Fractal analyses and geometrical models of fracture surfaces in rock.

Abstract

Fracture propagation in rock often produces complex patterns, such as the branching patterns of fracture networks, or the irregular radiating patterns on fracture surfaces. These patterns often appear complex because they are unpredictable in detail, yet predictable in the sense that smaller pieces of the pattern, when suitably enlarged, are statistically similar to larger pieces of the pattern. This property of statistical self-similarity can be quantified using fractal geometry. The fractal nature of rock fracture patterns is related to lithological properties and to the dynamics of the fracturing processes. Joint surfaces in homogeneous rocks display rough radiating ridges. A "plumose joint" surface was analyzed using the slit island method and found to have a fractal dimension (D$\sb{\rm f}$) of 2.24 $\pm$.14 (95%). Surfaces with similar fractal dimensions (2.2-2.5) are produced by a three-dimensional computer simulation of jointing. In the simulation, randomly-distributed defects cause local mis-orientations of the stress field and local deflection of the propagating fracture front. After passing through the defect the joint surface is re-oriented relative to remote stresses, and a planar radial fracture segment (i.e. inclusion hackle) is formed. Collectively, the numerous inclusion hackle form the plumose surface pattern. For the simulation results, D$\sb{\rm f}$ increases (i.e. the surface gets rougher) in proportion to the log of the defect density. The simulation also demonstrates a complex relationship between D$\sb{\rm f}$ of the propagating fracture front and D$\sb{\rm f}$ of the fracture surface. Shatter cones are conical fracture surfaces produced during high energy events such as meteorite impact and nuclear explosion. These fractures also display radiating surface features. Using a modified version of the slit island method, the fractal dimension of a shatter cone surface in limestone is estimated to be 2.24 $\pm$ 0.09. The observation of shingled convex fracture surfaces within the conical envelope surrounding the shatter cone surface is demonstrated to support the genetic model of Johnson and Talbot (1964). Striations on these fracture surfaces are reinterpreted as micro-fracture intersections. The measured fractal dimensions of the joint and shatter cone surfaces (i.e 2.24) are within the range reported for most fracture surfaces in metals (i.e. 2.1 $\sim$ 2.3).