The fractal dimension as a measure of the roughness of rock discontinuity profiles

Abstract

An important factor in the analysis of deformation and failure of rock masses along discontinuities is the roughness of the jointing surfaces. The fractal dimension is proposed as a method of objectively quantifying the roughness profile of such discontinuities. With this method much less engineering judgement and experience are required in determining the surface roughness,sb than is required for other methods such as the widely-used joint roughness coefficient (JRC) suggested by the International Society for Rock Mechanics [1] ( Int. J. Rock. Mech. Min. Sci. & Geomech. Abstr. 15, 319–368, 1978). The fractal dimension, suggested by Mandelbrot [2] ( Fractals—Form, Chance and Dimension, Freeman, San Francisco), describes the degree of variation of a curve, a surface or a volume has from its topological ideal. In this application, the length of the surface profile is measured stepwise along the curve with rulers of different lengths. From these data, the fractal dimension of the surface profile can be determined. An empirical equation was found relating the fractal dimension to the JRC value. Application of the fractal dimension to three independent studies (field and laboratory) all showed excellent agreement between the surface roughness and the fractal dimension, the rougher the surface, the higher the fractal dimension. The peak shear stress on large-block laboratory direct shear tests increased as the fractal dimension of the shear surface increased, which again verifies the usefulness of the concept.