Earthquake nucleation on (aging) rate and state faults

We obtain quasi-static, two-dimensional solutions for earthquake nucleation on faults obeying Dieterich's “aging” version of the rate and state friction equations. Two distinct nucleation regimes are found, separated by roughly a/b ∼ 0.5, where a and b are the constitutive parameters relating changes in slip rate V and state θ to frictional strength. When fault healing is unimportant (Vθ/D_c ≫ 1, where D_c is the characteristic slip distance for the evolution of θ), the nucleation zone spontaneously evolves toward a state of accelerating slip on a patch of fixed half length L_ν ≈ 1.3774(μ′D_c /bσ), where μ′ is the intrinsic stiffness of the medium and σ is the normal stress. This is the fixed length solution for which the stress intensity factor K = 0. Although this solution does not depend upon a/b explicitly, only for a/b < 0.3781 does healing remain unimportant as instability is approached. For a/b ≳ 0.5 and a wide range of slow loading conditions, Vθ/D_c ultimately approaches a quasi-constant value near 1, and the nucleation zone takes on the appearance of an expanding slip-weakening crack. A fracture energy balance indicates that in this regime the nucleation length asymptotically approaches π−1[b/(b − a)]2(μ′D_c /bσ), a result that is consistent with the numerical simulations despite considerable complexity asa approaches b. This suggests that nucleation lengths can sometimes be much larger than those found by Dieterich (e.g., by a factor of 100 for a/b = 0.95). For surfaces this close to velocity neutral, nucleation might produce signals detectable by surface seismometers for values of D_c at the upper end of the lab range (100 μm). However, the attributes of the aging law that give rise to such large nucleation lengths may be nonphysical; additional laboratory experiments are needed to address this issue