AGU, San Francisco, December 2012.

AGU2011 S53A-2469.

Interactions and triggering in a 3D rate-and-state asperity model.Pierre Dublanchet, Pascal Bernard and Pascal Favreau

Institut de Physique du Globe de Paris, 1 rue Jussieu 75005 Paris, France. Contact:dublanchet@ipgp.fr, bernard@ipgp.fr

1 - Introduction

We present a 3D rate-and-state model of fault that supportsthe observations of multiplets in which earthquakes occur oncoplanar asperities forced by surrounding aseismic creep.Fur-thermore, our mechanical model allows to compute syntheticcatalogs of seismicity, and therefore to adress the question ofthe relation between empirical laws characterizing seismicity(Omori decay, Gutenberg-Richter distribution) and friction onfaults. In particular, we focus on the behavior of aseismic bar-riers between neighbouring asperities during a seismic event,and we show how these barriers control the shape of the powerlaw decays characterizing the interaction of sources amongapopulation of asperities.

2 - Parkfield seismicity

Asperities on a creeping fault, from (Lenglinéet al., 2009):

−60 −40 −20 0−25

−15

−5

NW SE

x (km)

z (k

m)

1984−2006

Mw 6 2004

−400 −200 0 200 400

−400

−200

0

200

400

x (m)

z (m

)

0 5 10 15 200.5

1

1.5

2

2.5

t (years)

mw

a

b c

Earthquake statistics:

10−6

10−4

10−2

100

102

10−3

10−1

101

p* ∼0.8

p* ∼0.54

p* ∼0.81

dt/

pro

ba

bil

ity

de

nsi

ty

p* ∼0.8

p* ∼0.54

p* ∼0.81

p* ∼0.8

p* ∼0.54

p* ∼0.81

p* ∼0.8

p* ∼0.54

p* ∼0.81

p* ∼0.8

p* ∼0.54

p* ∼0.81

10−8

10−5

10−2

101

10−4

10−2

100

102

104

106

p* ∼ 0.56

p* ∼ 0.9

dt/

pro

ba

bil

ity

de

nsi

ty

100

102

104

106

108

1010

10−2

100

102

104

p ∼ 0.54

t (s)

r/r 0

100

102

104

106

108

1010

10−2

100

102

104

p ∼ 0.53p ∼ 0.83

t (s)

r/r 0

0 0.5 1 1.5 2 2.5 3 3.510

−4

10−3

10−2

10−1

100

b ∼ 1.07

m

n(m

w>

m)/

nm

ax

0 0.5 1 1.5 2 2.5 3 3.510

−3

10−2

10−1

100

b ∼ 0.86

b ∼ 1.79

m

n(m

w>

m)/

nm

ax

a b

c d

*

Global statistics:

events after 2004 Mw

6

events before 2004 Mw

6

all events

Local statistics:

(events before 2004 Mw

6)

Inte

rev

en

t ti

me

dis

trib

uti

on

Om

ori

law

Ma

gn

itu

de

-fre

qu

en

cy

d

istr

ibu

tio

n

3 - 3D rate-and-state asperity model

d

Velocity weakening

asperities (aw

,bw

and (a-b)w

0)

Fig 3: schematic diagram of the fault with velocity weakening(a − b = (a − b)w < 0) asperities embedded in a velocity

strengthening creeping area (a − b = (a − b)s > 0).

Rate-and-state friction with the slip law:

{

τ ixz = σ [µ0 + ai ln(vi/vp) + biΘi]

Θ̇i = −vidc

[Θi + ln(vi/vp)]

Quasi-dynamic stress interactions, damping parameterη:

τ ixz = τ∗

−

µw(δi − vpt) +

∑

jkij(δj − vpt) − η(vi − vp)

Length scale for nucleation: from (Ampuero & Rubin, 2008):

Lb =µdcbσ

In the following: vp = 3.15 cm.year−1, σ = 100 MPa,µ0 = 0.6,

dc = 0.3 mm, η = 107 Pa.s.m−1, aw = 0.0032, bw = 0.0067,

R = 30 m,h = 3 m andq = 1 so thath/Lb = 0.22.

4 - Seismic rupture of asperities.

x/Lb

y/L

b

t = 0 s

10 15 20

15

20

25

log

v/v

p

−2

0

2

4

6

8

x/Lb

y/L

b

t = 0.1007 s

10 15 20

15

20

25

log

v/v

p

−2

0

2

4

6

8

x/Lb

y/L

b

t = 0.1534 s

10 15 20

15

20

25

log

v/v

p

−2

0

2

4

6

8

x/Lb

y/L

b

t = 0.3009 s

10 15 20

15

20

25

log

v/v

p

−2

0

2

4

6

8

0 0.3 0.610

10

1012

1014

t (s)

dM

0/d

t (N

.m.s

−1

)

0 0.2 0.4 0.63.6

3.8

4

4.2x 10

13

t (s)

M0

(N

.m)

0 0.2 0.4 0.6−10

0

10

t (s)

τ−τ 0

(M

Pa

)

0 0.2 0.4 0.610

−5

100

105

1010

t (s)

v/v

p

τ-τ0

Fig. 4: Left (4 colored panels): sliding velocityv during an earthquakeaffecting a group of5 asperities.Right: cumulative momentM0 and

moment rateṀ0 released by the entire fault during a seismic event. Thetwo bottom panels indicate stress et velocity at the center of the five

asperities.as = 0.0042, bs = 0.001, Lb = 13 m.

5 - Synthetic statistics: randomdistribution of asperities.

-seismic moment:

M0(t)-moment rate:

dM0(t)/dt

Synthetic catalog:

-location (x,y,t)

-magnitude M

-stress drop Δσw

Earthquake if

dM0(t)/dt > μSvs

(vs=1 cm.s-1)

0 20 400

20

40

y/L

b

x/Lb

a-b0

Fig. 5: Construction of a synthetic catalog of seismicity from a knowndistribution of asperities.

100

104

108

10−4

104

1012

t (s)

r/r 0

p ∼ 0.3p ∼ 0.8p ∼ 0.3p ∼ 0.8p ∼ 0.3p ∼ 0.8p ∼ 0.3p ∼ 0.8

10−6

10−3

100

100

104

108

dt/

probability density

p* ∼ 0.49

p* ∼ 1.02

p* ∼ 0.49

p* ∼ 1.02

p* ∼ 0.49

p* ∼ 1.02

p* ∼ 0.49

p* ∼ 1.02

−1 0 1−1

−0.5

0

0.5

1

(a−b)s = 0.7e−3

(a−b)s = 1.7e−3

(a−b)s = 3.2e−3

(a−b)s = 5.2e−3

1.6 2 2.4 2.8

10−2

10−1

100

m

n(m

w>m)/nmax

b ∼ 1.74b ∼ 2.59b ∼ 1.74b ∼ 2.59b ∼ 1.74b ∼ 2.59b* ∼ 1.74b* ∼ 2.59

a b

c

Fig. 6: Synthetic earthquake statistics (a): omori law, (b): Intereventtime distribution, (c): magnitude frequency distribution.

7 - Conclusions

In this study, we were able to generate synthetic catalogs ofseismicitycharacterized by statistical laws with power decays similar to what isobserved in Parkfield. Moreover, we showed that realistic Omori lawand Gutenberg-Richter distributions occur only if the distribution of as-perities is characterized by interasperity distances broadly distributedaround a critical distance that depends on frictional properties of the in-terasperity creeping barriers. This large distribution indeed allows thepossibility of seismic ruptures affecting a large variety of fault surfaces,strong interaction between neighbouring sources, as well as indepen-dent rupture of asperities. The density of asperities is therefore a majorparameter controlling the statistical properties of seismicity.

6 - Critical density of asperities

(2)

(3)

(1)

∆τ (1)

rd

c

dc

R

(a-b)sσ

asperities (1) and (2)

are isolated

asperities (1) and (3) break in

a single seismic event

Δτ>(a-b)sσ

Fig 7: the rupture of the asperity (1) generates a stress per-turbation ∆τ . Large acceleration of creep occurs in thestrengthening regions experiencing large∆τ , that is at in-fracritical distances from (1) (d < dc).

0 0.5 10

1

2

3

4

d/2R

(a−

b)

* s/a

s

K.S.

far "eld

0 0.2 0.4 0.60

1

2

3

4

ρa

(a−

b)

* s/a

s

Fig 8: critical interasperity distanced (top) or density ofasperityρa (bottom). Dots: numerical results, red lines:theoretical prediction, based on (Kassir & Sih, 1966) solu-tion (K.S.).

ReferencesAmpuero, J.P., & Rubin, A.M. 2008. Earthquake nucleation

on rate and state faults: Aging and slip laws.J. geo-phys. Res, 113, B01302.

Kassir, MK, & Sih, G.C. 1966. Three-dimensional stressdistribution around an elliptical crack under arbitraryloadings.Journal of Applied Mechanics, 33, 601.

Lengliné, O., Marsan, D.,et al. 2009. Inferring the co-seismic and postseismic stress changes caused by the2004 Mw= 6 Parkfield earthquake from variations ofrecurrence times of microearthquakes.J. geophys.Res, 114, B10303.