Coarse Graining for Upscaling of Flow in Heterogeneous Porous Media

Abstract

This paper focuses on upscaling of permeability and flow in heterogeneous porous media. We develop a new upscaling method which considers the local permeability K(x) being a stationary random field of lognormal distribution and which is based on filtering procedures introduced in Attinger, Eberhard, and Neuss [Comput. Vis. Sci., 5 (2002), pp. 67--72]. The so-called coarse graining method is used to obtain an effective permeability tensor $K^{\text{eff}}(\lam)$ which depends on the given length scale $\lambda$. We formulate and extend the coarse graining method in Fourier space and give explicit results for the effective permeability tensor for a correct projector $P^\pm_\lambda$ in Fourier space. Furthermore, we develop a numerical upscaling scheme based on coarse graining which allows us to test the theoretical results. We compare the new method with simple upscaling methods such as arithmetic or geometric upscaling by evaluating fluxes $Q_\lambda$ and the solutions $u_\lambda$ of the flow equation itsel...