We examine the development of an instability of fault slip rate. We consider a slip rate and state dependence of fault frictional strength, in which frictional properties and normal stress are functions of position. We pose the problem for a slip rate distribution that diverges quasi‐statically within finite time in a self‐similar fashion. Scenarios of property variations are considered and the corresponding self‐similar solutions found. We focus on variations of coefficients, a and b, respectively, controlling the magnitude of a direct effect on strength due to instantaneous changes in slip rate and of strength evolution due to changes in a state variable. These results readily extend to variations in fault‐normal stress, σ, or the characteristic slip distance for state evolution, Dc. We find that heterogeneous properties lead to a finite number of self‐similar solutions, located about critical points of the distributions: maxima, minima, and between them. We examine the stability of these solutions and find that only a subset is asymptotically stable, occurring at just one of the critical point types. Such stability implies that during instability development, slip rate and state evolution can be attracted to develop in the manner of the self‐similar solution, which is also confirmed by solutions to initial value problems for slip rate and state. A quasi‐static slip rate divergence is ultimately limited by inertia, leading to the nucleation of an outward expanding dynamic rupture: asymptotic stability of self‐similar solutions then implies preferential sites for earthquake nucleation, which are determined by distribution of frictional properties.
SUMMARY The frictional properties of large faults are expected to vary in space. However, fault models often assume that properties are homogeneous, or nearly so. We investigate the conditions under which the details of variations may be neglected and properties homogenized. We do so by examining the behaviour of nonlinear solutions for unstably accelerating fault slip under frictional heterogeneity. We consider a rate- and state-dependent fault friction in which the characteristic wavelength for the property variations is a problem parameter. We find that homogenization is permissible only when that wavelength shows scale separation from an elasto-frictional length scale. However, fault models also often include property transitions that occur over distances comparable to the elasto-frictional length. We show that under such comparable variations, the dynamics of earthquake-nucleating instabilities is controlled by the properties’ spatial distribution.
<p>We model mechanics of an aseismic fault creep propagation and conditions when it may lead to the initiation of seismic slip. We do so by considering fault bounding medium to be elastically deformable and fault's interfacial strength to be slip rate- and state-dependent characterized by the steady-state rate-weakening. The fault is considered to be initially locked: a state of slip when interfacial slip velocity is considerably low and arbitrarily less than the steady-state sliding rate for given uniformly distributed prestress.</p><p>We find solutions for creep penetration into the fault under geologically relevant loading scenarios (e.g., that of a plate-bounding strike-slip faulting driven by the slip at depth, or that of a rate-weakening patch of a fault loaded by a creep on an adjacent rate-strengthening part due to, e.g., anthropogenic fluid injection). In all the cases, the creep makes its way as a self-similar traveling front characterized by high stress owed to the direct effect; however, the remaining creep profile exhibits a near steady-state sliding. This may imply that a choice from a set of rules for the evolution of state variable&#8212;with identical linearizations about steady-state sliding&#8212;has no bearing on the creep penetration. Further, we find that the prestress, close to or far from steady-state sliding stress, controls the rate and manner of the creep penetration.</p><p>We study slip propagation from an imposed dislocation accrued at a constant rate at one end of a homogeneous fault with the other end either at (1) the free surface of an elastic half-space or (2) strictly locked (buried) in the elastic full space. In both scenarios, no slip instability takes place over aseismic creep propagation distances relatable to the usual elasto-frictional nucleation lengthscale (e.g. Rubin & Ampuero 2005). Instead, in the first case creep propagation leads to the nucleation of the first and all subsequent dynamic events of the emerging cycle at/near the free surface after the creep traversed the entire length of the fault. In the second case, the creep front traverses nearly the entire length of the fault, but, instead of nucleating a dynamic event, the front arrests at some distance from the buried fault end, followed by the continual accumulation of aseismic slip without ever nucleating a dynamic event. These results may be owed to the physical and geometrical invariance of the considered homogeneous fault and may signal the essential role of fault strength heterogeneity, either that of the normal stress and/or frictional properties (Ray & Viesca, 2017, 2019), in defining its seismogenic character, i.e. under which conditions and where on the fault the earthquake slip instability can take place.&#160;</p>
We consider that a slip instability nucleates an earthquake. Past studies found blow-up solutions for diverging slip velocities. Prior stability analyses, considering heterogeneous frictional properties revealed that stable blow-up solutions can predictably dictate earthquake-nucleating instabilities. In this prior analysis, the focus remained mainly on the attraction to stable blow-up solutions. Here, we shift the focus of discussion towards unstable blow-up solutions, in particular, we seek frictional heterogeneities that can make all blow-up solutions to lose stability. (We consider variations in direct and evolution effect parameters and presume that slip scale and normal stress are uniform). We find that faults that include rate-strengthening regions can lack attractive blow-up solutions. In such scenarios, a rate-weakening fault that includes appropriate rate-strengthening regions can significantly delay the development of an instability compared to a fault that is entirely rate-weakening. That delay can be attributed to the loss of stability of the blow-up solutions. Owed to the non-existence of attractive blow-up solutions, the developing slip velocity fails to converge to a specific distribution at a single location. The near-chaotic transient dynamics may give rise to tremor-like activities.
<p>Mechanical models of slip development on geological faults and basal slip development in landslide or ice-sheets generally consider interfacial strength to be frictional and deformation of the surrounding medium to be elastic. The frictional strength is usually considered as sliding rate- and state-dependent. Their combination, elastic deformation due to differential slip and rate-state frictional strength, leads to nonlinear partial differential equations (PDEs) that govern the spatio-temporal evolution of slip. Here, we investigate how (synthetic) data on fault slip rate and traction<strong>&#160;can find the system of PDEs that governs fault slip development during the aseismic rupture phase and the slip instability phase.&#160;</strong>We first prepare (synthetic) data sets by numerically solving the forward problem of slip rate and fault shear stress evolution during a seismic cycle. We now identify the physical variables, for example, slip rate or frictional state variable, and apply<strong>&#160;</strong>nonlinearity identification algorithms&#160;within different time durations.&#160;We show that the nonlinearity identification algorithms can find the terms of the PDE that governs the slip rate evolution during the aseismic rupture phase and subsequent instability phase.</p><p>In particular, we use nonlinear dynamics identification algorithms (e.g., SINDy, Brunton et al., 2016) where we solve a regression problem, <strong>Ax=y.</strong> Here,&#160;<strong>y</strong>&#160;is the time derivative of the variable of interest, for example, slip rate.&#160;<strong>A</strong>&#160;is a large matrix (library) with all possible candidate functions that may appear in the slip rate evolution PDE. The entries in&#160;<strong>x</strong>, to be solved for, are coefficients corresponding to each library function in matrix&#160;<strong>A</strong>. We update&#160;<strong>A</strong>&#160;according to the solutions&#160;<strong>x&#160;</strong>so that&#160;<strong>A</strong>'s column space can span the dynamics we seek to find. To find the suitable column space for&#160;<strong>A,&#160;</strong>we encourage sparse solutions for&#160;<strong>x,&#160;</strong>suggesting that only a few columns in matrix&#160;<strong>A&#160;</strong>are dominant, leading to<strong>&#160;</strong>a parsimonious representation of the governing PDE.&#160;</p><p>We show that the algorithm successfully recovers the PDE governing quasi-static fault slip and basal slip evolution. Additionally, we could also find the frictional parameter, for example, a/b, where a and b, respectively, are the magnitudes that control direct and evolution effects. Moreover, the algorithm can also determine whether the associated state variable evolves as aging- or slip-law types or their combination. Further, with the data set prepared from distinct initial conditions, we show that the nonlinear dynamics identification algorithm can also determine the <strong>problem parameters&#8217; spatial distributions (heterogeneities)</strong>&#160;from fault slip rate and shear stress data.&#160;</p>
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