### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Method
- 3. Results
- 4. Discussion
- 5. Summary
- Acknowledgments:
- References
- Supporting Information

[1] Slow slip events (SSE) in many subduction zones incrementally stress the adjacent locked megathrust, suggesting that they could potentially either trigger or evolve into damaging earthquakes. We explore this with 2D quasi-dynamic simulations with rate-state friction, dilatancy, and coupled 1D pore-fluid and heat transport. Steady-state weakening friction allows transient slip to nucleate, but is inhibited by dilatant strengthening and destabilized by thermal pressurization. SSE spontaneously nucleate in Low Effective-Stress Velocity-Weakening (LESVW) regions. If the dimension of the LESVW is relatively small the SSE are trapped at its updip end, imparting a strong stress concentration in the locked zone. After several centuries SSE penetrate into the region of higher effective stress, where thermal pressurization eventually leads to dynamic rupture. For larger LESVW regions SSE tend to increase in length with time; ultimately higher slip speeds enhance thermal weakening, leading to dynamic instability within the SSE zone. In both cases the onset of the ultimate SSE is essentially indistinguishable from preceding events.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Method
- 3. Results
- 4. Discussion
- 5. Summary
- Acknowledgments:
- References
- Supporting Information

[2] In Cascadia SSE are accompanied by tectonic tremor and are referred to as Episodic Tremor and Slip (ETS). Because ETS there, and in other subduction zones, occur down-dip of the locked megathrust, each event incrementally stresses the zone where damaging quakes nucleate. This suggests that the occurrence of ETS may be useful in anticipating damaging earthquakes [*Mazzotti and Adams*, 2004].

[4] Earthquake nucleation is, however, time-dependent, which can cause the probability distribution of triggering times to be spread out relative to the inter-ETS period. In the theory of*Dieterich* [1994], the time for seismicity to decay to background levels following a step change in stress is denoted *t*_{a}. If *t*_{a} is relatively long compared to the duration of ETS events, the seismicity rate is not predicted to peak significantly during the transient. *Beeler* [2012] argues that this is likely to be the case (citing *t*_{a} of years to decades), such that earthquake probability is not significantly enhanced by periodic slow slip events.

[5] Nevertheless, triggered seismicity has been associated with SSE in some localities, including the Boso Peninsula of Japan, where slip events every ∼6 years have been accompanied by earthquakes as large as M 5.3 [*Ozawa et al.*, 2007]. SSE beneath Kilauea volcano are accompanied by swarms of small earthquakes [*Segall et al.*, 2006]. The timing of the GPS displacements relative to the swarm onset, as well as their spatial relationship, suggest that the earthquakes were triggered by static stress changes; the swarm events were thus referred to as “co-shocks” of the SSE [*Segall et al.*, 2006]. The temporal evolution of the swarms is consistent with the SSE induced stress changes and *t*_{a} on the order of 10 days (based on nearby aftershocks).

[6] Our simulations show that SSE can evolve directly into dynamic events, rather than trigger a remote nucleation site, a possibility that appears not to have received much attention, although see *Dragert et al.* [2004].

### 2. Method

- Top of page
- Abstract
- 1. Introduction
- 2. Method
- 3. Results
- 4. Discussion
- 5. Summary
- Acknowledgments:
- References
- Supporting Information

[7] We present models of dipping faults in an elastic half-space with depth variable friction and effective stress to represent SSE in subduction zones (Figure 1 in Text S1 in the auxiliary material). Deformation is plane-strain, with slip in the*x*-direction, and the fault centered on*y* = 0; *z* measures depth from the free surface. Fault slip *δ* is the integral of inelastic shear strain across the fault zone − *h*/2 ≤ *y* ≤ *h*/2. Inelastic deformation within the fault zone is approximated by a friction law, with shear resistance the product of friction coefficient *f* and effective normal stress *σ* − *p*(*y* = 0). We employ the radiation damping approximation of elastodynamics [*Rice*, 1993] such that the momentum balance on the fault is

Elastic Green's functions *G*_{τδ} and *G*_{σδ}give the changes in shear and normal stress, respectively, due to slip (slip-to-normal stress coupling arises due to the free surface).*τ*_{0}, *μ*, and *v*_{s}are respectively nominal stress, shear modulus, and shear-wave speed. Slip at a constant rate*v*^{∞} is applied below depth *z*_{pl} ∼ 52 km. The friction coefficient *f* is a function of slip speed *v* (integral of shear strain rate across the shear zone) and state variable *θ.*We employ the regularized form of the rate-state equations [*Rice et al.*, 2001],

and slip-law state evolution. The heat equation, with thermal diffusivity*c*_{th}, includes shear heating , where *c* is specific heat capacity and *ρ*density. The shear strain-rate is , where *g*(*y*) is a (Gaussian) shaping function, and ∫ *g*(*y*)*dy* = *h.*

[8] Changes in pore pressure *p* follow

where *c*_{hyd} is hydraulic diffusivity, *β* is compressibility of the fluid and pore space, *ϕ* is the inelastic component of porosity, and Λ is the thermal pressurization parameter [*Rice*, 2006]. Dilatancy/compaction acts as a fluid pressure sink/source, and thermal pressurization as a source. We employ a constitutive law for inelastic change in porosity *δϕ*, motivated by experiments [*Samuelson et al.*, 2009], that follows *Segall and Rice* [1995], but extended to account for a finite thickness shear zone [*Segall and Bradley*, 2012]. The latter reference also describes the numerical methods employed here.

[9] The distribution of frictional properties loosely follows *Liu and Rice* [2009]. The parameter *a* (‘direct effect’) is given by *a* = *α*(*T* + 273.15), where the nominal *α* = 3 × 10^{−5}, and *T* is taken from the *Peacock* [2009] geotherm. Lab experiments on gabbros [*He et al.*, 2007] show large scatter, but can be interpreted to show a transition from velocity weakening to strengthening behavior at *T* ∼ 510*°*C. We parameterize *a* − *b*, which gives the steady-state frictional response, as piecewise linear (Figure 1), with three sections: the shallow (low T) region is velocity strengthening; at intermediate depth *a* − *b* = − 3.5 × 10^{−3}; while at greater depth friction transitions to *a* − *b* > 0 at a depth of ∼45 km.

[11] We found that thermal pressurization overwhelms velocity strengthening friction at depth, causing dynamic ruptures to propagate well into the strengthening region. We thus suppress thermal pressurization at depth where *a* > *b.* An increase in shear zone thickness would reduce frictional heating; however, we simplify this by artificially increasing the heat capacity. For sufficiently high this is effective at stopping dynamic rupture.

### 3. Results

- Top of page
- Abstract
- 1. Introduction
- 2. Method
- 3. Results
- 4. Discussion
- 5. Summary
- Acknowledgments:
- References
- Supporting Information

[12] Numerical results fall into two classes, depending on the size of the LESVW (labeled *W*) relative to the drained critical nucleation dimension *L*_{c} = *d*_{c}*μ*′/(*σ* − *p*^{∞})(*b* − *a*), where *μ*′ = *μ*/(1 − *ν*). Accounting for incomplete drainage does not change this value significantly (see auxiliary material). An example where the LESVW is relatively small (but *W*/*L*_{c} ∼ 30–50) is shown in Figure 1a. For nearly a century following a dynamic event the fault is relatively quiescent, with creep penetrating updip into the locked zone, at which point slow-slip events (SSE) spontaneously nucleate. Note that, as in many geodetic inversions, there is a “transition zone” between creep at the relative plate velocity,*v*^{∞}, and the essentially locked fault. In the simulations this transition arises naturally, but is not static; the transition propagates updip as a function of time.

[13] As has been seen in previous simulations of this type [*Segall et al.*, 2010; *Liu and Rubin*, 2010; *Segall and Bradley*, 2012], the SSE are stabilized by dilatancy-induced decreases in pore pressure at the leading edge of the SSE (Figure 2b). This is followed by a very modest increase in pore pressure due to shear heating during the SSE. With time the SSE propagate updip until they hit the boundary where the background effective normal stress increases. In this example, the SSE remain trapped at the boundary of the low region for more than 200 years before a dynamic event occurs (Figure 1a).

[14] The behavior with larger LESVW (*W*/*L*_{c} ∼ 70) is shown in Figure 1b. Again there is a quiescent period following a dynamic event in which the transition between creeping and locked fault extends updip with time, followed by the spontaneous onset of SSE. The SSE generally propagate farther updip with time, but in this case a dynamic event nucleates before the SSE reach the updip limit of the low region.

[16] Nucleation of a dynamic event in the larger LESVW case is illustrated in Figure 4. As shown in Figure 1b the SSE generally propagate updip with time. This brings the SSE front into increasingly velocity weakening regions. In addition, the maximum slip speed during SSE is an increasing function of rupture length [*Segall et al.*, 2010], such that the slip speeds increase slightly with time. Although there is no clear increase in maximum slip speed over the last several SSE, increased slip speed enhances thermal pressurization. The onset of the ultimate SSE is unremarkable, and essentially indistinguishable from the preceding events. However, the slip speed eventually reaches a point where thermal pressurization becomes the dominant weakening mechanism [*Schmitt et al.*, 2011]. At this point slip accelerates to inertial speeds *within* the SSE zone, and then propagates updip.

### 4. Discussion

- Top of page
- Abstract
- 1. Introduction
- 2. Method
- 3. Results
- 4. Discussion
- 5. Summary
- Acknowledgments:
- References
- Supporting Information

[17] We have discussed three ways in which SSE can influence the occurrence of earthquakes: 1) static stress perturbations due to SSE cause nucleation of a dynamic event *updip* of the SSE zone; 2) SSE extend partially into the locked, higher region where enhanced weakening causes the rupture to accelerate; and 3) SSE extend gradually updip with time, ultimately initiating dynamic rupture *within* the SSE zone.

[18] Our numerical simulations do not exhibit mechanism 1, presumably because the model is too smooth (both in terms of fault geometry and/or material properties) to induce sufficient local stress concentrations. In the auxiliary material, we show that the stressing rate, , due to repeated SSE can be approximated by

where *l*is the half-width of the SSE zone,*ξ* = *x*′/*l* is normalized distance from the SSE zone center (*x*′ = 0). Equation (4) is in good agreement (Figure 5a) with the average rate of stress accumulation from the time the SSE first reach the updip end of the LESVW region (Figure 1a) to the time just prior to dynamic rupture. Equation (4) can be used to estimate the magnitude of an isolated stress heterogeneity within the locked zone required to trigger a dynamic rupture, assuming that the width of the heterogeneity is sufficiently large to permit unstable slip (that is, the width exceeds some multiple of *L*_{c}). Nucleation requires the shear stress to reach the static strength . Thus, for a stress heterogeneity of magnitude Δ*τ*, an estimate of the time to instability is , where *τ*_{res} ∼ 0.5 MPa is the residual stress after dynamic events. Figure 5b shows that stress concentrations Δ*τ* an appreciable distance from the top of the ETS zone would have to be of order 5 MPa (comparable to an earthquake stress drop) in order to nucleate an instability in ≤ 500 years. While stress concentrations of this scale could arise due to previous partial ruptures [*Lapusta and Rice*, 2003], non-planar fault geometry, or other factors, they do not arise in the simulations here.

[19] With mechanism 3, dynamic events nucleate within the SSE zone. Dynamic rupture within the SSE zone appears to be contradictory to observations from the Nankai region Japan, where the 1944–1946 Tonankai and Nankaido earthquakes apparently occurred updip of the ETS zone [*Dragert et al.*, 2004]. It is possible that fault properties strictly prohibit fast slip within the ETS zone, although that is not the case for the class of models here, and indeed mechanism 2 dynamic events also rupture into the ETS zone (Figure 1a). Increasing the shear zone thickness *h*within the LESVW region to suppress thermal pressurization does not prevent dynamic events from propagating through the ETS region, at least for the parameter ranges considered. We also examined drained models with a v-cutoff, such that friction becomes velocity strengthening above a critical threshold [*Shibazaki and Shimamoto*, 2007], and find that this too fails to prevent dynamic events from propagating through the ETS region. This occurs simply because the increase in friction *f* at very low yields a small increase in strength, one that cannot compensate for the large stress drop updip of the ETS zone. Understanding what limits fast slip from penetrating the ETS zone is an important outstanding mechanical problem.

[20] Mechanism 2 is likely the most intuitive. If SSE are limited in depth extent by a change in physical properties and/or background effective stress, the repeated SSE produce a stress concentration at the top of the ETS zone. Limited penetration of the SSE into the higher region causes thermal pressurization that is sufficient to allow rupture to accelerate to inertial limits. It is notable that there does not appear to be any obvious change in SSE behavior prior to the onset of a dynamic event, in terms of repeat time, maximum slip speed, or amount of slip in an SSE.

[21] We can use (4) to make a rough estimate of the recurrence time for dynamic events, assuming that the stress concentration at the top of the SSE region must reach the static strength over a critical nucleation dimension, which we take to be *κL*_{c}. Thus, where *ξ*^{∗} ≡ − 1 − *κL*_{c}/*l.* For the simulation in Figure 3 the ultimate SSE penetrates ∼ 700 m into the high region (*κ* ∼ 7). This corresponds to *ξ*^{∗} = − 1.01 and a predicted recurrence time of 290 years. This is of the order of the simulated recurrence interval of 300 years, although the simple estimate does not account for the more than 100 years before the SSE actually reach the updip boundary of the LESVW region.

[22] The critical dimension *W* for the transition in behavior between that observed in Figures 1a and 1b is controlled more by thermal pressurization than dilatancy. Properties are nonuniform so the dimensionless parameters are only approximately defined, but a reasonable estimate is *E*_{p} = 1.4 × 10^{−3}, *a*/*b* = 0.95 (see the auxiliary material for definitions). *Segall et al.* [2010] presented isothermal simulations that show for *E*_{p} = 0.001 and *a*/*b* = 0.9, dilatancy stabilizes slip to at least *W*/*L*_{c} > 20, and quite likely much more. Thus, it is not surprising that dilatancy can stabilize slip for *W*/*L*_{c} ∼ 30–50. For the case in Figure 1b the friction becomes more velocity weakening as slip propagates updip. *Schmitt et al.* [2011]compute a critical velocity for which, in the absence of dilatancy, thermal pressurization overwhelms rate-state friction and becomes the dominant weakening mechanism. The behavior is different for aging-law and slip-law friction, but the effect of dilatancy is to make nucleation more like aging law simulations. In this case*Schmitt et al.* [2011] find the critical velocity can be approximated by

where *q*(*a*, *b*) is a function of the rate-state parameters. Note that*v*_{crit} is independent of normal stress. For *b* − *a* ∼ 2 × 10^{−3}, *Schmitt et al.* [2011, Figure 13] find *v*_{crit} on the order of 10^{−5} m/s, which is comparable to the slip speed in Figure 4 at the point slip becomes unstable. Thus, the current simulations are consistent with previous experience and suggest that thermal pressurization induces the dynamic event, rather than *W*/*L*_{c} becoming too large for dilatancy to stabilize slip. We have confirmed this with a comparable simulation of the larger *W*/*L*_{c} case without thermal pressurization; as expected, SSE extend to the updip transition, where eventually the dynamic event nucleates.

### Supporting Information

- Top of page
- Abstract
- 1. Introduction
- 2. Method
- 3. Results
- 4. Discussion
- 5. Summary
- Acknowledgments:
- References
- Supporting Information

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