A NEW FREE-BOUNDARY PROBLEM FOR UNSTEADY FLOWS IN POROUS-MEDIA

We revisit the theory of filtration (slow fluid motion) through a hori zontal porous stratum under the usual conditions of gently sloping flu id height profile. We start by considering the model for flooding foll owed by natural outflow through the endwall of the stratum, which has an explicit dipole solution as generic intermediate asymptotics. We th en propose a model for forced drainage which leads to a new kind of fr ee boundary problem for the Boussinesq equation, where the flux is pre scribed as well as the height h = 0 on the new free boundary. Its qual itative behaviour is described in terms of its self-similar solutions. We point out that such a class of self-similar solutions corresponds to a continuous spectrum, to be compared with the discrete spectrum of the standard Cauchy problem for the porous medium equation. This diff erence is due to the freedom in the choice of the flux condition allow ed in our problem setting. We also consider the modifications introduc ed in the above models by the consideration of capillary retention of a part of the fluid. In all cases we restrict consideration to one-dim ensional geometries for convenience and brevity. It is to be noted how ever that similar problems can be naturally posed in multi-dimensional geometries. Finally, we propose a number of related control questions , which are most relevant in the application and need a careful analys is.