Steady Infiltration From Large Shallow Ponds

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Abstract
Exact quasi‐linear solutions are obtained for steady infiltration from a shallow half plane pond on the surface of an isotropic semi‐infinite porous medium. Three particular boundary value problems are considered, namely, constant Dirichlet boundary conditions in relative permeability and vertical flux, and a mixed boundary value problem. The boundary conditions on the surface of the porous media for these problems correspond physically, first, to fully saturated conditions under the pond and fully unsaturated conditions elsewhere; second, to a constant vertical flux under the pond and zero vertical flux elsewhere; and third, to fully saturated conditions under the pond and zero vertical flux elsewhere. Each problem has essentially the same diffusive boundary layer, or capillary fringe, connecting the fully saturated and unsaturated regions. The total flux per unit length of boundary due to capillarity in the first problem is infinite, and for the latter two problems, finite constants. Strong analogies between our results for a surface source and the corresponding results of Waechter and Philip for buried spherical and cylindrical sources are established. Finally, the exact results for the half plane pond are used in an approximate description of steady infiltration from a large shallow pond of arbitrary shape, subject to the minimum radius of curvature of the pond's perimeter being large relative to 2/α, where α is the inverse length scale introduced in quasi‐linear models. A boundary layer develops from capillarity at the pond's edge, away from which the solution is adequately described by a diffusion equation representing a balance between the vertical flow from gravity and the horizontal flow from capillarity. An approximate expression is obtained analytically for the total flow from large shallow ponds.