On numerical simulation of flow through heterogeneous formations

The conductivity of heterogeneous natural formations is modeled as a random space function (RSF). Consequently, the continuity equation and Darcy's Law become stochastic partial differential equations. Their solution by Monte Carlo simulations implies a partition of the flow domain in discrete blocks of random properties. The main aim of the study is to develop a methodology to derive the statistical moments of blocks conductivity as function of the given moments of the point values of conductivity and of the block size. The approach is similar to the one leading to effective conductivity, which is indeed a particular case for large block size. The general approach is illustrated for one-dimensional flow, which is solved exactly and by first-order perturbation expansion in the logconductivity variance.