The scattering analog for infiltration in porous media

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Abstract
This review takes the form of a set of Chinese boxes. The outermost box gives a brief general account of modem developments in the mathematical physics of unsaturated flow in soils and porous media. This provides the necessary foundations for the second box, which describes the quasi‐linear analysis of steady multidimensional unsaturated flow, which is an essential prerequisite to the analog. Only then can we proceed to the innermost box, devoted to our major theme. An exact analog exists between steady quasi‐linear flow in unsaturated soils and porous media and the scattering of plane pulses, and the analog carries over to the scattering of plane harmonic waves. Numerous established results, and powerful techniques such as Watson transforms, far‐field scattering functions, and optical theorems, become available for the solution and understanding of problems of multidimensional infiltration. These are needed, in particular, to provide the asymptotics of the physically interesting and practically important limit of flows strongly dominated by gravity, with capillary effects weak but nonzero. This is the limit of large s, where s is a characteristic length of the water supply surface normalized with respect to the sorptive length of the soil. These problems are singular in the sense that ignoring capillarity gives a totally incorrect picture of the wetted region. In terms of the optical analog, neglecting capillarity is equivalent to using geometrical optics, with coherent shadows projected to infinity. When exact solutions involve exotic functions, difficulties of both analysis and series summation may be avoided through use of small‐s and large‐s expansions provided by the analog. Numerous examples are given of solutions obtained through the analog. The scope for extending the application to flows from surface sources, to anisotropic and heterogeneous media, to unsteady flows, and to linear convection‐diffusion processes in general is described briefly.

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