Predicting rupture arrests, rupture jumps and cascading earthquakes


  • Y. Finzi,

    Corresponding author
    1. Earth Systems Science Computational Centre, School of Earth Sciences, University of Queensland, Brisbane, Queensland, Australia
    • Corresponding author: Y. Finzi, Earth Systems Science Computational Centre, School of Earth Sciences, University of Queensland, Brisbane, QLD 4072, Australia. (

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  • S. Langer

    1. Earth Systems Science Computational Centre, School of Earth Sciences, University of Queensland, Brisbane, Queensland, Australia
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[1] The devastation inflicted by recent earthquakes demonstrates the danger of under-predicting the size of earthquakes. Unfortunately, earthquakes may rupture fault-sections larger than previously observed, making it essential to develop predictive rupture models. We present numerical models based on earthquake physics and fault zone data, that determine whether a rupture on a segmented fault could cascade and grow into a devastating, multisegment earthquake. We demonstrate that weakened (damaged) fault zones and bi-material interfaces promote rupture propagation and greatly increase the risk of cascading ruptures and triggered seismicity. This result provides a feasible explanation for the outstanding observation of a very large (10 km) rupture jump documented in theMW7.8 2001 Kunlun, China earthquake. However, enhanced inter-seismic deformation and energy dissipation at fault tips suppress rupture propagation and may turn even small discontinuities into effective earthquake barriers. By assessing fault stability, identifying rupture barriers and foreseeing multisegment earthquakes, we provide a tool to improve earthquake prediction and hazard analysis.

1. Introduction

[2] Estimating the maximal magnitude and earthquake rupture length expected to occur along a fault-system is a fundamental step in seismic hazard analysis. Earthquakes rupturing several fault-segments where single-segment events were anticipated, have recently occurred in the Indian Ocean (M9.1), China (M7.9), Chile (M8.8) and Japan (M9) [Bilham, 2005; Shen et al., 2009; Lay, 2011; Stein and Okal, 2011]. The implications of under-predicting these earthquake-magnitudes are sadly portrayed by the horrific loss they inflicted (336,973 deaths and 6,177,000 houses destroyed according to the National Geophysical Data Center Significant Earthquake Database), and by the toppled seawalls designed to protect Japan's coastal communities and nuclear plants from anM ≤ 8 earthquake [Stein and Okal, 2011]. Fortunately, current knowledge in earthquake physics enables us to model rupture patterns based on observable fault properties. While these models cannot yield deterministic predictions of earthquake timing and location, they can quantify fault interactions and assess fault-system stability. That is, such models can be used to evaluate whether multisegment events are likely or not [Finzi and Langer, 2012]. Incorporating rupture models can therefore improve estimates of the maximal magnitude expected and contribute to short term predictions of aftershocks, swarms and regional earthquake sequences.

[3] Historic records show that large fault discontinuities constitute barriers that arrest propagating ruptures and limit the size of earthquakes. For instance, ruptures along strike-slip faults are expected to arrest at dilational step-overs wider than 4–5 km, but have the potential to cascade into multisegment earthquakes if the step-overs are narrower than 1–2 km [Wesnousky, 2006; Harris et al., 1991; Harris and Day, 1993]. This rule of thumb postulates that fault geometry may be a sufficient criterion for predicting rupture propagation. However, this simplification leads to two types of faulty predictions: Underestimated earthquake magnitude where a step-over is falsely considered an earthquake barrier and ruptures are expected to arrest (Figure 1, red stars), and overestimated earthquake magnitude where small barriers along segmented faults are overlooked (Figure 1, blue stars). A comprehensive compilation of earthquake observations of rupture jumps across very large step-overs and of ruptures arrested at very small step-overs is given in theauxiliary material. Inspired by such observations, Finzi and Langer [2012]introduced numerical simulations to determine how material damage in step-over zones may promote large rupture jumps. In that work we identified rigidity loss (within the step-over zone) and stress concentration along bi-material faults as potential mechanisms for promoting rupture jumps. In reality, the effect of damage on rupture propagation is far more complex and is significantly influenced by additional physical processes within the step-over (e.g. inter-seismic deformation and enhanced attenuation). Ignoring these processes, the model presented inFinzi and Langer [2012]could not simulate the arrest of earthquake ruptures at small step-overs. To achieve this and to better simulate rupture propagation along real fault-systems requires a more comprehensive understanding of deformation and stress distribution at the vicinity of damaged step-overs.

Figure 1.

Outstanding observations of earthquake rupture processes. Earthquakes that surprisingly ruptured multisegment or complex faults where only single- or double-segment events were previously observed (red stars with black outline). These include the recent disasters in Japan (2011), Chile (2010), China (2008) and the Indian Ocean (2004) that culminated in an estimated cost of $474 Billion. Earthquake ruptures that unexpectedly jumped step-overs 5 km wide or wider (red stars with white outline; labels indicate size of rupture jump in kilometers). And observations of rupture arrest at very small step-overs that constitute undetected earthquake barriers (blue stars with white outline; labels indicate size of step-overs).

[4] Fault step-overs are typically weakened (damaged) by distributed fractures, veins, and other deformation features. Recent studies indicate that enhanced damage and deformation at such fault zones consume a significant fraction of earthquake energy [Shipton et al., 2006] and may suppress propagating ruptures. The main processes contributing to this include: enhanced plastic deformation [Biegel et al., 2008; Ma and Andrews, 2010], fluid mobilization [Sibson, 1985], attenuation of radiated energy [Duan and Day, 2008], generation of surface area and rupture branches [Ando and Yamashita, 2007], and inter-seismic and dynamic stress changes [Harris and Day, 1999; Duan and Oglesby, 2005; Ma and Andrews, 2010]. In particular, releasing (dilational) step-overs are expected to constitute effective barriers due to their extensive damage structures and inter-seismic deformation [Cochran et al., 2009; Finzi et al., 2011]. This was recently supported by simulations [Ma and Andrews, 2010; Hok et al., 2010] and field observations [Duman et al., 2005] of rupture suppression at damaged fault zones. Nevertheless, identifying which step-over (or other fault structure) will arrest an earthquake and which will not remains a challenge as damage also reduces the resistance of rocks to deformation and introduces stress concentrations that enhance rupture propagation [Harris and Day, 1999; Duan and Oglesby, 2005; Ma and Andrews, 2010; Finzi and Langer, 2012].

2. Methods

[5] To determine how fault-damage in step-overs affects the stability of segmented fault-systems, we numerically simulate both inter-seismic deformation (during tectonic loading) and co-seismic rupture propagation into various step-over structures. We focus on rupture dynamics at the vicinity of releasing step-overs, where the sense of slip and fault configuration induces dilational stresses within the step-over. These step-overs are the most common structural features for which a sufficient number of rupture arrests are documented [Wesnousky, 2006; Sibson, 1985], and they often display extensive damage-zones (reaching the bottom of the seismogenic layer) and inter-seismic deformation [Finzi et al., 2009; Duman et al., 2005; Cochran et al., 2009].

[6] We use a model configuration and friction law that align with previous work [Langer et al., 2010; Finzi and Langer, 2012] and produce a magnitude 7 earthquake. Our 2D simulations consist of two parallel faults separated by a step-over zone 1,500 m to 10,000 m wide. The length of the first fault, on which rupture is nucleated, is 60 km and the length of the second fault is 40 km, as shown inFigure 2a(top). The step-over geometry in our simulations (width: overlap ratio of 1:1.5) is based on previous work [Harris et al., 1991; Harris and Day, 1993], where ratios of 1:1 and 1:4 have been used, and on the fact that large releasing step-overs tend to develop an inner active pull-apart basin with overlap comparable to the width of the step [Sylvester, 1988; Kim et al., 2004]. The faults are embedded in a homogeneous material as shown in Table 1, and a fixed (time-independent) damage level is prescribed within the step-over zone. While previous studies describe damage accumulation and healing during the seismic cycle [Finzi et al., 2011; Lyakhovsky and Ben-Zion, 2009], we simplify our models by assuming a constant damage level. This is supported by studies indicating that after a short post-seismic healing phase, damage-zones reach a steady state, which at step-over zones may consist of deep-rooted, highly damaged zones [Finzi et al., 2011]. In addition, at seismogenic depth, the accumulation of damage during a single seismic event is probably negligible at distances of several hundreds of meters off the fault-segment [Finzi et al., 2009], and the related dynamic energy dissipation is accounted for without updating damage levels (see Section 2.6).

Figure 2.

Destabilizing effect of a material interface and an abrupt termination of rupture. (a) Schematic diagrams of the model configurations used: A 4 km wide step-over and damage level ofα= 0.3 (top; This is the configuration used throughout the paper), faults not aligned with the material interface (the edge of the damage-zone) (center), and with increased cohesion at the tip of the first fault to get gradual termination of rupture (bottom). The three configurations were set to have the same size of damage zone and step-over, and similar overall potency (similar average slip);(b) Slip profiles on first faults, showing the reduced slip-gradient near the rupture termination site in the bottom model configuration; (c) Maximum co-seismic stress (CFS) on the second fault; (d) Maximum step-over width an earthquake is expected to jump for the three model configurations.

Table 1. Material Properties of Background Material
Rigidity modulusG (GPa)30
Lamé's first parameterλ (GPa)30
Poisson's ratioν0.25
Densityρ (kg/m3)2700
Shear wave velocityCS (m/s)3333
Primary wave velocityCP (m/s)5774

[7] Damage modifies the stress and strain characteristics of step-overs in three ways. First, the rigidity within the step-over is reduced (G = G0(1 − α/αcr), where 0 < α < 1 is a damage variable that correlates with crack density [Hamiel et al., 2006]). Secondly, we enable inter-seismic (stable) slip on highly stressed fault sections. The amount of slip (and corresponding stress-relaxation) is controlled by the damage-induced stress concentrations (Section 2.5). Finally, we introduce co-seismic energy dissipation in the step-over zone (Section 2.6). Systematically varying step-over size and characteristics, we evaluate the maximum step-over width a rupture is expected to jump. To do so, we record the maximal Coulomb Failure Stress (CFS) on the second fault and compare it to a reference stressing level determined at a distance of 4 km from the first fault in an undamaged step-over (Section 2.4). If the second fault experiences CFS>1, then rupture nucleates, the event becomes a multisegment earthquake, and the step-over is considered an ineffective barrier.

2.1. Two Phase Simulation

[8] Numerical solutions of dynamic fault rupture are achieved using the Finite Element Method (FEM) implemented in the esys.escript software [Gross et al., 2007]. Our numerical simulations consist of two main phases: (a) A quasi-static loading phase using the static, elastic deformation equationσij,j = 0 to apply a far field normal stress of σN= −200 MPa and a far-field shear stressτ = −69 MPa [Langer et al., 2010, 2012]. (b) The resulting stress field provides the initial conditions for a dynamic rupture phase [Olsen-Kettle et al., 2008; Langer et al., 2010, 2012] where rupture is initiated and observed as it propagates into the step-over zone (as inFinzi and Langer [2012]). The quasi-static loading procedure significantly reduces computation time which is critical as the parameter space studies presented required a total of ≈800 simulations (with an average run-time of 40–80 node-hours per simulation). An additional phase of stable slip is added to represent inter-seismic deformation at highly stressed step-overs (seesection 2.5 below).

[9] The finite element mesh is constructed using Gmsh [Geuzaine and Remacle, 2009] and consists of triangular elements with a grid step size of Δx = 100 m along the faults. The penalty method is used to enforce the contact boundary conditions [Perić and Owen, 1992; Olsen-Kettle et al., 2008]. The overall dimensions of the numerical model domain are varied with step-over size in order to maintain a 20 km wide border around the fault system and to minimize interactions between the faults and the model boundary. In addition, a buffer zone, 2 km wide, is set along the model boundaries with absorbing boundary conditions [Olsen-Kettle et al., 2008].

2.2. Friction Law

[10] The simulated friction law (velocity-weakening) and parameters follow widely used numerical techniques [Ampuero and Ben-Zion, 2008; Shaw and Rice, 2000] and yield a subshear pulse-like earthquake rupture on the first fault, with ratios of magnitude to rupture-length and average-slip to rupture-length in accord with historical earthquake observations [Wells and Coppersmith, 1994].

[11] The velocity-weakening friction law [Ampuero and Ben-Zion, 2008; Langer et al., 2012] is used with a dynamic friction coefficient of

display math

with parameter values as shown in Table 2, where Θ is the state variable for the weakening mechanism.

Table 2. Simulation Parameters Used in This Study
Maximum mesh grid step sizeΔx (m)100
Background normal stress across interfaceσN (MPa)−200
Background tangential stress along interfaceτ (MPa)−69
Static friction coefficientμs0.6
Direct effect coefficientαf0.01
Evolution effect coefficientβf0.41
Stress dropfs0.4
Andrews' parameter [Andrews, 1976]S1.7
Characteristic velocity scaleVC (m/s)0.7
Characteristic timescaleτC (s)0.3
Slip-weakening distanceDc (m)0.21
Length-scale forσ-regularizationδσ (1/m)0.2
Velocity-scale forσ-regularizationV* (m/s)2

[12] In simulations of damaged step-overs, the contrast in elastic moduli across step-bounding faults constitutes a bimaterial interface along which the solution of seismic slip is ill-posed [Prakash, 1998; Ranjith and Rice, 2001]. Introducing a fading memory of the normal stress response regularizes the problem [Cochard and Rice, 2000; Ampuero and Ben-Zion, 2008; Rubin and Ampuero, 2007]. An effective normal stress σ*:

display math

with the values for V* and δσ given in Table 2 is used to determine whether yielding occurs.

2.3. Nucleation Procedure

[13] To nucleate dynamic rupture, the static friction coefficient is lowered along a 1200 m nucleation patch near the far end of the first fault (marked as colored circles in Figure 2a). Following the nucleation procedures of Ampuero and Ben-Zion [2008], we permanently reduce the static friction in the nucleation site to μs = |τ|/|σN| − 0.001.

2.4. Analysis of Stress Transfer

[14] Our models yield time histories of fault slip and seismic waves from which we resolve stress increment tensors into shear (τ), normal (σN), and Coulomb Failure Stress (CFS) accounting for pore fluid pressure effects (σCFS = μσN + |τ|, where the effective coefficient of friction is μ′ = μ(1 − Bs) and Bs is the Skempton's coefficient [Harris and Day, 1993]). To evaluate whether a rupture jumps in a particular simulation, we record the highest value of CFS that occurs co-seismically anywhere along the second fault (i.e. the receiver fault on the un-ruptured side of the step-over; seeFigure 2). We then compare that value with a reference value observed (anywhere) along the second fault in a simulation with a 4 km wide, undamaged, step-over (representing a basic estimate of the minimum stress increase required to enable rupture jump in homogeneous dynamic and static rupture studies [Harris et al., 1991; Harris and Day, 1999; Wesnousky, 2006]). A receiver fault with a CFS value higher than the reference CFS is considered to be ruptured, whereas a step-over in which the receiver fault displayed lower CFS values is considered as an effective earthquake barrier that stabilizes the simulated fault-system [Finzi and Langer, 2012]. Figure 2cshows the highest CFS recorded along the receiver faults for different simulation set-ups. All CFS values in these figures are normalized using CFS = 0 for the background stress level and CFS = 1 to represent stressing levels sufficient to bring the reference model (4 km wide, undamaged step-over) to failure. In our analysis we consider rupture nucleation on the second fault as indication that the earthquake would most probably continue to propagate along the fault. While this is supported by preliminary simulations with similar stress conditions and friction parameters, it is not directly tested in the presented simulations (as the second fault here is locked to simplify stress analysis).

2.5. Inter-seismic Stress-Relaxation

[15] Far-field tectonic loading of a damaged (weak) step-over produces local stress distortions and stress concentrations along the bi-material interfaces bounding the step-over. This results in local stress enhancement along the receiver fault (Figure 3a, black line) and in the vicinity of the step-over (Figure 4b). Highly stressed fault sections in the proximity of such step-overs undergo inter-seismic slip and stress relaxation. This is numerically implemented by enabling stable slip on fault sections bounding the step-over and within 3 km of the step-over zone (relaxation outside the step occurs in simulations with high damage level and/or relaxation to below the background stress). To achieve this, the cohesion along the step-bounding fault sections is decreased to a negative value, and the dynamic friction coefficient is increased toμd= 0.598 (just below the static friction coefficient, to prevent dynamic rupture). We use this procedure to relax stresses either to pre-fixed levels (100 kPa or 1 MPa higher than the background CFS) or to levels lower than the background CFS by a fraction (0–25%) of the stress excess due to the damaged step-over (i.e., the difference between the highest CFS in the step-over zone and the background CFS). A short phase of artificially introduced viscosity is used to dampen the relaxation-induced stress waves in the model domain. Our inter-seismic relaxation procedure yields a smooth stress pattern without large stress peaks or sharp stress contracts along the faults bordering the step-over (Figure 3a). The stable-slip required to achieve these stress reductions is reassuringly small compared to the co-seismic slip (Figure 3b).

Figure 3.

Comparison of stress-relaxation limits and procedures. (a) Pre-seismic Coulomb Failure Stress (CFS) on the second fault in simulations without stress-relaxation (black curve), with stresses relaxed to 1 MPa and 100 kPa above the background stresses (blue and green curves, respectively) and with stresses relaxed by 110% of the maximum stress excess above background stress (i.e. ‘over-relaxed’ by 10%) (red curve). (b) Plots of inter-seismic slip in simulations with stress relaxation. (c.) The maximum step-over width an earthquake is expected to jump for the various relaxation procedures. As apparent from Figure 3c, the small differences in expected step-over widths indicate that the various stress-relaxation procedures and limits yield qualitatively similar effects on fault stability.

Figure 4.

Stress patterns along a segmented fault-system. (a) Maximum co-seismic Coulomb Failure Stress (CFS) in the reference simulation consisting of an undamaged 4 km wide step-over (hatched zone). The stressing level observed 4 km from the first fault is used as a threshold for triggering rupture (CFS = 1), and CFS = 0 corresponds to background stress. (b) Inter-seismic stress concentrations due to damage in a step-over (step width 4 km,α= 0.3). (c) Co-seismic stress patterns around a damaged step-over (α= 0.3) showing amplification of stress along the material interfaces bounding the step-over and of radiated stress waves. (d) Co-seismic stress patterns in a simulation incorporating weak dissipation within the damaged step-over. In these rupture simulations, a Skempton's coefficient ofBs = 0.5 is applied.

2.6. Energy Dissipation

[16] Damage and deformation at geometrically complex fault zones consume a significant fraction of earthquake energy [Shipton et al., 2006]. While some processes reduce the potential strain energy available prior to the seismic stage (e.g. quasi-static plastic strain and damage accumulation), others consume a fraction of the rupture energy (e.g. branching, cracking and strain in the fault gouge and rupture front) or attenuate the radiated seismic energy (e.g. wave interaction with off fault damage, off-fault plastic strain and dynamic fluid flow). Incorporating physical descriptions of all these processes in a dynamic rupture model is beyond the scope of our work and would introduce many unconstrained model parameters. (It would also exhaust the computational resources available for this work.) As damage enhances many of the dissipation processes above, and as recent estimates of energy dissipation due to damage range from 1–50% [Chester et al., 2005; Wilson et al., 2005; Ma et al., 2006] we assume that damage-related dissipation has an important role in rupture dynamics near step-over zones.

[17] A generalized form of energy-dissipation is therefore implemented in our simulations as a dampening force suppressing the motion of elements perturbed by the co-seismic stress waves. The dampening procedure is based on numerical schemes of velocity-dependent dampening [Mora and Place, 1994] modified to include both damage-dependent and damage-independent components. The body force applied in this procedure is equivalent to drag acting at the element integration points within the step-over. The dampening force (per unit volume) is formulated as follows:Fd = −vpkρV/L · (A + 1/n) with vp being the particle velocity, ρ the density and V/Lgiving a reference dissipation timescale (the time it would take a seismic-wave to travel through the maximum step-over in our simulations,L ≈ 10 km in our simulations). Parameters A, Bare weight functions controlling the relative effect of damage independent and damage-dependent contributions, respectively, andk is a constant scaling factor (k = 4 was chosen to adjust plausible values of the parameters A, Bso they range between 0 and 1). This procedure enables a qualitative representation of the total effect of different energy dissipating mechanisms such as wave attenuation, off-fault inelastic strain, damage accumulation, heat and slip on a pre-existing fault gouge. The general effect of this dampening procedure is the reduction of the magnitude and extent of peak CFS stress radiating from the tip of the source fault (Figures 4c and 4d). Detailed description of the dissipation method and how it correlates with commonly used attenuation measures is laid out in the auxiliary material.

[18] Realizing there is currently not enough information on the interactions between damage and the various energy-dissipation processes and on the fraction of energy dissipated, we consider various damage-dissipation functions (portrayed by a wide range ofA, B, n parameters). As a basic constrain to the dissipation parameters, we determined the range of A and B for which increasing dissipation increases stability (rejecting large A and Bvalues that result in complete attenuation of the waves within the step-over; see black line inFigure 5a). An additional constrain is derived by assuming that damage does not enhance dissipation (i.e. B = 0) and comparing the wave attenuation resulting from the applied dissipation force with measures of attenuation often used in seismic hazard studies. We find that for the range of realistic (inter-seismic) damage levels (0 <α < 0.6), parameter A values between 0.4 and 0.5 correspond to 4–6% attenuation (of high frequency seismic waves; see auxiliary material for full derivation).

Figure 5.

Rupture dynamics as a function of damage and dissipation. (a) Six damage-dissipation relations (sets ofA, B, n parameters) overlaying results of maximum rupture jumps (shaded background; 110% inter-seismic relaxation). The black contour indicates maximum feasible dissipation values. (b) The maximum rupture jumps predicted for the six parameter sets shown inFigure 4a. For each parameter set, step-overs wider than indicated by the curve constitute effective earthquake barriers. These results demonstrate the stabilizing effect of dissipation (compare with reference line inFigures 2d and 3cwhich represent simulations without dissipation). The open star represents conditions that may have enabled a 10 km rupture jump in Kunlun earthquake (China, 2001), and the solid star represents a potential earthquake barrier 2.8 km wide. (c) Maximum co-seismic stress values on the second fault for the various damage-dissipation relations, for a 4 km step-over withα = 0.3 (compare with reference line in Figure 2c).

[19] Figure 5presents the various dissipation functions considered in our study and the maximum distance a rupture may jump in simulations within the range of plausible dissipation, as a function of the damage in the step-over (Figure 5b). In the dissipation functions used, we assumed that energy-dissipation is primarily related to damage-dependent (damage-enhanced) processes (i.e.A < B). We also note that our parameter space diagram (Figure 5a) may be used to evaluate the maximum width of a step-over with any damage level, any damage-dissipation relation, and any velocity-damage relation (assuming an inter-seismic stress relaxation of 110%, as discussed above). This may be done by superimposing a damage-dissipation function on the phase diagram with account of the velocity effect on the dissipation force. For example, a damage independent process (e.g. attenuation, formation of new damage) could be represented by dissipation parametersB = 0 and A = 0.1 (or generally A < 1), resulting in a normalized dissipation force of ((1 − α) · (0.1 + 0 · α1/n))/3; where the division by 3 is introduced to normalize the dissipation force of the strongest dissipation used in our study to a value of 1 (as shown in Figure 5a). Figure 5c shows the calculated values of maximum CFS stress along the second fault in simulations applying the various dissipation functions (with CFS = 1 representing the threshold for rupture nucleation on the second fault, as in previous figures).

3. Results: Step-Over Stability and Rupture Dynamics

[20] Our step-over model described above has been used to determine how rupture propagation is affected by step-over damage and associated processes of stress concentration, inter-seismic relaxation and dynamic energy dissipation. Our simulations provide important insights into the mechanisms controlling rupture propagation and arrest. A central result is that damage destabilizes the simulated fault-system as greater damage within the step-over zone enables larger rupture jumps.

3.1. Damage-Related Destabilizing Mechanisms

[21] We identify three mechanisms that contribute to this effect of damage: rigidity reduction, stress concentration along bi-material faults [Ampuero and Ben-Zion, 2008] and increased slip gradient at the rupture termination site on the first fault [Oglesby, 2005]. These mechanisms enhance co-seismic slip along the final section of the first fault and concentrate both static and dynamic stresses along the second fault (bringing it closer to failure).

[22] To evaluate the relative importance of the latter two mechanisms we explored three different fault geometries. Figure 2compares results from a reference case (top panel, damage indicated by blue shaded area; this configuration is similar to that used throughout the paper); a step-over zone with a similar-sized damaged area, but shifted in order to separate the bimaterial interfaces from the faults (central panel, red); and a step-over configuration (bottom, green) with an extended fault and increased static friction within its final section (between 55 km and 62 km). The last configuration is tailored to obtain less abrupt rupture termination on the first fault and evaluate the effect of slip-gradient on the stress transferred onto the second fault. As the model configuration with faults not along material interfaces (red) appears most stable and yet it displays an abrupt termination of rupture on the first fault (Figure 2), we conclude that the dynamic stress-concentration related to bi-material interfaces is a more dominant destabilizing mechanism than the abruptness of rupture termination. Still, as the reference model (blue) is less stable than the one with gradual rupture termination (green) we support the hypothesis ofOglesby [2005] and Elliott et al. [2009]that slip-gradient at rupture termination affects the ability of step-overs to arrest propagating earthquake ruptures. Simulations with different properties of the increased-friction zone confirm that higher slip-gradient at rupture termination on the first fault decreases step-over stability and enables larger rupture jumps.

[23] These results, describing the destabilizing effect of damage in step-overs, demonstrate that realistic levels of fault zone damage [Finzi et al., 2011] may enable rupture jumps over step-overs wider than 10 km (Figure 2d, blue curve). Furthermore, we show that such large rupture jumps (>10 km) are plausible even with significant inter-seismic relaxation (Figure 3c) and with enhanced dissipation within the damage zone (Figure 5b, with weak or medium dissipation). We also note that rupture jumps may occur over larger discontinuities along-strike, as apparent inFigure 4c. These results indicate that high damage levels within step-overs zones can make a fault-system susceptible to large cascading earthquakes with multisegment ruptures. As material contrasts across large strike-slip faults and subduction faults have been shown to affect deformation patterns [Ma and Beroza, 2008], we expect that the destabilizing effect of bimaterial interfaces may play an important role in rupture dynamics along such plate boundaries.

3.2. Stabilizing Mechanisms

[24] In some simulations (with relatively high damage levels and large step-overs, but without stress relaxation), stress near highly damaged step-overs is sufficiently amplified to induce spontaneous rupture nucleation at the step-over. While natural earthquakes do often nucleate at step-overs [Reasenberg and Ellsworth, 1982], in many cases stress concentration in these locations induces significant seismic and aseismic deformation during the inter-seismic stage. The local stress relaxation associated with such deformation stabilizes damaged step-overs (Figure 3). To illustrate the effect of various degrees of stress-relaxation, we explored simulations with conditions ranging from the un-relaxed case (where rupture nucleates due to tectonic loading) to the over-relaxed step-over (with stress shadows due to enhanced inter-seismic deformation at the step-over). For each case we ran numerous simulations with a varying damage level and step-over width to determine the maximum rupture jumps.Figure 3shows results from simulations with relaxation to 1 MPa and 100 kPa, and relaxation by 110% of peak inter-seismic stress values. While the relaxed simulations show similar maximum rupture jumps (Figure 3c), we note that considering significant over-relaxation (i.e. strong stress shadows due to very active inter-seismic deformation) is expected to greatly stabilize the fault zone (e.g. relaxation of ≥150%). However, this would only be relevant for very unique fault zones where the main fault sections bounding a step-over display significant triggered or continuous slip throughout the seismic cycle (possible examples include the Southern Calaveras and Parkfield fault zones [Wills et al., 2008]).

[25] Earthquake ruptures are significantly inhibited by dynamic energy dissipation mechanisms such as co-seismic damage accumulation, rupture branching, porosity evolution and damage-enhanced attenuation. To evaluate the range of plausible dissipation parameters we ran simulations with zero damage and increasing dissipation. We found that rupture jump, which is expected to decrease with increasing dissipation, reaches a minimum atA= 4.5 and increases for greater dissipation. We interpret this as indication that over-suppression of the seismic waves in the step-over induces a sharp material interface along the second fault which in turn promotes re-nucleation and rupture jump over the step-over. As the dampening force is applied incrementally at every time step during which the stress-waves are traveling within the step-over, the total amount of dissipation depends on the size of the step-over and the velocity of waves (in addition to theA, B, n parameters which tune the magnitude of incremental dissipation force). Repeating the above procedure for a range of damage levels, we found the maximal dissipation which has a stabilizing effect on ruptures (indicated as the maximum dissipation line, black line in Figure 5a).

[26] To demonstrate the stabilizing effect of dissipation on rupture propagation, we constructed several parameter space diagrams showing the maximum rupture jump as a function of damage level and dissipation for different degrees of stress-relaxation (e.g.Figure 5aincorporates 110% stress-relaxation). The six curves inFigure 5arepresent six damage-dissipation functions used to represent low, mid and high dissipation effectiveness.Figure 5b presents the maximum rupture jumps expected from simulations with the dissipation functions considered in Figure 5a, and Figure 5c shows stress profiles on the second fault from simulations consisting of a damaged (α= 0.3) step-over 4 km wide (applying 110% relaxation and dissipation parameters as inFigure 5a). The approach of constructing a complete parameter space diagram enables direct evaluation of the maximum width of a step-over with any damage level and any damage-dissipation relation. To do so one simply assumes a damage-dissipation function and overlays it on the phase diagram (Figure 5a), then graphically records the maximum rupture jump values (shaded background colors) corresponding to damage levels along the damage-dissipation curve.

[27] As clearly portrayed in Figure 5, dissipation can have a dominant stabilizing effect on rupture dynamics. In fact, our simulations show that in dilational step-overs where pore fluids and inter-seismic deformation are observed and significant energy dissipation is expected, ruptures may arrest at steps as narrow as 1.8 km (Figure 6, lower blue curves α≤ 0.5). As dilational step-overs may be associated with porosity increase [Hamiel et al., 2005; Finzi et al., 2012], the above results were obtained from simulations with damage-related porosity and dynamic pore pressure changes. The effect of porosity evolution is implemented by adjusting the velocity in simulations as a function of density (assuming local porosity increase <4% and fluid filled pores). While the effect of realistically small porosity changes is small,Figure 6demonstrates that pore pressure changes can significantly stabilize a fault-system with releasing step-overs. It is important to note that rupture arrest may occur at much narrower step-overs in our simulations (depending on stress state, friction parameters etc) however, the latter result indicates that the stabilizing mechanisms discussed above may reduce the maximum distance a rupture can jump by a factor of 2.2 (compared to the 4 km reference we chose in our analysis).

Figure 6.

Effect of the pore pressure changes on step-over stability. Red curves show how the reference results of damage-dependent rupture jumps (black curve here, ‘reference’ curve inFigure 2d) are affected by pore pressure changes represented by Skempton's coefficients of Bs = 0.5 (dashed) and Bs= 1 (solid lines). Results with stress-relaxation (110%) and energy dissipation (blue:A = 0.5 B = 2.5 n = 5; green: A = 0.1 B = 0.5 n = 4) are also plotted based on simulations with Skempton's coefficients Bs = 0.5 and 1 (dashed and solid lines, respectively).

[28] Finally, our simulations consist of a 2D step-over model that simplifies structural complexities and may overlook some 3D effects. For example, our discussion of the role of porosity and pore pressure changes in rupture-dynamics does not account for poroelastic bimaterial effects [Dunham and Rice, 2008]. It is also important to note that we prescribe uniform damage in simulated step-overs, and therefore we probably over-estimate the stability of highly damaged step-overs. Incorporating a dynamically evolving damage model would result in larger rupture jumps. This is because damage is expected to localize and form a linking fault on which earthquake ruptures would propagate through the step-over [Oglesby, 2005; Finzi et al., 2011]. As step-over structures may evolve and secondary faults can connect the main segments of a fault-system, we suggest that seismic monitoring and structural analysis of step-over zones can yield important information on the ‘intrinsic stability’ of a strike-slip fault.

4. Discussion

[29] Referring back to the observations of large rupture jumps and unprecedented earthquakes in Figure 1, historic seismic records and simplified models prove to be inadequate in predicting rupture segmentation. Neglecting off-fault material properties, such models predict a maximum jump of 4–5 km [Harris and Day, 1999; Harris et al., 2002; Wesnousky, 2006]. To simulate the 10 km rupture jump documented in the 2001 Kunlun, China earthquake (Figure 1 and Table A1 from the auxiliary material), such models require either very large stress heterogeneities [Duan and Oglesby, 2005, 2006] or incorporation of unmapped linking faults [Antolik et al., 2004]. Our simulations indicate that significant damage within the ruptured step-over (as observed byXu et al. [2002]) may have enabled the large rupture jump. Specifically, we obtain very large rupture jumps (>9 km) in simulations with α ≥ 0.4, moderate stress relaxation and weak dissipation (unfilled star in Figure 5). Similarly large rupture jumps are predicted by simulations accounting for weak dissipation and the effect of pore pressure changes (Bs = 0.5, α ≥ 0.6 in Figure 6).

[30] In addition, our work shows that observations of fault zone material properties are essential for assessing fault-system stability. Evaluating stress patterns based on step-over geometry and material properties is proposed as a possible alternative to the commonly used ‘back-slip’ method [Olsen et al., 1997] which introduces arbitrary stress conditions to reproduce co-seismic observations. Obviously, dynamic models should incorporate heterogeneous initial stress patterns due to previous earthquakes (where such stresses are known), but invoking arbitrary pre-stresses to explain observed rupture dynamics may be counter-productive and obscure other rupture controlling mechanisms.

[31] Finally, while our simulations focus on predicting rupture jumps along strike-slip faults, we suggest that physics-based, rupture models can be used to predict the extent and magnitude of complex thrust earthquakes along subduction zones where material heterogeneities have been shown to control fault segmentation and seismicity [Shen et al., 2009; Sparkes et al., 2010]. Furthermore, as subduction faults juxtapose very different lithologies, the destabilizing effect of bimaterial interfaces probably plays an important role in rupture dynamics along such plate boundaries [Ma and Beroza, 2008].

5. Conclusions

[32] Our work illustrates and analyzes the important role material properties play in the dynamic propagation of earthquake ruptures. We conclude that a small damaged step-over may be as good an earthquake barrier as a larger undamaged step-over. However, large step-overs with significant damage induce stress and strain patterns that promote rupture jumps across large distances. As numerical rupture models are beginning to identify observable factors that could favor cascading ruptures and remotely triggered seismicity, their predictions may be testable as we accumulate more structural and geological data from major ruptures. We therefore suggest that such rupture models integrating fault zone observations should be used to assess fault stability and stress interactions that control earthquake size.

[33] Some pioneering hazard analyses do assess the probability of multisegment ruptures and the stability of step-overs [Field et al., 2009], but they still excessively rely on fault segmentation models and specialists' opinions. Other advanced models integrate ongoing observations of seismicity and deformation [Panza et al., 2011], but they do not implement rupture models and fault zone data to evaluate possible earthquake scenarios. Nevertheless, we need to go farther. The potentially catastrophic implications of underestimating earthquake magnitudes in heavily populated regions such as California, Turkey, Japan and China call for the implementation of physics-based models in a fundamental revision of earthquake size prediction and seismic hazard assessment.


[34] This research is supported by University of Queensland and the ARC Discovery grant DP120102188. Computations were performed on the Australian Earth Systems Simulator, an SGI ICE 8200 EX supercomputer. We thank reviewers S. Day and R. Harris for their thoughtful reviews and advise. We also thank E. Hearn, H. Muhlhaus, G. Rosenbaum, O. Dor, L. Olsen-Kettle, P. Nuriel, V. Boros, J. Fenwick and A. Papon for their comments and support.