Analytical Solutions for Linear Counter-Current Spontaneous Imbibition in the Frontal Flow Period

Analytical solutions have been derived in this article for counter-current spontaneous imbibition in a semi-infinite linear porous medium. The derivation is based on six fundamental equations, i.e., the existence of only two fluid phases, the continuity of flow, the generalized Darcy’s law for the wetting phase, the generalized Darcy’s law for the nonwetting phase, the relation between the capillary pressure generated by an interface and the difference in pressure across the interface between the nonwetting and wetting phases, and the conservation of mass. The mechanisms revealed by the solutions are simple and clear, and most of them are newly discovered. The solutions specify flow characteristics in three aspects. First, flow decelerates according to the square root of time. Second, the imbibed wetting phase volume is proportional to the square root of the product of porosity, suction capacity, and piston efficiency, while the sweeping distance is proportional to the square root of the ratio of suction capacity to the product of porosity and piston efficiency. Third, the volume of imbibed wetting phase is related to the area of the sample cross section, while sweeping distance does not. Suction capacity and piston efficiency are two measurable physical properties during counter-current spontaneous imbibition. An important step has been taken to extend the solutions to the cases where the saturation profiles are characterized by truncated fronts common in the productive reservoirs. It has been proven that the results calculated by analytical algorithms and the corresponding high-precision numerical methods match each other very closely. Previous study has already shown that the numerical simulation results can be highly consistent with the experimental performance measures. Therefore, with the advantages of clarity, accuracy, and speed, the analytical solutions provided in this article are very useful in increasing mechanism comprehension, enhancing engineering calculations and promoting wettability determination.