On the effective friction law of a heterogeneous fault

Numerical simulation of the rupture process is usually performed under an assumption of scale invariance of the friction process although heterogeneous fault properties are shown by both direct observations of surface crack geometry and slip inversion results. We investigate if it is possible to define an effective friction law for a finite fault with a small-scale heterogeneity, that is, with a distribution of narrow segments with a resistance to rupture higher than the rest of the fault. We consider a model where the local boundary condition corresponds to a linear slip-dependent friction law. We define the effective slip-dependent friction law by analogy with the theoretical spectral solution for the initiation phase in the case of a homogeneous infinite fault. We use finite difference simulations to test the validity of this approach. The results show a surprisingly good agreement between the calculations for the complete heterogeneous fault model and for a homogeneous fault with an effective friction law. The time of initiation and the average of the slip velocity on the fault are well predicted by the effective model. The effective friction law exhibits a nonlinear slip dependence with an initial weakening rate different from the one of the local laws. This initial weakening rate is related to the geometry of the heterogeneity and can be obtained by an eigenvalue analysis. The effective law shows a kink at a slip that corresponds to the average slip on the fault for which the stress concentration of the strong segments is sufficient to trigger their rupture. While based on a rather simple model of a fault, these results indicate that an effective friction can be defined and used for practical purposes. The heterogeneity of a fault tends to decrease the initial weakening rate of the local weak patches. Since the initial weakening rate controls the initiation duration, this last point indicates that the duration of initiation expected from actual heterogeneous faults is much larger than the one deduced from small-scale laboratory measurements. The actual fracture energy is not conservative in the rescaling of the friction law.