An examination of scale-dependent electrical resistivity measurements in Oracle granite.

Abstract

Geotechnical characterization of crystalline rock is often dependent upon the influence of the rock's fracture system. To test ensemble fracture behavior in situ, a series of cross-hole and single-hole electrical conductivity measurements were made within saturated Oracle granite. The tests were conducted with a point source and a point reference electrode and employed electrode separations ranging from 8 inches to over 100 feet. A volume of rock approximately 50 x 50 x 150 feet was tested (as bounded by the vertical test borings). Analysis of the data in terms of an equivalent homogeneous material showed that the effective electrical conductivity increased with electrode separation. The cross-hole data indicate that the rock can be treated as a non-homogeneous, isotropic material. Further, the spatial variation of measured conductivities along a line can be fit to a fractal model (fractional Brownian motion), implying that the scale-dependence is a result of a fractal process supported by the fracture system. Scale-dependence exists at the upper scale limit of the measurements, hence a classical representative elemental volume was not attained. Two directions were taken to understand the scale-dependence. The rock mass is treated in terms of a disordered material, a continuum with spatially varying conductivity. First, a percolation-based model of a disordered material was examined that relates the conductivity pathways within the rock to the backbone of a critical percolation cluster. Using the field data, a fractal dimension of 2.40 was derived for the dimensionality of the subvolume within the rock that supports current flow. The second approach considers an analytic solution for a non-homogeneous, isotropic material known as the alpha center model (Stefanescu, 1950). This model, an analytic solution for a continuously varying conductivity in three dimensions, is a non-linear transform to Laplace's equation. It is employed over a regular grid of support points as an alternative to spatially discretized (piece-wise continuous) numerical methods. The model is shown to be capable of approximating the scale-dependent behavior of the field tests. Scaling arises as a natural consequence of the disordered electrical structure caused by the fracture system.