DARCY'S LAW DURING UNSTEADY FLOW

Louis M. LAUSHEY(*) and Laxmidas V. POPAT (Junior Author) (**)

ABSTRACT

Darcy's Law is shown to be incomplete for laminar unsteady flow through porousmedia. Experiments and theory indicate that the total derivative of the water tableelevation, with both distance and time, must be used for the gradient of the static head.

Corrections to Darcy's Law are required when dldX(dhiet) and d/dt (dh/dt) arenot zero. Darcy's Law is correct for the quasi steady state of dhldt not zero if the twopreviously stated derivatives are zero.

Experiments seem to confirm that

dX dX dt. dX

dh dh

ÔX dt

d_fdhdX\dt

d_(dhdt\dt

RÉSUMÉ

L'équation de Darcy est incomplète pour l'écoulement laminaire non permanentà travers des matériaux poreux.

Les expériences et la théorie indiquent que pour obtenir l'inclinaison de la charged'eau, il faut employer la dérivée complète de l'élévation de la nappe aquifère avec letemps et la distance.

Il faut corriger la loi de Darcy avec d/dX(ôh/ôt) et ô/dt {dh/dt) différents de zéro.La loi de Darcy est correcte si dhjôt n'est pas nulle, mais les deux précédentes dérivées

sont égales à zéro.D'après les expériiences il vient que

dX

dh_ dh ét_

dX at dX= - K

dh_ dh_

ÔX dt

fex\ôtô (dh

~ôt\dt

INTRODUCTION

Darcy's Law specifies the proportionality of the velocity to the piezometric slopefor steady laminar flow through porous media. Many experimenters have verifiedDarcy's Law for many ground water flow phenomena.

The proportionality of the instantaneous velocity to the instantaneous piezometricslope is often assumed to be valid during unsteady flow too. That Darcy's Law can beextended to unsteady flow apparently has never been verified. It will be shown that thevelocity and piezometric slope are not always proportional during unsteady flow.

This paper is concerned only with flow phenomena for which Darcy's Law wasfound to be correct for steady flow. Nevertheless, it is relevant to review the researchwhich has disclosed some factors which deny Darcy's Law for even steady flow. Theexperiments to be reported purposely avoided all of the following situations of steadyflow, non-Darcy behavior reported by previous researchers.

(•) Head, Department of Civil Engineering University of Cincinnati, Cincinnati,Ohio, U.S.A.(*•) Graduate Student University of Cincinnati, Cincinnati, Ohio, U.S.A.

284

Previous Research

Jacob (M has stated that the departure from laminar (Darcy) flow begins at values ofReynolds Number between 1 and 10. Todd (2) presented the work of Rose (3) and otherinvestigators to confirm the applicability of Darcy's Law for almost all ground watermotion, which he states occurs at Reynolds Number less than unity except in uncon-solidated aquifers or in those containing large diameter solution openings.

Deviations from Darcy's Law have been observed by Swartzendruber (4) at verylow Raynolds Numbers, much less than the critical, when the porous media consistedof or contained clay. He suggested the possible mechanisms giving rise to the non-Darcybehavior, including the non-Newtonian liquid effect caused by clay-water interaction.

Several investigators have reported on the time variation of the pressure and velocityat a point resulting from time or space-dependent properties of the fluid. Yih (5> hasshown how to determine the velocity and pressure fields for steady flow of a fluid ofconstant density, but a space-dependent viscosity. De Josselin de Jong (6) showed howto determine the instantaneous velocity and pressure fields for transient flow of anincompressible fluid which varies in both density and viscosity. Knudsen (7) developedthe equations and boundary conditions which govern the instantaneous pressure fieldin which the fluid is continuously varying in density.

Darcy's Law for steady flow requires that the inertia terms (the kinetic energy of theflow) be considered negligible. Scheidegger (8) has written that the inertia terms, evenunder laminar flow conditions, can cause a deviation from Darcy's Law.

Previous investigators, however, have not apparently questioned Darcy's Law forunsteady flow when the inertia terms are negligible, and when the fluid and soil prop-erties are constant with time and space.

Experiments on unsteady flow have been conducted at the University of Cincinnatisince 1965. The first experiments were made in a sand model 20 feet long, 2 feet wide,and 4 feet deep. These tests represented an unconfined aquifer. Initial results demon-strated that the slope of the water table, as represented by the piezometric readings inopen manometers, was not proportional to the velocity at the cross-section where theslope was measured. Exact quantitative results were inconclusive because of the influ-ence of the capillary water and the variable amounts of water yielded or released in theunsteady zone. These studies have been reported by Tsai (9). The tests in a model of aconfined aquifer were undertaken to avoid the difficulties of capillarity and the variablepercentage of water stored or released instantly in the unsteady zone.

Objectives

The aims of this study were:

1. To investigate Darcy's Law for unsteady laminar flow, and if invalid,

2. To propose the correct relationship that defines the instantaneous velocity as afunction of the derivatives of the piezometric surface with both distance and time.

Significance of the Research

Almost all analyses of flow through porous media are based on the continuityequation. In its simplest form, it is:

dq Ôh A— + n— = 0dX at

where q = Vh

285

then,.dh , ôV oh n

V — + h hn-=OdX ÔX dt

At this point, the Darcy substitution is usually made, as follows:

dX

— = -K —dX dX2

After substituting and performing several algebraic steps, one would get the frequentlyused equation for unsteady flow:

d2h2 _2ndhex2 ~ K dt

Obviously this fundamental equation would be incorrect if the Darcy substitutionabove should not be proper for unsteady flow. Other similar forms'of this equation, orlinearized simplifications, would also be incorrect for unsteady flow.

EXPERIMENTAL APPARATUS

The experiments were made in a closed plastic tube filled with sand to simulate aconfined aquifer. Figure 1 is a sketch of the apparatus.

The inside diameter of the circular tube was 4.75 inches, and the length was 44 inches.The tube was filled with a compacted uniform Ottawa-type sand with rfio = 0.011",6̂0 = 0.0175", and a uniformity coefficient Cu = 1.59. The sand was compacted

carefully to insure uniformity. The sand was bonded to the wall of the tube to preventchanneling along the wall. Entry and exit from the sand column was through a cushion-ing reservoir and screen at each end of the tube.

The sand was always maintained full of water and under pressure by head and tailreservoirs in which the water levels were never allowed to drop below the top of thesand. The piezometric head in the sand and the levels in the reservoirs were measuredby manometers. Calculations showed that the effect of inertia and viscosity (friction) inthe manometers during unsteady flow was so small that the instantenaous heights of thewater columns measured the instantaneous piezometric heads with an error of less thanone percent. All manometer readings were recorded photographically, and read byestimation from the photographs to the nearest 0.01 inch.

The flow rates during steady runs were obtained by timing the volume of water thathad overflowed from the tail reservoir. The flow rates during unsteady runs weremeasured by observing the rates of depletion or accumulation in the head and tailreservoirs. The cross sectional areas were: head reservoir, 2.40 square inches; tailreservoir, 4.00 square inches. During unsteady flow, the supply and waste valves wereclosed while the levels in the reservoirs were approaching each other. The instantaneousdischarge rates were obtained from the slopes of plots of the volumes in the reservoirsagainst time.

286

Every effort was made to purge the sand of entrapped air and fully saturate all of thepores. Before any experiments were conducted, water was passed through the sand for along period of time to remove the air through the exit valve and through the discon-nected manometer taps. During the experiments, the temperature was always main-tained constant. Thermometers were used to measure the temperature continuously ateach end of the sand tube.

Fig. 1 — ApparatusCIC2: Cushioning Sections; SiS»: Screens; MiM> ... M;' Pin Holes for Manometers;

V: Control Valves.

Prior to each experiment, sodium sulphite was added to the water and passedthrough the tube for a considerable length of time to absorb free oxygen. Only whenmany steady state experiments, at different discharges, gave consistent and reliablevalues for the permeability were the unsteady tests initiated. Diligent attention wasalways given to the need to keep the sand full of water at all times, and at constanttemperature, and free of air bubbles.

Experimental procedure

Steady Flow Tests

Many steady flow tests were made to determine the coefficient of permeability to ahigh degree of accuracy. These runs were performed over a wide range of flow rates toobtain a confirmation of Darcy's Law during steady flow. Darcy's Law was found to bevalid during steady flow: the permeability was not a variable, the piezometric gradientwas proportional to the flow rate, and the piezometric slope was essentially constantalong the tube during each test. The maximum flow rate tested was at least as large asthat measured during the unsteady tests performed later. In none of the tests was theReynolds Number as great as unity.

287

Careful analysis of the data disclosed that there was a small variation in permeabil-lity along the length of the tube. After attempts to make the total sand column abso-lutely uniform in porosity, the permeability was calculated separately for each of thefour 10-inch long sections between the 5 manometers.

Unsteady Flow Tests

For verification of Darcy's Law during unsteady laminar flow, steady state runswere made immediately before and after the unsteady phase. The purpose was toinsure that the temperature, viscosity, air content, compressibility, and porosity wereindentical during both the steady and the unsteady portions of the experiment.

Unsteady flow was accomplished by starting with unequal water levels in the headand tail reservoirs. The instantaneous velocities through the sand tube were calculatedfrom the changing instantaneous water levels in the two reservoirs. The instantaneousreservoir levels, and the instantaneous piezometric heads in the manometers along thesand tube, were photographed at regular time intervals varying from 15 seconds to1 minute.

To observe any non-Darcian effects during unsteady flow, tests were made withboth falling and rising piezometric elevations. The testing procedure and observationsof two typical tests are described below.

Run # 4 — Falling Head

The permeability was first determined over a wide range of steady flow rates. Thepermeability of each 10-inch section was calculated by dividing the velocity (measureddischarge rate divided by gross area of the tube) by the piezometric slope along thesection. The outlet and inlet were then closed. After all manometers had come to restat a constant height, the outlet valve was opened to effect a draining situation. Thevelocity along the sand tube at any time was determined from the fall velocity of thewater in the head reservoir. The fall velocity in the head tank was determined from theslopes of the plotted height-time curve, the instantaneous heights being obtained veryaccurately from photographs taken approximately every 15 seconds. A clock with asweep-second hand was included in the photographs to give an accurate time that waskeyed to the manometer readings.

The bulk velocity in the sand tube was calculated using the continuity requirement:the product of the known area and measured velocity in the head tank had to equal theproduct of the known area and unknown velocity in the sand tube. Continuity alsorequired, of course, that at any instant, there had to be the same average velocity ateach cross-section along the length of the sand tube.

Run # 7 — Rising Head

Rising head tests were made as continuations of falling head tests. After the systemwas stabilized at room temperature, a falling head test was made as just described.After about 8 minutes, the outlet valve was closed, and the inlet valve was opened toadmit more water to the system. The water levels then rose in both reservoirs and in allof the manometers.

Calculations of the instantaneous velocities, and the recording of the piezometriclevels were made as described previously. However, the flow rate and velocity had to becalculated from the rate of rise in the downstream reservoir. When the downstreamlevel reached the overflow port, steady flow soon resulted. A full complement of read-ings were then made, to check Darcy's Law and the previously determined steady flowpermeability.

288

EXPÉRIMENTAL RESULTS

Figure 2 is a plot of the data observed during a typical falling head experiment(Run #4). This plot shows that Darcy's Law was not applicable to this unsteadylaminar flow. It shows that the instantaneous velocity could not be proportional to theinstantaneous piezometric gradient. For the Darcy proportionality to exist, each plot

10 20

Distance X (inches)

Fig. 2 — Falling Head Test.

of depth (at each time) should have been perfectly straight over the length of the tube.The observed curvature would erroneously suggest that the velocity at any time wasnot constant along the length of the tube—an impossibility if continuity is to be satisfied.Further evidence of the lack of proportionality of the velocity to the gradient of thepiezometric head derives from the magnitude of the slopes. At the beginning of theunsteadiness (/ = 0.25 min. to t = 2.5 min.,), the gradients increased with time, whenin fact the discharge was actually measured to be decreasing with time.

Figure 3 shows the relative error that would result from calculating the velocity bythe Darcy Law. The data are from the same experiment shown by figure 2. The relativeerror [( V- Vs)/Vs] decreased with dhjdt as might be expected. However, at later time

289

periods when dh/ôt became practically constant with time, the relative error decreasedto zero. The non-zero intercept (when t is very large and dh/dt is constant with time)indicates that dh/di alone is not enough to invalidate Darcy's Law. What is necessary isthe presence of second derivatives: djdt(dhldX) and dldtidhldt). Figures 4 and 5 showclearly that only when these two derivatives are zero will the Darcy Equation becorrect for unsteady flow.

c 5

£ 4oc

0 Sect 1& S t c i lQ Sect 3

>

o.z 0.4 o.c as 1.0

Fig. 3 — Falling Head Test.

Figure 6 is a plot of the data obtained during a typical rising head test (Run #7B).This plot also indicates that during unsteady flow the instantaneous velocity was notproportional to the instantaneous slope of the piezometric surface. As in the fallinghead test, the curvature and changing ch/cX would suggest an untenable non-zerodVjdXby Darcy's Equation.

Figure 7 is a plot of the data obtained from another falling head test (Run #7A).It is included only to show that although there is a considerable dh/dt, the plots are allstraight lines because d/dt(dhldt) and ôlôX(dhjôt) are practically zero.

Many similar observations in other unsteady experiments indicated that dh/dt doescontribute to the non-Darcian effects, but the rise or fall of the piezometric head is notsufficient by itself to invalidate Darcy's Law. Further verification was provided by datafrom a test in which ôh/ôl increased for a while to a maximum and then decreased asequilibrium was approached. Interestingly, at the maximum rate of rise (maximum

290

Sh/dt), the relative error (V- V,)/Vs was almost zero, since both dldt(dh/dX) andd/dt(dhldt) had become zero.

0.5

.£ 0.S

0,1

o Sect. 1a Sect zQ Sect. S

/

0.Z 0.4 0.G

V-Vs

10

Fig. 4 — Falling Head Test.

The explanation for the above phenomenon is that a quasi steady state of flowexists when dh/dt is not zero, but the two second derivatives are equal to zero. Frommomentum considerations, the difference of the hydrostatic forces on each end of acontrol volume remains constant with time just as it would in steady flow.

THEORY FOR UNSTEADY FLOW

The experimental results can be explained by a theory that will be derived fromenergy principles. The use of energy concepts should be appropriate because theenergy gradient is so closely associated with the flow rate. For example, the flow rate isdirectly proportional to the energy gradient in Darcy steady flow. Figure 8 shows acontrol volume in a confined aquifer and the notation for the derivation.

The energy per second entering the control volume is (yQH)The energy per second leaving the control volume is (yQH) + d/dX(yQH)dXThe rate of change of energy within the control volume is djdt[yAvAXH]The rate of friction loss in heat production is (yQS/iiX)

291

Conservation of energy requires that the difference between the rates of inflow andoutflow of energy (including heat production) must equal the rate of accumulation ofenergy within the control volume. The energy balance equation is:

— (yQH) dX dr + - (yAv dX H ) dt + yQSf dX dt = 0dX dt

dX dt

ndH tIdQ ôAvl  dH nc,Q— + H \ — + \ + Av— + QS f = Q

dX IÔX dt J Ôt f

50

"s40

30

2 0

10

o Seci lA Stctz0 Sect. 3

e

B A,

1

' A

/

1/

0 0.2 0.4

Ratio

0.G 1.0

Fig. 5 — Falling Head Test.

For a confined aquifer, both ôQ/dX and dAv/dt must be zero. Fortunately, however,their sum must always be zero, whether the aquifer is confined or unconfined. The sumis, of course, the continuity equation, equal to zero when there is no infiltration or leak-

292

age, and where (dQldX)dX is the difference of the rates of inflow and outflow of fluidvolume, (dAojdt)AX[% the rate of accumulation of volume of fluid in the pores of thecontrol volume.

Therefore, for any aquifer, either confined or unconfined, with or without a con-stant cross-section along its length, the very general equation is:

+ ASÔX dt

Dividing by Q, and recognizing that Av/Q = nA\VA = n\V

8H ndH1 — — i f

dX v dt '

-aö

70

i

50<

30

10 SecUo

~——<

Sec

Time (̂ wiin*

" ^ - ^

ion 2

ites)

Sad ion 310 20 30

Distance X finches)

Fig. 6 — Rising Head Test.

We can make the usual assumption that the instantaneous heat produced in frictionduring unsteady flow at any given velocity is equal to that which would occur at thesame velocity during stady flow. And since the flow is laminar, we can be sure that the

293

70

Distance X (jnches)

Fig. 7 — Falling Head Test.

ic Htai

Fig. 8 — Control volume — Energy Balance.

294

rate of heat production is proportional to the instantaneous velocity. It follows that5/ is proportional to V, and then V = - KS/ where K is the permeability and theconstant of proportionality.

Previous investigators have derived equations for the coefficient of permeability inthe form:

DeWiest (9) points out that the intrinsic permeability cd2 is characteristic of the soilmedium alone, and yl/i is dependent only on the fluid. It is reasonable than that thepermeability should not depend on the velocity, nor on the fact that the velocity mightbe changing with time at a point. For these reasons, the steady state permeabilityshould be used in any unsteady flow equations. Any concept that the inherent permea-bility varies because of the unsteadiness of the flow rate is incongruous.

The equation then becomes dH/dX+ nlV{dHjdt) = - Sf = -VjKand

+ \ (i)ex vdt]

If one wishes to neglect the velocity head, and it certainly is negligible in ground waterflow, then H = h.

Equation 1 suggests that an unsteady instantaneous velocity is proportional to thetotal derivative of the piezometric head, and that a steady velocity is proportional tothe partial derivative (with distance only) of the piezometric head.

Equation 1 is expressible as a total derivative

4dX LdX dt d

where dt/dx = n]V, and of course r>IV is the reciprocal of the velocity through thepores.

When the flow is steady, oH/dt = 0, and equation 1 reduces to the Darcy Equationfor steady flow.*The error in using Darcy's Law for unsteady flow then is:

dXJ dt

AGREEMENT OF THEORY WITH EXPERIMENTS

The relative error in applying Darcy's Law can be identified easily by the ratio V\Vt,where V is the measured instantaneous unsteady velocity and Vs is the correspondingDarcy velocity calculated from the instantaneous piezometric slope and the steadypermeability. It was observed that at instants of maximum unsteadiness this ratiodeviated considerably from unity. This ratio, V\VS, was greater than unity in fallinghead experiments, and less than unity in rising head experiments. For all tests, thevelocities were in the laminar range, and the Reynolds Numbers were far less than one.Table I presents sample values from several of the experiments.

295

Run

4FallingHead

IBRisingHead

1AFallingHead

6BRisingHead

Sec.

1

1

3

2

Timemin.

0.250.500.751.01.52.02.50

0.500.751.00

678

0.250.500.75456

TABLE I

Effect of Unsteadiness on Darcy'

Vmeasured

in/min.

0.420.370.340.3160.280.2640.250

0.650.670.66

0.4170.3840.360

0.250.330.4150.5130.4750.438

ys

K(dh/dX)in/min.

0.182 2.320.2160.2360.2420.2480.2500.250

.75

.44

.29

.12

.05

.00

0.945 0.6650.775 0.850.77 0.85

0.4070.3490.262

0.326 C

.02

.13.37

1.7660.396 0.830.474 . 0.8750.55 0.930.532 0.8950.511 0.855

j Law

NRReynolds No.yd/.u

.049

.043

.040

.0368

.0326

.0306

.029

.081

.083

.082

.054

.050.047

.028

.037

.0465

.0575

.053

.049

yd/fin

0.1470.1300.1200.1100.0980.0910.087

0.240.250.246

0.160.150.14

0.084.11.14.17.16.147

Remarks

Temp. 20 °Cdio = .011"Porosityn = 33%

Temp. 22.8 °C

Temp. 22.8 °C

Temp. 17.8 °C

Proposed Unsteady Equation

The term d//dA" in Equation 1 was impossible to determine analytically. The deter-mination of the velocity through the pores is exceedingly complex from any theoreticalapproach. Although values of dr/d-Y were calculated by Equation 1 from the observeddata, the following relationship was found to be more satisfactory.

d fdhAt_ dX\dtdX JL(d-h

ct\dt

296

By cross multiplying, the interpretation is

- — dt = — - — cLYdt\dtj dX\dtJ

The left side is the change in dh/dt over a time increment; the right side is the samechange in dh/dt over an increment of distance. Then dXfdt might be interpreted as thewave velocity ofdh/dr moving over a distance AX during a time Jr.

Alternatively, the following justification is proposed: writing that

dX\dt) dX\ôt) dt\ ôl )dX

Ifdhßt is constant along the path of a particle; that is, d/dX(dh/dt) = 0

dt_dX

(

dX\dt

ct\dt

The interpretation of dXjdt then is that of a superimposed reverse velocity of such amagnitude that moving along the path of a particle the local dhjdt appears to be con-stant.

The proposed unsteady flow equation then is:

V = - Koh dhdX ct

d_/dhdX\dt

6 foh~Jt\dt

(2)

Summary-Table II

The agreement of the experimental data with equation 2 is summarized in table II,where only several of the experimental runs are presented for brevity since all otherdata led to the same conclusions.

Column 11 shows that the ratio V\ Vs approaches unity as the unsteadiness decreasesand equilibrium is neared. The maximum value of the ratio was 3.58 during rapidfalling heads. The minimum value was 0.61, recorded during rising heads.

Column 12 shows the ratio of V/V«, the measured velocity divided by the velocitycalculated from equation 2. This ratio was always near unity, thus suggesting that it ismore correct than V[ Vt during unsteady flow.

CONCLUSIONS

Darcy's Law is shown to be incomplete for laminar unsteady flow through porousmedia. Corrections to Darcy's Law are required when d/dX(dfildt) and d/dl(dh/dt) arenot zero.

297

oo TABLE IISummary of Typical Data

1Time(min.)

0.250.500.751.001.251.502.002.50

0.250.500.751.001.251.501.752.002.5

2V

(in/min.)

0.4520.3820.3460.3200.3000.2900.2700.260

0.490.650.670.660.6550.6500.6450.6410.632

3dh/oX

-0.187-0.287-0.330-0.345-0.356-0.360-0.368-0.367

-1.288-1.54-1.28-1.26-1.23-1.20-1.17-1.142-1.10

4dh/dt

(in/min.)

Run-15.4- 6.5- 4.20- 3.10- 2.60- 2.20- 1.85- 1.80

Run42.5151.007.704.502.601.400.900.550.50

5d/d X (ch/dt)

# 4 Falling0.700.400.120.080.050.030.0120.004

# 7B Rising———

-0.120-0.140-0.115-0.107-0.100-0.08

6d/dt (.dh/dt)

-lead at Section—12.03.522.561.801.20.40.25

Head at Sectior———

- 9.00- 5.20— 3.10- 1.40- 0.80- 0.40

7

(in/min.)

2 Steady0.1250.1920.2200.2310.2380.2420.2460.250

8 9dX/dl

K = 0.67 in/m—

0.990.570.3840.2740.1980.1060.074

i 1 Steady K = 0.6150.7900.9450.7750.770.7520.7350.7170.700.670

0.380.3120.1350.1300.1330.1160.1020.0840.0715

302932.036.040.033.262.5

in/m—

—7537.227.013.208.005.00

10

v2(in/min.)

—0.3390.3180.300.2850.2750.2790.26

11VIVs 1

S.58.75.57.38.27.19

12

nvt

—.17.09.07.05.05

.10 0.97

.07

— 0.62— 0.675— 0.85

.00

——

0.698 0.85 0.950.681 0.87 0.970.680 0.885 0.960.66 0.900.634 0.915 10.608 0.94 1

.00

.01

.04

ACKNOWLEDGEMENTS

This paper is based on the Junior Author's M.S. Thesis, submitted in June, 1967 tothe Graduate School of the University of Cincinnati. The considerable help of Mr.Y.J. Tsai is acknowledged, and reference is made to his M.S. Thesis of 1966 at theUniversity of Cincinnati, "Laboratory Studies of Unsteady Ground Water Flow".

NOTATION

Q (or q) Flow rate;nVVsKhHdiodsoCu

yAvAd

NB

Pt

Porosity;Velocity of fluid;Darcy's steady velocity;Coefficient of permeability;Piezometric height above datum;Total head above datum;Effective grain diameter, or size smaller than 10 percent in weightGrain size smaller than 60 percent in weight;Uniformity coefficient;Specific weight of water;Area of woids in soil cross section;Total area of soil cross section;Average grain diameter;Viscosity of fluid;Reynolds Number;Fluid density;Time.

BIBLIOGRAPHY

0) JACOB, C. E., Flow of Ground Water, Engineering Hydraulics, (H. Rouse, editor),John Wiley, New York, 1950.

(2) TODD, David K., Ground Water Hydrology, John Wiley, New York, 1959.(3) ROSE, H.E., An Investigation of the Laws of Flow of Fluids through Beds of

Granular Materials, Proc. Inst. Mech. Engrs., Vol. 153, 1945.C) SWARTZENDRUBER, Dale, Non-Darcy Flow Behavior in Liquid-Saturated Porous

Media, Journal of Geophysical Research, Vol. 67, No. 13, December, 1962.(5) YIH, C , Flow of a Non-Homogeneous Fluid in a Porous Medium, J. Fluid Mech.,

Vol. 10, 1961.(6) DE JOSSEUN DE JONG, G., Singularity Distributions for the Analysis of Multiple-

Fluid Flow through Porous Media, J. Geophysical Research, Vol. 65, 1960.(7) KNUDSEN, W.C., Equations on Fluid Flow through Porous Media — Incompres-

sible Fluid of Varying Density, / . Geophysical Research, Vol. 65, 1960.(8) SCHEIDEGGER, A.E., The Physics of Flow Through Porous Media, Macmillan,

New York, 1960.(9) TSAI, Y.J., Laboratory Studies of Unsteady Ground Water Flow, Master of

Science Thesis, Dept. of Civil Engineering, Univ. of Cincinnati, 1966.(10) DEWIEST, Roger J.M., Geohydrology, pp. 169-170, John Wiley, 1965.

299