On an asymptotic expansion for Carreau fluids in porous media

Porous-media flow of polymers with Carreau-law viscosities and their application to enhanced oil recovery (EOR) is considered. Applying the homogenization method leads to a nonlinear two-scale problem. In case of a small difference between the Carreau and the Newtonian case an asymptotic expansion based on the small deviation of the viscosity from the Newtonian case is introduced. For uni-directional pressure gradients, which is a reasonable assumption in applications like EOR, auxiliary problems to decouple the micro- from the macrovariables are derived. The microscopic flow field obtained by the proposed approach is compared to the solution of the two-scale problem. Finite-element calculations for isotropic and anisotropic pore-cell geometries are used to validate the accuracy and speed-up of the proposed approach. The order of accuracy has been studied by performing the simulations up to the third-order expansion for the isotropic geometry.