Distribution of hydraulic conductivity in single scale anisotropy

Abstract

The cluster statistics of percolation theory are used to find the distributions of hydraulic conductivity, K, of anisotropic (truncated) random fractal media. Rescaling of variables to transform anisotropic to isotropic media also produces deformations of, for example experimental volumes, and the resulting non-equidimensional shapes may generate interesting size effects on K. Previously, the most likely value of K was obtained by comparing the correlation length from percolation theory with system dimensions, a procedure analogous to those developed for hopping conduction in disordered systems to calculate the longitudinal conductivity of thin films. The result probably explains the frequent tendencies of measurements of K in anisotropic fracture networks and agricultural soils to increase with the scale of measurement, similarly to how the longitudinal conductivity of a thick film would be larger than the corresponding conductivity of a thin film (three- rather than two-dimensional conduction). However, the same procedure applied to the conductivity in the perpendicular direction (analogous to the transverse electrical conductivity of a thin film) shows a diminishing function of spatial scale. Collectively, these ‘scale effects’ disappear if the shape of the experimental volume is selected to maintain the relationships of conduction in the various directions as the scale of the experiment is increased analogously to equidimensional volumes in isotropic media. The increase in K is, thus, merely due to an increase in the dimensionality of conduction from one to three with increasing system size. The paper, thus, provides a solid argument against a common assumption in the porous media communities that the connectivity of highly conducting regions of a medium should increase with increasing scale of measurement.

Acknowledgments

Thanks are due Dr Toby Ewing for constructing figure 1. This research was supported in part through DOE grant, DE-FG02-05ER64067.