Steady Two‐ and Three‐Dimensional Flows in Unsaturated Soil: The Scattering Analog

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 the Watson transform, are thus available for the solution and understanding of problems of unsaturated flow. 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 , with s 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. The paper deals specifically with steady infiltration from circular cylindrical and spherical cavities. The asymptotic methods prove remarkably accurate, even far from the limit. The results replace, and explain, previous semiempirical estimates of the limiting behavior. One notable result is that the depth of the penumbra (effectively wetted region) for the cylinder is 128 times the depth for the sphere, confirming and supplementing previous studies. An odd byproduct is that we correct a long‐standing classical result in scattering theory. The scope for extending these methods to flows in other geometries, to heterogeneous soils, and generally to linear convection‐diffusion processes, is indicated briefly.