Wetting Moisture Characteristic Curves Derived from Constant-rateInfiltration into Thin Soil Samples1

K. M. PERROUX, P. A. C. RAATS, AND D. E. SMILES2

ABSTRACTAlthough procedures are available for obtaining the wetting mois-

ture characteristics of soils, they often involve lengthy equilibrationperiods. A simple and quick laboratory method is described for ob-taining the wetting moisture characteristic using constant flux infil-tration into a thin section of soil. A theoretical analysis of the processis tested by comparing predicted and observed times for pressureheads to develop in a fine sand sample. The technique is demonstratedusing a fine sand, a loam, and a silty clay loam, and comparison ismade with data obtained by other methods.

Additional Index Words: uniform rate of wetting, linearization ofDarcy's law.

Perroux, K. M., P. A. C. Raats, and D. E. Smiles. 1982. Wettingmoisture characteristic curves derived from constant-rate infiltra-tion into thin soil samples. Soil Sci. Soc. Am. J. 46:231-234.

THE relationship between tensiometer pressure hand volumetric water content 0 is a key element

in studies of equilibrium and transport of water inunsaturated soils. Buckingham (1907) determined thewetting curve by maintaining a water table at the baseof a long, vertical soil column, and, after equilibriumwas believed to have been reached, sampling the watercontent profile. Buckingham's method is restricted tolong, homogeneous soil columns. In an extension ofBuckingham's theory, Haines (1930) determined 6(h)by exposing the base of relatively thin samples of soilto water at a sequence of increasing pressures andmeasuring the initial water content, the uptake ofwater for each increment of pressure, and the watercontent at saturation.

This paper describes a rapid and precise method forobtaining a wetting curve by supplying a small con-stant flux of water to the top of a thin section of soilfully occupying a cell of known volume, and mea-suring the pressure head at the bottom with a pressuretransducer. The method depends upon the notion thatthe increase in moisture content of the soil can beinferred from the influx, provided no water is lost fromthe sample. If the sample is thin and the rate of ap-plication of water is low enough, the pressure headdifference across the sample will tend toward zero.This condition implies that the measured pressurehead and a mean water content are applicable to thewhole sample.

THEORYOne-dimensional, vertical transport of water in unsatu-

rated soils is governed by the balance of mass equation:ae/dt = - [i]

1 Contribution from CSIRO, Australia. Received 4 Feb. 1981.Approved 9 Oct. 1981.

^ Experimental Officer, Visiting Research Fellow, and Chief,CSIRO Division of Environmental Mechanics, P.O. Box 821, Can-berra, A.C.T. 2601, Australia. Dr. Raats' permanent address isInstitute for Soil Fertility, P.O. Box 30003, Haren-Gr., TheNetherlands.

and Darcy's law:

+ k(h),01

[2]

where / is time, z is the vertical coordinate defined positivedownwards, v is flux, and k(h) is the hydraulic conductivitypressure head relation. We seek the solution of Eq. [1] andEq. [2] for a sample of finite depth L, subject to an initialcondition:

h = hn - z, t = 0, 0 L, [3]

where hn is the initial pressure head at the soil surface, andflux boundary conditions at both ends of the sample:

v = 0,t > 0, z = 0, [4]

> 0, z = L. [5]After the moisture front reaches the bottom of the sample,de/dt will tend to approach a constant value equal to vJL.A similar assumption was used by Passioura and Cowan(1968) in an analysis of radial flow of water to plant roots,and by Passioura (1977) in deducing soil water diffusivitiesfrom one-step outflow experiments. Equation [1] then re-duces to:

dvdz

dOAt ' [6]

where 0 is the average water content of the column. At timet the average increment in water content A0 is given by:

A0 = (v0t)/L = [ft (9 - 6n)dz]/L, [7]

and 0 will tend to 0[z - L/2] whenever v0 and L are small.The theory that follows demonstrates the effects of v0,

L, and k(h) on the evolving water content profiles.Integration of Eq. [6] using boundary condition Eq. [4]

gives:

[8]

To calculate from Eq. [2] and Eq. [8] the evolution of pres-sure head profiles, we consider the class of soils with anexponential dependence of the hydraulic conductivity uponthe pressure head:

k(h) = /3 exp (ah),where a and j8 are constants.

Introducing the matrix flux potential <f> defined by0 = /*_, k(h)dh,

and using Eq. [9] in Eq. [2] gives:3</> , j.v = - — + a<b.dz

[9]

[10]

[11]

Substituting Eq. [11] into Eq. [8] gives this linear differentialequation:

[12]

The solution of Eq. [12] is:v 2 / 1 _ . Z _ J _a \ L aL)

Setting z = 0 in Eq. [13] gives an expression for the in-tegration constant c:

[13]

231

232 SOIL SCI. SOC. AM. J., VOL. 46, 1982

Table 1—The physical properties of soils used in the experiment.Initial

Fine Coarse Bulk moistureSoil Clay Silt sand sand density content 6n

———————— % ———————— kgnr3 cm'/cm3

Bungendorefine sand 1.5 0.9 73.0 24.6 1.58 xlO3 0.050

Glebe loam 16.6 21.2 34.2 28.0 1.55 xlO3 0.266Brindabella

silty clayloam 33.0 26.0 25.0 16.0 1.16xl03 0.166

From constant-rate pump 1 bar ceramic

plates

Soil

To pressure transducerFig. 1—A cross-section of the soil cell. Soil diameter = 25.4 mm;

soil depth = 12.9 mm.

-h(m)

1.4 -

1.2 -

1.0 -

0.8

0.6

0.4

0.2

\.V -\.

II!•I

. V0 = 4.84x10 msec"'

A V=4.84«10~7 ••

I.\

0.1 O3 0.4

Fig. 2—The wetting experimental points 0(h) for fine sand obtainedby imposing the fluxes indicated. The solid line is a 0(/t) curve fora column (Smiles et al., 1981), and the open circles and dashedline are derived using the method of Haines (1930).

-(-7-1a \aL[14]

where h0 is the pressure head at the soil surface. IntroducingEq. [14] into Eq. [13] and using Eq. [9] and Eq. [10] tosolve for h gives:

[15]

Equation [15] describes pressure head profiles as a functionof z, L, a, /3, and ha.

MATERIALS AND METHODSThree soils were used in the experiment: a fine sand

(Bungendore), an aggregated (<2-mm fraction) silty clayloam (Brindabella), and an undisturbed loam (Glebe). Therelevant physical properties of the three soils are given inTable 1 . The fine sand and silty clay loam were packed intothe cell shown in Fig. 1. For the loam, undisturbed samples25.4 mm in diameter were obtained from the field andtrimmed to accurately fit the cell. A constant flux was im-posed at the top of the samples through a 1-bar ceramicplate, using a "Unita" constant- rate syringe pump. Thepressure head at the bottom of the sample was measuredcontinuously using a tensiometer consisting of a 1-bar ce-ramic plate connected to a pressure transducer.

The sets of 0(h) data were obtained for the fine sand usingfluxes of 4.84 x 10~8 and 4.84 x 10~7 m sec~'. For thesilty clay loam and loam, a single flux of 4.84 x 10~8 msec"1 was used. Alternative wetting moisture characteristicswere obtained for the three soils using thin samples on aHaines apparatus (Haines, 1930). A separate sample wasused for each pressure increment. For the fine sand 0(h)data were available for columns at equilibrium with a watertable for h = - 0.5 m to 0 (Smiles et al. , 1981). Data obtainedduring measurement of k(ff) relations (Perroux et al., 1981)provided a dynamic 0(/z) relationship for the silty clay loam.

1.5

1.0

-h

(m)

0.5

0.4

Fig. 3—Wetting (Hh} data for Brindabella silty clay loam (<2 mm).The squares are for the method of constant flux infiltration (v0= 4.84 x 10~" m sec"'); the dots are for the method of Haines(1930); and the solid line is for a dynamic (Xh} relationship (Perrouxet al., 1981).

PERROUX ET AL.: WETTING MOISTURE CHARACTERISTIC CURVES DERIVED FROM CONSTANT-RATE INFILTRATION 233

1.5

1.0

-h

(m)

0.5

1.5

0.2 0.3e

0.4

Fig. 4—The 0(/t) data for Glebe loam (undisturbed samples). Thesquares and curve are for the method of constant flux infiltration(v0 = 4.84 x 10"* m sec"1); the dots are for the method of Haines(1930).

These data were obtained by constant flux infiltration intolong, 50 mm in diameter columns; pressure heads weremeasured by a tensiometer-pressure transducer system andmoisture foments by gamma ray attenuation.

RESULTS AND DISCUSSIONFigures 2, 3, and 4 show 0(h) relationships for the

three soils using the constant flux method as well asstatic equilibrium methods. Also shown in Fig. 3 isthe independently obtained dynamic B(h) relation.

The fine sand data of Fig. 2 show good agreement,

0.5

Fig. 5—The fine sand wetting hydraulic conductivity-pressure headrelationships k(h) and fitted exponential relations.

even at the higher flux, between the alternative meth-ods except at high water content. It appears that whenwater is applied at a constant rate, less air is entrappedin the sand than when a column of sand is allowedto come to equilibrium with a water table.

In Fig. 3 the constant flux data agree well with thedynamic 6(h) relation for the silty clay loam but notwith the static equilibrium data obtained from theHaines method. The disparity between moisture char-acteristics for aggregated soils, subjected to differentpressure increments when wetted, has been demon-strated by Davidson et al. (1966), who attributed thisto a difference in the water content distribution be-

-h (m)1.0 1.5

24.021.6

17.214.2

9.74.

V = 4.84*10 msec

3.8

5

L/2

10

z L(mm) o

Fig. 6—Predicted pressure head profiles for the fine sand for imposed fluxes of (a) 4.84 x 10~* m sec'1 and (b) 4.84 x 10 7 m sec'1. Thecurves are labeled with the predicted time in hours for the profile to develop.

234 SOIL SCI. SOC. AM. J . , VOL. 46, 1982

1 /„, 0 f-—————L

V0 = 4,84 «10"8msec1

15 20

s /.x

./

•'

// V0 = 4.84 x 10 m sec~

0 1X> 2.0Measured time (hrs)

Fig. 7—Predicted and observed times for the soil at the bottom ofthe fine sand sample to reach pressure heads which correspond tothe profiles shown in Fig. 6. The straight line represents exactcorrespondence.

tween the small pores within aggregates and the largepores between aggregates. The disparity here is, how-ever, considerably greater than that reported by thoseauthors.

The undisturbed samples of Glebe loam exhibit con-siderable heterogeneity at the scale of sampling usedin this study (compare Haines data of Fig. 4). The0(h) data obtained by the constant flux method fallcentrally within the range of scatter.

Figure 5 shows a wetting k(h) relationship for thefine sand obtained by combining the k(0) relationshipof Talsma (1974) and the 6(h) relationship of Fig. 2.A single straight line does not describe the relationshipso it has been divided into three linear regions in theranges of h: 0 to -0.3 m, -0.3 to -0.6 m, and -0.6to -1.5 m. Lines have been fitted for these rangesas shown in the figure.

Figures 6a and 6b show the predicted evolution ofpressure head profiles in the fine sand for fluxes of4.84 x 10~8 m sec'1 and 4.84 x 10~7 m sec"1. Theseprofiles were calculated using Fig. 5 and Eq. [15]. Thetimes at which they are realized were calculated byintegrating the corresponding water content profilesand using Eq. [7]. At early times the difference be-tween the measured pressure head at the bottom ofthe sample and that at the top of the sample is sig-nificant, particularly at the higher flux. The differencediminishes as the pressure head increases until it isnegligible at -0.5 m for the higher flux and -0.7 mfor the lower flux.

For the fine sand a comparison between the pre-dicted and actual times taken for the soil at the bottomof the sample to reach a particular pressure head isshown in Fig. 7. The pressure heads for which the

comparisons are made correspond to the profilesshown in Fig. 6a and 6b. The predicted times werecalculated from Eq. [7] after integrating the watercontent profiles derived from the pressure head pro-files of Fig. 6a and 6b, and the wetting 6(ti) relationshipof Fig. 2. The points lie near the 1 to 1 line indicatinggood agreement between actual and predicted times.

The analytical approach, used to predict the h(z)profiles with respect to time and flux, is confirmed bycomparison.

We reiterate, however, that the analysis merelyconfirms our insights, based on experience of watermovement in soils, that the method should work. Withno a priori information on the properties of a particularsoil, the justification for the method lies in corre-spondence between sets of data obtained at substan-tially different low flow rates.

The method can be used for all nonswelling soilsprovided that the initial pressure head of the soil isnot beyond the normal operating range of tensiometers(—0.7 m). Its use with undisturbed samples is limitedby the need to shape thin samples to fit the cell andby the size of the cell in relation to the scale of soilheterogeneity.

Finally, we note that the analysis presented in thispaper can be applied to other situations in which pres-sure head profiles develop in shallow soil profiles dur-ing constant-flux infiltration. For example, it may beused in the case of constant-flux infiltration into lay-ered soils in which a fine soil overlies a coarser ma-terial. For that case, the time taken for the pressurehead at the interface to rise to a value in excess ofthe water entry value of the coarser material couldbe calculated.

CONCLUSIONSThe method provides a simple and quick method

for the measurement of the laboratory wetting curveof a soil sample. Accurate measurement can be ob-tained by using very low fluxes and/or correcting themeasurements of the pressure head by use of Eq. [14].The latter approach can be used only if reasonableestimates of a and /3 can be made.