A Stochastic Analysis of the Scale up Problem for Flow in Porous Media

We present a numerical study of the scale up problem for the fractional ow function in the Buckley-Leverett equation for ow in porous media. The scale up problem is to deene an averaged equation by local spatial averages, mapping from a micro-physical description to a meso-physical description and from a ne discretization grid to a coarser one. Scale up leads to the closure problem, which is the deenition of the nonlinear terms in the averaged equation, as these terms are not respected by the averaging process. If the micro-physics is speciied by a geostatistical probability ensemble, we see that the scaled up and This paper is dedicated to the memory of our colleague and friend, Paulo Paes-Leme. 1 closed fractional ow function is also a random variable. Numerical studies presented here show that its variance is greatly reduced relative to the variance in the geostatistical micro-physics which deenes it. Properties of the scaled up fractional ow are given in terms of the geostatistical parameters which deene the ensemble of permeabilities.