Corner and start-up singularities in porous flow

We consider the penetration of liquid into a porous medium which is, in part, impervious to the liquid. Corners arise at points where the medium becomes impervious and numerical difficulties can arise due to a conflict in the boundary conditions. In addition there are start-up problems associated with numerically modelling the flow problem because the flow domain is initially empty. We consider some techniques for exploiting similarity techniques to develop accurate numerical algorithms which avoid the above difficulties.