Steady infiltration in unsaturated soil from a buried circular cylinder: The separate contributions from top and bottom halves

Waechter and Philip (1985) obtained the asymptotic expansion of the mean infiltration rate for large s from a buried circular cylinder using a scattering analog. Here s(= αl/2) is defined as the ratio of the characteristic length l of the water supply surface (in fact, its radius) to the sorptive length 2α−1 of the soil and a satisfies the relationship K(ψ) = K(0) eαψ, where K is the hydraulic conductivity, and ψ is the moisture potential. This exact solution cannot be used directly to obtain the separate contributions to the mean infiltration rate from the top and the bottom halves of the cylinder; our analysis is based on a new class of special functions derived from the modified Bessel equation with a forcing term. In this paper, we obtain the separate asymptotics for the two halves for large s to make a comparison with the results of the trench problem (Waechter and Mandal, 1993). The asymptotic expansions for top and bottom halves are (2/π)(0.69553s−2/3) and (2/π)(1+0.30066s−2/3), respectively, whereas for a semicircular trench, the mean infiltration rate is given by (2/π)(1+0.30066s−2/3).