Geophysical Research Letters
Supporting Information for
Rupture propagation behavior and the largest possible earthquake induced by
fluid injection into deep reservoirs
Valentin S. Gischig
ETH Zürich, Swiss Competence Center for Energy Research (SCCER-SoE),
Sonneggstrasse 5, NO F27, CH-8092 Zürich, +41 44 632 36 48, gischig@erdw.ethz.ch
Contents of this file
Text S1
Table S1
Introduction
Here I outline the governing equations for hydro-mechanical processes and rate-andstate frictional behavior used in the numerical code CFRAC (Text S1), as well as the
parameters used to produce modeling results (Table S1).
Text S1
Numerical model setup
The numerical model CFRAC [McClure, 2012] is designed to address problems related
to fluid injection into a fractured rock mass and the associated induced seismicity. A fully
hydro-mechanically coupled flow model is solved for fractures, which can either open or
accommodate slip.
The transient pressure disturbance  p (i.e. above the ambient water pressure p 0 in the
undisturbed fracture) is computed using the fracture flow model described by the
following equations. The mass balance equation:
1
 ( E )
 q  s ,
t
(1)
where q is the mass flow rate, s is a source term,  fluid density, and E the void
aperture (or mechanical) of the fracture. The equation is coupled to Darcy’s equation
assuming the cubic law for fracture flow:
q
bw e 3 p
12 xi
(2)
where p is fluid pressure,  is fluid viscosity, e the hydraulic aperture, and bw the
model width in the off-plane direction. Both E and e are a function of effective normal
stress  n acting on the fracture (which itself is a function of fluid pressure p ), and on
slip distance D that has occurred along the fracture. The following equation written for
the mechanical aperture E has the same form for e with corresponding constants:
E
Eo
 Edil1
 D tan
 E res
1  9 n /  Enref
1  9 n /  Enref
(3)
Here, E 0 ,  Enref , E res , and  Edil1 are fracture properties. More detailed explanation of
these equations is given by M W McClure and R Horne [2011]. Values chosen in this
study are given in Table S1. Parameters used to describe the hydraulic aperture e
(equivalent to Equation 3) were chosen such that initial permeability is about 10-18 m2,
similar to the value estimated for the Basel reservoir[Gischig and Wiemer, 2013; Häring
et al., 2008]. The reference mechanical aperture E 0 was chosen to be ten times larger
than the hydraulic one. The dilation angle edil1  2.5 ensures strong coupling between
slip and hydraulic aperture (and hence the pressure field). For simplicity, the mechanical
aperture is assumed to not change with slip, i.e.  Edil1  0 .
As a consequence of the pressure disturbance  p the fracture either opens if p   n' 0 (
 n' 0   n  p 0 is the effective normal stress on the undisturbed fracture) or starts
slipping.
Slip D
is
computed
based
on
the
force
equilibrium
equation
  v   f ( n' 0  p) , where v is slip velocity,  is the radiation damping coefficient , 
is shear stress, and  f is the friction coefficient.
 f is computed using rate-and-state frictional behavior [Dieterich, 1979; M McClure
and R Horne, 2011; Ruina, 1983; C H Scholz, 1998; Segall, 2010]:
 f  f o  a ln
v
v
 b ln 0
v0
dc
(4)
2
Here, f o is the steady-state friction at a slip velocity v0 , which is a reference velocity. d c
is the characteristic distance scale.  is the state variable that evolves as:

v
 1
t
dc
(5)
At steady state the friction coefficient becomes:
 f  f o  (a  b) ln
v
v0
(6)
f o , v0 , d c , a and b are fracture properties. These are usually derived from laboratory
friction experiments (e.g. [Kilgore et al., 1993; Marone, 1998]. On fault-scale only few
attempts of directly estimating these parameters are reported in literature [Guglielmi et
al., 2015], more often they can only be derived indirectly from seismological
observations [Tse and Rice, 1986]. Equation 6 indicates that the constants a and b
define whether friction is velocity-weakening ( a  b  0 ) or velocity-strengthening (
a  b  0 ). The constants are usually on the order of 0.01 [Kilgore et al., 1993]. The
value of a  b can be as low as -0.01 within the seismogenic crust [C H Scholz, 1998].
Higher, but still negative a  b values on the order of -0.001 are more commonly
observed in laboratory experiments on bare granite surfaces [Kilgore et al., 1993].
Lower values for a  b enhance slip velocity and rupture propagation. The ratio a/b
determines the nucleation style (localized or extended, see Rubin and Ampuero [2005]).
Laboratory values for the weakening distance d c are in the range 10 µm (bare granite
surfaces) and 100 µm (gouge material) [Marone, 1998].C.H. Scholz [2002] mentions that
d c may depend on the typical roughness scale of the fault, and thus may be substantially
larger for natural faults than for laboratory samples. The value defines the size of the
nucleation patch in case of velocity-weakening, i.e. the half-width of the fault patch on
'
which slip accelerates prior to rupture propagation is on the order of L  Gd c / b n , e.g.
[Rubin and Ampuero, 2005] ( G is the shear modulus). Hence, in the case of pressureinduced slip, it also defines how far and how long a pressure front has to propagate until
rupture propagates unstably.
In this study, the values for a , b and d c are chosen such that unstable sliding can
occur. Additionally, to demonstrate that uncontrolled rupture propagation is possible, I
chose conditions that are favorable for rupture propagation. Thus, the lowest possible
value for a  b  0.01 was chosen that is still realistic for the seismogenic brittle crust [C
H Scholz, 1998]. For this, we chose a value of a = 0.01 that corresponds to laboratory
values [Kilgore et al., 1993], and set b = 0.02. Hence, the ratio a / b  0.5 is in the lower
range for laboratory values [Rubin and Ampuero, 2005]. However, the transition of
uncontrolled to pressure-controlled rupture propagation depends much more on the
difference a  b than on a / b that defines the nucleation style. The value d c  100m is
realistic for laboratory experiments on gouge material, on the higher side for bare granite
surfaces, and probably on the lower side for in-situ conditions along faults [C.H. Scholz,
2002]. Since lower values of d c imply smaller widths of the nucleation patch, rupture
3
propagation occurs earlier during injection. Again, I chose a value that may be on the
lower side for the seismogenic crust to promote rupture propagation.
f o and v0 are reference values. v0 is commonly set to a low value (here 10-6 m/s). f o is
set to 0.85 so that the friction coefficient at steady state and at slip velocities of v0
corresponds to the Mohr Coulomb friction coefficient used to derive the stress state of
the Basel reservoir ([Häring et al., 2008], see below). The initial value of  (t  0) was
set to a high value 30 years (≈2.6∙106s) so that the ratio v / d c  1 , with v(t  0) in the
range of 10-10 m/s at stress states typical for our study. This is termed the ‘no-healing’
condition [Rubin and Ampuero, 2005], and was chosen, because healing is considered
negligible for the time-scales of injection as used in this study.
Generally, the rate-and-state friction constants were chosen here such that uncontrolled
rupture propagation is promoted. Larger (but still negative) value for a  b and larger
values for d c would result in lower slip velocities and slower rupture propagation. Indeed,
additional simulations with different parameters for a  b and d c showed that the
transition from uncontrolled to pressure-controlled rupture propagation would shift to the
left (towards more critical stress states) in Figure 2b. A full sensitivity analysis focusing
on the complex interaction between rate-and-state constants as well as hydraulic
constants, on the one hand, and rupture propagation behavior, on the other hand, goes
beyond the scope of this research letter. However, it has to be addressed in further
research in the field of induced seismicity.
Stress calculation
The stress state of the Basel reservoir was computed analogous to Häring et al. [2008].
Induced earthquakes in the reservoir showed predominant focal mechanisms that are
consistent with a strike-slip stress regime. Hence, the intermediate principle stress can
be assumed to be vertical and corresponds to the weight of the overburden. Using
 2 ( z)  0.0249 z (Equation 1 in Häring et al. [2008]) gives  2 (4500m)  112MPa .
Similarly, hydrostatic pressure is
p0 (4500m)  45MPa assuming fluid density of
  1000kg / m 3 . No pressure-limiting behavior was observed during water injection at
up to 30 MPa wellhead pressure (or 75 MPa downhole pressure). Hence, no hydraulic
fracturing occurred implying that  3  75MPa . Here it was set to  3  76MPa in order
to avoid fracture opening at injection pressures of up to 30 MPa. Finally, the maximum
principal stress can be computed assuming that the rock mass is critically stressed and
that the Coulomb friction coefficient is f C  0.85 .
 1  p0
 [( f C2  1)1 / 2  f C ]2
 3  p0
(7),
which gives .  1  185MPa .
Shear and normal stress along different fault orientations were calculated using the
equations:
4
n 

1   3
2
1   3
2

1   3
2
sin( 2 )
cos( 2 )
(8)
(9)
 is the angle between the fault normal and the direction of  1 . Optimal
orientation (i.e. the most critically stress situation) is  opt   / 4  arctan( f C ) / 2 .
where
Because rate-and-state frictional behavior is not a failure criterion formulation of slip
initiation, I use the Mohr-Coulomb failure criterion with friction f C to define the modeled
stress states in terms of understress and hence criticality (proximity to the MohrCoulomb failure limit). Understress is defined as ( P   ) /  P , where  P  f C   n  S 0 ,
which is the Mohr-Coulomb failure criterion (here S 0  0MPa) . To obtain values of
understress between 0 (critically stressed) and 1 (non-critically stressed), I chose fault
orientations  between  opt and  / 2 (i.e. fracture orientation between optimal
orientation and normal to  3 ). Note that f C  f 0  0.85 . My computations showed that
prediction of the slip initiations by the rate-and-state friction model is in agreement with
the onset of slip predicted by the Mohr-Coulomb failure criterion (Figure 2).
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Table S1. Model parameters used for the CFRAC models.
E0
 Enref
Void aperture constant
1200 µm
Reference normal stress
25 MPa
E res
Residual void aperture
2 µm
 Edil1
e0
 enref
Dilation angle (void aperture)
0°
Hydraulic aperture constant
120 µm
Reference normal stress
25 MPa
eres
Residual hydraulic aperture
0.2 µm
 edil1
Dilation angle (hydraulic
aperture)
Fluid density
Fluid viscosity
Off-plane model width
2.5°
Radiation damping coefficient
Nominal friction coefficient
3 MPa∕(m∕s)
0.85
So
Cohesion
0 MPa
dc
100 µm
a
b
v0
Characteristic displacement
scale
Rate-and-state parameter
Rate-and-state parameter
Reference velocity
G
ν
Shear modulus
Poisson’s ratio
15 GPa
0.25


bw

fo
1000 kg/m3
0.001 Pa s
100 m
0.01
0.02
10−6 m∕s
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