Exact self-similar solution of countercurrent capillary imbibition

Consideration is given to the one-dimensional countercurrent capillary imbibition of a wetting phase (water) into a semi-infinite porous medium. A model based on consideration of the function of movement of self-similar saturation profiles is proposed. An equation is derived for it, and boundary conditions are determined. The solution of the equation gives the imbibition volume, the saturation and imbibition-flux profiles, and the parameters at entry into the medium and on the saturation front. Calculation of the imbibition indices for a sandstone sample is performed.