A new multiscale scheme for computing statistical moments in single phase flow in heterogeneous porous media

In this work we develop a new multiscale procedure to compute numerically the statistical moments of the stochastic variables which govern single phase flow in heterogeneous porous media. The technique explores the properties of the log-normally distributed hydraulic conductivity, characterized by power-law or exponential covariances, which shows invariance in its statistical structure upon a simultaneous change of the scale of observation and strength of heterogeneity. We construct a family of equivalent stochastic hydrodynamic variables satisfying the same flow equations at different scales and strengths of heterogeneity or correlation lengths. Within the new procedure the governing equations are solved in a scaled geology and the numerical results are mapped onto the original medium at coarser scales by a straightforward rescaling. The new procedure is implemented numerically within the Monte Carlo algorithm and also in conjunction with the discretization of the low-order effective equations derived from perturbation analysis. Numerical results obtained by the finite element method show the accuracy of the new procedure to approximated the two first moments of the pressure and velocity along with its potential in reducing drastically the computational cost involved in the numerical modeling of both power-law and exponential covariance functions.