A study of a complex system: Development of crack populations

Using a model of populations of cracks we construct new solutions for the study of the spatiotemporal evolution of normal faults. The physics incorporated in the model are slip-weakening friction, strain-hardening rheology, and heterogeneous yield strength. We investigate how the populations evolution depends on these three effects. We asymptotically approximate the growth of faults as a long time-scale phenomenon, thereby avoiding modeling the short time-scale earthquakes and show that this assumption is valid. This implies that faults that creep and faults with earthquakes have indistinguishable spatiotemporal evolutions. We find that the distribution of lengths gradually transitions from a power law at low strains to an exponential at high strains. At low strain the self-similarity of the cracks results in a power law distribution as is typical of self-organized critical systems. As brittle strain increases, stress shadowing increases which reduces the crack nucleation rate. Crack coalescence rate, conversely increases with brittle strain, leading to the annihilation of smaller cracks in the production of fewer larger cracks. Both of these processes progressively starve the population of small cracks and bloat it with intermediate size cracks, leading the initial power law distribution to be distorted until it eventually takes the appearance of an exponential distribution. This agrees with natural fault populations observed in low and high strain settings. We perform clay extension laboratory experiments to test the model and find very good agreement. These results indicate that actual fault populations can range from power law to exponential size-frequency distributions. We then consider the effects of fault interactions and boundary conditions on the vertical structure of the faults by comparing the model with field data observations of geological faults. Their complicated vertical structure is the accrued result of stress field interactions. We also find that the propagation velocity of faults decreases with length as the fault population reaches the saturation regime. Our results suggest that the complex three-dimensional organization of faults is due to the time evolving fault interactions.