Effective friction law for small-scale fault heterogeneity in 3D dynamic rupture

We address the problem of modeling dynamic rupture on multiscale heterogeneous faults in 3D. Under the assumption of slip-weakening friction, we numerically construct effective friction laws that integrate the effects of small-scale heterogeneity during the rupture. This homogenization process is based on the description of the initial phase of the rupture by the dominant unstable spectral mode. Its dynamics is influenced by the geometry of the fault, the static friction heterogeneities and the friction law. We first define a periodic small-scale heterogeneous model, introducing heterogeneity in the distribution of the static friction coefficient. We then describe a method for constructing this effective friction law. Applying this new law homogeneously on the fault permits to reproduce the dynamic evolution of the heterogeneous fault. Furthermore, we show that the effective friction law can be used to replace small-scale heterogeneities in two-scale heterogeneous models, while preserving their effects. We study three kinds of two-scale models, with growing complexity: first periodic at both scales, then periodic only at small scale, and finally irregular at both scales. This homogenization method can be adapted to the case where the heterogeneity is introduced in the initial stress rather than in the static friction value. Finally, we show in a simple example that the effective friction law permits to reproduce the transition between subshear and supershear rupture propagation, originally produced by heterogeneities on the fault.