Equivalent hydraulic conductivity, connectivity and percolation in 2D and 3D random binary media

Abstract

The equivalent hydraulic conductivity (Keq) relates the spatial averages of flux and head gradient in a block of heterogeneous media. In this article, we study the influence of connectivity on Keq of media samples composed of a high conductivity (k+) and a low conductivity (k−) facies. The k+ facies is characterized by a proportion p, and also by two connectivity parameters: a connectivity structure type (no, low, intermediate, high), and a correlation integral scale lc. The probability distribution of Keq, and the critical value of p at which percolation occurs (pav), are studied as a function of these connectivity parameters. The distribution of log(Keq) is Gaussian in all cases, so the results are presented in terms of the geometric mean (〈Keq〉) and the variance (σlog(Keq)2). Both quantities show a data collapse if expressed as a function of p−pav (for the variance σlog(Keq)2, notably, even if 2D and 3D data are plotted together). In 3D, when a connectivity structure exists, Keq is always greater than when no structure exists, and increases (while pav decreases) as lc increases. The same is observed in 2D, except for the low connectivity structure type (i.e. when the k+ facies is disconnected), that shows an unprecedented behaviour: Keq is greater in the absence of structure, and decreases (pav increases) as lc increases. Our results show that any influence of connectivity on Keq is well accounted for simply by a shift in the percolation threshold pav, and then, suggest that Keq is controlled mainly by the proximity to percolation.