Bernabé, Y.

Fitzenz, D.

Hickman, S.

Earth-prints Repository

 Interpreting lab creep experiments: 
from single contacts ...
	 	 	 	 ... to interseismic fault models
Delphine Fitzenz, EOST Strasbourg
Steve Hickman, U.S.G.S. Menlo Park
Yves Bernabé, EOST, now at MIT
• Can we use lab experiments to constrain 
large-scale fault models? what are the 
limitations?
• What is the best way to infer microphysical 
processes from macroscopic measurements?
• How can we account for observational errors, 
model uncertainties, and potential 
deformation regime changes during each 
experiment? 
Motivation
Our Bayesian inference scheme
∂ϕ
∂t
= θ0 × σθ1eff × exp(−θ2/(RT ))× exp(θ3ϕ)
When the grain-size effects can be neglected, 
and we make no assumption on the mechanisms, 
we can choose a creep law of the type:
Choice of the model m
Observations
Experimental or simulated porosity time series
Apparent 
activation 
energy 
Stress 
exponent
*
The five steps of the inference
A: Choose the parametrization making the problem 
as linear as possible   
B: When a factorization appears in the new Bayesian network, 
adopt a hierarchical inference scheme
D: Eliminate the nuisance parameters (marginalization step)
E: Revert from the new parametrization to the original 
parameters !i  
C: Calculate the joint probability density function according to the 
graph structure (the joint pdf is proportional to the posterior pdf).
*Prerequisite
- Integration of the creep law: we want to use the data, 
not a subproduct of it
- Reparametrization such that the problem is as linear 
as possible 
(work with Gaussians) Fitzenz & al., MaxEnt 05
Application to real 
experimental creep data
10
20
2000000
Niemeijer et al., EPSL 2002
50-150 MPa, 400-600oC
Po
ro
sit
y,
 %
Time, s
Fitzenz & al., JGR 07
Global data set: 88 points
!j={6.40 10-11; 2.55; 37 kJ/mol; 0.53}, 
with std. dev. of 3.0 10-11, 6.2 10- , 1.6 
103, and 9.4 10-3 resp.
Subset "/"0 ∈ [0.5,0.8] : 53 points
!j={4.2 10-13; 3.56; 57.6 kJ/mol; 0.71}, 
with std. dev. of 2.3 10-13, 0.11, 2.6 kJ/mol, 
and 0.016 resp.
Interpretation and questioning
Most likely mechanisms : 
compaction rate-controlled by dissolution 
+ cataclasis or stress corrosion
Stress exponent 3.5
Apparent activation energy 60 kJ/mol  
However: 
The relation between contact stress and applied stress 
may be affected by changes in grain packing
We need to perform a test on a more controlled 
system: could we identify pure pressure solution?
Validation procedure: 
simulated pressure solution
Single contact model by Bernabe and Evans 2007 
based on the evolution of the contact shape
	 	 	 	 	 	 		 	 - Stress exponent 1
	 	 - Apparent activation energy: 
	 	 	 between 72 kJ/mol  (contact dissolution) 
	 	 	 and 15 kJ/mol (interface diffusion)
	 	 throughout the simulated experiments
Data transformed into porosity time series by assuming 
a simple cubic packing geometry + Gaussian noise
Challenge:
Analyze data whose behavior is driven by the contact area 
without knowing how contact area evolves through time
Identifying transitions
x = realization of a random variable
hypothesis testing: x is normally distributed, 
variance σ2, mean μ
|x-μ|>2σ: less than 5% likely to have 
occurred by chance
For each experiment, we estimate the maximum number 
of points n belonging to the same deformation regime 
using a χ2 test
!
, 
%
t
1  
... t
i-1
t
i
pdf of !obs
with !(t)=m(T,"eff,#,!0,t)
Σn (Φ-Φobs)2
noise variancex=
10 simulated experiments
Exp # σn (MPa) T (K)
0 5.482 508
1 2.564 508
2 0.495 508
3 0.080 508
4 1.591 10-2 508
5 0.495 574
6 0.495 542
7 0.495 508
8 0.495 472
9 0.495 435
201 points/experiment, 
Gaussian noise with std dev 0.005
After Bernabe & Evans 
2007
Identifying transitions
Po
ro
sit
y,
 %
Time, s
Identifying transitions
Po
ro
sit
y,
 %
Time, s
The 2 deformation regimes 
(at 3σ)
Regime 1: 	 Stress exponent 0.97 +/- 4.4 e-3
	 	 	 	 	 Apparent act. energy 56.9 kJ/mol 
	 	 	 	 	 +/- 0.3 kJ/mol
Regime 2: 	 Stress exponent 1.2 +/- 5.4 e-3
	 	 	 	 	 Apparent act. energy 32.8 kJ/mol 
	 	 	 	 	 +/- 0.1 kJ/mol
- Stress exponents close to 1 as assumed in Bernabe & Evans
- Both activation energies between 72 kJ/mol  (contact dissolution) 
  and 15 kJ/mol (interface diffusion);
- For the early stage, Bernabe & Evans obtained 58 kJ/mol with a   
 different method (knowing the contact area as a function of time);
- The late times are better approached with our method.
Validation procedure: 
simulated pressure solution
Conclusion:
- With stress exponents close to 1 and a decreasing apparent 
activation energy from above 57 kJ/mol down to below 33 
kJ/mol, we can infer that the deformation is pressure 
solution, mostly controlled by dissolution at the beginning, 
and with a larger contribution of interface diffusion as 
contacts grow.
- The times (and porosities) of the transition are T- and σn-
dependent.
Time to failure using the creep law 
inferred using the Niemeijer et al. data
! at t
Time to failure;
!, Pf, !, "
n
 
at failure
Coulomb 
Failure 
model
{#m}
Fluid 
pressurization 
model: 
dPf/dt;
 d!/dt;
d"
n
/dt
! at t
obs
{#m}
obs
Shear stressing 
model: 
 
d!/dt
Pf at 
t
Pf at 
t obs
! at 
t
!at t 
obs
"
n
 at 
t
"
n
 at 
t obs
T at tT at t
obs
Friction 
coe$cient 
Creep parameters
Time to failure;
!, Pf, !, "
n
 
at failure
obs
Fault properties
Model of the interseismic period
fixed random observed
Monte Carlo 
sampling
through
Results in terms of
time to failure
*
*
Time to failure distributions
• undrained
• 1 fault element
• t=0, 12% porosity
• z=3km
• {!j} from previous 
inversion
• (static) friction=0.6
• d!/dt=2.5 bar/year
• d"n/dt=0
• Coulomb failure:
#!friction*("n-Pf)
 0
1
2
3
4
5
6
log(time to failure)
18.017.0 20.019.0 21.0
0
0.01
0.06
0.02
0.03
no
rm
ali
ze
d 
nu
m
be
r o
f o
cc
ur
re
nc
e
0.04
0.05
Fixed initial porosity of 12%
Gaussian initial porosity
Gaussian pdfs for reference
 
a
Uncertainties 
on the compaction 
law
+ Uncertainties on porosity 
after an earthquake
+
Conclusions
1. We developed a Bayesian inference scheme that we 
validated using simulated pressure solution experiments;
2. The analysis of simulated experiments can also guide 
the design of future experiments;
3. We applied this method to real compaction data 
obtained at hydrothermal conditions;
4. We showed how to propagate the uncertainties to time 
to failure distribution;
and future directions
This type of Bayesian framework seems promising for 
future efforts to compute earthquake probability models 
including the “known” fundamental physics, different 
types of data, their observational errors, and the model 
uncertainties when they exist. 


Interpreting lab creep experiments: from single contacts to interseismic fault models

http://www.earth-prints.org/bitstream/2122/2632/1/Fitzenz_EuroRock07.pdf

Interpreting lab creep experiments: from single contacts to interseismic fault models

Abstract

Similar works

Full text

Available Versions

Earth-prints Repository