Stress- and State-Dependence of Earthquake Occurrence: Tutorial 1 Jim Dieterich University of...

Stress- and State-Dependence of Earthquake Occurrence: Tutorial 1

Jim Dieterich

University of California, Riverside

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

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

Time-dependent strengthening

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


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


Woodroughened surface, #40

1 MPa normal stress100 µm

.60

.70

µ

µ

µ

µ

µ

µ

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


fast slow fast slowConstitutive law

# 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

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)

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

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

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

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

Spring-slider simulation withrate- and state-dependent friction (blue curves)

Westerly granite, =30 MPa

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

Contact stresses

Indentation yield stress, y

Acrylic 400 MPaCalcite 1,800 MPaSL Glass 5,500 MPa

Quartz 12,000 MPa

Increase of contact area with time

Elapsed Time

1 s

100 s

10,000 s

50 µm

Acrylic plastic Dieterich & Kilgore, 1994, PAGEOPH

Time dependent friction & Contact area

Dieterich and Kilgore, PAGEOPH, 1994

Velocity step & Contact area

Dieterich and Kilgore, PAGEOPH, 1994

€

ss =Dc

V

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()

Contact evolution with displacement

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

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

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

Confined Unstable Slip

Confined stick-slip in biaxial apparatus satisfies the

relation for minimum dimension for unstable slip

Earthquake nucleation on uniform fault

Earthquake nucleation on a uniform fault

Dieterich, 1992, Tectonophysics

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 )


c

0 300

4

3

2

1

0

-1

-2

-3

-4

-5

-6

-7

δ i



c

0 300

4

3

2

1

0

-1

-2

-3

-4

-5

-6

-7

δ i



c

0 300

4

3

2

1

0

-1

-2

-3

-4

-5

-6

-7

δ i



c

0 300

4

3

2

1

0

-1

-2

-3

-4

-5

-6

-7

δ i



c

0 300

4

3

2

1

0

-1

-2

-3

-4

-5

-6

-7

δ i



c

0 300

4

3

2

1

0

-1

-2

-3

-4

-5

-6

-7

δ i



c

0 300

4

3

2

1

0

-1

-2

-3

-4

-5

-6

-7

δ i



c

0 300

4

3

2

1

0

-1

-2

-3

-4

-5

-6

-7

δ i



c

0 300

4

3

2

1

0

-1

-2

-3

-4

-5

-6

-7

δ i



c

0 300

4

3

2

1

0

-1

-2

-3

-4

-5

-6

-7

δ i



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

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

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

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)

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

⎡

⎣ ⎢

⎤

⎦ ⎥

Accelerating slip prior to instability

Time to instability - Experiment and theory

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

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

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)

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

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

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

Stress- and State-Dependence of Earthquake Occurrence: Tutorial 1 Jim Dieterich University of...

Documents

Transcript of Stress- and State-Dependence of Earthquake Occurrence: Tutorial 1 Jim Dieterich University of...