Crustal earthquake instability in relation to the depth variation of frictional slip properties

Recent stability studies using constitutive relations of the type found by Dieterich, Ruina, and others to describe frictional slip of rocks in the laboratory have provided a new explanation of the depth cutoff of shallow crustal earthquakes. The class of friction laws discussed has the property that the sliding stress depends on normal stress, temperature, slip rate, and slip history. For sliding at a fixed slip rate V and fixed environment (e.g., normal stress, temperature, etc.) the shear strength τ evolves toward a steady state value τ ss ( V ). Stability analysis show that for d τ ss ( V )/ dV < 0 (i.e., velocity weakening), steady state sliding is unstable to any perturbation in systems of sufficiently low stiffness and is unstable to sufficiently large perturbations in systems of higher stiffness. Hence a surface with d τ ss ( V )/ dV < 0 is potentially unstable. Conversely, d τ ss ( V )/ dV > 0 (velocity strengthening) implies stable steady state sliding, at least for a broad class of constitutive laws more fully described in the paper. Experiments by Dieterich and by Tullis and Weeks on Westerly granite with mature sliding surfaces indicate that d τ ss ( V )/ dV is negative at room temperature, whereas higher‐temperature experiments by Stesky show that d τ ss ( V )/ dV becomes positive above approximately 300°C. Therefore, in the case of the earth, where temperature increases with depth, the above observations seem to suggest that the depth cutoff of crustal earthquake activity can be understood in terms of the variation of the frictional response with depth, from a regime with d τ ss ( V )/ dV < 0 to one with d τ ss ( V )/ dV > 0. This is not inconsistent with, but rather refines, the suggestion by Sibson and others that the depth cutoff is due to a transition from brittle friction to ductile flow. Further, our results show definitively that the shallow depth confinement of seismicity is compatible with a model in which deformation is localized to a fault zone extending well below the seismogenic depth. The depth confinement does not require a model showing transition to a zone of broadly distributed creep flow beneath the seismogenic zone. Following Mavko, a two‐dimensional quasi‐static strike‐slip fault model is analyzed but using different numerical procedures and a depth variation of frictional properties based on the laboratory data mentioned above and the Lachenbruch‐Sass depth variation of temperature for the San Andreas fault. The resulting predictions of such features as the confinement of crustal earthquakes to shallow depths, the development of locked patches, the recurrence time for the seismic cycle, the seismic stress drop and displacement, etc., are generally in agreement with the observed characteristics of large‐scale strike‐slip earthquakes along the San Andreas fault. Calculations based on the laboratory data predict, for example, that stick‐slip nucleates around 5–7 km, that large seismic motion occurs around the nucleation depth and above but diminishes gradually to zero at 13–15 km depths, that rapid postseismic creep occurs over another 3–4 km depth, and that at greater depths, steady slip consistent with the average plate velocity is only modestly perturbed by the earthquakes occurring above. The most uncertain parameter is the slip weakening distance L for evolution of fault surface state; we illustrate how features of the predicted earthquake cycles vary with L over the range for which calculations are feasible.