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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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Self-similar fault slip in response to fluid injection
There is scientific and industrial interest in understanding how geologic faults respond to transient sources of fluid. Natural and artificial sources can elevate pore fluid pressure on the fault frictional interface, which may induce slip. We consider a simple boundary value problem to provide an elementary model of the physical process and to provide a benchmark for numerical solution procedures. We examine the slip of a fault that is an interface of two elastic half-spaces. Injection is modelled as a line source at constant pressure and fluid pressure is assumed to diffuse along the interface. The resulting problem is an integro-differential equation governing fault slip, which has a single dimensionless parameter. The expansion of slip is self-similar and the rupture front propagates at a factor $$\lambda$$ of the diffusive length scale $$\sqrt {\alpha t}$$ . We identify two asymptotic regimes corresponding to $$\lambda$$ being small or large and perform a perturbation expansion in each limit. For large $$\lambda$$ , in the regime of a so-called critically stressed fault, a boundary layer emerges on the diffusive length scale, which lags far behind the rupture front. We demonstrate higher-order matched asymptotics for the integro-differential equation, and in doing so, we derive a multipole expansion to capture successive orders of influence on the outer problem for fault slip for a driving force that is small relative to the crack dimensions. Asymptotic expansions are compared with accurate numerical solutions to the full problem, which are tabulated to high precision.
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- Award ID(s):
- 1653382
- PAR ID:
- 10402050
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 928
- ISSN:
- 0022-1120
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/jfm.2021.825
