Theoretical investigation of two-ends-open free spontaneous imbibition

Two-ends-open free spontaneous imbibition refers to a laboratory core experiment, with one end face exposed to the wetting phase and the other end exposed to the non-wetting phase. Spontaneous imbibition leads to the production of non-wetting phase both co-currently and counter-currently. This paper extends previous work on systems of infinite length and presents the exact one-dimensional semi-analytic solution for such a system (Lagrangian approach) and validates the solution with numerical simulation (Eulerian approach). The methodology solves the partial differential equation of unsteady state immiscible, incompressible flow with arbitrary relative permeability and capillary pressure functions using a fractional flow concept. The solution strategy uses a Lagrangian discretization in both the temporal variable and the water saturation to solve for the instantaneous and time averaged normalized water fluxes. The approach avoids the evaluation of implicit integral solutions but instead combines an explicit calculation for the total fluid flux along with a shooting algorithm for the water imbibition to satisfy both the flow and pressure boundary conditions. The wetting phase is spontaneously imbibed into the core with the initial water inlet flux being close to infinite. As the imbibition front propagates, and before it reaches the system outlet, the ratio of the co-current non-wetting phase (oil) flux to the inlet water flux increases from zero to a finite value below one which is dependent on the intrinsic properties of the system. This indicates that the production of non-wetting phase at the inlet will not cease before the front reaches the outlet, irrespective of the length of the system. The time for the front to reach the outlet, the final ratio of co-current oil flux to inlet water flux, and the variation of total produced volume from both ends, along with their sensitivities with respect to the absolute permeability, system length, and free water saturation, are analyzed in this study as well. The results also indicate that the solution is independent of system length and permeability when expressed in terms of dimensionless time. Unlike previous studies, we have not assumed self-similar solutions or treated the flow as purely co-current or counter-current. The boundary conditions for the system analyzed here are easily achievable in the lab and have been discussed in the literature. The results from this study could be used to serve as a benchmark for numerical simulations, or in applications such as the interpretation of laboratory data for relative permeability and/or capillary pressure, or improved interpretation of laboratory to field relationships through scaling group analysis.