Asymptotic solutions for self-similarly expanding fault slip induced by fluid injection at constant rate

TL;DR

This paper derives asymptotic solutions for the self-similar expansion of fault slip caused by fluid injection, revealing how rupture growth depends on initial stress conditions and fluid diffusivity.

Contribution

It introduces a unified asymptotic framework for fault slip evolution under fluid injection, including both marginal and critically stressed limits, with detailed perturbation and boundary layer analysis.

Findings

Rupture radius grows as rac{rac{4\u00a0",

Series expansions converge in marginal pressurization but diverge in critically stressed conditions, requiring composite solutions.

Error estimates and optimal truncation strategies are provided for the asymptotic solutions.

Abstract

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 $0<\lambda<\infty$, 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 ($\lambda\rightarrow 0$), 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.