Efficient analytical upscaling method for elliptic equations in three-dimensional heterogeneous anisotropic media

Abstract

We propose an efficient analytical upscaling method to compute the equivalent conductivity tensor for elliptic equations in three-dimensional space. Our approach uses perturbation expansion and Fourier analysis, and considers heterogeneity, anisotropy and geometry of coarse gridblocks. Through low-order approximation, the derived analytical solution accurately approximates the central-difference numerical solution with periodic boundary conditions. Numerical tests are performed to demonstrate the capability and efficiency of this analytical approach in upscaling fluid flow in heterogeneous formations. We test the method in synthetic examples and benchmark cases with both Gaussian random fields and channelized non-Gaussian fields. In addition, we examine the impact of each parameter on the upscaled conductivity, and investigate the sensitivity of the variance and correlation lengths to the coefficients. We also indicate how to extend this approach to multiphase flow problems.