Numerical Model of Seismic Rupture

JOURNAL

OF GEOPHYSICAL

RESEARCH,

VOL. 97, NO. Bll,

PAGES 15,291-15,295, OCTOBER

10, 1992

Numerical Model of Seismic Rupture STEFAN B. NIELSEN AND ALBERT TARANTOLA Institut de Physique du Globe de Paris, France

Simplenumericalmodelsof the cellularautomatontype havebeenproposedrecently, as an analogyfor seismicfaults. Thoseshowedinterestingfeaturesof spontaneous rup-

ture evolution(Lomnitz-Adler and Lemus-Diaz, 1989)or evenseismicrecurrence (Bak andTang,1989).It is possible to incorporate realistictheologyandtensoriMphysics into this kind of model, to extendit to a portion of crust insteadof a singlefault-planeand to simulate its time evolution. A numerical model based on physicsof continuum media,

instead of a cellular automaton,can simulatedynamic rupture in two dimensions.The mediumcan be submittedto increasingload whichis introducedby an imposeddisplacement at the boundaries,spaceand time being discretizedby a finite differencescheme.

The equationof dynamicsis usedat eachtime step. A rupturecriterionallowseachnode in the grid of the mediumto fail if a thresholdis reached.The brittle thresholdis allowedto fluctuatefrom point to point in the medium accordingto a chosendistribution. Failureis representedby a lossof shearstiffnessin the rock elementconcerned.Hence we distinguish twostatesin our medium(the unbroken,viscoelastic theologyallowingfor a slightattenuationand the brokenstate,similarto a viscousfluid) the viscosityallows introductionof a kind of dynamicfriction at the brokenpoint. When failure takesplace at a point, elasticstressis then locallyreleasedand radiates,sometimescausingrupture of neighborpointsandlocalizationof fracturealongfissures or faults. This givesrealistic syntheticseismogramsand sourceslip history. INTRODUCTION

rmann and Rouz, 1990]. But thesemodelsare devel-

With respectto seismicprediction,it is interestingto opped in the static limit, whereasdynamicsare determake a mode] capable of simulating spontaneousrupminant in the earthquake phenomena. ture evolution, or even whole sequencesof earthquakes We believe that studying spontaneousevolution of distributed in space and time, to study seismicitypatrealistic seismicEarth model could lead to preciousinterns. This has already been done but by adopting sight in the mechanismsof earthquakerecurrence,fault physicallysimplifiedmodelsthat couldnot affordmore developmentand in the influenceof all-scalevariations than a qualitative comparisonwith real E•rth. The of medium properties and regionalstressconditionson first remarkableattemptin suchdirection[Burridgeand earthquake'ssize, shape, and dynamics. Knopoff,1967] was basedon a one-dimensional analThe main questionsraised by such approach were as ogy with a systemof springs•nd masses.Somemore follows: physicalmodelshavebeenproposed[Mikumoand Miy1. Is it possibleto obtain localisationof fracture in atake,1979],but they did not allow for properradia- model with spontaneousevolution and realistic tensotion of elastic waves;furthermore, the fault plane was rial physics? imposed.Bak and Tang[1989],describea simplemodel 2. If so, is it possibleto includedynamics,taking into usingonly a scalarfieldandcompareits behavior(mag- account wave propagation, 3. Does the result look realistic? nitude-frequency relation) to regionalseismicity. Furthermore,usualnumericalfault modelswith sponIn a first attempt to answerthe questionsabove, we taneousfracture propagationsufferfrom the fact that present a model which simulatesone rupture and its rupture is investigatedon a given plane, whether the dynamic propagation. surroundingmedium is taken to be three-dimensional Tug MOD or not. The assumptionthat faulting in the Earth ocOur unbroken medium is assumed to be elastic and curson a plane is oversimplified.Prescribinga rupture slightly attenuating;the constitutivestress-strainrelaplane is a deterministicapproachwhich preventsmodtion is a convolution in time with a relaxation function, eling a totally spontaneous evolutionof a seismicevent in a medium where only the rheologicalpropertiesare linear up to the threshold of rupture. This is a more given. Suchmodelsmust now be enhancedwith more generalapproachthan a simpleHooke'slaw, allowingto realistic mechanics.It is alsoimportant to try to incor- model complexmedia, suchas attenuating,viscoelastic porate in the modelsmore sophisticatedand realistic or viscousmedia, accordingto the choiceof relaxation friction laws, that may play a cardinal role in the oc- function' currenceand dynamicsof earthquakes. In another approach to rupture phenomenain gent) teral, fracture in disorderedmedia is beinginvestigated

/:

usingnumericalmodelsfor fractureon a lattice [HerCopyright 1992 by the American Geophysical Union. Paper number 92JB00205.

•Pi)• beingthe relaxationfunctionof the medium, whichcan includeattenuationand anisotropy[Tarantola, 1988],aiI the stress,el5the strain(in the elastic casethat reduces to •bijkl(x,t)Our numerical solution of the problem is computed

0148-0227/92/ 92JB-0205$05.00 15,291

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NIELSENAND TAIt.ANTOLA:NUMERICALMODEL OF SEISMICRUPTUR. E

by a finite difference schemein space and time with a

generates viscous resistance instead of friction. This is

staggeredgrid [Virieux, 1986]. The convolutionwith

acceptableif we considerthat a rock element between two grid nodes has the same thickness as the "fault

the relaxation

function

is then

discrete

in time

and is

calculated at every node of the grid where stressand deformation

are defined.

The

relaxation

function

can

gouge"betweentwo fault lips. A fault then develops assoonasa fewneighborelementsare broken,allowing

representeither elastic, attenuating, or viscousmedia. slip alongtheir direction of alignement. This point is essential, as viscoustheology shall allow The mediumis loadedrapidly and homogenously unbrokenpointsto be shearedin the directionof fault slip. til the threshold is reached somewhere inside the mediSince we deal with rupture, we must introduce a um. We load the mediumby imposinga displacement kind of nonlinearity in our medium behavior. We do speedparallel(shear)or perpendicular (compression) this by using a "rule" defining failure as a transition to two opposite boundaries, so that the load is increasfrom solid to viscoustheology. This rule could possi- ing homogeneouslyinside our medium until it breaks bly take into account the history of the medium. For somewhere. Then the loadingis "frozen"(sincethe reoverthe time the momentwe took a stress-based criterion (general- gionaltectonicstressincreaseis negligible

ly knownas the Coulombcriterion),whichsaysthat if the maximum shear stress surpassesa threshold at a point then this point breaks. Hencethe theologyis linear up to the threshold of rupture. The Mohr-Coulomb theory predicts failure along a plane of orientation • when the stresseson this plane satisfy the inequality

I •rr8 I -P•8• -> So, where•rreand •re• are shearand normal stresseson the plane. However, we are not investigatinga givenplanein the bulk of the medium,but any plane at a given point in space. Hence we use the maximum

shear stress r and the mean normal

stress •

to checkfor rupture: I r I -Pa > So, wherep is the internal friction and So the rock's local strength. The strength of the medium can then be allowed to fluctuate from point to point accordingto a chosendistribution. When a point of the medium breaks, we re-

of a dynamicrupture)andwelet the ruptureevolvedynamicallyby mere elasticradiationof stresscausedby rupture. PRELIMINARY RESULTS: EXAMPLE Or DYNAMIC RUPTURE PROPAGATION FROM A SINGLE FLAW

If we introducea preexisting flaw in our medium(a brokenpoint or a smallfissureas an alignementof brokenpoints),the stressconcentration inducedby the flaw will causefailure to start at the tips of the crack,then propagate. Thus we see the developmentof a single fault, and we can follow its history of rupture in time (Plates1 to 3). The strength probability distribution chosenis a box-

car(constantprobabilitybetweentwobounds),with no

placerocktheology(elastic)by fluid theology(viscous) value correlationbetweenpoints. The mediumis about at that point. Such a body has no shear stiffnessbut 2000 m side, space sampling of the grid is 8 m and only shearviscosity.This is equivalentto assumingthat time iterationsare 1 ms. The pattern looksquite comthe fault is not a flat surface but has a finite thickness plex;hencethere is a complexsourcehistory,dueto the (of the orderof the spacesamplingstepof the grid), in- "barriers" or strongpatchesof the mediumwhich slow sidewhichsheardeformationrate (insteadof slip rate) down the rupture propagationmomentaneously, caus-

Plate 1.

Vertical acceleration wave field induced in the me-

dium aroundrupture. The rupturehasjust startedto propagate.

Plate2. Propagating wavefieldof acceleration duringprogressof rupture. The fault traceis slightlyvisibledipping diagonally in the middle of the wave field.

NIELSEN AND TAP.,ANTOLA: NUMERICAL MODEL OF SEISMIC RUPTURE

15,293

Plate 4. Final vertical displacement induced by the rupture in the medium. The fault trace is clearly visible between

Plate 3. Propagating wave field of accelerationafter rup-

purple (negativeor downwarddisplacementin the plots) and green(positive)lobes.

ture has stopped.

ing higher frequences,inhomogenityin slip along the fault, and bifurcationsor changesin fault orientation. In Plate 4 we see the displacementinduced in the medium by the slip on the fault, and Figure 1 shows the growth of the dislocationalongthe fault with time. In this example, slip is occurringalong a fault that developsup to 400 m in about 0.05 s. The speedof rupture propagationis in this caseabout 80% the speedof P wave. The fact that rupture propagates at supershear velocity is apparently due to the rupture criterion

that is concordantwith the well-knownlaw fc •x Vr/L (wherefc is the frequencyat whichthe spectrumamplitude drops, Vr is the global rupture propagationspeed and L is the final fault length [Hanksand Mc Guire, 1981]),thus a cornerfrequencyof about 5Hz. The ac-

celeration spectrumroll-offdependence is onw2. After the rupture stopped and the wavesradiated, there is stressstrongly concentratedat the tips of the fault, and in somepatchesof the fault where a barrier or a changein fault orientation preventedfree slip on the fault.

which is based here on absolute locM'stress. If we take a criterion based on the difference between stress at the

Rupture stop is imposed here as a limit time after which rupture is not allowedany more. The crackstays

crack tip and inside the crack, as defined by Aki and

"open"oncebroken(no healingcondition)evenafter the rupture growth has stopped. This leavesa slight surfacewave on the crack lasting sometime after the end of rupture but rapidly vanishingby radiation and

Richards[1980, p. 905], our simulationyeldsa rupture velocity of the order of S wave propagation. We found out that rupture speedwas sensitiveto the type of strengthdistributionin the medium, droppingwhen the strength valueswere sparse. Figure 2 plots crackspeedsobtainedin a simulation for different strengths. The strength probability distribution

is a box car.

The box cat's central value is

called"mean,"the width is called"range"(on the plot, M and R). In this simulationthe meanof the distribu-

tionis keptat l0s Pa, whiletherangevaryesfrom10• to 1.3 x l0 s. We seethat the crackspeedsdrop as the distributionrange approachesand surpasses the mean. The syntheticseismograms obtained(Figure 3) display a cornerfrequencyin their spectrum(Figure 4)

friction.

The boundaryconditionsat the edgesof the lattice do not play a role during the rupture sincethey are further than the distance coveredby dynamic wavespropagation within the rupture time interval (the mechanical processis "blind" to the medium outside the causal cone). To investigatelattice influencein this model,we did the followingtest: we simulated twice a crack propagation that lasted 0.02 s. The first computationwas diseretizedon a grid samplingevery 8 m, and the seeond was diseretizedevery 4 m. The time step was also

Fig. 1. Sliphistoryon the fault: horizontaldimension represents the fault andthe tracesrepresent slipat increasing times along the third axis.

15,294

NIELSEN AND TARANTOLA: NUMERICAL MODEL OF SEISMIC RUPTURE

Pr*ob•bility O_

•

e

S•reng•h

!

, t

•

-

c}

t

0.5 strength

dispersion (R/M)

Fig. 2. Speedof crackpropagationin a two-dimensional simulationas a functionof strengthdispersion.The strength probabilitydistributionis a box car. Crackspeeddropswhenthe strengthvaluesare rangingin an intervalR of the sameorder as the mean M, i.e., when the strengthsare very sparse. decreased from 10-a s to 5 10-4 s so that the ratio of

spaceversustime samplingwas the same in both simulations. Hence the fault was sampledtwice as much in the secondcomputationthan in the first one;we can comparethe synthetic seismogramsthus obtained and see the effect of a changein the lattice. The two accelerogramsare presentedin Figure 5. They are quite similar, except that the one generated with the finer grid showsoscillatioinsof higher amplitude. DISCUSSION AND CONCLUSIONS

Fig. 3.

Synthetic seismogramat a few hundred meters of

thesource (m/s:•versus seconds).

uation.

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NIELSEN AND TARANTOLA: NUMERICAL MODEL OF SEISMIC RUPTURE

15,295

mic Earth, such as the relation between parameters of the medium and the seismicoccurrence,recurrenceand size, to understand the physicsunderlying seismicity problemsand to study aspectsof fault developmentand sourcedynamics. Acknowledgments. Many thanks to Pascal Bernard for numerous constructive discussions,and to Jean-Paul Montagnet for his encouragement. This work performed in the

SeismologicalLaboratory of Paris IPG was partially spon0

0.1

0.2

soredby the INSU (DBT program),by MRT with a grant for oneof us(S.N.) andby the FrenchMinistryof Education, which provided us accessto the Connection Machine.

time (s) Fig. 5. Accelerograxnsobtained for a crack propagation lasting 0.02 s in a homogeneousmedium. Comparison between computation on two different scalesof grid: solid line obtained with a samplingin spaceand time of 8 m and 0.001

s; dashedline obtainedwith 4 m and 0.0005s (twiceasmany samplingpoints).

Possibly,a three-dimensionalversionof the model wouldgivea morerealisticfrequencycontent(the high frequencies are presumablymoreattenuatedby destructive interferenceon a surface rupture than on a lineic

rupture)andradiationpattern. Suchquantitativecomparisonsshouldbe morerelevantin the far field, but our model is up to now limited in spaceto only a few times the dimension of the source. It is not clear at present

REFERENCES

Aki, K., and P.G. Richards, Quantitative Seisinology,vol. II, 893-905, W. H. Freeman, New York, 1980. Bak, P., and C. Tang, Earthquakes as a self-organizedcritical phenomenon, jr. Geophys. Res., 9•, 15,635-15,637, 1989.

Burridge, R., and L. Knopoff, Model and theoretical seismicity, Bull. Seimol. Soc. Am., 57, 341-371, 1967. Hanks, T.C. and R.K. McGuire, The character of highfrequency strong ground motion, Bull. Seismol. Soc. Am., 71, 2071-2095, 1981. Herrmann, H.J., and S. Roux, Statistical modelsfor the/racture o.fdisorderedmedia, North-Holland, New York, 1990. Lomnitz-Adler, J., and P. Lemus-Diaz, A stochastic model for fracture growth on a seismicfault, Geopys. J. Int., 99, 183-194, 1989.

Mikumo, T., and T. Miyatake, Earthquake sequenceson a frictional fault model with non-uniform strengths and relaxation times, Geophys. J. R. Astron. Soc., 59, 497-

522, 1979. what is the influenceof a spatiallowercutoff(discretisationof the medium)and highercutoff(limitation of Tarantola, A., Theoretical background for the inversion of waveforms including attenuation, Pure App. Geophys., the mediumsize). 1œ8, 365-398, 1988.

We have not been able up to now to generate a cata-

Virieux, J., P-$V wave propagation in heterogeneousmedia

loguewith sequences of eventsin time, sinceoncefrac- velocity-stressfinite-difference method, Geophysics,51, 889-9017 1986. ture starts, it is does seldom stop spontaneouslybut propagatesthroughoutthe model. We must imposean S. B. Nielsenand A. Tarantola, Laboratoire de Sismologie, arbitrary time limit for rupture propagationif we want Institut de Physique du Globe, 4 Place Jussieu, F-75252 it to stopat an acceptablesize. We alsodid not succeed Paris Cedex 05, France. in introducinga kind of healing of the broken points, without provokinga disturbinginstability of the rup(ReceivedJanuary23, 1991; ture evolution. We have good hopesthat this model revised July 3, 1991; can be appliedto investigateproblemsof the true seisacceptedJanuary23, 1992.)