Filtration law for polymer flow through porous media

This paper deals with filtration laws for polymeric flow in porous media. General global filtration laws that are obtained by mathematical homogenization suffer from the fact that the (sufficiently approximate) numerical solution of the homogenized equations is in many cases even more expensive than the solution of the original fine-scale problem. To overcome this problem, we study the problem of decoupling the micro- from the macroscale for homogenized versions of quasi-Newtonian models. We consider the case of share-dependent viscosity obeying either a Carreau or a power law. Specifically, we study in detail the Taylor approximation of the macroscopic filtration law written in the form of a nonlinear Darcy's law. We show that the usual engineer's approach to approximate the permeability function is a first order approximation (for the Carreau law). For the power law, we show that the usual engineer's approach needs to be used with care and is appropriate only in some simple cases. Additionally, we provide some simple results of numerical experiments that illustrate the theoretical findings.