13 research outputs found
Examination of the rate-state friction equations under large perturbations from steady sliding: A theoretical and experimental study.
No full textThe laboratory derived rate-state friction (RSF) relationships are the most widely used
constitutive equations for fault friction in numerical models of fault mechanics. But even
after more than three decades of these being first proposed, we are far from certain about the identity of the ‘proper’ set of these equations which describe all laboratory friction data. In
fact, the two most popular choices of the ‘state’ evolution component of RSF represent two
end-member physical pictures of how frictional strength evolves – with time even without
slip (Aging law) or only with slip (Slip law). Yet both these view points have traditionally
been inferred to be independently supported by different classes of friction experiments
which (sometimes) access similar portions of the RSF parameter space. We present a set of
comprehensive studies which establish, both theoretically and with inversion of laboratory
data, that in fact all the widely used experimental protocols provide evidence that friction
dominantly evolves with slip even when the interface is sliding at the lowest slip rates
accessed by these experiments.
We examined these state evolution laws under a diverse range of sliding conditions –
up to 3.5 orders of velocity steps on both initially bare rock and gouge, up to 3X10^4 s
long holds on initially bare rock performed using machine stiffnesses differing by 1.5 orders
of magnitude and 5% normal stress steps on initially bare rock carried out at an order of
magnitude different sliding rates. For all of these experimental regimes, the widely used
Aging law generally performed worse than the Slip law, even in those parts of the parameter
space where conventional RSF wisdom would have predicted it to find strong support.
Additionally, across all these experiments, more recent prescriptions of state evolution
were generally found to fit the data only as well as the Slip law even with the freedom of
extra parameters. We argue that these findings contradict the traditional view that the state
variable is a proxy for the ‘quantity’ of true contact area alone, it is likely that some measure
of the ‘quality’ of contacts contributes significantly to state evolution as well
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Frictional response to velocity steps and 1-D fault nucleation under a state evolution law with stressing-rate dependence
No full textA new state evolution law has recently been proposed by Nagata et al. (2012) that includes a dependence upon stressing rate through a laboratory derived proportionality constant c. It has been claimed that this law, while retaining the time‐dependent healing of the Dieterich (or Aging) law, can also match the symmetric response of the Ruina (or Slip) law to velocity step tests. We show through analytical approximations and numerical results that the new law transitions between the responses of the traditional Aging and Slip laws in velocity step‐up/step‐down experiments when the value of c is tuned properly. Particularly, for c=0, the response is pure Aging, while for finite, nonzero c one observes Slip law type behavior for small velocity jumps but Aging law type response for larger jumps. The magnitude of the velocity jump required to see this transition between aging and slip behaviors increases as c increases. In the limit of c≫1 the response becomes purely Slip law type for all geologically plausible velocity jumps. We also present results from detailed analytical and numerical studies of the mechanism of rupture nucleation on 1‐D faults under this new state evolution law to demonstrate that the style of nucleation can also be made to switch from Aging‐type (expanding cracks) to Slip‐type (slip pulses) by adjusting the value of c as indicated by the velocity step results
A fractal model of earthquake occurrence: Theory, simulations and comparisons with the aftershock data
No full textThree-dimensional aseismic ruptures driven by fluid injection
No full textInjection-induced seismicity is usually observed as an enlarging cloud of seismic events that grows in a diffusive manner around the injection zone. These observations are commonly interpreted as the triggering of instabilities in pre-existing fractures and faults due to the direct effect of pore pressure increase (Shapiro, 2015), whereas poroelastic stressing is usually associated with the occurrence of seismic events beyond the plausible zone affected by pore pressure diffusion (Segall and Lu, 2015). However, an alternative triggering mechanism based on the elastic transfer of stress due to injection- induced aseismic slip has been recently proposed (Viesca, 2015; Guglielmi et al, 2015). Previous studies have shown that in critically stressed faults, the aseismic rupture front can outpace fluid diffusion (Garagash and Germanovich, 2012; Bhattacharya and Viesca, 2019), and in turn be the primary cause that controls the evolution of seismicity as it has been recently inferred from in-situ experiments of fluid injection (Duboeuf et al., 2017) and recent cases of injection-induced earthquakes (Eyre et al, 2019). Despite the great relevance of aseismic slip on injection-induced seismicity, the conditions that control the three-dimensional propagation of aseismic ruptures are still poorly constrained. This is in part due to the challenge of solving such a 3D moving boundary problem in which both fault slip and rupture shape are unknown. Here, we study the mechanics of injection-induced aseismic ruptures on a planar fault characterized by a strength equal to the product of a constant friction coefficient and the effective normal stress. We systematically track the temporal evolution of the rupture area relative to the evolution of the pressurized zone and focus on the effect of the initial stress state and injection scenario. For injection at constant flux, we derive a semi-analytical solution for circular ruptures (for a Poisson’s ratio equal to zero), which gives the ratio between the rupture radius and a nominal pore pressure front location, which we named as amplification factor λ. This amplification factor is a function of a unique dimensionless parameter that depends on the initial fault stress criticality and the fluid-induced overpressure. Then, we generalize the semi-analytical solution to the case of non-circular ruptures (for any value of the Poisson’s ratio) by solving numerically for the spatiotemporal evolution of fault slip using a fully implicit boundary-element-based solver with quadratic triangular elements. We show that the rupture front is nearly elliptical and the rupture area Ar evolves in a self-similar diffusive manner such that Ar(t) = 4παλ2t, where α is the fault hydraulic diffusivity and λ is the amplification factor for circular ruptures. The rupture area is greater than the nominal pressurized area if λ > 1. The semi-analytical solution for the rupture area provides a unique opportunity for verifying numerical hydro-mechanical solvers. After, we investigate numerically the case of circular and non-circular ruptures driven by injection at constant pressure instead of constant flux. We show that the self-similar property of the rupture growth is lost under this injection scenario.GE
A fractal model of earthquake occurrence: theory, simulations and comparisons with the aftershock data
No full textOur understanding of earthquakes is based on the theory of plate tectonics. Earthquake dynamics is the study of the interactions of plates (solid disjoint parts of the lithosphere) which produce seismic activity. Over the last about fifty years many models have come up which try to simulate seismic activity by mimicking plate plate interactions. The validity of a given model is subject to the compliance of the synthetic seismic activity it produces to the well known empirical laws which describe the statistical features of observed seismic activity. Here we present a review of one such, purely geometric, model of earthquake dynamics, namely The Two Fractal Overlap Model. The model tries to emulate the stick-slip dynamics of lithospheric plates with fractal surfaces by evaluating the time-evolution of overlap lengths of two identical Cantor sets sliding over each other. As we show later in the text, some statistical aspects of natural seismicity are naturally captured by this simple model. More importantly, however, this model also reveals a new statistical feature of aftershock sequences which we have verified to be present in nature as well. We show that, both in the model as well as in nature, the cumulative integral of aftershock magnitudes over time is a remarkable straight line with a characteristic slope. This slope is closely related to the fractal geometry of the fault surface that produces most of thee aftershocks. We also go on to discuss the implications that this feature may have in possible predictions of aftershock magnitudes or times of occurrence
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Does fault strengthening in laboratory rock friction experiments really depend primarily upon time and not slip?
No full textThe popular constitutive formulations of rate‐and‐state friction offer two end‐member views on whether friction evolves only with slip (Slip law) or with time even without slip (Aging law). While rate stepping experiments show support for the Slip law, laboratory‐observed frictional behavior near zero slip rates has traditionally been inferred as supporting Aging law style time‐dependent healing, in particular, from the slide‐hold‐slide experiments of Beeler et al. (1994). Using a combination of new analytical results and explicit numerical (Bayesian) inversion, we show instead that the slide‐hold‐slide data of Beeler et al. (1994) favor slip‐dependent state evolution during holds. We show that, while the stiffness‐independent rate of growth of peak stress (following reslides) with hold duration is a property shared by both the Aging and (under a more restricted set of parameter combinations) Slip laws, the observed stiffness dependence of the rate of stress relaxation during long holds is incompatible with the Aging law with constant rate‐state parameters. The Slip law consistently fits the evolution of the stress minima at the end of the holds well, whether fitting jointly with peak stresses or otherwise. But neither the Aging nor Slip laws fit all the data well when a − b is constrained to values derived from prior velocity steps. We also attempted to fit the evolution of stress peaks and minima with the Kato‐Tullis hybrid law and the shear stress‐dependent Nagata law, both of which, even with the freedom of an extra parameter, generally reproduced the best Slip law fits to the data
Frictional State Evolution During Normal Stress Perturbations Probed With Ultrasonic Waves
No full textThree-dimensional fluid-driven stable frictional ruptures
No full textWe investigate the quasi-static growth of a fluid-driven frictional shear crack that propagates in mixed mode (II+III) on a planar fault interface that separates two identical half-spaces of a three-dimensional solid. The fault interface is characterized by a shear strength equal to the product of a constant friction coefficient and the local effective normal stress. Fluid is injected into the fault interface and two different injection scenarios are considered: injection at constant volume rate and injection at constant pressure. We derive analytical solutions for circular ruptures which occur in the limit of a Poisson's ratio ν=0 and solve numerically for the more general case in which the rupture shape is unknown (ν≠0). For an injection at constant volume rate, the fault slip growth is self-similar. The rupture radius (ν=0) expands as R(t)=λL(t), where L(t) is the nominal position of the fluid pressure front and λ is an amplification factor that is a known function of a unique dimensionless parameter T. The latter is defined as the ratio between the distance to failure under ambient conditions and the strength of the injection. Whenever λ>1, the rupture front outpaces the fluid pressure front. For ν≠0, the rupture shape is quasi-elliptical. The aspect ratio is upper and lower bounded by 1/(1-ν) and (3-ν)/(3-2ν), for the limiting cases of critically stressed faults (λ≫1, T≪1) and marginally pressurized faults (λ≪1, T≫1), respectively. Moreover, the evolution of the rupture area is independent of the Poisson's ratio and grows simply as Aᵣ(t)=4παλ²t, where α is the fault hydraulic diffusivity. For injection at constant pressure, the fault slip growth is not self-similar: the rupture front evolves at large times as ∝(αt)⁽¹⁻ᵀ⁾ᐟ² with T between 0 and 1. The frictional rupture moves at most diffusively (∝√(αt)) when the fault is critically stressed, but in general propagates slower than the fluid pressure front. Yet in some conditions, the rupture front outpaces the fluid pressure front. The latter will eventually catch the former if injection is sustained for a sufficient time. Our findings provide a basic understanding on how stable (aseismic) ruptures propagate in response to fluid injection in 3-D. Notably, since aseismic ruptures driven by injection at constant rate expands proportionally to the squared root of time, seismicity clouds that are commonly interpreted to be controlled by the direct effect of fluid pressure increase might be controlled by the stress transfer of a propagating aseismic rupture instead. We also demonstrate that the aseismic moment M₀ scales to the injected fluid volume V as M₀ ∝ V³ᐟ².GE
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Critical evaluation of state evolution laws in rate and state friction: Fitting large velocity steps in simulated fault gouge with time‐, slip‐, and stress‐dependent constitutive laws
No full textThe variations in the response of different state evolution laws to large velocity increases can dramatically alter the style of earthquake nucleation in numerical simulations. But most velocity step friction experiments do not drive the sliding surface far enough above steady state to probe this relevant portion of the parameter space. We try to address this by fitting 1–3 orders of magnitude velocity step data on simulated gouge using the most widely used state evolution laws. We consider the Dieterich (Aging) and Ruina (Slip) formulations along with a stress‐dependent state evolution law recently proposed by Nagata et al. (2012). Our inversions confirm the results from smaller velocity step tests that the Aging law cannot explain the observed response and that the Slip law produces much better fits to the data. The stress‐dependent Nagata law can produce fits identical to, and sometimes slightly better than, those produced by the Slip law using a sufficiently large value of an additional free parameter c that controls the stress dependence of state evolution. A Monte Carlo search of the parameter space confirms analytical results that velocity step data that are well represented by the Slip law can only impose a lower bound on acceptable values of c and that this lower bound increases with the size of the velocity step being fit. We find that our 1–3 orders of magnitude velocity steps on synthetic gouge impose this lower bound on c to be 10–100, significantly larger than the value of 2 obtained by Nagata et al. (2012) based on experiments on initially bare rock surfaces with generally smaller departures from steady state
