Euclidean and fractal models for the description of rock surface roughness

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Abstract
Accurate description of the topography of rock surfaces is essential because surface roughness affects frictional strength, the flow of fluids in joints and fractures, the seismic behavior of faults, and the formation of gouge and breccia in fault zones. Real rock surfaces can be described using self‐similar and self‐affine fractal models of surface roughness. If a surface is self‐similar, a small portion of the surface, when magnified isotropically, will appear statistically identical to the entire surface. If a surface is self‐affine, a magnified portion of the surface will only appear statistically identical to the entire surface if different magnifications are used for the directions parallel and perpendicular to the surface. At least two parameters are required to describe a fractal model; one parameter typically describes how roughness changes with scale, while the other specifies the variance or surface slope at a reference scale. The divider method and the spectral method are in common use to determine the best fit fractal model from surface profile data. Power spectra from self‐similar surfaces have slopes of −3 on log‐log plots of power spectral density versus spatial frequency, while spectra from self‐affine surfaces have slopes other than −3. Power spectra can be interpreted with greater facility if dimensionless amplitude to wavelength ratios are contoured on plots of power spectral density versus frequency. The topography of many natural rock surfaces, including both fractures and faults, is approximately self‐similar within the 6.5 order of magnitude wide wavelength band of 10 μm to 40 m. Within smaller wavelength bands, natural rock surfaces may exhibit self‐affine behavior.

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