Are fault growth and linkage models consistent with power-law distributions of fault lengths?

It has recently been recognized that fault lengths (L) in natural populations follow power-law scaling. Such power-law scaling is observed in a wide range of tectonic settings in regions that have experienced differing amounts of total strain, and exhibit faults over a very large range of dimensions. In this paper we explore possible constraints On fault growth and linkage required to maintain power-law length scaling during progressive deformation. We first consider a fault growth model in which individual faults in a population grow by an amount Delta L proportional to L(F) during slip increments (earthquakes), which have a recurrence interval tau proportional to L(E). If an initial power-law length distribution is assumed for the population, it is found that the growth model exponents must be related by F - E = 1 in order to continually maintain the same scaling. If the requirement of constant moment release rate through time is also imposed, this implies that for large faults E = 2, which leads to a loss of power-law scaling with increasing strain, unless F = 3. Current mechanical models for growth of single faults by tip propagation propose E greater than or equal to 1 and F = 1. Thus single-fault models are not consistent with observed power-law scaling. In a second model, fault lengths increase by growth specified by the first model, unless a nearby fault is encountered, in which case the two faults link. With this model, it is possible to produce a power-law distribution from a fault or flaw population that initially does not have a power-law distribution. Once a power-law distribution is developed, fault linkage causes the power-law exponent (C) to decrease as fault strain increases.