Multiscale Method for Oseen Problem in Porous Media with Non-periodic Grain Patterns

Accurate prediction of the macroscopic flow parameters needed to describe flow in porous media relies on a good knowledge of flow field distribution at a much smaller scale—in the pore spaces. The extent of the inertial effect in the pore spaces cannot be underestimated yet is often ignored in large-scale simulations of fluid flow. We present a multiscale method for solving Oseen’s approximation of incompressible flow in the pore spaces amid non-periodic grain patterns. The method is based on the multiscale finite element method [MsFEM Hou and Wu in J Comput Phys 134:169–189, 1997)] and is built in the vein of Crouzeix and Raviart elements (Crouzeix and Raviart in Math Model Numer Anal 7:33–75, 1973). Simulations of inertial flow in highly non-periodic settings are conducted and presented. Convergence studies in terms of numerical errors relative to the reference solution are given to demonstrate the accuracy of our method. The weakly enforced continuity across coarse element edges is shown to maintain accurate solutions in the vicinity of the grains without the need for any oversampling methods. The penalisation method is employed to allow a complicated grain pattern to be modelled using a simple Cartesian mesh. This work is a stepping stone towards solving the more complicated Navier–Stokes equations with a nonlinear inertial term.