Imbibition with swelling: Capillary rise in thin deformable porous media

The imbibition of a liquid into a thin deformable porous substrate driven by capillary suction is considered. The substrate is initially dry and has uniform porosity and thickness. Two-phase flow theory is used to describe how the liquid flows through the pore space behind the wetting front when out-of-plane deformation of the solid matrix is considered. Neglecting gravity and evaporation, standard shallow-layer scalings are used to construct a reduced model of the dynamics. The model predicts convergence to a self-similar behavior in all regions except near the wetting front, where a boundary layer arises whose structure narrows with the advance of the front. Over time, the rise height approaches the similarity scaling of t(1/2), as in the classical Washburn or BCLW law. The results are compared with a series of laboratory experiments using cellulose paper sheets, which provide qualitative agreement.