# two-dimensional water infiltration from a trench in unsaturated soils1

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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., New

Haven. 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 DISCUSSIONPositions of the wetting front at three different times are

drawn 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. 1Schematic 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. 2Movement 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. 3Movement 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. 4Movement 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 c

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