Stochastic analysis of high-permeability paths in the subsurface

Abstract

Subsurface fluids may travel along paths having a minimum permeabilility greater than the effective permeability of the rock. This may have an important impact on contaminant migration. A stochastic approach related to percolation theory is advanced to address the question of what is the probability that a high permeability path extends across a given volume of the subsurface. The answer is sought numerically through subdividing the volume of interest into a three-dimensional grid of elements and assigning a random permeability to each element. Four permeability processes are considered: 1) Stationary with independence between grid elements; 2) Stationary and autocorrelated; 3) Nonstationary due to conditioning on measured values; and 4) Random rock volume included in grid. The results utilizing data from fractured granites suggest that in large grids, at least one path having a minimum permeability in excess of the "effective" rock permeability will cross the grid. Inclusion of autocorrelation causes an increase in the expected value of the minimum permeability of such a path. It also results in a significantly increased variance of this permeability. Conditioning on field permeabilities reduces the variance of this value over that obtained by unconditional, correlated simulation, but still produces a variance greater than that obtained when independence was assumed. When conditioning is performed, the mean of the minimum permeabilities along these paths is dependent on the principal axis of the path. Finally, including a random rock volume by allowing the length of the grid to be random increases the variance of the minimum permeability.