What can we learn about dynamic triggering in the the lab?

Lockner and Beeler, 1999

Questions…

• Is the relationship between periodic stress loading and timing of failure threshold or time dependent?

(i.e. Coulomb or rate & state)

• What amplitudes and frequencies cause correlated failure?

• How can these results be extended to dynamic triggering in the real world?

(seismic waves, earth tides, seasonal stress variations)

Experimental design

•30 highly polished cut in a 190 x 76 mm core of Westerly granite

• Teflon “shim” minimizes resistance to lateral movement

•Polyurethane jacket undergoes elastic recovery when pressure removed, sample “jumps back”

Experimental procedure

• 3 background rates (Vr): 0.001 0.01 and 0.1 m/s

• Cosine periodic forcing: 2.8 hrs to 10 sec

• Amplitude ranges: 1.1 – 11 m (0.07- 0.7 MPa Shear or 0.04- 0.4 MPa Normal)

• is constant at 0.68

• Confining pressure constant at 50 MPa ???

Experimental procedure

• 1.5 to 2 mm of pre-shortening to reach steady state

• 14 mm of axial shortening, yields ~ 50 events (peak stress constant… stress drop increasing??)

• Fault gouge continually removed, repolished

Long period forcing…Vr=0.01 m/s ,

period ~ 38 min m~ 0.12 MPa

Short period forcing…Vr=0.001 m/s , period ~ 50 secm~ 0.07 MPa

Lockner and Beeler, 1999

• Determine the phase of each failure (), and determine the offset from the peak stress (phase shift, )

• Create a PDF use “cosine weighting functions” and normalizing by average seismicity rate (area = 1)

Analyzing results …um = UmCos(t)

Lockner and Beeler, 1999

•PDF’s are fit to:

P()=P0 + PmCos(-)

Where….

P0= probability w/out forcing (1/2)

Pm= amplitude

phase shift

(why no frequency term??)

• Define a metric:

When Pm/P0 > 1 , the correlated EQ signal is above the random noise

Analyzing results …um = UmCos(t)

Lockner and Beeler, 1999

Assessing “correlation” between phase of failure and forcing period

“random walk” statistics

Coulomb failure model predicts:

• R =R0(Ŝ/S0) (Seismicity rate is proportional to stressing rate)

• (Rmax-Rmin)/R0 = (Ŝmax-Ŝmin)/Ŝ0 = (2m Ŝ0

where… Ŝ is seismicity rate, m is max shear stress amplitude, is frequency

• Thus, if (Rmax-Rmin)/R0 = 1 is threshold for correlation of seismicity, then…

• m = (1/Ŝ0/2) is a line predicting correlation…

m =

(1/Ŝ0/2

We can plot the line representing predicted Coulomb failure

• Notice that required m for correlation decreases at high frequency () m

Beeler and Lockner, 2003

• Periodic displacement: um = UmCos(t) (differentiate to get periodic stressing rate)

Where… Um is amplitude of periodic displacement (in m)

• Full stressing rate (Vt): Vt = Vr- UmSin(t)

Where… Vr is the backround rate, and is frequency

•Um/Vr < 1 -stressing rate is always positive (compressive)

•Um/Vr > 1 - stressing rate becomes negative for part of the cycle

When in the cycle does Coulomb model predict failure?

Lockner and Beeler, 1999

When …. Um/Vr > 1

Peak seismicity migrates towards the peak stress because the stress “shadow”Is becoming larger and larger

When… Um/Vr < 1

Peak seismicity rate corresponds to peak stressing rate at = -90

When in the cycle does Coulomb model predict failure?

Beeler and Lockner, 2003

m=0.69

m=0.35

m=0.17

Observed in Experiments Coulomb Model

Coulomb model doesn’t explain experimental observations at short period ….

Both examples ~~ Vr = 0.01 m/s , period = 25 sec , = .68 , n =86 MPa

Lockner and Beeler, 1999

• Dieterich (1987) did numerical simulations showing that if faults obey the rate and state model, then seismicity rate is…

What does the rate and state model predict?

Ra = (Rmax-Rmin)/R0 = 2m/A1n = Pm/P0

1 yr cycle

Dieterich (1987)

Thus, Ra is Dependent on: Shear stress amplitude (m), and the constituitive parameter A1 such that…

Ra is Independent of: frequency (), characteristic length (Dc), stiffness (K), and parameter B

When Ra = 1… m = A1n / 2

What does the rate and state model predict?

Ra = 2m/A1n = Pm/P0

We expect different responses at differentpoints in frequency-amplitude space

100% probabilityof correlation usingCoulomb model

R&S model(independentof period)

Coulomb :(dependenton period)

Model predictions for Vl= 0.1 m/s m

Beeler and Lockner, 2003

• Dependence on both frequency () and amplitude (m)

This study observes…

Lockner and Beeler, 1999

The response mode is determined by relationship of nucleation time (tn) to period (tw)

Coulomb (if tn < tw)

Rate & State(if tn > tw)Nucleation

Time (tn) m

Beeler and Lockner, 2003

Positive slope at high frequencyis not predicted by rate & state model…

PositiveSlope!!

Perhaps a lower value of A1 is necessary to explain this?

• Dieterich (1987) R&S theory predicts:

Ra/2 = m/A1n = Pm/P0

Ra is seismicity rate, m is shear stress, A1 is the R&S constituitive parameter

• Plugging in experimental observations to back-calculate A1 gives:

A1 = 0.001 -0.004 (measured)

A1= 0.003 – 0.009 (known)

But… R&S model can correctly back Calculate the parameter A1….

m=0.69

m=0.35

m=0.17

Observed in Experiments Coulomb Model Rate & State Model

And…R&S model correctly predicts onset ofCorrelation as function of and m

All three examples ~~ Vr = 0.01 m/s , period = 25 sec , = .68 , n =86 MPa(note different phase predictions for Coulomb vs. R&S)

Lab results suggest we should see dynamic triggering of EQ’s by seismic waves

• Three studies have observed a threshold stress amplitude for dynamic triggering:

Anderson et al., 1994 - 0.6 - 0.9 MPa ( 350 > n < 50) Hill et al., 1993 - 0.1 - 0.2 MPa ( 350 > n < 50) Gomberg and Davis, 1996 - 0.003 - 0.012 MPa ( 50 > n < 0)

This Study (experimental) - 0.4 – 1 MPa (n ~ 50 MPa)

• But… do experimental results extrapolate to higher confining pressures (higher n at depth)

• Do results under estimate the stress amplitude necessary to induce EQ correlation… assuming is constant?

Implications for dynamic triggering by earth tides

• Because earth tides have a longer period than nucleation time, they fall in the Coulomb regime, and stresses are not high enough

• This study: Lowest amplitude is ~ 0.05 MPa Longest period is ~ 2.8 hours

• Earth tides: Amplitudes ~ 0.001 – 0.004 MPa Period ~ 12.5 hours

• Assuming linear relationship between seismicity correlation and stress amplitude, > 20,000 EQ’s are required to see a correlation between earth tides and seismicity

• What assumption are required to extrapolate to frequencies and amplitudes of earth tides?(i.e. constant A1 constant )

Key results…• Premonitory slip is observed in lab, suggesting failure is a time

dependent (R&S) process

• Two distinct linear relationships between m and required for correlation, dependent on nucleation time (tn)

• Coulomb or R&S models work for long period forcing, only R&Sworks for short period

• Because period of earth tides is much shorter than nucleation times, R&S model is appropriate, tides don’t exert enough shear stress amplitude to cause failure in R&S model

• Seismic waves may impart enough shear stress, earth tides do not, except in rare cases (~1% caused by earth tides…)

Some questions…

• Is the confining pressure (50 MPa) in this experiment reasonable… and does it allow extrapolation of this data set to real EQ’s ?

• Do the high frequencies used in this study allow extrapolation to triggering by earth tides?

• Does the lack of fault gouge accumulation cause these results to underestimate the necessary stress amplitudes for EQ intitiation?