GEOPHYSICAL
RESEARCH
LE•fERS, VOL. 18,NO.5, PAGES893-896,MAY 1991
FAULT STEPS AND THE DYN•C
R•
PROCESS: 2-D NUMERICAL
SPONTANEOUSLY PROPAGATING SHEAR FRA•
SIMULATIONS
OF A
RuthA. HarrisandRalphJ. Archuleta
Department
of Geological
Sciences,
Universityof CaliforniaatSantaBarbara
StevenM. Day
Department
of Geological
Sciences,
SanDiegoStateUniversity
•11gr&9i.
Faultsteps
mayhavecontrolled
thesizesof the
1966Parkfield,1968 BorregoMountain, 1979 Imperial
Valley,
1979CoyoteLakeandthe 1987Superstition
Hills
earthquakes.
Thisp.roject
investigates
theeffect
offaultsteps
MAP VIEW
ofvarious
geometries
on thedynamicruptureprocess.We
haveuseda finite differencecodeto simulatespontaneous
COMPRESSIONAL STEP
DILATIONAL STEP
rupture
propagation
in twodimensions.
Weemploy
a slip-
weakening
fracturecriterionas the conditionfor rupture
propagation
andexamine
howrupture
ononeplane
initiates
rupture
onparallelfaultplanes.Thegeometry
of thetwo
parallel
faultplanes
allows
forstepover
widths
of0.5to10.0
....
stepover
• •
•
width
overlap
krnandoverlapsof-5 to 5 kin. Our resultsdemonstrate
that
thespontaneous
rupture
onthefarstfaultsegment
continues
to
propagate
ontothe second
fault segment
for a rangeof
geometries
forbothcompressional
anddilational
faultsteps.
Amajor
difference
between
thecompressional
anddilafional
cases
is.that a dilationalsteprequiresa longertime delay
between
therupturefrontreachingthe endof the firstfault
/
SIDE VIEW
IIII
...... .v..-..-..-.
.......
II
•.•.".... •' "i
.ft/tttt/tt/t//ttt/tttt///ttttl
segment
and initiatingruptureon the secondsegment. Fig. 1. Rightandleft stepsin a left-lateralverticalstrike-slip
Therefore
ourdynamics.tudyimpliesthata compressional
step fault. Whentwoof thefaultsegments
areslippingatthesame
willbejumpedquickly,whereas
a dilational
stepwill causea
time,a rightstepis a compressional
stepanda left stepis a
timedelayleadingto a lowerapparent
rupturevelocity.We
di!afional
step.Forright-lateral
fauks,fightsteps
aredilational
alsofind that the ruptureis capableof jumpinga wider andleft stepsarecompressional.
The stepover
widthis the
dilational
stepthancompressional
step.
perpendicular
distance
between
thetwofaultsandtheoverlap
is thealong-strike
distance
of faultcrossover.
Whenthetwo
Introduction
faultendsdo notpasseachothertheoverlapis negative,as
Faultsegmentation
is observedat manydifferentlength
shownfor thecompressional
step.
scales
in the field, from centimetersto kilometers,but how
doesit affectthe dynamicruptureprocess?It hasbeen
proposed
thatonetypeof fault segmentation,
thefaultstep
(Figure
1) mayhavecontrolled
thesizes(magnitudes)
of the
1966Parkfield,1968BorregoMountain,and1979Imperial
Valley,earthquakes
[Sibson,1986], the 1979CoyoteLake
earthquake
[Reasenberg
andEllsworth,1982;Sibson,1986],
andthe1987Superstition
Hills earthquake
[Rymer,1989].
However,
to dateno onehasgivenlimitsto thegeometrical
Previous Studies
Sibson[1986] presenteda numberof casestudiesand
proposed
thatfault stepsmightaffecttheruptureprocess;
but
to dateno dynamiccalculations
(spontaneous
rupture)have
beenmadefor discontinuous
faults consistingof two or more
non-cop!anar
segments.Rodgers[1980]studiedthe static
problemof fault stepsusingdislocationtheorybut did not
includethe importantfactorof fault interaction.Segalland
Cananearthquake
'jump'a 1 km stepor a 5 km step?If
wecoulddete•ne which geometries
stopa propagating Pollard[1980] analyzedthe 2-D (planestress)quasi-static
parameters
thatcouldaffecttheruptureprocess.
earthquake
rupture,
thenwe mightalsodetermine
thesizeof
anexpected
earthquake.
Thelengthof fatfitrupture
isdirectly
problem
bylookingattheeffectsof pre-existing
echelon
shear
crackson theregionalstressfield andincludingtheelastic
proportional
to seismicmoment.A rupturewhichpropagates interactions between the cracks. Their static solution showed
onthe
through
manyequal-length
faultsegments
willproduce
a much thatfor left stepsin right-lateralshear,normal-traction
cracktipsincreases,
therebyinhibiting
frictional
larger
earthquake
(e.g.theM8 1857Ft. Tejonearthquake) overlapping
Forrightsteps
infightlateralshear,
thestatic
solution
than
onewhichonlypropagates
through
oneor twoof the sliding.
suggests
thatslidingis enabledby a reduction
in thenormalsegments
(e.g.the M5 1966Parkfieldearthquake).The
problem
is determining
the minimumstepdistance
which tractions. Basedon theseresults,Segall and Pollard [1980]
proposed
thattheleft(compressional)
stepsshould
bea barrier
cannot
be•'urnp ed by a dyn amicall
rupture
yp ropagating
. As
.
to
fault
slip
and
could
stop
an
earthquake.
aninitial
approach
weexamine
thecaseofvertical
strike-shp
faults
inalinearly
elastic
medium.
Weusea two-dimensional Mavko [1982] alsomodeledfault interactionusing2-D
quasi-static
calculations.
He usedhisresultsto successfully
finitedifference
program
to simulate
the spontaneouslypredict
the
creep
records
nearHollisfer,CA. Aydin and
propagating
shearfracture.
Schultz[1990] lookedat the fault interactionproblemin an
attempt
tounderstand
therelationship
between
faultstepand
overlaps
for en echelon
strike-slip
faultsaroundtheworld.
Usinga quasi-static
study,
theyconcluded
thatfaultinteraction
isresponsible
fortheobserved
enechelon
faultgeometr?.
Copyright
1991 by the AmericanGeophysicalUnion.
One essentialelement not included in the quas!-static
Papernumber91GL01061
studies
is thetimedependence
of faultrupture.Although
the
0094-8534/91
/ 91GL-0106!
$3. 00
893
Harrisetal.' Rupture
Propagation
across
FaultSteps
894
28 km
quasi-static
studiesgive insightinto processes
whichoccur
over a long time scale,they do not includethe stresswaves
andtime dependentstressconcentrations
generated
duringthe
earthquakeruptureprocess. It is the stressfield duringthe
dynamicrapturethatloadsthe nextfault segmentandsatisfies
a rupturecriterionthatdetermines
if a M6 earthquake
cascades
into a M8 earthquake. To examinehow the dynamicstress
field affects rupture on an adjacentparallel segmentwe
considera spontaneouslypropagatingshearfracturein an
a)
o
4--$--•
FAULT
2
FAULT 1
T= 3.4 sec
/
28 km
elastic me•um.
b)
tlover
Method
starts
10.9
10.7
8.4
7.9
To studythe dynamicfault-stepproblemwe use a 2-D
finite differenceprogram[Day and Skholler,writtencomm.,
1990] which is a 2-D versionof the 3-D methodusedby Day
6.1
[1982]. The newprogramalsoaccommodates
non-coplanar
faultsegments.The 2-D caseimpliesthatwe aresolvingthe
problemof planestrain. The two shearcrackshavehalflengthsof 14 kin. We initiatetherupturein themiddleof the
5.1
4.6
4.4
3.4
3.2
first fault segmentthen allow it to spontaneously
propagate
usinga slip-weakening
fracturecriterion[Ida, 1972;Day,
3.1
3.4
3.6
1982; Andrews, 1985] and a Coulomb friction law. Both
staticand dynamicfriction are proportionalto the normal
3.9
stress.When the fault first beginsto slip its strengthdrops
linearlyfroma staticyieldstrength
to a dynamicyieldstrength.
The slipdistanceoverwhichthisoccursis the slip-weakening
criticaldistance,d0. We usea do of 10 cm [Day, 1982]. At
theendof thefirst fault the staticyield stressis veryhighand
therupturecannotbreakthrough. Threepossiblethingscan
happen: 1) The rupturecan die at the endof the first fault
segment.This leadsto the shortest
rupturelengthandsmallest
earthquake. 2) The rupture can trigger the secondfault
segment,but then run out of energyand stoppropagating.
Thisleadsto a slightlylargerearthquake.3) Therupturecan
triggerthe secondfault segment,thencontinueto propagate.
Thislastcaseproducesthe largestearthquake
sincetherupture
lengthis longest.
Results
We presentthe resultsfrom a set of 2-D caseswhose
parameters
arelistedin Table 1. Eachof thesesimulations
is
for two shearcracks,simulatingtwo left-lateralverticalstrikeslipfaultsof inf'miteverticalextent.Eachtestcasesimulates
a
100 bar stressdrop (a very large earthquake)usinga static
coefficientof frictionof 0.75 (an 'average'valuefrom Byeflee
[1978]) and a dynamic coefficientof friction of 0.3. The
initial shearstressis 200 bars,emulatinga 'weak'fault setting
[Bruneet al., 1969]. It takes2.9 secondsfor the ruptureto
first reach the end of the first fault segment(a supershear
rupturevelocity as predictedby Das and Aki [1977] and
Andrews [1985]). Rupture of the secondsegmentthen
dependsuponon its geometricalrelationship
to theinitial fault
segment. Figure 2 summarizesthe locationand time of the
firstpoint(s)to ruptureon thesecondsegmentfor thecaseof 5
km overlap. Note that the dilationalstepsgenerallytrigger
laterthanthe compressional
stepwith equalstepoverwidth.
Furthermorethelocationsof theinitialpointof ruptureon the
offsetsegmentdifferfor the two geometries.
TABLE
1. Simulation variables
Initial shearstress(bars)
Initial normal stress(bars)
Static coefficient of friction
Dynamiccoefficientof friction
P-wavevelocity(lcm/s)
S-wavevelocity(lcm/s)
Density
(g/cm
3)
Grid size (km)
200.
333.
0.75
0.30
6.000
3.464
2.7
0.25
4.3
never
starts
"t
'
10
DISTANCE(KM)ALONGSTRIKEFROMMIDPOINTOF FAULT
1
Fig. 2a). Map view of two faultsat 3.4 seconds
for thecase
of a dilational step (left step in left-lateral shear)andthe
parameters
listedin Table 1. Both faultsare 28 km long.
Stepover
widthis 1 km, overlapis 5 km. Opencircleindicates
pointwhererupturefirstnucleated
onfault 1 at 0 seconds.
At
2.9 seconds
therupturefu'streachedthe endof fault 1. At 3.4
seconds
thepointmarkedby thesolidcircleonfault2 starts
to
rupture.After3.4 seconds,
therupturepropagates
bilaterally
on fault 2. b) Summary(mapview) of the resultsfrom19
simulationsof fault stepsin left-lateral shear. For each
simulation
onlytwo faultsexist,asdepictedin a). Fault1is
drawnwitha heavydarkline. All of thefault2'sareshown
bythelightparallellines.Eachsolidcircleindicates
thepoint
wherea fault2 is initiallytriggered.Thetimestotherightof
thefigurearethetriggertimesfor eachfault2.
To explainthe locationandtimingof the triggering
we
analyzethestress
perturbation
dueto rupturepropagation
on
thefirstfaultsegment.
It is important
torealizethatthesingle
faultstudyonlygivesinformation
on whena pointonthe
second fault exceeds the static friction level and does not
determine whether the stress concentration is sufficient to
induce
propagation
onthesecond
segment.Ourmodels
do
includethe rupturepropagationon the secondsegment,
therefore
wepresent
thesinglefaultstudy(Figure3) onlyas
anaidto understanding
theruptureinitiationproblem.
Therupture
canstartonthesecond
faultsegment
when
Coulombfriction has been exceeded(when the shearstress
on
thefault exceedsthe normalstresstimesthe coefficient
of
staticfriction).Because
boththeshearstress
fieldandthe
normalstress
field in the mediumareperturbed
bythe
propagating
ruptureon thefirstfault segment,
ruptureOccurs
onthesecond
segment
whereandwhentheinitialshearstress
(x0) plusthechange
in theshear
stress
(Ax(t))exceeds
the
staticcoefficient
of friction(g) timestheinitialnormal
stress
(On0)plusthechange
in thenormalstress
(AOn(t)):
I '•0+ A'•(t)I > It I On0q-AOn(t)
I
(1)
Harrisetal.: Rupture
Propagation
across
FaultSteps
This
implies
thatrupture
canoccur
whenandwhere
thestress
difference,
as is lessthanzero:
s(t) = g l (o0 + Aon(t)[ - I x0+ A•(t) I < 0
(2)
Equations
(1)and(2)assume
acompressional
stress
field
(øn0
+'XOn(0
< 0 ) sothatthefaults
remain
closed.
Plots
ofas(t)
duetorupture
ofthefarst
faultse.gment
are
presented
fora few timeslices(Figure3). Theregtons
with
895
negativecontoursindicatewhere a secondparallel segment
couldbeginto rupture.As time increases,
thepotentialrupture
regionexpandsandthisexplainsthe time dependent
rupture
nodepatternpresented
in Figure2.
Discussion
The purposeof thepresentstudyhasbeento explorethe
effectof a fault stepon the dynamicruptureprocess.We
assumed
verticalstrike-slip
faultssetin anotherwise
linearly
896
Harrisetal.: Rupture
Propagation
across
FaultSteps
elastic
material.It ispossible
thatfluidsplayanimportant
role numerical
simulations
have
been
presented
forspontaneously
in thestaticruptureprocess[e.g.Sibson,1986]however,the
propagating
shear
fractures
encountering
a faultstep.
results
of thisstudyneglect
fluids.It is interesting
tonotethat
References
Sibson
usedfluidpressure
toexplaina timedelayoccasionally
observed
atdilational
faultsteps.We alsopredict
a timedelay
(although
atamuchshorter
timescale)
fordilational
steps
even Andrews,
D. J., Dynamicplane-strain
shearrupture
witha
though
wehaveneglected
theeffectsof fluidpressure.
slip-weakening
friction
law
calculated
by
a
boundary
Anotherimportantfactorin the numericalsimulations
is
integralmethod,Bull.Seisin,$oc,Am,,75. 1-21,1985.
thatweassume
thatthetwoparallelfaultsegments
arevertical. Aydin,A., andR. A. Schultz,Effectof mechanical
interaction
At the presenttime we have limited informationaboutwhat
on the development
of strike-slipfaultswith echelon
happensto neighboringstrike-slipfaultsat depth[e.g.
patterns,J. of Struct,G½ol,,12, 123-129, 1990.
Kadinsky-Cade
andBarka,1989]. We do notknowif they Brune,J. N., T. L. Henyey,andR. F. Fox,Heatflow,
mergeor if they remain separateentitiesthroughoutthe
stress,
andrateof slipalongthe SanAndreas
fault,
seismogenic
regime.Ix)calseismicity
hasdelineated
separate
California,J. Geophys.Re•,, 2.4, 3821-3827, 1969.
faultstrands
for the SanAndreasfaultat Parkfield[Eatonet
Byedee,
J.D., Friction
of rocks,
PureAppl.Geophys,
JJ.6,
a!., 1970] and the CoyoteCreek fault in the southernSan
615-626, 1978.
Sacinto
faultzone[H•Iton, 1972].
Das,S., andK. Aid, A numericalstudyof two-dimensional
Onepointof thisstudyhasbeento determine
if it is even
spontaneous
rupturepropagation,
Geophys.
J. R, as•
possible
fora rupture
to 'jttrnp'
a faultsteponcetheseparation
Soc., •, 591-604, 1977.
betweentwo faultsis greaterthana certaincriticaldistance. Day, S. M., Three-dimensional
simulation
of spontaneous
For example,ourresultsfor the specificcase(Table1) of a
rupture:theeffectof nonuniform
prestress,
Bull.Seisrn.
100barstressdrop(andsupershear
rupturevelocity)show
Soc. Am., 72, 1881-1902, 1982.
thata compressional
stepwith a stepover
widthof 3 km will
Eaton,J.P., M. E. O'Neill, andJ. N. Murdoch,Aftershocks
neverbejumped,whereasa di!ationalstepwitha stepover
of the1966Parkfield-Cholame,
California,
earthquake:
a
widthof 3 km couldbejumped.Knuepfer[1989]notesthat
detailedstudy,Bull Seism.Soc.Am,,!•O.,1151-1197,
1970.
in hisdatacollected
fromfield observations
of strike-slip
faults,norupturehaseverjumpeda compressional
step(or Hamilton,R. M., Aftershocksof the BorregoMountain
doublebend)widerthan5 km andnorupture
hasjumpeda
earthquakefrom April 12 to June 12, 1968, in The
dilationalstep (or double bend) wider than 8 km. These
BorregoMountainearthquake
of April 9, 1968,IL•
observations,
thatruptures
jumpcompressional
stepswith
Geol. Sure. Prof. Par•.787, 31-54, 1972.
narrower
stepover
widths,is predicted
by ourstudy.We also Hanna,W. F., S.H.Burch,andT. W. Dibblee,Jr.,Gravity,
notethatfor dilational
stepswithnegative
overlap
thereis a
magnetics,andgeologyof the SanAndreasfault areanear
very narrow range of stepover widths for a second
Cholame,California,U.S. Geol.Surv.Prof.Paper646-_
'triggerable'
fault segment.We thereforeexpectthatmost
C, 29 pp. andmaps,1972.
dilational
steps(in verticalstrike-slip
faults)havepositive Ida,Y., Cohesive
forceacross
thetip of a longitudinal
shear
overlap
eventhoughtheymaybemapped
ashavingnegative
crackandGriffith'sspecificsurface
energy,J. Geophy•,
overlap.An exampleof thisis takenfrom the 1966Parkfield
earthquakewhere the stepoverwidth betweenthe fault
segments
is about1 km. Hannaet al. [1972] map the
dilationalfaukstepat Parkfieldwitha negative
overlap,but
aftershocks
of the 1966earthquake
[Eatonet al., 1970]showa
positive
overlap.Additionaldataarepresented
byAydinand
Schultz[1990]whocompiledmeasurements
of strikeslip
faultsaroundtheworld. Beyond0.5 km stepover
widthall of
theirdilationalfault stepsexhibitpositiveoverlap. This
observation
is alsopredicted
by ourresults.
Another
important
resultof thisstudyis therelative
timing
Res., 84, 3796-3805, 1972.
Kadinsky-Cade,K., and A. Aykut Barka, Effects of
restraining
bendsontheruptureof strike-slip
earthquakes,
U.S. Geol.Surv.Open-FileRep. 89-315, 181-192,1989.
Knuepfer,P. L. K., Implicationsof thecharacteristics
of endpointsof historicalsurfacefaultrupturesfor thenature
of
fault segmentation,
U.S. Geol. Surv.Open-FileRep,89315. 193-228, 1989.
Mavko, G. M., Fault interactionnearHollister,California,L
Geophys.Res.,87, 7807-7816, 1982.
Reasenberg,P., and W. L. Ellsworth, Aftershocksof the
of therupturenodeson the secondfault. We haveseenthat
CoyoteLake earthquakeof August6, 1979: a detailed
study,L Geophys.Res.,87, 10637-10655,1982.
thedilafional
stepis to temporarily
ttiedelaytherupture. Rodgers,D. A., Analysisof pull-apartbasindevelopment
produced
byenechelon
strike-slip
faults,in Sedimentation
Compressional
stepsarejumpedfor a narrowerstepover
width,butwhentherupturedoescontinue
topropagate,
it is
in Oblique-Slip
MobileZones,ed.by Ballance,
P.F.,and
not delayedas muchat the step. This impliesthat if the
H. G. Reading,Int. Assoc.Sediment.,
Spec.Publ.4, 27-
although
a widerdilationalstepcanbe jumped,theeffectof
optimalgeometries
existfor botha compressional
stepanda
dilational
stepthentherupture
will takethefaster
path,which
isusuallythecompressional
step.Ourresults
alsoimplythat
the averagerupturevelocityin a large earthquake
which
rupturesthrougha dilational step will be lower than the
average
velocityfor anearthquake
whichruptures
through
a
compressional
step.
41, 1980.
Rymer,M. J., Surfacerupturein a fault stepoveronthe
SuperstitionHills fault, California, U.S. Geol. Su•
O•en-File Rep. 89-315, 309-323, 1989.
Segafl.P., andD. D. Pollard,Mechanics
of discontinuous
faults,J. Geor•hvs,.
Re•,, 85, 4337-4350, 1980.
Sibson,R. H., iq.tiptureinteractions
with faultjogs,in
To summarize,
wehavemodelled
theeffects
offaultsteps
Earthcmake
Source
Mechanics,
Geophysical
Monographs,
on the dynamicruptureprocess. The purposewas to
Maurice
EwingSeries,
6, American
Geophysical
Union,
157-167, 1986.
determine
if a spontaneous
rupturecan'jump'fromonefault
segmentto anotherparallelfault segmentwith specified
distance
andoverlapbetween
thetwofaultsegments.
If we
RalphArchuletaandRuthHarris,Institutefor Crustal
canunderstand
whatstopsor allowsa ruptureto continue Studies,
University
of California,SantaBarbara,
CA 93106
propagating
alongthelengthof a faultzone,thenwe might
Steven
Day,Department
of Geological
Sciences,
San
alsodetermine
thepotential
magnitude
of theearthquake.
We
DiegoStateUniversity,
SanDiego,CA 92182
haveshown
examples
of geometrical
cases
wheretherupture
doesjumpa stepandcontinue
propagating
onthesecond
fault
segment.
We havealsoshown
cases
wherethestepterminates
(ReceivedNovember7, 1990;
the rupture on the first fault. This is the first time that
accepted
January14, 1991.)