Groundwater and Seepage

MODEL ANALYSIS OF HYDRAULIC CONDUCTIVITY OF AN AQUIFER

by

H. Y. Fang

R~ D. Varrin

December 1968

Fr i t zEn gin eer; n9 Lab 0 rat 0 r y Rep 0 r t No. 3,4 1 • 2

TABLE OF CONTENTS

SYNOPSIS ..

INTRODUCTION .

HYDRAULIC CONDUCTIVITY AND TRANSMISSIBILITY ..

PAGE;

3

STEADY FLOW - EQUILIBRIUM EQUATION . . .. .... 5

NONSTEADY FLOW - NONEQUILIBRIUM EQUATION . . . . . .• 7

MODEL STUDY . . . . . . . . . . . . . . . . . . . 8

SUMMARY AND CONCLUSION S•.

ACKNOWLEDGEMENTS • .

REFERENCES .

. . . . . . . . . . . . . . . . . . . . . . .APPENDIX

FIGURES

Description of the Horizontal

Viscous-Flow Model •...

Computation of Hydraulic Conductivity

Model

Thiem·s Equilibrium Equation.

Theis· Nonequilibrium Equation

. . . .

8

11

11

12

13

15

18

19

21

22

Model Analysis of Hydraulic Conductivity of an Aguifer

H. ¥. Fang 1 and Robert D. Varr; n2

S Y N 0 PSI S

This paper presents a study of steady and nonsteady

radial well flow using a horizontal viscous-flow model. The

fundamental behavior of ground-water movement during pumping

tests ;s demonstrated. The validity of the classical equi­

librium and nonequilibrium well flow equations for computing

the hydraul; c conducti vi ty of an 'aqui fer ; s di scussed.

The analysis demonstrates that there is a close agree­

ment between flow quantities predicted by the existing

theories and those obtained in the model. It;s suggested

that using the nonequilibrium equation for analyzing pump­

ing test data ;s the more logical and time-saving method.

1. Director, Geotechnical Engineering Division, Fritz Engi­neering Labor~tory, Lehigh University

2. Director, Water Resources Center, University of Delaware

INTRODUCTION

Hydraulic conductivity (or coefficient of permeabil­

ity), K, is one of the most important engineering properties

of soil. It;s a measure of the capacity of soil to trans­

mit wate~. In order to reliably determine hydraulic conduc­

tivity in the field, pumping tests are commonly used.

Two types of equations are generally used to analyze

field pumping test data. One is the Thiem equilibrium equa­

tion (Thiem, 1906), and the other is the Theis nonequilibrium

equation (Theis, 1935). The equilibrium equation Can be

used to determine the hydraulic conductivity of an aquifer

if the rate of discharge of a p~mped well is known, and if

the drawdown in the observation wells at various known dis­

tances from the pumped well after the cone of depression has

been stabilized is established~ The nonequilibrium equation

permits this determination when the rate of discharge of a

pumped well ;s known, and when the drawdown as a function of

time is determined for one or more observation wells at given

distances from the pumped well.

The equilibrium equation has been widely applied for

hydraulic conquctivity determinations in the field of soil

mechanics. In ground-water hydrology, the nonequilibrium

equation is commonly used. However, the validity of these

-2-

two equations has not yet been fully verified by laboratory

studies.

Among the various types of analog models which are used

in the laboratory for studying ground-water movement are the

sand model (Hall, 1935), the membrane analogy (Hansen, 1935),

the viscous-flow analogy (Todd, 1954), and the electrical

analogy (Zee, Peterson and Bock, 1957). In the study of

radial well-flow phenomena, the viscous-flow model appears

to be highly promising (Santing, 1957; DeWiest, 1965; Varrin

and Fang, 1967).

Recently, a new type of horizontal viscous-flow model

with Jtinfinite ll areal extent has been developed by DeWiest

(1966) and Varrin and Fang (1967). A conformal mapping tech­

nique has been applied in the model design in order to con­

sider the lIinfinite" extent of an aquifer. This type of

model has been found to be a useful device in analyzing many

types of ground-water flow problems, especially for analyzing

steady or nonsteady radial well flows.

The major objectives of this study were: to demonstrate

the fundamental behavior of ground-water movement during a

pumpin,9 test; to verify the validity of the classical equi­

librium and nonequilibrium well-flow equations; to compare

the test results on the basis of various curve fitting methods

which have been applied to the Theis nonequilibrium solution;

and to attempt to clarify the differences in hydraulic con­

ductivity units and nomenclature as used by soils engineers

and ground-water hydrologists.

-3-

HYDRAULIC CONDUCTIVITY AND TRANSMISSIBILITY

Various names have been given to K, including effective

permeability (Muskat, 1937), coefficient of permeability

(Terzaghi, 1943), seepage coefficient (Polubarinova-Kochina,

1962), and hydraulic conductivity (DeWiest, 1965; Wit, 1966).

In the field of soil mechanics, the term coefficient of per­

meability is commonly adopted. In ground-water hydrology

the term hydraulic conductivity is widely used. In this

paper, hydraulic conductivity is used throughout.

The hydraulic conductivity of an aquifer is influenced

by the properties of the fluid in the aquifer. The fluid

influence may be expressed by the ratio of its unit weight

to its dynamic viscosity. P. C. Nutting (1930) was among

the first to recommend the term physical permeability, k,

for which

k = K ll/Y

where K = hydraulic conductivity, [LIT]

~ = dynamic viscosity of fluid, [FT/L 2] or [MILT]

y = unit weight of fluid, [MIL 2 T 2 J

( 1 )

Other names have also been used for k, such as trans­

mission constant (Muskat, 1937), specific permeability

(Todd, 1959), and intrinsic permeability (DeWiest, 1965).

The term intrinsic permeability ;s used throughout this text.

-4-

In the field ef soil mechanics, the centimeter-gram­

second system of units is commonly used for hydraulic con­

ductivity. In the field of ground-water hydrology, the

foot-gallen-day system is used. For example, the U. s.Geological Survey used the me1nzer unit as a measure of

hydraulic conductivity (Wenzel, 1942). The meinzer unit is

defined as the flow of water ;n gallons per day through a

cross-section of aquifer 1 ft. thick and 1 mile wide under

an hydraulic gradient of 1 ft. per mile at field temperature.

The unit of intrinsic permeability, k, ;s usually extremely

small, so-that the darey has been adopted as a more practical

unit. Conversio'n factors for the meinzer and darcy units

are shown in the Appendix.

In 1935 Theis introduced the term coefficient of trans­

missibility, T, which is expressed as the rate of flow in

gallons per day through a 1 ft. wide vertical strip of the

aquifer under a hydraulic gradient of 1 ft. per ft. at the

prevailing water temperature. The relationship between

hydraulic conductivity, K, and the coefficient of transmiss­

ibility, T, is as foll-ows:

K = Tlb

where b is the thickness of confined aquifer and has the

dimensions of length [L].

( 2 )

-5-

STEADY FLOW .- EQUILIBRIUM EQUATION

The equilibrium equation was developed by Gunter Thiem

of Germany in 1906 for the determination of hydraulic con­

ductivity, K. The equation was based on the following

assumptions: (a) that the aquifer is homogeneous and iso­

tropic with respect to hydraulic conductivity, and of infi­

nite areal extent; (b) that the hydraulic conductivity is

independent of time; (c) that the flow ;s laminar and steady;

(dl. that the discharging well penetrates and receives water

from the entire thickness of the permeable, water-bearing

stratum; and (e) that the well is pumped continuously at a

constant rate until the flow of water to the well is stab;-

lized.

Using plan polar coordinates with the well as the origin,

the radial flow equation for a well completely penetrating

a confined aquifer is given by (see Figure 1):

Q = A v :::; 21frbK dh/dr ( 3)w

where Qw = well discharge, [L 3 IT]

A = cross-sectional area perpendicular to flow

= 27Trb, [L 2]

b = thickness of confined aquifer, [L]

r = radial distance to a'ny point from axis of well, [L]

v = flow velocity, [LIT]

h = head at any point in the aquifer at time, t, [L]

dh/dr = i = 'hydraul ; c grad; ent, [d~men.6,[onle..6~].

-6-

Separation of variables gives the following differential

equat'ion:

Qw drdh = 2bK r ( 4)

The boundary conditions for Eq. 4 are at the well h = hwand r = rw' and at the edge of the area of well influence

h = H, and r = R. Integrating Eq. 4 between limits as in­

dicated:

H R

dh =Qw dr

27TbK rhw rw

H - hw =Qw 1n R ( 5 )21fbK r w

Thus, the hydraulic conductivity, K, can be calculated as

( 6)

Since any two points will define the drawdown curve,

Eq. 6 can be written in terms of drawdowns measured in two

observation wells. For this case, the equation for hydraulic

conductivity, K, becomes:

( 7 )

where 51 = H hi

52 = H h 2

-7-

r1

and r2

refer to the radial distance from axis of the

pumped well to observation wells 1 and 2 respectivelys and

sand s refer to the drawdown at observation wells 1 and1 2

'2 s respectively.

NONSTEADY FLOW - NONEQUILIBRIUM EQUATION

The partial differential equation in plane polar co­

ordinates governing nonsteady well-flow in an incompressible

confined aquifer of uniform thickness is:

( 8)

*where S = storage coefficient, [dlmen~~onle~~]

t = time since the flow started, [L]

The terms h, r, and T have been defined previously.

Theis (1935) obtained a solution for the above equation

based on the analogy between ground-water flow and heat con­

duction. By assuming that the well is replaced by a math­

ematical sink of constant strength, that after pumping begins s

*Storage coefficient ;s defined as the volume of water that

an aquifer releases or takes into storage per unit surface

area of the aquifer per unit change in the component of head

normal to that surface (Theis, 1935).

...... 8-

h approaches Hand r approaches 00, the solution is:00

where u = r2S

4Tt

s = H - h

u

-ueu du (9 )

Eq. 9 is known as the nonequilibrium of Theis equation. This

equation permits determination of the aquifer constants:

storage coefficient, S; transmissibility, T; and hydraulic

conductivity, K.

The exponential integral in Eq. 9 has been assigned the

symbol W(u) which ;s called the tlwell function of uH• Eq. 9

may be somewhat too complicated for engineering purposes,

but several investigators have developed approximate solu­

tions. Included amongst these are the Theis method of super­

position, the Theis recovery method and the Cooper and Jacob

method.

Following a description of the experimental procedure

used in the model study, each of the above approximate solu­

tions is illustrated using experimental data obtained from

the model. The results are compared with those obtained

from the Thiem equilibrium solution.

MODEL STUDY

Description of the Horizontal Viscous-Flow Model

The first horizontal viscous-flow model was developed

by H. S. Hele-Shaw in 1897-1899. Since then the model has

-9-

been refined by many investigators (Todd, 1959; Bear, 1960;

Sternberg and Scott, 1964; DeWiest, 1965), and shown to be

useful for analysis of almost any' two-dimensional ground­

water flow problem, whether steady or not. In 1956 ,this

type of model was further improved to include three­

dimensional flow (Bear and Kruysse, 1956; Santing, 1957).

Recently, by using a conformal mapping technique to take

fully into consideration the infinite extent of an ideal

aquifer, a further improvement has been made on the model

(DeWiest, 1966; Varrin and Fang, 1967). In this paper, only

the horizontal type of viscous-flow model is discussed.

Since the complete description of the horizontal viscous­

flow model has been reported by Varrin and Fang (1967), a

brief description of this model is presented here:

The horizontal viscous-flow model consists of two

closely spaced parallel plates and may simulate a portion of

an aquifer, either phreatic or confined. The interspace be­

tween the plates represents the aquifer, with the val,ue of

the hydraulic conductivity being influenced by the width of

the interspace as well as 'the properties of the permeant and

the temperature of the test. Storage capacity is achieved

by means of vertical tubes or vessels on top of the upper

plate each of which ;s connected to the interspace.

In this study,. the interspace between the plates was

1 mm. and the diameter of each storage vessel was 1 em. The

model was divided into interior and exterior regions. The

-10-

interior region was·l meter by 1 meter and represented a pro­

totype aquifer 10,000 by 10,000 meters. The exterior region

represented an aquifer of infinite areal extent. Of course,

it is physically impossible to model exactly this condition,

but the technique of conformal mapping can be used to extend

the model aquifer a considerable distance with a modest in­

crease in size. Thus, the resulting model tends to approach

the condition of an aquifer of infinite areal extent. Details

of the conformal mapping technique are described by Nehari

(1952) and DeWiest (1966). The- interior region and the ex­

terior region of the aquifer are shown in Figure 2. The plan

view of both regions ;s shown in Figure 3.

The time scale for the model containing 96% aqueous

glycerine as the fluid was determined by dimensional analysis

to be 1 :10,900. Therefore, as an example, one hour of labo­

ratory testing simulates 15 months of field testing.

For the laboratory nonequilibrium pumping test to demon­

strate the fundamental behavior of ground-water movement, the

experimental procedure adopted was as follows:

1. Aquifer and storage vessels were filled with

glycerine to a certain level. The head at each

storage vessel was recorded.

2. At the same time, glycerine was added at a con­

stant rate to the exterior part of the aquifer

(cloverleaf shape). This is important, because

glycerine. flows from the exterior part of the

-11-

aquifer to the interior part and a constant head

must be maintained at lIinfinityli.

3. Once the pumping test started, the head dis­

tribution in each of the storage vessels was re­

corded as a function of time. Millimeter scales

attached to the storage vessels facilitated the

reading of the head changes.

4. The discharge from the pumping well and the

temperature of the glycerine used in the model

were recorded.

5. The pumping test was continued until a near

steady state condition was reached.

Several tests were performed for various discharges,

Qw• Results of these tests are shown in Figures 5 and 6.

The variation of drawdown with distance from the pumped

well and with time is shown in Figure 5. It is indicated

that the drawdown decreases as distance from the pumped

well increases. Also, as time since pumping started in­

creases, the drawdown increases. Figure 6 shows the linear

relationship between the drawdown and the discharge for a

given well for all values of time until the boundary of the

aquifer is reached.

Computation of Hydraulic Conductivty

Model

From the Navier-Stokes equation (Harr, 1962; DeWiest,

1965) for flow between plates, the equation for the

-12-

transmissibility, T, of the model is found to be

where 9 = acceleration of gravity = 981 cm/sec 2

= kinematic viscosity of the fluid,

= 5.29 cm 2 /sec for 96% aqueous glycerine at 20°C

b = thickness of the interspace between the plates,

= 1 mm

T=-' .9.b 312 V

From Eq. 2:

K = I = -' .9. b 2b 12 v

( 10)

( 11 )

Substituting these known values in Eq. 11, gives

K = 15.4 X 10- 2 em/sec

Eq. 11 indicates that the hydraulic conductivity value is

influenced by the viscosity of the fluid. The viscosity of

the fluid is 'inversely proportional to the temperature (see

Figure 7). Therefore, the hydraulic conductivity value is

dependent upon the temperature. The relationship between

hydraulic conductivity and the temperature is shown in

Figure 8. Since the model scaling is based on glycerine at

20°C, a correction should be made if the temperature during

the t est va r i ,e d .

Thiem's Equilibrium Equation

From Eq. 7:

In(r /r )2 1

S 1 - S 2

-13-

where Qw = discharge from pumping well = O. 18 em /sec.

b = 1 mm; r1

= 7.06 em; r2 = 21 . 02 em.

s 1 = 5.6 em; and s 2 = 3.8 em. ( From ~igure 9)

The refore , K = 17.3 X 10- 2 em/sec (at 23°C)

Theis' Noneguilibrium Equation

A. Theis Method of Superposition (1935)

A log ar; t hmi c plot 0 f W( u) ve r sus U, know n as a II ty Pe

curve" is prepared. Theoretical values of W(u) for a wide

range of u have been prepared by Wenzel (1942) and may be

found in any standard ground-water text book. Values of

drawdown, 5, are plotted against values of r 2 /t on logarithmic

paper of the same' size as the type curve. The observed data

curve is superimposed on the type curve. An arbitrary point

is selected on the co~ncident segment, and the coordinates

of this matching point are recorded (see Figure 10). With

values of W(u), u, S, and r 2 /t thus determined, T and K can

be obtained from Eq. 9.

For u = 0.07; W(u) = 1; s = 1.05 em.

t = 100 sec; and Q = 0.22 cm 3 /sec.

Eq. 9 yields:

T = 16.6 X 10- 3 cm 2 /sec.

K = 16.6 X 10~2 em/sec.

B. Theis Recovery Method (Theis, 1935; Wenzel, 1942)

If a well is ·pumped at a constant rate and then shut

down, the head will recover from its lowest value at time

-14-

when pumping is stopped to attain value hi at time t l

from the time of shutdown. If H is the initial value of the

head before pumping started, then H - hi = s' is called the

residual drawdown. If a semilog plot of 5' versus tIt' is

prepared, then T may be calculated by the following equation:

2.30 QwT = 47f~S'

where ~Sl = slope of the semi log plot

From Figure 11 , IJ.s' = 2.3-8 cm.

Q = 0.22 cm 3 /sec.

Therefore, T = 17 .5 x 10- 3cm 2 /sec.

K = 17.5 X 10- 2 em/sec. (at 23°C)

( 12 )

c. Coo per andl J acob Met hod

A simplified form of the Th~is equation was developed

by Cooper and Jacob (Cooper and Jacob, 1946; Jacob, 1950).

The equation for computing T and K ;s as follows:

T = 2.30 Q4rr.6s

From Figure 12, Q = 0.22 em 3 /sec.

65 = 2.4 em.

Therefore, T = 16.7 X 10- 3 cm 2 /sec.

K = 16.7 X 10- 2 em/sec.

( 13)

A summary of hydraulic conductivity values computed

from the above methods are tabulated as follows:

-15-

Method

Model

Thiem equilibrium equation

Nonequilibrium equation

Theis method

Theis recovery method

Cooper and Jacob method

Hydraulic Conductivityem/sec.

15.4 X 10- 2 at 20°C

17.3 x 10- 2 at 23°C

16.6 x lO-2'-at 23°C

17.5 x 10-2

at 23°C

16.7 x 10- 2 at 23°C

From the model analysis, within the experimental limits,

the equilibrium and nonequilibrium equation yielded the same

hydraulic conductivity. However, with the equ;'libr;um equa­

tion, the straight line portion of drawdown curve should be

used.

SUMMARY AND CONCLUSIONS

The model analysis of hydraulic conductivity of an aqui­

fer can be summarized as follows:

1. A new type of horizontal viscous-flow model

wi th II i nfi ni tell areal extent has been found to

be a useful device in analyzing steady and non­

steady radial well flows.

2. The principal advantages of the analog model

are:

a. A time scale in the model whereby 1

second in the laboratory represented

3 hours in nature.

-16-

b. The simulation of a very large homogeneous

and isotropic aquifer (24,000 meters by

24,000 meters).

3. From a model pumping test, the following rela­

tionships were shown:

a. Drawdown decreased as distance from the

pumped well increased.

b. As time since pumping started increased,

the drawdown increased until equilibrium

was reached.

c. A straight-line relationship existed

between the drawdown and the discharge

for a given well for all values of time

until the boundary of the model aquifer

was reached.

4. The Thiem equilibrium equation and various methods

based on the Theis nonequilibr;um equation were

used to determine the hydraulic conductivity of

the model aquifer. Within the experimental

limits, the equilibrium and nonequilibrium equa­

tions yielded the same hydraulic conductivity.

5. The duration of the laboratory pump test required

to establish equilibrium was about two hours or

the equivalent of about 3 years in nature. In

contrast, hydraulic conductivity was determined

from the nonequilibrium equation after a few

-17-

minutes in the laboratory (several days in nature).

The advantage of calculating hydraulic conductivity

from the nonequilibrium methods was easily dem­

onstrated.

-18-

ACKNOWLEDGEMENT

The work upon which this publication is based was sup­

ported in part by funds provided by the United States Depart­

ment of the Interior as authorized under the Water Resources

Research Act of 1964, Public Law 88-379.

Sincere appreciation is extended to the staff members of

the Envirotronics Corporation for their technical assistances.

Professor T. J. Hirst provided constructive criticisms

and numerous suggestions.

-19-

REFERENCES

Hansen, V. E., 1935. Unconfined ground-water flow to mul­tiple wells. Trans. American Society of Civil Engineers,v. 118.

Bear, J. and Van Overstraaten Kruysse, 1956. Report oninvestigations of ground-water movement and related prob­lems in Zealand Flanders by means of a horizontal siltmodel. Israel Institute of Technology, Israel.

Bear, J., 1960. Scales of viscous analogy models for groundwater studies. Journal Hydraulic Div., American Societyof Civil Engineers, v. 86, no. HY2, pp. 11-25.

Cooper, H. H., Jr. and C. E. Jacob, 1946. A generalizedgraphical method for evaluating formation constants andsummarizing well-field history. Trans. American Geo­physical Union, v. 27. pp. 526-234.

DeW i est, R. J. M., 1965. Ge0 hydr 0 logy . John Wi 1ey andSons, 366 p. '

DeWiest, R. J. M., 1966. Hydraulic model study of nonsteadyflow to mu1tiaquifer wells. Journal of Geophysical Re­search, v. 71, no. 20, pp. 4799-4810.

An investigation of steady flow towardLa Houi11e Blanche, v. lOs pp. 8-35.

Hall, H. P., 1935.a gravity well.

Harr, M. E., 1962. Groundwater and Seepage. McGraw-Hill,316 p.

Hele-Shaw, H. S., 1897. Experiments on the nature of thesurface resistance in pipes and on ships. Trans. INA,v. 39, pp. 145-156.

He1e-Shaw s H. S., 1898. Investigation of the nature ofsurface resistance of water and stream-line motion undercertain experimental conditions. Trans. INA, v. 40,pp. 21-46.

Jacob, C. E., 1950. Flow of ground water. (Eng. Hydraulic)John Wiley and Sons, pp. 321-386.

Leonards, G. A., 1962. Foundation Engineering. McGraw­Hi 11 .

Muskat, M, 1937. The Flow of Homogeneous Fluids throughPorous Media. McGraw-Hill, 736 p.

Nehari, Zeev, 1952. Conforma 1 Mappi n9. McGraw-Hi 11 ,396 p.

Nutting, P. G., 1930. Physical analysis of oil sands. Bull.American Association of Petroleum Geologists, v.' 14,pp. 1337-1349.

Polubarinova-Kochina, P. Va, 1962. Theory of Ground WaterMovement. Princeton University Press.

Santing, G. A., 1957. A horizontal scale model, based on theviscous flow analog, for studying ground-water flow in anaquifer having storage. lASH, General Assembly, Toronto,pp. 105-114.

Sternberg, Y. M. and V. H. Scott, 1964. The Hele-Shaw Model­A research device in ground-water studies. GroundWater, v. 2, no. 4, pp. 33-37, October.

Terzaghi, K., 1943. Theoretical Soil Mechanics. John Wileyand Sons.

Theis, C. V., 1935. The relation between the -lowering ofthe piezometric surface and the rate and duration of dis­charge of a well using ground-water storage. Trans.American

l

Geophysical Union, v. 16, pp. 519-524.

Thiem, Gunther, 1906. Hydro1ogische Methoden, Gebhart,Leipzig, 56 p.

Todd, D. K., 1959. Ground Water Hydrology. John Wiley andSons, pp. 314-316.

Varrin, R. D. and H. Y. Fang, 1967. Design and constructionof a horizontal viscous-flow model. Ground Water, v. 5,no. 3, pp. 35-41, July.

Wenzel, L. K., 1942. Methods for determining permeabilityof water-bearing materials with special reference to dis­charging well methods. USGS Water-Supply Paper 887.

Wit, K. E., 1966. Apparatus for measuring hydraulic con­ductivity of undisturbed soil samples. ASTM, STP 417,pp. 72-83.

Zee, C. H., D. F. Peterson and R. O. Bock, 1957. Flow intoa well by electric and membrane analogy. Trans. AmericanSociety of Civil Engineers, v. 122, pp. 1088-1112.

-2.:1:-

APPENDIX

Conversion Factors

From Darcy's Law

Q = K A !lL

where Q = discharge, [L 3 IT]

K = hydraulic conductivity, [LIT]

A = cross-sectional area, [L 2 ]

h/L = hydraulic gradient, [dimen~ionle~~]

or

Hydraulic Conductivity

Q = K A ~t expressed in general terms

and

Intrinsic Permeability

K = QA(*> ________(A)

-k - K 11 = u Q/ A- y y (dh/dL)

From eq~ation (B), the darey is defined as:

_____(8)

1 oarcy = 1 centipoise x 1 cm 3 tsec./cm. 2

1 atmosphere/l em.

= 0.987 x 10- 8 cm 2 = 1.062 x 10- 11 ft 2 •

The meinzer is defined as:

1 meinzer ~ 5.5 x 10- 2 gal/day (for water at GOoF).

Qw

t [ OBSERVATION lWELLS---.......... ...-.----------------.... ~---- ............."....---.................

S BFACt;._~2 --­~..........-

ICO:~DEPRESSION

H

~----- r2 ------.-.t

.. I I · I • •

. I I ·. III I •

. ' I I · .'.. I I .'. · ·

....

........--------R-------..-...

Figure 1 Steady Flow to a Well in a Confined Aquifer

TO Q)

TO CD

44

u 0

TO ell

'i

2.4 v -2.0 ~-2.

EXTERIOR DOMAIN

-1.0 '7777 "7FJ !"'" '/77/ '7r7,1

·0 ~I

~~

INTERIOR DOMAIN~ e V=0.5~ u· 1.5.I

~~

~~~STORAGE VESSEL---r--OWELL I I I 1 Iv-O v-O v 1l 2.4• u-I.O u=o

vu

TO Q)

, -1.0.-',0

Figure 2 Relationship of Interior and Exterior Domains ­Horizontal Viscous-Flow Model

........1

I7~' I~.... "'-

;:;OUs IN :-l IUPPER PLATE 'I

'IIIiIII

JlI

06 02

010 07 03

o

INTERIOR DOMAIN OF AQUIFER

o

o

o 013 011 08 04

015 014 012 09 05

o

o

TUBE CON NECT IONS INSTORAGE LOWER PLATE ONLYVESSELS (40 TUBES TOTAL)

~--~o-~~~~

EXTERIORDOMAIN

\

VALVE

SUPPLY TANK

CROSS TUBE BETWEEN WELLS

EXTERIOROOMAIN

/

INTERIOR DOMAIN OF AQUI FER

TU8E CONNECTIONS INLOWER PLATE ONLY STORAGE(200 TUBES TOTAL) VESSELS

"£~~~-----j---10 0 0 0

Ia 20 10 0 0

B 30 70 100 0 0

lI 40 10 110 130 0

50 90 120 140 150

(

I

_{ol

Figure 3

(b)

Plan View of Model (Left portion only used)

Figure 4 Horizonal Viscous-Flow Model

OBSERVATION WELLS

uQ)

-.J~-.J ~"W E3: <.)

o 00w ·CL 0~ II

::>0...10

.oz

<.0.oz

o...-.oz

rt)a--.oz

10

.oz

DISTANCES FROM PUMPING WELL

rl = 7 .06 emr6 = 21.02 emrIO = 35.40 emr13 = 45.60 emr15 = 63.70 em

I '·-t = 0o I I I I t, 0- t = 1500 sec.n~ t = 3000 sec.=~t = 7200 sec.

"t = 15000 sec.Eo 4

...en

z~ 8oo~<ta:::°12

16 . , , , I , ' , ,

o 10 20 30 40 50 60

DISTANCE FROM PUMPING WELL, em70

Figure 5 Drawdown V5. Distance from Pumping Well

0, ii'

TIME = 4000 sec.

TIM E r100 sec.

54 I I A! i "'"""0 200 s~c. I

2 1 I I I I I I I I I

...(J)

z3: 6 1000 sec.g I~ 2000 sec.a:: I~ I I I8 I I I I I .........o I

10 I I I I I I I I I I

0.15 0.20 0.25 0.30 0.35 0.40

DISCHARG E Q, cm3 jse.c.0.45 0.50

Figure 6 Drawdown vs. Discharge

700 .--------~-----_r_-----_,_-----..,

96 % AQUEOUS GLYCEROL

""CD 600 .....-------+---T-en.-oa.....cCDu.. 500 '-------4-----~-_+_-----__+_-----__t

>I-CJ)

ooen; 400 I---------+------+---"r-----t--------t

o-:tc:(z~ 300 ~-----+-------+------~-----__t

200 '-- ~ ....L.- ......L_ .....

15 20 25 30 35

TEMPERATURE, °C

Figure 7 Viscosity versus Temperature for Aqueous Glycerol

35 ...----------.oyo------.....--------------.

u-..J=»«0=C 151---------+-------f-------+-------I)-:x:

>= 25 t--------+-----.-f--I------+--------1

I-->-I-o:::>oz020r-------t----+---+-------+---------I(.)

.o~30~------+------+-----I----t------~

"Eo

C\J,o

Figure 8 Effect of Temperature on Hydraulic Conductivity

353025

TEMPERATURE. °C

2010 ~--____.....-_-------------..L.-------'

15

100,000

5,= 5.6 em

10,000

OBSERVATION WELL NO.2i (r2 = 21.02Cm)

REACHAPPROX.

OBSERVATION WELL NO.1I ( r1 =7.06 em)

AQUIFER NO. 1

6 I I I I , I I I II 'I' I , "'! I I I ! I I" I I!' I I I!! I I I I I I I I I'

1 10 100 1000

TI ME AFTER PUMPIN-G STARTED IN SECONDS

Figure 9 Thiem's Equilibrium Method

o , iii i" ,

5 I I I I'I.e: I

T =17.3 x 10-3 cm2 /sec. AT 23°C

U)

a:IJJ~ 2 I I I !\. ~I l§2=3.8 Cry'l_i i

:EI­ZLLIo~ :3 I I I '\. I ~ I I

Z3too3t~ 4/ I I I' I I I~ Q = O. 18 cm3 /sec. '1i\n

10 i , , ,

o

o

o

o0°0°

0°ooI n 0 Q MATCH POINT

oo

000000

00 0 0

s = 1.05 em

t = 100 sec.

W(u) = 1

u = 0.07

0 00 0

0°

0 0

AQUIFER NO.1

z3too~c(~

o

VJEIIJ.1­LLI~

I­ZWoZ

Q = 0.22 cm3/sec.

T = 16.6 X 10-3 em2/sec. AT 23°C

K = 16.6 X 10-2 ~cm/sec.

10,0000.1' I I I I I I I. I I "" , I , , 1 "" I I I I , ,

10 100 1000

TIME AFTER PUMPING STARTED IN SECONDSFigure 10 Theis· Nonequilibrium Method

AQUIFER NO. 1

alog tit' =

2 I I ,- I I

6 i , 0----,

z

0' ii'i" , , , , 'i.. "" ! , , , ,

1 -10 100 1000

TIME AFTER PUMPING STARTED/TIME AFTER PUMPING STOPPED IN SECONDS

Figure 11 Theis· Recovery Method

z 7 I I3= 31 I f:.S' = 2.25 emoo~c:(0:o..Jftl:)Q

enI&J0: 1 I , I I I

~ 51 Q = 0.22 cm3/sec. I I jlT ·l&JI-lJJ I T = 17.5 X 10-3 cm2/sec. AT 25 °c2~ I K = 17.5 X 10-2 em/sec. Iz 4 :J..] I

1&Jo

o

~ log t = 1

As =2.4 em·

21 '0::1 Q = 0.22 cm3 Jsec.

T = 16.7 X 10-3 cm2/sec. AT 23°C

K = 16. 7 X 10-2 em jsec.

6 I I I'I:c: I

AQUI FER NO. 1

I

Z3':oo~c::(

0::o

en«11.1I-IIJ 3 t-I----------------

:EI­ZI.1JU

2: 4 I I fA I I

10,0007 I I I I I I , I I I " I , I , I , I "I I I I I I I

10 100 1000

TIME AFTER PUMPING STARTED IN SECONDSFigure 12 Jacob's Modified Nonequilibrium Method