of 32
1
Geophysical Research Letters
Supporting Information for
Constraining Fault Friction and Stability with Fluid
-
Injection Field Experiments
S
tacy
Larochelle
1
, N
adia
Lapusta
1,2
,
Jean
-
P
aul
Ampuero
3
, and F
rédéric
Cappa
3,4
1
Division of Geological and Planetary
Sciences, California Institute of Technology, Pasadena,
California 91125, USA.
2
Division of Engineering and Applied Science, California Institute of Technology, Pasadena,
California 91125, USA
3
Université Côte d’Azur, IRD, CNRS, Observatoire de la Côte
d’Azur, Géoazur, 06560 Sophia
Antipolis, France
4
Institut Universitaire de France, Paris, France
Contents of this file
Text S1 to S
2
Table S1
Figures S1 to S
29
Introduction
T
he following supporting information
describes long
-
term simulations ju
stifying our
choice of initial conditions as well as further discussion on
the
lengthsca
le
"#
derived in
the main text.
We also provide additional figures
that illustrate the evolution of all
variables in the simulation and how certain combinations of parameters control the
simulation results.
2
Text S1.
Long
-
term simulations
without tectonic or fluid pressure loading
In the models presented in this study,
w
e prescribe initial conditions that are
consistent with a dormant fault by starting with a highly healed fault (i.e., high initial
value
of the state variable
%&%
). This choice of initial conditions is justified by the long
-
term simulations without tectonic or fluid pressure loading shown in Figures
S4
-
S
7
.
T
he
initial values
affect some initial behavior/slip of the fault but
, long
-
term, the fault
heals
under the near
-
constant values of shear stress, with
a power
-
law decrease in slip rate as
well as an increase in state variable over time; at long times, the value of the state
variable is
approximately equal to the healing time of the fault
. This be
havior can be
predicted analytically: When the fault is well below steady
-
state (
V
/
()
1
),
̇
~
1
and thus
~
. Moreover, with shear stress being almost constant, the rate
-
and
-
state
friction coefficient is fixed and
̇
=
̇
/
+
=
0
, implying that
9
:
"
.
The initial
conditions in the
intermediate
-
and hi
gh
-
friction
cases in this study are consistent with
this behavior. In the
low
-
friction
case, although we do prescribe a high initial state
variable and a low initial slip rate, the fault
needs to be
initially above steady state to
match the measured slip b
ehavior at the injection size and therefore not consistent with
the behavior described above.
Text S
2
.
풂풄
:
Estimate of slipping zone length
at
slip acceleration
I
n the main text, w
e derived an estimate of the slipping zone length at
the time
of slip acceleration (beginning of Stage 3).
I
f
(
)
remained cons
tant throughout the
simulation,
Eq.
(
11
)
would reduce to
"#
()
/
which is similar to the condition for
acceleration
<
:
(where
is stiffness) in the spring
-
block slider model
(Dieterich,
1992; Helmstetter & Shaw, 2009)
and to the condition
>
:
for acceleration on
continuum fault segments that are far above steady
-
state
(Rubin & Ampuero, 2005)
.
Eq.
(
11
)
is also similar to the findings
for
seismic slip nucleation
in slip
-
weakening
friction
models
(Uenishi & Rice, 2003; Viesca & Rice, 2012)
except that
"#
depends on
pressure; specifically on the maximum value of pressure (at the injection site). The fact
that this lengthscale does not depend
-
at least to first order
-
on the extent
or shape of
3
the pore pressure distribution is also consistent with prior findings
(Uenishi & Rice,
2003; Viesca & Rice, 2012)
. At the same time,
"#
is different from some of the
discussed critical lengthscales, since it does not signify the transition to dynamic,
inertial
ly
-
controlled earthquake slip, but rather corresponds to the beginning of the
different quasi
-
static slip stage in this particular experiment. The existence of
"#
is
linked to the two
-
stage quasi
-
static slip process in the field experiment which the
simulations are trying to emulate. The associated evolution of the friction coefficient
-
with sharp increase to a peak value, then near
-
linear decrease vs. slip with the slope of
, and then near
-
constant value
-
is likely related to the relatively rap
id increase of the
pore pressure at the injection site compared to the timescale of state variable evolution
considered in this work.
To demonstrate that Eq.
(11)
holds, in Figures
S
12
and S1
4
(A
-
C)
we show 3
simulations in which
"#
is
increased compared to the
intermediate
-
friction
case by
increasing
(pink), increasing
()
(yellow) or decreasing
(turquoise) while keeping
J
constant. Figures
S13 and
S1
4
(D
-
E)
show simulations in which both
J
and
"#
are
increased by increasing
(pink) or
%&%
(yellow). Figures S1
3
and S14
(F)
also show a case
(turquoise) in which both
J
and
"#
are kept the same as in the
intermediate
-
friction
reference case but
"#
is delayed due to th
e decreased hydraulic diffusivity
which
controls how fast the slipping zone expands during Stage
2
.
In all cases, the onset of
Stage
3
is delayed compared to the intermediate
-
friction reference case. Thus,
parameters
,
()
,
,
J
and
have a primary control on the onset of Stage
3
observed
in all simulations shown in this work.
As for the amplitude and slope of the slip acceleration, four parameters
-
,
,
and
-
have been identified to play a key role in controlling
them as shown in
Figures S15 to S
19
.
4
Table S1.
Model parameters for the three cases presented in Figures 2
-
4 in the main text.
Properties
Symbol
Low
Friction
Intermediate
Friction
High
Friction
Total fault length [m]
NON
250
250
250
Frictional
interface length [m]
PQ
200
200
200
Initial shear stress [MPa]
STS
2.15
2.15
2.15
Initial normal stress [MPa]
STS
4.00
4.00
4.00
Initial coefficient of friction
STS
0.5375
0.5375
0.5375
Reference coefficient of friction
0.481
5
0.5500
0.6000
Reference slip rate [m/s]
10
-
6
10
-
6
10
-
6
Direct effect frictional parameter
0.01500
0.01125
0.01125
Evolutionary effect frictional parameter
0.01600
0.01600
0.01600
Critical slip distance [
m]
()
16.75
16.75
16.75
Hydraulic diffusivity [m
2
/s]
0.04
0.20
0.85
Initial state variable [s]
STS
1.21e12
2.38e12
7.00e12
Shear modulus [GPa]
10
10
10
5
Figure
S1.
Temporal
evolution
of
pore
pressure,
slip
and
slip
rate
and
evolution
of
friction
as
a
function
of
slip
as
in
Figure
2
AB
in
the
main
text
b
ut
for
the
exact
pressure
history.
The
simulated
slip
rate
is
similar
but
noisier
and
harder
to
interpret
.
6
Figure S2.
Spatial
and
t
emporal
evolution o
f pore pressure and slip
as Figure 3 in the main text
but for the exact pressure history as in Figure S1 and including the depressurization stage.
7
Figure S
3
.
Temporal
evolution of several quantities at the injection site for the
prolonged injection
s
imulations
(Figure 4)
with domain sizes of 250 m (solid lines) and 300 m (dashed lines).
From top
to bottom: the normalized effective normal stress, slip, no
rmalized slip rate (
UV&
= 10
-
2
m/s),
state variable, friction coefficient, norma
lized shear stress and closeness to steady state at the
injection site.
Changing the domain size slightly changes the timing
of the dynamic events
but not
the overall behavior.