Modelling the flow of power-law fluids through anisotropic porous media at low-pore Reynolds number

The flow of power-law fluids through fibrous media at low-pore Reynolds number is investigated using the homogenization method for periodic structures with multiple scale expansions. This upscaling process shows that the macroscopic pressure gradient is also a power-law of the volume averaged velocity field. To determine the complete structure of the macroscopic flow law, numerical simulations have to be performed on representative elementary volume of porous media. In this paper, this has been achieved on 2D periodic arrays of parallel fibers with elliptical cross section of different aspect ratios. It is found that macroscopic flow models already proposed in the literature fail in reproducing numerical data within the whole volume fractions of fibers and aspect ratios ranges. Consequently, a novel methodology is proposed to establish the macroscopic tensorial seepage law within the framework of the theory of anisotropic tensor functions and using mechanical iso-dissipation curves. This methodology is illustrated through our numerical results.