UPSCALING OF STEADY FLOW IN THREE-DIMENSIONAL HIGHLY HETEROGENEOUS FORMATIONS

Determining the velocity field V(x) by accurate numerical solutions of flow through heterogeneous formations of three-dimensional random structures requires a fine-scale discretization by a dense grid. With l(m) the maximal cell size needed to ensure an accurate solution and I-Y the logconductivity integral scale, l(m)/I-Y = 1/5 is commonly adopted for logconductivity variance sigma(2)(Y) <= 1. To ease the numerical burden, the actual employed l/I-Y values are usually larger, requiring upscaling of the parameters K-G (conductivity geometric mean), I-Y and sigma(2)(Y) (logconductivity variance), which characterize the isotropic medium. With the upscaled velocity field (V) over tilde (x) defined as the space average of V(x) over blocks of size L, the underlying upscaled (Y) over tilde (x) is generally smoother (sigma(2)(Y) < sigma(2)(Y) ) and of larger correlation scale (I-(Y) over tilde > I-Y) than the fine-scale one. These properties allow for an accurate numerical solution of (V) over tilde (x) with the coarse discretization. The aim of the present study is to determine the dependence of the upscaled parameters (K) over tilde (G), I-(Y) over tilde, sigma(2)((Y) over tilde) upon K-G, I-Y, sigma(2)(Y), and L/I-Y for highly heterogeneous formations (the problem was solved in the past at first-order in sigma(2)(Y) < 1). 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 L/I-Y and ensuing structural parameters for particular l/I-Y values selected by numerical analysts.