Interpreting two-dimensional cuts through broken geologic objects: Fractal and non-fractal size distributions

Abstract

Numerical simulations and physical fragmentation experiments confirm the theoretical prediction that the fractal dimension of a two-dimensional (2-D) cut through a set of three-dimensional objects with fractal dimension D is approximately equal to D– 1. This leads to a size distribution in two-dimensional cuts that is skewed strongly toward larger objects compared to the three-dimensional distribution. Three-dimensional shape (aspect ratio) does not significantly affect the resulting 2-D size distribution except for highly nonequant objects, such as prolate ellipsoids with aspect ratios of 10 or more. In contrast, fragmentation of an object by breakage along persistent fractures results in a non-fractal distribution of sizes and far fewer small objects than predicted by fractal statistics. Powdering a rock by extensive crushing also results in non-fractal size distributions because particles are reduced to sizes on the order of 1 μm, a comminution limit below which further brittle fracture is difficult. Natural examples of fragmental objects observed in two-dimensional cuts, such as crushed rocks, breccias, and xenoliths, are generally consistent with a three-dimensional fractal dimension near 2.5 over one or two orders of magnitude in size. However, a limestone breccia from Death Valley exhibits a non-fractal size distribution consistent with fragmentation of a strongly jointed rock. Mafic enclaves in Yosemite National Park have a restricted size range of about one order of magnitude and a three-dimensional fractal dimension of ∼3.1, consistent with other enclave swarms. The restricted size range of enclaves may reflect the apertures of mafic dikes that fed them.