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, ℓm/ I Y = 1/ 5 is commonly adopted for logconductivity variance σ²_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 σ²_Y (logconductivity variance), which characterize the isotropic medium. With the upscaled velocity field Ṽ(x) defined as the space average of V(x) over blocks of size L, the underlying upscaled Ỹ (x) is generally smoother (σ² Ỹ < σ²_Y) and of larger correlation scale (I Ỹ ≥ IY ) than the fine-scale one. These properties allow for an accurate numerical solution of Ṽ(x)) with the coarse discretization. The aim of the present study is to determine the dependence of the upscaled parameters K̃ G, I Ỹ , σ² Ỹ upon K_G, I Y , σ²_Y , and L/ I Y for highly heterogeneous formations (the problem was solved in the past at first-order in σ² _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.