Transmissivity and conductivity of single fractures

A fracture can be seen as a void space between two rough surfaces in partial contact. Transmissivity and conductivity can be determined numerically by solving the Stokes and Laplace equations between these two surfaces. These problems were first solved and published by the same authors between 1995 and 2001. Updated, more complete and precise results are presented here. Each surface of a fracture can be schematized as a random surface oscillating around an average plane, characterized by the probability density and autocorrelation function C(u) of the heights; their standard deviation is the roughness sigma. The two surfaces are separated by a mean distance b_m and their heights are correlated with an intercorrelation coefficient theta. The two major classes for C(u), namely the Gaussian and the self-affine autocorrelations with Hurst exponent H, are both characterized by a length scale l_c, which is a typical scale for the surface features in the Gaussian case and a cut-off length in the self-affine case. Gaussian surfaces are statistically homogeneous while the mean properties in the self-affine case are size-dependent. Systematic calculations were performed for these two classes, with recent emphasis put on the Gaussian fractures. The results are modeled as functions of b_m/sigma, l_c/sigma and theta. Cubic law applies for large b_m/sigma in terms of the aperture reduced by the hydraulic thickness of the surface rugosity. Another cubic law applies in the opposite limit of tight fractures with an offset depending on theta and a prefactor which depends on theta and l_c. A transition takes place between these two regimes. It is also shown that the Reynolds approximation may overestimate the true transmissivity by almost an order of magnitude. Similar calculations were performed for conductivity. The whole work is summarized by a series of master curves and models which can be used to estimate the properties of real fractures.