Determination of Soil-Water Diffusivity for Anisotropic Stratified Soils1

B. L. SAWHNEY, J.-Y. PARLANCE2, AND N. C. TURNER3

ABSTRACTThe unsaturated soil-water diffusivity of an anisotropic soil can be

described by a second-order tensor. In the particular case of a strati-fied soil, the diffusivity tensor is defined by the two values of the dif-fusivity in the principal directions, one normal and one parallel to thesoil layer. The standard method of Bruce and Klute then requires theuse of two soil columns, one for each direction, to define the diffusivitytensor. The present method makes use of a two-dimensional similaritysolution first derived for an isotropic medium and extended here to astratified soil. It is then possible to obtain the diffusivity tensor of thestratified soil from one experiment only. As an illustration of themethod, the diffusivity tensor of a mica layer is measured and theresult is used to analyze infiltration from a finite trench.

Additional Index Words: unsaturated soil, similarity solution.

I T HAS BEEN KNOWN for a long time that stratified soils areusually anisotropic so that water movement varies greatly

with the direction of the flow. The physical concept of watermovement in anisotropic media is clear even though experi-mental observations are limited. Historical developmentsand a general discussion of anisotropic permeability can befound elsewhere (Bear, 1972; Cisler, 1972 a; 1972b).

Recently (Turner and Parlange, 1975), a method wasproposed to measure the absorptive properties of a soil intwo dimensions. The method differs from the standardBruce and Klute (1956) method in that the latter considerswater movement in one direction, whereas in the two-dimensional method the movement is radial, i.e., in all di-rections around the source. Hence if the soil is anisotropic,the shape of the wetting front will not be a circle, but anellipse with longer axis parallel to layering in the soil.

THEORYConsider a stratified soil. The principal directions of the dif-

fusivity tensor are: x, normal to the stratification and y, z, parallelto the stratification. The diffusivity tensor is then determined bythe two functions Dxx(6) and£>TO(0) since D2Z(0) = Dm(0), where6 is the water content. The Bruce and Klute method then requirestwo soil columns, one taken normal to the direction of stratifica-tion and the other parallel to it. With the present method a soillayer must be taken parallel to the x-y plane. When this layer ishorizontal, i.e., in the absence of gravity, water movement obeysthe diffusion equation

where t is the time. Note that Eq. [1] can also be written in termsof the conductivity tensor, Kfj, instead of the diffusivity tensor Dtj(Bear, 1972; Cisler 1972a, b). Since the matric potential, </>, andthe water content are both scalar properties the usual relation,Ktid<j> = Dtid6, still holds.

Reichardt, Nielsen, and Biggar (1972) showed that the soil-

'Contribution from The Connecticut Agric. Exp. Stn., New Haven.Presented before Div. S-1, Soil Science Society of America, Chicago, 111.,14 Nov. 1974. Received 29 Apr. 1975. Approved 25 Aug. 1975.

"Associate Soil Chemist and Mathematician, respectively.3Senior Research Scientist. CSIRO Division of Plant Industry, P.O. Box1600, Canberra, A.C.T. 2601, Australia.

water diffusivity of isotropic materials obeys a scaling law (withinthe usual experimental variability). It is assumed here that such ascaling law also applies to the various coefficients of the dif-fusivity tensor, or,

DVV/DXX = X* [2]

where X is a constant independent of 6. The diffusivity tensor isnow defined simply by the function Dyv(ff) and the number X.

The solution of Eq. [1] requires boundary conditions, i.e.

[3]

where 00 is the uniform initial moisture content. Then at time zerowater is injected at a constant rate along a line source situated zix= y = 0. In that case a single similarity variable enters theproblem,

= [v2

[4]

and 0 is a function of <|/ only, at any time and in any directionaround the source. Measurement of the water profile, t|/(0), yieldsthe diffusivity coefficient, Dyu (Turner and Parlange, 1975) as

[5]

Eq. [4] shows that the lines of equiconcentration are ellipses ofsemiaxes i///"2 and <|/fl/2X in the y and x directions, respectively.Hence the ratio of the distances travelled by the front in the y and*directions yields X.

Once the diffusivity tensor is determined, two dimensionalmovements of water can be analyzed. For instance, the two-dimensional infiltration of water from a finite trench consideredearlier for isotropic soils (Selim and Kirkham, 1973; Turner andParlange, 1974; Sawhney and Parlange, 1974) will be strongly in-fluenced by the anisotropy in the layered soil. Use of Eq. [5] forthe measurement of the diffusivity tensor and its application to theinfiltration of water from a finite trench in a stratified medium arenow illustrated.

MATERIALS AND METHODSBiotite micaowas suspended in water and poured in small incre-

ments into a 60 cm long, 60 cm wide and 4.5 cm deep plexiglasschamber held vertically for the "trench" experiment and tilted at45° for the "tensor" experiment. Water was allowed to drainslowly through small perforations at the bottom of the chamber,permitting parallel orientation of the mica flakes. The mica wasallowed to dry at room temperature until 00 was < 1%. One-thirdof the mica particles ranged between 1 to 2 mm, one-third between0.5 mm to 1 mm, and the remaining < 0.5 mm. As the micaplatelets were fairly large, stratification of layers could be ob-served and the alignments of particles could be checked visually.Heterogeneity in anisotropic media results not only from variationsin density, as in isotropic soils, but also from variations in particlealignment.

The density of the dried mica was 1.3 and its porosity was about57%. This amount of compaction permitted reasonable rates ofwater movement in the x and y directions.

For measurement of soil-water diffusivity tensor, the chamberwas placed horizontally and the plexiglass side facing up wasremoved. Water was injected at the center of the chamber througha tygon tubing (0.25 cm diameter) 0.5 cm above the mica surface.

SOIL SCI. SOC. AMER. J., VOL. 40, 1976

60,

Fig. 1—60 by 60 cm box and wetting fronts at various times. Positionsof sampling are indicated by crosses, + .

The rate of water flow was adjusted at 7.3 ml min"1 using a con-stant flow piston pump (Milton Roy Co., Philadelphia, Pa.). Thismethod of injection is convenient and avoids some of the difficul-ties encountered with other methods, as discussed earlier (Turnerand Parlange, 1975). At selected time intervals, the position of thewetting front was marked on the plexiglass chamber. At the end ofthe experiment, soils samples were withdrawn at various positionsand their moisture contents determined gravimetrically.

For measurements of two-dimensional infiltration from a finitetrench, a trench was dug by removing mica in a corner of thechamber following the procedure described earlier (Sawhney andParlange, 1974). The chamber was then placed vertically, waterwas introduced in the trench, and a water level of 1 cm was main-tained. The position of the water front was recorded at varioustime intervals.

RESULTS AND DISCUSSIONFigure 1 shows the top view of the wetting front in the x-y

plane at different times. The average value of \ for this caseis about 2.6, and is given by the ratio of the*two distancesfrom the center to the front along the stratification and nor-mal to it. Although the shape of the wetting fronts changedwith time due to some heterogeneity in the packing, theyremain close to the ellipsoidal shape as predicated from thetheory. The average water content was about 0.25, which issignificantly lower than saturation that occurs at about 0.57.Figure 2 shows the moisture profile, B, as a function of ijj2/i//2

0, where <//<, is the maximum value of </» . Positions ofsampling are indicated in Fig. 1. Ratio of the distances oftwo points from the center along a radial line is equal to theratio of their x or y coordinates and hence to the ratio of theiri/>s according to Eq. [4]. Therefore, for each position, i///i//0is given by the ratio of the distance of the sampling pointfrom center of the chamber (a in Fig. 1) to the distance ofthe wetting front from the center along the radial line pass-ing through that point (b in Fig. 1). In principle one mustchoose the best fitting curve, e.g., the solid line in Fig. 2.

0.3

eS / c n

0.2 •

0.1

0.5

Fig. 2 — Water content as a function of the reduced variable i^2/ «J<20,where </» is defined by Eq. [4] and ^i0,is the maximum value of \ji.

Measurements were carried out at 2.5 hours, • , and 5 hours, • .The solid line is chosen by curve fitting and is given by Eq. [7] and[8].

This function i/»2(0) can then be used in Eq. [5] to yieldDyv. A purely analytical interpretation can also be given ap-proximately; the approximation involved is far less than thevariability in the data (Turner and Parlange, 1975). For in-stance in the present case we can postulate that

Dm=A exp Be [6]

following Reichardt et al. (1972). Then the profile, i.e., theapproximate solution of Eq. [5], is taken as

j'/ift = exp - [(411 x 4.5/7.3) J * Dm (a) da\. [7]

The best choice for A and B, yielding the solid line in Fig.2, corresponds to

Daa = 0.0384 exp 14 0 cm2 mnr [8]

in Eq. [7]. It is clear from Fig. 2 that the values of the dif-fusivity Dm are reliable only if 6 =s 0.35. Extrapolation ofEq. [8] to much larger values of 6 will yield values of Dyawhich are often too small (Stroosnijder and Bolt, 1975). Tomeasure the values of Dyy for 6 =s 0.35, the flow rateshould be larger than the one used here. No attempt wasmade to measure Dyy at large 6, since the observationswere presented only as an illustration of the method, and notas a study of mica diffusivity.

Stratification becomes apparent when water movement isnot one-dimensional so that the variation of permeabilitywith direction has a direct effect on water flow. A practicaland important example is the infiltration of water from sep-tic tank trenches. If the soil is isotropic, the main feature ofthe flow is that the lateral movement is about half as fast asthe downward movement (Selim and Kirkham, 1973;Turner and Parlange, 1974; Sawhney and Parlange, 1974).Since stratification is normally horizontal, i.e., parallel tothe bottom of the trench, the horizontal coordinate is effec-

SAWHNEY ET AL.: SOIL-WATER DIFFUSIVITY FOR ANISOTROPIC STRATIFIED SOILS 9

.^——-_^_I————^^——^-^—i ^——»'i ———— ^———— ————————•——^——i ^ ^

6O cm. 1 9 c m .

^————————————————————————————————————————————————————————————————-———— <—————————————————————————————»

/ ^ ~ ~ ~ ~ ~ " 7 / / /- ,- ______________________ o

I / / • / • / f i WTER — 3\ \ \ • \ V N. • ' min.

~ — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — " " " 1 f

Fig. 3—In right hand corner of the box, a 19-cm trench with 1 cm ponded water is indicated. Wetting fronts are given for different times. Dottedlines are the predicted shapes of the front, taking X = 2.6.

lively stretched by the factor X. Hence if X is > 2, the hori- the diffusivity tensor of a stratified soil. The method is two-zontal movement will be X/2 times faster than the down- dimensional and is particularly appropriate for anisotropicward movement. Figure 3 illustrates the situation when soils since the effect of anisotropy is most striking for two-water flows into the stratified mica layer. As expected, the and three-dimensional flows of water. The method is illus-lateral movement is much more pronounced than with an trated using sedimented layers of mica. The knowledge ofisotropic soil. The dotted lines represent the theoretical the diffusivity tensor can then be used to analyze two-wetting front at three times. The downward position of the dimensional movements of water. Observation of waterfront is assumed to be given by Sf1/2/0.57, i.e., neglecting movement from a trench exemplifies the effect of stratifica-gravity effects. The sorptivity 5 is obtained from the posi- tion as well as the quantitative variations in flow pattern in-tion of the front at 1 minute and is about equal to 0.57 X 5.4 troduced by relatively small changes in particle orientation,cm min"1'2. The predicted positions at 3 minutes and 6minutes can then be calculated. The position of the wettingfront after 6 minutes is much closer to the source thanpredicted. In fact, the front hardly moves downwards after15 minutes. The predicted lateral movement is also shownand corresponds to a parabola with its apex at the level ofthe water in the trench and situated at a distance X/2 = 1.3time greater than the downward distance. Again at 1, 3, and6 minutes there is at least qualitative agreement between theobserved and calculated positions of the front. Afterwardsthe front moves laterally at a speed which is greater than 1.3time the downward speed. This particular run, shown inFig. 3, illustrates the great difficulty in obtaining homoge-neous anisotropic soils. Direct observation of the micaplatelets showed that just below the position reached by thefront after 6 minutes the platelets were extremely wellaligned, and were much more closely packed than above, sothat the downward movement is drastically reduced. On theother hand the lateral movement still occurs through a lesspacked layer and hence is free to proceed. Note that theposition of the front above the water level is not indicated,except at 72 minutes, since it was not possible to observe itaccurately before that time.

In summary, we have presented a method to determine