Flow-rate-dependent Equilibrium Saturation-distributions Through Hysteresis in Two-Phase Flow in Porous Media

Two-phase flow systems in porous media are inherently complex and can exhibit both path- and rate-dependence in their behavior. Mathematical models may be developed that include explicit representations of path dependence (via hysteretic constitutive functions) and rate dependence (for example, via boundary conditions). We report on such a model in this paper. We use a set of quasi-analytical solutions and simulations to study the combined impact of hysteresis and flow rate on the equilibrium distribution of two fluids in a one-dimensional vertical column. We use boundary conditions to control the flow rate in the system, and assign different properties to each of the fluids, including different densities. We consider drainage in an initially water-saturated column, and compare the equilibrium profile for creeping, capillary-dominated invasion and an advection-dominated Buckley-Leverett displacement as well as displacement processes of injection rates between the two extreme cases. All of these injections are followed by a capillary-driven redistribution. The bounding cases permit quasi-analytical solutions while mixed displacement processes require numerical solutions. We find that the differences in the solutions are significant and should be included in analyses of these kinds of problems. We also consider these results in light of upscaling and the design of effective models. We present a model that is able to capture the rate-dependent equilibrium saturation distributions in the column based on the injected volume and the history of a dynamic apparent saturation for low and intermediate injection rates. Based on these calculations we are preparing experiments to corroborate general hysteresis models or evaluate their limitations. We will report on our findings.