JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 99, NO. Bll, PAGES 21,983-22,000, NOVEMBER
10, 1994
Modeling of periodic great earthquakes on the San
Andreas fault: Effects of nonlinear crustal rheology
Ze'ev Reches
Department of Geology, Hebrew University of Jerusalem,Jerusalem,Israel
Gerald
Schubert
Department of Earth and Space Sciencesand Institute of Geophysicsand Planetary Physics,
University of California, Los Angeles
Charles Anderson
EngineeringSciencesg• Applications,Los Alamos National Laboratory, Los Alamos, New Mexico
Abstract.
with
• finite
We •n•lyze the cycle of gre•t e•rthqu•kes •long the S•n Andre•s f•ult
element
numerical
model
of deformation
in • crust with
• nonlinear
viscoelastictheology. The viscouscomponentof deformationh•s •n effective
viscosity that depends exponentially on the inverse •bsolute temperature •nd
nonlinearly on the she•r stress;the elastic deformation is linear. Crustal thickness
•nd temperature •re constrained by seismic •nd he•t flow d•t• for C•liforni•.
The models •re for •nti plane strain in • 25-km-thick crustal l•yer having a very
long, vertical strike-slip f•ult; the crustal block extends 250 km to either side
of the f•ult. During the e•rthqu•ke cycle that lasts 160 years, • constant plate
velocityVp/2 - 17.5mm yr-• is •ppliedto the b•seof the crust•nd to the
vertical end of the crustal block 250 km •w•y from the f•ult. The upper half
of the fault is locked during the interseismicperiod, while its lower half slips at
the constant plate velocity. The locked part of the fault is moved abruptly 2.8 rn
every 160 years to simulate great earthquakes. The results •re sensitiveto crustal
rheology. Models with quartzite-like rheologydisplay profound transient stagesin
the velocity, displacement, and stressfields. The predicted transient zone extends
about 3-4 times the crustal thicknesson each side of the fault, significantly wider
than the zone of deformation in elastic models. Models with di•base-like rheology
behave similarly to elastic models •nd exhibit no transient stages. The model
predictions •re compared with geodetic observationsof f•ult-p•mllel velocities in
northern •nd central C•liforni• •nd local r•tes of she•r strain •1ongthe S•n Andre•s
fault. The observationsare best fit by models which are 10-100 times less viscous
than a quartzite-like rheology. Since the lower crust in California is composedof
intermediate to mafic rocks, the present result suggeststhat the in situ viscosity
of the crustal rock is orders of magnitude less the rock viscositydetermined in the
laboratory.
Introduction
are followed by aftershock activity that lasts for months
to years;the energyreleaseof the aftershocksdecaysexThe elasticreboundmodelof Reid[1910]impliesthat ponentially with time. While the locations and mechaelasticstrain is accumulatedduring a long interseismic nismsof aftershockshave been analyzed in terms of elaset al., 1988],the duration
period that is followedby local yieldingand fast fault tic models[e.g.,Oppenheimer
of
aftershock
activity
and
the
mode
of its decay require
slip during an earthquake. This processrepeats itself
in irregular periods known as earthquakecycles.Reid's a viscoelastictheology. Additionally, postseismicdefor-
[1910]model for a perfectelasticmediumcannotex-
mation, revealedby geodeticmeasurements
[Thatcher,
Papernumber94JB00334.
nonlinearviscoelastictheology(Figure 1) [e.g., Kirby,
1983].Nevertheless,
geodeticobservations
of crustalde-
of creepeventsthat lasthours
plain the time-dependent behavior exhibited by actual 1975],andthe occurrence
earthquakes. For example, almost all large earthquakes to weeks along parts of the San Andreas fault suggest
that the crust cannot be regarded as a simple elastic
solid.
Finally, it is widely accepted that continental
Copyright1994by the AmericanGeophysical
Union.
crustal rocks below 10-15 km deform according to a
0148-0227/94/94JB-00334505.00
21,983
21,984
RECHES ET AL.: PERIODIC GREAT EARTHQUAKES ON SAN ANDREAS
Shear stress
and rheologiesare perhaps too simplifiedto accountfor
real crustal
mechanics.
Li and Rice [1987]alsoanalyzedthe earthquakecycle within
Brittle-elastic
5
an elastic-viscoelastic
crust. Their
model in-
cludes an upper elastic layer of thickness H, with a
long strike-slipfault that penetrates to depth D, where
D < H. The elastic layer is underlain by a viscoelas-
(faulting)
tic (Maxwell) layer of thicknessh. Analytic solutions
Transition
zone
Nonlinear, viscoelastic
(dislocation flow)
to the deformation during the earthquake cycle were
obtained by using a modified Elsasserapproximation.
This model is compared below with the numerical results of the presentmodeling.
We investigate here the cycle of great earthquakes
along the San Andreas fault with a finite elementcode
(ABAQUS). The rock rheologyis assumedto be temperature dependent and nonlinear; crustal thickness
and temperature are constrained by seismic and heat
flow data. Lyzengaet al. [1991]have alsouseda fi-
20-
nite element approach to study deformationduring a
Crust
25Mantle
Figure 1. General rheologyof the continentalcrust
[afterBraceandKohlstedt,1980].
formation betweenearthquakesare frequentlymodeled
by dislocationsin an elastic half-space.These disloca-
series
ofe•':•thquakes
along
a model
SanAndreas
fault
in a medium with nonlinear, depth-dependentrheology. The focus of their study was the influenceof fault
strength on the great earthquake recurrencetime. Details of the ABAQUS numericalprocedureare givenin
Appendix A, and a comparisonbetween the numerical
modelandthe•:analytical
modelof Li andRice[1987]is
givenin AppehdixB.
The
Numerical
Model
Approach
Our model for investigatingthe effectsof nonlinear
tionsmustbe placedat greatdepths(below10 km) to crustal rheology on periodic earthquake behavior asobtaingoodfits to geodeticdata [e.g.,Prescottand Yu, sumes a very long vertical fault that penetrates a uni19861.
form, isotropicviscoelastic
crust (Figure 2). At great
Severalviscoelastic
modelshavebeenproposedto explain these and other shortcomingsof the elasticmod-
distancefrom the fault a constantvelocity directedparallel to the fault is specified;this velocity corresponds
els [Nut and Mavko,1974;Savageand Prescott,1978; to the plate velocity. During an earthquakecycle the
Thatcher, 1983; Turcotteet al., 1984;Li andRice, 1987; fault is locked, and at the end of the cycleit is abruptly
Lyzengaet al., 1991]. Nut and Mavko [1974]consid- moved to match the far-field plate displacementaccuered a long strike-slipfault within an elasticlayer that mulated during that cycle.
We have determined the deformation and stresses in
overliesa viscoelastichalf-space. In their model, the
depth of the fault D is equal to or smaller than the the crustal block during the great earthquakecycleby
thicknessof the elastic layer H. The earthquakecy- using the finite element code ABAQUS. This code accleswere modeledby uniform slip alongthe fault and curately modelsthree-dimensional,nonlinear,large desequential loading of the elastic layer and the under- formation effectsin solids,includingheat transfer. The
lying viscoelasticmedium,followedby time-dependent codefeatures a robust element library comprisingone-,
relaxationof thelinear(Maxwell)viscoelastic
substrate. two- and three-dimensionalfinite elements, linear and
The substrate transfers shear stresses back to the elasnonlinear material models, and accurate time-stepping
tic layer, generatingtransientstressand velocityfields. algorithmsand solutiontechniquesfor solvingnonlinear
Thatcher[1983]comparedpredictionsof the Nut and equations. Details of the formulation and the numerical
Mavko[1974]andelastichalf-space
modelswith geode- proceduresare presentedin Appendix A.
tic observationsalong the San Andreas fault but was
We employ eight-node linear isoparametricfinite elonly able to provide weak constraintson the crustal ements(or bricks) to representthe spatial variation
structure and the intensity of elastic-viscoelasticcou- of the displacement
and stressfields (Figure 3) (Ap-
pling. The difficultyin obtainingstrongerconstraints pendixA). With the linear isoparametricelement,disprobablystemsfrom two sources.The availablegeode- placementsvary linearly along the edgesof each eletic observationsare limited, in time and location,and ment. Strainandstressare determined
at oneintegrathey includelargeintrinsicnoise[e.g., Thatcher,1983, tion point (stressand strain are constantwithin each
Figures12 and 13]. Also, the analyzedconfigurationselement).
RECHES ET AL.- PERIODIC GREAT EARTHQUAKES ON SAN ANDREAS
25
21,985
CRUST
Figure 2. A modelfor crustaldeformationduringthe earthquakecyclealongthe SanAndreas
fault, California.The modelis an antiplanestrainapproximation
for a longstrike-slipfault. The
crust is assumedto havea temperature-dependent,
nonlinear,viscoelastic
rheology.The locked
part of the San Andreasfault is 12.5 km deep.
Model
Parameters
Crustal
and
fault
dimensions.
The thickness of
the California crust along the central San Andreas sys-
tem rangesfrom 24 km to 26 km [Oppenheimerand
Eaton, 1984; Fuis and Mooney,1990],with 25 km as a
representativethickness. Seismicactivity along the San
Andreas is mostly restricted to the upper 10-15 km,
with local zonesof deeper seismicity.It is generallyac-
ceptedthat the depthof seismicity
corresponds
to the
depth of the portion of the San Andreas which slips
during great earthquakes. We therefore use 25 km as
the total crustal thicknessand 12.5 km as the depth of
the lockedzone (Figure 2).
The assumptionof a very long fault (Figure 2) reducesthe problem to an antiplane condition. The deformation is restricted to a plane perpendicularto the fault
and this deformation is defined in terms of a singleun-
surementsindicate that the North American plate and
the Pacific plate move at a relative velocity of up to
46 mm yr-• [Ward, 1990],suggesting
that part of the
relative motion is accommodated by distributed deformation in a 200-400 km wide zone along the San Andreas fault. We consider only the component of the
plate velocity that correspondswith slip along the San
Andreas,i.e., vp- 35 mmyr-•.
Sieh e! al. [1989] have shownthat time intervals
between great earthquakesin central California range
from 44 to 332 years with the two most frequent intervals at about 250 and 50 years. Our numerical calculations are for an earthquake cycle time of 160 years, the
sameas in Li and Rice [1987],to facilitate benchmarking with their results(AppendixA).
Our model earthquake cycle is sketchedin Figure 4.
The constant
velocityvp•2 (17.5mmyr-1) isappliedto
knowndisplacementcomponentu(y, z, t).The antiplane all faces of the model except the locked portion of the
strain condition is enforcedas describedin Appendix A. fault (from z - 0 to z - -12.5 km) for the interseismic period of 160 years, yielding a total displacement
We analyze the deformation within a crustal block that
extendslaterally from the fault at y = 0 to y = 250 kin,
and vertically from the ground at z = 0 to z = -25 km
(Figures2 and 3). The lockedportionof the fault is the
x - z plane from z = 0 to z = -12.5 km. Three of the
faces of the analyzed block slip at the constant fault-
of 2.8 m. Then the locked part of the fault is moved
abruptly 2.8 m, and the cycle is repeated.
The calculationswere carriedout by steppingforward
in time; approximately 20 time steps composeeach cycle, with eachrun consistingof five to sevencycles.As
parallelhorizontalvelocityVp/2,wherevpis the long- solutionsfor the nonlinearrheologyconvergeafter three
term slip rate along the fault. These facesare the downward continuation of the fault, from z = -12.5 km to
z = -25 kin, the base of the crustal block z = -25 km,
and the far sideof the blockat y = 250 km (Figure2).
The fault itself, from z = 0 to z = -12.5 km, is locked
(zerofault-paralleldisplacement)
throughoutthe earthquake cycle except for a very brief period of time at the
end of each cycle when the fault is abruptly displaced
to catch up to the end at y = 250 km.
Long-term plate motion and loading history.
The long-term slip rate along the San Andreas fault
to four cycles we present below results for the fifth or
sixth cycle.
Temperature.
The thermal propertiesof the crust
in the vicinity of the San Andreas fault are based
on the analysis of surface heat flow data by Lachen-
bruchand Sass[1980]. They find that the mean surface heat flow along the San Andreasis 72 mW m-2.
They suggestthat in spite of the wide scatter in the
heatflowversusheat productiondata [Lachenbruch
and
Sass, Figure 10], the mantle heat flow in California
may be comparablewith that of the Basinand Range
wetakeTs - 10øC,
is about35 mm yr-• according
to displaced
Holocene (qm- 58 mW m-2). Accordingly,
features[Lisowskie! al., 1991]. Recentgeodeticmea- qs- 72 mWm -• q,•-58mWm -• k-2.5Wm -•
21,986
RECHES ET AL' PERIODIC GREAT EARTHQUAKES ON SAN ANDREAS
earthquake cycle with the experimentally determined
rheologicalproperties of quartzite, quartz-diorite, and
diabase. As shown below, only models with viscosity
lower than the experimental viscosity of quartzite fit
the field observations.
Therefore
the numerical
results
presentedhere are for five sets of rheologicalparameters which do not necessarilycoincidewith experimental
parameters of a particular rock. These sets are as fol-
lows(Table 2): NR2D is identicalto the experimental
rheology of quartzite. NR2H and NR2J have the same
Q as NR2D (quartzite) and valuesof As that are 100
Log effectiveviscosityat 1 MPa
shearstress(log Pa s)
Figure 3. The NR2 mesh of elements used in the
ABAQUS calculations. The mesh includes 440 nodes
and 190 three-dimensional
10
elements.
K-1 [Lachenbruch
and $ass,1980],and hr = 10 km
[TurcotteandSchubert,
1982]in equation(A7).
20
25
30
5
Rheology. We have assumedthat the rheologyof
the entire crust can be representedby a singlenonlinear,
temperature-dependentviscoelasticlaw, as presentedin •
equations(A5) and (A6). Then, the effectiveviscosity
r/when Vxyis the only nonzerostresscomponentis
1-n
15
10
Q
vf3-)
l+nAs
2o
The effective viscositiesof six crustal rock types are
25
plottedin Figure5a for Vxy= i MPa, the temperaturedepthprofileequation(A7), and the valuesof A, n, and
Q after Kirby and Kronenberg[1986](Table 1). For Ao
these conditions,r/ranges over 8 ordersof magnitude
Log effectiveviscosityat 1 MPa
shearstress(log Pa s)
(or more) from Westerlygranite, the least viscous,
diabase,the mostviscous(Figure5a).
10
We carried out numerical simulations of the great
15
20
25
30
Very weak
Fault slips
160yr•
quartzite
•
5
• 10
quartzite
••
fault
Locked
•
Diabaseat20
MPa
20
2•
Bo
Time
Figure 4. Loadinghistory of the presentmodei.
Figure 5. The effectiveviscosityof (a) crustalrocks
and (b) modelmaterialsversuscrustaldepth. The viscosityis calculatedfor the flow parametersgivenin Table 1 for rocks(laboratoryparameters)and in Table 2
for model materials, the geothermalprofileof California
(seetext), and a shearstressof 1 MPa.
RECHES ET AL.: PERIODIC
GREAT EARTHQUAKES
ON SAN ANDREAS
21,987
Table 1. Nonlinear Viscosity Parameters of Crustal Rocks
Rock
Type
A,s-•
n
MPa-"
Quartzite(dry)
Quartzite(wet)
Westerlygranite(wet/dry)
kJ m
oQ1.
•
2.58x 10-4
6.79x 10-a
6.65x 10-5
2.4
2.26
2.73
155
153
127
Quartz-Diorite
1.29x 10-a
2.4
219
Anorthosite
Diabase
3.16 x 10-4
2.00 X 10 -4
3.2
3.4
238
260
Selectionfrom Kirby and Kronenberg
[1986].
and 10 times larger, respectively,than the A• value for
quartzite. NR2I and NR2K have the same A• as NR2J
and values of Q which are 0.9 and 1.1 times the value
of Q for NR2J, respectively. All sets of rheologicalparameters have the same value of n as quartzite.
The effective viscositiesof these five rheologicalparameter sets are plotted in Figure 5b for the same temperature distribution and shear stressas used in calculating r/for the laboratory-determinedrheologicalpa-
rameters(Figure 5a). In general,•/(NR2D)m •/(NR2I)
and r/(NR2H)mr/(NR2K); thusnumericalsolutionsfor
these two pairs of rheologicalmodels turn out to be approximately the same. These resultsimply that a factor
of 10 changein the multiplication factor A• is equivalent
to about a 10% changein the activationenergyQ.
Below we present the results for the rheologicalparameters NR2D and NR2H, and we also discussthe resuits for NR2J. For conveniencein recognition we rename the rheologicalmodels: NR2D is termed "quartzite," NR2J is termed "weak quartzite" and NR2H is
termed "very weak quartzite."
Results
The model results presented here are for the rheological parameter sets "quartzite" and "very weak
quartzite" (Table 2) as discussed
above. The results
include the vertical and horizontal
distributions
of fault-
weak quartzite" model. Surfacevelocityis plotted as
a function of distance from the fault at several times
duringthe earthquakecycle. The spatialvariationsin
surfacevelocityoccurwithin an •150-km-wide zoneon
either side of the fault. During a post-seismicperiod of
m50years,the surfacevelocityseveraltensof kilometers
awayfrom the fault exceedsthe long-termplate velocity.
On the other hand, during a preseismicperiod of m40
years, the surfacevelocityat distancesup to •50 km
from the fault is significantlybelow the plate velocity.
For example, the velocity at a distanceof •30 km away
from the fault drops from 1.65 times the plate velocity 10 years after the earthquaketo about 0.45 times
the plate velocity156 yearsafter the earthquake(i.e., 4
yearsbeforethe next earthquake).Thus the variations
in surface velocity by this model display a profound
postseismictransient mode.
Figure 7a showsthe depth distributionof the faultparallel velocityat a distanceof 3.8 km from the fault
and at several times in the earthquake cycle for the
"very weak quartzite" model. The velocity variations
with depth and time during the postseismictransient
are particularly large below about 10 km depth in the
relatively low viscositylower crust. For example, the
velocity at a depth of about 18 km changesfrom •1.4
times the plate velocity 10 years after the earthquake
to m0.3 times the plate velocity 156 years after the
earthquake. From the surfaceto ..•10 km depth, the
upper crustmovesat an approximatelyconstantvelocity at any stage of the earthquakecycle. Prior to the
earthquake, the velocity decreasesapproximatelylin-
parallel velocity,fault-parallel displacement,and faultparallel shear stressduring the earthquakecycle. The
velocity, displacement,and shear stressare calculated
at five timesduringthe cycle(usually10, 40 or 50, 72,
121, and 156 or 160 yearsafter the earthquake)and are earlyfromthe platevelocity(17.5mm yr-1) at 25 km
plotted in Figures 6-9.
depth to about 0.07 times the plate velocityat 15 km
depth, while the crustabove15 km movesat a relatively
Fault-Parallel Velocity
constantvelocity of 0.05-0.07 times the plate velocity.
Variations in the fault-parallel velocity during the Thus the upper crust is partially decoupledfrom the
earthquake cycle are shownin Figure 6a for the "very mantle by flow within the lower crust.
Table 2. NonlinearViscosityParametersof the ModelsStudiedin This Paper (Equations(A5) and (A6))
Model Name on ABAQUS
As,
MPa-" s-1
n
(Model Name Used Here)
NR2D ("Quartzite")
NR2H ("Very weakquartzite")
2.58• 10-4
2.58x 10-2
2.4
2.4
NR2I
NR2J ("Weakquartzite")
NR2K
Q,
kJ mol -•
155
155
2.58 x 10-s
2.4
140
2.58x 10-s
2.4
155
2.58 x 10-•
2.4
170
Comments
quartzite rheology
0.01 times quartzite viscosity
0.1 times quartzite viscosity
21,988
RECHES ET AL.: PERIODIC
•
GREAT EARTHQUAKES
ity drops from m0.85 times the plate velocity 10 years
after the earthquake to •-,0.7 times the plate velocity 4
3o
•
ß-"
yearsbeforethe next earthquake(Figure8a). The zone
Very
weak
quartzite
of deformation in this model is •75 km wide, approximately half of the deformation zone width in the "very
weak quartzite" model.
Figure 9a showsthe depth distribution of the faultparallel velocity at a distance of 3.8 km from the fault
and at several times in the earthquake cycle for the
"quartzite" model. Temporal variationsin velocity are
relatively small at all depths. The fault-parallel velocity
decreases
approximatelylinearly from the plate velocity
•0•37yr
•
g 20
10
156yr
g'•
(17.5mmyr-1) at 25km depthto •0.2 timestheplate
0
A.
o
•"
ON SAN ANDREAS
50
lOO
15o
200
velocity at ,•10 km depth; the crust above 10 km moves
at the roughly constant velocity of 0.2 times the plate
velocity.
0.6
Very weak quartzite
Fault-Parallel
Displacement
The displacementsfor models NR2H ("very weak
quartzite")and NR2D ("quartzite")are shownin Figures 6c and 8c, respectively,as a function of distance
from the fault plane at severaltimes during the earthquake cycle. As the fault is locked during the entire
--'4
cycle(excludingthe shorteventof the earthquake),the
fault is the referencepoint for the displacements.The
displacementat the fault is constant and zero, and the
displacementat distanceof 250 km at a given time is
the integrated plate velocity since the earthquake. For
1lyr
/
r.•
-o.4
the quartziterheology(Figure 8c), spatialvariationsin
0
Be
50
100
150
200
fault-parallel surface displacementare confinedto distances < 75 km on either side of the fault, but for "very
weak quartzite" (Figure 6c) the variationsextendfarther from the fault to distances of about •200
Very
weak
quartzite-----'"'"'"•16øyr=
2.25
km.
During an early postseismicperiod (•40 years af-
ter the earthquake for the weakened quartzite rheology, Figure 6c, and a somewhat shorter time for the
121yr'"
72yr-'
41yr-"
o
-0.75
0
Ce
50
100
150
Distance from fault 0cm)
200
quartziterheology,Figure 8c), the displacement
near
the fault is inverted from the senseof the long-term
or far-field displacement; i.e., there is left-lateral displacementin contrast to the long-term right-lateral slip
along the San Andreasfault. The inverted displacement
during the early postseismicperiod reflectsthe velocity deficit near the fault during the late stagesof the
earthquake cycle. When the earthquake occurs,material closeto the fault plane, whosefault-parallel velocity
and displacementhave been retarded by the lockedfault
duringmostof the earthquakecycle(comparetimesof
160 and 11 yearsin Figures6c and 8c), movesaheadof
Figure 6. (a) Fault-parallelsurfacevelocity,(b)sur- material farther from the fault plane. The late stage
faceshearstress,and (c) fault-parallelsurfacedisplace- velocity deficit is smaller and concentrated closer to
ment versus distance from the fault for model NR2H
the fault for the strongerrheology(compareFigures6a
("very weakquartzite"in Table 2). The shearstress and 8a), makingthe inverteddisplacement
smallerand
does not include the ambient stress. The results are for
confined nearer the fault.
Thus the inverted sense of
five times sincethe last great earthquake.
displacement reflects the viscoelastic character of the
model. Note that the fault-parallel velocity itself is alTemporal variations in fault-parallel surface velocity ways in the right-lateral senseand not inverted at any
during the earthquake cycle are rather different for the stage of the earthquake cycle.
Figures 7c and 9c show the variationswith depth, at
"quartzite"model (Figure 8a). The velocityvariations
with time are relatively small; for example, at a dis- severaltimes during the earthquake cycle, of the faulttance of •25 km away from the fault, the surfaceveloc- parallel displacement at 3.8 km from the fault for the
RECHES ET AL.- PERIODIC GREAT EARTHQUAKES ON SAN ANDREAS
Fault-parallelvelocityat 3.Skinfrom fault (mm/yr)
0
3
6
9
12
15
1•
21
24
Very weak quartzite
-lO
21,989
"very weak quartzite" and "quartzite" models, respectively. The left-lateral displacementearly in the earthquake cycle occursover the entire thicknessof the crust
for both rheologies. The upper •10 km of the crust
movesrelatively rigidly in contrast to the strong decoupling that occurswithin the lower •15 km of the crust
for both the quartzite and weakenedquartzite rheologies.
Fault-Parallel
Shear
Stress
The fault-parallel surface shear stress is plotted in
Figures 6b and 8b as a function of distancefrom the
fault at several times during the earthquake cycle for
the weakenedquartzite and quartzite rheologies,respectively. The shear stress in these figures indicates only
-15
-2o
the deviation from the ambient tectonic stress; the lat-25
Ae
Fault-parallelshearstressat 3.8km from fault (MPa)
•.g
o,
•.6
•.2
•
•
0
11yr
0.2
41yr
0.4
72yr
0.6
0. g
121yr160yr
ter might be significantly larger. The ambient stress
cannot be calculated in the present modeling as the
strength of the San Andreasfault was not specified.
The shear stress variations throughout the earth-
quakecycleincreasewith proximityto the fault (Figures6b and 8b). The variationsin shearstressthat are
earthquake related are the differencebetween the shear
stressjust beforethe earthquake(the 160-yearcurvesin
Figures6b and 8b), and the shearstressa shortperiod
after the earthquake(the 11-yearcurvesin Figures6b
and 8b). Figure6b (weakenedquartzite)indicatesthat
the shear stress next
to the fault
is 0.53 MPa
at the
end of the cycle and -0.33 MPa at 11 years after the
earthquake; thus the stress change is 0.86 MPa. Similarly, Figure 8b indicates a stresschangeof 1.33 MPa
(0.93 MPa before and -0.3 MPa 11 years after the
earthquake).
-2o
Very weak quartzite
-25
Be
Fault-parallel
displacement
at 3.Skmfromfault (m)
-0.5
0
0
0.5
1
1.5
2
2.5
•
Very weak quartzite
The shear stress also changessign during the earthquake, from right-lateral shear before the earthquake
(the 41- to 160-yearcurvesin Figure6b) to left-lateral
shearafter the earthquake(the 11- to 41-yearcurvesin
Figure6b). As the ambientshearstressis unknown,it is
not clear if this sign changeis sufficientto significantly
affect the state of stress in the crust. The right-lateral
shear stress during the late stages of the earthquake
cyclecorresponds
to right lateral displacement
(41- to
160-yearcurvesin Figure 6c), whereasthe left-lateral
shear during the early stages of the earthquake cycle
corresponds
to left-lateral displacement(11- to 41-year
curvesin Figure 6b).
1
-2O
11yr
58yr••N•
Figures 7b and 9b showthat shearstressesat 3.8 km
from the fault are relatively constant with depth in the
upper •10 km of the crust at all stages of the earthquake cycle. Below •10-15 km there is a reversal in the
sign of the shear stress;while left-lateral shear prevails
in the upper crust during the first •40 years of the cycle, right-lateral shear prevails in the lower crust for the
same period and vice versa for the later stagesof the
cycle.
Figure ?. (a) Fault-parallelvelocity,(b)shear stress,
and (c) fault-parallel displacementversusdepth for Field Observations
modelNR2H ("very weakquartzite"in Table 2), at Predictions
a distance of 3.8 km from the San Andreas fault. Ambient stresses are not included. The results are shown
for five elapsedtimes sincethe last great earthquake.
and
Model
We compare the results of the model calculationswith
three sets of geodetic data for the San Andreas system
21,990
"•
RECHES ET AL'
PERIODIC
GREAT EARTHQUAKES
ON SAN ANDREAS
18
Fault-Parallel
15
uredfault-parallel
velocities
in thecentralTransverse
10y•
•
12
9
Velocity
We
first
compare
the
numerica
results
with
the
mea
Ranges[Lisowskiet al., 1991]and north of San FranFault-parallel
velocityat 3.8kmfromfault (mm/yr)
0
3
6
9
12
18
Quartzite
-5
3
15
ß
••
156yr
0
0
Ae
50
100
150
200
11yr
-2O
0.6
-25
•60yr
Shearstressat 3.8km from fault (MPa)
121yr
-1.2
0.3
'72yr
-0.8
-0.4
0
0.4
0.8
1.2
0
-'58
-/ ,
-0.3
0
B©
! ' ,
50
•
100
11yr
Quartzite
,
i
58yr
72yr
121yr160yr
,
150
2OO
160yr-2O
2.5
Quartzite
121yr-25
1.5
Fault-parallel
displacement
at 3.8kinfromfault (m)
-0 5
58yr--
0
0.5
1
0.5
2
2.5
Quartzite
11yr--
-5
Quartzite
1.5
-0.5
o
Co
50
lOO
150
200
Distancefrom fault (km)
Figure 8. Similar to Figure 6, for model NR2D
("quartzite"modelin Table 2).
(Figure 10). We use the fault-parallelvelocitiesmeasuredin northernCalifornia[Prescottand Yu, 1986],the
fault-parallelvelocitiesin centralCalifornia[Lisowskiet
al., 1991],and the rate of shearstrainfor varioussites
alongthe SanAndreassystem[Thatcher,1983].In this
-2O
11yr
-25
41yr
72yr
121yr•
0yr
selectionof geodetic data sets we follow the work of L i
Figure 9.
and Rice [1987].
("quartzite"modelin Table2).
Similar to Figure 7, for model NR2D
RECHES ET AL.: PERIODIC
GREAT EARTHQUAKES
I
ON SAN ANDREAS
21,991
,Locked
sections
.Cape
Mendecino
...... Slipping
sections
•01N'l'•
ß40 N
San
Francisco
••woo•
c•w NORTH
AMERICA
PLATE
36•N •o• • • Cholame
• •
•
B
Figure 10. Simplifiedmap of the San Andreasfault with the two lockedsegmentsanalyzedhere
[after Turcotteand Schubert,1982]. The calculatedfault-parallelvelocitiesare comparedwith
geodetically determined velocitiesat the locationson the map. Profile A, north of San Francisco
Bay,velocitiesafter the 1906SanFrancisco
earthquake[Prescottand Yu, 1986];profileB, central
TransverseRanges,velocitiesafter the 1857Fort Tejon earthquake[Lisowskiel al., 1991].
ciscoBay [Prescottand Yu, 1986]. The modelsare
viscositymodel ("quartzite") has relativelylittle vari-
"quartzite," "weak quartzite," and "very weak quart-
ation in fault-parallel surface velocity with time dur-
zite"
of Table
2.
ing the earthquakecycle(Figure 8a). The modelwith
The fault-parallel velocities in the region north of lower viscosity("weak quartzite")has a greatertime
San Francisco Bay were measured during 1972-1982 variationin fault-parallelsurfacevelocity(seeFigure6a
by Prescottand Yu [1986](Figure11). They regarded for the "very weak quartzite" model);the low-viscosity
these as steady state velocities since they were essen- model evolvesfrom a high velocity postseismicstate to
tially constant during the survey period some 70 years a lower-velocity preseismic state. At 72 years after the
after the 1906 San Francisco earthquake. The mea- earthquake the intermediate velocity state of the lowsurements
are consistent
with a monotonic
increase
viscosity model happens to coincide with that of the
in the magnitude of the velocity with distance from high-viscositymodel.
The observationsof fault-parallel surface velocities in
the fault, with the velocity approachingan asymptotic
plate tectonic speed at distances greater than about the central Transverse Ranges compiled by L isowski et
50 km from the fault. Local velocity deviations have al. [1991]are shownin Figure 12. It is assumedthat
been attributed to features such as The Geysers ther- these velocitiesare related to the 1857 Fort-Tejon earthmal anomaly [Prescottand Yu, 1986]. Prescottand quake. Figure 12 also displaysthe model velocitiescalYu [1986]showedthat the data in Figure ll couldbe culated 121 years after the earthquake. The field data
matched by deformation in an elastic half-spacewith an are fit well by the velocities of the low viscosity models
appropriate distribution of dislocations. Several of the NR2J and "very weak quartzite"; the viscoelasticmodel
dislocationsin their set extend from depths of 10 km to of Li andRice[1987]alsoprovides
a goodmatchto the
infinity (seediscussion).
observations.
Figure l l also showsthe fault-parallel surfacevelocities versus distance
from
the fault
for the three
mod-
els of "quartzite," "weak quartzite," and "very weak
quartzite" at 72 years after the earthquake. The three
model
curves
all fit
the
observations
which
have
rel-
atively large uncertainties. An interesting feature of
this plot is the coincidence of the velocities for mod-
els "quartzite" and "weak quartzite" even though these
modelsdiffer by a factor of 10 in viscosity.The higher-
Figures 11 and 12 emphasizethat models with different crustal viscosities yield different velocity distributions. These distributions could be distinguishable
by geodeticobservationseven a long time after the last
great earthquake, for example, 121 years in Figure 12.
These velocity differencesthat are associatedwith different viscositiesprobably reflect the degreeof coupling
between the lower and upper crust. In our model, the
base of the crust is subjected to a constant plate veloc-
21,992
RECHES ET AL'
PERIODIC
GREAT EARTHQUAKES
Weakquartzite
(NR2J)
ON SAN ANDREAS
1.6
15
Very weakquartzite(NR2H)
Quartzite
(NR2D)I 1.2
.
Li&Rice
model • 0.8
Very weakquartzite(NR2t0
Weakquartzite(NR2J)
0.4
Quartzite(NR2D)
-15
25
-45
-30
-15
0
15
30
50
75
100
125
45
Time after greatearthquake(years)
Distancefrom SanAndreas (km)
Figure 11. Fault-parallel surfacevelocity along pro-
Figure 13. Rates of fault-parallel shear strain near
the San Andreas fault for various times after the great
file A in Figure 10. Vertical bars (2 standarddevi- earthquake. Vertical and horizontalbars (2 standard
ations in length) indicate the geodeticvelocitiesfor deviationsin length)arethe strainratesestimatedfrom
the period 1973-1984 as determined by Prescott and geodeticdata by Thatcher[1983]. Solid curvesare the
Yu [1986]. Solid curvesare the calculatedvelocities calculated velocities 123 years after the great earth72 years after the great earthquake for models NR2D quake,for modelsNR2D ("quartzite"), NR2J ("weak
("quartzite"), NR2J ("weak quartzite"), and NR2H quartzite"),and NR2H ("very weakquartzite").
("very weak quartzite"). The curvesfor NR2D and
NR2J coincide(seetext).
Rate
ity of vp/2. The effectiveviscosityof the modelcrust
at i MPa shearstressvariesstronglywith depth,from
approximately1030Pas at the surfaceto ,•,10TMPas at
of Shear
Strain
Thatcher[1983]calculatedthe shearstrainratespar-
allel to the SanAndreasfault ('•v in the presentconvention),fromgeodeticmeasurements
at differentstages
thebase(for the "quartzite"model).A lowercrustwith of the earthquakecycle. Figure 13 displaysthesegeodesucha low viscositycannotapplylargeshearstresses
to tic strain rates versuselapsedtime sincethe last major
the uppercrust(Figures7b and9b);it essentially
gen- earthquake. After about 50 years following the eartherates a zone of partial decouplingbetweenthe upper
crust and the mantle.
quake, the measuredstrain rates approachan approxi-
matelyconstantvalueof 0.3 x 10-• yr-•. Duringthe
transient stage of •50 years followingthe earthquake,
the measuredstrain rates are highly variable and have
large uncertainties.
5
0
Quartzite
and
weak
quartzite
-5
(NR2D
and
NR2J)
ß
%
-10
-15
-20
-25
-30
0
v •
•
•
•
•
10
20
30
40
• ,
50
•
•
60
70
Figure 13 alsoshows'•v predictedby "quartzite,"
"weak quartzite," and "very weak quartzite" models;
these rates were calculated for the ground surfaceat
a distance of 6 km away from the fault. For periods
of more than about 50 years after the earthquake, the
three models fit the measurementsabout equally well.
However, the situation is different for the first 50 years
after the earthquake during which none of the models
providesa good fit to the measuredstrain rate. Models "weak quartzite" and very weak quartzite fit the
data better than the "quartzite"model(Figure13), in
80
Distancefrom SanAndreas (km)
Figure 12. Fault-parallel surface velocitieson both
sidesof the San Andreasfault alongprofile B in Fig-
ure 10. Vertical bars (two standard deviationsin
length) indicate the geodeticvelocitiesas presentedby
Lisowskiet al. [1991]. Solid curvesare the calculated
velocities123 yearsafter the great earthquakefor modelsNR2D ("quartzite"),NR2J ("weakquartzite"),and
NR2H ("very weakquartzite").
accord with the above deductionsfor the fault-parallel
velocities(Figures11 and 12).
Summary: Comparison of Observations
and
Predictions
The good agreementshownbetweengeodeticobservations and predictions of the "weak quartzite" and
"very weak quartzite" models(Figures11-13) is significant for several reasons. First, the model parameters represent thermal and mechanicalpropertiesof the
RECHES ET AL.: PERIODIC
GREAT EARTHQUAKES
crust in the vicinity of the San Andreas fault and the
agreementis found without parameterfitting and without local, ad hoc modifications. Second,the same model
fits, in general, the velocity observationsin several regions,eachin a differentstageof the earthquakecycle.
Third, the comparisonbetween predicted and observed
surface velocities indicates some decouplingwithin the
lower crust. In particular, the estimated in situ viscosity of the lower crustal rocks near the San Andreas is
similar to the viscosityof one of the lessviscouscrustal
rockstested in laboratory experiments(i.e., Westerly
granite,seeFigure 5).
Discussion
Composition of the Lower Crust
and in Situ Rheology
The seismicstructure of the lithosphereand the composition of its lower crust in central and southernCalifornia were recently examined by Fuis and Mooney
[1990]. Their profile for centralCalifornia [Fuis and
Mooney,1990,Figure8.5] showsthat the seismicveloc-
ON SAN ANDREAS
21,993
the presentmodel)are 10-14 to 10-15 s-1 (strainrates
that are comparable to the experimental strain rates
could last for periods of days after the earthquake at
the immediateproximityof the fault). The secondreason is related to the size of the sample. The lower crust
could
include
narrow
shear
zones
that
accommodate
large strain and broad zonesof relatively small strain.
Laboratory measurementsof small samplescannot account for the large-scalecomplexity of suchsystems.A
third option is related to the deformation mechanisms
active in laboratory testing of lower crustal rocks. For
example, pressuresolution creep cannot be studied experimentally due to its slow rate, yet it might be effective under
crustal
conditions.
The estimated effective viscosity of the lower crust is
very low, and such low-viscosity could lead to decoupling between the mantle and the crust and to seismic
attenuation. This low viscositylayer probably doesnot
affect postglacialrebound analysesbecauseof the wavelength differences:the postglacialrebound requiresflow
on continental scale which is accommodatedprimarily
in the mantle. Geodetic observationsof earthquake dis-
placement are a good way to estimate the viscosityof
ity exceeds
6.0 km s-1 (approximately
granitevelocity) the lower crust due to the similarity in relaxation time
at a depth of 5 km in the Salinian block west of the
San Andreas(Santa Cruz Mountains/GabilanRange)
and at depth of 15 km in the Diablo Range, east of
the San Andreas. Detailed velocity-depth profiles for
these two regions are presentedin Figure 14. For the
and length scales.
Limitations
of the
Model
While the present model provides good fits to field
SalinianblockFuisandMooney[1990,p. 216]suggested observationsof earthquake related deformation along
two possiblecompositionsof layer 4, at depth of 10 to the San Andreasfault (Figures11-13), so do previous
25 km (Figure14a): "... gneiss
of intermediate
[diorite] models with dislocations in elastic layers or linear viscomposition.
•. [or]in an alternativemodel,wheremid- coelasticlayers [e.g., Prescottand Yu, 1986; Lisowski
dle and lower crust are separated as layers 4a and 4b, et al., 1991; Li and Rice, 1987]. Modelswith dislocalayer4b may be reasonablyinterpretedasgabbro"(Fig- tions in elastic layers frequently fit the geodetic obserure14a). FortheDiabloRange,FuisandMooney[1990] vationsby employing "very deep" dislocationsthat may
showed that the velocities of the lower crust at depths extend thousandsof kilometers in depth for the elastic
of 15-29 km are compatible with a gabbroic composi- half-spacecalculationor by using ad hoc setsof shallow
tion (Figure 14b). The structurein southernCalifornia dislocationsthat generate pseudo-viscousdeformations.
is more complicated and data are more limited. For the Some of the linear viscoelastic models are based on the
PeninsularRanges,the seismicvelocityis 5.9 km s-x
to a depthof 8 km and 6.8 km s-1 from 8 km to 26 km
[Fuis and Mooney,1990]. AlthoughFuis and Mooney
[1990]mentionseveraldifferentinterpretationsof the
seismic velocity profiles, none of the alternate models
include a granitic compositionfor the lower crust.
We showedthat the geodetic observationsalong the
San Andreas can be matched only if the rheology of
the lower crust is weaker than the experimentally de-
generalizedElsasserapproximation[Li andRice,1987].
The fitting of the geodetic data in this approximation
is obtained by adjusting a central parameter that is not
measurableeitherin situ or in the laboratory(Appendix
B).
Crustal rocks, however, have a nonlinear and temperature-dependent rheology that is incorporated into
the present modeling. Yet, while the present model
examinesthe complete range of the nonlinear viscosity
of crustal rocks, it is restricted in other parameters.
terminedrheologyof quartzite(or similarto the rheologyof a lessviscousrock suchas Westerlygranite). Theseparametersare crustalthidkness(25 kin); depth
The effective viscositiesof our "quartzite" to "very weak of lockedfault (12.5 kin); period of earthquakecycle
quartzite" models are 3-6 ordersof magnitude lessthan (160 years);slip rate (35 mm/yr); constantvelocityat
the effectiveviscositiesof quartz-diorite(intermediate) the base of the crust and below the fault; earthquake
and diabase(Figure 4b), the rocksanticipatedin the magnitude(great, M > 8); shearmodulus/Poisson's
lower crust.
ratio (35 GPa/0.25);fault geometry(vertical,long,twoThis discrepancy is probably related to the intrinsic dimensional,antiplane);constantcrustallithologyand
differencebetween in situ viscosity and laboratory viscosity due to a few reasons. First, the strain rates in
mostlaboratorytestsrangefrom10-6 s-1 to 10-s s-1,
whereasthe regionalstrain rates in the field (and in
constant thermal parameters. These parametersare not
necessarily constant, and the effect of their variations
cannot be easily evaluated. We regard the investigation
of these parameters as a task for future study.
21,994
RECHES ET AL.' PERIODIC GREAT EARTHQUAKES ON SAN ANDREAS
Velocity (km/sec)
4
5
6
7
8
Layer
2 '• •
..................................................
' --.
Vdocltyprofile
Layer
3 •,
10
9
---Alternate
model
'•..............
'1'2'"'"
0•'-•
•-QZ
monzonlte
Layer
a
iT •
--•--
15
Depth
(lcm)
Gabbro
4•
20
25
A. Salinian
block
3O
o
Velocity profile
5
10
--o-
Metmnorphosed
graywacke
'--•--
Gabbro
Depth
•5
(kin)20
25-
B. D/able Range
I
3O
4
5
6
I
I
7
8
9
Velocity (km/sec)
Figure 14. Velocity-depthcurvesfor centralCalifornia.(a) Salinianblock. (b) DiabloRange.
Heavy lines are seismicvelocities;light lines with symbolsare velocitiesfrom laboratory mea-
surements
(two differentgeotherms
wereassumed
below10 km) [afterFuis and Mooney1990,
Figure 8.5].
Conclusions
play profound transient stagesin the velocity,displace-
ment, and stressfields (Figures 6-9). On the other
We analyzed the cycles of great California earthquakesby usingfinite element computationsfor a temperature-dependent, nonlinear viscoelasticcrust. The
known crustal thicknessand crustal temperature for
California and the slip rate of the San Andreas fault
were used in the computations. The analysisleads to
the following conclusions:
1. The viscousrheologyof the crust stronglyaffectsthe transient behaviorof large earthquakes.Models with a rheologysimilar to granite or quartzite dis-
hand, modelswith stiff diabase-likerheologydisplayinsignificanttransient stages,and in this respect,models
with diabase rheology resembleelastic models. Since
large earthquakesdisplay transient behavior, reflected
in the aftershock activity and the postseismicdeformation, elastic models of the crust are probably inadequate
to describethe earthquake cycle.
2. The models which best fit the geodeticdata of
California have rheologicalparameters that are similar to the experimentally determined valuesof Westerly
RECHES ET AL.' PERIODIC GREAT EARTHQUAKES ON SAN ANDREAS
21,995
velocity directed parallel to the fault is specified. Durthe crustal rocks consideredby Kirby and Kronenberg ing an earthquakecycle the fault is locked,and at the
[1986].Sincethe lowercrustof Californiais moremafic end of the cycle it is abruptly movedto match the far-
granite and wet quartzite;theseare the least viscousof
then Westerly granite (diorite to gabbrocomposition field plate displacementaccumulatedduringthat cycle.
[FuisandMooney,1990]),we conclude
that the in situ Under such assumptionsthe problem can be modeled
viscosityof crustalrocksis significantlysmallerthan the as antiplane strain with the time-varying displacement
viscosity determined in the laboratory measurements. field of the form
We estimatethat the in situ effectiveviscosityis 10-2
to 10-6 of the viscosityinferredfrom laboratorymea-
u - u(y,z,t),
v-O,
w- O
(A1)
surements.
3. The model calculations
indicate that the effective
viscosityat the base of the crust might be as low as
where t is time, x is the coordinate directed horizontally
along the fault, y is the horizontal coordinatenormal to
10• Pa s. In the presentanalysisthe baseof the crust the fault, z is the verticalcoordinate(Figure2), and u,
moves at the uniform plate velocity assumingperfect
couplingbetweenthe mantleand the crustalbase(Figure 2). However,sincethe lowercrust cannotsupport
high shear stresses,the basal plate motion is not fully
v, w are displacementsalong x, y, z. For antiplane
strain
the strain
rates can be written
as
i
02u
-0, •z - O,4/zy=20yOt'
transmitted to the upper crust, and a zone of partial,
transient decoupling developswithin the lower crust
i
02u
= 20zOt'
(Figures7b and 7c). Someof the geodeticobservations
- O. (A2)
made •70 to 120 years after a major earthquake, which
The only nonzero strain rate componentsare the comis relatively late in the cycle,are in agreementwith such
ponents•zy and •zz. From the elasticrelationsand
decoupling
(Figures11 and 12).
4.
The models that better fit the geodetic data
the assumption of additivity of the elastic and viscous
v) we obtain the
(Figures11-13) displaytransientstagesthat include strain rates (denotedby superscripts
large spatial variations in the displacements,veloci-
constitutive relations for our problem
ties and shearstresses(Figures6 and 7). Thesespa-
i
i
tial changesoccur within a deformationzone along the
fault; the width of this zone dependson crustal viscos-
(A3)
ity. It is about 75 km (on either sideof the fault) for
the "quartzite"model(Figure 8), and it is morethan
150 km for the "very weak quartzite"modelFigure6).
Equations(A2) and (A3) must be supplemented
with
the equilibriumequationsfor the shearstresses
r•y and
Thesewidths are significantlylarger than the 50 km dis!
= 0
(a4)
tancecalculatedfor a modelwith diabaserheology(not
Oy
Oz
shownhere). Further, thesewidths are constantdur- and the relationship between the viscous strain rates
ing the earthquake cycle, in contrast to the widening and the shear stresses
of the deformationzonepredictedby Thatcher[1983].
The
widths
of the
deformation
zones
in our
calcula-
ßv - •3•._•Ase_q/aT
7zy
q'zy,
tions suggest that geodetic data should be measured
over distances of 200 km from the San Andreas
(A5)
fault to
'•
adequately represent the transient deformation.
5. The transient stage of 50 years for the least vis-
-
3•'•-lAse-q/aTv•,z
(A6)
,
cousmodel ("very weak quartzite" in Figures6 and
•- x/•(v•y
+ V•z
)•/2 Asand
naretheological
7) displayssomenew featuresnot predictedby elastic where
constants,Q is activation energy,R is the gas constant,
models. First, the surface velocity away from the fault
and T is temperature. The quantity • is recalculated
exceedsthe long-termplate velocityby asmuchas 65% for each increment of the model deformation. Equations
(Figures6a and7a). Second,thesenseof thefault-para- (A5) and (a6) expressthe fact that the viscousmodeof
llel displacement near the fault is inverted with respect
to the long-term and far-field displacement;i.e., transient left-lateral displacement is predicted in contrast
to the long-term right-lateral slip of the entire fault
system(Figures6c and 7c). Third, the fault-parallel
shear stressis also inverted with respect to the long-
deformation of crustal rocks is controlled by nonlinear,
temperature-dependent
dislocationflow [Kirby, 1983].
Kirby andKronenberg
[1986]expressed
the nonlinear
viscosity by
-Q
5- A
-
term shear(note the left-lateralshearin Figures6b and
In this equationthe rock parametersn and Q are the
7b).
sameasin (A5) and (A6). The relationsbetweenA and
As weredeterminedfor n - 2.4 for threecases.If V•y >>
Appendix
A' Numerical Methods
Our model assumesa very long vertical fault that
V•z, then A •0.6As; if Vzy= v•, thenA •As; andif
V•y = v•/3, then A •2.2As. Our solutionsare not
penetratesa uniform, isotropicviscoelasticcrust (Fig- sensitiveto a factor of 2 changein the A coefficient(see
ure 2). At great distancefrom the fault a constant the sectionon results).We thus assumethat A
21,996
RECHES ET AL.: PERIODIC GREAT EARTHQUAKES ON SAN ANDREAS
Temperatureis assumedto vary with depthaccording 0.1 year. ABAQUS monitorsthe stabilityof eachstep
internally,and if the solutionexceedsthe selectedtolerance,the programautomaticallyswitchesto implicit
to equation(4-31) of TurcotteandSchubert
[1982],
T-T•q-qrn•q-(q•-q,•)-•1-exp •
time integration(backwarddifference)whichis unconditionallystable. The sizeof the time incrementduring
automatictime steppingis controlledby a user-supplied
accuracy
parameterwhichis 10-5 in the presentcomputations. This valueis the maximumpermitteddif-
where Ts is surface temperature, qs is surfaceheat flux,
q,• is the mantle heat flux, hr is the characteristiclength
scalefor the decreaseof crustal radiogenicelementcon- ferencebetween the creep strain at the beginningof a
centration with depth, k is thermal conductivity, and z computationincrementand the creepstrainat the end
is depth. The temperature profile is assumedconstant of that increment. If this value is exceeded anywhere
in time with no dependenceon fault movement. In our in the mesh, the time incrementis reducedautomatimodel, the viscous constants As, n, and Q, and the cally. The actual maximumdifferences
in creepstrain
elastic
constants
(7 and y are assumed to be the same
rangefrom 10-6 to 10-•4 duringvariousstepsof the
for the entire crust. Thus depth variationsin rheology computations.
The highervalueof 10-6 indicates
that
reflect only the variations of temperature with depth.
the maximum error of the stress is 0.025 MPa. The
We used eight-node linear isoparametric finite el- corresponding
maximum error of the displacementis
ements (or bricks) to representthe spatial variation 0.01 mm; and the velocity error is negligibleaccord-
of the displacement and stress fields. With the lin- ingly.
ear isoparametric element, displacementsvary linearly
In a typicalcalculationovera 160-yearearthquakecyalong the edgesof each element. This elementhas one cle that usesthe nonlinearviscousrheologyof quartzite,
integrationpoint, and the stressand strainhavea single the number of increments is 20 with 2-3 iterations in
value within the element.
The stress and strain values
each increment. The shortest step is the initial userat the nodesare determinedby averagingfrom the inte- definedstep of 0.1 year, and the longestautomaticstep
grationpointsin the neighboring
elements(maximumof was•25 years. The numberof equationssolvedin such
four elementsin the presentmodel). StandardGalerkin model is •800, and the total solutiontime was •380 s
procedures,using linear isoparametricshape functions of Cray YMP cpu timeø
as weightingfactors, are usedto developthe discretized
equationsanalogousto (A1)-(AC). The resultingsystem of linear algebraic equationsis then solvedafter incorporating the equationsrepresentingthe applied displacement boundary conditions. ABAQUS uses Newton's method becausethis method convergesfaster than
Appendix B' Comparison With the
Model of Li and Rice [1987]
We have validated the application of the ABAQUS
alternatemethods(modifiedNewtonor quasi-Newton finite element code to problems involving earthquake
related deformation by comparing numerical and anmethods)[e.g.,Kikuchi,1986].
The antiplanestrain condition(Figure3) is enforced alytic [Li and Rice, 1987]resultsfor a simplemodel
of the great earthquake cycle along the San Andreas
fault. The model is illustrated in Figure B1. An elasOu/Ox = 0 is enforcedby constraining
the fault-parallel tic layer of thicknessH overliesa viscoelasticlayer of
displacementsat correspondingfront and back surface thicknessh. Both layers are infinite in the horizontal
x, y directions.A vertical fault alongthe x-z plane exnodes to be identical in the ABAQUS calculation.
The ABAQUS solutions are derived in increments. tends from the surface to a depth L within the elastic
layershave
First, a displacementincrement is applied to the bound- layer (L _<H). The elasticand viscoelastic
aries of the model. This displacementincrement is asso- shear moduli G and G•, respectively;Poisson'sratio u
ciated with a selectedtime increment and is thus equiv- is the same in both layers. Far from the fault in the
alent to a velocity. Second,the program calculatesthe 4-y directionsthere are imposedfault-parallelvelocities
by setting the v, w displacementcomponentsat all
nodes of the slice to zero, while the requirement that
elasticstrain (linear) associated
with the displacement 4-vp/2.The fault is lockedoverits entireextentduring
and the associated stresses. These stresses
the earthquake
cycletimetoy;aftertimetoyhaselapsed
are substitutedinto (A5) to determinethe incremental the fault is displacedsuddenlyto catchup to the accu-
increment
viscosity and the associated viscous strain rates. This
mulated displacementsfar from the fault. The cycle
procedureis iterateduntil equilibrium(A4) is achieved repeats indefinitely.
within the selected accuracy parameter. In ABAQUS,
equilibrium is measured by ensuring that the out-ofequilibrium forces are smaller than the specifiedforce
tolerance. In the present computationsthe force toler-
anceis 5 x 109N;this valueis smallwith respectto the
averageforceat the nodeswhichis 3-4x 10•2N.
ABAQUS provides a choiceof an automatic time increment or a user-specified,constant time increment;we
used the automatic time increment option. The computation starts with a user-defined initial time step of
We have solved for the displacementsand stresses
in the elasticand viscoelastic
layersexactly (to within numericalaccuracy)with the finite elementcode
ABAQUS. Numerical accuracyis mainly affectedby
both the finenessof the computational mesh and its
overall size. The mesh is illustrated in Figure B1. Becauseof the symmetry about the fault planeit sufficesto
consideronly y > 0. There is only one row of elements
alongthe fault-paralleldirection,a simplification
made
possibleby the antiplanestrain nature of the problem.
RECHES ET AL.: PERIODIC
GREAT EARTHQUAKES
ON SAN ANDREAS
6.4
kin
L=H=22.5km,h=13.9km,G=1.05
x 105MPa,
G•=G/10,v=0.25,
rl=2.1
x 10• MPay
VELOCITY
V
P
ELASTIC
LAYER
VISCOELASTIC
LAYER
FAULT
990 nodes, 448 elements,anti-plane
Figure B1. The modelusedto benchmark
the ABAQUSsolution
versus
the Li andRice[1987]
solutionfor greatearthquakes
alongthe SanAndreasfault. (top) Modelconfiguration
following
the Li and Ricesetting;(bottom) computational
meshusedin the ABAQUS numericalsolution
of the model at the top.
49]
I
42
10 yr
//
/
I
\
average
velocity t /
(mm/yr)
281//
14
]//J/' 147yr
7
-
0
78
156
224
312
Distancefrom fault (kin)
FigureB2. Thickness-averaged
fault-parallel
velocity
of theelastic
layer(normalized
by vr)
versusdistanceperpendicular
to the fauk (normalizedby H) at differenttimesduringthe earthquake cycle. Resultsare shownfor the ABAQUS finite elementmodel and the generalized
Elsassermodelof Li and Rice [1987].Parametersfor the numericalfinite elementmodelinclude
G1/G- 0.1. Forthe generalized
Elsasser
modelb- 139km. Otherparametervaluesaregiven
in Appendix B.
21,997
21,998
RECHES ET AL.: PERIODIC
GREAT EARTHQUAKES
ON SAN ANDREAS
28-
21-
Fault-parallel
surfacevelocity
(mm/yr)
14
7
0
78
0
156
2•4
312
Distance from fault (kin)
Figure B3. Fault-parallel surfacevelocity as a function of distancenormal to the fault at
different times during the earthquake cycle. Results are from the numerical finite element model.
Parameter valuesand normalization are the same as in Figure B3.
The thicknessesof the elasticand viscoelasticlayersare
H
=
22.5 km and h =
13.9 km.
The mesh extends
312.5 kTMin the y direction perpendicular to the fault
parallelvelocityvp/2. The fault planein the elastic
layeris heldfixedduringtcyandis displaced
parallelto
itselfand horizontallyby an amounttcyvp/2 duringa
and contains 990 nodes and 448 elements. We consider
very short time interval at the end of a cycle to catch
up with the far end. The numerical model is exercised
parametervaluesare tcy -- 160 years,G = 105 GPa, over a number of cycles to produce a quasi-steady soG1 = 10.5 GPa, u = 0.25, viscosityr/= 210 GPa years lution. Before discussingthe results of the numerical
(seeFigureB1), andvp/2 = 17.5mm yr-1. The choice calculation, we briefly describethe analytic solution of
the caseof a through-goingfault, i.e., L - H. Other
of G and G1 is discussedfurther below.
The boundary conditions for the numerical calculation are as follows. The end of the model far from and
parallel to the fault is moved at the constant horizontal
Li and Rice [1987]with whichwe compare.
Li and Rice [1987]presentedan analytic,albeit approximate solution to the problem described above.
Their approach, based on a generalization of the E1-
fault-parallelvelocityvp/2 for a time tcy. The vertical sassermodel [Elsasset,1969;Rice, 1980;Lehneret al.,
velocity and the fault perpendicularvelocity are zero 1981], determinesapproximatelayer-average
displaceon this end. The top surfaceis stressfree. The bottom ments and stressesin the elastic layer. The viscoelastic
surfaceand the extensionof the fault plane downward layer is not treated explicitly; its affect on the elastic
into the viscoelasticlayer are also movedwith the faultlayer is approximately accountedfor by a couplingpa-
averagevelocity
(mm/yr)
14
7
i
0
78
i
156
i
224
i
312
Distance from fault (kin)
Figure B4. Similar to FigureB3 but for G- G1 =35 GPa and b- 13.9 km. In this casethe
generalizedElsassersolutiondoesnot providea goodmatchto the exactnumericalsolutionat
early times in the earthquakecyclewhen elasticbehaviordominatesthe solution.
RECHES ET AL.: PERIODIC GREAT EARTHQUAKES ON SAN ANDREAS
rameter b. The value of b is not uniquely determinedbut
dependson the thicknesses
of the layersand their theologicalproperties.The generalized
Elsassetequation
for the thickness-averaged
fault-parallel displacementu
in the elasticlayeris [Li and Rice, 1987]
21,999
The approximate analytically determinedvaluesof
thickness-averaged
fault-parallel velocityin the elastic
layerasa functionof distance
fromthefault (FigureB4}
agreescloselywith the exactnumericalsolutionat times
late in the earthquakecyclewhen viscousrelaxationeffects are important. However,the approximateanalytic
result is not a goodrepresentationof the exactsolution
-- =
+bH
(B1) early in the earthquakecyclewhen the elasticresponse
Ot
Oy
of the coupledlayers dominates. The inability of the
analytic solutionto match the exact solutionis probaIt is the solutionof the approximateequation(B1) bly a consequence
of the choiceof the b value. There
with which we comparethe exact numericalsolution. is no generalprescriptionfor determiningb;indeed,the
The solutionof (B1) dependson b, and there is no "correct" value of b can only be obtained a posteriori
way to determineb for arbitrary valuesof H, h, G, by fitting the exact numericalsolution.This is a basic
G1, and •/. If G1 << G, then b is approximatedby limitation of the generalizedElsassermodel.
hG/G1 [Lehnere! al., 1981]. For this reasonwe took
In summary, we have benchmarkedthe numericalfiG1/G = 0.1 asnotedabove.The corresponding
value nite element code against the approximateanalytic so-
Ou(HGhO) 02u
of b is then 139 km and the associated relaxation time
tr = brl/Gh is 20 years.
FigureB2 showsthe thickness-averaged
fault-parallel
velocityin the elasticlayer(normalized
by vp)at different timesduringthe earthquakecycleversusperpendicular distancefrom the fault (normalizedby the elastic
layerthickness)
for the Li andRice[1987]modeland
our numerical model. The numerical model was run for
lution of Li and Rice [1987]with a choiceof modelparameter valuesfor which the analytic solution is closeto
beingexact. We then usedthe exactnumericalsolution,
with another choice of model parameters, to demonstrate the approximate nature and limitations of the
analytic solution.
Acknowledgments.
The authors are grateful to Nan-
five cyclesand the resultsare plottedfor correspondingnette Anderson for her help in the preparation of this paper.
times in the fifth cycle, i.e., 648.7 years= 4 x 160 years
The thorough reviews of Roger Denlinger, Stephen Hickman,and James Savagecontributed to significantimprovement of the manuscript. The work was supportedby grants
+8.7 years.Agreementbetweenthe numericaland approximateanalyticmodelsis excellentat all timesduring the earthquakecycle. The small disagreement
at
largedistances
from the fault late in the cycle(Figure B2, time = 142.5 years) is a consequence
of the
from the Institute of Geophysics and Planetary Physics at
the Los Alamos National Laboratory, the NASA Geodynam-
numericalmodel not being quite long enough;i.e., the
veloped by HKS Inc., Pawtucket, Rhode Island.
numerical model should have extended somewhat farther from the fault.
The behaviorof grounddeformationduringthe earthquakecycleis illustratedin Figure B3 by the plot of
normalizedfault-parallel surfacevelocityversusnormalized distance from the fault for different times during
ics program(NAGW 2646),andthe U.S.-IsraelBi-national
ScienceFund grant (86-183). The ABAQUS codewas de-
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