  Hysteresis in the pressure-saturation relation in unsaturated porous media,
which is due to surface tension on the liquid-gas interface, exhibits strong
degeneracy in the resulting mass balance equation. Solutions to such degenerate
equations have been recently constructed by the method of convexification even
if the permeability coefficient depends on the hysteretic saturation. The model
is extended here to the case that the solid matrix material is viscoelastic and
that the system is coupled with a gravity driven moisture flux. The existence
of a solution is proved by compact anisotropic embedding involving Orlicz
spaces with respect to the time variable.
