Upscaling of Steady Flow in Three-Dimensional Highly Heterogeneous Formations
Aldo Fiori, Gédéon Dagan, Igor Janković · Multiscale Modeling and Simulation · 2011
Determining the velocity field [Formula: see text] by accurate numerical solutions of flow through heterogeneous formations of three-dimensional random structures requires a fine-scale discretization by a dense grid. With [Formula: see text] the maximal cell size needed to ensure an accurate solution and [Formula: see text] the logconductivity integral scale, [Formula: see text] is commonly adopted for logconductivity variance [Formula: see text]. To ease the numerical burden, the actual employed [Formula: see text] values are usually larger, requiring upscaling of the parameters [Formula: see text] (conductivity geometric mean), [Formula: see text] and [Formula: see text] (logconductivity variance), which characterize the isotropic medium. With the upscaled velocity field [Formula: see text] defined as the space average of [Formula: see text] over blocks of size [Formula: see text], the underlying upscaled [Formula: see text] is generally smoother [Formula: see text] and of larger correlation scale [Formula: see text] than the fine-scale one. These properties allow for an accurate numerical solution of [Formula: see text] with the coarse discretization. The aim of the present study is to determine the dependence of the upscaled parameters [Formula: see text], [Formula: see text], [Formula: see text] upon [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] for highly heterogeneous formations (the problem was solved in the past at first-order in [Formula: see text]). This is achieved in an approximate manner with the aid of the multi-indicator model we developed in the past, and results are checked using accurate numerical simulations of three-dimensional flow. The solution may serve to determine an upscaling block size [Formula: see text] and ensuing structural parameters for particular [Formula: see text] values selected by numerical analysts.