# Spatio-Temporal Evolution of Earthquakes and Faults

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```					       A generalized law for aftershock
behavior in a damage rheology model
1. University of Southern California
2. 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
Main observed features of aftershock sequences:
1. Aftershocks occur around   2. Aftershock decay rates can be
the mainshock rupture zone    described by the Omori-Utsu law:

DN/Dt = K(c + t)-p
However, aftershock decay rates can
also be fitted with exponential and
other functions (e.g., Kisslinger, 1996).

3. The frequency-size statistics of

logN(M) = a - bM

4. The largest aftershock magnitude
is typically about 1-1.5 units below
5. Aftershocks behavior is    that of the mainshock (Båth law).
NOT universal!
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.
Non-linear Continuum Damage Rheology (1) Mechanical aspect:
sensitivity of elastic moduli to distributed cracks and sense of
peak
stress
yielding

Stress

a=0                 Strain       0 < a < ac
s                                     s

Tension                             Tension

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

The elastic energy U is written as:

1 2                                    I1
U =  I 1  I 2 - I 1 I 2              =
2                                       I2
Where  and  are Lame constants;    I1= ekk
 is an additional elastic modulus   I2= eijeij
U           I2                 I1 
sij =      =  -      I1ij   2 -      eij
eij       I1 

         I2 

Non-linear Continuum Damage Rheology (2) Kinetic aspect associated
with damage evolution
peak
stress
yielding

Stress

Strain
0 < a < ac
a = ac
s                              s

Tension                     Tension
Compression               e     Compression             e
This is accounted for by making the
elastic moduli functions of a damage
state variable a(x, y, z, t), representing
crack density in a unit volume, and
deriving an evolution equation for a.
Thermodynamics
Free energy of a solid, F, is
F = F(T, eij, a)
T – temperature, eij – elastic strain tensor,
a – scalar damage parameter

= F  TS  = s ij e ij -  i J i
dU d           1
Energy balance
dt dt          

Entropy balance         dS         Ji 
= - i    G
dt        T
Gibbs equation
F            F
dF = -SdT        de ij     da
e ij         a
The internal entropy production rate per unit mass, G, is:
Ji       1            1 F da
G = - 2  i T  s ij e ij -         0
T        T            T a dt
da
da/dt > 0                           = Cd  I 2  - 0 
dt
 > 0
I1            Strain
weakening                             =             invariant
Shear                                                I2            ratio
Stress
                       = tan () sn
 = 0               I1=ekk
I2= eijeij
da/dt < 0     da            a
 < 0          = C1  exp( )  I 2  -  0 
dt            C2
healing
(strengthening)

Normal Stress
sn
Rate- and state-dependent
friction experiments constrain
parameters c1 and c2.                D

0.1                1               10
V / V0

D = D/s

For details, see Lyakhovsky
0     0.1    0.2       0.3   0.4   0.5
et al. (GJI, 2005)
LOGe (s/s0)
Non-linear Continuum Damage Rheology (3) damage-related viscosity

10 years creep experiment
on Granite beam at room
temperature

Ito & Kumagai, 1994

For typical values of
shear moduli of granite
(2-3 * 1010 Pa)
Viscosity = 8 x 1019 Pa s
The Maxwell relaxation time
is as small as
tens of years
Non-linear Continuum Damage Rheology (3) damage-related viscosity

Stress-strain and AE locations
for G3 (Lockner et al., 1992)
600
Z
500       G3 data
Simulation
400
Y
Stress, MPa

300

1
=      , a 0

X                               200                       Cva

100             1/Cv) = 5·1010 Pa, Cd = 3 s-1

0
0   2      4        6        8      10   12   14

Strain ( x103 )
Berea sandstone under 50 MPa confining pressure
250

200
Differential stress (MPa)

150

100

50                                           Accumulated
irreversible
strain
0
-1          -0.5        0        0.5         1         1.5
0 = 1.4 1010 Pa,                                             Strain %

Cv = 10-10 Pa-1,                                                         Data from Lockner lab. USGS
Model from Hamiel et al., 2004
R = 1.4
Aftershocks: 1D analytical results for uniform deformation
For 1D deformation, the equation for positive damage evolution is

da/dt = Cd (e2-e02),                                                  (1)
where e is the current strain and e0 separates degradation from healing.
The stress-strain relation in this case is

s = 20(1 – a)e,                                                      (2)
where 0(1–a) 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
failure relation with exponent 1/3 for a system-size brittle event).
For positive rate of damage evolution (e > e0), we assume inelastic strain
before macroscopic failure in the form

e = (Cv da/dt) s                                                      (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,

2de/dt = –e                                                                       (4)
de                         da
Using this condition in (2) and (3) gives            = -C v  0 1 - a   e      (5)
dt                         dt
1         2
and integrating (5) we get                 e = A  exp  R1 - a                (6)
2          
 1           2
where R = d/M = 0Cv and             A = e s  exp - R1 - a s     is integration
 2            
constant with a = as and e = es for t = 0.

Using these results in (1) yields exponential damage evolution

da
dt
                               
= C d  e s2 exp R1 - a  - R1 - a s  - e 0
2             2    2
(7)
Scaling the results to number of events N

Assuming that a is scaled linearly with the number of aftershocks N

a = a s  fN                                                                  (8)

we get

f
dN
dt
                                      
= Cd  e s2 exp R1 - a s - fN  - R1 - a s  - e 0
2             2    2                     (9)

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

f
dN
dt

= Cd  e s2 exp- 2fNR1 - a s  - e 0
2

(10)
If also the initial strain induced by the mainshock is large enough so that

e 02 << e s2 exp - 2fNR 1 - a s 
(11)
the solution is (the Omori-Utsu law)
dN         Cd e s2
=
dt 2fR1 - a s Cd e s2t  f                                       (12)
      C d e s2
For t = 0                     N0 =
f
dN         
N0                 
N0               1
so      =                   =                                             (13)
 t  1 2fR1 - a N t  1 2fR1 - a N
dt 2fR1 - a s N 0              s

0              s

0

The parameters of the Omori-Utsu law are

k=
1                                             dN/dt = K(c + t)-p
2fR1 - a s 
1           k
c=                 =
2fR1 - a s N 0 N 0
    

and p = 1

analytical results first with e0=0, as=0 and then with finite small values.

f
dN
dt
                                      
= Cd  e s2 exp R1 - a s - fN  - R1 - a s  - e 0
2             2    2    (9)
Events rate vs. time for several values of R = d/M with e0=0, as=0)
Timescale of fracturing
Material property R =
Timescale of stress relaxation
180
Small R:
160                                                              •expect long
R = 0.1                                active
Number of aftershocks per day

140
aftershock
120                                                              sequences

100

R=1
80
Modified Omori
60
law with p = 1

Large R:                                  40
•expect short
R = 10
diffuse                                   20

sequences                                 0
0   10   20   30     40    50     60   70   80   90   100
Time (day)
Changing the power-law parameters, we can fit the other lines !!!
Events rate vs. time for several values of R = d/M with finite e0,as)

Timescale of fracturing
Material property R =
Timescale of stress relaxation

150
Omori
p=1
125
Number of aftershocks per day

100
R = 0.1

Omori
75
p=1

R = 0.3
50                                 Omori
p = 1.2

25
R = 10                       R=1

0
0        20             40             60     80       100
Time (day)
Imposed damage
3-D numerical simulations                       (major fault zone)

1-7 km

35 km
Newtonian
Sedimentary cover         viscosity
Damage visco-elastic rheology
Crystalline     plus power law viscosity
Crust          (based on diabase lab data)
50 km
Upper mantle
Damage visco-elastic rheology
plus power-law viscosity
(based on Olivine lab data)
100 km                      y
x
z
In each layer the strain is the sum of damage-elastic, damage-related
inelastic, and ductile components: e t = e e  e i  e d
ij      ij       ij       ij

Initial stress = regional stress + imposed mainshock slip on a fault extending
over 50 km ≤ y ≤ 150 km, 0 ≤ z ≤ 15 km with fixed boundaries
Differential Stress (MPa)
0   100          200            300    400   500
0
Initial regional stress for
20 oC/km – heavy line
10                           s = s n                  30 oC/km – dash line
40 oC/km – dotted line

Strain rate = 10-15 1/s
20
Brittle-ductile
e = A0 e -Q / RT n

transition at 300 oC
30
Moho

40

50
Simulations with fixed r = 300 s
Effects of R (sediment thickness = 1 km, gradient                                                                                                              20 oC/km )

200                                                                                      200                                                                    100
R=0.1                                                                               R=1

Number of events
Number of events

R=2

Number of events
150                                                                                                                                                               80
150

60
100                                                                                      100
40
50                                                                                   50
20

0                                                                          0                                                                      0
0        20        40        60        80        100         4060 2080
0                                       100         0       20   40                                      60     80   100
Time (days)                                                                                                Time (days)                                                  Time (days)
50                                                                                       50

Increasing R values:
45                                                                                       45
40                                     R=3                                               40
Number of events

R=10
Number of events

35                                                                                       35
30
25
30
25
•diffuse sequences
20                                                                                       20
15                                                                                       15                                                            •shorter duration
10                                                                                       10
5                                                                                        5
0                                                                                        0                                                            •smaller # of events
0        20        40        60        80         100                                    0     20      40   60      80   100
Time (days)                                                                            Time (days)
4                            R=0.1
3.5                           R=1
R=2
3
R=3
2.5
Log(Number)

R=10
2

1.5

1

0.5

0
3   4               5           6
Magnitude

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

Large R values (R > 3): Narrow range of event sizes
Effect of Sediment thickness (R = 1, gradient                                                                                20 oC/km )

200                                                       250                                                            100

S=1 km                                                   S=4 km                                                    S=7 km
Number of events

Number of events

Number of events
200                                                            80
150
150                                                            60
100
100                                                            40
50
50                                                        20

0                                                         0                                                          0
0   20     40   60     80   100                           0   20    40   60      80   100                           0   20    40   60      80   100
Time (days)                                               Time (days)                                               Time (days)

Increasing thickness of weak sediments: diffuse sequences, shorter
duration, smaller number of events (similar to increasing R values)
Effect of thermal gradient and R (sediment layer 1                            km )

0

-5

-10
Depth (km)

-15

-20

-25

-30                                      R = 0.1
-35                                      T = 20 C/km

-40
0   30   60   90   120   150   180 210   240 270   300

Time (days)

Increasing thermal gradient and/or R: thinner seismogenic zone
The maximum event depth decreases with time from the mainshock
Observed Depth Evolution of Landers aftershocks
(Rolandone et al., 2004)

“Regional” depth
(1283 events)                                       d95
d5%

JV

HypoDD
Johnson Valley Fault
11944 events      Hauksson (2000)

Depth of seismic-aseismic transition increases following Landers EQ
and then shallows by ≤ 3 km over the course of 4 yrs.
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          s = 2  e seis
can be estimated as

The rate of gradual inelastic strain can          de i dt = -aCvs / 2

be estimated as

The inelastic strain accumulation (aseismic creep) is    e i = Cvs / 2

R    Slip ratio
Seismic slip              e seis    1
=        =                     0.1     90 %
Total slip               e total 1  R                 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.

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