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, and validates the solution with numerical simulation.

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 as a function of saturation and time. The solution strategy uses backward finite differences on both the temporal variable and water saturation to solve for the instantaneous and average normalized water fluxes. The approach avoids the evaluation of implicit integral solutions and applies iterations on the flux ratio to satisfy both the flow and pressure boundary conditions. The solutions are obtained through two shooting processes for both normalized water flux and the pressure profile at all temporal steps.

The wetting phase is continuously imbibed into the core with initial flux being close to infinity. As the imbibition front propagates, the ratio of the co-current non-wetting phase 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 ending 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 wettability are analyzed in this study as well. The results also indicate the solution is independent of system length and permeability in its dimensionless form.

Unlike previous literature, 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, in future applications such as relative permeability and capillary pressure estimation, or improved interpretation of lab to field relationships through scaling group analysis.


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