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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