Shock propagation in a flow through deformable porous media

We consider a one dimensional incompressible flow through a porous medium undergoing deformations such that the porosity and the hydraulic conductivity can be considered to be functions of the flux intensity. The medium is initially dry and we neglect capillarity, so that a sharp wetting front proceeds into the medium under the influence of a given pressure on the surface. We study the problem of the continuation in time of the solution in presence of singularities, under the assumption that the porosity is a non increasing function of the volumetric velocity. This assumption implies that the hyperbolic equation expressing the conservation law can degenerate.