Flow in rock fractures: The local cubic law assumption reexamined

We investigate the validity of applying the “local cubic law” (LCL) to flow in a fracture bounded by impermeable rock surfaces. A two-dimensional order-of-magnitude analysis of the Navier-Stokes equations yields three conditions for the applicability of LCL flow, as a leading-order approximation in a local fracture segment with parallel or nonparallel walls. These conditions demonstrate that the “cubic law” aperture should not be measured on a point-by-point basis but rather as an average over a certain length. Extending to the third dimension, in addition to defining apertures over segment lengths, we find that the geometry of the contact regions influences flow paths more significantly than might be expected from consideration of only the nominal area fraction of these contacts. Moreover, this latter effect is enhanced by the presence of non-LCL regions around these contacts. While contact ratios of 0.1–0.2 are usually assumed to have a negligible effect, our calculations suggest that contact ratios as low as 0.03–0.05 can be significant. Analysis of computer-generated fractures with self-affine walls demonstrates a nonlinear increase in contact area and a faster-than-cubic decrease in the overall hydraulic conductivity, with decreasing fracture aperture; these results are in accordance with existing experimental data on flow in fractures. Finally, our analysis of fractures with self-affine walls indicates that the aperture distribution is not lognormal or gamma as is frequently assumed but rather truncated-normal initially and increasingly skewed with fracture closure.