Linearized unsteady multidimensional infiltration

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Abstract
The parameters in the linearized multidimensional unsteady equation for unsaturated flow in homogeneous soils can be chosen so that small time flow rates are given correctly, with the quasi‐linear steady solution as the large time limit. Matched in this way, the linearized equation offers insight into the time course of multidimensional infiltration. The Laplace transform reduces the equation and its governing conditions into forms similar to those for quasi‐linear steady flows. The problems of unsteady infiltration from buried spherical and cylindrical cavities are solved. In many cases the solutions are approximately reducible to the product of the steady solution and a function only of time and radial coordinate. Theorems show that approximate product solutions apply not only to these configurations but generally to multidimensional cavities of arbitrary shape. Multidimensional infiltration from finite supply surfaces involves two characteristic lengths and two characteristic times, one set deriving from the capillarity gravity interaction and the other from the capillarity‐geometry interaction. We explore the influence of the two characteristic times on the time scales of approach to steady moisture distribution near the supply surface and to steady discharge. The analysis provides explanation of the relatively rapid approach to the steady state observed in three‐dimensional infiltration from small supply surfaces.

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