A formal solution to a classical initial-boundary value problem in groundwater hydraulics
Classical descriptions, dating to the early 1950s, of the fully saturated Darcian flow of pumped groundwater through layered, confined geological media with leaky interfaces have avoided what has been recently described as an ''analytically intractable'' problem (Freeze and Cherry). The main results due to the work of Hantush, Jacob, Neuman, and Witherspoon, reported during the past 20+years, have circumvented the ''intractability'' by using various clever averaging procedures to take account of the leaky interfaces, and have obtained closed form solutions to problems which, in effect, have averaged away certain complications due to inhomogeneous driving terms. For many applications, these solutions have been quite sufficient. The work here describes a Green's kernel which integrates with the boundary leakage inhomogeneity in the original initial-boundary value problem, thus supplying a computable Fourier expansion for the solution in each layer which avoids modifying the original differential equations as performed in the classical methods. It is seen that the indicated kernel arises from a consideration of a two point boundary value problem for a linear, inhomogeneous ordinary differential equation, B"0[u] = f on (0, ~), where B"0 is the modified Bessel operator of order zero. The condition as r -> + ~ involves a somewhat different manipulation which eventually takes account of the leaky boundary contribution over the entire interface. The specific integral equations obtained here are soluable from the practical standpoint, and the author has obtained useful physical information not available from earlier works.