The present paper studies the infiltration of an incompressible liquid in an initially dry (or partially dry), deformable sponge-like material made of an incompressible constituent in the slug-dow approximation having in mind the application to some industrial processes involving dow through sponge-like materials and, in particular, some composite materials manufacturing processes. The resulting initial-boundary value problem is of Stefan type, with suitable interface conditions and evolution equations describing the position of the interfaces delimiting the saturated region within the porous material. Different models are then suggested in the saturated region, depending on the importance of the inertial terms and on the constitutive equation for the stress. Comparison of the simulation with known experimental results is satisfactory.