An analytical effective tensor and its approximation properties for upscaling flows through generalized composites

Abstract

The direct numerical simulation of flows through composites is difficult due to the fine-scale heterogeneity in the media and also due to the complexity of the dynamic systems. Thus, flow based on upscaling of the effective diffusion has become an important step in practical simulations of flow through composites. In this paper, an analytical form for computing the effective diffusion tensor is proposed when the variable coefficient is locally isotropic, K( x) = k( x) I, where k(x) is a scalar function, not necessarily periodic, and may be defined as a step function. Currently, similar results are obtained by numerical means and require a special (re)design of a numerical code and may be computationally demanding. The proposed analytical upscaled tensor approximates the one obtained from the periodic cell-problem in the context of classical homogenization theory. The diagonal values are computed by an averaging procedure of the well known Cardwell and Parsons bounds. The off-diagonals are derived from the rotation of an angle related to the center of mass of the unit cell or representative elementary volume (REV). When comparing the proposed form with various known analytical and numerical results, it shows to be accurate within less than 3% on average. The comparisons include realistic reservoirs description and analytical results, such as the concentric spheres of Hashin–Shtrikman and the two and four-phase checkerboard geometries. Moreover, convergence results corroborate theoretical ones from classical homogenization literature. Convergence is obtained by performing successive refinement from an initial coarse description of a given heterogeneous medium, which include the SPE10 comparative solution project.