ON SOLVING THE UNSATURATED FLOW EQUATION

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Abstract
We present a new quasi-analytical technique for solving the flow equation. It has affinities with Parlange's method, but offers the following advantages: freedom to choose an initial assumed flux-concentration relation, F1, greatly improves the possible accuracy of the first approximation, and the higher approximations preserve integral continuity and therefore behave more stably. The first of these advantages is of practical importance, but the second is more basic. This paper treats only solutions subject to concentration conditions; the related technique for solutions subject to flux conditions will be developed in a later paper.The technique is studied analytically and numerically for one-dimensional sorption subject to constant concentration conditions. It is found to be convergent for a wide range of shapes of the diffusivity function. For the unfavorable case of the ‘linear’ soil, the mean error is 3 percent after two iterations and 1 percent after three. For absorption in Yolo light clay the corresponding figures are 0.57 percent and 0.07 percent.The general iterative scheme for one-dimensional infiltration subject to constant con—centration conditions is presented. Three choices of F1 should yield useful first approximations: (A) F1A = limt→oF (t is time) = Fabs, the F for the analogous absorption process; (B) F1B = limt→∞F = θ; and (C) F1c, an interpolation function which is exact in the limits as t → 0 and t → ∞. F1A should lead to a good lower bound for the infiltration rate function q(t), F1B an upper bound, and F1C a close upper bound for all except very large t, and the quality of the estimates of moisture profiles should be comparable. Detailed calculations for Yolo light clay bear out these expectations; the three estimates are wholly consistent with the power series solution of Philip (1957b). The error of the approximation based on F1C increases from 0 percent at t = 0 to about 1 percent at t = 106 sec. This first approximation is accurate enough to render iteration unnecessary for most purposes. Parallel calculations confirm the nonconvergence of Parlange's method when applied to infiltration.General iterative schemes are given also for two- and three-dimensional sorption subject to constant concentration conditions. We present a new quasi-analytical technique for solving the flow equation. It has affinities with Parlange's method, but offers the following advantages: freedom to choose an initial assumed flux-concentration relation, F1, greatly improves the possible accuracy of the first approximation, and the higher approximations preserve integral continuity and therefore behave more stably. The first of these advantages is of practical importance, but the second is more basic. This paper treats only solutions subject to concentration conditions; the related technique for solutions subject to flux conditions will be developed in a later paper. The technique is studied analytically and numerically for one-dimensional sorption subject to constant concentration conditions. It is found to be convergent for a wide range of shapes of the diffusivity function. For the unfavorable case of the ‘linear’ soil, the mean error is 3 percent after two iterations and 1 percent after three. For absorption in Yolo light clay the corresponding figures are 0.57 percent and 0.07 percent. The general iterative scheme for one-dimensional infiltration subject to constant con—centration conditions is presented. Three choices of F1 should yield useful first approximations: (A) F1A = limt→oF (t is time) = Fabs, the F for the analogous absorption process; (B) F1B = limt→∞F = θ; and (C) F1c, an interpolation function which is exact in the limits as t → 0 and t → ∞. F1A should lead to a good lower bound for the infiltration rate function q(t), F1B an upper bound, and F1C a close upper bound for all except very large t, and the quality of the estimates of moisture profiles should be comparable. Detailed calculations for Yolo light clay bear out these expectations; the three estimates are wholly consistent with the power series solution of Philip (1957b). The error of the approximation based on F1C increases from 0 percent at t = 0 to about 1 percent at t = 106 sec. This first approximation is accurate enough to render iteration unnecessary for most purposes. Parallel calculations confirm the nonconvergence of Parlange's method when applied to infiltration. General iterative schemes are given also for two- and three-dimensional sorption subject to constant concentration conditions. © Williams & Wilkins 1974. All Rights Reserved.

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