The Soil Water Diffusivity Near Saturation1

B. E. CLOTHIER AND R. A. WoooiNO2

ABSTRACTMany laboratory porous materials are considered to have a zone of

tension-saturation in their water retentivity curve near saturation. Thisimplies that the soil water diffusivity, D, is infinite in such a region. Herethree new methods of measuring the diffusivity near saturation are pre-sented. Experiments were carried out on a fine sand and all methodsfound D to be finite, yet large, having a value of approximately 1 to 2X 10~3 m2 s '. Since the near-saturated conductivity of the sand is ofthe order 10~5 m s~' this implies that the water retentivity curve nearsaturation has a finite but large slope of order 102 m. This was com-patible with the measured retentivity data. The theory of one-dimen-sional absorption implies that the slope of the Boltzmann variable withrespect to volumetric water content near saturation equals twice D di-vided by the sorptivity. The slope calculated this way was compatiblewith that found in absorption experiments.

Additional Index Words: conductivity, linearization, pressure poten-tial, retentivity curve, absorption.

Clothier, B.E., and R.A. Wooding. 1983. The soil water diffusivity nearsaturation. Soil Sci. Soc. Am. J. 47:636-640.

TRADITIONALLY for soil and other porous media thevolumetric water content-dependent diffusivity

function D(0) has been inferred from one-dimensionalabsorption experiments (Bruce and Klute, 1956). How-ever the diffusivity data so derived have been found in-adequate when used critically in experiments with othergeometries or boundary conditions. Substantial modifi-cation of D(6) has sometimes been necessary simply topreserve such a basic constraint as continuity (Bresler etal., 1971). Bruce and Klute (1956) recognized the lim-itations of their approach stating that "the method leavesmuch to be desired in the way of precision. There isunfortunately no standard against which the diffusivityvalues can be checked." More recently, Morel-Seytouxand Khanji (1975) commented that it "may be very dif-ficult to measure accurately the diffusivity at high watercontents." In this paper we provide three methods formeasurement of the diffusivity near saturation.

When used in analytical or quasi-analytical solutionsof water flow (Fujita, 1952; Parlange, 1971) the diffu-sivity occurs in an integral sense, thus values of D(0) nearsaturation are of greatest import (Hanks and Bowers,1963). Recently Smiles et al. (1980) argued that manysoils near saturation, especially laboratory ones, have aso-called "tension-saturated zone" (Philip, 1957) in whichthe change in pressure potential ^ with 6 is effectivelyinfinite. The length of this zone is often defined by \ps,some potential at which the soil, when wetting, first ap-pears to attain effective saturation at water content B,.Here, 6, is equivalent to the satiated water content ofMiller and Bresler (1977). This implies that D for \f*s <\(/ < 0 is infinite, since D = K d\j//dd where K is thehydraulic conductivity.

In a porous medium, allowed to imbibe freely to near1 Contribution from the Plant Physiology Division, DSIR, Palmer-

ston North, and Applied Mathematics Division, DSIR, Wellington, NewZealand. Received 6 Dec. 1982. Approved 17 Mar. 1983.2 Scientists, DSIR.

effective saturation, it is unlikely that the entrapped airis at atmospheric pressure (Collis-George and Bond, 1981)therefore d\[t/dO, and so D will reflect this. We show thatfor our soil D(\(/) for \[/s< \l/ < 0 is finite, so that d\j//d0is also finite here, albeit large. In our laboratory soil andwe suspect others, there is no zone of true tension-satu-ration, there being a range of pore sizes near saturationthat do imbibe in the range \f/S:\l/s.

THEORYLinearization

The one-dimensional soil water flow equation,

a0/3r = 8(Z)30/3z)/9z-(rfA:/</0)30/az , [1]

is of the nonlinear diffusion-convection type, where / is timeand z the depth (positive downwards). It is possible to reduceEq. [1] to a linear form by introduction of the matric fluxpotential 0, defined by

fJga

[2]

where the subscript n refers to the antecedent condition. If thehydraulic conductivity function of the soil is described by

[3]

where Ks is the saturated conductivity, then Eq. [1] becomes(Gardner, 1958)

Z)(32<f>/3z2) - £>a(3cj>/3z) . [4]

This equation is used as the basis for finding a value for thediffusivity (D) near saturation in three experiments describedbelow. It is anticipated that since all the experiments are ford~6s the assumption of a constant D will not be limiting.

Experiment IConsider a column of soil with water supplied at the upper

surface at flux density /, and free-water dripping through itsbase. Following Gardner (1958) the steady-state profile of pres-sure potential can be found from Eq. [4] to within a constantof integration as

- J/K,) - [5]

At fluxes near the saturated hydraulic conductivityfor some distance below the soil surface d^//dz is very small.Solution of Eq. [5] permits judicious selection of appropriatelengths of soil column and applied water fluxes whereby it ispossible to monotonically wet the soil to obtain a region of near-constant potential so that the water content is as close as re-quired to its saturated value 6S.

If, however, the water flux is applied as a series of discretedrops, changes in both water content and pressure potentialabout their respective mean values will be propagated down thecolumn. Confounding hysteretic effects should be minimal overthe narrow range of 6(^ds) involved. Suppose that the 0 of Eq.[4] is composed of the steady-state profile *(z) and a smallsuperimposed fluctuation Ti(u>,z) exp(/'u>f) representing a sinu-soidal travelling wave of angular frequency o>. Then solution ofEq. [4] gives that

636

CLOTHIER & WOODING: THE SOIL WATER DIFFUSIVITY NEAR SATURATION 637

= D(3>)(d2x/dz2 - [6] the base will then provide a value for the diffusivity at the meanpotential of -[z -

For the range in which i£ and hence D(<b) varies slowly withrespect to z, the solution of Eq. [6] has the approximate form

X = A (z)exp /(ut - VcoM/2£>z) ,

so the amplitude A at depth z is

A (z) = A0expl (a/2 - Ju/2DM )z j ,

where

M = (1 + a4D2/16w2)'/2 + a2£>/4co ,

[7]

[8]

[9]

and A0 is the amplitude at the surface.Over the range of interest near saturation ($, < ^ <0) a is

small so that M can be taken equal to unity. Also, a/2 may beneglected in Eq. [8] provided that a\/£>/2a><Kl. If this is thecase then the effect of gravity may be ignored and Eq. [7] willreduce to the solution given by Carslaw and Jaegar (1959, §2.6).From a general order of magnitude consideration; u> = 1 s~',a = 10 m~', and D = 10~3 m2 s~', so M—1^10~2 anda\//)/2a>==ilO~1, justifying the omission of gravity.

The exponential factor in Eq. [8] represents the amplitudedecay. Since differentiation of this equation, with a neglected,gives

'} , [10]

for any frequency, the decline in the amplitude of the pressurepotential fluctuations induced by the droplets can be used tocalculate the diffusivity at that mean potential. The use of pe-riodic variations in water content to infer D was suggested byGardner (1964), and pressure head fluctuations are also mootedfor use in hydrogeology to find aquifer diffusivity by Black andKipp (1981).

Alternatively the exponent in Eq. [7] can be written in termsof v, the velocity of propagation through the soil of featuressuch as the maxima of pressure potential waves (Carslaw andJaegar, 1959). This gives the diffusivity as

D = c2/2coand provides another means of inferring D.

[11]

Experiment IIConsider a semi-infinite column of soil initially in equilibrium

with a free water surface at its base. It is now convenient toconsider the datum of z = 0 at the base, with z positive up-wards, so that Eq. [4] becomes

3<f>/3r = £>(32<J>/3z2) + £a(3<j>/3z) . [12]In terms of <£ the condition that describes this initial equilibriumstate of ^ = — z, is

<J> = <f>0exp( - az), z>Q, ; = 0, [13]where <f>0 = Ks/a. If at time zero the free water surface israised by a small height *, then a change in water content andpressure potential will be propagated up the column. Theboundary condition for this step change is

), z = Q, t>0. [14]In terms of pressure potential the solution of Eq. [12] subjectto Eq. [13] and Eq. [14] is

(4/(z,0 + z}/¥ = (erfc{(z + aDt)/2jDt}+ exp( - oz)erfc { (z - aDt)/2JDt } )/2 . [15]

Measurement of the transient in pressure potential in the soilat height z following a step change in the water potential at

Experiment IIIAs an alternative to the semi-infinite requirement of the the-

ory in experiment II consider a column of finite length, L. Ifthe soil is initially at equilibrium with a free water surface atits base and at time zero the potential at the base is changedto *, then initial and boundary conditions [13] and [14] effec-tively describe this. Solution of Eq. [12] for these conditionsgives the transient in the pressure potential at any height z inthe sample (0 < z < L) as

= 1 + 2exp(AZ) f { gn/(A2 + Q2)}

({Asin Qn(l-Z)+QncosQn(l-Z)}/{(A+Ocosa-as ina ,} ) , [16\

where Z = z/L, A = aL/2, and Qn is the nth root of theequation Asingn+gncosgn. The pressure potential at the topof the soil core (z = L, Z = 1) is then

= 1 +2exp(A) a,exp{-(A2+ Q2)Dt/L2}/n= 1

((A2 + Qn2) { (A + 1 )cos a - a«n Qn } ) . [1 7]

If, after the soil has attained the new equilibrium, the watertable is returned to the base of the column, then Eq. [17] willnow give the transient in the pressure potential [$(L,t) + *+ L]/^. The experimental traces of potential with time willprovide a value for D. Comparison of the two experimentalhKO + L]/^ and [$(t) 4- * + L]/ty curves will test theassumption of a constant diffusivity.

If gravity is ignored A = 0, and Eq. [17] reduces to (Carslawand Jaeger, 1959, §3.4)

" n= 1,3,5 . . .

X exp(-nWDt/L2) . [18]This equation will apply for short columns or soils with smalla.

In a subsequent paper (Scotter and Clothier, 1983) this tech-nique is developed further to permit measurement of the un-saturated diffusivity of field cores.

MATERIALS AND METHODSThe soil used in all experiments was the fine sand-textured

C horizon of Manawatu fine sandy loam, a Dystric FluventicEutrochrept. Water was added to the air-dried soil to bring itto fln=0.05, a water content suitable for packing. The soil waspacked to a bulk density of 1 400 ± 30 kg m~3 m~', in acryliccolumns of diameter 34.5 mm.

The soil water pressure potential was measured by tensiom-eter of diameter 10 mm and sealed with a membrane of thinnylon mesh having 60-ji wide pores. The pressure transducer(Validyne DP15TL) scanned the tensiometers via a hydraulicswitch (Scanivalve Co., WO602/1P-24T). Measurement of astep change in free water showed the tensiometer-scanning valve-transducer system to have a time constant of < 0.2 s.

The water retentivity curve of the soil (Fig. 1) measuredusing the capillary-rise technique has been given by Clothierand Scotter (1982). Considering other published \l/(6) data thissoil has, by the usual definition, a tension-saturated zone witha $s of about —0.25 m. The wetting hydraulic conductivityrelationship K(\{/) was obtained using a soil column of length

638 SOIL SCI. SOC. AM. J., VOL. 47, 1983

Volumetric Water Content , 6

-0.1

-0.2

E -0.3»

"~o

1 -0.4£0

I

£ -0.5

-0.6

0 0.2 0.3 0.4 0.5u ' I D '

1a 1

-I/"* «, Dl' 4 ?

a 11 a

D'

aa

. D

aa

aD

"%aa aD D

DDDa a a Capillary rise q>(9)

DnD

aD

-

'<" 5ElO"5

*

^••g

I0

u1J

10"

i i i i i i i i

K(4i) , (welting)^''

^''

^^

- ^.^

a = 3 m" 1

•

•

•

_•

•

-0.4 -0.3 -0.2 -O.I 0

Pressure Potential ty , mFig. 2— The wetting hydraulic conductivity curve, Kty). Also shown is

Eq. [3] with a = 3 or1.Fig. 1—The wetting retentivity curve, 0(i/<), of the fine sand, (Clothier

and Scotter, 1982). The broken line is the prediction by Eq. [23] for-0.2 < ^ < 0 m, given 9, = 0.4.

0.60 m, onto which a Unita constant-infusion pump drippedwater at a prescribed flux. Once water came out of the base,tensiometers were used to measure the equilibrium dty/dz nearthe top of the column so that Darcy's law could be used todetermine K at the mean potential. The K(\l/) function is shownin Fig. 2 along with Eq. [3] for a = 3 m~'. Equation [3] isseen to provide an adequate description over the region of in-terest, ty, < ^ < 0.

In experiment I water was supplied at the prescribed flux tothe top of the 0.6-m long soil column by the infusion pump.Variation in the period of droplets landing was obtained byselection of an appropriate pump outlet orifice. The frequencyrange 0.8 < o> < 3.9 rad s~' was covered. Tensiometers wereinstalled at six depths (20, 35, 55, 90, 135, and 190 mm) belowthe soil surface once the wet front passed the deepest level. Theexperiment did not begin until water dripped out of the baseand the steady-state potential profile (Eq. [5]) established. Twoflux densities of / = 8.91 X 1Q-* and 1.34 X 10~5 m s"1 wereused to achieve uniform pressure potentials in the measurementzone of —0.23 and —0.04 m of water, respectively. It is con-venient here to express ^ in units of head of water.

In experiment II the acrylic column from experiment I wasused. However the column was inverted so that now the pres-sure potential was measured at heights of 45, 65, 100, 145, and200 mm above the free water surface. Once the equilibriumprofile (Eq. [13]) was attained the free water surface was raiseda height S^ =15 mm to initiate the experiment.

For experiment III a 75-mm long column of soil was placedon a membrane of 60-n nylon mesh from which was hanginga water column of adjustable height. The hanging column wasinitially arranged to provide ^ = 0 at the base of the soil col-umn (z = 0) and the soil allowed to come to equilibrium (Eq.[13]). The experiment began by lowering the adjustable watercolumn to provide ̂ = — 0.15 m at z = 0. A flat-bed tensiom-eter placed on the top of the soil (z = L) was used to measurethe transient in the pressure potential and so give the draining

diffusivity. Following attainment of this new equilibrium thewater table was returned to ^ = 0 and a wetting diffusivitymeasured.

RESULTSExperiment I

The pressure potential fluctuations induced by thedroplets (Fig. 3) appear to rapidly become sinusoidal asthe higher harmonics disappear (Carslaw and Jaegar,1959). The sinusoidal boundary condition is then a rea-sonable approximation. The amplitude-depth relation-ships obtained at the two mean pressure potentials of—0.23 and —0.04 m are shown in Fig. 4 for a range offrequencies. The diffusivity was found for each experi-ment using Eq. [10] and a log-linear least squares regres-sion. For I = -0.4 m, D = 1.7 (±0.3) X 10~3 m2 s"1;and for ̂ = -0.23 m, D = 1.9 (±0.2) X 10"3 m2 s~'

PRESSURE POTENTIAL FLUCTUATIONS:I MINUTEI —— » ——— f

ly = - somm -> |t j4j- ti::!; H:l:::; 4 t-Iul" i4-4iA4 AIA4 frill i:Hi4iiWUl|M

DEPTH (mm) : 20 , 3 5

^

50

Fig. 3—A chart-trace of tbe fluctuations in pressure potential measuredat three depths for a flux density of 1.34 X 10~5 m s-1.

CLOTHIER & WOODING: THE SOIL WATER DIFFUSIVITY NEAR SATURATION 639

t< 10'' .

ICP .

Depth , mFig. 4—The amplitude of pressure potential fluctuations with depth for

various frequencies o>. The solid lines are for J/K, for 0.67 and thebroken lines for J/K, = 0.45.

a diffusivity value not significantly different. At the meanpotential of —0.23 m it was found that pressure potentialmaxima were propagated between the depths of 0.035and 0.19 m at a velocity of 5.13 (±0.14) X 10~2 m s~l

when w = 0.73 s-1. Equation [11] gives a D of 1.80(±0.01) X 10~3 m2 s~' by this alternative means.

Experiment IIThe measured transients in pressure potential at var-

ious heights above the water table following a step changeimposed at z = 0 are presented in Fig. 5. Also shown isthe time course at these depths as predicted by Eq. [15]with D = 2 X 10~3 m2 s"1 and a = 3 m"1 given that^ = 0.015 m. There is fair agreement especially atgreater heights (z > 0.10 m) above the free water sur-face. Near z = 0, definition of height becomes difficult,since the tensiometers themselves are 0.01 m in diameterand the step change ^ is a significant fraction of theactual height z. If gravity is ignored the right-hand sideof Eq. [15] becomes simply erfc(z/\/2Z)F). However thissolution was found inadequate because of the significantcontribution of the second term in Eq. [12] at anythingbut small distances above the water table.

Experiment IIIThe transient in pressure potential at the top of a short

column (L = 75 mm) following step changes in *ff offrom 0 to —0.15 m and —0.15 m to 0 are shown in Fig.6. Equation [17], with D = 1.5 X 10~3 m s~l and a =3 m"1 and Eq. [18] with the same D are both found toadequately simulate the results from the experiments upto t/L2 = 300 s m~2. There is, however, evidence of a

1.0

0.8

0.6

0.4

0.2

0

Experiment

Eq.(15) ———

: * = 0.015m.—: D = 2 x 10-3m2s-1

a = 3m-1

10 20t,s.

30 40

Fig. 5 — The normalized transient pressure potential response at variousheights, z, in a long column following a step change * in the freewater surface (solid line). The broken line is Eq. [15].

long tail; a well-known trait of outflow experiments. Thishas been related to slow changes in the volume of en-trapped air (Peck, 1969). That the simpler gravity freeexpression (Eq. [18]) differs little from the full equation(Eq. [17]) is due both to the low value of a over thepotential range and the small column length L. In factA (i.e., aL/2) is only 0.1125. The similarity of both ex-perimental curves suggests D changes minimally over 0to —0.15 m and that hysteresis is minimal.

DISCUSSIONThe value of the diffusivity near saturation (—0.23 <

\l/ < —0.04) has been found by four different techniquesto give a common value of between 1 to 2 X 10~3 m2

s~', with an arithmetic mean of 1.8 X 10~3 m2 s"1. Theimplication of this value is discussed in relation to boththe one-dimensional absorption profile and the water re-tentivity curve.

Bruce and Klute (1956) showed that D(B) could befound from one-dimensional absorption profiles, x(0), us-ing

D(9)= - C8\dO [19]

IT+ 1

o»

1.0

0.8

0.6

0.4

0.2

' Experiment: L = 0.075 m, * = 0.15 m.Eq.(17) • • • • • • • • : D = 1.5 x 10-3m2s-1,« = 3m-1

Eq.(18) —• —• —: D = 1.5 x 1Q-3m2s-1

500 1000 1500

Fig. 6—The normalized transient pressure potential response at the topof a short column following the raising (solid line) or lowering (brokenline) of the water table at the base. Equation [17] (dotted line) andEq. [18] (dashed line) are also shown.

640 SOIL SCI. SOC. AM. J., VOL. 47, 1983

CD

Boltzmann variable 10 A , m s"̂Fig. 7—The one-dimensional absorption data of Clothier and Scottei

(1982). The shaded area is the sorptivity and a line of d\/dO of -2.4m s w near saturation is shown.

where X is the Boltzmann similarity variable, X = x(S)t~>A,t being the elapsed time. Close to the saturated watercontent Os we can rewrite Eq. [19] to give

dXd9 --2D/S,

where the sorptivity S is

S = (e*\d9 .Je..

[20]

[21]

The X(0) absorption similarity profile for this soil, as pre-sented by Clothier and Scotter (1982), is shown in Fig.7. From Eq. [21] S = 1.47 X 10~3 m s"* so that fromEq. [20] d\/dO near saturation is —2.4 m s~w, for D =1.8 X 10~3 m2 s"1. A line with this slope is shown inFig. 7 to be consistent with the experimental data. Howthis diffusivity near saturation can be combined with atraditional Bruce and Klute diffusivity analysis (Eq. [19])is outlined in another paper (Clothier et al., 1983).

By definition, the diffusivity is given byD(9) = K(9)(dt/d9], [22]

so that near saturation, if D is considered constant, in-tegration of Eq. [22] gives the water retentivity curve as

9 = 9, + D 1 {*K<W ,Jo * > & [23]

From the measured K(\}/) relationship (Fig. 2), with D= 1.8 X 10~3 m2 s-1, the predicted 0ty) for 0 < ^ <— 0.2 is shown in Fig. 1 for Os = 0.40. The slope of this

line is of the order 102 m which is compatible with the\l/(6) found by the capillary-rise technique.

The exceptionally high water contents very near \j/ =0 and X = 0 are ignored. These are probably a result ofthe unrepresentative \j/(6) that occurs near the imbibitionsurface of a porous medium. During wetting, near thesurface, there is a higher probability of air escape andso the soil wets to a greater extent (Philip, 1957).

From measurements presented here of a sand near sat-uration, in the potential range (—0.2 < ^ < 0), theconductivity was found to halve whereas the diffusivityremained effectively constant. These results suggest awetting water retentivity curve for this sand that is verysteep near saturation, but that a true zone of tension-saturation does not exist. This implies that near satura-tion there are indeed some pores that do wet or drain inthe potential range — 0.2 < \l/ < 0 m.