The seismic cycle

• The elastic rebound theory.

• The spring-slider analogy.

• Frictional instabilities.

• Static-kinetic versus rate-state friction.

• Earthquake depth distribution.

The elastic rebound theory (according to Raid, 1910)

The spring-slider analog

Frictional instabilities

The common notion is that earthquakes are frictional instabilities.

• The condition for instability is simply:

• The area between B and C is equal to that between C and D.

€

dF

du>K

Frictional instabilities are commonly observed in lab experiments and are referred to as stick-slip.

Frictional instabilities

Brace and Byerlee, 1966

From laboratory scale to crustal scale

Figure from http://www.servogrid.org/EarthPredict/

Frictional instabilities governed by static-kinetic friction

Str

ess

Slip

Time

The static-kinetic (or slip-weakening) friction:

stre

ss

slipLc

static friction

kinetic friction

experiment Constitutive law

Ohnaka (2003)

Frictional instabilities governed by rate- and state-dependent friction

€

τσ=μ =μ∗+ A ln

V

V *

⎛

⎝ ⎜

⎞

⎠ ⎟+ B ln

θV *

DC

⎛

⎝ ⎜

⎞

⎠ ⎟

and

dθ

dt=1−

θV

DC−αθ

B

dσ /dt

σ ,

were:• V and are sliding speed and contact state, respectively.• A, B and are non-dimensional empirical parameters.• Dc is a characteristic sliding distance.• The * stands for a reference value.

Dieterich-Ruina friction:

Frictional instabilities governed by rate- and state-dependent friction

Sta

t e [

s]

loading point (I.e., plate) velocity

The evolution of sliding the speed and the state throughout the cycles. An earthquake occurs when the sliding speed reaches the seismic speed - say a meter per second.

According to the spring-slider model earthquake occurrence is periodic, and thus earthquake timing

and size are predictable - is that so?

The Parkfield example

Mag

nitu

de

Year

2004

A sequence of magnitude 6 quakes have occurred in fairly regular intervals.

The next magnitude 6 quake was anticipated to take place within the time frame 1988 to 1993, but ruptured only on 2004.

So the occurrence of major quakes is non-periodic - why?

The role of stress transfer

• Faults are often segmented, having jogs and steps.

• Every earthquake perturb the stress field at the site of future earthquakes.

• So it is instructive to examine the implications of stress changes on spring-slider systems.

Animation from the USGS site

Stein et al., 1997

The effect of a stress step

The effect of a stress perturbation is to modify the timing of the failure according to:

That means that the amount of time advance (or delay) is independent of when in the cycle the stress is applied.

€

Δtime =Δstress

dstress /dtime .

The effect of a stress step

stat

e [t

]

The effect of a stress step is to increase the sliding speed, and consequently to advance the failure time.

The effect of a stress step

The ‘clock advance’ of a fault that is in an early state of the seismic cycle (I.e., far from failure) is greater than the ‘clock advance’ of a fault that is late in the cycle (I.e., close to failure).

In summary:

• The effect of positive and negative stress steps is to advance and delay the timing of the earthquake, respectively.

• While according to the static-kinetic model the time advance depends only on the magnitude of the stress step and the stressing rate, according to the rate-and-state model it depends not only on these parameters, but also on when in the cycle the stress has been perturbed.

• Thus, short-term earthquake prediction may be very difficult (if not impossible) if rate-and-state model applies to the earth.

What are the conditions for instabilities in the spring-slider system?

stre

ss

slipLc

static friction

kinetic friction

€

slope =σ N (μ static −μkinetic)

Lc

Thus, the condition for instability is:

€

σN (μ static −μkinetic)

Lc>K

The static-kinetic friction:

€

slope =σ N (b− a)

Dc

The condition for instability is:

Thus, a system is inherently unstable if b>a, and conditionally stable if b<a.

€

σN (b− a)

Dc>K

The rate- and state-dependent friction:

What are the conditions for instabilities in the spring-block system?

How b-a changes with depth ?

Scholz (1998) and references therein

• Note the smallness of b-a.

The depth dependence of b-a may explain the seismicity depth distribution

Scholz (1998) and references therein

But a spring-slider system is too simple…

• Fault networks are extremely complex.• More complex models are needed.• In terms of spring-slider system, we need to add many more springs and sliders.

Figure from Ward, 1996

System of two blocks

€

k1y1 + kc (y1 − y2) = FS1k2y2 + kc (y2 − y1) = FS2

€

m1

d2y1

t 2+ k1y1 + kc (y1 − y2) = FD1

m2

d2y2

t 2+ k2y2 + kc (y2 − y1) = FD2

During static intervals:

During dynamic intervals:

Several situations:

To simplify matters we set:• • •

We define:

€

=kck

and β =FS1FS2

.

€

=0 versus α → ∞

and

β =1 versus β ≠1 .

€

m1 = m2 = m

€

k1 = k2 = k

€

FS1 /FD1 = FS2 /FD2 = φ

System of two blocks

Turcotte, 1997

Next we show solutions for:asymmateric ( )

€

β ≠1symmateric ( )

€

β =1

Were:

€

Yi = ky i FSi

Breaking the symmetry of the system gives rise to chaotic behavior.

Summary

• Single spring-slider systems governed by either static-kinetic, or rate- and state-dependent friction give rise to periodic earthquake-like episodes.

• The effect of stress change on the system is to modify the timing of the instability. While according to the static-kinetic model the time advance depends only on the magnitude of the stress step and the stressing rate, according to the rate-and-state model it depends not only on these parameters, but also on when in the cycle the stress has been perturbed.

• Breaking the symmetry of two spring-slider system results in a chaotic behavior.

• If such a simple configuration gives rise to a chaotic behavior - what are the chances that natural fault networks are predictable???

Recommended reading

• Scholz, C., Earthquakes and friction laws, Nature, 391/1, 1998.• Scholz, C. H., The mechanics of earthquakes and faulting, New-York: Cambridge Univ. Press., 439 p., 1990.• Turcotte, D. L., Fractals and chaos in geology and geophysics, New-York: Cambridge Univ. Press., 398 p., 1997.