A generalized law for aftershock behavior in a damage rheology

model Yehuda Ben-Zion1 and Vladimir Lyakhovsky2

1. University of Southern California2. Geological Survey of Israel

Outline• Brief background on aftershocks• Brief background on the employed damage

rheology• 1-D Analytical results on aftershocks• 3-D Numerical results on aftershocks• Discussion and Conclusions

3. The frequency-size statistics of aftershocks follow the GR relation:

logN(M) = a - bM

2. Aftershock decay rates can be described by the Omori-Utsu law:

N/t = K(c + t)p

5. Aftershocks behavior is NOT universal!

1. Aftershocks occur around the mainshock rupture zone

Main observed features of aftershock sequences:

However, aftershock decay rates can also be fitted with exponential and other functions (e.g., Kisslinger, 1996).

4. The largest aftershock magnitude is typically about 1-1.5 units below that of the mainshock (Båth law).

Existing aftershock models:

•Migration of pore fluids (e.g., Nur and Booker, 1972)

•Stress corrosion (e.g., Yamashita and Knopoff, 1987)

•Criticality (e.g., Bak et al., 1987; Amit et al., 2005)

•Rate- and state-dependent friction (Dieterich, 1994)

•Fault patches governed by dislocation creep (Zöller et al., 2005).

Is the problem solved?

The above models focus primarily on rates.

None explains properties (1)-(5), including the observed spatio-temporal variability, in terms of basic geological and physical properties. This is done here with a damage rheology framework and realistic model of the lithosphere.

Str

ess

Strain

yielding

peakstress

Tension

Compression

0

Tension

Compression

0c

Non-linear Continuum Damage RheologyNon-linear Continuum Damage Rheology (1) Mechanical aspect: sensitivity of elastic moduli to distributed cracks and sense of loading.

This is accounted for by generalizing the strain energy function of a deforming solid

ij

2

1ij1

1

2

ijij

I

I2I

I

IU

U I I I I

1

2 12

2 1 2

The elastic energy U is written as:

Where and are Lame constants ; is an additional elastic modulus

I1= kk I2= ijij

2

1

I

I

Str

ess

Strain

yielding

peakstress

Tension

Compression

0c

Tension

Compression

c

Non-linear Continuum Damage RheologyNon-linear Continuum Damage Rheology (2) Kinetic aspect associated with damage evolution

This is accounted for by making the elastic moduli functions of a damage state variable (x, y, z, t), representing crack density in a unit volume, and deriving an evolution equation for .

Gibbs equation

 The internal entropy production rate per unit mass, , is: 

Thermodynamics 

Free energy of a solid, F, is

F = F(T, ij, ) 

T – temperature, ij – elastic strain tensor,

– scalar damage parameter 

Energy balance 

iiijij Je1

TSFdt

d

dt

dU

Entropy balance 

T

J

dt

dS ii

dF

dF

SdTdF ijij

0dt

dF

T

1e

T

1T

T

Jijiji2

i

ShearStress

Normal Stress

n

d/dt > 0 > 0

weakening(degradation)

d/dt < 0 < 0

healing(strengthening)

= tan () n

0

02d ICdt

d

022

1 I)C

exp(Cdt

d

2

1

I

I

Strain invariant

ratio

I1=kk

I2= ijij

0.1 1 10

V / V0

0 0.1 0.2 0.3 0.4 0.5

LOGe (/0)

Rate- and state-dependent friction experiments constrain parameters cparameters c11 and c and c22.

For details, see Lyakhovskyet al. (GJI, 2005)

10 years creep experiment on Granite beam at room

temperature

Ito & Kumagai, 1994

Viscosity = 8 x 1019 Pa s

For typical values ofshear moduli of granite

(2-3 * 1010 Pa)

The Maxwell relaxation timeis as small astens of years

Non-linear Continuum Damage RheologyNon-linear Continuum Damage Rheology (3) damage-related viscosity

0

100

200

300

400

500

600

0 2 4 6 8 10 12 14

Strain ( x103 )

Stre

ss, M

Pa

G3 data

Simulation

0,1

vC

Cv) = 5·1010 Pa, Cd = 3 s-1

Stress-strain and AE locations for G3 (Lockner et al., 1992)

X

Z

Y

Non-linear Continuum Damage RheologyNon-linear Continuum Damage Rheology (3) damage-related viscosity

0

50

100

150

200

250

-1 -0.5 0 0.5 1 1.5

Strain %

Dif

fere

nti

al s

tres

s (M

Pa)

Berea sandstone under 50 MPa confining pressure

Data from Lockner lab. USGS

Model from Hamiel et al., 2004

Accumulated irreversible

strain

0 = 1.4 1010 Pa,

Cv = 10-10 Pa-1,

R = 1.4

What about aftershocks?

Aftershocks: 1D analytical results for uniform deformation

For 1D deformation, the equation for positive damage evolution is

d/dt = Cd (2-2), (1)

where is the current strain and 0 separates degradation from healing.

The stress-strain relation in this case is

= 20(1 – )(2)

where 0(1–) is the effective elastic modulus of a 1D damaged material with 0 being the initial modulus of the undamaged solid.

(Ben-Zion and Lyakhovsky [2002] showed analytically that these equations lead under constant stress loading to a power law time-to-failure relation with exponent 1/3 for a system-size brittle event).

For positive rate of damage evolution ( > 0), we assume inelastic strain before macroscopic failure in the form

e = (Cv d/dt) (3)

For aftershocks, we consider material relaxation following a strain step. This corresponds to a situation with a boundary conditions of constant total strain.

In this case the rate of elastic strain relaxation is equal to the viscous strain rate,

2d/dt = –e(4)

Using this condition in (2) and (3) gives (5)

and integrating (5) we get (6)

where RR = = dd//MM == CCvv and is integration

constant with = s and = s for t = 0.

Using these results in (1) yields exponential damage

evolution

(7)

dt

dC

dt

dv

10

212

1RexpA

21

2

1exp ss RA

20

222 11exp ssd RRC

dt

d

Scaling the results to number of events N

Assuming that is scaled linearly with the number of aftershocks N

(8)

we get

(9)

If N is small (generally true), so that (N)2 can be neglected

(10)

If also the initial strain induced by the mainshock is large enough so that

(11) the solution is (the Omori-Utsu law)

(12)

Ns

20

222 11exp sssd RNRCdt

dN

20

2 12exp ssd NRCdt

dN

ss NR 12exp220

tCR

C

dt

dN

sds

sd2

2

12

For t = 0

so (13)

The parameters of the Omori-Utsu law are

and p = 1

We now return to the general exponential equation (9) and examine analytical results first with 0=0, s=0 and then with finite small values.

(9)

2

0sdCN

sRk

12

1

0012

1

N

k

NRc

s

00

0

0

0

121

1

12112 NRtNR

N

tNR

N

dt

dN

sss

20

222 11exp sssd RNRCdt

dN

dN/dt = K(c + t)p

0 10 20 30 40 50 60 70 80 90 100

Time (day)

0

20

40

60

80

100

120

140

160

180

Nu

mb

er o

f af

ters

hoc

ks

per

day

R = 10

R = 1

R = 0.1

Events rate vs. time for several values of R = d/M with0=0,

s=0)

Changing the power-law parameters, we can fit the other lines !!!

Small R:•expect long active aftershock sequences

Large R:•expect short diffuse sequences

Modified Omorilaw with p = 1

Material property R =Timescale of fracturing

Timescale of stress relaxation

0 20 40 60 80 100

Time (day)

R = 0.1

R = 0.3

R = 1R = 10

Omori p = 1

Omori p = 1

Omori p = 1.2

0

25

50

75

100

125

150

Nu

mb

er o

f af

ters

hoc

ks

per

day

Events rate vs. time for several values of R = d/M with

finite0,s)

Material property R =Timescale of fracturing

Timescale of stress relaxation

In each layer the strain is the sum of damage-elastic, damage-related inelastic, and ductile components:

Initial stress = regional stress + imposed mainshock slip on a fault extending over 50 km ≤ y ≤ 150 km, 0 ≤ z ≤ 15 km with fixed boundaries

Damage visco-elastic rheology plus power-law viscosity

(based on Olivine lab data)

100 km

100

km

Moho

Damage visco-elastic rheology plus power law viscosity

(based on diabase lab data)

Basem

ent

Sedimentary cover

CrystallineCrust

Upper mantle

1-7 km

35 kmFre

e su

rface

50 km

Imposed damage (major fault zone)

xy

z

Newtonianviscosity

dij

iij

eij

tij

3-D numerical simulations

0

10

20

30

40

50

0 100 200 300 400 500

Differential Stress (MPa)

Moho

Initial regional stress for temperature gradients20 oC/km – heavy line30 oC/km – dash line40 oC/km – dotted lineStrain rate = 10-15 1/s

Brittle-ductile transition at 300 oC

ns

nRTQeA /0

Effects of Effects of R R (sediment thickness = 1 km, gradient 20 oC/km )

0

100

200

0 20 40 60 80 100Time (days)

Nu

mb

er o

f ev

ents

0

50

100

150

200

0 204060 80 100

Time (days)

Nu

mb

er o

f ev

ents

0

20

40

60

80

100

0 20 40 60 80 100Time (days)

Nu

mb

er o

f ev

ents

05

101520253035404550

0 20 40 60 80 100Time (days)

Nu

mb

er o

f ev

ents

05

101520253035404550

0 20 40 60 80 100Time (days)

Nu

mb

er o

f ev

ents

150

50

R=0.1 R=1 R=2

R=3 R=10

Simulations with fixed r = 300 s

Increasing R values:

•diffuse sequences

•shorter duration

•smaller # of events

0

0.5

1

1.5

2

2.5

3

3.5

4

3 4 5 6

Magnitude

Log

(Nu

mb

er)

R=0.1

R=1

R=2

R=3

R=10

Small R values (R < 1): Power law frequency-size statistics

Large R values (R > 3): Narrow range of event sizes

0

50

100

150

200

0 20 40 60 80 100Time (days)

Nu

mb

er o

f ev

ents

0

50

100

150

200

250

0 20 40 60 80 100Time (days)

Nu

mb

er o

f ev

ents S=4 km

Effect of Sediment thickness (R = 1, gradient 20 oC/km )

S=1 km

0

20

40

60

80

100

0 20 40 60 80 100Time (days)

Nu

mb

er o

f ev

ents S=7 km

Increasing thickness of weak sediments: diffuse sequences, shorter duration, smaller number of events (similar to increasing R values)

-40

-35

-30

-25

-20

-15

-10

-5

0

0 30 60 90 120 150 180 210 240 270 300

Time (days)

De

pth

(k

m)

R = 0.1 T = 20 C/km

Effect of thermal gradient and R (sediment layer 1 km )

Increasing thermal gradient and/or R: thinner seismogenic zone The maximum event depth decreases with time from the mainshock

JV

Depth of seismic-aseismic transition increases following Landers EQ and then shallows by ≤ 3 km over the course of 4 yrs.

Johnson Valley Fault

d95

d5%

11944 events

“Regional” depth(1283 events)

(Rolandone et al., 2004)

Observed Depth Evolution of Landers aftershocks

HypoDDHauksson (2000)

The parameter R controls the partition of energy between seismic and aseismic components (degree of seismic coupling across a fault)

The brittle (seismic) component of deformation can be estimated as

The rate of gradual inelastic strain canbe estimated as

The inelastic strain accumulation (aseismic creep) is

seis 2

2/ vi Cdtd

2/ vi C

Rtotal

seis

1

1

Seismic slip

Total slip

R Slip ratio

0.1 90 %

1 50 %

2 33 %

10 10%

Main Conclusions

•Aftershocks decay rate may be governed by exponential rather than power law as is commonly believed (see also Dieterich, 1994; Gross and Kisslinger, 1994, Narteau et al., 2002)

•The key factor controlling aftershocks behavior is the ratio R of the timescale for brittle fracture evolution to viscous relaxation timescale.

•The material parameter R increases with increasing heat and fluids, and is inversely proportional to the degree of seismic coupling.

•Situations with R ≤ 1, representing highly brittle cases, produce clear aftershock sequences that can be fitted well by the Omori power law relation with p ≈ 1, and have power law frequency size statistics.

•Situations with R >> 1, representing stable cases with low seismic coupling, produce diffuse aftershock sequences & swarm-like behavior.

•Increasing thickness of weak sedimentary cover produce results that are similar to those associated with increasing R.

Thank you

Key References (on damage and evolution of earthquakes & faults):

Lyakhovsky, V., Y. Ben-Zion and A. Agnon, Distributed Damage, Faulting, and Friction, J. Geophys. Res., 102, 27635-27649, 1997.

Ben-Zion, Y., K. Dahmen, V. Lyakhovsky, D. Ertas and A. Agnon, Self-Driven Mode Switching of Earthquake Activity on a Fault System, Earth Planet. Sci. Lett., 172/1-2, 11-21, 1999.

Lyakhovsky, V., Y. Ben-Zion and A. Agnon, Earthquake Cycle, Fault Zones, and Seismicity Patterns in a Rheologically Layered Lithosphere, J. Geophys. Res., 106, 4103-4120, 2001.

Ben-Zion, Y. and V. Lyakhovsky, Accelerated Seismic Release and Related Aspects of Seismicity Patterns on Earthquake Faults, Pure Appl. Geophys., 159, 2385 –2412, 2002.

Hamiel, Y., *Liu, Y., V. Lyakhovsky, Y. Ben-Zion and D. Lockner, A Visco-Elastic Damage Model with Applications to Stable and Unstable fracturing, Geophys. J. Int., 159, 1155-1165, doi: 10.1111/j.1365-246X.2004.02452.x, 2004.

Ben-Zion, Y. and V. Lyakhovsky, Analysis of Aftershocks in a Lithospheric Model with Seismogenic Zone Governed by Damage Rheology, Geophys. J. Int., in press, 2006.