stress- and state-dependence of earthquake occurrence: tutorial 1 jim dieterich university of...
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Stress- and State-Dependence of Earthquake Occurrence: Tutorial 1
Jim Dieterich
University of California, Riverside
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Approach
The formulation is based on the premise that earthquake nucleation controls
the time and place of initiation of earthquakes. Hence, processes that alter
earthquake nucleation times control changes of seismicity rates.
For faults with rate- and state-dependent friction, the relationship between
nucleation times and stress changes is highly non-linear.
Tutorial 1 reviews some features of rate- and state-dependent friction and
earthquake nucleation that form the basis of the model.
Tutorial 2 reviews the derivation of the constitutive formulation and outlines
some applications of the model.
Constitutive formulation for earthquake rates
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Experimental Conditions – Rate & State
• Wide range of rocks and rock forming minerals– Bare surfaces and gouge layers
• Also glass, wood, paper, plastic, gelatin, metals, ceramics, Silicon in MEMs devices
• Contact times <1s - 106s (indirect ~4x107s) • V= mm/yr - cm/s (servo-controlled tests)• V≥100m/s (shock impact)• T=20°C - 350°C• Nominal =1 MPa - 300 Mpa, Contact stresses to
12GPa• Dry, wet, hydrothermal
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Time-dependent strengthening
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Response to steps in slip speed
100 µm.70
.75
Granite #60 surface
15 MPa normal stress
100 µm.65
.70
Granite#60 surface, 1mm gouge
10 MPa normal stress
100 µm.62
.67 Soda-lime glass#60 surface
5 MPa normal stress
Acrylic plastic #60 surface
2.5 MPa normal stress100 µm.70
.85
100 µm.020
.025
Teflon on steelpolished surface
30 MPa normal stress
Woodroughened surface, #40
1 MPa normal stress100 µm
.60
.70
µ
µ
µ
µ
µ
µ
10µm/s 1µm/s 10µm/s 1µm/s
10µm/s 1µm/s 10µm/s 1µm/s
1µm/s 0.1µm/s 1µm/s 0.1µm/s
1µm/s 0.1µm/s 1µm/s 0.1µm/s
2µm/s20µm/s 2µm/s .2µm/s
10µm/s 1µm/s 10µm/s 1µm/s
fast slow fast slowConstitutive law
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# 240 surface
# 30 surface
Fault slip, m
Thickness = 0.42 m
2.0 m
Hydraulic flatjacks & Teflon bearings
1.5 m
Displacement-weakening at onset of rapid slip
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Rate- and state-dependent formulation
Coefficient of friction:
State variable:
For example:
τσ
=μ =μ0 +AlnVV*
⎛ ⎝
⎞ ⎠ +Bln
θθ*
⎛ ⎝
⎞ ⎠
dθ =dt−θDc
dδ −αθBσ
dσ
At steady state, d/dt=0 and
θss =Dc
Vμss =const. +(A−B) lnV
θ =G(t,δ,σ,Dc),
Dc
Displacement, δ
Slip speed
,Friction µ
V1 V2 V1
A (ln V1/V2) B (ln V1/V2)
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Log
Coe
ffici
ent o
f fric
tion
€
=0 + A lnV
V *
⎛
⎝ ⎜
⎞
⎠ ⎟+ Bln
θ
θ *
⎛
⎝ ⎜
⎞
⎠ ⎟
€
ss = const.+ (B− A) lnθss
Constant V (high)
Constant V (lo
w)
ssB
B-A
x V1
€
ss =Dc
V
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Log
Coe
ffici
ent o
f fric
tion
€
=0 + A lnV
V *
⎛
⎝ ⎜
⎞
⎠ ⎟+ Bln
θ
θ *
⎛
⎝ ⎜
⎞
⎠ ⎟
€
ss = const.+ (B− A) lnθss
Constant V (high)
Constant V (lo
w)
ssB
B-A
€
ss at V1
x
During slip evolves toward ss
V1
€
ss =Dc
V
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Log
Coe
ffici
ent o
f fric
tion
€
=0 + A lnV
V *
⎛
⎝ ⎜
⎞
⎠ ⎟+ Bln
θ
θ *
⎛
⎝ ⎜
⎞
⎠ ⎟
€
ss = const.+ (B− A) lnθss
Constant V (high)
Constant V (lo
w)
ssB
B-A
€
ss at V1
x
During slip evolves toward ss
V1
€
ss =Dc
V
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Log
Coe
ffici
ent o
f fric
tion
€
=0 + A lnV
V *
⎛
⎝ ⎜
⎞
⎠ ⎟+ Bln
θ
θ *
⎛
⎝ ⎜
⎞
⎠ ⎟
€
ss = const.+ (B− A) lnθss
Constant V (high)
Constant V (lo
w)
ssB
B-A
€
ss at V1
x
During slip evolves toward ss
V1
€
ss =Dc
V
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Log
Coe
ffici
ent o
f fric
tion
constant V (high)
constant V (lo
w)
ss
Time dependent strengthening
Slip
a
b
c
d
ab
c
d
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Log
Coe
ffici
ent o
f fric
tion
constant V (high)
constant V (lo
w)
ss
Velocity steps
Slip
a
b
c
d
a
b
c
d
a
V1 V1V2
V1
V2
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Spring-slider simulation withrate- and state-dependent friction (blue curves)
Westerly granite, =30 MPa
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Imaging contacts during slip
Schematic magnified view of contacting surfaces showing isolated high-stress contacts. Viewed in transmitted light, contacts appear as bright spots against a dark background.
Acrylic surfaces at 4MPa applied normal stress
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Contact stresses
Indentation yield stress, y
Acrylic 400 MPaCalcite 1,800 MPaSL Glass 5,500 MPa
Quartz 12,000 MPa
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Increase of contact area with time
Elapsed Time
1 s
100 s
10,000 s
50 µm
Acrylic plastic Dieterich & Kilgore, 1994, PAGEOPH
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Time dependent friction & Contact area
Dieterich and Kilgore, PAGEOPH, 1994
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Velocity step & Contact area
Dieterich and Kilgore, PAGEOPH, 1994
€
ss =Dc
V
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Interpretation of friction terms
Time and rate dependence of contact strength terms
Indentation creep: c() = c1 + c2ln()
Shear of contacts: g(V) = g1 + g2ln(V)
= c1 g1 + c1g2ln(V) + c2g1ln() + c2g2ln(V+)
c=1/ indentation yield stressg=shear strength of contacts
Bowden and Tabor adhesion theory of friction
Contact area: area = c
Shear resistance: = (area) (g), / = =cg
(Drop the high-order term) = 0 + A ln(V) + B ln()
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Contact evolution with displacement
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SUMMARY – RATE AND STATE FRICTION
• Rate and state dependence is characteristic of diverse materials under a very wide range of conditions
• Contact stresses = micro-indentation yield strength (500 MPa – 12,000 MPa)
• State dependence represents growth of contact area caused by indentation creep
• Other process appear to operate at low contact stresses
• Log dependence thermally activated processes.
• Power law distribution of contact areas
• Dc correlates with contact diameter and arises from displacement-dependent replacement of contacts
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Critical stiffness and critical patch length for unstable slip
K
€
ξ =B− APerturbation from steady-state sliding, at constant [Rice and Ruina, 1983]
€
Kc =ξσ
Dc
€
ξ ≈0.4BApparent stiffness of spontaneous nucleation patch (2D) [Dieterich, 1992]
€
lc =Gη
Kc
=GηDc
ξσEffective stiffness of slip patch in an elastic medium
€
K ≡Δτ
d=
Gη
l
crack geometry factor, ~ 1G shear modulus
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Large-scale biaxial test
Thickness = 0.42 m
2.0 m
Hydraulic flatjacks & Teflon bearings
1.5 m
Strain gage
Displacement transducer
Lc = G Dcξ
L c =.75m
G=15000MPa=0.5Dc=2µm=5MPaξ=.4B=0.004
Minimum fault length for unstable slip
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Confined Unstable Slip
Confined stick-slip in biaxial apparatus satisfies the
relation for minimum dimension for unstable slip
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Earthquake nucleation on uniform fault
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Earthquake nucleation on a uniform fault
Dieterich, 1992, Tectonophysics
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Earthquake nucleation – heterogeneous normal stress
Dieterich, 1992, Tectonophysics cPosition ( 1 division = 200,000 D )
Log ( Slip speed / D )
c
0 300
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
δ i
cPosition ( 1 division = 200,000 D )
Log ( Slip speed / D )
c
0 300
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
δ i
cPosition ( 1 division = 200,000 D )
Log ( Slip speed / D )
c
0 300
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
δ i
cPosition ( 1 division = 200,000 D )
Log ( Slip speed / D )
c
0 300
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
δ i
cPosition ( 1 division = 200,000 D )
Log ( Slip speed / D )
c
0 300
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
δ i
cPosition ( 1 division = 200,000 D )
Log ( Slip speed / D )
c
0 300
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
δ i
cPosition ( 1 division = 200,000 D )
Log ( Slip speed / D )
c
0 300
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
δ i
cPosition ( 1 division = 200,000 D )
Log ( Slip speed / D )
c
0 300
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
δ i
cPosition ( 1 division = 200,000 D )
Log ( Slip speed / D )
c
0 300
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
δ i
cPosition ( 1 division = 200,000 D )
Log ( Slip speed / D )
c
0 300
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
δ i
cPosition ( 1 division = 200,000 D )
Log ( Slip speed / D )
c
0 300
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
δ i
cPosition ( 1 division = 200,000 D )
Log ( Slip speed / D )
c
0 300
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
δ i
2Lc
Lc =G η Dc
ξ σ
Lc
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SPRING-SLIDER MODEL FOR NUCLEATION
€
(t) − Kδ
σ= μ 0 + A ln ˙ δ + Blnθ
K
(t)
δ
During nucleation, slip speed accelerates and greatly exceeds steady state slip speed
€
Dc
>>1˙ δ
⇒ dθ
dδ≈ −
θ
Dc
, θ =θ0e−δ / Dc
€
dθ
dδ=
1˙ δ
−θ
Dc
Evolution at constant normal stress
€
(t) − Kδ
σ= μ 0 + A ln ˙ δ + Blnθ0 −δ
B
Dc
Log
Coe
ffic
ien
t of
fric
tion
ssB
B-A
€
ss at V1
xV
1
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SPRING SLIDER MODEL FOR NUCLEATION
€
(t) − Kδ
σ= μ 0 + A ln ˙ δ + Blnθ0 −δ
B
Dc
€
exp˙ τ t
Aσ
⎛
⎝ ⎜
⎞
⎠ ⎟
0
t
∫ dt = expHδ
A
⎛
⎝ ⎜
⎞
⎠ ⎟
0
δ
∫ dδ€
˙ δ Re-arrange by solving for
Where:
€
˙ δ 0 =θ0−B / A exp
τ 0 /σ( )−μ 0
A
⎡
⎣ ⎢
⎤
⎦ ⎥
€
H = −K
σ+
B
Dc
Model parametersInitial condition
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SPRING SLIDER MODEL FOR NUCLEATION
€
δ =−A
Hln
˙ δ 0Hσ˙ τ
1−exp− ˙ τ t
Aσ
⎛
⎝ ⎜
⎞
⎠ ⎟
⎡
⎣ ⎢
⎤
⎦ ⎥+1
⎧ ⎨ ⎩
⎫ ⎬ ⎭ , ˙ τ ≠ 0
δ =−A
Hln 1−
˙ δ 0Ht
A
⎧ ⎨ ⎩
⎫ ⎬ ⎭ , ˙ τ = 0
€
˙ δ =1˙ δ 0
+Hσ
˙ τ
⎡
⎣ ⎢
⎤
⎦ ⎥exp
− ˙ τ t
Aσ
⎛
⎝ ⎜
⎞
⎠ ⎟
⎡
⎣ ⎢
⎤
⎦ ⎥−
Hσ˙ τ
⎧ ⎨ ⎩
⎫ ⎬ ⎭
-1
, ˙ τ ≠ 0
˙ δ =1˙ δ 0
+Ht
A
⎡
⎣ ⎢
⎤
⎦ ⎥
-1
, ˙ τ = 0
€
ti =Aσ
˙ τ ln
˙ τ
Hσ ˙ δ 0+1
⎡
⎣ ⎢
⎤
⎦ ⎥ , ˙ τ ≠ 0
ti =A
H
1˙ δ 0
⎛
⎝ ⎜
⎞
⎠ ⎟ , ˙ τ = 0
Slip
Slip speed
Time to instability
€
( 1/ ˙ δ → 0 )
Dieterich, Tectonophysics (1992)
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Solutions for time to instability
101010910810710610510410310210110010-110-210-310-410-5
Time to instability (s)
max
s
/=0
Dieterich, 1992, Tectonophysics
Slip speed (DC /s)
103
102
101
100
10-1
10-2
10-3
10-4
10-5
10-6
10-7
10-8
2D numerical model
Fault patchsolution
€
˙ τ /σ =10−2 / a
€
˙ τ /σ =10−3 / a
€
˙ τ /σ =10−4 / a
€
ti =Aσ
˙ τ ln
˙ τ
Hσ ˙ δ 0+1
⎡
⎣ ⎢
⎤
⎦ ⎥
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Accelerating slip prior to instability
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Time to instability - Experiment and theory
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1011101010910810710610510410310210110010-
1
10-
2
6
4
2
0
-2
-4
-6
-8
-10
-12
-14
-16
1 yr10 yr20 yr
Time to instability (seconds)
Log
(sl
ip s
pee
d)
m
/s
Effect of stress change on nucleation time
€
ti =Aσ
˙ τ ln
˙ τ
Hσ ˙ δ 0+1
⎡
⎣ ⎢
⎤
⎦ ⎥
€
˙ τ = 0.05 MPa/yr
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1011101010910810710610510410310210110010-
1
10-
2
6
4
2
0
-2
-4
-6
-8
-10
-12
-14
-16
5min
1 yr10 yr20 yr
~1hr~5hr
Time to instability (seconds)
Log
(sl
ip s
pee
d)
m
/s
Effect of stress change on nucleation time
€
ti =Aσ
˙ τ ln
˙ τ
Hσ ˙ δ 0+1
⎡
⎣ ⎢
⎤
⎦ ⎥
€
˙ τ = 0.05 MPa/yr
€
˙ δ = ˙ δ 0σ
σ 0
⎛
⎝ ⎜
⎞
⎠ ⎟
α / A
exp τ
Aσ−
τ 0
Aσ 0
⎛
⎝ ⎜
⎞
⎠ ⎟
= 0.5 MPa
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Use the solution for time to nucleation an earthquake
(1) , where
and assume steady-state seismicity rate r at the stressing rate
This defines the distribution of initial conditions
(slip speeds) for the nucleation sources
(2)
The distribution of slip speeds (2) can be updated at successive time steps
for any stressing history, using solutions for change of slip speed as a
function of time and stress.
€
t =Aσ
˙ τ ln
˙ τ
Hσ ˙ δ 0+1
⎡
⎣ ⎢
⎤
⎦ ⎥
˙ τ =const≠0˙ σ =0
H =B
DC
−Kσ
€
˙ τ r
t = n r , n is the sequence number of the earthquake source
€
˙ δ 0(n) =1
Hσ˙ τ r
exp˙ τ r n
Aσ r
⎛
⎝ ⎜
⎞
⎠ ⎟−1
⎡
⎣ ⎢
⎤
⎦ ⎥
Model for earthquake occurrence
Log (time to instability)
Lo
g (
slip
sp
ee
d)
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For example changes of with time are given by the nucleation solutions
and change of with stress are given directly from the rate- and state-
formulation
In all cases, the final distribution has the form of the original distribution
where
€
˙ δ 0(n) =1
Hσ
γexp
˙ τ r n
Aσ r
⎛
⎝ ⎜
⎞
⎠ ⎟−1
⎡
⎣ ⎢
⎤
⎦ ⎥
Evolution of distribution of slip speeds
€
˙ δ 0(n)
€
˙ δ 0(n)
€
˙ δ = ˙ δ 0σ
σ 0
⎛
⎝ ⎜
⎞
⎠ ⎟
α / A
expτ
Aσ−
τ 0
Aσ 0
⎛
⎝ ⎜
⎞
⎠ ⎟
€
dγ =1
Aσdt − γdτ + γ
τ
σ−α
⎛
⎝ ⎜
⎞
⎠ ⎟dσ
⎡
⎣ ⎢
⎤
⎦ ⎥
€
˙ δ =1˙ δ 0
+Ht
A
⎡
⎣ ⎢
⎤
⎦ ⎥
-1
, ˙ τ = 0
€
˙ δ =1˙ δ 0
+Hσ
˙ τ
⎡
⎣ ⎢
⎤
⎦ ⎥exp
− ˙ τ t
Aσ
⎛
⎝ ⎜
⎞
⎠ ⎟
⎡
⎣ ⎢
⎤
⎦ ⎥−
Hσ˙ τ
⎧ ⎨ ⎩
⎫ ⎬ ⎭
-1
, ˙ τ ≠ 0
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Earthquake rate is found by taking the derivative dn/dt = R
For any stressing history
€
dγ =1
Aσdt − γdτ + γ
τ
σ−α
⎛
⎝ ⎜
⎞
⎠ ⎟dσ
⎡
⎣ ⎢
⎤
⎦ ⎥
€
R =r
γ ˙ τ r
Evolution of distribution of slip speeds
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Coulomb stress formulation for earthquake rates
Earthquake rate ,
Coulomb stress
Assume small stress changes (treat as constants) ,
Note: . Hence,
Earthquake rate ,
R =r
γ ˙ τ rdγ =
1Aσ
dt−γdτ +τσ
−α⎛ ⎝
⎞ ⎠ dσ
⎡ ⎣ ⎢
⎤ ⎦ ⎥
dS=dτ −μdσ
€
−α⎛
⎝⎜
⎞
⎠⎟ (Aσ )
R =r
γ ˙ S rdγ =
1Aσ
dt−γdS[ ]
Dieterich, Cayol, Okubo, Nature, (2000), Dieterich and others, US Geological Survey Professional Paper - 1676 (2003)
€
eff =τ
σ−α
⎛
⎝ ⎜
⎞
⎠ ⎟≈ 0.3−0.4
€
= and 0 ≤ α ≤ μ