Parallel fault systems with evolving self-similar sliding zones

Catastrophic fault sliding is preceded by the development of sliding zones which grow further driven by the excess of the shear stress over friction at the loci of initiation. This growth is strongly affected by the interaction between the sliding zones. We propose a model of development of such zones based on two major simplifications. Firstly, each sliding zone is modelled as a disc-like shear crack driven by a pair of concentrated forces representing the excess of the shear stress over friction at the loci of initiation. Secondly, the interaction between these cracks is modelled based on the assumption that the distribution of their sizes is self-similar and the self-similarity is maintained in the process of their growth. We show that for parallel cracks the latter is only possible if the sliding zones are localised in a narrow layer. In this case the exponent and the prefactor of the distribution function are uniquely determined. The addition of new sliding zones does not change the distribution but rather increases the upper cut-off. This happens either by instantaneous growth of each added sliding zone to the maximum size producing the strongest microseismic event or by initiating a cascade of intermediate growth producing a series of smaller events. We determine the energy distribution associated with the cascade and the probability of hazardous events. We show that measuring the statistical properties of seismic energy alone is not sufficient for determining the parameters of the model; monitoring of fault deformation is also needed.