Analytical solution for upscaling hydraulic conductivity in anisotropic heterogeneous formations

Modern geological modeling techniques represent anisotropic heterogeneous formations by high-resolution grids, which can be computationally prohibitive. This motivates the upscaling process that scales-up properties defined at a fine-scale system to equivalent properties defined at a coarse-scale system. In general, analytical methods are very efficient but limited to assumptions and approximations, whereas numerical methods are more robust albeit more time-consuming. In this work, we developed an analytical method to approximate numerical solutions in a finite difference scheme with periodic boundary conditions for two-dimensional problem. Using perturbation expansion techniques and Fourier analysis, the method generates explicit formulas of tensorial equivalent conductivity considering heterogeneity and anisotropy of two-dimensional space, as well as geometry of gridblocks. It is applicable for various cases with different covariance/variagram models and a wide range of log-conductivity variances, correlation lengths, rotation angles, anisotropy ratios of fine grid conductivity, anisotropy ratios of fine grid size, and the number of fine gridblocks in a coarse gridblock. The analytical method matched well with the numerical method for the estimation of the conductivity tensor, hydraulic head, and discharge velocity. The coefficients in the analytical method need to be computed only once for any given statistics, which makes the proposed method much more efficient than the numerical method.