A two-scale percolation approach to compute the effective two-phase flow coefficients of heterogeneous porous media in the general case of a shear-thinning nonwetting fluid

Mechanistic simulators of the two-phase flow in pore networks (length scale~1 cm) have widely been used to determine the effective transport coefficients (e.g. capillary pressure curve-Pc, relative permeability curve of wetting phase-krw and nonwetting phase-krnw, resistivity index-IR) of macroscopically homogeneous porous media. However, in the classical 1-scale approach, very large networks, complicated algorithms and enormous computational effort are required to (1) simulate the up- scaled multiphase transport coefficients of macroscopically heterogeneous porous media (length scale~1 m), and (2) take into consideration the effects of buoyancy and viscous forces on Pc, krw, krnw, IR. In the present work, a two-scale percolation approach is developed to simulate the displacement of a Newtonian wetting fluid by a power law fluid in a heterogeneous porous medium. First, a gradient percolation model is developed by taking into account the power law rheology of the nonwetting fluid and the flow of wetting fluid along pore edges. In this manner, the small-scale Pc, krw, krnw, IR of each homogeneous unit are calculated. Then, the small-scale effective transport coefficients are fed as input data into a large-scale site-percolation model, where the pore sizes are replaced by the small-scale Pc curves, and instead of the critical pore pressure of penetration the critical breakthrough pressure is employed. The effects of the power law parameters, Bond number (Bo), capillary number (Ca), and contact angle (θe) on the small (homogeneous) and large (heterogeneous) scale displacement growth pattern as well as on the corresponding Pc, krw, krnw, IR are investigated. Finally, the calculated growth patterns and effective transport coefficients are compared to results of drainage experiments performed on glass-etched dual pore networks.