Spatio-Temporal Evolution of Earthquakes and Faults

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Spatio-Temporal Evolution of Earthquakes and Faults Powered By Docstoc
					       A generalized law for aftershock
     behavior in a damage rheology model
         Yehuda Ben-Zion1 and Vladimir Lyakhovsky2
            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
                              aftershocks follow the GR relation:

                                     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
loading.
                                             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
         (degradation)
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
What about aftershocks?
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
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 (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



We now return to the general exponential equation (9) and examine
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
                                                        temperature gradients
                                                        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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