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We examine the circular, self-similar expansion of frictional rupture due to fluid injected at a constant rate. Fluid migrates within a thin permeable layer parallel to and containing the fault plane. When the Poisson ratio$$\nu =0$$, self-similarity of the fluid pressure implies fault slip also evolves in an axisymmetric, self-similar manner, reducing the three-dimensional problem for the evolution of fault slip to a single self-similar dimension. The rupture radius grows as$$\lambda \sqrt {4\alpha _{hy} t}$$, where$$t$$is time since the start of injection and$$\alpha _{hy}$$is the hydraulic diffusivity of the pore fluid pressure. The prefactor$$\lambda$$is determined by a single parameter,$$T$$, which depends on the pre-injection stress state and injection conditions. The prefactor has the range$$0\lt \lambda \lt \infty$$, the lower and upper limits of which correspond to marginal pressurisation of the fault and critically stressed conditions, in which the fault-resolved shear stress is close to the pre-injection fault strength. In both limits, we derive solutions for slip by perturbation expansion, to arbitrary order. In the marginally pressurised limit ($$\lambda \rightarrow 0$$), the perturbation is regular and the series expansion is convergent. For the critically stressed limit ($$\lambda \rightarrow \infty$$), the perturbation is singular, contains a boundary layer and an outer solution, and the series is divergent. In this case, we provide a composite solution with uniform convergence over the entire rupture using a matched asymptotic expansion. We provide error estimates of the asymptotic expansions in both limits and demonstrate optimal truncation of the singular perturbation in the critically stressed limit.
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The Slow Slip of Viscous Faults
Abstract We examine a simple mechanism for the spatiotemporal evolution of transient, slow slip. We consider the problem of slip on a fault that lies within an elastic continuum and whose strength is proportional to sliding rate. This rate dependence may correspond to a viscously deforming shear zone or the linearization of a nonlinear, rate‐dependent fault strength. We examine the response of such a fault to external forcing, such as local increases in shear stress or pore fluid pressure. We show that the slip and slip rate are governed by a type of diffusion equation, the solution of which is found using a Green's function approach. We derive the long‐time, self‐similar asymptotic expansion for slip or slip rate, which depend on both timetand a similarity coordinateη = x/t, wherexdenotes fault position. The similarity coordinate shows a departure from classical diffusion and is owed to the nonlocal nature of elastic interaction among points on an interface between elastic half‐spaces. We demonstrate the solution and asymptotic analysis of several example problems. Following sudden impositions of loading, we show that slip rate ultimately decays as 1/twhile spreading proportionally tot, implying both a logarithmic accumulation of displacement and a constant moment rate. We discuss the implication for models of postseismic slip as well as spontaneously emerging slow slip events.
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- Award ID(s):
- 1653382
- PAR ID:
- 10371351
- Publisher / Repository:
- DOI PREFIX: 10.1029
- Date Published:
- Journal Name:
- Journal of Geophysical Research: Solid Earth
- Volume:
- 124
- Issue:
- 5
- ISSN:
- 2169-9313
- Page Range / eLocation ID:
- p. 4959-4983
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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