A fast semi-analytical homogenization method for block heterogeneous media.

Direct numerical simulation of flow through heterogeneous media can be difficult due to the computational cost of resolving fine-scale heterogeneities. One method to overcome this difficulty is to coarse-grain the model by decomposing the domain into a number of smaller sub-domains and homogenizing the heterogeneous medium within each sub-domain. In the resulting coarse-grained model, the fine-scale diffusivity on each sub-domain is replaced by an effective diffusivity, calculated from the solution of an appropriate boundary value problem over the sub-domain. However, in simulations in which the heterogeneous sub-domain geometries evolve over time, the effective diffusivities need to be repeatedly recomputed and may bottleneck a simulation. In this paper, we present a new semi-analytical method for solving the boundary value problem and computing the effective diffusivity for block heterogeneous media. We compare the new method to a standard finite volume method and show that the equivalent accuracy can be achieved in less computational time for several standard test cases. We also demonstrate how the new method can be used to homogenize complex heterogeneous geometries represented by a grid of blocks. These results indicate that our new semi-analytical method has the potential to significantly speed up coarse-grain simulations of flow in heterogeneous media.