We provide a first approach of the mechanisms of liquid imbibition in a porous medium from a wet paste in contact with this substrate. Through Magnetic Resonance Imaging (MRI) we first show that, in contrast with intuition, the liquid can invade the substrate even if it has a larger pore size than the paste, which induces a lower capillary pressure in the substrate. This phenomenon happens because the paste can easily shrink. We then show that the imbibition stops when the capillary pressure in the substrate balances the stress needed to further contract the paste. The dynamics of the process then mainly results from the competition of these two effects plus the pressure gradient associated with the liquid flow through the paste. This in particular shows that the liquid penetration in a porous medium, from a poultice in contact with this medium, may be controlled by adjusting the poultice characteristics.