Modeling the dynamic rupture propagation on heterogeneous ... · plane shear crack for which the...

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ANNALS OF GEOPHYSICS, VOL. 48, N. 2, April 2005

Key words dynamic rupture – fault constitutive law –fault friction – stress heterogeneities

1. Introduction

Modeling the dynamic propagation of earth-quake ruptures requires the adoption of a con-stitutive relation, which governs the tractionevolution within the cohesive zone (Ida, 1972;

Andrews, 1976a; Okubo and Dieterich, 1984;Cocco and Bizzarri, 2002). In particular, theRate- and State-dependent friction laws (RS)derived from laboratory experiments (Di-eterich, 1979a,b; Ruina, 1983) have been wide-ly used in numerical simulations of earthquakeruptures, both at the laboratory scale (Bizzarriand Cocco, 2003 and reference therein) and forreal-world faults (Guatteri et al., 2001, 2003).The frictional stress in RS laws depends on theslip rate and on the state variable. This class ofconstitutive laws includes an evolution law forthe state variable and involves a friction de-pendence on time. Since RS laws account forfault restrengthening after dynamic failure, theycan be used to simulate repeated seismic events

Modeling the dynamic rupturepropagation on heterogeneous faults

with rate- and state-dependent friction

Elisa Tinti, Andrea Bizzarri and Massimo CoccoIstituto Nazionale di Geofisica e Vulcanologia, Roma, Italy

AbstractWe investigate the effects of non-uniform distribution of constitutive parameters on the dynamic propagation ofan earthquake rupture. We use a 2D finite difference numerical method and we assume that the dynamic rupturepropagation is governed by a rate- and state-dependent constitutive law. We first discuss the results of severalnumerical experiments performed with different values of the constitutive parameters a (to account for the di-rect effect of friction), b (controlling the friction evolution) and L (the characteristic length-scale parameter) tosimulate the dynamic rupture propagation on homogeneous faults. Spontaneous dynamic ruptures can be simu-lated on velocity weakening (a < b) fault patches: our results point out the dependence of the traction and slip ve-locity evolution on the adopted constitutive parameters. We therefore model the dynamic rupture propagation onheterogeneous faults. We use in this study the characterization of different frictional regimes proposed byBoatwright and Cocco (1996) based on different values of the constitutive parameters a, b and L. Our numeri-cal simulations show that the heterogeneities of the L parameter affect the dynamic rupture propagation, controlthe peak slip velocity and weakly modify the dynamic stress drop and the rupture velocity. Moreover, a barriercan be simulated through a large contrast of L parameter. The heterogeneity of a and b parameters affects thedynamic rupture propagation in a more complex way. A velocity strengthening area (a > b) can arrest a dynam-ic rupture, but can be driven to an instability if suddenly loaded by the dynamic rupture front. Our simulationsprovide a picture of the complex interactions between fault patches having different frictional properties and il-lustrate how the traction and slip velocity evolutions are modified during the propagation on heterogeneousfaults. These results involve interesting implications for slip duration and fracture energy.

Mailing address: Dr. Elisa Tinti, Istituto Nazionale diGeofisica e Vulcanologia, Via di Vigna Murata 605, 00143Roma, Italy; e-mail: [email protected]

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Elisa Tinti, Andrea Bizzarri and Massimo Cocco

(Tse and Rice, 1986; Rice, 1993). Furthermore,they have been applied to model preseismic andpostseismic processes, such as earthquake nu-cleation (Dieterich, 1992; Lapusta and Rice,2003) and afterslip (Marone et al., 1991).

Another constitutive relation, the Slip-Weak-ening law (SW), is also currently used in dy-namic modeling of earthquake ruptures; accord-ing to this constitutive law the traction dependsonly on the slip (Ida, 1972; Andrews, 1976a,b;Ohnaka and Yamashita, 1989). Although thesetwo constitutive formulations are considered al-ternative (see Ohnaka, 2003), they both providea physically correct description of the dynamicpropagation of an earthquake rupture (see Biz-zarri et al., 2001 and references therein). Recentstudies have proposed analytical relations to as-sociate the constitutive parameters of these twoformulations (Cocco and Bizzarri, 2002; Biz-zarri and Cocco, 2003). The main difference be-tween these two constitutive laws is that SWprescribes the traction evolution with slip,whereas RS do not and traction spontaneouslyevolves with time and/or slip driven by the statevariable evolution (Bizzarri and Cocco, 2003;Cocco et al., 2004). It has to be pointed out thatRS laws have been derived from laboratory ex-periments at low slip velocity (<1 cm/s) and areusually assumed (as in the present study) to bevalid also at high slip velocities (∼1 m/s), such asthose observed during large earthquakes. How-ever, other mechanisms might affect fault fric-tion at high slip velocity such as frictional heat-ing (see Fialko, 2004 and references therein),thermal pressurization (see Andrews, 2002 and

references therein) or mechanical lubrication(Brodsky and Kanamori, 2001).

Numerous recent investigations have shownthat the rupture history imaged on the fault planeduring moderate-to-large magnitude earthquakesis quite complex: slip is non-uniformly distrib-uted on the fault and rupture velocity and slip du-ration can change dramatically during the dy-namic rupture propagation. The heterogeneity ofslip and the variations of rupture speed are cer-tainly associated with the complexity of faultgeometry, the non-uniform distribution of pre-stress (Day, 1982; Peyrat et al., 2001) and thevariations of frictional properties on the faultplane (Boatwright and Cocco, 1996). The shortslip duration and the healing of slip can also beassociated with stress heterogeneity (Beroza andMikumo, 1996; Day et al., 1998). Conversely, ina homogeneous configuration the healing of sliphas been modeled by appropriately modifyingthe constitutive relation (Perrin et al., 1995;Zheng and Rice, 1998; Cocco et al., 2004); thisallows a sudden traction re-strengthening. How-ever, it is important to point out that, although theconstitutive laws which include self-healing ofslip are physically reasonable, they have notbeen corroborated with laboratory experiments.

The goal of this study is to model the dynam-ic propagation of an earthquake rupture on a het-erogeneous fault using RS constitutive laws. Thisimplies that we have to describe and represent thefrictional heterogeneity in terms of non-uniformdistributions of RS constitutive parameters alongour 2D fault model. Several studies focused onthe investigation of the effects of spatial hetero-

Table I. Frictional behaviors proposed by Boatwright and Cocco (1996). A= aσneff and B= bσn

eff.

Regime Description A and B Seismicity Strain Release

Velocity Strong Seismic B >> A Main shocks Episodic dynamic slipWeakening (VW) (S-VW) and some aftershocks

Weak Seismic B − A > 0 Interseismic, foreshocks, Creep and intermittent(W-VW) B − A ≤ 0.05 MPa main shocks dynamic slip

and aftershocks

Velocity Compliant A − B > 0 Some aftershocks Creep and forcedStrengthening (VS) (aseismic) A − B ≤ 0.1 MPa dynamic slip

Viscous A >> B None Stable sliding(aseismic)

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Modeling the dynamic rupture propagation on heterogeneous faults with rate- and state-dependent friction

geneities of the constitutive properties (see Tseand Rice, 1986; Rice, 1993; Boatwright and Coc-co, 1996 among many others). In particular,Boatwright and Cocco (1996; BC96 in the fol-lowing) discussed the frictional control of crustalfaulting by using a RS law and a single degree offreedom spring-slider dynamic system. They pro-posed that the fault response to the dynamicstress perturbations can differ depending on thevariability of the constitutive parameters. Theyproposed four frictional fields separating the Ve-locity Weakening regime (VW) into strong andweak fields and the Velocity Strengtheningregime (VS) into compliant and viscous response(see table I). BC96 associated these four friction-al regimes with different values of the constitu-tive parameters and with observed features ofseismicity and strain release as depicted in tableI. Several authors have suggested that fault fric-tional parameters can vary with depth (Blanpiedet al., 1991, 1995; Rice, 1993; Lapusta et al.,2000 among many others). BC96 proposed thatthese frictional parameters can also change alongthe fault strike, suggesting that frictional proper-ties may control crustal faulting. In this study weused our 2D finite difference algorithm to inves-

tigate the rupture propagation on a fault modeledthrough the different regimes proposed by BC96.We study how the interaction between velocityweakening and velocity strengthening portions ofa 2D fault can affect the dynamic rupture propa-gation.

2. Methodology

We solve the fundamental elastodynamicequation neglecting body forces for a 2D in-plane shear crack for which the displacementand the shear traction depend on time and onespatial coordinate. We solve the fully dynamic,spontaneous problem by using a finite differ-ence method where the fault is simulated as aplane of split nodes in the grid (the Traction-at-Split-Nodes – TSN, numerical technique). Theapproach is described in Andrews (1973) and inAndrews (1999) and the details of the numeri-cal implementation are in Bizzarri et al. (2001).Our procedure allows the adoption of differentconstitutive laws. In this study, we use the RSlaw with slowness evolution equation derivedby Dieterich (1986), which has been discussed

Table II. Model and constitutive parameters used for the simulations on a homogeneous fault. This set of pa-rameters refers to a laboratory-scale experiment.

Parameter Reference case Strong seismic regime Weak seismic behavior(figs. 1a-d, 2 and 3) (fig. 4)

λ = µ 27 GPa 27 GPa 27 GPavP 5196 m/s 5196 m/s 5196 m/svS 3000 m/s 3000 m/s 3000 m/sµ* 0.56 0.56 0.56a 0.012 0.0085 0.015b 0.016 0.016 0.016L 1×10−5 m 1×10−5 m 0.8×10−6 m

σ neff 100 MPa 100 MPa 100 MPa

vinit 1×10−5 m/s 1×10−5 m/s 1×10−5 m/sΨnucl 1×10−4 s 1×10−4 s 1×10−4 s

Ψoutside the nucleation Ψss(vinit) Ψss(vinit) Ψss(vinit)Nucleation region [−1.5 m, 1.5 m] [−1.5 m, 1.5 m] [−2.0 m, 2.0 m]

∆x1 0.01 m 0.01 m 0.01 mFault region [−10 m, 10 m] [−10 m, 10 m] [−10 m, 10 m]

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in detail by many authors (Gu et al., 1984; Tseand Rice, 1986; Roy and Marone, 1996)

.

ln lna vv b

L

t Ldd

v

v

Ψ

Ψ Ψ

1 1

1

** *

n

eff= - + + +

= -

x n va ak k; E

(2.1)

In (2.1) v is the slip velocity, Ψ is the state vari-able (which provides a memory of previous slipepisodes), µ* and v* are reference values for thefriction coefficient and for the slip velocity, re-spectively. a, b and L are the constitutive param-eters: a represents an instantaneous rate sensi-tivity, that is the direct frictional response to achange in slip velocity, L is the characteristiclength that together with b controls the evolutionof state variable toward the steady state. σn

eff isthe effective normal stress; in the literature the

parameters A = aσneff and B = bσn

eff are also oftenused. The length-scale parameter L is differentfrom the characteristic slip-weakening distanceDc (see Cocco and Bizzarri, 2002). We will showthe relation between L and Dc in the next section.

All the parameters used to perform the sim-ulations of dynamic rupture propagation pre-sented in this study are summarized in tables IIand III: λ and µ are the Lamè constants; vP andvS are P- and S-wave velocities, respectively;vinit is the initial value of slip velocity; ∆x is thespatial grid size. The adopted discretization andmedium parameters guarantee sufficiently ac-curate numerical experiments: the simulationspresented and discussed in this paper have neg-ligible grid dispersion (i.e. the accuracy fre-quency in space is 50 KHz). Moreover, all thestability and convergence conditions (see Biz-zarri and Cocco, 2003) are satisfied.

Table III. Model and constitutive parameters used for the simulations on a heterogeneous fault.

Parameter Heterogeneity of L Barrier-healing Finite slip duration Complex rupture(figs. 5a,b and 6) (fig. 7a,b) (fig. 8a,b) propagation (fig. 9a,b)

λ = µ 27 GPa 27 GPa 27 GPa 27 GPavP 5196 m/s 5196 m/s 5196 m/s 5196 m/svS 3000 m/s 3000 m/s 3000 m/s 3000 m/sµ∗ 0.56 0.56 0.56 0.56a1 0.012 0.012 0.015 0.014a2 0.012 0.012 0.015 0.015a3 0.012 0.012 0.012 0.012b1 0.016 0.016 0.014 0.016b2 0.016 0.016 0.014 0.013b3 0.016 0.016 0.016 0.016

L1=L2 1.5×10−5 m 1×10−3 m 1 ×10−5 m 1×10−5 mL3 0.9×10−5 m 1 ×10−5 m 1 ×10−5 m 1×10−5 mx1 5 m 5 m 7 m 6 mx2 5 m 5 m 7 m 4 m

σ neff 100 MPa 100 MPa 100 MPa 100 MPa

vinit 1×10−5 m/s 1×10−5 m/s 1×10−5 m/s 1×10−5 m/sΨnucl 1×10−4 s 1×10−4 s 1×10−4 s 1×10−4 s

Ψoutside the nucleation Ψss(vinit) Ψss(vinit) Ψss(vinit) Ψss(vinit)Nucleation region [−1.5 m, 1.5 m] [−1.5 m, 1.5 m] [−1.5 m, 1.5 m] [−1.5 m, 1.5 m]

∆x1 0.01 m 0.01 m 0.01 m 0.01 m Fault region [−10 m, 10 m] [−10 m, 10 m] [−50 m, 50 m] [−50 m, 50 m]

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3. Dynamic rupture propagationon a homogeneous fault

We present in this section the simulation ofthe dynamic rupture propagation on a homoge-neous fault model with a uniform distributionof constitutive parameters. This fault modelcan be considered as a reference model for thediscussion that will be presented in the follow-ing of this paper; this reference model was pre-viously described by Bizzarri and Cocco(2003). The parameters adopted for this simu-

lation are listed in table II. The model repre-sents a velocity weakening fault for which thevalue of b – a is 0.004 (i.e. B – A = 0.4 MPa).The results of this simulation are shown in fig.1a-d and they represent a good example to dis-cuss the typical behavior of a spontaneous dy-namic rupture governed by RS. The nucleationstrategy is described in details by Bizzarri et al.(2001); the nucleation patch is quite small andthe nucleation stage is relatively short.

Figure 1a shows the spatio-temporal evolu-tion of slip velocity, while the bottom panels

Fig. 1a-d. a) The spatio-temporal evolution of slip velocity during the dynamic rupture propagation on a ho-mogeneous fault. The initial and constitutive parameters used for these calculations are listed in table II. The bot-tom panels show the traction evolution as a function of state variable (b), slip velocity (c) and slip (d) for a tar-get point P1. Ψinit in (b) represents the initial value of the state variable; v0 in (c) is the slip velocity when thedisplacement reaches the critical slip weakening distance Dc

eq (d).

a

b c d

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display the traction evolution for a target pointP1 as a function of state variable (fig. 1b), slipvelocity (fig. 1c) and slip (fig. 1d). We summa-rize in the following the major features emerg-ing from this simulation:

i) The rupture speed increases during thedynamic propagation and remains for this con-figuration sub-shear (vcrack ∼ 2200 m/s), al-though in general it might accelerate to a super-shear value, asymptotically approaching P-wave speed (Bizzarri et al., 2001).

ii) Peak slip velocity increases during thedynamic propagation with an increasing spatialdistance from the nucleation; the largest in-crease occurs when the crack accelerates tohigher rupture velocities (figs. 1a and 2).

iii) Simulations for a homogeneous config-uration show no healing of slip: slip velocitydoes not drop to zero during the rupture propa-gation (figs. 1a and 2). All points have roughlythe same final slip velocity value (v2), whichrepresents the velocity at the new steady stateafter the dynamic stress release (fig. 2).

iv) The dynamic traction evolution shows acharacteristic slip-weakening behavior (fig. 1dfor a target point P1; see also Cocco and Biz-zarri, 2002; and Bizzarri and Cocco, 2003). Theweakening phase is denoted in this figure by thelabels B and D (the latter identifies the point inwhich dynamic traction reaches the kineticstress value, τf

eq, see fig. 1d). The slip requiredfor traction to drop is called «equivalent» slipweakening distance Dc

eq.v) The phase diagram shown in fig. 1c points

out that the shear stress reaches the peak value(the maximum stress value τu

eq, i.e. the equivalentyield stress, indicated in panel d and characteriz-ing the point B in fig. 1b-d) earlier than slip ve-locity (label C), in agreement with Tinti et al.(2004).

vi) Figure 1b points out that during dynamicpropagation the state variable evolves from the as-sumed initial steady-state value (Ψinit) to a newone (Ψss= L / v). This represents the well-knownself-damping behavior of the state variable. Biz-zarri and Cocco (2003) and Cocco et al. (2004)have shown that it is the state variable evolution(line BC in panel b) that drives the slip accelera-tion (same line in panel c) and most of the tractiondrop during the weakening phase (see panel d).

vii) The duration of the weakening phase(line BD in panels c and d) defines the break-down zone duration, as described by Ohnakaand Yamashita (1989).

viii) Simulations performed with a homo-geneous configuration of RS friction with aslowness evolution law (eq. (2.1)) yield a con-stant weakening rate (see fig. 1d), provided thatthe spatial and temporal discretization is accu-rately selected to resolve the state variable evo-lution and the fast slip acceleration.

The considerations presented above arecharacteristic of the rupture propagation on a

Fig. 2. Temporal evolution of dynamic traction andslip velocity for several selected points located at dif-ferent distances from the nucleation. Dashed linesrepresent the envelopes of τ u

eq and τ feq values. The

constitutive parameters are the same used for the pre-vious figure and are listed in table II.

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velocity weakening homogeneous fault. It isimportant to point out that the peak slip veloci-ty, the yield stress, the dynamic stress drop, therupture velocity, the slip weakening distanceare strongly affected by the values of a and b.Moreover, the values of τu

eq and τ feq, whose an-

alytical expressions as a function of RS consti-tutive parameters are computed by Bizzarri andCocco (2003), do not show relevant variationsduring the rupture propagation: we observethat, as the crack grows, τu

eq slightly increaseswhile τ f

eq remains nearly constant (see fig. 2).

3.1. The cohesive zone

The concept of cohesive zone was original-ly introduced by Barenblatt (1959a,b) for a ten-sile crack and subsequently by Ida (1972) forshear cracks in order to remove the physicallyunrealistic singularity of dynamic stress at thecrack tip and to avoid an unbounded energyflux at the rupture front. The cohesive zone isthe region of shear stress degradation near thecrack tip; it is located just behind the rupturefront and it is also named breakdown zone(Ohnaka and Yamashita, 1989). Different phys-ical processes can be responsible of the shearstress degradation within the cohesive zone,and we refer to them as breakdown processes.The most important physical quantity charac-terizing the breakdown process is the character-istic slip-weakening Dc. This parameter repre-sents the amount of slip required to drop the dy-namic traction from the upper yield stress (τu

eq)to the kinetic friction level (τ f

eq) and to absorbfracture energy (see fig. 1d). Dc is an input pa-rameter imposed a priori in the well known SWconstitutive law (Andrews, 1976a,b). The spa-tial extension of the cohesive zone scales withDc and both these parameters are important toassess the required resolution of numerical ex-periments (see Bizzarri et al., 2001).

As briefly mentioned above, numerical simu-lations performed by adopting RS constitutivelaws display a slip-weakening behavior of dy-namic traction (see fig. 1d). Bizzarri and Cocco(2003) refer to this parameter as the equivalentslip-weakening distance (d0

eq, here indicated asDc

eq) and suggest that it is related to the charac-

teristic length scale parameter (L) of RS formula-tion (eq. (2.1)) by the following scaling relation:

lnD L vv

bLc

n

u feq

eff

eq eq

0

init, ,

-

v

x xb

_l

i(3.1)

where v0 is the slip velocity value when the dis-placement reaches Dc

eq (see fig. 1c) and vinit isthe initial sliding velocity. This scaling relationwas obtained under the assumption that the rup-ture starts from a steady state value of the statevariable, but can be also generalized for an ar-bitrary initial condition of the state variable (seeeq. (9) in Bizzarri and Cocco, 2003).

We show in fig. 3 the slip-weakening curvesfor a set of neighboring points resulting fromthe simulation illustrated in figs. 1a-d and 2 (i.e.the reference model). This figure reveals that, inour simulations, Dc

eq does not depend on thedistance from the nucleating patch. We infer avalue of Dc

eq larger than the adopted L parame-ter and the ratio Dc

eq/L ranges between 15 and20, in agreement with Cocco and Bizzarri(2002), Bizzarri and Cocco (2003) and Lapustaand Rice (2003).

Fig. 3. Slip-weakening curves calculated for thereference model shown in the previous figures andlisted in table II. The different traction evolutions asa function of slip have been plotted for differentpoints located at different distances from the nucle-ation patch.

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We observed the slip-weakening curves re-sulting from several simulations performed byadopting different values of the L parameter(ranging between 8 µm to 15µm): the inferredDc

eq value increases for increasing L in agree-ment with the scaling relation expressed in eq.(3.1).

All these simulations were performed withparameter values representative of the laborato-ry scale (L ∼ 10 µm and Dc

eq ∼ 0.15 µm). The in-ferred Dc

eq values for actual earthquakes rangebetween several centimeters to meters (see Tin-ti et al., 2004 and references therein), suggest-ing that Dc may be a significant fraction of themaximum slip on the fault. However, the scal-ing of the constitutive parameters from labora-tory to real fault dimension is still an openquestion and different opinions exist about thereliability of these extremely large values of Dc

for real earthquakes (see Guatteri and Spudich,2000; Guatteri et al., 2003).

3.2. The velocity weakening frictional regime

Because we are interested in modeling thespontaneous dynamic propagation of an earth-quake rupture, we first consider the propaga-tion in a velocity weakening frictional regime.This frictional behavior is characterized by(B–A)>0 and relatively small values of the Lparameters. We present in this section the re-sults of several simulations performed on a ho-mogeneous fault. The model parameters usedfor these simulations are listed in table II. Theconvergence and stability conditions of nu-merical experiments are discussed in detail inBizzarri and Cocco (2003). We consider a setof parameters representative of the laboratoryscale and we assume a constant normal stressσn

eff (σneff= σn –pfluid; where pfluid is the pore fluid

pressure).The Velocity Weakening (VW) behavior

(A< B) represents the unstable regime that caus-es dynamic slip episodes on the fault plane, dy-namic stress drop and the emission of seismicwaves. For instance, we can associate VWpatches with intermediate depth fault portions,like those located between 3 to 15 km along theSan Andreas Fault.

3.2.1. Strong velocity weakening fault

We first model a VW regime characterizedby a relatively large difference of constitutiveparameters (B–A ≥ 0.6 MPa) and, according toBC96, we refer to it as a strong VW regime.The results of the simulations are shown infig.4a,b and the input and constitutive parame-ters are listed in table II. The strength parame-ter was introduced by Das and Aki (1977a,b)to quantify how a fault area is unstable andready to fail [S = (τu–τo)/(τo–τf)]. For a 2D faultgoverned by a linear SW law, Andrews(1976a,b) showed that S expresses also a limitto discriminate if a crack can (S<1.77) or can-not (S > 1.77) propagate with super-shear rup-ture velocity. For RS constitutive laws it is pos-sible to define a Seq equivalent to S by using thevalues of parameters τ u

eq, τo and τ feq (Bizzarri

and Cocco, 2003). The simulation shown in fig.4a,b, performed using the RS constitutive lawsfor a strong seismic behavior (B – A = 0.75MPa), is associated with a value of equivalentparameter S eq<1.77.

We can clearly see in fig. 4a that the rupturefront bifurcates; this was previously obtained byOkubo (1989) and Bizzarri et al. (2001). Thespatio-temporal plot shown in this figure hasbeen drawn with a resolution smaller than thatadopted for numerical calculations to better il-lustrate the increase of peak slip velocity and thejump in rupture front speed. Figure 4b shows thetemporal evolutions of slip velocity: black linesindicate slip velocity time histories at those faultpositions located before the acceleration of therupture front to super-shear speed; the coloredlines show slip velocity at those points wherethe crack is bifurcated. The increase of peak slipvelocity occurs within the cohesive zone and itis more evident for those points located beforethe rupture front bifurcation. After the bifurca-tion (x ≥ 4.5 m), peak slip velocity increases dur-ing the propagation of the external front (propa-gating at super-shear rupture speed), but it isnearly constant for the slow internal front.Moreover, these simulations illustrate that whenthe rupture jumps to a super-shear rupture ve-locity, the slip velocity drops to zero and thusslip heals before to re-accelerate in the internalrupture front. Thus, a strong VW regime allows

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the simulation of earthquake ruptures in agree-ment with theoretical experiments performedwith a slip-weakening law (see Bizzarri et al.,2001 and references therein) and with laborato-ry experiments (Rosakis et al., 2000).

The strong seismic areas have a large break-down stress drop (defined as the difference be-tween the yield and the frictional stress ∆τb==τu

eq – τ feq) and very large coseismic slip: in a

strong VW patch, slip and peak slip velocity arethree times larger than those obtained for the ref-erence model (fig. 1a-d) at the same distancesfrom the nucleation. The rupture velocity is closeto 2.6 km/s immediately after the nucleation andthe external front moves at about 4.5 km/s afterthe bifurcation. The separation between the twofronts increases going far from the nucleation,because of the different rupture velocities.

Fig. 4a,b. Dynamic rupture propagation on a strong velocity weakening fault. The model and constitutive pa-rameters used for these calculations are listed in table II. Panel a) shows the spatio-temporal evolution of slip ve-locity. We have drawn this plot with a reduced resolution in order to emphasize the increase in peak slip veloc-ity. Panel b) shows the time histories of slip velocity in different positions along the fault: black lines identifypoints located before the point where crack accelerates to a super-shear rupture velocity, while coloured curvesidentify those points located in the region where the crack has bifurcated.

a

b

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3.2.2. Weak velocity weakening fault

According to BC96 we define a weak faultas a fault characterized by a small differencebetween A and B (B – A < 0.05 MPa). The re-sults extrapolated by comparing figs. 1a-d and4a,b are representative of simulations character-ized by values of B and A parameters selectedto have (B – A) > 0.2 MPa. We experienced a se-vere difficulty in achieving a spontaneous nu-cleation and then spontaneous dynamic rupturepropagation with 0 MPa<B – A < 0.1 MPa per-forming many numerical experiments. We pointout here that decreasing (B – A) we increase thedimension of the nucleation patch, which is de-fined as (Dieterich, 1992)

( )b a

Lc

n

eff, =- vhn

(3.2)

where η is a geometric constant that dependson the crack type. Therefore, to avoid increas-ing the size of the nucleation patch we are con-fined to using a smaller L. We obtained a spon-taneous nucleation using (B – A) = 0.1 MPa andL = 0.8 × 10−6 m (see table II for the whole set ofadopted parameters). As expected, under theseconditions the slip velocities within the nucle-ation patch (where the instability is promoted)are much larger than those inferred during the dy-namic propagation within the weak fault portion.Our modeling results suggest that: i) peak slip ve-locity in weak regions can be even more than 10times smaller than that simulated for the refer-ence model shown in fig. 1a-d; ii) the breakdownstress drop is decreased by 30%, although yieldstress and kinetic friction level are both de-creased; iii) the equivalent slip weakening dis-tance decreases because both the breakdownstress drop and L decrease; iv) the inferred rup-ture velocity is smaller than that of the referencemodel and we never observed a crack bifurcation;v) despite the smaller values of the dynamic pa-rameters we still retrieve an evident slip weaken-ing behavior of the traction versus slip curves.

We believe that these results and the in-ferred behavior of a weak fault portion arephysically reasonable because, according toBC96, we expect that weak fault zones undergodynamic instabilities only during micro-earth-

quakes or when they are loaded by a dynamicrupture front propagating in an adjacent strongweakening area. We will model this latter be-havior in the next section.

4. Numerical representation of frictionalheterogeneities

The goal of this section is to model the dy-namic rupture propagation on a 2D heteroge-neous fault. Because in our approach the spon-taneous dynamic propagation is governed by aRS friction law, we represent the source hetero-geneities in terms of non-uniform distributionof constitutive parameters. As described above,we follow the findings of BC96, who proposedthat both the velocity weakening and the veloc-ity strengthening regimes (see Scholz, 1990,1998) are separated into two fields dependingon the values of constitutive parameters and theresponse to external loading (see table I). Asshown in the previous section an earthquakerupture can spontaneously nucleate only withina Strong Velocity Weakening area (S-VW)characterized by sufficiently large values of theparameters (B – A) and L. However, a dynamicrupture can also propagate in a Weak VelocityWeakening area (W-VW). Nucleation in theseregions occurs only for very small earthquakesor when it is forced by a sudden external stresschange. We recall here that in their originalclassification BC96 used the A and B parame-ters, which in the present study correspond toA = aσn

eff and B = bσneff.

The Velocity Strengthening (VS) behavior(defined by the condition A>B) models a stablesliding, i.e. an aseismic slip. By definition, it isimpossible to obtain spontaneous rupture propa-gation for a homogeneous case. The VS behav-ior was proposed to describe several portions ofthe San Andreas Fault characterized by creepevents (see Scholz, 1990). A VS area can becharacterized by the thickness of unconsolidatedsediments (like Southern California) or by un-consolidated gouge within the fault (like in Cen-tral California; Marone et al., 1991). Accordingto BC96 the velocity strengthening areas cancontribute to the arrest of an earthquake rupture.In particular, compliant areas are velocity

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strengthening fault regions that slip aseismicallybut can be driven to instability if they are suffi-ciently loaded by an abrupt stress increase due tothe rupture propagation in an adjacent velocityweakening area (BC96). The VS behavior ischaracteristic of the RS laws and it cannot besimulated using the SW law.

The two «intermediate» fields proposed byBC96, weak and compliant, have frictional veloc-ity dependencies that are close to velocity neutral:they can modulate both tectonic loading and thedynamic rupture process. Aftershocks can occuron compliant areas around a high slip patch, butmost of the stress is diffused through aseismic slip.

Fig. 5a,b. Dynamic rupture propagation along a heterogeneous fault: the adopted constitutive parameters arelisted in table III. The fault is represented as two patches having different values of the L parameter (L1 and L3

in table III): the external one has a larger value (1.6 times the inner one). Panel a) shows the spatio-temporal evo-lution of slip velocity. Panel b) displays the time histories of slip velocity in different positions along the fault:solid lines identify slip velocity computed for points located in the inner region (low L), while dashed lines iden-tify slip velocity computed for those points located in the external region (high L).

a

b

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In the following sections we will presentand discuss the results of different simulationsof dynamic crack propagation on a heteroge-neous fault representing frictional heterogene-ity in terms of a non-uniform distribution of ei-ther the L or B and A parameters.

4.1. Heterogeneous distribution of the parameter L

In this section we will present two numeri-cal experiments in which we consider differentvalues of the characteristic length L along thefault; all models and constitutive parameters arelisted in table III. We recall here that the L pa-rameter controls the state variable evolution andaffects the size of nucleation zone (eq. (3.2)).Therefore, with the increasing L value, the faultspends a longer time to reach an extensionequal to the critical length ,c and to initiate thespontaneous dynamic rupture propagation. Infig. 5a,b we show the results of a numerical ex-periment in which a fault patch with L = 9 µm issurrounded by a region with a greater L (L = 15µm). In panel a the spatio-temporal evolution of

slip velocity is illustrated. In panel b we presentthe time histories of slip velocity for differentpoints located along the fault. Figure 5a,b pointsout that the rupture penetrates within the regionhaving a greater L. As previously noted (fig. 1a-d), during the propagation the peak slip velocityincreases as the rupture front moves far awayfrom the nucleation patch and as soon as thecrack tip encounters the region with a greater L,peak slip velocity is immediately reduced, buteven in this region it starts to grow again. As inthe reference model, v2 (i.e. the velocity at thenew steady state) is almost the same for allpoints. We have plotted in fig. 6 the slip-weak-ening curves calculated for different points lo-cated along the fault strike for the same simula-tion shown in fig. 5a,b. Because the state vari-able evolution is different for different values ofthe L parameter, the dynamic traction also dis-plays a quite diverse behavior. When the ruptureenters the region characterized by a greater L,the value of Dc

eq and consequently the fractureenergy increase. Thus, fig. 6 corroborates thelinear relation existing between the equivalentslip weakening distance Dc

eq and the L parame-ter (eq. (3.1)). The kinetic friction level is nearly

Fig. 6. Slip-weakening curves calculated for the simulation shown in the previous figure, whose model andconstitutive parameters are listed in table III. The traction evolutions as a function of slip have been plotted forseveral points located on the two patches having different values of L parameter at different distances from thenucleation patch.

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the same in the two regions. We emphasize thatthe slip-weakening curves maintain the expectedtrend also during the propagation along a het-erogeneous fault but the dynamic physical quan-tities (such as dynamic stress drop or fractureenergy) depend on the constitutive parameters.

The simulation presented above is character-ized by a small contrast of the L parameter in thetwo regions. We show in fig. 7a,b the results ofa numerical experiment in which the variation ofL is more pronounced: we simulate the dynam-ic propagation along a fault where the inner re-gion has L = 10 µm and the external region hasa higher L value (L = 1 mm). In this case therupture is unable to penetrate the external patch.This configuration was named «barrier model»by Bizzarri et al. (2001). In fact, we observe thatduring the propagation within the inner regionthe rupture behavior is identical to the referenceconfiguration of fig. 1a. When the rupture frontapproaches the high-L region (L is 1 mm, thusthe contrast is 1000), a back-propagating heal-ing front causes the crack arrest, as the rupturedoes not have enough energy to break the barri-er and propagate inside the high-L area. Thisprocess is known as a «barrier-healing». Slip ve-locity time histories of different fault points areplotted in fig. 7b: we can clearly observe the in-

crease of the peak as rupture propagates awayfrom the nucleation patch. The back propagatingfront causes a remarkable decrease of the slipvelocity with a consequent healing. In this caseour simulation yields a slip velocity time histo-ry with a finite duration.

4.2. Heterogeneous distribution of (B–A)

In this section we will present and discusssome numerical experiments in which a veloc-ity weakening zone is surrounded by velocitystrengthening region. In the first simulation wecompute the dynamic propagation of an earth-quake rupture along a fault on which a veloci-ty weakening patch is adjacent to a velocitystrengthening region. The initial and constitu-tive parameters are listed in table III. Figure8a,b shows the results of these calculations: thedynamic rupture nucleates and propagateswithin the velocity weakening zone, thus itpenetrates within the velocity strengthening re-gion for a small distance before being arrested.The rupture arrest is gradual and it generateshealing of slip only when rupture is stopped in-side the VS region. In panel b we plot the slipvelocity time histories, which show the expect-

Fig. 7a,b. Dynamic rupture propagation along a heterogeneous fault: the adopted constitutive parameters are list-ed in table III. The fault is represented as two patches having different values of the L parameter (L1 and L3 in tableIII): the external one has a larger value (1000 times the inner one). Panel a) shows the spatio-temporal evolution ofslip velocity. Panel b) displays the time histories of slip velocity in different positions along the fault.

a b

340


ed behavior (similar to that one of fig. 2) untilthe crack tip penetrates within the VS zone;thus, peak slip velocity gradually decreasesand heals. The healing phases (i.e. the way inwhich slip velocity drops to zero) in fig. 7a,band fig. 8a,b are evidently different: the first,due to a barrier, is more evident and abruptthan the latter.

Figure 9a,b shows the results of a numericalexperiment in which two VW regions are sepa-rated by a VS patch located between them. Theinitial and constitutive parameters are listed intable III. As expected, the rupture initially ac-celerates within the VW region and it partiallypenetrates within VS area. Slip velocity is pro-gressively attenuated and the crack tip deceler-

Fig. 8a,b. Dynamic rupture propagation along a heterogeneous fault: the adopted constitutive parameters arelisted in table III. The fault is represented as two patches having different values of the a and b parameters (a1,b1 and a3, b3 in table III): a velocity strengthening area is adjacent to the velocity weakening patch where therupture nucleates. Panel a) shows the spatio-temporal evolution of slip velocity. Panel b) displays the time his-tories of slip velocity in different positions along the fault.

a

b

341


ates. In this area, the rupture velocity is verylow and the crack is almost arrested. Because ofthe small dimension of the VS region, the crackis able to propagate within the VS patch and itre-accelerates again when it reaches the exter-nal VW region. Figure 9b shows the time histo-ries of the slip velocity. It emerges that slip ve-locity has a finite duration only during the rup-

ture propagation within the internal VW region,while in the external VW region it does not re-turn to zero (i.e. no healing of slip).

These simplistic simulations provide a pic-ture of the complex interactions between faultpatches having different frictional properties.Our 2D simulations illustrate how the tractionand slip velocity evolutions are modified during

Fig. 9a,b. Dynamic rupture propagation along a heterogeneous fault: the adopted constitutive parameters arelisted in table III. A narrow velocity strengthening patch separates two velocity weakening areas (identified bythe values of the parameters (a1, b1), (a2, b2) and (a3, b3) in table III). The rupture nucleates within the inner ve-locity weakening patch. Panel a) shows the spatio-temporal evolution of slip velocity. Panel b) displays the timehistories of slip velocity in different positions along the fault.

a

b

342


the propagation on heterogeneous faults. Thesecalculations propose stimulating implicationsfor slip duration and fracture energy. We willdiscuss these issues in the following sections.

5. Implications for slip duration

The simulations presented in the previoussections confirm that the dynamic rupture prop-agation on a homogeneous fault governed by aRS friction law (the slowness formulation de-fined in eq. (2.1)) does not show the healing ofslip (as clearly shown in figs. 2 and 4a,b). Thisis in agreement with the results of Perrin et al.(1995) and Bizzarri et al. (2001). Although dif-ferent regularizations or modifications of theconstitutive formulations have been proposed toobtain the healing of slip and/or self-healingpulses (see Cocco et al., 2004 and referencestherein), even source heterogeneities can pro-duce a finite duration of slip velocity and shortslip durations (see Beroza and Mikumo, 1996;Day et al., 1998). In the present study we test-ed two different heterogeneous configurations;the first is based on the variation of the L pa-rameter and the other on the interaction be-tween VW and VS patches represented bychanging the values of A and B.

A strong contrast of the L parameter repre-sents a barrier and produces a «barrier healing»(see Zheng and Rice, 1998; Bizzarri et al.,2001) similarly to the arrest on the crack-likerupture propagation. Figure 7b shows that theslip duration depends on the distance from thebarrier and it is shorter for closer distances tothe barrier. Actually, the slip velocity time his-tories of the points closest to the barrier have ashorter duration than those located closer to thenucleation region. The interaction between VWand VS still yields finite slip durations (see fig.8a,b). It is important to emphasize that the timehistories of slip velocity simulated for hetero-geneous configurations do not resemble a self-healing pulse, although they have a finite dura-tion. The latter is usually defined as a pulsepropagating along the fault with nearly constantslip duration; in this case healing is independentof the crack arrest phase. In other words, whilewe simulated finite and relatively short slip du-

rations, we are unable to generate self-healingpulses. These results are consistent with thefindings of Perrin et al. (1995) and corroboratethe outcomes of Day et al. (1998), who sug-gested that source heterogeneity yields healingof slip. An important implication emergingfrom these results is that healing of slip doesnot require traction re-strengthening: total dy-namic traction remains at the kinetic frictionlevel also when slip velocity tends to zero.

6. Implications for fracture energy

We computed fracture energy for the differ-ent simulations performed in this study. Thefracture energy is defined as the amount of en-ergy that the crack spends to advance and in-crease its length. It has been defined as

G du21

f0

=3+

-x x# _ i (6.1)

and is measured as the area below the tractionversus slip curve shown in figs. 3 and 6 andabove the minimum traction τ f. For a dynamicrupture propagation on a homogeneous faultgoverned by a SW law the fracture energy G isconstant and known a priori over the wholefault surface [G=1/2(τu−τ f)Dc]. This is not thecase when rupture propagation is governed byother constitutive laws or when the constitutiveparameters are not uniform on the fault.

In order to quantify the fracture energychanges during the crack growth, we plot in fig.10a-d the computed G as a function of the posi-tion along the fault for four different configura-tions investigated in this study. Figure 10ashows the resulting values for the homogeneousconfiguration (the reference model shown infig. 1a-d), which reveals an increasing trend asthe rupture propagates far away from the nucle-ation patch. This is in agreement with the resultsshown in fig. 2, which illustrate that the maxi-mum yield stress increases moving far awayform the nucleation patch, while τ f

eq slightly de-creases. Figure 10b-d displays the resulting val-ues computed for three heterogeneous configu-rations. In particular, fig. 10b corresponds tothe non-uniform distribution of the parameterL: as the rupture penetrates in the region with

343


higher L, the computed fracture energy increas-es jumping to a larger constant value; this is al-so expected looking at the results shown in fig.6, which displays the increase in Dc

eq.Figure 10c refers to the rupture arrest within a

VS area shown in fig. 8a,b: after an initial in-crease of G with increasing distance from the nu-cleation patch, consistent with fig. 10a, we noticea decrease of G within the VS area. This is due tothe decrease of dynamic stress drop and peak slipvelocity shown in fig. 6. In the VS patch the totalamount of work spent to allow the crack advanceis lower than that in the VW patch. Finally, fig.10d shows the fracture energy estimated in dif-ferent fault positions during the rupture propaga-tion in the heterogeneous fault modeled in fig.9a,b: the propagation within the VS area pro-duces a sudden drop in G, which is followed byan increase when it starts to propagate again with-in the external VW region.

The calculations summarized in fig. 10a-dpoint out that frictional heterogeneity explains

the variability of fracture energy on the faultplane, which is associated with both the varia-tions of slip and breakdown stress drop.

7. Conclusive remarks

One of the main goals of this study is to ex-tend the results of BC96 to investigate the dy-namic rupture propagation on a 2D fault with aheterogeneous distribution of constitutive pa-rameters. We used the rate- and state-dependentformulations to characterize fault hetero-geneities following the findings of BC96, whoproposed to split the velocity weakening andthe velocity strengthening regimes into fourdistinct frictional fields. Our results corroboratethe conclusions of BC96 demonstrating that avelocity strengthening area can arrest and canalso be driven to a dynamic instability by anearthquake rupture propagating in the adjacentfault patch. We have represented numerically

Fig. 10a-d. Fracture energy values in different positions along the fault for four configurations investigated inthis study: a) refers to the reference model shown in fig. 1a-d; b) refers to the heterogeneous L distribution shownin figs. 5a,b and 6; c) and (d) refer to the heterogeneous distribution of the a and b parameters shown in figs.8a,b and 9a,b, respectively.

a b

c d

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the fault heterogeneity by assigning differentvalues of L or (B−A) parameters along the faultline. Our simulations show that the interactionbetween the propagating dynamic rupture frontand the heterogeneous fault patches depends onthe values of the constitutive parameters. Inparticular, we have shown that a variation of theL parameter can modify the peak slip velocityor can arrest the rupture propagation dependingon the value of the L contrast. The heterogene-ity of the L parameter does not modify thebreakdown stress drop nor does it contribute tothe variations of rupture velocity if the contrastis smooth. On the contrary, the heterogeneity ofthe distribution of the difference (B−A) affectsthe dynamic rupture propagation in a morecomplex way: dynamic stress drop and strengthexcess strongly depend on B and A parameters.Moreover, rupture can penetrate within a veloc-ity strengthening area and the heterogeneousdistributions of B and A yield complex time his-tories of slip velocity.

We propose that frictional heterogeneitiescan explain the observed complexity of slip dis-tribution and the variability of rupture velocityduring earthquakes. Our results have importantimplications for slip durations (i.e. local risetime) and fracture energy. In particular, our re-sults are consistent with the findings of Perrinet al. (1995) and Bizzarri et al. (2001). A largecontrast in the L parameter represents a barrierthat produces a crack-like solution with vari-able but finite slip durations. Because the varia-tions of constitutive parameters affect both thecritical slip-weakening distance (see Bizzarriand Cocco, 2003 and references therein) andthe breakdown stress drop, the inferred fractureenergy varies along the fault. We have shownthat the increase in the L parameter results in afracture energy increase and that heterogeneousdistribution of (B−A) yields evident variationsof fracture energy along the fault. Because wemodel here a laboratory fault, our estimates(∼103 J/m2) of fracture energy cannot be com-pared with those inferred for real earthquakes.

In this study we mainly focused on the inter-actions between the propagating dynamic rup-ture front and the heterogeneous fault patches.According to BC96, the response of fault patch-es having different frictional properties to a con-

stant tectonic load also controls the pattern ofseismicity and the behavior of crustal faults.Thus we conclude that fault frictional propertiesand their variations on the fault plane play an im-portant role in characterizing crustal faulting andthe mechanical properties of major fault zones.

Acknowledgements

We thank Stefan Nielsen for helpful discus-sions. We thank Antonio Rovelli, Maria ElinaBelardinelli and Peter Moczo for their detailedreviews of this paper.

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(received December 02, 2004;accepted March 15, 2005)

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