Similarity solution for linear counter-current spontaneous imbibition

Recovery of nonwetting phase (NWP) by counter-current spontaneous imbibition (COUCSI) of a wetting phase (WP) has been investigated. Experimental results cover a range of viscosity ratios of over seven orders of magnitude. An analytical solution for linear counter-current spontaneous imbibition based on similarity is presented. The mathematical model is based on: (1) Darcyʹs law for each phase, (2) the imbibition capillary pressures acting as the only driving force, (3) a back pressure given by the drainage interfaces associated with production of NWP at the open face, and (4) continuity of counter-current flow. An important consequence of the similarity solution is that the ratio of the WP flow rate at any location to the WP flow rate at the inlet of the sample is a function only of saturation.