The evolution of earthquake-nucleating slip instabilities under spatially variable steady-state rate dependence of friction

Abstract:

Following laboratory rock friction experiments, fault strength under sub-seismic slip speeds is thought to depend on a slip rate- and state-dependent friction. Laboratory-measured temperature dependence of the frictional properties and their implied variation with depth form the basis for current models of the seismic cycle. However, scant attention has been paid to the role such heterogeneity has on determining the location and manner in which an earthquake nucleating slip instability develops. Recent work demonstrates that a slip instability on a fault with rate-and-state friction (in which state evolution follows the aging law) occurs as the attraction of a dynamical system towards a fixed point (Viesca, this meeting). Based on this development, we find that the location of that fixed point may be determined if a heterogeneous distribution of the relative rate-weakening parameter a/b is known. (Rate-weakening occurs for 0<a/b<1, and rate-strengthening for a/b>1). That this arises can be deduced considering that (i) the problem that determines the fixed points is equivalent to finding the equilibrium solution for a linearly slip-weakening crack, and (ii) heterogeneities in the parameter a/b have analogy in the equivalent problem to heterogeneities in the background stress. Physically, instability develops where rate-weakening is strongest. We examined the influence such a heterogeneity has on the fixed point attractor (and hence on the instability development) by considering the scenario of a rate-weakening patch embedded within a rate-strengthening region with in-plane or anti-plane slip conditions. Specifically, we solve for fixed points under a rate-weakening heterogeneity within |x|<H of the simple form a(x)/b = (a/b)_m + (1-(a/b)_m)*|x|/H and rate-strengthening behavior (a/b>1) outside. Additionally, a linear stability analysis reveals the effect of heterogeneity on the stability of the fixed points of the dynamical system. The heterogeneity parameters (a/b)_m and H enter as bifurcation parameters indicating a transition in the classification of the fixed point from asymptotically stable to unstable at critical values of (a/b)_m and H. The results are further verified by full numerical simulation of the system of slip acceleration and state evolution under this heterogeneity.