Asymptotic solutions for self-similarly expanding fault slip induced by fluid injection at constant rate
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916953587712000 |
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| author | Viesca, Robert C. |
| author_facet | Viesca, Robert C. |
| contents | 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 $ν=0$, 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 $λ\sqrt{4α_{hy} t}$, where $t$ is time since the start of injection and $α_{hy}$ is the hydraulic diffusivity of the pore fluid pressure. The prefactor $λ$ is determined by a single parameter, $T$, which depends on the pre-injection stress state and injection conditions. The prefactor has the range $0<λ<\infty$, the lower and upper limits of which correspond to marginal pressurization 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 pressurized limit ($λ\rightarrow 0$), the perturbation is regular and the series expansion is convergent. For the critically stressed limit ($λ\rightarrow \infty$), 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_13828 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic solutions for self-similarly expanding fault slip induced by fluid injection at constant rate Viesca, Robert C. Fluid Dynamics Geophysics 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 $ν=0$, 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 $λ\sqrt{4α_{hy} t}$, where $t$ is time since the start of injection and $α_{hy}$ is the hydraulic diffusivity of the pore fluid pressure. The prefactor $λ$ is determined by a single parameter, $T$, which depends on the pre-injection stress state and injection conditions. The prefactor has the range $0<λ<\infty$, the lower and upper limits of which correspond to marginal pressurization 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 pressurized limit ($λ\rightarrow 0$), the perturbation is regular and the series expansion is convergent. For the critically stressed limit ($λ\rightarrow \infty$), 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. |
| title | Asymptotic solutions for self-similarly expanding fault slip induced by fluid injection at constant rate |
| topic | Fluid Dynamics Geophysics |
| url | https://arxiv.org/abs/2401.13828 |