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A new free boundary problem for unsteady flows in porous media

We revisit the theory of filtration (slow fluid motion) through a horizontal porous stratum under the usual conditions of gently sloping fluid height profile. We start by considering the model for flooding followed by natural outflow through the endwall of the stratum, which has an explicit dipole solution as generic intermediate asymptotics. We then propose a model for forced drainage which leads to a new kind of free boundary problem for the Boussinesq equation, where the flux is prescribed as well as the height h=0 on the new free boundary. Its qualitative 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 difference is due to the freedom in the choice of the flux condition allowed in our problem setting. We also consider the modifications introduced in the above models by the consideration of capillary retention of a part of the fluid. In all cases we restrict consideration to one-dimensional geometries for convenience and brevity. It is to be noted however 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 analysis.