Analytical solution for groundwater recharge on a hill

We present an exact solution to the problem of two-dimensional flow with a free surface from a recharge area on a hill with vertical faces to valleys on either side of the hill. The valleys are at different elevations, and the question we address is to what extent the difference in elevation of the valleys affects the amount each valley receives from the recharge. We find that the difference in elevation has a major effect, a result that may be of interest in farming, where similar conditions exist. We solve the problem by means of the hodograph method and conformal mapping. The domain in the hodograph plane is constructed according to the boundary conditions, which leads to a domain bounded by a combination of straight lines and a circular arc. The domain in the hodograph plane is transformed into a domain bounded by straight lines, which makes conformal mapping of the reference upper half plane onto this domain possible, by means of the Schwarz–Christoffel transformation. The solution is obtained by the use of a second function, the Zhukovski function, which is mapped onto the same reference plane. The results are presented in terms of graphs of discharges into the valleys for both isotropic and anisotropic aquifers; flow nets are presented as well.