A 3D filtration law for power-law fluids in heterogeneous porous media

In this paper, a 3D filtration law for power-law fluids flowing in heterogeneous porous media is derived through stochastic homogenisation. The filtration equation for isotropic porous media is first considered, at the local Darcy-scale. This equation possesses a single flow parameter, which depends on the space variables. The spatial variation of this parameter is modelled by a stationary random field and therefore arbitrarily heterogeneous and anisotropic in character. The stochastic homogenisation technique is then applied for averaging the interplay between rock and fluid parameters. A simple analytical and tractable formula is derived which expresses the importance of both rheological and porous medium related parameters on the mean flow. In order to validate the formula, comparisons are made with numerical experiments for 2D flows. The new law is found to be in good agreement with numerical experiments.