nondestructive, and rapid response measure- ment of the soil water suction and water con-. ⢠During 1966, Research Civil Engineer, U.S.. Water Conservation ...
VOL.
2, NO.
4
WATER
RESOURCES
RESEARCH
FOURTH
QUARTER
1966
An Instantaneous ProfileMethod[or Determining the HydraulicConductivity o[ Unsaturated Porous Materials K.
K.
WATSON
School of Civil Engineering The University of New South Wales, Kensington,N. S. W., Australia• Abstract. The hydraulic conductivity-water content relationship, as used in the differential equahon describing unsteady flow in unsaturated porous materials, is determined from measurements obtained during the drainage of an initially saturated sand column. The approach utilizes instantaneous profiles of the macroscopicflow velocity, the potential gradient, and the water content. The derived relationship exhibits a direct proportionaltry between the instantaneous velocity and the correspondingpotential gradient, thus lending strong support to the contention that Darcy's Law is applicable to flow in unsaturated porous materials of the size range considered in this study. (Key words: Permeability; porous mediums; soil moisture)
INTRODUCTION
It is now well known that two basic hydrologic characteristicsof a porous material must be definedexperimentallybeforeit is possibleto carry out numerical analyses of water movement in the unsaturated phase for that material.
These
characteristics
are the soil water
suetion-water content relation and the hydraulic conductivity-water content relation. In general, the first relationship presentsno great difficulty in its determination, at least when the water movement processis solely of a sorption or desorption nature. When the problem to be solved (e.g., the infiltration-redistribution sequence and its eyelie repetition) requires complete hysteresisdefinition, the necessaryprimary and secondary scanning curves can still be determined quite readily, but more sophisticated equipment is required. Watson [1965a] has discussedin detail equipment designedfor this purpose, in which the water content is determined from gamma-ray absorption measurements,and the soil water suctionis measured with a tensiometer-pressuretransducer system. This arrangement permits the simultaneous, nondestructive, and rapid response measurement of the soil water
suction and water
tent and provides a chart record of these changes against a time base. The use of measuring equipment of this type in conjunction
with a suitable soil column assembly,in which selectedreversals of flow can be programmed, enables the complete hysteresis relationship to be obtained. Although the response of the equipment is rapid, it is nevertheless necessary to make corrections for the time lag of the recorded outputs relative to the imposedinputs. Watson [1965b, 1965c] has considered the responsecharacteristicsof both the soil water suction and the water content meas-
uring systems.In the former referencea short, yet adequate,descriptionof the instrumentation is also included.
The experimental determination of the hydraulic conductivity-water content relation is more intractable. The difficult nature of its precise determination is borne out by the paucity of reliable data on this relation in the literature.
If, for the purposeof this study, attention is confined to methods applicable to laboratory columns,as distinct from outflow type experiments and field methods,we find the most prominent method
to be that
of Childs and Coilis-
George [1950]. This steady-state method not only provided the required experimental relationship but in addition gave proof of the applicability of Darcy's Law (keeping in mind the water content dependenceof the relationship) to steady-stateflow in unsaturated porous
con-
• During 1966, Research Civil Engineer, U.S. Water Conservation Laboratory, Soil and Water Conservation Research Division, Agricultural Research Service, United States Department of Agriculture, Phoenix, Arizona. 709
710
K.K.
WATSON
materials. Other column-typemethodsare those of Moore [1939] and Youngs [1960; 1964]. Since I)arcy's Law is an equation associated with steady-stateflow, it is consistentto find the proportionality in the Darcy expression (i.e., the hydraulic conductivity) by steadystate methods. However, once this has been determined,it has been usual to formulate the differential flow equation applicableto the analysis of unsaturated unsteady flow, and then to use in the analysisthe hydraulic conductivity relationship as determined by the steady-state methods. In such an approach there is inherent the reasonable assumption that the dynamic effects present in the unsteady flow are small and do not affect appreciably the 'steady-state'hydraulic conductivity;however, this position is, as yet, without direct experimental proof. In the absence of such proof, there is considerableattraction in the idea of determining an 'instantaneous' hydraulic conductivity in which the proportionality between the velocity and the potential gradient in an unsteady flow regime is considered at particular instants of time. This paper presentssuch an approach for a particular set of boundary conditions.
0.3500 ñ 0.0005 cc/cc and the bulk density 2.080 ñ 0.002 gm/cc. The temperaturein the laboratorywas controlledby a room conditioning unit that regulatedthe temperatureto 69øF _
1.5øF.
Since the flow condition was that of drainage of the saturated column to atmosphereat its
base,the base of the columnwas constructed so that air at atmosphericpressurewas always maintainedduring an experimenton the underside of the screen (100 mesh) supportingthe sand. The lowest elevation in the sand column
was thus a fixed piezometricsurfaceat atmospheric pressure;this surfacewas conveniently taken as the datum in gradient measurements. A steady-state condition was achieved at the start of the experimentby supplyingexcess
water (distilled and boiled) to the surfaceof the column such that a very small head of water (1 mm) was maintained there. This condition was continued for at least 10 minutes,
during which time the rate of volume outflow from the bottom of the column was measured.
Under these conditions the hydraulic gradient was known and, from the rate of volume out-
flow,the saturatedhydraulicconductivitycould be determined.In the above work, care was taken to ensure that the sand surface received
EXPERIMENTAL
The porousmaterial usedin the experimental work was a sand fraction (300•-150•) sieved from Botany sand. The sand grains in this fraction were characterized by a well-rounded shape. A saturated column, 57 cm in height, was prepared by depositingthis material under water in an acrylic column of cross section 15 cm by 10 cm and maintaining careful compaction control. The measuringequipment used, that discussedby Wats'on [1965a], made available a precise means of checking the column bulk density at each elevation by gamma-ray absorptionmeasurements.It was not possible to obtain the necessarysoil water suction and water content measurementsfrom a single column owing to the speedwith which the sand
drainedand, accordingly,replicatecolumnshad to be prepared.In forming thesecolumns,careful controlusingthe gamma-ray equipmentwas exercisedto obtain not only a uniform density throughout the column, but also a bulk density correspondingto that in the first column. The saturated water content of the column was
a minimum of disturbance. The steady-state saturated flow conditions just described also allowed the tensiometersto reach an equilibrium condition before drainage began. The drainage processcommencedthe moment the ponded water disappearedthrough the upper soil surface.
The experimentalinformation obtainedfrom the chart records of the soil water suction and
water content changeshas been summarizedin Figures i and 2. Figure i indicatesthe variation of soil water vations
suction with time at several ele-
in the column for the first 20 minutes
of drainage. In a similar manner, Figure 2 gives the water content changeswith time at the same column elevations.Not all the experimental
information
has been
summarized
in
these figures,sincesoil water suctionand water contentreadingswere alsotaken at intermediate elevations to those shown and at elevations of
less than 45 cm. In addition, the total period of readingwas in excessof 20 minutes,being usually I to 2 hours,with somecolumnsbeing permitted to drain to equilibrium. It was not
HydraulicConductivity
Z= 57,.., ........ u
711
•
40
• 3o 25
20
0
2
4
6
8
10
12
14
16
18
20
TIME (minutes)
Fig. 1. Variationof soilwatersuctionwith time at severalcolumnelevations.
possible to recordsoilwatersuctionandwater cordedoutput from the tensiometer-transducer contentchanges at the uppersurfaceof the soil system. In the caseof the water content meascolumn,but readingswere taken 2 mm below urements, the ratemeter output record rethe surface, and these were extrapolatedto
flectedsmall-periodfluctuationsdue to the ran-
givethe surface(z = 57 cm) valuesshownby
dom nature
of
radioactive disintegrations.
However,thesevariationsare inherentin radioFigure I do not representmeancurvesthrough isotopecountingand did not representirreguscatteredpoint data but are, rather, the direct larities of water content change; the mean representationof the extremely smooth re- curve through these statistical fluctuationswas, the dashed line. The smooth curves given in
0.40
0.35
030
025
0.20
0'15
0'10
0 05
_
I
0
2
I
4
6
I
8
10
I
12
14
I
16
18
20
TIME (minutes)
Fig. 2. Variation of moisturecontentwith time at severalcolumnelevations.
712
•.
•.
WATSON
in each case, a smooth curve, which yielded,
TOTAL 4
2
following appropriate calculations, the smooth
POTENTIAL 6
(cm) 8
10
12
14
continuousdata of Figure 2. Although it does not form a part of this study, it should be noted that a dynamic draining moisture characteristicis obtainable by plotting, for any elevation, the soil water suction at successivetimes against the water
contents at those same times. Whenthisproc-
ess was carried out for different elevations,
usingthedataof Figures I and2, theone
moisturecharacteristic wasdefined. Sincethe moisture characteristicis very sensitiveto den-
sityvariations, theabove resultindicated that
consistent
densities were achieved
within
indi-
vidual columns andbetween replicates. INSTANTANEOUS
PROFILE
ANALYSIS
In essence,the method of instantaneousprofiles consistsof determining down the column the profilesof the macroscopicflow velocity, the potential gradient, and the water content at any instant
of
time
after
the
commencement
of drainage. Once these are known for a particular time, it is then possible to find the instantaneoushydraulic conductivityfor each elevation by dividing the appropriate ve-
locity value by the potential gradient value. Since the water content profile is known at the same time, a seriesof points on the instantaneous hydraulic conductivity-water content relation
is available.
The equation of continuity for an unsaturated porous material in a one-dimensionalflow system may be stated as
Fig. 4. Instantaneous total potential profiles.
where v = flow velocity (cm/sec); w = volumetric water content (cc/cc); z = elevation above the datum plane definedas positive in the upward direction.
The curves presentedin Figure 2 give the vari-
(Ow/Ot)z=
ations
of water
content
with
time
for several
columnelevations.Using this information,it is VELOCITY (crn/sec) -0001
-0002
-0003
- 0 (:X:)4
-0.005
-000•o
i
a straightforward matter to find the relation
betweenOw/Ot and z at severalrequired times.
In this analysis,profilesat timesof 1, 3, 5, 10, and 20 minutes
have
been considered. Since
Ow/Ot : -- (Ov/Oz), the velocity profilesare obtainedby integratinggraphicallywith respect to z the Ow/Otprofile curves.This integration
has been carriedout for the abovetimes,and the resultingvelocity profilesare given in Figure 3. These profilesrepresentthe instantaneous velocities down the column at the times stated.
Fig. 3. Instantaneous velocity profiles.
For consistencywith the sign convention adopted,thesevelocitiesare negative.It should be noted that the velocityis zero at the surface
713
Hydraulic Conductivity POTENTIAL
0'1
02
GRADIENT
0'3
04 ,
0-5 ,
0 G
07
,
has been tabulated in Table 1, together with data definingthe dynamicmoisturecharacteristic for the draining condition. DISCUSSIOl•
The differentialequationrepresenting the flow of water in unsaturatedporousmaterialsmay be written
=
(2)
As mentioned in the Introduction, the water
Fig. 5. Instantaneous potentialgradientprofiles. contentdependenthydraulicconductivityK in this expression is usuallydeterminedby steadystate methods.However, to be fully precise, and rises to a maximum value at the drainage the K relationshipshouldbe determinedunder
front, whichis definedas the planeseparating the conditionsto which the equationis relevant, the partly drainedupperprofileand the stillnamely,transientconditions,as in this study.
saturatedlower profile.The velocityis constant
throughoutthe still-saturatedzoneof the column.
The first part of the next step is the determination of the soil water suctionprofile val-
Accordingly,the instantaneoushydraulic conductivity-water content relation, as given in Figure 7, representsexperimentaldata for unsteadyflow analysisin whichno assumptions regardingthe negligibilityof dynamiceffectsare
uesat 1, 3, 5, 10, and 20 minutesfrom Figure involved. 1. Sincethe total potential• is equal to the Althoughit is not evident from Figure 7, negativesuctioncomponent • and the gravita- it can be shown that the superposedpoints on tional componentz, the total potentialprofiles the curve not only occurat the differenttimes at the abovetimesmay be readily plotted and as stated, but, most importantly, at widely are presentedin Figure 4. These curves may differentgradientvalues.This fact makesthe then be differentiatedgraphically to give the conclusionsdeducible from the analysis of funpositivepotentialgradient(0•/0z) profilesas damentalsignificance, for it indicatesthat, for shown in Figure 5. any given water content, the instantaneous From Figures3 and 5, it is a relativelysimple hydraulicconductivityhas a uniquevalueindematter to determine the instantaneoushydraupendentof the gradient.This relationshipcan lic conductivityfor any elevationand time by be stated alternatively as follows: the relation dividing the velocity value at that point in between velocity and gradient at any water space-time by the corresponding potentialgradcontentunder unsteadyflow conditionsis linear, ient value, due note being taken of the signs. The final requirementis the water contentprofiles at the selected times. These profiles are read direcfiy from Figure 2 and are presented
in Figure 6 for times of 1, 3, 5, 10, and 20 minutes and for equilibrium.The last step in the methodis the plotting of the curve relating the values of the instantaneous hydraulic con-
ductivity and the water content.This relationshipis givenin Figure7, wherethe conductivity scaleis logarithmicto enablethe lower conductivities to be presentedaccurately.It shouldbe noted that, in Figure 7, the points plotted for the different elapsedtimes form a single relationship with only small experimental departures occurring.The relation given in Figure 7
WATERCONTENT( cc,,/cc) 0
57 i55
0 05
010
' '.'.
0'15
0'20
0 25
' _ . _' .
0.30
0'35
-Iro•n •
• 50
35 I I I I I I I Fig. 6. Instantaneouswater content profiles.
714
x.x.
WATSON
0'40
•.030
0'20
0.10,=,-..-
(•0001
00003 0.00060001
0003
INSTANTANEOUS HYDRAULIC
•
5rain
• ©
10 rain 20 rnin
OOOCo 001
003
CONDUCTIVITY (crn/sec.)
Fig. 7. The water content-instantaneous hydraulic conductivityrelation showingthe computed values.
the slopebeing the instantaneoushydraulic conductivity. This latter sentence,when related to steady-state flow, represents the formal statement necessaryto show the validity of Darcy's Law for flow in unsaturated porous materials. However, cautionmust be exercisedin drawing
expressionis valid, giving a direct proportionality betweenany instantaneous velocity and the corresponding potential gradient. Realizing the small order of magnitude of possibledynamic effects,and noting that it is somewhat difficult to imagine the physical factors that
too hurried
would causethe changefrom unsteady flow to steadyflow to result in a lossof linearity in the velocity-gradientrelationship,it would appear safe to infer that the work in this paper lends strong support to the contentionthat Darcy's Law is applicableto unsaturatedflow in porous materials of the size range considered. It shouldbe noted that the instantaneousprofile method provides K values over a wide range. For example, at 20 minutes the values derived vary from 0.00015 to 0.01860 cm/sec.
a conclusion from
the results of
this study. What can be said quite definitelyis that an unsteadyflow equivalent of the Darcy TABLE 1. Hydrologic Characteristicsof Sand Fraction Obtained during ]Primary Draining Cycle
Soil Water Suction,
Water Content,
Hydraulic Conductivity,
cm of water
cc/cc
cm/sec
This 0 39 40 40 41 41 42 42 43 43 44 44 45 46 47 48 49 50
00 80* 00 5O 00 5O 00 .50 ß00 .50 .00 5O 00 00 00 00 00 ß00
0.3500 0.3500 0.3410 0.3182 0.2955 0.2727 0.2500 0.2295 0.2215 0.1905 0.1740 0.1595 0.1450 0.1225 0.1045 0.0930 0.0845 0.0775
0 01860 0 01860 0 01730 0 01410 0 01100 0 00860 0 00640 0 00445 0 00390 0.00205 0 00150 0 OO1OO 0 00066 0 00033 0 00016 0 00009 0.00006 0.00005
* The soil wa•er suction of 39.80 cm represents the air-entry value of the sand fraction. No change of water content occurred until this value of suction was reached.
fact
obviates
one of the difficulties
in
steady-state measurements of hydraulic conductivity, where a small constant flow to the surface for the lower conductivity values is difficult
to maintain.
It will be realized that the
time required for the compilation of sufficient data for the calculation
of the w-K
curve is a
function of the pore size of the porousmaterial and will increase with decreasing pore size. A convenientcheck on the computedvelocity profilesis made possibleby comparingthe computed velocity in the still saturated zone at the selected profile times with the outflow rate per unit
area at the same times as measured
from
the volume outflow at the base of the column.
Acknowledgmentß The researchwork described in this paper was made possibleby researchfunds provided by the Water Research Foundation of Australia, Ltd.
Hydraulic Conductivity REFERENCES
715
Watson,I•. I•., Correctionof ratemeteroutputin water content measurements using gamma-ray
Childs, E. C., and N. Coilis-George,The permeability of porousmaterials, Proc. Roy. Soc. (London), 201 A, 392-405, 1950.
Moore, R. E., Water conductionfrom shallow water tables, Hilgardia, 12, 383-401, 1939. Watson, K. K., The measurement of non-con-
tinuous unsteady flow in long soil columns, Moisture Equilibria and Moisture Changes in Soils beneath Covered Areas, pp. 70-77, Butterworth, Sydney, 1965a.
Watson, K. K., Some operating characteristicsof a rapid responsetensiometersystem,Water Resources Res., 1, 577-586, 1965b.
absorption,Australian Road Res., 2, 34-42, 1965c.
Youngs,E. G., The hysteresiseffect in soil moisturestudies,Trans.Intern. Congr.Soil Sci., 7th Congress,Madison, 1, 107-113, 1960.
Youngs,E.G., An infiltration methodof measur-
ing the hydraulicconductivityof unsaturated porousmaterials,Soil Sci.,97, 307-311,1964.
(Manuscript receivedFebruary 24, 1966.)
