volume of the zone through which the 

water table moves, is the specific yield . Except in aquifers of low 

porosity the volume of water involved in gravity drainage or refilling 

will ordinarily be so many hundreds or thousands of times greater 

than the volume attributable to compressibility that for practical 

purposes it can be said that the coefficient of storage equals the specific 

yield . The conventional method of measuring change in head by 

observing change in water-level elevation in a well evidently identifies 

the vertical change in position of the water table. In other words, 

head change equals vertical movement of the water table. It can 

be seen that the volume of the zone through which the water table 

moves is equal to the area of aquifer surface over which the head 

change occurs, multiplied by the head change, multiplied by the cosine 
THEORY OF AQUIFER TESTS 77 

Unit cross-sectional area 

A . ARTESIAN AQUIFER 

Unit cross-sectional area 

Water table 

B. WATER-TABLE AQUIFER 

FIGURE 18 .-Diagrams for coefficient of storage . 

of the angle of inclination of the water table. The product of the last 

two factors is the component of head change acting normal to the 

aquifer surface. The importance of interpreting correctly the phrase 

"component of head change" which appears in the definition of the 

storage coefficient cannot be overemphasized . 

Examine figure 18B, which depicts, in schematic fashion, ahorizontal 
78 GROUND-WATER HYDRAULICS 

water-table aquifer. Again a unit prism of the aquifer is shown, and 

it is assumed that the water table is lowered a unit distance, x. 

Usually the water that is thereby released represents, for practical 

purposes, the gravity drainage from the x portion of the aquifer 

prism. Theoretically, however, a slight amount of water comes 

from the portion of the prism that remains saturated, in accord with 

the principles discussed for the artesian case . 

Summary statement.-For a water-table aquifer, regardless of its 

attitude, the water released from or taken into storage, in response 

to a change in head, is attributed partly to gravity drainage or refilling of the zone through which the water table moves, and partly to 

compressibility of the water and aquifer material in the saturated 

zone. The volume of water thus released or stored, divided by the 

product of the area of aquifer surface over which the head change 

occurs and the component of head change normal to that surface, 

correctly determines the storage coefficient of the aquifer. Usually 

the volume of water attributable to compressibility is a negligible 

proportion of the total volume of water released or stored and can be 

ignored . The storage coefficient then is sensibly equal to the specific 

yield. The storage coefficients of water-table aquifers range from 

about 0 .05 to 0.30. 

ELASTICITY OF ARTESIAN AQUIFERS 

It has long been recognized that artesian aquifers have volume 

elasticity . D. G. Thompson, though not the first to publish on the 

subject, apparently was among the first in the Geological Survey to 

recognize this phenomenon . In studying the relation between the 

decline in artesian head and the withdrawals of water from the Dakota 

sandstone in North Dakota, Meinzer (Meinzer and Hard, 1925) came 

to the conclusion that the water was derived locally from storage. 

He found that the withdrawals could not be accounted for by the 

compressibility of the water alone, but might be accounted for by 

the compressibility of the aquifer. 

INTERNAL FOROW 

The diagram in figure 19A shows the forces acting at the interface 

between an artesian aquifer and the confining material . These forces 

may be expressed algebraically as 

s1=sm+Sk 

where s, is the total load exerted on a unit area of the aquifer, s is 

that part of the total load borne by the confined water, and sk is that 

part borne by the structural skeleton of the aquifer. Assume that 

the total load (s t ) exerted on the aquifer is constant . If s. is reduced, 
--------------- --------------- ---------------

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

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

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

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

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

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

THEORY OF AQUIFER TESTS 79 

as a result of pumping, the load borne by the skeleton of the aquifer

increases and there is slight distortion of the component grains of 

material . At the same time, the water expands to the extent permitted by its elasticity. Distortion of the grains of the aquifer 

skeleton means that they will encroach somewhat on pore space

formerly occupied by water. 

Conversely, if s. is increased, as in response to cessation of pumping, 

the piezometric head builds up again gradually approaching its 

original value, and the water itself undergoes slight contraction . 

With an increase in s there is an accompanying decrease in s x and 

the grains of material in the aquifer skeleton return to their former 

Confining mo>'eriol_--

Grains of material 

constituting aquifer 

skeleton 

A MICROSCOPIC VIEW OF FORCES ACTING AT INTERFACE BETWEEN 

ARTESIAN AQUIFER AND CONFINING MATERIAL 

Pumped well 

Abrupt lowering Dashed lines denote of water level bowed confining loyer, representing piezometric highly exaggerated surface, lower aquifer 

B . DIAGRAM SHOWING THE EFFECT ON A DEEP ARTESIAN AQUIFER OF PUMPING FROM A SHALLOW ARTESIAN AQUIFER 

FIGU$Z 19:Diagrams for elastic phenomena in artesian aquifers . 
80 GROUND-WATER HYDRAULICS 

shape. This releases pore space that can now be reoccupied by water 

moving into the part of the formation that was influenced by the 

compression. 

TRANSMISSION OF FORCES BETWEEN AQUIFERS 

It has been observed in some places that a well pumping from an 

aquifer affects the water level in a nearby well that is screened in a 

deeper or shallower artesian aquifer. Consider the case shown in 

figure 19B. The well screened in the upper aquifer, for convenience 

depicted as artesian, is pumped and the water level in the well screened 

in the lower aquifer abruptly declines when pumping begins . As 

pumping continues, the water level of the lower aquifer ceases to 

decline and gradually recovers its initial position. Altbough not 

proved, a logical explanation of this phenomenon may be as follows 

When the well in the upper aquifer begins pumping, there is a lowering 

of the pressure head in the vicinity of the pumped well . The decrease 

in pressure head unbalances the external forces that were acting on 

the upper and lower surfaces of the confining layer separating the two 

aquifers . In seeking a new static balance the confining layer will 

be bowed upward slightly thereby creating additional water storage 

space in the lower aquifer. The abrupt lowering of water level in 

the observation well represents the response to the newly created 

storage space and the reduction in the forces s, and s,, in the lower 

aquifer. The subsequent water-level recovery, approaching the 

initial position, represents filling of the new storage space and return 

to the original pressure head as water in the lower aquifer moves in 

from more remote regions. 

An interesting phenomenon that has been observed a few times, but 

for which no completely satisfactory explanation has yet been given, 

is that where pumping a well in one artesian aquifer causes a rise in 

the water level in a nearby well producing from a different artesian 

aquifer. (See Barksdale, Sundstrom, and Brunstein, 1936 ; and 

Andreasen and Brookhart, 1952.) 

EFFECTS OF CHANGES IN LOADING 

EXCAVATIONS 

"Blowthroughs" may occur if deep excavations are made in the 

confining materials overlying an artesian aquifer that has a high 

artesian head. This is shown diagramatically in figure 20. An 

excavation lowers the total load on part of the aquifer, which means 

that the forces s,, and s, (see fig. 19A), are now the dominant forces 

acting on the remaining layer of confining material, separating the 

bottom of the excavation from the top of the aquifer. Thus this 

layer will be bowed upward and if it is incompetent to contain the 
THEORY OF AQUIFER TESTS 81 

FiaVR$ 20 .-Excavating in material overlying artesian aquifers, 

bowing forces, it will rupture in the form of "blowthroughs" or "sand 

boils." It is the practice in Holland (Krul and Liefrinck, 1946), 

where this situation is commonly encountered, to install relief wells 

to lower the artesian head until an excavation is refilled . 

MOVING RAILROAD TRAINS 

It is a frequent observation that a passing railroad train affects the 

water levels in nearby artesian wells . This is another demonstration 

of the elasticity of artesian aquifers . The fluctuation of water 

level in a well on Long Island, N.Y., produced by a passing railroad 

train is shown in figure 21A. As the train approaches the well, an 

additional load is placed on the aquifer. This load tends to compress

the aquifer, causing a rapid rise in water level that reaches a maximum 

when, or shortly after, the locomotive is opposite the well . As the 

aquifer becomes adjusted to the new loading, the water level declines 

toward its initial position. When the entire train has passed the well, 

the aquifer expands and the water level in the well declines rapidly 

and reaches a minimum shortly after the train has left the well . The 

water level then recovers toward its initial position as the aquifer 

again becomes adjusted to this new condition of loading. The time 

required for this cycle of events is commonly a few minutes. Thus 

the fluctuations in water levels caused by the passing of a train usually 

appear as vertical lines on water-stage recorder charts because the 

time scale ordinarily used is too small to record the fluctuations in 

any greater detail . 

The diagram in figure 21B shows, schematically, the effect of an 

instantaneously applied load on the pressure distribution within an 

elastic artesian aquifer and on the compression and subsequent ex­
82 GROUND-WATER HYDRAULICS 

c a v 

v +0 .03­

u " 0 .02-

-wvv 

>~ +0 .0 11 

~, 0 

-0 .02J 

1-- - 0 .031 0 

b, 19391 

A . WATER-LEVEL FLUCTUATION IN WELL 5-201, LONG ISLAND, N .Y ., PRODUCED 

BY EAST-BOUND FREIGHT TRAIN, MARCH 21, 1938 

Land 

surface 

Confining

material 

Aquifer 

B. SCHEMATIC DIAGRAM SHOWING EFFECT OF INSTANTANEOUS APPLICATION 

AND SUBSEQUENT REMOVAL OF A SINGLE CONCENTRATED LOAD AT THE 

LAND SURFACE ON THE PRESSURE DISTRIBUTION AND ON THE COMPRESSION 

AND SUBSEQUENT EXPANSION OF AN ELASTIC ARTESIAN AQUIFER 

FIGURE 21 .-Effects of changes in loading on artesian aquifer. 

pansion of the aquifer after removal of the load. In discussing figure 

21B Jacob (1939) says : 

The upper diagrams a, b, c, and d show the distribution of pressure and the 

deflection of the upper surface of the aquifer at the respective times indicated on 

the time-pressure and time-distribution curves. The hydrostatic pressure in the 

aquifer is plotted as a full line, the upper limit of the confining layer arbitrarily 

being adopted as a base. The deflection curve for the upper surface of the aquifer 

is plotted as a dashed line . (The lower surface of the aquifer is assumed fixed .) 

These quantities are, of course, grossly exaggerated and are obviously plotted to 

quite different scales . The length of the arrows indicates the relative magnitude 

of the velocity of flow at various distances from the load. The lower diagram 

. . . shows, by the heavy full line, the change in pressure produced by the load, 

and, by the heavy dashed line, the deflection of the upper surface of the aquifer, 

plotted against time. 
THEORY OF AQUIFER TESTS 83 

CHANGES IN ATMOSPHERIC PRESSURE 

It has often been observed that water levels in wells tapping artesian 

aquifers respond to changes in atmospheric pressure . An increase 

in the atmospheric pressure causes the water level to decline, and a 

decrease in atmospheric pressure causes the water level to rise . The 

diagrams shown in figure 22 will aid in explaining why this phenomenon 

is observed in artesian wells and why it ordinarily is not observed in 

water-table wells. 

IDEALIZED SECTION OF WATER-TABLE AQUIFER 

B, IDEALIZED SECTION OF ARTESIAN AQUIFER 

FiouRB 22.-Effect of atmospheric pressure loading on aquifers . 
84 GROUND-WATER HYDRAULICS 

Referring to diagram A, figure 22, the force Apo, representing the 

change in atmospheric pressure, is exerted on the free water surface in 

the well . The same force Apo is also exerted simultaneously on the 

water table because there is direct communication between the 

atmosphere and the water table through the unsaturated pore space of the soil . Thus the system of forces remains in balance and there is 

no appreciable change in water level in the well with changes in atmospheric pressure . Some water-table wells exhibit barometric fluctuations if the soil is frozen or saturated with water. But either of these 

conditions is, in effect, only a special case of the artesian condition. 

Referring to diagram B, figure 22, the force Apo, which again represents the change in atmospheric pressure, acts on the free water surface in the well and also on the layer of material confining the artesian 

aquifer. Jacob (1940) in discussing this situation, reasons that barometric fluctuations in a well are an index of the elasticity of the aquifer. 

In other words the confining layer, viewed as a unit, has no beam 

strength or resistance to deflection sufficient to withstand or contain 

any sensible part of an applied load. Thus in effect any charges in 

the atmospheric pressure loading on a confining layer are transmitted 

through it undiminished in magnitude. The forces acting at a point 

at the interface between the aquifer and the confining layer may then 

be drawn as shown in the inset sketch . Observe that the change in 

atmospheric pressure, Apo, is now accommodated by a change in stress 

in the skeleton of the aquifer, As,,, plus a change in the water pressure

in the aquifer, Op, applied over the percentage b of the interface, 

where the water is in direct contact with the confining layer. It is 

evident, therefore, that in an artesian situation there will be a pressure

differential between an observation well where the water is directly

subject to the full change in atmospheric pressure, and a point out in 

the aquifer where the water is required to accept only part of the 

change in atmospheric pressure. Thus barometric fluctuations will 

be observed in the well . Some wells near the outcrop of an artesian 

aquifer or near a discontinuity in the confining layer will show little or 

no response to atmospheric pressure changes. 

Although not associated with the elasticity of artesian aquifers, it is 

interesting to note that the phenomena of blowing and sucking wells, 

which exhibit a pronounced updraft or downdraft of air at the well 

mouth, may also be related to changes in atmospheric pressure. In 

areas where such wells have been noted, a bed of fine-grained, relatively impermeable material usually lies some distance above the 

water table, thereby effectively confining, in the intervening un­
THEORY OF AQUIFER TESTS 85 

saturated pore space, a body of air that can communicate with the 

atmosphere only through wells. 

The barometric efficiency of an aquifer may be expressed as 

BE='-

8b 

where s,, is the net change in water level observed in a well tapping the 

aquifer and sb is the corresponding net change in atmospheric pressure, 

both expressed in feet of water. It is frequently convenient to determine the barometric efficiency by plotting the water-level changes as 

ordinates and the corresponding changes in atmospheric pressure as 

abscissas on rectangular coordinate paper. The slope of the straight 

line drawn through the plotted points is the barometric efficiency. 

TIDAL FLUCTUATIONS 

OCEAN, LAKE, OR STREAM TIDES 

Water levels in wells near the ocean or near some lakes or streams 

exhibit semidiurnal fluctuations in response to tidal fluctuations . In 

wells tapping water-table aquifers, the water-level response to tidal 

fluctuations is due to actual movement of water in the aquifer. However, in wells tapping artesian aquifers that are effectively separated 

from the body of surface water by an extensive confining layer, the 

response is due to the changing load on the aquifer, transmitted 

through the confining layer with the changing tide . Thus with the 

rise of the tide the load on the aquifer is increased, which means that 

in the aquifer there will be compensating increases of the water pressure and of the stress in the skeleton . Accordingly, the water-level 

rise in the well is but a reflection of the increased pressure head in the 

aquifer caused by the tidal loading. 

An artesian well that responds to tidal fluctuations should also 

respond to changes in atmospheric pressure, because the same mechanism in the aquifer produces both types of response. 

The tidal efficiency of an aquifer may be expressed as 

TE=L-S t 

where s. is the range of water-level fluctuation, in feet, in a well tapping the aquifer, and s= is the range of the tide, in feet, corrected for 

density when necessary. There is a direct relation between the tidal 

efficiency or the barometric efficiency and the coefficient of storage. 

This relation will be discussed in a later section of this report . 
86 GROUND-WATER HYDRAULICS 

Jacob (1950, p. 331-332) has derived expressions relating the tidal 

and barometric efficiency and the elasticity of an artesian aquifer. 

The two pertinent equations are 

«/e0

TE=1 +a/B# 

and 

where a is the bulk modulus of compression of the solid skeleton of the 

aquifer, 0 is the bulk modulus of compression of water (reciprocal of 

the bulk modulus of elasticity), and 6 is the porosity of the aquifer. 

If these two equations are added it is evident that the sum of the 

barometric efficiency and the tidal efficiency equals unity, that is, 

BE+TE=1 

EARTH TIDES 

It has been observed that earth tides, which are caused by the 

forces exerted on the earth's surface by the sun and the moon, may 

produce water-level fluctuations in artesian wells. Water-level fluctuations due to earth tides were apparently first observed by Klonne 

(1880) in a flooded coal mine at Dux, Bohemia. Such fluctuations in 

wells were first observed by Young (1913) near Cradock, South Africa . 

After the water levels have been adjusted for changes in atmospheric 

pressure the "high" water levels have been observed near moonrise 

and moonset and the "low" water levels near the upper and lower 

culminations of the moon. 

For a well near Carlsbad, New Mexico, and for a well at Iowa City, 

Iowa, Robinson (1939) showed that the water-level fluctuations, after 

adjustment for changes in atmospheric pressure, are coincident with 

the earth tides. The low water levels showed a tendency to precede 

the culmination of the moon, suggesting that the tide in the well 

precedes the culmination of the moon . 

According to Theis (1939), the possible effects of tidal forces acting 

either directly upon the water in the aquifer or upon the aquifer 

itself by varying the weight of the overburden could not account for 

the observed water-level fluctuations . The explanation for these 

water-level fluctuations is probably the distortion of the earth's 

crust. In this regard Theis (1939) states : 

As the crust of the earth in any given area rises and falls with the deformation 

of the earth caused by the tidal forces the crust is most probably alternately 
THEORY OF AQUIFER TESTS 87 

expanded and compressed laterally-expanded when the earth bulges up and 

compressed when it subsides . Water in an artesian aquifer making up part of 

the crust shares in this deformation. In localities distant from points of outflow 

of the water, it is in effect confined without possibility of outflow within the period 

of tidal fluctuations . The slight hydraulic gradient imposed by the tidal distortion is too small to cause effective release of pressure . Hence the aquifer is 

essentially sealed with respect to its included fluid. With the expansion of the 

aquifer incident to the tidal bulge the hydrostatic pressure falls and with its 

compression incident to tidal depression the hydrostatic pressure rises. 

EARTHQUAKES 

Fluctuations in water levels due to earthquakes have been observed 

in many wells equipped with water-stage recorders. Veatch (1906, 

p. 70) was apparently one of the first hydrologists in this country to 

recognize that some water-level fluctuations might be in response to 

earthquake disturbances . Subsequent investigators who have published papers on the subject include H. T. Stearns (1928), Piper (1933), 

Leggett and Taylor (1935), LaRocque (1941), Parker and Stringfield 

(1950), and Vorhis (1953) . An earthquake may be defined as a 

vibration or oscillation of the earth's crust caused by a transient 

disturbance of the elastic or gravitational equilibrium of the rocks at 

or beneath the land surface. Earthquakes are classified as shallow 

or deep depending on the vertical position, relative to the land surface, 

of the source of the disturbance. Shock waves, propagated by an 

earthquake, travel through the earth and along the earth's surface. 

Because the earth is an elastic body it is first compressed by the shock 

waves and subsequently it expands after the sho,-k wave is dissipated. 

Where an aquifer is included in the segment of the earth affected by 

the shock waves of an earthquake there will first be an abrupt increase 

in water pressure as the water assumes part of the imposed compressive 

stress, followed by an abrupt decrease in water pressure as the imposed stress is removed. In attempting to adjust to the pressure 

changes, the water level in an artesian well first rises and then falls. 

The amounts of the rise and fall of the water level, with respect to the 

initial position, are approximately the same. Cases have been recorded, however, where the water level did not return to its initial 

position (Brown, 1948, p. 193-195) . This is presumably due to 

permanent rearrangement of the grains of material composing the 

aquifer. Fluctuations of greater magnitude have been observed in 

wells in limestone aquifers than in wells in granular material . 

The following table gives the types of shock waves caused by 

earthquakes, the approximate average velocities at which they travel 

and the path they take. 
$$ GROUND-WATER HYDRAULICS 

TABLE 1 .-Approximate average velocities and paths taken by different types of shock 

waves caused by earthouakes 

[After Byerly, P.,1933, p.155] 

Approximate average velocity 

Type of shock wave Path (km per (ft per se )c sec) ( min) r I (hr~er 

Deep-seated--_____---------__----------------------- 5 .5 18, 000+ 2051 112, 030+ Chord Surface-----------------_----------__---------------- 3 .2 10, 500+ 120+ 7,200+ Arc 

COEFFICIENT OF STORAGE AND ITS RELATION TO ELASTICITY 

The coefficient of storage is a function of the elasticity of an artesian 

aquifer. Jacob (1950) has expressed the relation as 

S=7o0m(Q+e 

where in this instance yo is the specific weight of the water at a stated 

reference temperature, and 0, m, (3, and a are as defined earlier in 

this report . This formula assumes no leakage from or into contiguous beds. 

Digressing momentarily, the specific weight, y, of a fluid at a 

stated reference temperature is defined as its density, p, multiplied by the local acceleration due to gravity, g. Stated another way, it is the 

weight per unit volume that takes into account the magnitude of the 

local gravitational force. The manner in which specific weight is 

related to such more commonly used properties as mass, weight, and 

density can be developed in the following fashion. First it should be 

recognized that in the English engineering system the unit of mass is 

termed a "slug" . It is the mass in which an acceleration of 1 ft per 

sec per sec. is produced by a force of 1 lb. Thus 1 slug of mass is 

approximately equal to 32.2 lb of mass . The mass, M, in slugs, of 

any substance is determined from the relation 

M=W 

9 

where W is the weight of the substance in pounds . Inasmuch as 

weight is dependent on the local gravitational force it obviously varies 

with location . Thus, the fraction W/g takes into account in both 

numerator and denominator the local force of gravity, which shows 

that mass is an absolute property that does not change with location . Density is defined as mass per unit volume, V. That is 

Density V 
THEORY OF AQUIFER TESTS 89 

Thus density also is an absolute property that does not change with 

location . (Slight changes occur with change in temperature but 

ordinarily, within the range of ground-water temperatures encountered, 

no adjustment for this factor is necessary.) If the original relation 

given for determining mass, 111, is solved for the weight, W, and if both 

sides of the rewritten equation are divided by the volume, V, there 

follows 

The brackets on the left side of the equation are now seen to include 

the fraction equivalent to specific weight and the brackets on the 

right side include the fraction equivalent to density. That is : 

7=pg 

Returning to the discussion at hand, it is important to point out 

that considerable study remains to be done regarding the elastic 

behavior of artesian reservoirs . It would be worthwhile, for example, 

to compute the values of the bulk modulus of compression («) of the 

solid skeleton of the various artesian reservoirs where coefficients of 

storage have been determined from various types of aquifer tests. 

R. R. Bennet,t (oral communication) has supplied the following data 

for the principal aquifer in the Baltimore, 'Id., area: 

S=0.0002 

9=0.30 

m=100 feet=1200 inches 

Recognizing that for water 

62 .4 

70=62.4 lbs per cu ft=1728 (or 0.0361 lb per cu in) 62 

and 

Q 33X10'6 (sq in per lb) 3001000 

appropriate substitution of all known quantities is made in Jacob's 

equation for the storage coefficient and the value of a is computed. 

Thus 

0.0002=(0 .0361)(0 .30)(1200) L

0.0000033+ 0 30] 

or 

a=0.00000363 sq in per lb or 3 .63 X 10-6 sq in per lb 

Conversely, it is possible to determine the coefficient of storage if a is 
90 GROUND-WATER HYDRAULICS 

known. Jacob (1941) determined from pumping tests that the modulus of compression («) of the Lloyd sand member of the Raritan for 

mation on Long Island, N.Y., was 2 .1 X 10--8 in per lb . In a location 

where this sand has a porosity (0) of 0.30 and is confined to form an 

aquifer having a thickness (m) of 50 feet the coefficient of storage 

may be computed as 

2.1X10-1 S= (0 .0361) (0 .30) (50X 12) [](3 .3X10-8)+ 0.30 

S=6.7X10-5 

The coefficient of storage is related to barometric and tidal efficiency . 

As stated previously, Jacob (1940) showed mathematically that the 

sum of the barometric efficiency and tidal efficiency must equal unity 

(BE+TE=1). He showed further than when b=1 the storage coefficient is related to the barometric efficiency as follows, 

S=('Yoem#) [BE]1

using the same terminology as before and assuming no leakage from or 

into contiguous beds . By observing tidal fluctuations in a well, 

screened in the Lloyd sand member of the Raritan formation on Long 

Island, Jacob (1940) could then determine tidal efficiency and compute 

the coefficient of storage. His computations for well Q-288, near 

Rockaway Park, where the tidal efficiency was determined as 42 

percent, are as follows : 

_1 _ 1 _ 1 -1.72. BE 1- (TE) 1-0.42 

Ascribing a thickness of 200 feet and a porosity of 0.35 to the Lloyd 

sand member in the vicinity of Rockaway Park, Long Island, and substituting in the foregoing equation for storage coefficient, there 

resultsS= (0 .0361) (0 .35) (200 X12) [3001000] (1'72)' 

S=1.7X10-4 . 

This value for the coefficient of storage is comparable to the values 

determined from pumping tests. 

Jacob (1941) observed that the compressibility of the Lloyd sand 

member, as computed from a discharging-well test, was about 2% 

times that computed from tidal fluctuations and reasoned that this 
THEORY OF AQUIFER TESTS 91 

disparity might be due, in part, to the range of stress involved . He 

states­

* * * during the pumping test of 1940 the head in the Lloyd sand declined to 

a new low over a considerable area in the vicinity of the pumped wells, and consequently the stress in the skeleton of the aquifer reached a new high. It is to be 

expected that the modulus of elasticity would be smaller for the new, higher 

range of stress than for the old range over which the stress had fluctuated many 

times. 

AQUIFER TESTS-BASIC THEORY 

WELL METHODS-POINT SINK OR POINT SOURCE 

CONSTANT DISCHARGE OR RECHARGE WITHOUT VERTICAL LEAKAGE 

EQUILIBRIUM FORMULA 

Wenzel (1942, p. 79-82) showed that the equilibrium formulas 

used by Slichter (1899), Turneaure and Russell (1901), Israelson 

(1950), and Wyckoff, Botset, and Muskat (1932) are essentially 

modified forms of a method developed by Thiem (1906), as are the 

formulas developed by Dupuit (1848) and Forchheimer (1901) . 

Thiem apparently was the first to use the equilibrium formula for 

determining permeability and it is frequently associated with his 

name. The formula was developed by Thiem from Darcy's law and 

provides a means for determining aquifer transmissibility if the rate 

of discharge of a pumped well and the drawdown in each of two 

observation wells at different known distances from the pumped 

well are known . The Thiem formula, in nondimensional form, can be 

written as 

T= Q log, (r2/rl) 

27r(sl - s2) 

where the subscript e in the log term indicates the natural logarithm. 

In the usual Geological Survey units (see p. 73), and using common 

logarithms, equation 1 becomes 

T_527 .7Q logo (r2lrl) ~ (2) 81 -82 

where 

T=coefficient of transmissibility, in gallons per day per foot, 

Q=rate of discharge of the pumped well, in gallons per minute, 

rl and r2 =distances from the pumped well to the first and second 

observation wells, in feet, and 

sl and s2 =drawdowns in the first and second observation wells, in feet . 

The derivation of the formula is based on the following assumptions : 

(a) the aquifer is homogeneous, isotropic, and of infinite areal extent ; 

(b) the discharging well penetrates and receives water from the entire 

thickness of the aquifer ; (c) the coefficient of transmissibility is constant at all times and at all places ; (d) pumping has continued at a 
92 GROUND-WATER HYDRAULICS 

uniform rate for sufficient time for the hydraulic system to reach a 

steady-state (i .e ., no change in rate of drawdown as afunction of time) 

condition ; and (e) the flow is laminar. The formula has wide application to ground-water problems despite the restrictive assumptions 

on which it is based . 

The procedure for application of equation 2 is to select some convenient elapsed pumping time, t, after reaching the steady-state condition, and on semilog coordinate paper plot for each observation 

well the drawdowns, s, versus the distances, r. By plotting the values 

of s on the arithmetic scale and the values of r on the logarithmic scale, 

the observed data should . lie on a straight line for the equilibrium 

formula to apply. From this straight line an arbitrary choice of 

s1 and 82 should be madeand the corresponding values of rl and r2recorded. 

Equation 2 can then be solved for T. 

Jacob (1950, p. 368) recognized that the coefficient of storage 

could also be determined if the hydraulic system had reached a steadystate condition (see assumption d, above), for thereafter the drawdown 

is expressed very closely by the nondimensional formula 

2.25Tt 

s_ Q 

-4rT loge r2 S (3) 

or, in the usual Survey units and using common logarithms, 

264Q, 0.3Tt 

8=-T 0910 r2S, (4) 

Thus after the coefficient of transmissibility has been determined, the 

coordinates of any point on the semilogarithmic graph previously 

described can be used to solve equation 4 for the coefficient of storage. 

NONEQIIILIBRIIIM FORMULA 

Theis (1935) derived the nonequilibrium formula from the analogy 

between the hydrologic conditions in an aquifer and the thermal 

conditions in an equivalent thermal system. The analogy between 

the flow of ground water and heat conduction for the steady-state 

condition has been recognized at least since the work of Slichter (1899), 

but Theis was the first to introduce the concept of time to the mathematics of ground-water hydraulics . Jacob (1940) verified the derivation of the nonequilibrium formula directly from hydraulic concepts . 

The nonequilibrium formula in nondimensional form is 

u e. du, s-4Q fr23/4Tj u 

where u=r2Sl4Tt, and where the integral expression is known as 

an exponential integral . 
THEORY OF AQUIFER TESTS 93 

Using the ordinary Survey units equation 5 may be written as 

e_U 114.6Q 

T f.870S/Tt u , 

where 

u=1.87r2S/Tt, 

s=drawdown, in feet, at any point of observation in the vicinity 

of a well discharging at a constant rate, 

Q=discharge of a well, in gallons per minute, 

T=transmissibility, in gallons per day per foot, 

r=distance, in feet, from the discharging well to the point of 

observation, 

S=coefficient of storage, expressed as a decimal fraction, 

t=time in days since pumping started . 

The nonequilibrium formula is based on the following assumptions : 

(a) the aquifer is homogeneous and isotropic; (b) the aquifer has 

infinite areal extent ; (c) the discharge or recharge well penetrates 

and receives water from the entire thickness of the aquifer; (d) the 

coefficient of transmissibility is constant at all times and at all places ; 

(e) the well has an infinitesimal (reasonably small) diameter ; and 

(f) water removed from storage is discharged instantaneously with 

decline in head. Despite the restrictive assumptions on which it is 

based, the nonequilibrium formula has been applied successfully 

to many problems of ground-water flow. 

The integral expression in equation 6 cannot be integrated directly, 

but its value is given by the series 

eu" du=W(u)=-0.577216-log, u+u .,J 1'7rIS/Ti 

2 3 4 -22 ! -+- .3! -44! . . . . . (7) 

where, as already indicated, 

_1 87r2S 

u Tt 

The exponential integral is written symbolically as W(u) which 

is read "well function of u." Values of W(u) for values of u from 

10-1 b to 9.9, as tabulated in Wenzel (1942), are given in table 2 . In 

order to determine the value of W(u) for a given value of u, using 

table 2, it is necessary to express u as some number (N) between 1 .0 

and 9 .9, multiplied by 10 with the appropriate exponent . For example, when u has a value of 0.0005 (that is, 5 .O X10-4 ), W(u) is 
94 GROUND-WATER HYDRAULICS 

determined from the line N=5.0 and the column NX10-4 to be 

7.0242 . 

Referring to equations 6 and 8, if s can be measured for one value 

of r and several values of t, or for one value of t and several values of r, 

and if the discharge Q is known, then S and T can be determined . 

Once these aquifer constants have been determined, it is possible, 

theoretically, to compute the drawdown for any time at any point on 

the cone of depression for any given rate and distribution of pumping 

from wells. It is not possible, however, to determine T and Sdirectly 

from equation 6, because T occurs in the argument of the function and 

again as a divisor of the exponential integral . Theis devised a convenient graphical method of superpoFition that makes it possible to 

obtain a simple solution of the equation. 

The first step in this method is the plotting of a type curve on 

logarithmic coordinate paper. From table 2 values of W(u) have 

been plotted against the argument u to form the type curve shown in 

figure 23 . It is shown in two segments, A-A and B-B, in order that 

the portion of the type curve necessary in the analysis of pumping 

test data could be plotted on a sheet of convenient size . Curve B-B 

is an extension of curve A-A and overlaps curve A-A for values of 

W(u) from about 0.22 to 1 .0 . 

Rearranging equations 6 and 8 there follows 

S-[114.6Q] 

T W(u) (9) 

or 

log s=Clog 11 TQ]-{-log W(u) (9a) 

and 

z 

t -[1 .87S] u (10) 

or 

log t=[log 1 (l0a) .g7S]--log u 

If the discharge, Q, is held constant, the bracketed parts of equations 

9a and 10a are constant for a given pumping test, and W(u) is related to 
3 gapote eucto(u)costat dscage 

O 

1-4 

O 

10d 

H 

sn 

u 

FIGURE .-Logarithmic 
~Q~VN~O~C+~NCD N-N ~ .A W 00

0 8 -I O2 9Vm00.&M- W OUDV m mN 

C. . . . .. . .. . . .rrrrN-.-~ rrN r ~ N

+rN . . . . . . . . . . . . . . . . . . . . . . . . . . .


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98 GROUND-WATER HYDRAULICS 

u in the manner that s is related to r1lt . This is shown graphically

in figure 24 . Therefore, if values of the drawdown s are plotted against 

r2/t, or 1/t if only one observation well is used, on logarithmic tracing 

paper to the same scale as the type curve, the curve of observed data 

will be similar to the type curve. The data curve may then be superposed on the type curve, the coordinate axes of the two curves being 

held parallel, and translated to a position which represents the best 

fit of the field data to the type curve. An arbitrary point is selected 

anywhere on the overlapping portion of the sheets and the coordinates 

of this commwu poizit. on both sheets are rewrdecl. It is often cocuagnient to select a point whose coordinates are both 1 . These data are 

then used with equations 9 and 10 to solve for T and S. 

FIGURE 24 .-Relation of W(u) and u to a and r2/t . 

A type curve on logarithmic coordinate paper of W(u) versus 1/u, 

the reciprocal of the argument, could have been plotted . Values of 

the drawdown (or recovery), s, would then have been plotted versus 

t, or tlr2 and superposed on the type curve in the manner outlined 

above. This method eliminates the necessity for computing 1/t values 

for the values of s. 

MODIFIED AONEQUILIBRIUM FORMULA 

It was recognized by Jacob (1950) that in the series of equation 7 

the sum of the terms beyond logeu is not significant when u becomes 

small. The value of u decreases as the time, t, increases and as r 

decreases. Therefore, for large values of t and reasonably small 
THEORY OF AQUIFER TESTS 99 

values of r, the terms beyond logeu in equation 7 may be neglected . 

When r is large, t must be very large before the terms beyond log,u in 

equation 7 can be neglected. Thus the Theis equation in its abbreviated or modified nondimensional form is written as 

__ Q loge 4Tt_0 s .5772) 41rT rYS 

=4Qloge 2.25Tt 

which is obviously identical with equation 3. In the usual Survey 

units, then, this equation will be identical with equation 4, all terms 

being as previously defined. 

In applying equation 4 to measurements of the drawdown or 

recovery of water level in a particular observation well, the distance 

r will be constant, and it follows that 

at time tl, s1= 2

TQ (loglo0.3Tt1)~ 

at time t2, s2 =2

TQ (log r2 S2 )~ 

and the change in drawdown or recovery from time tl to t 2 is 

. s2-s1= 2TQ(loglo tl) 

Rewriting this equation in form suitable for direct solution of T, 

there follows 

Z,-264Q(loglo t21t1) ~ (11) 

s2 -81 

where Q and T are as previously defined, tl and t 2 are two selected 

times, in any convenient units, since pumping started or stopped, and 

s l and 82 are the respective drawdowns or recoveries at the noted 

times, in feet . The most convenient procedure for application of equation 11 is 

to plot the observed data for each well on the semilogarithmic coordinate paper, plotting values of t on the logarithmic scale and values 

of s on the arithmetic scale. After the value of u becomes small 

(generally less than 0.01) and the value of time, t, becomes great, 

the observed data should fall on a straight line . From this straight 

line make an arbitrary choice of tl and t2 and record the corresponding 

values of sl and 82 . Equation 11 can then be solved for T. For 
100 GROUND-WATER HYDRAULICS 

convenience, t l and t2 are usually chosen one log cycle apart, because 

then 

logio t2 t=1 

and equation 11 reduces to 

T=2s4Q (12) As 

where As is the change, in feet, in the drawdown or recovery over one 

log cycle of time. 

The coefficient of storage also can be determined from the same 

semilog plot of the observed data. When s=0, equation 3 becomes 

s=0=4Qloge 2.25Tt 

r2s 

Solving for the coefficient of storage, S, the equation in its final form 

becomes 

S, -2.25Tt 2 (13) r

or, in the usual Survey units, 

S-0.3Tt o 

r2 (14) 

where S, T, and r are as pre,,iously defined and to is the time intercept, 

in days, where the plotted straight line intersects the zero-drawdown 

axis . If any other units were used for the time, t, on the semilog plot, 

then obviously tO must be converted to days before using equation 14. 

Lohman (1957) has described a simple method for determining S 

using the data region of the straight-line plot without extrapolating 

to the zero-drawdown axis . 

THEIS RECOVERY FORMULA 

A useful corollary to the nonequilibrium formula was devised by

Theis (1935) for the analysis of the recovery of a pumped well . If 

a well is pumped, or allowed to flow, for a known period of time and 

then shut down and allowed to recover, the residual drawdown at 

any instant will be the same as if the discharge of the well had been 

continued but a recharge well with the same flow had been introduced 

at the same point at the instant the discharge stopped . The residual 

drawdown at any time during the recovery period is the difference 

between the observed water level and the nonpumping water level 
THEORY OF AQUIFER TESTS 101 

extrapolated from the observed trend prior to the pumping period,

The residual drawdown, s', at any instant will then be 

s1= 114T.6 Q[f' 

e-u du-f' , -e (15) z 2 .s7r SITI u 1 .s7r s/TC 'uu duJ 

where Q, T, S, and r are as previously defined, t is the time since 

pumping started, and t' is the time since pumping stopped. The 

quantity 1 .87r2S/Tt' will be small when t' ceases to be small because 

r is very small and therefore the value of the integral will be given closely by the first two terms of the infinite series of equation 7, 

Equation 15 can therefore be written, in modified form, in the usual 

Survey units, as 

264Q

T= s710910 F

t (16) 

The above formula is similar in form to, and is based on the same 

assumptions as, the modified nonequilibrium formula developed by

Jacob, and it permits the computation of the coefficient of transmissibility of an aquifer from the observation of the rate of recovery 

of water level in a pumped well, or in a nearby observation well 

where r is sufficiently small to meet the above assumptions. 

The Theis recovery formula is applied in much the same manner as 

the modified nonequilibrium formula. The most convenient procedure is to plot the residual drawdown, s', against tlt' on semilogarithmic coordinate paper, s' being plotted on the arithmetic scale and 

tlt' on the logarithmic scale. After the value oft' becomes sufficiently

large, the observed data should fall on a straight line . The slope of 

this line gives the value of the quantity log, 0 (tlt') Is' in equation 16. 

For convenience, the value of tlt' is usually chosen over one log cycle 

because its logarithm is then unity and equation 16 then reduces to 

T-264 Q 

As' (17) 

where Os' is the change in residual drawdown, in feet, per log cycle of 

time. It is not possible to determine the coefficient of storage from 

the observation of the rate of recovery of a pumped well unless the 

effective radius, r, which is usually difficult to determine, is known. 

The Theis recovery formula should be used with caution in areas 

where it is suspected that boundary conditions exist. If a geologic

boundary has been intercepted by the cone of depression during pumping, it maybe reflected in the rate of recovery of the pumped well, and 

the value of T determined by using the Theis recovery formula could 

be in error. With reasonable'care the recovery in an observation well 
102 GROUND-WATER HYDRAULICS 

can be used, of course, to determine both transmissibility and storage, 

whether or not boundaries are present. 

APPLICABMTY OF METHODS TO ARTESIAN AND WATER-TABLE AQUIFERS 

The methods previously discussed have been used successfully for 

many years in determining aquifer constants and in predicting the 

performance of both water-table and artesian aquifers . The derivations of the equations are based, in part, on the assumptions that the 

coefficient of transmissibility is constant at all times and places and 

that water is released from storage instantaneously with decline in 

head . It should be recognized, however, that these and many other 

idealizations are necessary before mathematical models can be used 

to analyze the physical phenomena associated with ground-water 

movement . Thus the hydrologist cannot blindly select a model, turn 

a crank, and accept the answers. He must devote considerable time 

and thought to judging howclosely his real aquifer resembles the ideal . 

If enough data are available he will always find that no ideal aquifer, 

of the type postulated in the theory, could reproduce the data obtained 

in an actual pumping test . He should understand that the dispersion 

of the data is a measure of how far his aquifer departs from the ideal. 

Therefore, he must plan his test procedures so that they will conform 

as closely as possible to the theory and thus give results that can 

safely be applied to his aquifer. He must be prepared to find out, 

however, that his aquifer is too complex to permit a clear evaluation of 

its coefficients of transmissibility and storage. He must not tell 

himself or the reader that "the coefficient of storage changed" during

the test but must realize that he got different values when he tried to 

apply his data, inconsistently, to an ideal theoretical aquifer. 

Thus there is little justification for the premise that the storage

coefficient of a water-table aquifer varies with the time of pumping,

inasmuch as such anomalous data are merely the results of trying to 

apply a two-dimensional flow formula to a three-dimensional problem. 

The nonequilibrium formula was derived on the basis of strictly radial 

flow in an infinite aquifer and its application to situations where 

vertical-flow components occur is not justified except under certain 

limiting conditions . As the time of pumping becomes large, however, 

the rate of water-level decline decreases rapidly so that eventually the 

effect of vertical-flow components in water-table aquifers are minimized. 

If the drawdowns are large compared to the initial depth of flow, 

it is necessary to adjust the observed drawdown in a pumping test of a 

water-table aquifer before the nonequilibrium formula is applied. 

According to Jacob (1944, p. 4) if the observed drawdowns are 

adjusted (reduced) by the factor s2/2m, where 8 is the observed draw­
THEORY OF AQUIFER TESTS 103 

down and mis the initial depth of flow, the value of Twill correspond 

to equivalent confined flow of uniform depth, and the value of S will 

more closely approximate the true value. He adds that when the 

drawdowns are adjusted the nonequilibrium formula can be used with 

fair assurance even when the dewatering is as much as- 25 percent of 

the initial depth of flow. 

Where the discharging well only partially penetrates the aquifer it 

mayalso be necessary to adjust the observed drawdowns. Procedures 

for accomplishing this have been described by Jacob (1945) . 

INSTANTANEOUS DISCHARGE OR RECHARGE 

"BAMEB" METHOD 

Skibitzke (1958) has developed a method for determining the 

coefficient of transmissibility from the recovery of the water level in a 

well that has been bailed . At any given point on the recovery curve 

the following equation applies : 

,_= V (18) 

s 47rTt ler,,ZS/4TtI 

where 

s'=residual drawdown, 

V=volume of water removed in one bailer cycle, 

T=coefficient of transmissibility, 

S=coefficient of storage, 

t=length of time since the bailer was removed, 

r,,, =effective radius of the well . 

The effective radius, r , of the well is very small in comparison to 

the extent of the aquifer. As r,o is small, the term in brackets in 

equation 18 approaches unity as t increases. Therefore for large 

values of t, equation 18 may be modified and rewritten, in consistent 

units, as 

V V 

s 19) 47rTt=-12.57Tt' 

where s', T, and t have units and significance as previously defined, 

and where V represents the volume of water, in gallons, removed 

during one bailer cycle. If the residual drawdown is observed at 

some time after completion of n bailer cycles then the following 

expression applies 

s

,_ 1 (20) 12.57T 

where the subscripts merely identify each cycle of events in sequence . 
104 GROUND-WATER HYDRAULICS 

Thus V3 represents the volume of water removed during the third 

bailer cycle and to is the elapsed time from the instant that water 

was removed from storage to the instant at which the observation 

of residual drawdown was made. 

If approximately the same volume of water is removed by the 

bailer during each cycle, then equation 20 becomes 

1 s,_ V 1 11 . (21) 12.57T ltt2+t+ 3

. . 

t 

The "bailer" method is thus applied to a single observation of the 

residual drawdown after the time since bailing stopped becomes large. The transmissibility is computed by substituting in equation 21 the 

observed residual drawdown, the volume of water Vconsidered to be 

the average amount removed by the bailer in each cycle, and the 

summation of the reciprocal of the elapsed time, in days, between 

the time each bailer of water was removed from the well and the time 

of observation of residual drawdown. 

"SLUG" METHOD 

Ferris and Knowles (1954) discuss a convenient method for estimating the coefficient of transmissibility, under certain conditions . 

This is done by injecting a given quantity or "slug" of water into a 

well . Their equation for determining the coefficient of transmissibility is the same as the equation derived by Skibitzke for the bailer 

method, inasmuch as the effects of injecting a slug of water into a 

well are identical, except for sign, with the effects of bailing out 

a slug of water. Thus equation 19 has direct application, only s' 

now represents residual head, in feet, at the time t, in days, following 

injection of V gallons of water. 

As used in the field, this method requires the sudden injection of a 

known volume of water into a well and the collection thereafter of 

a rapid series of water-level observations to define the decay of the 

head that was built up in the well . An arithmetic plot of residual 

head values versus the reciprocals of the times of observation should 

produce a straight line whose slope, appropriately substituted in 

equation 19, permits computation of the transmissibility . 

Suggested equipment for use in injecting a slug of water into a 

well, and for making the rapid series of water-level observations 

required immediately thereafter, is shown schematically in figure 25 . 

The duration of a "slug" test is very short, hence the estimated 

transmissibility determined from the test will be representative only 

of the water-bearing material close to the well . Serious errors will 
THEORY OF AQUIFER TESTS 105 

Rope or wire 

Cover plate with fixed "eye" 

Gasket secured to cover plate 

-Flange and nipple making 

watertight connection 

with bottom of drum 

Surface 

A . APPARATUS FOR MAKING "SLUG"TEST 

Pipe 

Lead filler 

B . PLAN FOR PERCUSSION INSTRUMENT FOR RAPID 

MEASUREMENT OF WATER LEVELS 

FIGU$B 28.-Suggested equipment for a "slug" test . 

be introduced unless the observation well is fully developed and 

completely penetrates the aquifer. Use of the "slug" test should 

probably be restricted to artesian aquifers of small to moderate 

transmissibility (less than 50,000 gallons per day per foot) . 
106 GROUND-WATER HYDRAULICS 

CONSTANT HEAD WrMOIIT VERTICAL SAGE 

Controlled pumping tests have proved to be an effective tool in 

,determining the coefficients of storage and transmissibility . In the 

usual test the discharge rate of the pumped well is held constant, 

whereas the drawdown varies with time. The resulting data are 

analyzed graphically as previously described. Jacob and Lohman 

(1952) derived a formula for determining the coefficients of storage 

and transmissibility from a test in which the discharge varies with 

time and the drawdown is held constant . The formula, based on 

the assumptions that the aquifer is of infinite areal extent, and that 

the coefficients of transmissibility and storage are constant at all 

times and all places, is developed from the analogy between the hydrologic conditions in an aquifer and the thermal conditions in an 

equivalent thermal system . The formula is written as 

Q=2rTs.G(a), (22) 

where m 

G(a)=4 xe- z [2+tan-1 Yo(x)] dx (23) Jo 

and 

a_- Tt (24) r-2 S 

Using the customary Survey units, equations 22 and 24 are rewritten 

in the form 

Q=Ts.G(a) (25) 229 

and 

0.134 Tt a= (26) r,~2 S 

where Q, T, and t have the units and meaning previously defined and 

where 

s.=constant drawdown, in feet, in the discharging well, 

r.=effective radius, in feet, of the discharging well . 

The terms Jo(x) and Yo(x) are Bessel functions of zero order of the 

first and second kinds respectively. 

The integration required in equation 23 cannot be accomplished 

directly so it is necessary to replace the integral with a summation and 

solve it by numerical methods. In this fashion values of G(a) for 

values of a from 10-4 to 10'2 , have been tabulated by Jacob and Lohman, (1952), and are given herewith in table 3. The term G(a) is 
THEORY OF AQUIFER TESTS 107 

here designated as the "well function of a, constant-head situation ." 

This table is used in the same manner as table 2, which gives values 

of W(u) versus u. 

It is seen from equations 25 and 26 that if Q can be measured for 

several values of t and if the constant drawdown, s, and the effective 

radius, r., are known, S and T can be determined . It is not possible 

to determine S and T directly, however, since Toccurs both in the 

argument of the function and as a multiplier of G(a). A convenient 

graphical method, similar to that used in solving the nonequilibrium formula, makes it possible to obtain a simple solution . 

The first step in this method is the plotting of a type curve on logarithmic coordinate paper. From table 3, values of G(a) were plotted against the argument a to form the type curve shown in figure 26. 

It is shown in several segments in order that the entire type curve 

may be plotted on a sheet of convenient size . 

Rearranging equations 25 and 26 there follows : 

Q 229 G(a) 

or 

log Q= log-'29]-}-log G(a), (27) 1

and 

t= rWS 

0.13Tor 

log t=Clog 0r

.1`°3SST

]-{-log a. (28) 

If the drawdown, sm, is held constant, the bracketed parts of equations

27 and 28 are constant for any given test and log G(a) is related to 

log a in the same manner that log Q is related to log t. (Note the 

similarity in form between equations 27 and 28 and equations 9a and 

10a.) Therefore if values of the discharge, Q, are plotted against corresponding values of time, t, on logarithmic tracing paper to the same 

scale as the type curve, the curve of observed data will be similar to 

the type curve. The data curve may then be superposed on the type 

curve, the coordinated axes of the two curves being held parallel, and 

translated to a position that represents the best fit of the data to the 

type curve. An arbitrary point is selected on the overlapping portion of the sheets and the coordinates of this common point on both sheets 

are used with equations 25 and 26 to solve for Tand S. This graphical solution is similar to that used with the Theis nonequilibrium formula . 
FIGURE 26.--Logarithmic graph of well ftInctlon G(a)-coustan$ 4rawdown . 
THEORY OF AQUIFER TESTS 10 9 

TABLE 3.-Values of G(a) for values of a between 10-! and 10 12 

[From Jacob and Lohman, 1952, p. 561] 

10-+ 10-S 10-2 10-1 1 10 l0= 101 

1-------------------- 56.9 18.34 6.13 2.249 0.985 0.534 0.346 0.251

2____________________ 40.4 13.11 4.47 1.716 .803 .461 .311 .232 3____________________ 33.1 10.79 3.74 1.477 .719 .427 .294 .222

4____________________ 28.7 9.41 3.30 1.333 .667 .405 .283 .215 25.7 8.47 3.00 1.234 .830 .389 .274 .210 8____________________ 5-------------------- 23.5 7.77 2.78 1.160 .602 .377 .268 .206 21 .8 7.23 2.60 1.103 .580 .367 .263 .203

7-------------------- 20.4 8.79 2.48 1.0.57 .562 .359 .258 .200 8-------------------- 19.3 6.43 2.35 1 .018 .547 .352 .254 .198 9-------------------- 10------------------- 18.3 8.13 2.25 .985 .534 .346 .251 .196 

104 100 100 107 100 100 1010 1011 

1____________________ 0.1984 0.1608 0.1360 0.1177 0.1037 0.0927 0.0838 0 .0764 2-------------------- .1841 .1524 .1299 .1131 .1002 .0899 .0814 .0744 a-------------------- .1777 .1479 .1288 .1106 .0962 .0883 .0801 .0733

4-------------------- .1733 .1449 .1244 .1089 .0968 .0872 .0792 .0726

5.A __________________ .1701 .1426 .1227 .1076 .0958 .0884 .0785 .0720 8-------------------- .1675 .1408 .1213 .1088 .0950 .0857 .0779 .0716

7____________________ .1654 .1393 .1202 .1057 .0943 .0851 .0774 .0712 8-------------------- .1636 .1380 .1192 .1049 .0937 .0846 .0770 .0709 9-------------------- .1621 .1369 .1184 .1043 .0932 .0842 .0767 .0706

10 ------------------- .1608 .1360 .1177 .1037 .0927 .0838 .0784 .0704 

Jacob and Lohman (1952) showed that for large values of t, the 

function G(a) can be replaced by 2/W(u), and it has already been 

shown (see discussion, p. 99) that the approximate form of W(u) is 

given by 2.30 log,, (2 .25Tt/Sr.2) . Making this substitution for G(a)

in equation 22, there follows 

Q__ 47rTs,o/2 .30 

logo (2 .25Tt/r.2S) 

or, rearranging terms, 

2.30 t 2.30 2.25T (29) Q-47rT 1°g r2+ 47rT log S 

It should be evident from the form of equation 29, that if arithmetic 

values of the variable s./Q are plotted against logarithmic values of 

the variable tlr"2 the points will define a straight line . The slope of 

2 this line, in equation 29, is the prefix of the variable term log . (tlr. )

In other words, 

Slope of straight-line plot=A log 

2.30 2 =4T 

Once the slope of the graph is determined, therefore, the coefficient of 

transmissibility may be computed from the relation 

T_2.30A(log tlr,o2). (30) 4,rA(s /Q) 
GROUND-WATER HYDRAULICS 

If the slope is measured over one log cycle then the term 0 log (t/r.2 )

equals unity and equation 30 is further simplified to the form 

_ 2.30 

T-47r0 (s~,/Q) (31) 

The coefficient of storage could then be found by substituting in 

equation 29 the computed value of T and the coordinates of any convenient point on the straight-line plot . However, the computation is 

greatly simplified by noting that for the point where the straight-line 

plot intersects the logarithmic time axis (that is, where s /Q=0),

equation 29 becomes 

S= 2.25T(t/r,,2)o . (32) 

In the usual Survey units, equations 31 and 32 are written 

6 T o(s/Q) (33) 

and 

S=0.3T(t/r 2 ) o. (34) 

Thus equations 33 and 34 are applied through the simple device of 

a semilogarithmic plot where values of ,,,IQ are plotted on the arithmetic scale against corresponding values of t/r.

2 on the logarithmic

scale. 

The methods that have been outlined in this section are useful in 

determining the coefficient of transmissibility but should be used with 

caution in determining the coefficient of storage because it is often 

difficult to determine the effective radius of the pumped well . 

CONSTANT DISCHARGE WITH VERTICAL LEASAGE 

"L$AHY AQIIIFRR" FORMULA 

A problem of practical interest is that of an elastic artesian aquifer

that is replenished by vertical leakage through overlying or underlying semipermeable confining beds . In most places the confining beds 

only impede or retard the movement of ground water rather than 

prevent it . It is often true that this retardation of ground-water 

movement is sufficient so that the Theis equation (which assumes 

impermeable confining beds) can be applied. Nevertheless there will 

be occasions when departure of the test data from the predictions of 

the Theis equation will require investigation of the ability of the 

confining beds to transmit water. 

As an example of the magnitude of flow through material of low 

permeability, consider a semipermeable confining bed, 50 feet thick, 

consisting of silty clay that has a permeability of 0.2 gallon per day 

per square foot . Such a material is listed by Wenzel (1942, p. 13, 
THEORY OF AQUIFER TESTS 11 1 

lab. no. 2,278) as including about 49 percent (by weight) clay and 

about 45 percent silt. Assume that the confining bed is saturated 

and that in some manner there is established and maintained a head 

differential of 25 feet between the .top and bottom surfaces of the bed. 

The rate of percolation, related to this head differential, through the 

confining bed is computed from the previously given (see p. 73) variant 

of Darcy's law, 

Qd=P'IA, 

where, in this example, 

Qd=discharge in gallons per day through specified area of confining

bed, 

P'=vertical permeability of confining bed=0.2 gallon per day per 

square foot, 

I=hydraulic gradient imposed on confining bed=25/50=0.5 foot 

per foot . 

A=specified area of confining, bed through which percolation occurs . 

Thus, through a confining-bed area of one square foot, 

Qd=0.2 X0.5 X 1=0.1 gallon per day, 

or, through a confining-bed area of one square mile, 

Qd=0.2 X0.5 X5,280 X5,280=2,800,000 gallons per day. 

It is known that the cone of depression created by pumping a well 

in an artesian aquifer grows rapidly and thus in a relatively short 

time encompasses a large area . As shown by the above computations, 

the total amount of vertical seepage through confining beds may be 

quite large, even though the permeability of these formations is 

relatively small. If the confining bed in turn is overlain by an aquifer of appreciable storage and transmitting capacity, the radius of the 

cone of influence developed by a well pumping from the artesian aquifer will be determined by the hydrologic regimen of the artesian 

aquifer, the confining bed, and the leakage-source aquifer. 

The first detailed analysis and solution of the leaky-aquifer problem 

was developed by DeGlee l (1930) and later supplemented- by Steggewentz and Van Nes (1939) . 

In these analyses, assumptions related to the physical flow system 

are : (a) the artesian aquifer is bounded above or below by a semipermeable confining bed, (b) the aquifer, when pumped is supplied by 

leakage through the confining bed, the leakage being proportional to 

the drawdown, and (c) the aquifer and confining bed are independently

homogeneous and isotropic. It is also assumed that the water level 

in the aquifer supplying water to the semipermeable bed is maintained 

(}lee, (3. J. de, 1930, over grondwaterstroomingen bei wateronttrekking door mittel van putten (On 

ground-water currents through draining by means of wells] : Delft [Netherlands] Tech . Hogeschool thesis. 

I 
GROUND-WATER HYDRAULICS 

at or very near static level through the interval of pumping. The 

solution developed is for the steady-state condition, wherein it is 

assumed that the drawdown is zero at r= co . 

Jacob (1946) also analyzed this problem, verifying the solution 

for steady flow and also developing a solution for the transient state. 

His final steady-state equation, in nondimensional form, for the drawdown in an infinite artesian aquifer has the form 

Ko (x) (35) 

or, in the usual Survey units, 

229 QKo (x) s- (36) T 

where 

br x=- (37) a 

and 

a=~T/S 

b= P /m'S 

T=coefficient of transmissibility of the artesian aquifer in gallons 

per day per foot, 

P'=coefficient of vertical permeability of the semipermeable confining bed, in gallons per day per square foot, 

S=coefficient of storage of the artesian aquifer, 

Q=rate of withdrawal by the pumped well, in gallons per minute, 

m'=thickness of the semipermeable confining bed, in feet, 

r=distance from the pumped well to the observation well, in feet, 

s=drawdown in the observation well, in feet . 

The symbol Ko(x) is a notation widely but not universally used to 

identify the modified Bessel function of the second kind of the zero 

order. In order to avoid any misunderstanding of its present usage 

it is identified as follows : 

Ko(x)=-[0.5772+logs (xl2)]Io(x) 

+(1/11)2 (x/2) 2+ (1/21) 2 (x/2) 4 (1 +1/2) 

+(1/31)2(x/2)1(1+1/2+1/3+ . . ., (38) 
THEORY OF AQIIIFER TESTS ].13 

where 

Io(x)=1+(x/2)2/(1!)2+(X/2)4/(202+(x/2)6/(3!)2+ . . . . (39) 

The notation Io (x) is used to represent the modified Bessel function 

of the first kind of zero order. Values of the function Ka (x) over 

the range of interest for most ground-water problems are given in 

table 4. 

Equations 36 and 37 may be rewritten in the following form 

log s=log C

ZZT Q~-}-log Ko(x) (40) 

log r=log Cb]~--log x (41) 

The bracketed portions of equations 40 and 41 Ynclude all the terms 

that have been assumed constant in the derivation . It follows then 

that the variable s is related to r in the same manner that Ko(x) is 

related to x. Thus the form of equations 40 and 41 once again sug­ gests the same convenient method of graphical solution that has 

already been described for resolving the Thess formula. Atype curve 

for use in solving equations 36 and 37 is prepared by plotting on 

logarithmic graph paper the values given in table 4 . In figure 27 

curve AA is in part a duplication of the lower part of curve BB and 

in part an extension of that curve into the next lower log cycle. The solution of equations 36 and 37 thus requires plotting the field 

observations of s and r, at some particular time t, on Logarithmic graph paper, using the same size of logarithmic scale adopted for the 

type curve. The data curve is superposed on the type curve, the 

coordinate axes of the two curves being held parallel, and translated 

to the position that represents the best fit of the field data to the type 

curve. When the match position is found, the amount of shift or 

translation from the s scale to the Ka(x) scale is measured by the 

bracketed term of equation 40, and the translation between the r 

scale and the x scale is represented by the bracketed member of 

equation 41. An arbitrary point is selected on the data curve and the 

coordinates of this common point on both the data curve and the 

type curve are recorded . These coordinates, when substituted in 

equations 36 and 37, permit computation of the coefficient of transmissibility, T, of the artesian bed, and the value of x, which has 

inherent in it the coefficient of vertical permeability of the leaky confining bed. 
0 

m 

d w

Q e b n m f

a 

b 

A

0 

m 

N 

e 

U 

a 

0

V Y ba 

(V 

4, 1~k 'I 11 p w 

a

w h k ch v

D 

Y 0 P4 LV b s

U 

> 

(i{O3f B aAJn:)) 

O 

O 

_ O 
THEORY OF AQUIFER TESTS 

TABLE 4.-Values of Ko(x), the modified Bessel function of the second kind of zero order, for values of x between 10-1 and 9 .9 

[Data for plotting type curve (fig . 27) used in solving equations 36 and 37. Values of Ko(z) in the interval
 0 .015zS1 .00 taken from tables in Commerce Dept . (1952, p . 36-80) . Values of Ko(z) in the interval
 1 .OSz59 .9 taken from Gray, Mathews, and MacRobert (1931, p. 313-315)] 

N z-N(100) z-N(10- 1) z-N(10-2) N z-N(100) z-N(10-1) z-N(10- 9) 

1 .0 0.4210 2.4271 14 .7212 5.5 0 .002139 .8466
 1. 1 .3656 5.6 .001918 1.2 .3185 5.7 .001721
 1 .3 .2782 6.8 .001544
 1 .4 .2437 5.9 .001386


1 .5 .2138 2.0300 6.0 .001244 .7775 2 .9329
 1 .6 .1880 6.1 .001117
 1 .7 .1655 6.2 .001003
 1 .8 .1459 6.3 .0009001
 1 .9 .1288 6.4 .0008083


2 .0 .1139' 1 .7527 4.0285 6.5 .0007259 .7159
 2 .1 .1008 6 .6 .0006520
 2 .2 .08927 6.7 .0005857
 2 .3 .07914 6 .8 .0005262
 2 .4 .07022 6 .9 .0004728


2 .5 .06235 1 .5415 7 .0 .0004248 .6605 2.7798
 2 .6 .05540 7 .1 .0003817
 2 .7 .04926 7 .2 .0003431
 2 .8 .04382 7 .3 .0003084
 2 .9 .03901 7 .4 .0002772


3 .0 .03474 1 .3725 3.6235 7 .5 .0002492 .6106
 3 .1 .03095 7 .6 .0002240
 3 .2 .02759 7 .7 .0002014
 3.3 .02461 7 .8 .0001811
 3.4 .02196 7.9 .0001629


3.5 .01960 1.2327 8.0 .0001465 .5653 2 .6475
 3.6 .01750 8.1 .0001317
 3.7 .01563 8.2 .0001185
 3.8 .01397 8.3 .0001066
 3.9 .01248 8.4 .00009588


4.0 .01116 1 .1145 3 .3365 8.5 .00008626 .5242
 4.1 .009980 8.6 .00007761
 4.2 .008927 8.7 .00006983
 4.3 .007988 8.8 .00006283
 4.4 .007149 8.9 .00005654
 4.5 .006400 1 .0129 9.0 .00005088 .4867 2.5310
 4.6 .005730 9.1 .00004579
 4.7 .005132 9 .2 .00004121
 4.8 .004597 9 .3 .00003710
 4.9 .004119 9 .4 .00003339


5.0 .003691 .9244 3.1142 9 .5 .00003006 .4524
 5.1 .003308 9 .6 .00002706
 5 .2 .002966 9 .7 .00002436
 5 .3 .002659 9 .8 .00002193
 5 .4 .002385 9.9 .00001975


1 Whenz-0, go(z)-ae . 

In application it is not possible to determine either a or b from field 

observation of steady flow, but their ratio can be determined from the 

definition of x : 

x=r(b/a)=r~PTS, S=r P'/TM' " (42) 

The vertical permeability of the leaky bed can thus be determined 

from equation 42 if the bed thickness, m', is known. However, S,

the coefficient of storage for the artesian aquifer cannot be determined 

as it is removed from the b/a ratio by cancellation . Hantush (1955)

has designated the ratio P'/m' as the "leakage coefficient," and 
116 GROUND-WATER HYDRAULICS 

Hantush and Jacob (1955) have described in considerable detail their 

development of equations for the nonsteady-state solution to the 

foregoing problem . 

The preceding discussion has stipulated that equations 36, 37, 

and 42 are properly applied only to steady-state conditions . This 

means that enough time must have elapsed for the drawdown to have 

stabilized throughout the region for which the plot of s versus r is to 

be made. The manner in which the drawdown stabilizes at observation 

points at selected distances from the discharging well is shown on a 

semilogarithmic plot by Hantush and Jacob (1955, fig. 1) . In effect 

their plot shows individual time-drawdown curves because values of 

drawdown divided by a constant are plotted against values of the 

logarithm of time multiplied by a constant. Of special interest is 

the fact that for all the curves, regardless of the represented distance 

from the discharging well, the drawdown stabilizes or levels off at 

the same value of time. 

Assuming, therefore, that the requirement of stabilized drawdown 

has been met, an important feature of the logarithmic type curve 

(fig . 27) should be recognized . Note that the curve is drawn only for 

values of x greater than 0.01. Thus the matching of a logarithmic 

plot of s versus r against the leaky-aquifer type curve is appropriate

only if the observed data and computed results can be shown to 

yield values of x (which is directly related to r) that are greater than 

0.01 . Actually the critical value of x is about 0 .03, as can be demonstrated in the following manner. 

In the tables of the Bessel functions (U.S . Department of Commerce, 

1952) the following relation applies for small values of x: 

Ko(x)=E'o(x)+F'o(x) logio (x) .

The tables show that for values of x ranging from 0 to about 0.03 

the values of Eo (x) and Fo (x) are 0 .116 and -2 .303 respectively . 

Substituting these equivalents in the above relation yields 

Ko (x)=0.116-2 .3031og1o (x), 

which, by substitution from equation 37 and conversion to the natural 

logarithm, becomes 

KO (x)=0.116-log, (br/ca) . 

If equation 35 is rewritten in terms of the difference in drawdown 

between two points at radii rl and r2 (where r2 >r,) on the cone of 
THEORY OF AQUIFER TESTS 

depression, and the foregoing relation for Ko (x) substituted therein, 

there follows the expression 

s1-s2 2Q CC0.116-loge_ 1r)_(0. 116-loge br2 

or 

S1-S2=2Q 109, 1 

which is the familiar Thiem equilibrium formula previously presented 

in the form of equation 1 . The conclusion to be drawn is that in the 

region x<0.03 a logarithmic plot of s versus r exhibits only the effects 

of radial flow through the aquifer toward the discharging well ; the 

leakage effects are not significant enough to influence the shape of the 

curve. Although the leaky-aquifer type curve could be extended 

readily into this region of low x values, its curvature is insensitive to 

leakage and is too slight to permit a matching that would be definitive 

of the x value needed for computing the leakage coefficient. 

The nature of the abbreviated relation for Ko(x), presented in the 

preceding discussion, suggests a simple means for analyzing the steadystate drawdown data within the region x<0.03 . Rewriting equation 

35 in terms of this special relation for Ko (x) produces 

s=2Q [0 .116-loge (brla)], 

or, in the usual Survey units and the common logarithm, 

s_ T 

229Q 0.116-2.3031oglo burl . (43) 

Recognizing that s and r are the only variables in equation 43, obviously a semilogarithmic plot of s versus log r produces a straight 

line . If ro is the intercept of this straight line at the zero-drawdown 

axis, appropriate substitution in equation 43 yields 

br°=0.116 logg 10 a 2.303 

or 

b 1 .12 (43a) 
GROUND-WATER HYDRAULICS 

The analysis of steady-state test data for a leaky aquifer can thus 

be summarized in the following three simple procedures : 

1 . Select for plotting only the drawdown data which are within 

the region where drawdowns have levelled off. 

2 . Use equations 36, 37, and 42 with a logarithmic plot of s versus r, 

matched to the leaky-aquifer type curve (fig . 27), only if the observed data and resulting computations produce values of x greater 

than 0.03. 

3 . Use equations 2, 4, and 43a with a-semilogarithmic plot of s versus 

log r if the data and resulting computations produce values of x 

less than 0.03. 

The earliest observations of drawdown in each observation well, 

when s is small, should conform to the Theis nonequilibrium type 

curve for the infinite (nonleaky) aquifer if the rate of leakage from the 

confining bed is comparatively small. The coefficient of storage for 

the artesian aquifer can be determined under these conditions from 

the earliest observations of drawdown (Jacob, 1946, p. 204) . The 

computed coefficient of transmissibility should be checked by comparing the value obtained from matching the earliest data to the 

nonequilibrium type curve with the value obtained by matching the 

later data to the steady-state leaky-aquifer type curve . If consistency 

of the T values is not obtained, then the leakage may be causing too 

much deviation at the smaller values of t to permit application of the 

Theis nonequilibrium formula. 

VARIABLE DISCHARGE WITHOUT VERTICAL LF.ASAGE 

By R. W. STALLMAN 

CONTIRIIOIISLY VARYING DISCHARGE 

The rate at which water is pumped from a well or well field commonly varies with time in response to seasonal changes in demand. 

For instance, the pumping rate, as shown by records of daily or 

monthly discharge, is often found to be varying continuously. Where 

this element of variability is recognized in ground-water problems, the 

analytical methods that are described in the preceding sections of this 

report are not applicable without some modification or approximation . 

Exact equations could perhaps be developed for the case of continuously varying discharge, but the cost of analysis, in terms of time and 

effort, would likely be prohibitive considering that a separate and 

specific solution would be required for each problem . It is considered 

more expedient, therefore, to utilize the existing analytical methods, 

rendering them applicable to the field situation by introducing tolerable approximations of the field conditions. As an example, consider a situation where the pumping rate in a well (which may also 

represent a well field) tapping an artesian aquifer varies continuously 

with time in the manner indicated by the smooth curve shown in 
THEORY OF AQUIFER TESTS 

A 

Time ­

0 3 0o I

1 _ 

Q20o 

d 

0 

d I 

o~ QO I 

I I 

L

0 I 1 

U 1 N I 1 

tl r2 

tG 

Time 

c 

0 

0 

d 

ar 

d 

v 

0 

3 

I 1 

I 

I 

I I 

f I 

I 

I I 

t3 i 

f 

-

Time -s 

F>iovxa 28 .-Nomenclature for continuously varying discharge. 

figure 28A . This smooth curve may be approximated by the series 

of steps shown, and the analysis of each step may be undertaken 

starting with conventional theory and equations. Thus the Theis 
__ 

120 GROUND-WATER HYDRAULICS 

nonequilibrium formula (eq. 6) can be used to construct a type curve 

for analyzing the observed drawdowns caused by the stepped pumping 

rates indicated in figure 28A and B. The drawdown, s, at distance r 

from the pumped well, at any time t, is 

S =Si+s2+S3 . . . . . . .+S. (44) 

where the subscripts refer to the OQ values of figure 28B. The zero 

reference time to is chosen arbitrarily so that the effects of the antecedent rate of pumping, Qo, are established as a regular trend that 

can be projected or extrapolated with certainty, as shown in figure 

28C, over the time span occupied by the stepped pumping rates. 

Applying the Theis nonequilibrium formula to define each of the 

'drawdown components given in equation 44, there follows, 

s`1 T.6 

(AQ1W(u)1+AQ2W(u)2+AQ3W(u)3 . . . . . +AQnW(U),J- (45) 

The corresponding u values are 

�11 . 87r2S . _ 1 . 87r2 S. _1 . 87r2S. 2-1 T(t-t1)' 

u2 

T(t -t2) ' u'T(t-t3)' 

. . . ; 87r2S '1,6n-T(t-t,,) . (46) 

Therefore, 

Ft- t ' , t ­t1 . t- t1 

u2-u1 It-t31 ~, u3=u1 t-t. 

Inspection of equations 46 and 47 should indicate that virtually an 

infinite number of type curves can be constructed for solving equation 

45. For practical purposes, however, only a family of curves need 

be constructed. 

It can be seen from equations 46 that the relation between the u 

values is dependent on the value of t selected . For any given value 

of t, the values of u are proportional to the constant 1 .87r2S/T. 

Therefore the family of curves must be constructed using t and n 

1 .87 r2SI Tas independent variables and Z OQW(u) as the dependent 

variable . This is accomplished by first assuming several values of 

1 .87 r2SITfor a particular value oft. Values of u1 are then computed 

for that t for the assumed values of 1.87r2SIT using the first of equations 46. Equations 47 are then used to compute values of ul, 

u3 . . . u,t for each assumed value of 1 .87r 2 SIT. These in turn 

determine (see table 2) the corresponding W(u) values, which are 

used to compute the quantity in brackets (the sum of all the OQW(u) 

terms) in equation 45. Thus a set of values is produced for the sum 
THEORY OF AQUIFER TESTS 

of the AQW(u) terms, corresponding with the assumed values of 1 .87 

r2SIT and all are related to one assumed value of t . This computing 

procedure is repeated for each value of t in a whole set of t values 

selected to span a time range that will permit drawing the family of 

type curves, shown schematically in figure 29, through the same time 

a 

4 

o, 0 

log t 

FIOIIBE 29.-Schematic plot of family of type curves for problems involving continuously varying discharge . 

interval covered by the drawdown observations in the aquifer. The 

field-data plot of log s versus log 1/t is superposed on the family of 

type curves, taking care that the logarithmic time scales of the two 

graphs are exactly matched . The data plot is then shifted along the n 

OQW(u) axis until the position is found where the curvature of the 

data plot is identical with an underlying type curve or with an interpolated type curve position . It follows that this serves to identify

the data curve with a specific value of 1 .87r2S/T. Values of s and n 

r, OQW(u) are read from a point common to both graphs and entered 

in equation 45 to solve for T. The computed value of T can then be 

used with the value of 1 .87r 2SIT to solve for S. Should several 

observation wells at different radii be available, it may be more convenient to construct a type curve suitable for matching with the 

observed drawdown profile. For a selected observation time, values 
122 GROUND-WATER HYDRAULICS 

n 

Of 7, OQW(u) and 1.87r2S/Tare taken from figure 29 and used to conn 

struct a new type curve by plotting log AQW(u) against log (1 .87 

r2S/r. This new logarithmic type curve, drawn for a selected time, 

t, can be matched with a logarithmic data plot of s versus r2, drawn 

for the same time t. 

INTERMITTENT OR CYCLIC DISCHARGE 

Analysis of drawdown data by means of the methods described in 

the preceding section is likely to require a large amount of calculation . 

However, for certain specific kinds of discharge variations the analysis 

can be simplified considerably. The detailed solutions of two specific 

cases have been described by Theis and Brown (1954). One of the 

problems solved was that of computing the drawdown occurring in a 

well being operated in a regular cycle of pumping at a constant rate 

for a given time interval, then resting for a given time interval . 

Their final equation, in the usual Survey units, for drawdown in the 

pumped well after n cycles of operation is 

264Q 1 .2 .3 . . . . . . . . n 

sn= T togoo (1-p)(2-p)(3-p) . . . . (n-p) (8) 

where p is the fractional part of the cycle during which the well is 

pumped. In part, the simple form of equation 48 was obtained by 

utilizing the semilog approximation (eq. 4) of the Theis nonequilib 

rium formula. Many regular operational cycles are easily generalized 

and analysis may lead to a final expression comparable, in simplicity, 

to equation 48 . 

CHANNEL METHODS-LINE SINK OR LINE SOURCE 

CONSTANT DISCHARGE 

NONSTEADY STATE, NO RECHARGE 

As early as 1938 Theis (Wenzel and Sand, 1942, p . 45) had developed 

a formula for determining the decline in artesian head at any distance 

from a drain discharging water at a uniform rate. In 1949 Ferris 

(1950) derived a formula that can be shown to be identical with the 

one derived by Theis. The development is based on the following 

assumptions : (a) the aquifer is homogeneous, isotropic, and of semiinfinite (bounded on one side only by the stream) areal extent ; (b) 

the discharging drain completely penetrates the aquifer; (c) the aquifer is bounded by impermeable strata above and below ; (d) the flow 

is laminar and unidimensional ; (e) the release of water from storage 

is instantaneous and in proportion to the decline in head ; and (f) the 

drain discharges water at a constant rate . 
THEORY OF AQUIFER TESTS 123 

Slightly modifying the form used by Ferris, the drain formula can 

be written nondimensionally, as 

-U2 z 

Qax e

-1

+ 2 

fo

2 VMS s- e-uadu (49) 2T 11 

where 

u=x V4Tt' 

or 

2- x2S 

u (50) 4Tt~ 

Ferris suggested that the quantity in brackets be written symbolically, 

for convenience, as D(u) which is to be read "drain function of u," and 

in this report the subscript q will identify it with the constant discharge situation. Equations 49 and 50 can therefore be rewritten 

in abbreviated form, in the usual Survey units, as 

s_720Q6 D(u~, (51) 

and 

1 .87x2S u2- (52) Tt ' 

where 

s=drawdown, in feet, at any point in the vicinity of the drain discharging at a constant rate, 

Qn =constant discharge (that is, base flow) of the drain, in gallons per 

minute per lineal foot of drain, 

x=distance, in feet, from the drain to the point of observation, 

t=time, in days, since the drain began discharging, 

and S and Thave the meaning and units already defined. 

From inspection of equations 51 and 52 it follows that if s can 

be measured at several values of t, and if x and Qt are known, then S 

and T can be determined. However, the occurrence of two unknowns 

and the nature of the drain function make an exact analytical solution 

impossible and trial solution most laborious. A graphical solution 

of superposition, similar to the one devised by Theis for solution of 

his nonequilibrium formula, affords a simple solution of equation 51 .

The first step in constructing the type curve is to assume values of 

u and compute the corresponding values for D(u), from equation 49, 

which can be done easily with the aid of published tables (U.S . Natl. 

Bur. of Standards, 1954). Values of D(u), and u2 for values of u 

from 0 .0510 to 1 .0000 are given in table 5. These data are then used 
720 06 x D(u) 9 I .o s . D(u), F9 T 

rL 8 

C7 

-6 

FiouRE 30.-Logarithmic graph of the drain function D(u) a for channel method-constant discharge. 

Tf 

0.1 1 2 3 5 6 7 8 

.001 0.01 

v 2 
THEORY OF AQUIFER TESTS 125 

to prepare a type curve on logarithmic coordinate paper by plotting 

values of u or u2 against values of D(u) Q . Such a type curve is shown 

in figure 30 and, for convenience in subsequent computation, values 

of D(u), have been plotted against values of u2 . 

Rearranging equations 51 and 52 and taking the log of both sides, 

there follows : 

log 8= [log 720~°x]+log D(u) Q , (53) 

and 

logt2=Clog (54) 1 .87 S]-}-log u2. 

For a given test, the bracketed parts of equations 53 and 54 are constant and log D(u), is related to log u2 in the manner that log s is related to log (x 2 /t) . Therefore, if values of the drawdown, s, are 

plotted versus x2lt on logarithmic tracing paper having the same log 

scale as the type curve for the drain formula, the curve of observed 

data will be similar to the type curve. The data curve may thus be 

superposed on the type curve, with the coordinate axes held parallel, 

and translated to the position where the observed data coincide or 

make the best fit with the type curve. When this matching position 

has been found, an arbitrary point is selected, common to both curves, 

and the coordinates of this common point are used to solve equations 

53 and 54 for Tand S. 

TABLE 5 .-Values of D(u)a, u, and u2 for channel method-constant discharge

formula .-

(Data for plotting type curve (fig . 30) for equation 51 . After Ferris (1950)1 

u 

0.0510 

u2 

0.0026 

D(u), 

10.091 

u 

.2646 

u9 

.070 

D(u), 

1 .280 

.0600 .0036 8.437 .3000 .090 1.047 

.0700 .0049 7.099 3317 .110 .8847 

.0800 .0064 6.097 :3605 .130 .7641 . 0900 .0081 5.319 .4000 .160 .6303 

.1000

.1140 

.1265 

.010

.013 

.016 

4.693 

4.013 

3.531 

.4359 

.4796 

.5291 

.190 

.230 

.280 

. 5327 

.4310 

.3516 

.1414 .020 3.069 .5745 .330 .2895 

.1581 .025 2.657 .6164 .380 .2426 

.1732 

.1871 

.2000

.2236 

.2449 

.030 

.035 

.040 

.050

.060 

2.355 

2.120

1.933

1.648

1 .440 

.6633 

.7071 

.7616

.8124 

.8718 

.440 

.500 

.530 

. 660 

.760 

.1996

.1666 

1333 

:1084 

.08503 

. 9487 .900 .06207 

1 .0000 1 .000 .05026 

Despite the restrictive assumptions upon which it is based, the 

drain formula, as it has been called, has been applied successfully in 

determining the coefficients of transmissibility and storage of an 

aquifer and in estimating the pickup by or leakage from drains . 
126 GROUND-WATER HYDRAULICS 

Discussion and comparison of various ways of plotting the type 

curves for the drain function D(u), and the well function W(u) (see

figures 30 and 23) are given by Warren (1952, written communication) . 

CONSTANT HEAD 

NONSTEADY STATE, NO RECHARGE 

By R. W. STALLMAN 

The decline in artesian head at any distance from a stream or drain, 

whose course may be approximated by an infinite straight line, subsequent to a sudden change in stream stage, can be found by borrowing

the solution to an analogous heat-flow problem (Ingersoll, Zobel, and 

Ingersoll, 1948, p. 88) . It is assumed that (a) the stream occurs 

along an infinite straight line and fully penetrates the artesian aquifer ; 

(b) the aquifer is semi-infinite in extent (bounded on one side only by

the stream) ; (c) the head in the stream is abruptly changed from zero 

to so at time t=0; (d) the direction of ground-water flow is perpendicular to the direction of the stream ; and (e) the change in the rate 

of discharge from the aquifer is derived from changes in storage by

drainage after t=0. Substituting ground-water nomenclature in the 

heat-flow equation, the distribution of drawdown in the artesian 

aquifer is found to be 

2so x 

so-8 2 Tcls e-'~'du,

=~J0 

or 

s ~f .so [I-- 2 Tt/S e-u'du -soD(u)n, (55) '~ o 

where D(u),, replaces the quantity in brackets and represents the drain 

function of u for the constant head situation, and where 

u2_X

2S 

(56) 4Tt~ 

In the foregoing expressions x is the distance from the stream or drain 

to the point at which the decline in artesian head, s, is observed or 

known, and so is the abrupt change in stream stage at t=0. Other 

symbols are as previously defined. (Note : In equation 55 the integral

expression and its coefficient constitute what mathematicians have 

labelled the error function, written as "erf ." The bracketed portion

of equation 55 is identified as the complementary error function, written 

as "cerf" .) 

The relation expressing the discharge from the aquifer, per unit 

length of stream channel, Qb, resulting from the change in stream stage, 
THEORY OF AQUIFER TESTS 127 

can also be found in texts on heat flow (Ingersoll, Zobel, and Ingersoll, 

1948, p. 90) . When written using ground-water notation, and multiplying by 2 to account for the water contributed from both sides of 

the stream the equation has the form 

(57) 

Equations 56 and 57 now afford a means for evaluating the two unknowns Tand S, inasmuch as the ratio S: Tis determined from equation 

56 and the product ST is obtained from equation 57. 

Comparing equations 55 and 56 the use of the method of superposed 

graphs, described in previous sections, is again indicated as the most 

logical means of solution because log s evidently varies with log xllt in 

the same manner that log D(u),, varies with log u2. Thus the solution 

of equation 56 for the ratio S : Twill evidently require matching a logarithmic data plot of values of 's versus corresponding values of x2 lt 

(or simply ilt if only one observation well is available) to a logarithmic 

type curve prepared by plotting values of D(u),, versus corresponding 

values of u2. Such a type curve is shown in figure 31, prepared from 

the drain function values given in table 6 . 

If equations 56 and 57 are rewritten using the usual Survey units 

(except for Qb which is the base flow in gallons per minute per foot of 

stream length), they become 

_1 .87x 2 S 

u2_ ' (58) Tt 

and 

Qb=2 .15X10-3s0 tT(59)

TABLE 6.-Values of D (u) b, u, and ua for channel method-constant head formula 

[Data for plotting type curve (fig . 31) for equation 55. Prepared by R. W. Stallman . Values of 

D(u)g,, for selected values of ul or u, were computed with the aid of U .S. Natl . Bureau of Standards tables 

(1954)] . 

u u3 D(u)a u u2 D(u)A 

0.03162 0.0010 0.9643 0 .6325 0.40 0 .3711 

.04000 .0016 .9549 .7746 .60 .2733 

.05000 .0025 .9436 .8944 .80 .2059 

.06325 .0040 . 9287 1 .000 1 .00 .1573 

.07746 .0060 .9128 1.140 1 .30 .1069 

.08944 .0080 .8994 1.265 1 .60 .0736 

.1000 .010 .8875 1 .378 1 .90 .0513 

.1265 .016 .8580 1 .483 2.20 .0359 .1581 .025 .8231 1 .581 2.50 .0254 

.2000 .040 .7730 1 .643 2.70 .0202 

.2449 .060 .7291 1.732 3 .00 .0143 

.2828 .080 .6892 1.789 3 .20 .0114 

.3162 .10 .6548 

.4000 .16 .5716 

.5000 .25 .4795 
io T 

e 5 S o o (u) 6 

where a -

_ 3 OIuJn = I _ 2 

~ 

_ _ du 

z 

- 1 .87 

T 

0.01 5 6 7 

0.01 0 .1 

u 2 

1 .0 10 .0 

FioURE 31 .--Logaritbmic graph of the drain function D(u)g, for channel method-constant head . 
THEORY OF AQUIFER TESTS 129 

Equations 55 and 57 define the changes in head and flow which occur 

in the aquifer if the stream stage is abruptly changed. Therefore 

the stipulation of no recharge implies only that the rate of recharge 

must be constant for a sufficient interval of time, so that the regional 

water-level trends can be extrapolated with accuracy throughout the 

period in which the changes in the aquifer are being observed. 

Water-level data from wells near the stream may, in some field 

situations, yield erratic results, depending on the flow pattern in the 

aquifer in the vicinity of the stream. For example, if the stream 

only partly penetrates the aquifer the flow lines in the aquifer will 

obviously bend upward as they approach the stream, thereby producing vertical components of flow. Thus the smaller the distance x, 

between observation well and stream, the greater the errors inherent 

in the observed water levels . This in turn means that as x decreases, 

the error in the computed value of SIT increases . It should also be 

realized that instantaneous or abrupt lowering of stream stage is 

seldom possible, which means that the determination of a reference 

or zero time is difficult. Thus observations made a short time after 

the stream stage is lowered may be somewhat unreliable . In general, 

therefore, it would seem prudent to favor the data collected at comparatively large values of x and large values of t to provide the most 

reliable basis for analysis. 

Where it is known that the stream or drain penetrates only a part 

of the aquifer thickness, the following adjustment procedure, though 

not yet proven by field trial, may offer a means for determining more 

realistic values for T and S. It should be evident that in a field 

situation of this kind, the change in head in the stream channel is 

not as effective in producing head changes throughout the aquifer 

as when the stream is fully penetrating. Near the stream, groundwater levels adjust quickly to changes in stream stage, but part of 

the adjustment is caused by the bending of the flow lines. It can 

be assumed, however, that at some relatively short distance xo away 

from the stream the bending of the flow lines in the aquifer will be 

small enough so that the effects on the head values may be neglected . 

Thus, for distances greater than xo the flow lines may be considered 

parallel-that is, flow is essentially one-dimensional . The change 

in head in the aquifer, at the distance xo, may therefore be considered 

as an effective value of so and it is related to the changes in head 

throughout the aquifer, for all distances greater than xo, in the manner 

described by equations 55 and 56 . 

In effect this reasoning means that the real partly penetrating 

stream, in which the stage was abruptly changed an amount so, is 

being replaced (at the distance xo) by a theoretical fully penetrating 

stream in which the stage change may be regarded as essentially 
-----------------------------------------------------------------------

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

130 GROUND-WATER HYDRAULICS 

SECTION VIEW OF IDEAL AQUIFER SUBJECT TO UNIFORM ACCRETION, 

BOUNDED BY PARALLEL STREAMS 

Water table 

0 a x 2' a 

, FIoW 

---=--=Confining ~-_-=.material -----_-____-_

(At1er Jacob, 1943) 

B. NOMENCLATURE FOR MATHEMATICAL ANALYSIS OF PROFILE SHOWN IN SKETCH 

FIGTIRE 32.-Section views for analyzing steady-state flow in hypothetical aquifer of large thickness with 

uniform accretion from precipitation . 

abrupt but of a lesser magnitude which shall be termed an effective 

value of so. 

This socalled effective value of so can be computed from equation 

55 after superposing the data and type curves in the manner 

already described. The critical distance x0 can also be computed, 

using the coordinates for a point common to the matched data and 

type curves, if more than one observation well is available . For 

each observation well the ratio SIT is computed, using the distance 

from the well to the real stream channel as a first estimate in equa­
THEORY OF AQUIFER TESTS 131 

tion 56. If the SITvalues thus determined are not alike, the equation 

for u2 is adjusted to read as follows : 

u2-1 .87(x -x°)2 S. 

Tt 

An estimate of x° is then made, and SIT values for each observation 

well are recomputed using the effective distance to the stream (x=x°) . 

If the data and field conditions are sufficiently ideal to permit an 

accurate analysis, several assumed values of x° will indicate the one 

that will produce the closest agreement in the computed values of 

SIT. 

It is pointed out that it is difficult to assess the true value of these 

adjustment procedures, inasmuch as the opportunity for applying 

them to a specific field problem has not yet been afforded . 

STEADY STATE, UNIFORM RECHARGE 

A problem of considerable practical interest is that of estimating 

the base flow of streams, or the effective average rate of ground-water 

recharge, from the shape of the water table. Consider the case of an 

aquifer bounded on two sides by fully penetrating parallel streams of 

infinite length as shown in figure 32A. It is assumed that the aquifer 

is homogeneous and isotropic, and that the aquifer is recharged at a 

rate of accretion, W, that is constant with respect to time and space. 

Flow is therefore one-dimensional and a ground-water divide is created at distance a, midway between the streams (see figure 32B) . 

Jacob (1943, p. 566) has given the equation of steady-state profile as 

(2 

°-(2T) ax a2 J~ 

or 

T ax x2 (60) Wh° -2h' 

where 

W=constant rate of recharge to the water table ; 

a=distance from the stream to the ground-water divide ; 

x=distance from the stream to an observation well ; 

h° =elevation of the water table, at the observation well, with 

respect to the mean stream level. 

It is frequently convenient to express the rate of recharge, W, in 

inches per year, while a, x, and h° are expressed in feet, and T is in 

the usual Survey units. Equation 60 is then rewritten in the form 

2 

T=1.71(10 -3) W(ax (61) -2ho 
132 GROUND-WATER HYDRAULICS 

In the absence of artificial withdrawal of water from an aquifer, 

the net recharge must equal the natural discharge, provided changes 

in storage are insignificant. Although it is recognized that under 

natural conditions there are variations of W in time, it should be 

apparent that if average or mean ground-water levels are used in 

equation 61, a figure for Wequivalent to an effective average accretion 

rate will result . 

Equation 61 can be solved if the average (in time) contribution, 

Qt, to the base flow of the stream, per unit length of channel can be 

determined from stream-flow measurements. If Qb is expressed in 

gallons per minute per foot of stream channel, then 

W=8.44(105) Qb~ 

2a' 

or 

W=4.22(105) b(62) 

Equation 62 permits determination of Wwhich can be used in equation

61 in computing the value of T. 

In many field situations, of the type postulated here, general 

appraisal of the occurrence of ground water may indicate that the 

ground-water divide is parallel to the stream course, although the 

distance a will be unknown. If two observation wells are available, 

it is possible to compute a value for a and a value for the ratio 586T1W. 

If a larger number of observation wells is available, a graphical so 

lution may be used to solve for the values of a and 586 T/W. The 

procedure requires observation of the distances, x, from the stream to 

the individual observation wells and the corresponding values of ho. 

Using the data from one observation well, and arbitrarily selecting 

several values of a, the corresponding values of 586T/W are computed 

after appropriate substitution in equation 61 . The computed values 

of 586T/W are plotted against the corresponding assumed values of 

a and a smooth curve is drawn through the plotted points. In similar 

fashion a curve is drawn on the same graph for each observation well . 

The coordinates of the single point at which all the curves intersect 

give the particular values of a and of 586 T/W which will satisfy all 

the available data. 

SINUSOIDAL HEAD FLUCTUATIONS 

Werner and Noren (1951) and later Ferris (1951) independently 

analyzed the problem of fluctuations of water levels in wells in response 

to sinusoidal changes in stage of nearby surface-water bodies. The 

solution to this general type of problem has long been available in 

other fields of science and as Ferris indicates it may conveniently be 
THEORY OF AQUIFER TESTS 133 

found in reference works on heat flow such as the text by Ingersoll,

Zobel, and Ingersoll (1948, p . 46-47) . Translated into ground-water 

terms, the solution requires an isotropic semi-infinite artesian aquifer

of uniform thickness in full contact, along its one boundary, with a 

surface-water body that may be considered an infinite line source . Within the aquifer the change in water storage is assumed to occur 

instantaneously with, and at a rate proportional to, the change in 

pressure . Ferris (1951, p . 3) shows that the equation for the range

of ground-water fluctuation in an observation well, a distance x from 

the aquifer contact with the surface-water body, whose stage is changing sinusoidally, has the nondimensional form 

(63) 

or, in the usual Survey units, 

s,=2so e - l . az~lsltoT~ (64) 

where 

s,=range 6f ground-water stage, in feet ; 

so=amplitude or half range of the surface-water stage, in feet ; 

x =distance from the observation well to the surface-water 

contact with the aquifer ("suboutcrop"), in feet ; 

t o =period of the stage fluctuation, in days ; 

S=coefficient of storage ; and 

T=coefficient of transmissibility, in gallons per day, per foot . 

For convenience equation 64 can be written 

2.1 ~YltoT= -logio (sr/2so) (65) x 

The form of equation 65 suggests a semilogarithmic plot of the 

log of the range ratio, s,/2so, for each observation well, versus the 

corresponding distance, x, as an expedient method of analyzing the 

observed data . The right-hand member of equation 65 represents the slope of the straight line that should be defined by the plotted data. If the change in the logarithm of the range ratio is selected 

over one log cycle of the plot, equation 65 becomes 

2.1VS/t,T=- 1 

or 

.4(Ax)ZS T-4.4 4 (66) to 

If the field conditions fulfill the assumptions made in deriving equation 
134 GROUND-WATER HYDRAULICS 

66, the straight line drawn through the data on the semilogarithmic 

plot should pass through the origin of the coordinate axes-that is, 

should intercept a value of rl2so=1 at a value of x=0. For the case 

of a stream of substantial width, partially penetrating the artesian 

aquifer, this intercept will usually be found at negative values of x, 

indicating an "effective" distance offshore to the suboutcrop . This 

effective distance mayor may not have physical significance depending 

on the nature of the flow field in the vicinity of the stream. For 

example, if the stream does not cut through the upper confining bed, 

stage changes in the surface-water body may still provide, through the 

changes in loading that are involved, a source of sinusoidal head 

fluctuation in the artesian aquifer along the stream location . Unless 

the loading changes are completely effective (100 percent tidal 

efficiency) in producing a stage range of 2s o at x=0 in the aquifer, 

the straight-line intercept with the x axis will again be at some negative 

value of x, and will in part be an indication of the efficiency with 

which the aquifer skeleton accepts the changes in loading. However, 

it should be observed that regardless of the exact value of so in the 

aquifer, at x=0, the slope of the data plot described is unaffected. 

Therefore the T value computed by means of equation 66 will be 

correct regardless of the actual value in the aquifer of so. 

It is seen from equation 66 that it is necessary to know S in order 

to solve for T. Frequently, when the coefficient of storage is not 

known, it is possible to make a reasonable estimate of its value by 

studying the well logs and water-level records. 

The lag in time of occurrence of a given maximum or minimum 

ground-water stage, following the occurrence of a similar surfacewater stage, is given by Ferris (1951) as 

_x t os tl- (67) 2~rT' 

where t, is the lag in time. Solving this equation for the coefficient 

of transmissibility, and rewriting in terms of the usual Survey units, 

there follows 

2 

T=0.60t oS (tl) (68) 

The only variables in equation 68 are evidently x and tl . Thus an 

arithmetic plot is suggested with the value of the distance, x, for each 

observation well, plotted against the corresponding value of the time 

lag, t. The slope of the straight line that should be defined by these 

plotted points will then give the value of xlt,, which appears to the 

second power in equation 68. If the straight line that is drawn 
THEORY OF AQUIFER TESTS 135 

through the plotted points should intersect the zero-timelag axis 

at a negative value of x it may be an indication of the effective distance 

offshore to the suboutcrop. 

In those situations where the aquifer is not fully penetrated or 

where it is under water-table (unconfined) conditions, the methods 

of analysis described in this section will be satisfactory if (a) the 

observation wells are far enough from the suboutcrop so that they are 

unaffected by vertical components of flow and (b) the range in fluctuations is only a small fraction of the saturated thickness of the 

formation . 

AREAL METHODS 

NUMERICAL ANALYSIS 

By R. W. STALLMAN 

The equations presented in the preceding sections of this paper 

were derived by means of the calculus. Darcy's law combined with 

the equation of continuity (Rouse, 1950, p. 326) yielded the basic 

differential equations that described states of flow. In turn, solutions 

to these differential equations were found that satisfy the boundary 

conditions of a particular problem . Certain generalizations were 

made in regard to the boundary conditions so as to provide rather 

specific equations, which can be used with convenience to obtain a 

solution to the field problem. Among these generalizations are the 

assumptions of a constant head or discharge at some point or line, 

homogeneity of the aquifer, simple geometric form or shape of the 

aquifer, and complete penetration of the well or stream . Certainly 

for many field problems these conditions are fulfilled to a sufficient 

degree that the available formulas can be used to obtain reliable 

approximations of the values of T, S, or W. However, the groundwater hydrologist frequently encounters problems for which the 

complicated boundary conditions cannot be expressed by simple 

mathematical relations . Furthermore, the complicated boundary 

conditions related to a given field problem seldom recur in nature 

in the same combination. Thus it could be poor economy to spend 

a large amount of time and energy deriving complicated analytical 

equations whose application might be limited to one problem . Under 

such circumstances it may be found more expedient to use numerical methods of analysis for the quantitative investigation . 

Numerical methods have been used in other sciences for some time for 

the same purpose. Basic formulas and procedures have been described by Southwell (1946, 1940) and many of his colleagues (Shaw, 

1953) . Scarborough (1950) and Milne (1953) have written extensively 

on the same subject and an application of numerical methods to 

ground-water investigations has been described by Stallman (1956) . 

The formal derivation of analytical equations for describing ground­
136 GROUND-WATER HYDRAULICS 

water movement involves application of the rules of calculus for 

integrating, or summing, an infinite set of infinitesimal changes in 

head between two or more points in the flow region under study. 

In lieu of application of the rules of calculus to perform this addition 

conveniently, it would be possible to accomplish the same thing 

simply by addition of the infinite set of infinitesimals . The latter 

method is, of course, impractical unless approximations are introduced . Thus an area may be calculated by considering that it is 

composed of small but finite parts, each having an area (Ax)(Ay) . 

By means of this more coarse subdivision of the area, the problem is 

reduced to the addition of a sensibly small and finite set of component 

parts instead of the infinite set of infinitesimal areas (dx) (dy) postulated by the orthodox calculus . In brief this constitutes the basis 

of numerical analysis as applied in finding solutions to differential 

equations : the substitution of finite entities for the differential forms 

that appear in the fundamental differential equations. 

Consider, for example, the differential equation describing twodimensional unsteady flow in a homogeneous and isotropic aquifer, 

subject to recharge at a steady rate of accretion W. It can be shown 

that this equation has the form 

a2h+a2h-

(T 

S bh W 

axeaye) b~T' 

(69) 

where h is the head at any point whose coordinates are x, y. Let the 

differential lengths dx and dy be expanded so that each can be considered equivalent to a finite length w, and similarly let dt be con 

sidered equivalent to At. A plan representation of the region of flow 

to be studied may then be subdivided by two systems of equally 

spaced parallel lines at right angles to each other . One system is 

oriented in the x direction and the other in the y direction and the 

spacing of the lines equals the distance w (see fig. 33) . A set of 5 

gridline intersections or nodes, selected in the manner shown on 

figure 33, is called an array. Accordng to Southwell (1946, p. 19) 

the first two differentials in equation 69 can b,: expressed, in terms of 

the head values at the nodes in the array, in the following fashion : 

_a 2h_h,+h,-2h, 

6X2' w2 

and 

a

a2h h2+h4-2h, ye ­ w2 

where the subscripts of h refer to the numbered nodes of figure 33 . 

Substituting these equivalent expressions for the first two differentials 
THEORY OF AQUIFER TESTS 137 

Enum EM 

2 

0 3 

4 

NOTES; 

Filled circles indicate boundary nodes selected 

to approximate irregular boundary of the area 

The grid spacing w is an approximation of the 

differential distances dx and dy 

The five numbered nodes constitute an array 

FIGURE 33.-Finite grid and nomenclature used in numerical analysis . 

in equation 69, and letting bhlat be considered equivalent to Oho,/At, 

there follows 

h 7 hi +h2+hs+h4-oho=[S

T ~A, T w2 , (70) 

where Oho, is the change in head which occurs at node 0 through the 

time interval At . Depending on the type of data available, equation 

70 can be used to compute the ratios S/T and W/T, or the head 

distribution in time and space, in an aquifer of given size and shape. 
138 GROUND-WATER HYDRAULICS 

Furthermore, the calculations involve only the use of simple arithmetic. 

In working out a solution where a finite difference equation such as 

equation 70 is being applied to a problem in which the head distribution is required, the primary aim of the computing methods is to find 

the particular head values at all the nodal points (like points 0-4 of 

figure 33) such that the finite difference equation is satisfied at all 

nodes simultaneously. This head distribution can be found either 

by "relaxation" or "iteration" methods of numerical analysis . The 

computations are begun by assuming head values for all the nodes in 

the flow system . These initial estimates will not ordinarily satisfy

the finite difference equation and will require revision so that the 

equation will be satisfied at all points in the grid . Refinement of the 

head values is continued until accuracy is considered to be sufficient 

for the needs of the problem . 

If the refinement is made using a routine adjustment procedure, it is termed an iteration method ; if it is made by approximation, it is 

termed a relaxation method. Consider the steady-state form of 

equation 70, which is 

hi +h,+h8+h4-4ho+ (71) 

Assume that it is desired to find the head distribution in an aquifer of 

known size and shape, the flow conditions (and/or head values) being known along its perimeter, and the values of Wand T, or their ratio,

being known . Substituting in equation 71 the initial set of assumed 

head values for nodes 0-4, and the known value of (W/T)w2 there 

follows 

hii+h2i+h31+h4i-4hoi+(T )w2=Ro, (72) 

where the subscript i is used to set apart or distinguish the head values 

that are initially assumed. The term Ro is the residual at node zero ; 

in other words, it is the remainder resulting from summation of the 

assumed and known values on the left side of the equation . Inasmuch 

as it is virtually impossible. to assume, an initial set of head values such 

that the summation indicated in equation 71 is zero, there will almost 

inevitably be a remainder or residual as shown in equation 72. This 

residual, Ro , may be thought of as an indication of the amount and 

direction of excess curvature on the piezometric surface defined by the assumed heads. The value of Ro is also an indication of the amount 

by which h0i must be changed so that in the next trial summation of 

head values the result will more closely approach the zero value 

required for complete satisfaction of equation 71 . 
THEORY OF AQUIFER TESTS 139 

It can easily be shown that if the head at the zero node is changed 

by an amount AN=-Ro /4, the residual at that node will be reduced 

to zero . However, it should be kept in mind that equation 71 is to 

be tried on each array in the grid net and that residuals will appear 

at many if not all of the other nodes . Thus a head adjustment at 

one node will affect residuals at other nodes, and conversely subsequent head adjustments at adjacent nodes will affect the residual at 

the first zero node. Accordingly, several circuits must be made 

through the net before the residual values are reduced to zero or 

nearly zero. Successive circuits of nodal head adjustments amounting 

to -Ro/4, applied regularly over the net, constitute an iterative 

process. With practice the computer will recognize that the distribution of residuals can be improved more efficiently by either a larger 

or smaller head adjustment than indicated by -Ro/4. Applying such 

improvement tempered by judgment gained from experience is a 

relaxation method . 

FLOW-NET ANALYSIS 

By R. R. BENNETT 

In analyzing problems of ground-water flow, a graphical representation of the flow pattern is of considerable assistance and often 

provides the only means of solving those problems for which a mathematical solution is not practicable. The first significant development 

in graphical analysis of flow patterns was made by Forchheimer (1930) . 

A "flow net," which is a graphical solution of a flow pattern, is 

composed of two families of lines or curves . One family of curves 

represents the streamlines or flow lines, where each curve indicates 

the path followed by a particle of water as it moves through the 

aquifer in the direction of decreasing head. Intersecting the streamlines at right angles is a family of curves termed equipotential lines, 

which represent contours of equal head in the aquifer. 

Although the real flow pattern contains an infinite number of flow 

lines and equipotential lines, it may be conveniently represented by 

constructing a net that uses only a few of those lines . The flow lines 

are selected so that the total quantity of flow is divided equally between adjacent pairs of flow lines ; similarly the equipotential lines 

are selected so that the total drop in head across the system is evenly 

divided between adjacent pairs of potential lines. 

The change in potential, or drop in head between two equipotential 

lines in an aquifer, divided by the distance traversed by a particle of 

water moving from the higher to the lower potential determines the 

hydraulic gradient . Recognizing that this movement of a water particle is governed in part by the proposition that the flow path adopted 

will be the one involving the least work-that is, the shortest possible 

path between the two equipotential lines in question-it follows that 
140 GROUND-WATER HYDRAULICS 

the direction of water movement is everywhere synonymous with 

paths that are normal to the equipotential lines. Hence the system of flow lines must be drawn orthogonal to the system of equipotential lines. 

A flow net constructed with the foregoing principles in mind is a 

pattern of "rectangles" in which the ratio of the mean dimensions of 

each "rectangle" is constant. If the net is constructed so that the 

sides of each rectangle are equal, then the net is a system of "squares." 

It should be recognized, however, that in flow fields involving curved 

paths of flow, the elemental geometric forms in the net are curvilinear 

and thus are not true squares ; however, the corners of each "square" 

are right angles and the mean distances between the two pairs of 

opposite sides are equal. If any one of these elemental curvilinear 

"squares" is repeatedly subdivided into four equal parts the subdivisions will progressively approach the shape of true squares. The proper sketching of a flow net by the graphical method is 

something of an art that is learned by experience ; however, the following points summarized from a paper by Casagrande (1937, p. 

136-137) may be helpful to the beginner: 

1 . Study the appearance of well-constructed flow nets and try to 

duplicate them by independently reanalyzing the problems they represent. 

2 . In the first attempts at sketching use only four or five flow channels. 3 . Observe the appearance of the entire flow net ; do not try to adjust details until the entire net is approximately correct. 

4 . Frequently parts of a flow net consist of straight and parallel lines, which result in uniformly sized true squares. By starting the sketching in such areas, the solution can be obtained more 

readily. 

5. In flow systems having symmetry (for example, nets depicting radial flow into a well) only a section of the net need be constructed, as the other part or parts are images of that section. 6. During the sketching of the net, keep in mind that the size of the 

square changes gradually; all transitions are smooth and, where 

the paths are curved, are of elliptical or parabolic shape. Taylor (1948) recommends a somewhat different procedure for 

sketching flow nets . This procedure, called the procedure of explicit 

trials, has been found to have value in developing intuition for flow 

net characteristics . In this method a trial equipotential or flow line 

is established and the entire net completed as if that trial line were 

correct . If the completed net is not correct, the initial trial line is 

resketched and a new net is constructed. The adjustment of the trial 

line is judged from the appearance of the entire net and how well it 

conforms to the boundary conditions of the system . 
THEORY OF AQUIFER TESTS 

For steady flow, with a particular set of boundary conditions, only 

one flow net exists . If at some subsequent time the boundary conditions are altered, then after sufficient time has elapsed to reestablish 

the steady state, a different flow pattern would be developed and 

again there would be only one possible solution for the new set of 

boundary conditions . Thus before attempting to construct a flow 

net, it is important that the boundary conditions be established and 

carefully described. For example, consider the aquifer shown in figure 34, bounded by an impermeable barrier paralleling a perennial 

stream . Line AB, designating the stream, is obviously an equipotential line along which the head is equal ; line CD marking the 

impermeable barrier, is evidently coincident with the limiting or 

boundary flow line . Accordingly, the equipotential lines will adjoin 

the barrier at right angles . The discharge through any channel or 

path of the flow net may be obtained with the aid of Darcy's law, 

one variation of which, as has been previously shown, may be written 

in the form 

Q=PIA. 

For convenience in applying this variation of Darcy's law, consider 

a unit width or thickness of the aquifer, measured normal to the 

direction of flow indicated by L in figure 34-that is, normal to the 

plane of the diagram. The preceding equation may then be rewritten 

for this unit thickness (in this example) of aquifer, and for one flow 

channel through the net as 

Aq=Plb, (73) 

where Oq represents the flow occurring between a pair of adjacent 

flow lines (one flow channel) and b is the spacing of the flow lines. 

If L represents the spacing, and oh the drop in head, between the 

equipotential lines, equation 73 becomes 

Aq=PAh CL) (74) . Inasmuch as the flow net (figure 34) was constructed to comprise a 

system of "squares" the ratio b/L is equal to unity and the same 

potential drop occurs across each "square" . It follows then from 

equation 74 that the same incremental flow, Oq, occurs between 

each pair of adjacent flow lines . If there are of flow channels, the 

total flow, q, through a unit thickness of the aquifer, is given by 

q=nfAq. (75) 
PP-1

N 

O 

O 

z 

d 

H 

d 

d

r 

Fie-URx 34.-Flow net for a discharging well in an aquifer bounded by a perennial stream parallel to an impermeable barrier. 
THEORY OF AQUIFER TESTS 143 

If there are nd potential drops, the total drop in head, h, is given by 

h=ndAh. (76) 

Substituting in equation 75 the values of Oq and oh given by equations 

74 and 76 respectively, there follows 

q=nr Ph. (77) d 

Noting that q represents the total flow through a unit thickness of 

the aquifer, the equation for total flow through the full thickness 

of the aquifer becomes 

Q=IPhm, (78) 

where Q=flow through full thickness of the aquifer in gallons per 

day ; 

nf=number of flow channels ; 

nd =number of potential drops ; 

P=coefficient of permeability of the aquifer material, in 

gallons per day per square foot ; 

m=saturated thickness of aquifer, in feet ; 

h=total potential drop, in feet ; and 

Pm=transmissibility of the aquifer, in gallons per day per foot . 

The pi eceding discussion of the graphical construction of flow nets 

concerns two-dimensional flow fields in homogeneous and isotropic 

media. The graphical construction of flow nets for three-dimensional 

problems generally is impracticable ; however, many ground-water 

problems of a three-dimensional nature can be reduced to two 

dimensions without introducing serious errors . 

For two-dimensional problems involving simple anisotropy, such 

as a constant difference between the vertical and horizontal permeability, or a directional area) transmissibility, the flow net can be 

constructed by the conventional graphical procedure (system of 

squares) provided the flow field is first transformed to account for the 

anisotropy . If the values of maximum or minimum permeability 

(or transmissibility) are designated P.Bx, and Pm,n, all the dimensions 

in the direction of P... must be reduced by the factor ~Pmt RIP.; or, 

all dimensions in the direction of P., . must be increased by the 

factor P.1P,Rin, . After the flow field is transformed, the net is 

constructed by graphical methods . Then the net is projected back 

to the original dimensions of the field. It will be found that the 
144 GROUND-WATER HYDRAULICS 

projected net generally will no longer be a system of squares, and the 

equipotential and stream lines will not intersect at right angles . 

For areally nonhomogeneous aquifers-that is, those comprising 

subareas of homogeneous and isotropic media but of different transmissibility-the flow pattern cannot, according to theory, be represented by a single system of squares. If the flow net were constructed 

so that each flow path conducted the same quantity of water, one 

subarea could be represented by a system of squares, but the nets 

in the other subareas would consist of rectangles in which the ratio 

of the lengths of the sides would be proportional to the differences 

in transmissibility . If the flow lines from one subarea enter another 

subarea at an angle, the flow lines (and equipotential lines) would be 

refracted according to the tangent law. The graphical construction 

of a flow net under such conditions is extremely difficult and, with the 

data that are available for most ground-water problems, is generally 

impossible . However, Bennett and Meyer (1952, p. 54-58) have 

shown that by generalizing the flow net for such an area into a system 

of squares and determining the quantity of flow by making an inventory 

of pumpage in each of the subareas, the approximate transmissibility 

of the subareas may be determined . Although such an application of 

the method departs somewhat from theory, it is likely that for many 

areas it provides more realistic, areal transmissibilities than could be 

obtained by use of pumping-test methods alone. Whereas pumping 

tests may provide accurate values of transmissibilities they generally 

represent only a small "sample" of the aquifer. Flow-net analysis 

on the other hand may include large parts of the aquifer, and hence 

provide an integrated and more realistic value of the areal transmissibility. Moreover, by including comparatively large parts of 

the aquifer, the local irregularities that may appreciably affect some 

pumping-test analyses generally have an insignificant effect on the 

overall flow patterns. 

The application of flow-net analysis to ground-water problems has 

not received the attention it deserves ; however as the versatility of 

flow-net analysis becomes more widely known, its use will become 

more common . Such a method of analysis greatly strengthens the 

hydrologist's insight into ground-water flow systems ; it provides 

quantitative procedures for analyzing and interpreting contour maps 

of the water-table and piezometric surfaces . 

For other illustrations of flow-net construction, see figures 36 and 

38 . 

THEORY OF IMAGES AND HYDROLOGIC BOUNDARY ANALYSIS 

The development of the equilibrium and nonequilibrium formulas 

discussed in the preceding sections was predicated in part on the as­
THEORY OF AQUIFER TESTS 145 

sumption of infinite areal extent of the aquifer, although it is recognized that few if any aquifers completely satisfy this assumption . In 

many instances the existence of boundaries serves to limit the continuity of the aquifer, in one or more directions, to distances ranging 

from a few hundred feet to as much as tens of miles. Thus when an 

aquifer is recognized as having finite dimensions, direct analysis of 

the test data by the equations previously given is often precluded . 

It is often possible, however, to circumvent the analytical difficulties 

posed by the aquifer boundary. The method of images, widely used 

in the theory of heat conduction in solids, provides a convenient tool 

for the solution of boundary problems in ground-water flow. Imaginary wells or streams, usually referred to as images, can sometimes 

be used at strategic locations to duplicate hydraulically the effects on 

the flow regime caused by the known physical boundary. Use of the 

image thus is equivalent to removing a physical entity and substituting 

a hydraulic entity. The finite flow system is thereby transformed by 

substitution into one involving an aquifer of infinite areal extent, in 

which several real and imaginary wells or streams can be studied by 

means of the formulas already given . Such substitution often results 

in simplifying the problem of analysis to one of adding effects of 

imaginary and real hydraulic systems in an infinite aquifer . 

An aquifer boundary formed by an impermeable barrier, such as a 

tight fault or the impermeable wall of a buried stream valley that cuts 

off or prevents ground-water flow, is sometimes termed a "negative 

boundary." Use of this term is discouraged, however, in favor of the 

more meaningful and descriptive term "impermeable barrier." A line 

at or along which the water levels in the aquifer are controlled by a 

surface body of water such as a stream, or by an adjacent segment of 

aquifer having a comparatively large transmissibility or water-storage 

capacity, is sometimes termed a "positive boundary." Again, however, use of the term is discouraged in favor of the more precise terms 

line source or line sink, as may be appropriate. 

Although most geologic boundaries do not occur as abrupt discontinuities, it is often possible to treat them as such . When conditions 

permit this practical idealization, it is convenient for the purpose of 

analysis to substitute a hypothetical image system for the boundary 

conditions of the real system. 

In this section, where the analysis of pumping-test data is considered, several examples are given of image systems required to 

duplicate, hydraulically, the boundaries of certain types of areally 

restricted aquifers. It should be apparent that similar methods can 

be used to analyze flow to streams or drains through areally limited 

aquifers. 
146 GROUND-WATER HYDRAULICS 

PFRFNNIA7. STREAMLINE SOURCE AT CONSTANT HEAD 

An idealized section through a discharging well in an aquifer 

hydraulically controlled by a perennial stream is shown in figure 35A. 

For thin aquifers the effects of vertical-flow components are small at 

relatively short distances from the stream, and if the stream stage is 

not lowered by the flow to the real well there is established the boundary condition that there shall be no drawdown along the stream position . Therefore, for most field situations it can be assumed for 

practical purposes that the stream is fully penetrating and equivalent 

A . REAL SYSTEM 

i 

Aquifer thickness m should be very large compared to resultant drowdown near real well 

B. HYDRAULIC COUNTERPART OF REAL SYSTEM 

FIGURE 35 .-Idealized section views of a discharging well in a semi-infinite aquifer bounded by a perennial 

stream, and of the equivalent hydraulic system in an infinite aquifer . 
THEORY OF AQUIFER TESTS 147 

to a line source at constant head. An image system that satisfies the 

foregoing boundary condition, as shown in figure 35B, allows a solution 

of the real problem through use, in this example, of the Theis non 

equilibrium formula. Note in figure 35B that an imaginary recharging 

well has been placed at the same distance as the real well from the line 

source but on the opposite side . Both wells are situated on a common line perpendicular to the line source . The imaginary recharge 

well operates simultaneously with the real well and returns water 

to the aquifer at the same rate that it is withdrawn by the real well . 

It can be seen that this image well produces a buildup of head everywhere along the position of the line source that is equal to and cancels 

the drawdown caused by the real well which satisfies the boundary 

condition of the problem. The resultant drawdown at any point on 

the cone of depression in the real region is the algebraic sum of the 

drawdown caused by the real well and the buildup produced by its 

image. The resultant profile of the cone of depression, shown in 

figure 19B, is flatter on the landward side of the well and steeper on 

the riverward side, as compared with the shape it would have if no 

boundary were present. Figure 36 is a generalized plan view of a 

flow net for the situation given in figure 35A. The distribution of 

stream lines and potential lines about the real discharging well and its 

recharging image, in an infinite aquifer, is shown. If the image 

region is omitted, the figure represents the stream lines and potential 

lines as they might be observed in the vicinity of a discharging well 

obtaining water from a river by induced infiltration . 

IMPERMEABLE BARRIER 

An idealized section through a discharging well in an aquifer 

bounded on one side by an impermeable barrier is shown in figure 

37A. It is assumed that the irregularly sloping boundary can, for 

practical purposes, be replaced by a vertical boundary, occupying

the position shown by the vertical dashed line, without sensibly 

changing the nature of the problem. The hydraulic condition imposed 

by the veritcal boundary is that there can be no ground-water flow 

across it, for the impermeable material cannot contribute water to 

the pumped well . The image system that satisfies this condition 

and permits a solution of the real problem by the Theis equation is 

shown in figure 37B. An imaginary discharging well has been placed 

at the same distance as the real well from the boundary but on the 

opposite side, and both wells are on a common line perpendicular to 

the boundary. At the boundary the drawdown produced by the 

image well is equal to the drawdown caused by the real well . Evidently, therefore, the drawdown cones for the real and the image 

wells will be symmetrical and will produce a ground-water divide at 

every point along the boundary line . Because there can be no flow 
0 

00 

CO 

v. 

O 

Recharging 

image \well d 

a 

y 

t=i 

x 

d 

a 

r 

c~ 

0 

X 

Q 

Fic-URE 36.-Generalized flow net showing stream lines and potential lines in the vicinity of a discharging well dependent upon induced infiltration 

from a nearby stream . 
THEORY OF AQUIFER TESTS 14 9 

Q 

r 

L > V > L < r 

r V r -7 -7 r

< J L C V < V 7 L, 

r r 7 -7 l" < r 

Impermeable or . ^ L. 

A . REAL SYSTEM 

NOTE : 

Aquifer thickness m should be very large com­ pared to resultant drawdown near real well 

B. HYDRAULIC COUNTERPART OF REAL SYSTEM 

FIGURE 37.-Idealized section views of a discharging well in a semi-infinite aquifer bounded by an impermeable formation, and of the equivalent hydraulic system in an infinite aquifer. 

across a divide, the image system satisfies the boundary condition of 

the real problem and analysis is simplified to consideration of two 

discharging wells in an infinite aquifer . The resultant drawdown 

at any point on the cone of depression in the real region is the algebraic sum of the drawdowns produced at that point by the real well 

and its image. The resultant profile of the cone of depression, 

shown in figure 37B, is flatter on the side of the well toward the 

boundary and steeper on the opposite side away from the boundary 

than it would be if no boundary were present. Figure 38 is a general­
v 

c 

0 

Q 

O 

O 

(i 

z 

Cy 

a 

H 

t~7 

C7 

a 

r 

n 

FiaUR$ 38 .-Generalized flow net showing stream lines and potential lines in the vicinity of a discharging well near an impermeable boundary . 
THEORY OF AQUIFER TESTS 151 

ized plan view of a flow net for the situation given in figure 37A. 

The distribution of stream lines and potential lines about the real 

discharging well and its discharging image, in an infinite aquifer, is 

shown . If the image region is omitted, the diagram represents the 

flow net as it might be observed in the vicinity of a discharging well 

located near an impermeable boundary. 

TWO IMPERMEABLE BARRIERS INTERSECTING AT RIGHT ANGLES 

The image-well system for a discharging well in an aquifer bounded 

on two sides by impermeable barriers that intersect at right angles is 

shown in figure 39 . Although the drawdown effects of the primary 

image wells, h and I2, combine in the desired manner with the effect 

Impermeable B barrier 

A A Discharging 

real well 

J r~rJ ~ 

Impermeable 

barrier 

NOTESImage wells, / , are numbered in the sequence 

in which they were considered and located 

Open circles signify discharging wells 

FIGURE 39.-Plan of image-well system for a discharging well in an aquifer bounded by two impermeable barriers intersecting at right angles. 
152 GROUND-WATER HYDRAULICS 

of the real well at their respective boundaries, each image well produces an unbalanced drawdown at the extension (reflection) of the 

other boundary. These unbalanced drawdowns at the boundaries 

produce a hydraulic gradient, with consequent flow across the extension of each boundary, and therefore do not completely satisfy the 

requirement of no flow across the boundaries of the real system. It 

is necessary, therefore, to use a secondary image well, 13, which balances the residual effects of the two primary image wells at the two 

extensions of the boundaries . The image system is then hydraulically 

in complete accord with the physical boundary conditions. The 

problem thereby has been simplified to consideration of four discharging wells in an infinite aquifer. 

IMPERMEABLE BARRIER AND PERENNIAL STREAM INTERSECTING AT 

RIGHT ANGLES 

The image-well system for a discharging well in an aquifer bounded 

on two sides by an impermeable barrier and a perennial stream which 

intersect at right angles is shown in figure 40. The perennial stream 

of figure 40 might also represent a canal, drain, lake, sea, or any other 

line source of recharge sufficient to maintain a constant head at this 

boundary. As before, the drawdown effects of the primary images,

h and I2, combine in the desired manner with the effects of the real 

well at their respective boundaries . However, discharging image well 

h produces a drawdown at the extension of the line source, which is 

a no-drawdown boundary, and recharging image well 1, causes flow 

across the extension of the impermeable barrier, which is a no-flow 

boundary. By placing a secondary recharging image well, Ia, at the 

appropriate distance from the extension of each boundary, the system 

is balanced so that no flow occurs across the impermeable barrier and 

no drawdown occurs at the perennial stream. Thus again the problem 

has been simplified to consideration of an infinite aquifer in which 

there operate simultaneously two discharging and tworecharging wells. 

The simplest way to analyze any multiple-boundary problem is to 

consider each boundary separately and determine how best to meet the 

condition of no flow or no drawdown, as the case may be, at that 

boundary. After the positions of the primary image wells have been 

established, the boundary positions should be reexamined to see if 

the net drawdown effects of the primary image wells satisfy all stipu 

lated conditions of no flow or no drawdown . For each primary image 

causing an unbalance at a boundary position, or extension thereof, it 

is necessary to place a secondary image well at the same distance from 

the boundary but on the opposite side, both wells occupying a common line perpendicular to the boundary. When the combined drawdown (or buildup) effects of all image wells are found to produce the 

desired effect at this boundary the same procedure is executed with 
THEORY of AQUIFER TESTS 153 

A 

B Impermeable c 

NOTES 

Image wells, / , are numbered in the sequence 

in which they were considered and located 

Open circles signify discharging wells 

Filled circles signify recharging wells 

Fiouaz 40 .-Plan of image-well system for a discharging well in an aquifer bounded by an impermeable

barrier intersected at right angles by a perennial stream . 

respect to the second boundary. Thus, the inspection and balancing 

process is repeated around the system until everything is in balance 

and all boundary conditions are satisfied, or until the effects of additional image wells are negligible compared to the total effect . 
154 GROUND-WATER HYDRAULICS 

TWO IMPERMEABLE BARRIERS INTERSECTING AT AN ANGLE OF 45° 

Although it is intended here to consider the particular image-well 

system required for analyzing flow to a well in a 45-degree wedgeshaped aquifer, it is appropriate first to comment briefly on some 

general aspects of image-well systems in wedge-shaped aquifers . By 

analogy with similar heat-flow situations it is possible to analyze the 

flow to a well in a wedge-shaped aquifer, and equivalent image systems 

can be constructed regardless of the wedge angle involved . However, 

closed image systems that are the simplest to construct and analyze 

occur when the wedge angle, 0, of the aquifer equals (or can be 

approximated as equal to) one of certain aliquot parts of 360 °. 

These particular values of 0 may be specified as follows (after Walton, 

1953, p. 17), keeping in mind that it is required to analyze flow to a 

single pumped well situated anywhere in the aquifer wedge : If the 

aquifer wedge boundaries are of like character, 0 must be an aliquot 

part of 180 °. If the boundaries are not of like character, 0 must be 

an aliquot part of 90*. 

Other simple solutions not covered by the above rule appear possible when 0 is an odd aliquot part of 360*, the pumped well is on the 

bisector of the wedge angle, and the boundaries are similar and impermeable. For any of the foregoing special situations it can be 

shown, with the aid of geometry, that the number of image wells, n, 

required in analyzing the flow toward the single real pumping well is 

given by the relation 

3600 

n- -1. (79) 0 

It can also be shown that the locus of all image-well locations, for 

a given aquifer-wedge problem, is a circle whose center is at the wedge 

apex and whose radius equals the distance from the apex to the real 

discharging well (see figure 45) . 

The image-well system for a discharging well in a wedge-shaped 

aquifer bounded by two impermeable barriers intersecting at an angle 

of 45* is shown in figure 41 . The real discharging well is reflected 

across each of the two boundaries which results in location of the two 

primary image wells h and 12 as shown. Considering boundary 1 

only, the effects of the real well and image well I,, are seen to combine 

so that, as desired, no flow occurs across that boundary. However, 

image well 12 will produce flow across boundary 1 unless image well 

Ia is added at the location shown . The system now satisfies the condition of no flow across boundary 1 . Repeating this examination 

process for boundary 2 only, it is seen that the effects of the real well 

and image well 12 combine, as desired, to produce no flow across 

boundary 2 . However, image wells h and I8 will produce flow 
THEORY OF AQUIFER TESTS 155 

across this boundary unless image wells I, and 15 are added as shown . 

The image system now satisfies the condition of no flow across boundary 2 . Reexamining, it is seen that image wells I4 and Is will produce 

flow across boundary 1 unless image wells IB and I, are added as shown. 

A final appraisal of the effects at boundary 2, shows that the entire 

system of image wells, plus the real well, satisfies the requirement of 

no flow across the boundary. Thus the flow field caused by a discharging well in this wedge-shaped aquifer can be simulated by a 

total of eight discharging wells in an infinite aquifer. The seven 

image wells have replaced the two barriers . The drawdown at any 

point between the two barriers can then be computed by adding the 

NOTES: 

Image wells, I, are numbered in the sequence 

in which they were considered and located 

Open circles signify discharging wells 

FiouRz 41-Plan of image-well system for a discharging well in an aquifer bounded by two impermeable 

barriers intersecting at an angle of 45 ° . 
156 GROUND-WATER HYDRAULICS 

effects produced at that point by the real well and the seven image 

wells . Each image well begins discharging at the same rate and at 

the same time as the real well. 

IMPERMEABLE BARRIER, PARATMEL TO A STREAM 

Shown in figure 42 is the image-well system for a discharging well 

in an aquifer bounded by an impermeable barrier and cut by a fully 

penetrating perennial stream parallel to the barrier. A recharging 

image well, I,, and a discharging image well, I2 , are placed as shown 

to satisfy respectively the conditions that no drawdown can occur 

along the line source, and no flow can occur across the impermeable 

barrier. Although these two primary image wells produce, in conjunction with the real well, the desired effects at their respective 

boundaries, each image well produces a residual effect at the opposite 

boundary which conflicts with the stipulated boundary conditions. It 

is therefore necessary to add a secondary set of image wells, 13 and 14, 

as shown, to produce effects that will combine properly with the 

residual effects of the primary images. Each image well in the secondary set will again produce residual effects at the opposite boundary, 

and similarly with each successively added image pair there will be 

residual effects at the boundaries . It should be evident, however, 

that as more pairs of image wells are added the effects of adding a 

new pair have lesser influence on the cumulative effect at each boundary. In other words it is only necessary to add pairs of image wells 

until the residual effects associated with addition of the next pair can 

be considered to have negligible influence on the cumulative effect at 

each boundary. It is seen in figure 42 that there is a repeating pattern in the locations of the image wells. Therefore, after the positions of the first images have been determined, it is possible to locate 

by inspection as many more as are needed for the practical solution 

of the problem. Once the required number of image pairs has been 

determined, the aquifer boundaries can be ignored and the problem 

analyzed like any other multiple-well problem in an infinite aquifer. 

If the two parallel boundaries are of like character-that is, if the 

perennial stream in figure 42 were replaced by an impermeable barrier 

or if the impermeable barrier were replaced by a perennial streamthe positions of the image wells would not be changed. In the first 

case, however, all the images would be discharging wells, and in the 

second case the image system would be an alternating series of recharging and discharging wells. 

TWOPARAi .LELIMPERMEABLEBARRIERS INTERSECTED AT RIGHTANGLES 

BY A THIRD IMPERMEABLE BARRIER. 

The image-well system for a discharging well in this type of areally 

restricted aquifer is shown in figure 43 . The positions of the images 

are determined as before by adding imaginary discharging wells so 
Average or effective position of 

impermeable barrier 

Notes 

Image wells,/, ore numbered in SECTION VIEW OF REAL SYSTEM 

the sequence in which they were I 

considered and located 

Open circles signify discharging 

wells 

Filled circles signify recharging 

wells 

REDUCED PLAN VIEW OF HYDRAULIC COUNTERPART OF REAL SYSTEM 

FioITRE 42 .-Image-well system for a discharging well !n an aquifer bounded by an impermeable barrier parallel to a perennial stream . 
Image pattern repeats to infinity I Image pattern repeats to infinity 

Note 

Open circles signify discharging wells 

Fiausa 43 .-Plan of image-well system fora discharging well in an aquifer bounded by two parallel impermeablebarriers intersected at right

angles by a third impermeable barrier. 
THEORY OF AQUIFER TESTS 159 

that, in combination with the real discharging well, the condition of 

no ground-water flow across any of the three boundaries is established. 

As shown in the figure, two parallel lines of discharging image wells 

are required, separated by twice the distance between the real well 

and the barrier that intersects the two parallel barriers . Theoretically 

the two lines of image wells extend to infinity in both directions from 

the real well . The practical analysis of a problem of this kind, however, requires the addition of only enough images so that the effect of 

adding the next image, in any of the directions involved, has a negligible influence on the cumulative effect at each of the boundaries. 

It is seen from figure 43 that there is a repeating pattern in the positions of the image wells, so that the locations of only the first few 

images are required to determine the locations of as many succeeding 

image wells as are needed . For the case of two parallel impermeable 

barriers intersected at right angles by a perennial stream, the image 

system would be the same as shown by figure 43 except that all images 

on the line reflected across the stream would be recharging wells. 

RECTANGULAR AQIIIFER BOUNDED DYTWO INTERSECTING IMPFRMEABLE 

BARR33MRS Pwg,AT"T "~rsNO PERIMNIAL STREAMS 

The image-well system for a discharging well in such an aquifer is 

shown by figure 44. The positions of the images are determined in 

the manner previously described . It is seen from figure 44 that there 

is again a repeating pattern that extends to infinity in all directions 

from the real well . Thus only the first few images need be located to 

determine the positions of as many succeeding images as are required 

in the practical solution of the problem . If the four boundaries in 

figure 44 were all impermeable barriers, all images would be discharging wells; and if the four boundaries were all perennial streams, the 

image system would be alternating series of recharging and discharging 

wells. 

APPLICABILITY OF IMAGE THEORY INVOLVING INFINITE SYSTEMS OF 

IMAGE WFJAA 

Referring to the three problems discussed in the three preceding

sections, it will be observed that in each situation the aquifer involved 

is limited in areal extent by two or more boundaries. Furthermore, 

the arrangement of the boundaries is such that at least two are parallel 

to each other, which means that analysis by the image theory requires 

use of an image-well system extending to infinity. 

It has been stated, in discussing the practical aspects of using an 

infinite image-well system, that the individual effects of image wells 

need be added only out to the point where the effect associated with the 

addition of the next more distant well (or wells, depending on the 

symmetry of the array) can be considered to have negligible influence 

on the cumulative effect . Although this criterion ostensibly provides, 
{-.A-' t B , `-A i B -,A i -l 

O O O O " " O O 

O O " " O O " " O O " " O O 

Notes : 

Impermeable 

barriers -~ I I 

Open 

Filled 

circles 

circles 

signify 

signify 

discharging 

recharging 

wells 

wells 

" " O O 

O 

0 " " O O " " G 

0

Y 

Y 

FtavsX i4.--Plan of image-wen system for a disebwging wenIn a rectangular aquifer enclosed by2 intersecting impermeable barriers and 2 intersecting

perennial streams. 
THEORY OF AQUIFER TESTS 161 

a reliable and practical means of terminating what would otherwise 

be an endless analytical process, closer scrutiny appears warranted. 

There is no reason to state categorically that this practical approach 

to a solution should never be tried . Undoubtedly there will occur 

situations wherein sensible results can be obtained . On the other 

hand it seems prudent to observe that if the process of algebraically 

summing the individual effects of an infinite system of image wells 

is terminated anywhere short of infinity, there is no simple way of 

determining what proportion of the infinite summation is represented 

by the partial summation. Although addition of the next image well 

(or wells) might have a negligible influence on the sum of all imagewell effects considered out to that point, there is no simple way of 

deciding whether the same may be said of the total influence represented by adding the effects of say the next 10 or 20 or 100 more distant image wells. Thus it would appear wise to keep in mind the 

possible limitations of any solution involving the use of an infinite 

system of image wells. 

COROLLARY EQUATIONS FOR APPLICATION OF IMAGE THEORY 

The nature and location of hydrologic boundaries of water-bearing 

formations in some cases can be determined from the analysis of pumping-test data. Considering the discussion in the preceding sections, 

it should be evident that in an aquifer whose extent is limited by one 

or more boundaries, a plot of drawdown or recovery data will depart 

from the form that would be expected if the aquifer were of infinite 

extent. Thus, in a problem involving a discharging well in a semiinfinite aquifer bounded by an impermeable barrier, some part of a 

time-drawdown plot may be steepened by the boundary effects. 

Conversely, if the boundary involved in the same type of problem were 

a perennial stream, a part of the time-drawdown plot may be flattened because of the boundary effects . 

Imagine a pumping test made in ail aquifer whose extent is limited 

by one or more boundaries . During the early part of the test, the 

drawdown data for observation wells close to the pumped well will 

reflect principally the pumping effects. As the test continues, however, 

there will very likely come a time for each observation well when the 

measured drawddwws reflect the net effect of the pumped well and any 

boundaries that are present. At distant observation wells boundary 

effects may arrive almost simultaneously with the effect of the real 

discharging well . Thus determination of the aquifer coefficients of 

transmissibility and storage should be based on the early drawdown 

data, as observed in a well near the pumped well, before the boundary 

effects complicate the analysis . Superposition and matching of a 

plot of these early data (s versus r2/t) on the Theis type curve permits 
GROUND-WATER HYDRAULICS 

drawing in the type-curve trace. Extension or extrapolation of this 

trace beyond the early data indicates the trend the drawdowns would 

have taken if the pumping had occurred in an infinite aquifer. The 

departure, 8t , of the later observed data from this type-curve trace 

represents effects of the boundaries on the drawdown. The subscript i 

refers to the image-well system substituted as the hydraulic equivalent of the boundaries. Usually it is convenient to note values of 8, at 

a number of points along the data curve and to replot these departures versus values of r.2/t on the same graph sheet that was used in determining the coefficient of storage and transmissibility from the early

data. The subscript r refers to the real discharging well . The latter 

part of the replotted departure data may again deviate from the 

type-curve trace if the cone of depression has intercepted a second 

boundary. As before, the departures can be replotted against corresponding values of r, 2 lt to form a second departure curve. This 

process should be repeated until the last departure curve shows no 

deviation from the type curve. The observed data array will then 

have been separated into its component parts which can be used to 

compute the distances between the observation wells and the image

wells. 

Inasmuch as the aquifer is assumed to be homogeneous (that is, the 

coefficients of transmissibility and storage are constant throughout

the aquifer) it follows from equation 8 that 

1 .87S u, a u, es-, (80) T r;/t r;/t 

where the subscripts r and i have the significance previously given. If on the plots of early drawdown data and first-departure curve a pair

of points is selected so that the drawdown component caused by the 

real well, 8 and the drawdown component caused by the image well, 

8j, are equal, it follows that u,=u{ . On the plots of observed early

drawdowns and first departures just described, 8, and s, obviously 

occur at different elapsed times, which can be labelled t, and t, 

respectively . Equation 80 can therefore be rewritten as follows : 

r2 r? 

ti .t* ) (81) 

or 

(82) 

Equation 81, known as the "law of times" in the physics of heat 

conduction, shows that at a given observation well location the times 

of occurrence of equal drawdown components vary directly and only 
THEORY OF AQIIIFER TESTS 163 

as the squares of the distances from the observation well to the 

pumped well and to its image. 

Referring to the data plots mentioned earlier in this section, note, for 

the pair of points selected, that values of s, and r,2/t, will be read from 

the early drawdown data while values of ss and r.

2/t, will be read from 

the first departure curve. Equation 82 can be made more useful, 

therefore, if it is rewritten in the form 

r'It, r,=r, 2/t, (83) 

Equation 83 now affords a ready means of computing the distance 

from an observation well to an image well . Similar analysis may be 

made of each departure curve constructed from the original drawdown 

data. 

Stallman (1952) has described a convenient method for computing 

ri when the observed drawdown in the aquifer represents the algebraic 

sum of the drawdown effects from one real well and one image well . 

If equation 6 is used to provide expressions for s, and s,, and W(u) is 

substituted as a symbolic form of the exponential integral, it is seen 

that the drawdown at the observation well is 

s=s,fs,= 114

T

.6 [W(u),±W(u)s] " 

From equation 80, 

or 

ut 

rs=r, u,

. (85) 

From equations 84 and 85, it can be seen that r i and the sum of the 

W(u) terms in equation 84 can be expressed in terms of r, and the 

ratio u{/u,. Thus for any given values of u,,/u,=g, a type curve 

can be constructed by plotting assumed values of u, against corresponding computed values of the bracketed portion of equation 

84 [which may be written in abbreviated form as Z W(u)] . The ., 

data plot, s versus t, will match this constructed type of curve if the 

observation well is located so that the ratio r jlr, equals the given 

value of K. However, if a family of type curves is drawn for a number of given values of K, the observed data plot, s versus t, for any 

observation well, can be compared with the set of type curves . Once 
164 GROUND-WATER HYDRAULICS 

the best matching curve is found, any convenient matchpoint is 

selected and the coordinate values, s, t, u,, Z W(u), and Kare noted. 

These values, substituted in equations 80, 84, and 85 provide the 

means for computing T, S, and r i . 

Stallman's set of curves is the familiar type-curve u versus W(u), 

used in conjunction with the Theis formula, with a series of appendage 

curves (two for each value of K) asymptotic to it . The trend of the 

appendage curve for a recharging image well is below, and for a dis 

charging image above, the Theis curve. Appendage curves could have 

been constructed by assuming values of ui instead of u, . In this 

event, however, the matching process would not be as direct inasmuch 

as the parent type curve, instead of occupying a single position, 

would shift along the u axis with each pair of appendage (K) curves . The appendage curves, computed by Stallman, are for ideal image

wells-those which are pumped or recharged at the same rate as the 

real well . The hydrogeologic structure which gives rise to the 

hypothetical image is not always ideal ; therefore the hypothetical 

images are not always ideal. For this case the method of plotting 

departures may yield an erroneous and misleading analysis . On the 

other hand, the deviations from ideality can be seen immediately 

if the observed data plot s versus t is matched to Stallman's set of 

type curves . Furthermore, for nonideal images, the most accurate 

selection of K is made by utilizing the portion of the appendage 

curve that is nearest the parent or Theis type curve. 

If little is known of the possible location of a local hydraulic boundary a minimum of three observation wells is required to fix the position 

of an image well, which in turn permits location of the boundary. 

After the distances from the individual observation wells to the image 

well have been computed, arcs are scribed with their centers at the 

observation wells and their radii equal to the respective computed

distances to the image well . The intersection of the arcs at a common 

point fixes the location of the image well, and the strike of the boundary is represented by the perpendicular bisector of a line connecting 

the pumped well and the image well . 

Another graphical method for locating a hydraulic boundary in 

the vicinity of a discharging well was devised by E. A. Moulder 

(1951, written communication, p. 61) . The geometry is shown in 

figure 45. A circle is scribed whose center is at a nearby observation 

well, O, and whose radius, r {, is equal to the computed distance from 

the observation well to the image well . The image well lies somewhere 

on this circle, say, at point I. Lines are drawn from the selected 

point I to the observation well and to the real discharging well, P. 

If point I is the image-well location and if A is the midpoint of the 
THEORY OF AQUIFER TESTS 165 

line IP, then point A lies on the boundary. It can be proved by 

geometry, that the locus of all points Adetermined in this manner is a 

circle, of radius BA or rj/2, with its center, B, located midway between 

the discharing well and the observation well . Moulder's method 

is particularly useful in aquifer-test situations where data from 

only one or two observation wells are available for locating a boundary 

position . If the approximate position of a suspected boundary is 

known before a pumping test begins, it is desirable to locate most 

of the observation wells along a line parallel with the boundary and 

passing through the pumped well . If feasible the range of distances 

from the observation wells to the pumped well should be distributed 

logarithmically to assure well-defined are intersections in the graphics 

of locating a point on that boundary. At least one observation well 

-%e 

c~ 

Locus of all possible locations 

of a point on the hydrologic 

boundary 

Fiovx$ 45 .-Geometry for locating a point on a hydrologic boundary, with reference to the locations of a 

discharging well and a nearby observation well . 
166 GROUND-WATER HYDRAULICS 

should be located close enough to the pumped well so that the early 

drawdown data, unaffected by the boundary, can be used in computing the aquifer coefficients of storage and transmissibility. 

APPLICABILITY OF ANALYTICAL EQUATIONS 

The assumptions used in developing the equations presented in this 

report include the stipulation that the aquifer is homogeneous and 

isotropic. Even though most naturally deposited sediments do not 

satisfy this condition, the equations may still be applied and the 

results qualified according to the extent of nonhomogeneity. It 

should be realized that homogeneity is a relative term with respect to 

time and space. As an illustration, consider an aquifer composed of 

two types of material-a fine sand and a very coarse sand. Assume 

that these materials occur individually in deposits having the shape 

of cubes one-eighth of a mile on a side, and that alternate rows of cubes 

(squares in plan view) are offset a distance equal to one-half the length 

of one side of the cube (that is, one-sixteenth of a mile). Let the fine 

and coarse sand occur in alternate cubes along the continuous rows, 

and assume that water occurs, in the aquifer thus created, under watertable conditions . Strictly speaking, this aquifer, of infinite extent, 

would now be described as nonhomogeneous . However, the areal 

extent of the portion of the aquifer sampled in a test would be significant in judging this element of the aquifer's description. For example, 

if a discharging well test is conducted in the center of one of the squares 

and if the test is terminated before the area of influence reaches the 

perimeter of the square, the test results probably would be considered 

excellent and the aquifer described as homogeneous . The results 

would in no way differ from the results to be expected if a similar test 

were made on an infinite "homogeneous" aquifer, composed of material 

identical to that occurring in the limited area here tested . 

As another example, again consider an aquifer test using a discharging well in the center of one of the squares of the hypothetical aquifer. 

The nearest of several observation wells is at a radius of 5 miles from 

the pumped well, and the test is run until the area of influence is 

described by a circle 10 miles in radius . Coefficients of transmissibility 

computed from data collected at all the observation wells should be in 

close agreement (although not equal to the values obtained from the 

previously described test), and again the hypothetical aquifer, even 

on the larger scale represented in this sample, would be adjudged 

homogeneous. This judgment relies upon the reasoning that, for the 

distances involved, the slightly meandering path of water, as it moves 

toward the well, may be described statistically as conforming to the 

concept of radial flow. For any case in which nonhomogeneity is so 
THEORY OF AQUIFER TESTS 167 

distributed that the flow field statistically fits the geometry of the 

mathematical model, the mathematical solution will provide a sound 

analysis . Conversely, when the flow field or a portion thereof is 

significantly distorted in the area of observation, the assumption of 

homogeneity is incorrect. Thus, for the hypothetical aquifer considered in the two preceding examples, the distorted condition is seen 

to exist if the area of influence of the discharging well were to extend to 

a radius of about ' 4 to 1 mile-that is, a little beyond the limits of one 

cube of the aquifer material. 

Often the field situation is encountered where a zone of relatively 

impermeable material such as a clay lens, of limited thickness and 

extent, occurs in an aquifer. It should be evident from the foregoing 

examples and discussion, however, that the presence of this clay lens 

in the flow field will have less influence on aquifer test results when 

the effects of the test encompass an area of large radius than when 

the area affected is of small radius . 

An important criterion, therefore, regarding the applicability of the 

equations discussed in this report, is the amount the flow field is distorted, as compared with the flow field that would have been observed 

in an ideal aquifer. 

It should be understood that the numerical results obtained by 

substituting aquifer-test data in an appropriate mathematical model 

indicate the transmissibility and storage coefficients for an ideal aquifer. 

The hydrologist must judge how closely the real aquifer resembles 

this particular ideal. It is usually recognized, for example, that in 

short pumping tests under water-table conditions the water does not 

drain from the smaller openings in the unwatered portion of the aquifer 

in a manner even approximating the instantaneous release assumed 

in devising the mathematical model. Similarly, in testing artesian 

aquifers it is recognized that the aquifer skeleton does not adjust 

instantaneously to the change in head, that considerable water is often 

contributed by intercalated clay beds, and furthermore that water 

leaks through the confining beds, which in the mathematical model 

have been assumed to be impermeable. However, these recognized 

departures from the ideal do not constitute grounds for abandoning, 

or rarely using, available analytical equations. Such departures simply add emphasis to the admonition that mere substitution of aquifertest data in an equation will not of itself assure anyone of establishing 

the correct hydraulic properties for that aquifer. The mechanics of 

applying any of the analytical equations in this report must be accomplished with sound professional judgment, followed by critical evaluation and testing of the results. 
168 GROUND-WATER, HYDRAULICS 

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Wenzel, L. K., and Sand, H. H., 1942, Water supply of the Dakota sandstone in 

the Ellendale-Jamestown area, North Dakota, with reference to changes between 1923 and 1938 : U.S . Geol . Survey Water-Supply Paper 889-A. 

Werner, P. W., and Noren, Daniel, 1951, Progressive waves in non-artesian 

aquifers : Am. Geophys. Union Trans., v. 32, no. 2, p. 238-244. 

Wyckoff, R. D., Botset, H. G., and Muskat, M., 1932, Flow of liquids through 

porous media under the action of gravity: Physics, v. 3, no. 2, p. 90-113. 

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Royal Soc. South Africa Trans., v. 3, pt . 1, p. 61-105. 
INDEX


Page 

Abstract-------------------------------------- 69 

Anisotropy, adjustment for, in flow-net construction----------------------- 143,144 

Aquifer, significance of geologic characteristics 70 

Aquifer boundary, determining location of__ 164-166 

Aquifer tests, the problem___________________ 70-78 

Aquifer worth, relation to storage and transmissibility characteristics_________ 70 

Aquifers, transmission of pumping effects between__ _ _____-_____-_____________ 80 

wedge-shaped -------------------------- 154-156 

Array, in numerical analysis, description of__ 136 

Artesian aquifers, elasticity of--- ____________ 78-88 

Atmospheric pressure effects on water levels 

in wells--------------------------- 83-85 

"Bailer" method of aquifer test_____________ 103,104 

Barometric efficiency of an aquifer------------ 85 

Bessel function, first kind, zero order_--------- 113 

second kind, zero order------------------- 112 

Blowthroughs-------------------------------- 80-81 

Boundaries of aquifers ---------------------- 145-161 

Coefficient of permeability, definition________ 72 

Coefficient of storage, artesian case------------ 76 

definition --------------------------------- 74 

relation to elasticity, of aquifer____________ 88-91 

water-table case -------------------------- 78 

Coefficient of transmissibility, definition____-- 72 

Complementary error function_--------------- 126 

Compressibility, of aquifer and water--------- 76,78 

Compression, bulk modulus of, for aquifer 

skeleton ----------------------- 86,88-90 

bulk modulus of, for water------------- 86,88-90 

Cone of depression -------------------------- 101,111 

Cone of influence----------------------------- 111 

Darcy'slaw---------------------------------- 71,73 

Drain function, constant discharge, D(u) a __ 123-125 

constant head, D(u)n------------------- 126-128 

Drawdown, residual ---------------- 100,101,103,104 

Earthquake effects on water levels in wells--- 87 

Earth-tide effects on water levels in wells_ ____ 86-87 

Effective radius of well ------------- 101,103,106,110 

Efficiency, barometeric, of an aquifer--------- 85 

barometric, related to elasticity of aquifer_ 86 

barometric and tidal, sum of-------------- 86 

tidal, of an aquifer------------------------ 85 

related to elasticity of aquifer--------- 86 

Elasticity of aquifer, relation to coefficient of 

storage--------------------------- 88-91 

Elasticity of artesian aquifers ----------------- 78-88 

Equipotential lines, in ground-water flow ----- 139-

144,148,150 

Page 

Finite aquifers, transformation of----------- 144-166 

Finite difference equation, example of ------ 137,138 

Grid, finite, diagram of----------------------- 137 

Ground water, storage of_____________________ 70 

Ground-water divide_______________________ 131, 132 

Ground-water flow, differential equation of___ 136 

pattern, graphical representation of_ ___ 139-144, 

148, 150 

Ground-water hydraulics--------------------- 70 

Ground-water studies, objectives_____________ 70 

Head of water. See Equipotential lines______ 75 

Hydrologic entities --------------------------- 70 

Hydraulic gradient, explanation of ----------- 73,139 

Image theory ------------------------------- 144-166 

Image well, distance from observation well- 162-164 

Image-well systems, closed, criteria for, in 

wedge-shaped aquifers____________ 154 

limitations of --------------------------- 159,161

Impermeable barrier----------- 145, 147, 149, 151-159 

Internal forces in aquifer---------------------- 78-80 

Iteration method of numerical analysis_____ 138,139 

Jacob modified nonequilibrium formula_ ____ 98-100 

Law of times---------------------------------- 162 

Leakage, coefficient of------------------------ 115 

vertical, through confining bed_________ 110,111 

Leaky-aqufferproblem --------------------- 111-118 

Leastwork, principle of, in ground-water flow- 139 

Limitationsof well methods of aquifer tests_ 102, 103 

Line source____________________ 145-147, 152, 156, 159 

Load on aquifers, effects of changes in -------- 80-88 

Node, in numerical analysis, description of--- 136 

Partial penetration of a well ------------------ 103 

Permeability, limitations of laboratory meas­ urement of ----------------------- 73-74 

Porosity, of aquifer material --------------- 86,88-90 

Railroad trains, effect on water levels in wells_ 81, 82 

Range ratio, of an observation well ----------- 133 

Recharge, uniform rate, determination of___ 131,132 

Recovery of water level in a pumped well__ 100, 101 

References cited ---------------------------- 168-171 

Relaxation method of numerical analysis___ 138, 139 

Residual, in numerical analysis ------------- 138,139 

Residual drawdown--------------- 100, 101, 103, 104 

Residual head-------------------------------- 104 

Sand boils------------------------------------ 81 

"Slug" method of aquifer test -------------- 104,105 

Specific weight of fluid, definition____________ 88-89 

Error function________________________________ 126 Specific yield, definition of-------------------- 76 

Explicit hials, procedure of, in flow-net con- Storage coefficient . See Coefficient of storage. 

struction------------------------- 140 Streamlines, in ground-water flow . 139-144, 148, 150 

Exponential integral -------------------------- 92,93 Symbols-------------------------------------- vi-vi 

173 
-----------------------------

174 INDEX 

r&M9w 

Terminology ---------------------------- m-m/ .u 

zumouone«muumooformula ---------------- 92-W 

Toeiv,ecvvery formula --------------------- IOD-102 

Tmom equilibrium formula------------------- 91,92 

Tidal efficiency of an aquifer------------------ 85 

Tides, surface water, effects vuwater levels in 

wells 85 

Transformation of flow field, in flow-net 

analysis------------------------ 143,144 

page 

Well function, o(a) ------------------------ 106-109 

W(u) ----------------------------------- 93-97 

Water levels, anomalous fluctuations of------- so 

Water levels, fluctuations of, caused by earth 

tides ------------------------------ M-87 

caused by earthquakes ------------------- 87 

caused by surface-water uoes ------ 85 

caused by atmospheric pressure ---------- 83-85 

caused by railroad trains ----------------- 81,82 

Well, partial penetration of------------------- 103 

Wells, blowing or sucking -------------------- 84 

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