Non-Newtonian flow in macroscopic heterogeneous porous media: from power-law fluids to transitional rheology.

Many natural or industrial fluids exhibit non-linear behavior. Non-Newtonian fluids can therefore be found in different applications related to porous or fractured media such as mud flows, oil recovery, hydraulic fracturing or foam injection clean-up. This work focuses on the understanding of the flow of these fluids in large scale heterogeneous porous media, and particularly in the coupling between flow heterogeneity and rheology. This work consider fluids that exhibit a change in behavior, where the viscosity changes above a certain velocity threshold. This change of viscosity affects drastically the inhomogeneity of velocity field. In this paper, we analyse some general mathematical properties for non-linear rheologies at the microscopic and macroscopic scale. Most non-linear rheologies imply the existance of two asymptotic limit, at low and high flow rate. The velocity field of both limits are obtained by solving a power-law rheology in the medium, which is analysed using a perturbation expansion. As an example, we will consider fluids which are Newtonian for low shear-rate but becomes shear-thinning or shear-thickening at higher shear rate. Three macroscopic flowing regimes are then observed. It will be shown that the limits at low and high flow-rate are governed by the properties of power-law fluids. The transition between these two regimes is the analysed. The range depends on the heterogeneity permeability field but also on the rheological parameters. The statistical properties of the flow field displays interesting statistical and geometrical properties such as multi-scale (fractal), similarly to other critical systems (percolation, avalanches, etc.)