arXiv subjects

Robert C. Viesca

Publications and source records attributed to Robert C. Viesca.

6 recordsLinked to original sources

arXiv2026

Linear and nonlinear stability of rate-and-state faults

Models of faults incorporating slip rate- and state-dependent friction have reproduced phenomena from spontaneous slow, aseismic slip to earthquake-generating dynamic rupture. Numerical explorations of model parameter space regularly show sudden transitions in behavior. However these boundaries are poorly constrained analytically, with commonly used scalings derived assuming unrepresentative conditions of uniform sliding on an infinite, homogeneous fault. In this work, we demonstrate that an analysis of linear stability can reflect model conditions. We examine two scenarios that move beyond the classical case: an asperity driven by the steady creep of its surroundings, and a finite fault experiencing a constant rate of shear loading. We identify the critical fault dimension $L_c$ at which point linear stability is lost. Beyond this linear regime, the non-linear nature of the friction law implies the loss of memory of loading conditions as instability progresses and the existence of universal solutions describing this process. We refine prior analyses of this non-linear instability and find the minimum fault size that can support self-sustaining, unstable acceleration towards dynamic rupture. We examine the role of the state evolution law and delineate conditions under which faults may be linearly stable but non-linearly unstable, requiring finitely large perturbations to trigger instability. On the basis of numerical solutions, approximate but accurate algebraic expressions for the transition boundaries are presented. These results provide means for careful model design and to delimit plausible regions of parameter space when considering physical observations of stable creep, aseismic (slow slip) or seismic transients.

physics.geo-ph↗

arXiv2025

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

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.

physics.flu-dyn↗

arXiv2024

Non-linear stability analysis of slip in a single-degree-of-freedom elastic system with frictional evolution laws spanning aging to slip

We present a non-linear stability analysis of quasi-static slip in a spring-block model. The sliding interface is governed by rate- and state-dependent friction, with an intermediate state evolution law that spans between aging and slip laws using a dimensionless parameter ε. Our results extend and generalize previous findings of Gu et al. (1984) and Ranjith and Rice (1999) that considered slip and aging laws, respectively. We examine the robustness of these prior results to changes in the evolution law, including the finding of unconditional stability of the aging law for spring stiffnesses above a critical value. Our analysis provides analytical trajectories of slip motion in a phase plane as function of dimensionless governing parameters. We investigate two scenarios: a spring-block model with stationary and non-stationary point loading rate. When the loading point is stationary, we find that deviations from the aging law lead to only conditional stability of the slider for spring stiffnesses above a critical value: finite perturbations can trigger instability, consistent with prior results for the slip law. We quantify these critical perturbations as a function of the governing parameters. We find that, for a given supercritical stiffness, the size of the perturbation required to induce instability grows as the state evolution law approaches the aging law. In contrast, when the point loading rate is stationary, our results suggest that there exists a maximum critical stiffness above which an instability can never develop, for any perturbation size. This critical stiffness is ε-dependent and vanishes as the slip law is approached: conditional stability is then expected in the slip law limit. Finally, we derive relations for an effective spring stiffness as a function of the elastic moduli and a characteristic fault dimension or a characteristic perturbation wavelength.

physics.geo-ph↗

arXiv2023

Frictional state evolution laws and the non-linear nucleation of dynamic shear rupture

We assess if a characteristic length for a non-linear interfacial slip instability follows from theoretical descriptions of sliding friction. We examine friction laws and their coupling with the elasticity of bodies in contact and show that such a length does not always exist. We consider a range of descriptions for frictional strength and show that the area needed to support a slip instability is negligibly small for laws that are more faithful to experimental data. This questions whether a minimum earthquake size exists and shows that the nucleation phase of dynamic rupture contains discriminatory information on the nature of frictional strength evolution.

cond-mat.soft↗

arXiv2021

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 modeled 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 $λ$ of the diffusive lengthscale $\sqrt{αt}$. We identify two asymptotic regimes corresponding to $λ$ being small or large and perform a perturbation expansion in each limit. For large $λ$, in the regime of a so-called critically stressed fault, a boundary layer emerges on the diffusive lengthscale, 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 to accurate numerical solutions to the full problem, which are tabulated to high precision.

physics.flu-dyn↗

arXiv2018

The slow slip of viscous faults

We examine a simple mechanism for the spatio-temporal 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 non-linear, 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 time $t$ and a similarity coordinate $η=x/t$, where $x$ denotes fault position. The similarity coordinate shows a departure from classical diffusion and is owed to the non-local 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 $1/t$ while spreading proportionally to $t$, implying both a logarithmic accumulation of displacement as well as a constant moment rate. We discuss the implication for models of post-seismic slip as well as spontaneously emerging slow slip events.

physics.geo-ph↗