Vertically integrated unconfined groundwater flow on sloping base

We consider unconfined flow in aquifers with a small sloping base, a magnitude of no more than a few percent at maximum and aim the analysis to large-scale regional modeling using analytic elements. We write the equations for the vertically integrated discharge, and find that the resulting two-dimensional discharge vector is rotational, as opposed to the specific discharge vector. We propose an approximate model for describing three-dimensional flow in an aquifer with a sloping base, and apply the approach to cases of flow between two parallel rivers and flow with a well in an aquifer bounded by a long river. We apply Wirtinger calculus and are able to estimate the error due to the approximation.