A generalized law for aftershock behavior in a damage rheology model
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Transcript of A generalized law for aftershock behavior in a damage rheology model
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.