Influence of fractal flaw distributions on rock deformation in the brittle field

Abstract The geometrical distribution of flaws plays a crucial role in the physical behaviour of geological materials under stress. Flaws are present in the earth on all scales, from microcracks to plate-rupturing faults. They may be distributed on one characteristic length scale (e.g. joints, ‘characteristic’ earthquakes), or more commonly exhibit scale-invariance over a specified range of sizes. Scale-invariance implies that the discrete length distribution in a finite range is a power law of negative exponent D , where 1 ≤ D < 3. Fault systems where motion is concentrated on a dominant fault (e.g. San Andreas) have D ≈ 1, but more diffuse fault systems have D near 2. D is one of the fractal dimensions of the fracture system. The length distribution of faults or microcracks may be inferred from the slope b of the log-linear frequency—magnitude distribution of earthquakes, or laboratory-scale acoustic emissions, since it can be shown that D = 3 b/c . The scaling factor c depends on the relative time constants of the seismic event and the recording instrument, and is usually equal to 3/2. b is found experimentally to be negatively correlated with the stress intensity on the dominant flaw, which depends in turn on the applied stress and the flaw length. Thus a fracture mechanics model of rock failure which includes a range of flaw sizes can be tested by seismic monitoring. We describe a fracture mechanics model of rock failure for a variety of styles of deformation, ranging from elastic failure to quasi-static cataclastic flow, and predict the time-dependence of D and the seismic b -value at different times up to and including failure. Critical coalescence of microcracks during dynamic failure (e.g. earthquake foreshocks) occurs when D = 1 ( b = 0.5); random processes (e.g. cataclastic flow, background seismicity) are associated with D = 2 ( b = 1); positive feedback in the concentration of stress on the dominant flaw (e.g. during strain softening and shear localisation) occurs when D < 2 ( b < 1); negative feedback in stress concentration (e.g. during the early stages of dilatancy), and where a highly diffuse fracture system is produced, occurs at low stress intensities and is associated with D > 2 ( b > 1). It has long been a goal of structural geologists to measure stress on rocks, since most geometrical signatures of deformation are strain-related. We show that stress is not usually as significant in rock fracture as stress intensity, and furthermore that the geometric signature of the length distribution of microcracks is well-correlated with the stress intensity.