The Imaginary Part of Rock Jointing

Abstract

The distribution of joint spacings in a granitic massive in Saudi Arabia is found to be well-described by a power-law with characteristic exponent μ ≃ 0.5. We compare the cumulative and density distributions and show how to correct the cumulative distribution for bias due to the finite sampling size. The exponent μ is close to those obtained for size distribution in fragmentation processes. We study simple models of fragmentation/jointing processes, which predict that the power law distribution must be decorated by a log-periodic modulation if the fragmentation involves a preferred ratio (even approximately so, i.e. with disorder) corresponding to an approximate discrete scale invariance. We corroborate this prediction by carrying out a more detailed analysis of the density distribution and find at least 6 log-periodic oscillations. This implies that the exponent μ possesses an imaginary part, embodying the existence of an average discrete scaling structure with preferred fragmentation ratio close to 1/2. The confidence level of this result is found better than 97 % from synthetic tests.