Upscaling of permeability of anisotropic heterogeneous formations: 1. The general framework

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Abstract
In numerical simulations of flow through heterogeneous formations, the domain is partitioned into numerical elements. The solution requires assigning physical properties to each numerical block. The process of transferring information from the scale of actual heterogeneity to that of the numerical elements is known as upscaling. We consider steady, one‐phase flow and the only property of interest is the permeability, which is regarded as a random space function, and the same is true for the upscaled conductivity. In this paper we establish the necessary and sufficient conditions to be satisfied by upscaling. The necessary conditions are expressed with the aid of the global response of the formation, e.g., the total flux caused by a constant pressure head drop applied on the boundaries. The requirement is that the expected value and the variance of the total flux are the same in the actual and upscaled domains. These are supplemented by local conditions which stipulate that space averages of flow variables over the numerical blocks in the upscaled domain tend to those in the actual formation when the elements size tends to 0. The result of this analysis is that the upscaled permeability must be such as to ensure the equality, in a statistical sense, between the space averaged dissipation in the two media. By assuming an unbounded domain, average uniform flow and stationarity, simplified relationships between the moments of the dissipation in the upscaled and actual media are derived. It is shown that the effective permeability of the two media is the same. The approach is illustrated for stratified formations for which exact composition rules are possible. Even in this simple case the upscaled permeability is a tensor, characterized at second order by two mean components and three covariance functions.