Two-Dimensional Water Infiltration from a Trench in Unsaturated Soils1

B. L. SAWHNEY AND J.-Y. PARLANCE2

ABSTRACTTwo-dimensional infiltration of water from finite trenches

was observed in three soil samples with different textures. Thevertical infiltration is slowed by the lateral infiltration and theslowing is inversely proportional to the width of the trench.Furthermore, the vertical infiltration is about twice as fast asthe lateral infiltration for all soils tesed. At the same relativedistance from the trench, the soil was drier beside than belowthe trench. Our observations and interpretations agree withnumerical simulations of Selim and Kirkham.

Additional Index Words: unsaturated flow.

A.THOUGH most investigations of water infiltration intosoils have been concerned with unidirectional flow,

understanding of two-dimensional flow in soils is essentialbecause of its role in a variety of problems wherever thesource is finite, including the movement of effluent and pol-lutants from septic tank trenches and interpreting data fromring infiltrometers.

Two recent studies (Selim and Kirkham, 1973; Turnerand Parlange, 1974) have considered the infiltration froma finite source with a corner. The first study (Selim andKirkham) is a numerical investigation when gravity effectsare important; the second study (Turner and Parlange) isa theoretical and experimental investigation when gravityeffects are not important. In spite of these differences, theshape of the wetting fronts are remarkably alike in bothcases.

In the present report, we have concerned ourselves witha quantitative description of the slowing of the verticalmovement of the front AB caused by the lateral flow BCD(Fig. 1); the extent of the vertical infiltration comparedwith the lateral infiltration, i.e. the ratio FA to CE; the ver-tical and lateral profiles in the y and r directions respec-tively; and finally interpretations of the numerical calcula-tions of Selim and Kirkham in the light of our results.

MATERIALS AND METHODSThe soils used were Merrimac fine sandy loam and Buxton

silty clay loam. The less than 2-mm fraction of air-dry soil waspacked in a 60-cm long, 60-cm wide and 4.5-cm deep plexi-glass chamber having 1 cm diam holes at different distancesfrom the top. Soil was poured into the chamber in small succes-sive additions and, after each addition, was packed with a longmetal spatula followed by tapping the chamber on the floor toobtain a uniformly packed soil column. In the top right handsection of the chamber, crushed stones were added instead ofsoil to provide a trench of a desired width, L, and of about 15cm depth, DE. The trench was separated from the soil by a3.2-mm thick masonite board to prevent soil from entering thetrench during packing. The board was subsequently removed.At the bottom of the trench 2 or 3 folded paper towels wereplaced to permit a uniform flow of water along the entire width

1 Contribution from The Connecticut Agr. Exp. Sta., NewHaven. Received 22 Apr. 1974. Approved 22 Aug. 1974.2 Associate Soil Chemist and Mathematician, respectively.

of the trench. Water level in the trench was maintained atabout 1 cm throughout with a Mariotte bottle. (Fig. 1)

At selected intervals, the position of the wetting front wasmarked on the plexiglass chamber and the volume of waterentering the soil was determined from the change in water levelin the graduated carboy used to deliver water into the trench.At the end of the experiment, soil samples were withdrawn atdifferent vertical and horizontal distances from the trench andtheir moisture contents determined gravimetrically. Mean bulkdensity of the soils was determined from the weight and volumeof the soil packed in the chamber, their diffusivity was meas-ured by the method of Bruce and Klute (1956), and their con-ductivity was deduced from measurements of the moisture con-tents by the "long column" version of the steady state method(Klute, 1972).

RESULTS AND DISCUSSION

Positions of the wetting front at three different times aredrawn in Fig. 2, 3, and 4 for Merrimac B21, Merrimac AP,and Buxton B21 respectively with trenches approximately31 cm wide. Flow patterns for narrower trenches, varyingfrom 15 to 23 cm in width, were similar except that thewidth AB was reduced.

The distance of the wetting fronts from the trench alongthe vertical wall, i.e., point A in Fig. 1, and in the lateral di-rection, i.e., point C in Fig. 1, as a function of the squareroot of time are shown in Fig. 5. All distances were normal-ized with respect to vertical distances at t = ISO2 sec, whichare given in Table 1. Normalized vertical movements of thewetting front from all wide and narrow trenches in threesoils were identical and are plotted together along a singleline while lateral movements of the wetting fronts are pre-sented separately for each soil. The results show essentiallya linear relationship between the movement of the wettingfront and the square root of time for both vertical and lat-eral flows.

Deviation from linear relationship between the verticalmovement of the front and the square root of time can becaused by two physical processes. In the first case, move-ment of the water toward the side of the trench (Fig. 3)tends to slow the vertical movement. Secondly, gravity tendsto accelerate the vertical movement. Linear relationship be-tween the movement of A and the square root of time ob-served in our experiments show that these two effects canceleach other unless each of them is separately negligible. Con-sequently, if an estimate of the effect of one of these factorscan be obtained and is not negligible, the other is alsoknown.

Although Turner and Parlange observed the slowing ofthe wetting front due to a corner in experiments where grav-ity was absent, an accurate estimate of the slowing effectwas not obtained because it involved estimating the effectwhich is a small quantity as the difference between twolarge numbers, i.e. the positions of the front from a sourcewithout and with a corner. Here, the slowing effect will beobtained directly from the gravity effect.

When the gravity correction is small as in the present ex-

867

868 SOIL SCI. SOC. AMER. PROC., VOL. 38, 1974

wot er

B AFig. 1—Schematic representation of two dimensional water

flow from a trench; L, width of trench; y, vertical distance;r, radial distance; ABCD, wetting front.

periments, the effect can be expressed by (Talsma, 1969;Talsma and Parlange, 1972)

Fig. 2—Movement of the wetting front in a sandy loam soil,Merrimac B21. The front moved vertically downward 12.1,27.9, and 44.0 cm in 30, 120, and 300 min, respectively.

[1]

where Ay is the contribution to the distance y travelled bythe front that can be attributed to gravity. 5 is the sorptivity;Ks is the conductivity at 6S, moisture content at saturation.Here, standard one-dimensional experiments gave for MB-21, S ?=* 0.6 cm min"*4 and Ks = 0.01 cm min"1. As seenfrom Eq. [1] the gravity correction increases with time, andafter 300 min when the position of the front was last ob-served, the gravity effect represented about 10% of thetotal movement of the point A. This contribution due togravity is not negligible since it exceeds appreciably thevariability of the data shown in Fig. 5.

In the case of MB21, gravity and corner effects canceleach other exactly. Hence, Ay/y must be numerically equalto the slowing down of A caused by the lateral flow. Thelateral diffusion of water does not depend explicitly on theconductivity since gravity does not affect the process toany extent. The slowing effect, Ay, must then be written asa function of time, (0S — 6J where 6t is the initial moisture

Fig. 3—Movement of the wetting front in a fine sandy loamsoil, Merrimac AP. The front moved vertically downward14.5, 24.0, and 36.3 cm in 90, 210, and 450 min, respectively.Arrows indicate the movement of water at various points asobserved with a dye. Note that the flow remains vertical awayfrom the corner and diverges from a vertical flow at pointscloser to the corner.

content, the width of the trench L, and the diffusive proper-ties of the soil, characterized here by the sorptivity. Theonly dimensionless quantity obtained by combining t, L andS is S2tL~2, or y2L~2 since y =* S(6S — e^1^. Hence thedimensionless Ay/L is a function of S2tL~2, and of course(0S —0j). Lateral diffusion of water per unit area increasesas f^ for a diffusion process, but the lateral diffusion takesplace across an area proportional to y, which is itself pro-portional to t^. Consequently the total lateral diffusion andAy must increase as t. Hence Ay/L must finally be propor-tional to SttL-2 or y2£~2, i.e. Ay/y = £y/L, the coefficientof proportionality £ being a function of (6S — 0{). Since(6S — 0j) is about 0.3 for MB21, the value of £ for MB21should apply generally to any soil when (0S — 0{) «* 0.3. Toevaluate £ for MB21, we simply used Ks, S, Os — 0f andL = 31 cm in Eq. [1], and obtain £ ~ 1/3 Ks S~2 L (6S — Oj= 0.07. Calculations using data in Fig. 2 indicate that £ isknown with a 10% precision only. When L was reduced to21.5 cm, the two effects still cancelled each other althoughslowing down should increase with decrease in L. However,because of increased 9S at lower L (see Table 1), the gravityacceleration also increased and £ remained about the same.We can thus obtain an estimate of the slowing down effect,

Fig. 4—Movement of the wetting front in a silty clay loamsoil, Buxton B21. The front moved vertically downward 8.2,18.2, and 36.0 cm in 90, 450, and 1890 min, respectively.

SAWHNEY & PARLANCE: TWO DIMENSIONAL WATER INFILTRATION FROM A TRENCH 869

, for all soils and trench sizes as long as (6S — 0j) ~- 0.3from the following equation

0.07 y/L. [2]

Equation [2] then represents the slowing of the downwardmovement of water caused by the lateral movement.

It is worthwhile to compare the slowing of vertical move-ment of the front predicted by using this equation withSelim and Kirkham's observations. In their case, the gravityeffect is much larger than the slowing effect. To evaluatethese effects, one can first estimate the cumulative infiltra-tion, M, i.e. the amount of water per unit area behind a one-dimensional front. One can still use the general expression

Table 1—Pertinent characteristics and movement of wettingfronts for different size trenches in three soil samples

Soil

MB21

MAP

BB21

^Density 0i

g/cc1.72 0.015

1.81 0.011

1. 26 0. 020

+e's

0.270.300.360.370.370.380.350.500.500 50

Trenchwidth——— cm31.021.531.031.517.517.515.030.531.023.0

y a tth

= 150see's

————50.054.531.033.034.036.028 017.516 517 0

Symbol

•O+•x4VD•TA

* 01, initial moisture content, SB, moisture content at the trench, given as volume ofwater per volume of soil.

= fJn.

[3]

0.582M = — — — l n |

2V [6]

(Eq. [3] of Talsma and Parlange, 1972), and with the par-ticular values for D and K used by Selim and Kirkham

D = 3.33 X 10-5 exp (29.346) cm2 [4a]

and K = 3.33 X 10~13 exp (54.116) cm min-1. [4b]

Straight-forward calculations yield

t = 0.582 [61/3/2

1 — KS 1[5]

which gives the cumulative one dimensional infiltration Mas a function of time since V itself is related to time by Eq.[5]. Ks is equal to 0.1871 cm min^1 and V is the one-di-mensional flux. Calculations for M at various times givenin Table 2 correspond fairly closely to the amount of waterbehind the front as calculated by Selim and Kirkham, indi-cating that in their case too the slowing down of the frontcaused by the corner is small. However, subtracting thevalues given by Eq. [6] from their observed values in orderto estimate the magnitude of corner effect is again not relia-ble since we expect Eq. [5] and [6] to have a precision ofabout 5% (Parlange, 1973), which is of the same order asthe slowing effect.

The slowing effect can be estimated more precisely by

Fig. 5—Relationship between the vertical distances, Y, of the point A from the source, normalized at t = ISO2 sec and i& for allsoils for both narrow and wide trenches (Uppermost curve), and between the lateral distance of the point C from the corner andt^ for the soils, MB21, MAP and BB21. Symbols for each experiment are given in Table 1. The positions of C are given for widetrenches only. The lines correspond to ratios of 1/1.7, 1/1.8 and 1/2.0 for C to A. Smaller ratio for MB21 indicates a greatercorner effect for this soil than the other two soils (see also Figs. 2, 3, 4 and 7).

870 SOIL SCI. SOC. AMER. PROC., VOL. 38, 1974

Table 2—Cumulative infiltration behind a one-dimensionalwetting front from a trench in the absence of corner effect,

for the soil used by Selim and Kirkham (1973)

Cumulativeinfiltration

sec.7.0

15.622.129.247.360. 164.967.587.5

cm3.786.007.50

-7.8812.3614.8215.6016.3519.95

comparing the positions of the wetting fronts calculated bySelim and Kirkham for trenches of two different widths. Lety15 be the value of y when the trench was 15 cm wide andy75 when it was 7.5 cm. Then Eq. [2] predicts the differ-ence between the corner effect for the two trenches orthe difference between the distance travelled by the twofronts, as,

Ay7.5 - Ay15 = y15 - y7.5 ̂ 4.5 X 1Q-3 [7]

where y is the position of the front for a wide trench. In thefollowing numerical application we replaced y by y15. Forinstance, at 50 min when y75 is about 32 cm and y15 isabout 37 cm, the right hand side of Eq. [7] is about 6 cm,which is approximately the same as their calculated differ-ence. At 100 min, y75 would be about 44 cm (if the fronthad not interacted with the bottom of the cell) and y15 = 56cm, and Eq. [7] again gives roughly the calculated dif-ference.

It has been observed qualitatively by Turner and Par-lange that the movement of the wetting front was not re-duced appreciably until the front had moved a distancelarger than the source's width. This observation is con-firmed by the present experiments and by numerical calcu-lations of Selim and Kirkham. The quantitative agreementobserved here between the experiments and the numericalcalculations suggests that Eq. [2] correctly describes theslowing down of the front in relation to the distance trav-elled by the front and the width of the trench in spite ofmany possible causes of error in determining such a smalleffect. For instance, the present experiments require thatour estimate of the gravity effect be correct. In particular,the chosen value of K, has some uncertainty since it wasmeasured in a separate one-dimensional experiment andslight differences in packing under the trench affect Ks. Itis also possible that the numerical calculations might haveoverestimated the corner effect. Note that at the corner ofthe source the flux is very large, theoretically becoming infi-nite. Obviously computer calculations must underestimatethe flux near the corner, and if enough water is not comingin at the corner, more water might have to be artificially di-verted from the region under the trench, to the side. Thispossibility is, no doubt, a speculation which might be re-solved by comparing the numerical solution with an exactanalytical solution of the same problem obtained for con-stant diffusivity (Chu, Parlange, and Aylor, 1974).

In our experiments (Fig. 5) the distances from C to thetrench seem to be larger than half the distances from A to

31cm

Fig. 6—Positions of the wetting fronts at * = 90 min. Rangeabout this observed position is indicated by the vertical barsfor all other times *, after scaling distances by(t/90)% forthe three soils MB21, MAP, and BB21. The dotted lines showthe positions predicted by the model.

the trench, which was predicted by the model (Turner andParlange, 1974) and vary slightly from soil to soil. It is notclear whether this movement along C, more rapid than pre-dicted, was indicative of the model being only a crude ap-proximation, or that some cracks developed at the cornerof the trench due to the very rapid movement of water inthat region even though we did not observe them. The nu-merical calculations of Selim and Kirkham also suggest theratio to be slightly more than 1:2.

The predicted movement of A is more rapid than themovement of C because of the radial nature of the solutionof the diffusion equation in a corner and not primarily be-cause of gravity. It is only after a long time when the frontis sufficiently far from the source that the solution is radialeverywhere and gravity causes water to move more rapidlydownwards than laterally. This asymtotic effect can be esti-mated (Parlange, 1972) and is actually small even forSelim and Kirkham's case.

Figure 6 summarizes the movement of the fronts for thethree soils. The fronts are drawn by dividing all distancesat each time by (f/90)%. Theoretically all the fronts forone soil should reduce to the front at 90 min. The range inposition of the normalized curves is indicated by bars. Thefront for MB21 is farthest from the source at 90 min, thenMAP and finally BB21, in the order of their sorptivities.The dotted lines are the positions predicted by the model(Turner and Parlange, 1974). Note that the part CD of thefront (see notations in Fig. 1) is a parabola like CB withthe distance from D to the corner being half the distancefrom C to the corner. The general shape of the fronts is thesame as predicted from the theoretical model.

Finally, the model also predicts that the soil in the sideregion behind the parabolic front is dryer than under thetrench. Let

[8]

where y is, as before, the distance from A to the trench andy (9) is the distance from a point where the moisture con-tent is 6, r is the distance from a point on the parabolic frontto the corner, and r (8) is the distance to the corner at apoint behind the parabolic front where the moisture content

SAWHNEY & PARLANCE: TWO DIMENSIONAL WATER INFILTRATION FROM A TRENCH 871

0.4

0.20.2 0.4 0.6 0.8 Y

Fig. 7—Relationship between the normalized moisture content6" and the normalized vertical distance Y for the wettingfronts in the three soils and seven trenches. Symbols for dif-ferent soils and trenches are indicated in Table 1. Solid sym-bols represent moisture contents measured under the extremeright of the trench and the primed symbols moisture contentsmeasured under the midportion of the trench. Note that thesepoints intermingle at random within the scatter of the obser-vations. The solid line is the theoretical one-dimensional pro-file for a diffusivity varying as exp 8 0".

is 0. The model also predicts that at least on the average,we should have

/?-Y2. [9]

Moisture profiles for all soils and trenches indicating nor-malized water contents, 6*, as a function of vertical dis-tances from the trench are shown in Fig. 7, where

e* = (e — ei)/(6s — 6i). [10]

A theoretical moisture profile (Parlange, 1973) correspond-ing to diffusivity varying as exp. 86*, known to hold formany soils (Reichardt et al., 1972) is shown by a solid lineand holds in our case as well. Similar moisture profiles indi-cating water contents along 45° incline in the lateral regionare shown in Fig. 8. Comparison of Fig. 7 and 8 show thatat the same relative distance from the trench, the soil on theside is drier than the soil under the trench. The solid line inFig. 8 is the theoretical profile drawn according to Eq. [9]where Y was given by the solid line in Fig. 7. The fairlygood agreement between theoretical profiles and observeddata lends credence to Eq. [9]. The earlier experiments ofTurner and Parlange did not permit a check of this relation,but the numerical calculations of Selim and Kirkham lendsupport to these results.

In conclusion, the present study provides an understand-ing of the movement of water in unsaturated soils from atrench, particularly of the movement around the trench cor-ner. However, possible effects of rapid movement of waterat the corner on the soil matrix remain to be investigatedfurther.

The results have a number of applications to field inves-tigations. For example, the shapes of the wetting fronts

Fig. 8—Relationship between the normalized moisture content0* and the normalized lateral distance R for the wettingfronts in the three soils. Symbols for each experiment areindicated in Table 1. The solid line is the theoretical profileaccording to Eq. [9] when the diffusivity varies as exp 8 6.

around the trenches clearly demonstrate that pollutantsfrom septic tank drainfields would move half as far in thehorizontal direction as in the -verbal direction assumingthat conductivities are the same in both directions. Further-more, since the moisture content of the region along the sideof the trench is less than the region underneath the trench,concentrations of pollutants per unit volume of soil alongthe side of the trench would also be proportionately less.The results also suggest that in-situ measurements of sorp-tivity can be made with a single ring infiltrometer when inaddition to the usual gravity correction, the lateral flow andthe slowing down of the vertical flow are taken into account.