Steady water flow around parabolic cavities and through parabolic inclusions in unsaturated and saturated soils

In this paper, we undertake the study of two closely related groundwater flow problems, both two-dimensional, steady and Darcian, and moreover involving parabolic inclusions. First we consider unsaturated flow for which the conductivity depends exponentially on the pressure. Second, we consider saturated flow and an inclusion with a differing, constant, conductivity from the exterior material. We apply the method of separation of variables, conformal mappings, and the Schwarz reflection principle. The distributions of the Kirchhoff potential, specific discharge, and flow net are derived in an explicit analytic form. We show the focusing/diverting properties of a parabola with a more/less permeable interior than the ambient medium, the location of the hinge point and separatrice, and the Maxwell–Philip uniformity of the flow in the interior zone.