Upscaled Coefficients for Immiscible Two-Phase Flow in Porous Media for Drainage and Imbibition Processes

In this work, we study immiscible two-phase flow in porous media through the upscaled model derived by Whitaker in 1994. This model contains two momentum equations for each fluid, which are coupled through four effective tensors, one for each phase and two crossed tensors between phases. Two tensors correspond to the effective permeability of phases, and the other two are named as viscous drag tensors. The four tensors are determined by solving associated tensorial closure problems in representative geometries of the porous medium. We have numerically solved the integro-differential equations composing the closure problems in a 2D cavity (unit cell) undergoing imbibition and drainage processes, as a first approximation. Thus, we have estimated the main directions of the effective tensors as functions of the wetting-phase saturation. The effective permeabilities follow trends similar to experimentally measured permeability relative curves, showing hysteresis for drainage and imbibition, although with some deviations from the experimental values. Meanwhile, the viscous drag tensors exhibit estimations of order 1, which are in agreement with the analytical predictions of Whitaker. The findings of this work are promising, as in future works, more realistic cells as: thin sections of rocks, SEM images, or 3D tomography of rocks, can be used to improve the numerical predictions of the effective tensors and elucidate thus the effect of microscale phenomena on the two-phase flow at larger scales.