-
PROCEEDINGS, Fourtieth Workshop on Geothermal Reservoir
Engineering Stanford University, Stanford, California, January
26-28, 2015 SGP-TR-204
1
Injection-Triggered Seismicity: An Investigation of
Porothermoelastic Effects Using a Rate-and-State Earthquake
Model
Jack Norbeck and Roland Horne
Department of Energy Resources Engineering, 367 Panama Dr.,
Stanford, California, USA, 94305
[email protected]
Keywords: injection-triggered seismicity, induced seismicity,
earthquake modeling
ABSTRACT Physical processes associated with injection-triggered
seismicity were investigated through the use of a numerical model.
We investigated the role of the following physical mechanisms on
causing triggered earthquake events: fluid pressurization within
the fault zone, poroelastically-induced stress due to fluid leakoff
into the rock surrounding the fault, and thermoelastically-induced
stress due to cooling of the reservoir rock. A model of a fault
that had a direct hydraulic connection to an injection well was
used to develop a numerical experiment. In the model, relatively
cold fluid was injected into the fault for a period of one day, and
then the well was shut-in. A rate-and-state friction framework was
used to model the earthquake nucleation, rupture, and arrest
processes.
Four simulations were performed in order to isolate the effects
of the different physical mechanisms. We observed that, depending
on which physical mechanisms were active, the overall behavior in
seismicity differed significantly between the four cases. For the
reservoir and fault parameters used in this study, it was observed
that the poroelastic and thermoelastic stresses were of the same
order of magnitude as the change in fluid pressure within the fault
zone. Of particular interest, the thermoelastic stresses introduced
significant levels of heterogeneity in the distribution of
effective stress along the fault, which ultimately led to a
markedly distinct character in the individual earthquake events and
overall seismic pattern. This study demonstrated that the physical
mechanisms investigated have the potential to control behavior
during injection-triggered seismicity. However, it should be stated
clearly that the results presented in this paper cannot be extended
generally to all scenarios related to injection-triggered
seismicity, and further parametric studies must be performed in
order to classify the range of geological and operational settings
over which each physical mechanism may be important.
1. INTRODUCTION Past experience has shown that
injection-triggered seismicity is an extremely important phenomenon
that must be considered when evaluating geothermal projects.
Seismicity related to injection activities has been observed at
nearly all geothermal sites, and relatively large earthquakes
(greater than magnitude 3) during reservoir stimulation treatments
have caused the cancellation of at least two geothermal projects.
If geothermal is to be adopted as a widespread renewable energy
technology, then the issue of injection-triggered seismicity will
have to be addressed head-on. Outside of the geothermal community,
this issue has also recently gained attention in the oil and gas
sector, as several earthquake events have been attributed to
wastewater disposal activities in the last few years. An increased
understanding of the fundamental physical processes that contribute
to injection-triggered seismicity will be helpful for designing
strategies to mitigate seismicity and for developing informed
regulatory policy.
Injection-triggered seismicity related to both geothermal and
oil and gas activities has been researched extensively in the past.
Field experiments at the Rangely Oil Field in the 1960s
demonstrated that seismicity could be correlated to fluid injection
(Gibbs et al., 1973). Significant levels of seismicity were
observed during wastewater disposal operations at the Rocky
Mountain Arsenal in the 1960s (van Poollen and Hoover, 1970).
Mossop (2001) performed a study that indicated that fluid injection
and extraction at The Geysers geothermal field could be correlated
to seismicity at the site. More recently, significant levels of
seismicity associated with stimulation of geothermal wells at sites
in Basel, Switzerland in 2006 and in St. Gallen, Switzerland in
2013 were determined to pose a high enough level of risk to nearby
communities that the projects were ultimately cancelled (Häring,
2008; SED, 2013).
In terms of the physical mechanisms associated with fluid
injection that trigger the earthquake events, it is widely accepted
that changes in the effective normal stress acting on the fault
controls the seismic behavior. Previous studies have focused
primarily on the role of increased fluid pressures caused by
injection as the main cause of seismicity. Ellsworth (2013)
discussed a simple conceptual model illustrating that increased
fluid pressure can bring faults closer to a state of failure.
Baisch et al. (2010) and McClure and Horne (2011) performed
detailed numerical simulations that showed that pore pressure
distributions in fault zones can cause unintuitive behavior to
occur, such as triggered events that occur following shut-in of the
injection well. While these conceptual models and numerical
examples demonstrate that fluid pressurization within fault zones
is indeed an important mechanism to consider, it is important to
recognize that a broader range of physical effects may be occurring
during fluid injection processes. For example, McClure and Horne
(2011) suggested that fluid redistribution following shut-in could
cause pressure near the edges of the fault to increase even after
injection ceases, which can cause earthquake events to nucleate at
the edges of the fault. However, the study assumed that the fault
was embedded in an impermeable rock, so that no fluid leakoff would
occur. In some geologic settings, perhaps the surrounding rock
would be permeable enough to allow fluid to leakoff and cause
pressure within the fault zone to dissipate relatively quickly.
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Norbeck and Horne
2
Studies performed by van Poollen and Hoover (1970) and Mossop
(2001) suggested that in some geologic settings, reservoir cooling
due to injection of relatively cold fluid could induce tensile
stresses near faults that are equal to or greater in magnitude than
the changes in pore pressure. These authors linked thermal stresses
to the seismicity observed at the Rocky Mountain Arsenal in
Colorado, USA and The Geysers in California, USA, respectively.
Ghassemi (2008) performed a study that suggested that poroelastic
effects due to fluid leakoff and thermoelastic effects due to
reservoir cooling both could have significant impacts in shear slip
behavior in geothermal reservoirs, but did not perform any detailed
earthquake simulations. Dempsey et al. (2014) performed a
fully-coupled porothermoelastic simulation of a reservoir
stimulation treatment at the Desert Peak geothermal site, and
demonstrated that both local and nonlocal stress changes could be
large enough to cause shear failure in certain locations throughout
the reservoir. Although it is difficult to obtain field data that
directly corroborates the dominance of a particular mechanism,
theoretical studies evidently suggest that there may be more to the
picture than fluid pressurization alone.
In this work, we sought to investigate a scenario in which three
different physical mechanisms (i.e., fluid pressurization,
poroelastic effect, and thermoelastic effect) each contribute
significantly to the earthquake behavior. The study used the
application of a numerical model. The remainder of the paper is
outlined as follows. In Section 2, we discuss the physical
mechanisms of interest and provide the mathematical background
necessary to address the problem. The numerical formulations of
fluid and heat flow, solid mechanics, and earthquake modules are
presented briefly. In Section 3, we describe a numerical experiment
of injection-triggered seismicity in order to identify the role of
the various physical mechanisms. Finally, we discuss the
implications of the numerical results and provide possible
directions for future related studies in Section 4.
2. PHYSICAL MECHANISMS CONTRIBUTING TO INJECTION-TRIGGERED
SEISMICITY Seismic events occur when the state of stress along a
fault plane is perturbed such that the fault experiences a rapid
and unstable reduction in its frictional resistance to shear
displacement. In the context of tectonic earthquakes, the loading
processes that cause these instabilities are most commonly
associated with the accumulation of strain due to regional tectonic
activity over relatively long periods of time (i.e., many years).
In the context of injection-triggered seismicity, the loading
mechanisms that arise as a consequence of fluid injection can be
very diverse and may develop over a broad range of time scales. In
this work, we identified that the following mechanisms can have a
significant impact on the frictional behavior of fault zones during
fluid injection processes: pore pressurization within the fault
zone, poroelastic stress changes due to fluid leakoff into the rock
surrounding the fault, and thermoelastic stress changes due to
reservoir cooling.
Traditionally, analyses of injection-triggered seismicity have
tended to focus on the role of pore pressurization within the fault
zone as the primary cause of seismicity. This conceptual model is
based upon the combination the principle of effective stress and a
Mohr-Coulomb-type shear failure criterion, which ultimately
indicates that increased fluid pressure promotes failure (Jaeger et
al., 2007; Zoback, 2007). It is clear that this is a very important
mechanism, but previous theoretical studies and field studies have
indicated that other physical processes can lead to stress
perturbations on the same order of magnitude as the changes in
fluid pressure. In this section, we present a brief mathematical
description of the processes that we believe are important to
consider when performing investigations of injection-triggered
seismicity. We also describe the reservoir simulator that we used
to perform the numerical experiment presented in Section 3.
2.1 Mathematical Description In this work, we made use of a
combination of the theories of porothermoelasticity in porous
media, fracture mechanics, and earthquake rupture. Here, we present
a description of the governing equations used to solve for the
poroelastic and thermoelastic deformations. In addition, we provide
an overview of the rate-and-state friction theory used to model
earthquake nucleation, rupture, and arrest. A detailed discussion
of the fracture mechanics theory applied in this work can be found
in McClure and Horne (2013).
2.1.1 Porothermoelastic Deformation in Porous Material In
continuum mechanics, the equations describing momentum balance for
the case of quasistatic deformation, neglecting body forces, are
(Jaeger et al., 2007):
∂σ ij∂x j
= 0, (1)
whereσ ij are the components of the stress tensor. Assuming a
linear elastic porous material that may be subjected to changes in
fluid pressure and temperature, Hooke’s law is:
(2)
where εij are the components of the strain tensor, Δp = p− p0 is
the change in fluid pressure from a reference state, ΔT = T −T0 is
the change in temperature from a reference state, G is the shear
modulus,Λ is Lame’s modulus, K is the bulk modulus,α is Biot’s
coefficient, β is the linear thermal expansion coefficient of rock,
and δij is the Kronecker delta function. Note that compression has
been taken as positive in this sign convention. In Eq. 2, it was
assumed that changes in fluid pressure and temperature result in
purely volumetric deformations. Assuming infinitesimal strains, the
strain-displacement relation is:
σ ij = 2Gεij +Λεkkδij +αΔpδij +3βKΔTδij,
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Norbeck and Horne
3
(3)
where ui are the components of the displacement vector.
Substitution of Eqs. 2 and 3 into Eq. 1, yields the equations of
motion for a body subjected to fluid pressure and temperature
perturbations:
G∂2ui∂xk∂xk
+ Λ+G( ) ∂2uk
∂xi∂xk= −α
∂∂xi
Δp( )−3βK ∂∂xi
ΔT( ). (4)
It is apparent from Eq. 4 that gradients in fluid pressure and
temperature in the domain act as body forces. In the subsurface,
rock is typically constrained from motion to a certain extent. As
the material attempts to deform subject to pressure and temperature
perturbations, changes in the solid stresses can be induced both
within the zones of the perturbation (local) and outside of the
zones of perturbation (nonlocal). These changes in solid stress can
act as stress perturbations to existing fault structures.
Mode-I (normal) and mode-II (shear) fault deformation can also
cause changes in the solid stresses. Then, the overall state of
stress at any point in the reservoir is the superposition of the
different physical effects:
(5)
whereσ ijR are the remote tectonic stresses,σ ij
M are the mechanically-induced stresses due to fault
deformation,σ ijP are the poroelastically-
induced stresses, andσ ijT are the thermally-induced
stresses.
Consider a fault that exists in a reservoir near an injection
well. One of the primary parameters that controls seismic behavior
of the fault is the component of the stress tensor acting in the
direction normal to the fault plane because shear strength of the
fault is a function of the effective normal stress. The effective
normal stress acting on the fault must reflect the combined effect
of the different physical mechanisms discussed above:
(6)
where p is the fluid pressure within the fault zone. In general,
the distribution of effective stress is not expected to be constant
over the fault surface. This philosophy of the superposition of
stresses caused by different physical mechanisms provides the basis
for the present study.
2.1.2 Fault Friction Under a Rate-and-State Friction Framework
For injection-triggered seismicity applications, it is not
sufficient to consider a single earthquake event. While relatively
large earthquakes are naturally of interest, the seismicity leading
up to and following large events can also be extremely important in
terms of characterizing the seismic behavior of a given site.
Earthquake sequences associated with fluid injection have often
indicated that many earthquakes can be triggered along the same
fault plane (e.g., see Horton, 2012). These observations correspond
to the fact that loading distributions that cause earthquake events
are changing continually during fluid injection processes.
Therefore, when attempting to model injection-triggered seismicity,
it is necessary to employ a framework that accounts for earthquake
nucleation, rupture propagation, rupture arrest, and fault
restrengthening in order to allow for the emergence of seismic
sequences. Rate-and-state friction is one theory that is able to
capture the full earthquake rupture cycle. When combined with a
rigorous treatment of elasticity, using rate-and-state theory to
model friction evolution provides a powerful tool to gain insight
into the physical mechanisms that trigger earthquakes. The theory
discussed in this section follows closely that described by McClure
(2012).
In this work, we considered a two-dimensional, mode-II shear
problem in plane strain conditions. The fault was assumed to be
perfectly planar. Stress transfer due to mode-I displacements were
neglected. Let the shear stress resolved on the fault due to the
in-situ state of stress be designated . In the context of the
mode-II problem, the elasticity equations can be expressed as:
τ x, t( ) = τ 0 −ηV x, t( )+Θ x, t( ), (7)
where is the shear stress distribution on the fault,η is a
radiation damping parameter used to approximate inertial effects,
andΘaccounts for the elastic stress transfer due to mode-II
displacement along the fault. TheΘ term is the mechanism for
dropping stress across the region of the fault that has previously
experienced slip and concentrating stress at the tips of the
slipping fault patch. In this work, wave-mediated stress transfer
was neglected, soΘwas assumed to occur quasistatically.
Assuming Mohr-Coulomb behavior, the shear strength of the fault
is:
τ s = fσ n + S, (8)
εij =12∂ui∂x j
+∂uj∂xi
"
#$$
%
&'',
σ ij =σ ijR +σ ij
M +σ ijP +σ ij
T ,
σ n =σ nR +σ n
M +σ nP +σ n
T − p,
τ 0
τ
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Norbeck and Horne
4
where S is the fault cohesion and the coefficient of friction, f
, is defined under the rate-and-state framework as (Segall,
2010):
(9)
where V is the slip velocity, Ψ is the state variable, and f0 ,
V0 , a , b , and d are constants derived from experiments. The
second term on the right hand side of Eq. 9 represents what is
called the direct effect, which captures an immediate
velocity-strengthening behavior that is commonly seen in laboratory
experiments. The third term represents the state evolution effect.
When b− a > 0 , it is possible to have unstable friction
weakening that can lead to earthquake nucleation and rupture. The
aging law for the state variable is:
(10)
When the shear strength is greater than the shear stress acting
on the fault (i.e., τ s > τ 0 ), then the shear displacements
are assumed to be zero, and the slip velocities remain very small
at the initial condition, V0 . During sliding, equilibrium must be
enforced such that:
τ 0 −ηV x, t( )+Θ x, t( ) = f x, t( )σ n x, t( ) (11)
Equation 11 can evolve in a highly nonlinear fashion, especially
during earthquake events, and must be solved numerically. As an
algebraic constraint, the slip velocity is equal to the time
derivative of the cumulative shear slip discontinuity,δ :
V =∂δ∂t. (12)
Under the rate-and-state friction framework, a requirement to
achieve an instability that cascades into an earthquake rupture is
that a suitable patch size of the fault must be perturbed. A
stability analysis for the mode-II shear problem can be performed,
and under certain limiting cases it can be shown that the critical
perturbation length is:
Lc =πGd
σ n b− a( ), (13)
where the effective stress is taken as constant along the fault.
This important parameter controls the maximum perturbation distance
before unstable seismic ruptures will occur.
2.2 Numerical Model To perform the numerical experiments
presented in this study, we extended the work of McClure (2012),
Norbeck et al. (2014), and Norbeck and Horne (2014) to incorporate
poroelastic and thermoelastic effects into a reservoir simulator
with earthquake modeling capabilities called CFRAC. McClure (2012)
and McClure and Horne (2013) provided the foundation for the model
by coupling fluid flow in faults to the mechanical deformation of
faults. The fault mechanics calculations were performed using a
boundary element method called the displacement discontinuity
method (Shou and Crouch, 1995). Norbeck et al. (2014) and Norbeck
and Horne (2014) extended the model to allow for fluid flow and
heat transfer interaction between faults and surrounding matrix
rock using an embedded fracture coupling strategy. In the present
work, we used the fluid pressure and temperature distributions that
resulted from the embedded fracture model in order to calculate
changes in solid stress cause by poroelastic and thermoelastic
effects. These stresses were then used to modify the boundary
conditions used in the fault deformation and fault friction
calculations.
As a first attempt to address the issue of poroelasticity and
thermoelasticity in CFRAC, an approach based on elastic potentials
was employed (Nowacki, 1986). The poroelastic and thermoelastic
potential fields can be described by the following Poisson
equations, respectively:
(14)
and
(15)
Here,ΦP is the poroelastic potential, ΦT is the thermoelastic
potential, andν is Poisson’s ratio. A finite difference scheme was
used to discretize Eqs. 14 and 15 to arrive at systems of linear
equations used to solve for the potential fields. The pressure and
temperature
f V,Ψ( ) = f0 + a lnVV0+ b lnΨV0
d,
∂Ψ∂t
=1− ΨVd.
∂2Φp
∂xi∂xi=1+ν1−ν$
%&
'
()α3K$
%&
'
()Δp,
∂2ΦT
∂xi∂xi=1+ν1−ν$
%&
'
()βΔT.
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Norbeck and Horne
5
distributions calculated during the fluid flow and heat flow
simulations were used to calculate the sourcing terms. Once the
potential fields were calculated, the induced stresses were
calculated, using numerical differentiation, as:
(16)
and
(17)
As a postprocessing step, the induced stresses were resolved
into normal tractions acting on the fault surface. The effective
stress was calculated using Eq. 6, and was applied as a boundary
condition in the fault deformation calculations. In the current
version of the numerical code, the poroelastic and thermoelastic
effects were not fully coupled to the fluid flow calculations,
because the matrix porosity was not considered to be a function of
the overall effective stress. This will be pursued in future
work.
The overall coupling strategy between the fluid flow and
mechanics calculations can be thought of as an explicit scheme (Kim
et al., 2011). As the first step, the fluid pressure from the
previous timestep was held constant and a Runge-Kutta timestep was
initiated. During this step, parameters related to the
rate-and-state friction model were updated. These parameters
included shear displacement, δ , slip velocity,V , the elastic
stress transfer,Θ , the state variable, Ψ , and the friction
coefficient, f . These variables, especially state and friction
coefficient, can change rapidly even during very small timesteps,
and so the Runge-Kutta method was used to provide the necessary
level of accuracy. Each of those parameters were held constant for
the remainder of the timestep. Next, a sequential iterative
coupling procedure was used to perform the fluid flow and heat flow
simulations. Upon convergence of this procedure, both fluid
pressure, p , and reservoir temperature, T , were obtained. The
fluid flow and heat flow calculations were performed using an
embedded fracture approach, which is based on conventional finite
volume discretization strategies (Li and Lee, 2008; Norbeck and
Horne, 2014). Finally, the pressure and temperature distributions
were used to calculate the poroelastic and thermoelastic stresses.
At this point, the effective normal stresses acting in the fault
surface were updated, and the solution proceeded to the next
timestep.
3. NUMERICAL SIMULATION OF INJECTION-TRIGGERED SEISMICITY The
purpose of this work was to lay the foundation for future
investigations on the relative impact of different physical
mechanisms associated with injection-triggered seismicity. We were
interested in determining which physical mechanisms of interest
contributed significantly to the seismic behavior during fluid
injection processes. We used a numerical model to perform a
simulation of injection-triggered seismicity along a fault. The
physical processes considered were: fluid pressurization within the
fault zone, poroelastically-induced stress due to fluid leakoff
from the fault into the surrounding rock, and
thermoelastically-induced stress due to cooling of the reservoir
rock.
3.1 Problem Description We constructed a model of a fault that
had a direct hydraulic connection to the injection well.
Appropriate conceptual models might include an open-hole well that
was drilled through a fault (e.g., Basel, Switzerland), a hydraulic
fracture that intersected a fault (e.g., Horn River Basin, Canada),
or an acid stimulation that wormholed into a fault (e.g., St.
Gallen, Switzerland). We considered a strike-slip vertical fault
that was well-oriented for shear failure in the given state of
stress. The fault existed at a depth of roughly 4 to 5 km, in a
reservoir that was initially at a temperature of 200 °C and a
pressure of 45 MPa. The reservoir rock was permeable, so that fluid
was able to leak off from the fault into the surrounding rock.
Fluid entered the center of the fault at a constant rate of 50 kg/s
and a constant temperature of 50 °C. The simulations lasted a total
of ten days. Fluid was injected for a period of one day, over which
the majority of the seismicity was observed. Following one day of
injection, the well was shut in and the behavior was monitored
continually for an additional nine days. The simulations were
essentially two-dimensional in the horizontal plane. An
illustration of the problem configuration is given in Fig. 1. Lists
of the parameter values used in the simulations are given in Tables
1 – 4.
We performed a total of four simulations, labeled Cases A – D,
in order to isolate the effects of the different physical
mechanisms. In all cases, fluid pressurization within the fault
zone contributed to a reduction in the overall effective stress. In
Case A, neither the poroelastic or thermoelastic effect were
considered. In Case B, the poroelastic effect was considered and
included in the effective stress calculations. In Case C, the
thermoelastic effect was considered and included in the effective
stress calculations. In Case D, both the poroelastic and
thermoelastic effect were considered.
In general, the following behavior was observed. As fluid
entered at the center of the fault, pressure gradients set up
within the fault zone such that pressure was highest near the
injection location and lowest near the edges of the fault.
Increased pore pressure within the fault zone had a destabilizing
effect in terms of the fault’s frictional resistance to shear
failure. Due to the difference in pressure within the fault zone
and the surrounding matrix rock, fluid leakoff occurred along the
fault. The elevated fluid pressure in the surrounding rock caused
the rock to attempt to expand. The poroelastic effect induced
compressive stresses, which had a stabilizing effect. The
magnitudes of the poroelastic stresses were always less than the
magnitude of the change in pressure within the fault zone, but the
two effects were of the same order. As cold fluid entered the
fault, the fault and surrounding rock experienced a cooling
phenomenon. The cooled region tended to be localized near the
injection point. As the rock surrounding the fault cooled and tried
to contract, the thermoelastic effect generated tensile stresses.
In contrast to the poroelastic effect, the thermoelastic effect
tended to have a destabilizing
σ ijP = 2G ∂
2Φp
∂xi∂x j−1+ν1−ν$
%&
'
()α3K$
%&
'
()Δp
+
,--
.
/00,
σ ijT = 2G ∂
2ΦT
∂xi∂x j−1+ν1−ν$
%&
'
()βΔT
+
,--
.
/00.
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Norbeck and Horne
6
effect. An important additional feedback from the coupled
processes was that changes in effective normal stress along the
fault caused the permeability of the fault to change, which in turn
affected the well’s injectivity.
Figure 1. Schematic of the model configuration. Cold fluid was
injected at a constant mass rate at the center of a vertical
strike-slip fault for a period of one day. The injected fluid was
150 °C colder than the initial temperature of the reservoir rock.
The fault was planar and had homogeneous frictional properties. The
initial shear and effective stress distributions along the fault
were homogeneous.
Table 1. Reservoir and fluid model parameters.
Table 2. Fault model parameters.
Parameter Value Unit Parameter Value Unit
L 500 m e0 0.0004 m
H 50 m S 0.5 MPa
θ 45 deg σ e,ref 50 MPa
70 MPa σ E,ref 50 MPa
11 MPa ϕe,dil 1 deg
E0 0.002 m ϕE,dil 1 deg
σ nR
τ 0
Parameter Value Unit Parameter Value Unit
200 C κ r 2.42 W ⋅m-1 ⋅o C-1
45 MPa κ f 0.6 W ⋅m-1 ⋅o C-1
m2
crT
816 J ⋅kg-1 ⋅o C-1
0.15 - cfT 4200 J ⋅kg-1 ⋅o C-1
ρ f ,0 1000 kg ⋅m−3 ρr 2650 kg ⋅m−3
µ 0.0005 Pa ⋅s β 2×10−5 C-1
crP 4.4×10−4 MPa-1 G 15 GPa
cfP
4.4×10−4
MPa-1 ν 0.2 -
α 0.8 - η 3.15 MPa ⋅m-1 ⋅s
T0
p0
k 1×10−14
φ0
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Norbeck and Horne
7
Table 3. Rate-and-state friction parameters.
Parameter Value Unit
f0 0.6 -
a 0.011 -
b 0.014 -
d 0.00005 m
Ψ0 50000 s
V0 1×10−9 m ⋅s-1
Table 4. Injection well controls.
Parameter Value Unit
mw 50 kg ⋅s-1
Tw 50 C
3.2 Simulation Results A concise summary of the results of the
four simulations is given in Table 5. Triggered seismicity was
observed for Cases A, C, and D. The earthquake sequences for these
cases are shown in Fig. 2. Zero events were observed for Case B.
The nucleation and arrest of each earthquake event was demarcated
based on thresholds in the slip velocity. The reported earthquake
magnitudes were calculated by integrating the cumulative slip over
the surface area of the slipping patch during each event. It is
difficult to have a strict criterion to determine when an event
begins and ends, and so the earthquake magnitude values should not
be interpreted as being extremely accurate. However, their relative
magnitudes can be used with confidence for comparison.
As expected, Case A demonstrated that increased fluid pressure
within the fault zone can trigger seismicity. The fact that zero
events occurred for Case B indicates that the poroelastic stresses
effectively negated the weakening that occurred due to fluid
pressurization within the fault zone. It is possible that the
poroelastic effect simply delayed the onset of seismicity, and if
injection was continued for a longer period of time then perhaps
some events may have eventually been triggered. In Case D, both the
poroelastic effect and the thermoelastic effect contributed to the
overall effective stress, and seismicity was observed. Considering
Cases A, B, and D together emphasizes a very important result of
this study: each of the three physical mechanisms investigated in
this study have the potential to influence the seismic behavior
during fluid injection processes.
Table 5. Summary of the shear slip behavior for the four
different simulations.
Case A B C D
Total Number of Earthquakes [-] 4 0 8 2
Maximum Earthquake Magnitude [-] 1.9 N/A 1.5 0.7
Maximum Cumulative Shear Slip [cm] 2.8 0.7 3.7 2.0
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Norbeck and Horne
8
Figure 2. Sequences of seismicity over the injection period of
one day for the different simulation cases. In Case B, zero seismic
events were observed. The events increased in magnitude
systematically because successively large patches of the fault
experienced slip. This can most likely be attributed to the fact
that the frictional properties of the fault were homogeneous, and
there was no random heterogeneity in either the fluid flow
properties of the fault or the stress conditions along the
fault.
Figure 3 shows a snapshot of the distribution of several
parameters of interest along the fault during the first earthquake
event observed in Case D. This figure gives a sense of the relative
magnitudes of the fluid pressure, poroelastic stress, thermoelastic
stress, and their combined impact on the effective normal stress.
Each of the physical mechanisms contributed significantly to the
overall effective stress. Fluid pressure was highest near the
injection point, and had a relatively steep gradient away from the
center of the fault. The poroelastic effect had a similar shape to
the pore pressure distribution, but was smaller in magnitude. The
temperature profile along the fault indicates that the cold fluid
perturbed a significant portion of the fault. However, a high
degree of cooling occurred only within a few tens of meters of the
injection point. The effect of this very localized region of
cooling is reflected in the thermoelastic stress profile. High
tensile (negative) stresses were generated over a very narrow
section of the fault close to the injection point. Directly outside
of the zone of tensile stresses, significant compressive stresses
were generated. These stresses correspond to the “nonlocal” elastic
stress transfer that must occur to accommodate the deformation that
occurred within the cooled zone. Perhaps the most significant
impact of the thermal stresses was to create a highly heterogeneous
state of stress along the fault patch. As opposed to Cases A and B,
where the effective stress always increased away from the wellbore,
the thermal stresses that were generated in Cases C and D caused
the effective stress distribution to be much more tumultuous. This
had a noticeable impact on the character of the individual seismic
events. Note that the step-like character of the distributions is a
discretization error associated with the embedded fracture model
used to calculate heat and mass transfer between the fault and the
surrounding rock. This error should not have significantly impacted
the overall trend in the results.
The temporal evolution of slip velocity and cumulative shear
slip for typical earthquake events observed in Case A (left panel)
and Case C (right panel) are illustrated in Fig. 4. The shading of
the lines becomes lighter as time progresses. The event in Case A
was triggered solely by increased fluid pressure within the fault
zone. The event in Case C was triggered by fluid pressure and an
additional thermal component. The two events showed markedly
different behavior.
In Case A, as fluid pressure continued to increase following a
previous event, slip velocities began rising rapidly near the edges
of the fault patch. Initially, the rupture fronts propagated from
the edges of the fault towards the center of the fault. As the two
fronts met near the center of the fault, the slip velocities
quickly reached levels approaching 1 m/s (typical slip velocities
of real earthquake events). At this point the seismic event was
occurring, which is also reflected by a sudden and large jump in
cumulative shear slip along the entire slipping patch. The rupture
was close to symmetric and propagated across a significant portion
of the fault. The rupture eventually arrested once it propagated
far enough into the zone of increasing effective stress. Contrast
that event with the event that had a thermal component. In this
case, the event nucleated near the center of the fault and the
rupture front propagated towards one edge of the fault. This event
was not symmetric, and as evidenced by the evolution of shear slip,
the event only ruptured along one half of the fault before
arresting. This resulted in an earthquake event that was
significantly smaller in magnitude.
We attribute the difference in behavior to the higher level of
stress heterogeneity introduced by the thermal stresses. The
additional tensile stress near the center of the fault caused the
rupture to nucleate near the center of the fault, instead of near
the edges. As the rupture attempted to propagate back across the
entire fault patch, it approached a small region of increased
effective stress due to the “nonlocal” compressive thermal stress.
This heterogeneity was evidently enough to cause the event to
arrest. Note that the general observation was that the
thermally-triggered earthquakes tended to cluster in pairs of
relatively small events that occurred in quick succession. The
pairs of events were associated with the first event rupturing one
side of the fault and arresting, and the subsequent event quickly
nucleating and rupturing across the other side of the fault.
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Norbeck and Horne
9
Figure 3. A snapshot of the distributions of important
parameters along the fault for Case D during the first earthquake
event. It is clear that the magnitudes of the three different
physical mechanisms were all of similar magnitude (compare a, c,
and d) and contributed significantly to the overall effective
stress distribution for this particular simulation. The dashed line
in (e) corresponds to the “critical” effective stress based on the
friction coefficient at the initial condition, which gives a rough
estimate of the effective stress at which slip can be expected to
occur.
(a) (b)
(c) (d)
(e)
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10
It is worthwhile to note that in all cases, a significant amount
of aseismic slip occurred. Recall that under the rate-and-state
friction framework, there is a critical length scale that must be
perturbed in order for further slip to cascade into an unstable
earthquake rupture (see Eq. 13). For perturbations less than the
critical length the fault is still able to slip, but the slip will
occur at extremely low velocities. For example, Table 5 indicates
that although zero earthquake events were recorded for Case B, the
maximum cumulative shear slip that occurred along the fault was 0.7
cm. This slip was entirely aseismic.
The critical length scale given in Eq. 13 depends on the
rate-and-state parameters, the shear modulus of the matrix rock,
and the effective stress acting on the fault. To arrive at that
expression, a stability analysis was performed assuming that the
effective stress along the fault is constant. In these simulations
of injection-triggered seismicity, the effective stress was not a
constant, and so it is not directly apparent that Eq. 13 is
applicable. The effective stress parameter appears in the
denominator of Eq. 13; using the initial effective stress state
most likely provides a conservative estimate of the critical
perturbation length because the effective stress decreased over
most of the fault during injection. Using the values listed in
Tables 1 – 3, the critical length was calculated to be Lc = 31.4 m.
In each of the four simulations, this value of critical length was
observed to provide an accurate prediction of the onset of
seismicity.
Figure 4. Temporal evolution of typical earthquake events
observed for Case A (left panel) and Case C (right panel). The
shading of the lines becomes lighter as time progresses. Of
particular interest, note that the event with a thermal component
had a significantly different character than the event triggered
purely by pressurization within the fault zone. The seismic events
in Case C tended to rupture across smaller patches of the fault,
and therefore produce smaller event magnitudes. We attribute this
behavior to the higher level of heterogeneity in effective stress
along the fault caused by the thermal stresses present in Case
C.
3.3 Discussion of Results For the set of geologic and
operational parameters used in this study, each of the physical
mechanisms that were considered contributed significantly to the
seismic behavior of the system. Many additional similar cases were
tested, and it was observed that the behavior was highly dependent
on the model parameters. Here, we will discuss the implications of
some of the more important model parameters.
First and foremost, one of the major assumptions in this study
was that the injection well had a direct hydraulic connection to
the fault. In some instances, this may be an appropriate conceptual
model. At many geothermal sites, engineers and geologists attempt
to target faults and drill directly through them, because it has
been observed that faults and fractures provide high-permeability
pathways through
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Norbeck and Horne
11
the otherwise impermeable crystalline rock that the reservoir is
comprised of. Since most geothermal wells are completed open-hole,
direct hydraulic connections with faults are often present. Another
example could be a hydraulic fracture that propagates into a fault.
There has been evidence to suggest that such a phenomenon has
occurred during reservoir stimulation treatments in shale gas
reservoirs in the United States, the UK, and Canada (Maxwell and
Rutledge, 2013). This conceptual model contrasts the case where an
injection well is located at some proximity to a fault, and is
unable to interact directly with the fault. For example, this was
evidently the case at a wastewater disposal site in Arkansas, USA,
where several earthquakes larger than M 4.0 were triggered in 2011
(Horton, 2012). At this particular site, the earthquakes were
triggered on a very large fault that existed in the basement rock
beneath the target injection aquifer. It is likely that it is
extremely important to consider the proper conceptual model when
interpreting triggered earthquake sequences at a particular field
site or when performing theoretical studies of triggered
seismicity, because the relevant physical mechanisms may manifest
themselves differently depending on the situation.
The rate and magnitude of fluid pressurization within the fault
zone is controlled by the injectivity of the system. The
injectivity is a function of the fault and matrix rock permeability
and storativity. In the model, the initial fault permeability was
extremely high. The matrix permeability was 10 md, which is
relatively high in the context of most geothermal reservoirs that
are located in crystalline basement rock. High values of matrix
permeability encourage fluid leakoff, which has several feedback
responses. The poroelastic effect becomes more prominent as fluid
diffuses further into the matrix rock. In addition, as fluid
leakoff rates increase, the cold injection fluid is advected
further into the matrix rock. This effect helps to promote the
thermoelastic response. Because the matrix permeability value used
in the simulations was relatively high, the roles of the
poroelastic and thermal stresses may have been exaggerated. Other
feedback mechanisms that contribute to the injectivity are changes
in fault storage and permeability in time. In these simulations,
fault storage and permeability were functions of the effective
stress and cumulative shear slip, which may have influenced onset
of seismicity.
In the model, the fluid entering the fault was 150 °C colder
than the initial reservoir temperature. At most geothermal fields,
fluid is typically injected at the surface either at ambient
temperatures or at slightly elevated temperatures if water is being
recirculated after exiting a power station. As the fluid flows down
the well, conductive heat transfer towards the well acts to heat
the fluid up, which in some cases can be significant. In oil and
gas wastewater disposal settings, the reservoir temperatures are
likely to be much lower than geothermal reservoirs, and the
temperature contrast might be significantly smaller in magnitude.
We performed an additional simulation very similar to Case D, but
with a temperature contrast of 100 °C. In this case, no earthquake
events were recorded, which indicated that the thermal effect was
not strong enough to overcome the poroelastic back-stress over the
injection period. The injection rates and duration of injection
will also have a large impact on the evolution of reservoir
temperature.
It was observed that the simulations tended to produce a
relatively small number of seismic events. For Cases A, C, and D,
the earthquake magnitudes within each case were relatively
constant. In real cases of injection-triggered seismicity,
earthquake sequences with many events exhibiting a
Gutenberg-Richter-type frequency-magnitude distribution are
commonly recorded. We attribute this inconsistency to the fact that
we modeled a perfectly planar fault with homogeneous frictional
properties and a homogeneous initial state of stress. Real faults
have complex geometrical features. Previous theoretical and
numerical studies of the dynamic earthquake rupture process have
shown that fault roughness and bends in faults can significantly
impact nucleation, arrest, and slip distributions (Dunham et al.,
2011; Fang and Dunham, 2013). Stress measurements using wellbore
image logs often indicate that the state of stress deviates
slightly around some mean value (Zoback, 2007). In Case D, we
observed that the relatively complex distribution of effective
stress that arose due to the thermal stresses caused the earthquake
events to arrest prematurely at locations of increased compression.
In future studies, it would be helpful to incorporate some degree
of heterogeneity in the fault geometry, initial state of stress,
and the frictional properties in order to obtain more realistic
simulation results. This study would have also benefited greatly
from a sensitivity study on the rate-and-state friction
properties.
4. CONCLUDING REMARKS The main purpose of this study was
achieved in the sense that the suite of simulations showed that
several different physical processes can influence earthquake
behavior during fluid injection into the subsurface. This is most
readily seen when considering Cases A, B, and D together. Most
studies of injection-triggered seismicity have focused on the role
of pore pressurization in the fault zone as the main mechanism for
causing seismicity. The results from Case A can therefore be
considered as a “base case.” In Case A, four seismic events were
triggered solely by increased fluid pressure in the fault. In Case
B, the poroelastic back-stress caused a clamping behavior to occur,
which largely negated the decrease in effective stress due to fluid
pressure. In this case, only a small amount of slip occurred,
completely aseismically, and no earthquake events were observed. In
Case D, all of the physical mechanisms (fluid pressure, poroelastic
stress, and thermoelastic stress) were included in the effective
stress calculation. Reservoir cooling induced tensile stresses in a
localized region near the injection point, which had a
destabilizing effect, and seismicity was again observed. These
observations imply that it may not always be obvious which physical
mechanisms are directly responsible for triggering earthquakes, and
it may be naïve to assume that any single mechanism dominates.
The conceptual model, reservoir parameters, and injection
conditions used in this study represent only one sample of an
extremely large set of possible scenarios. The results of this
study verified the hypothesis that several competing mechanisms may
contribute to injection-triggered seismicity in opposing or
reinforcing ways, and suggest that a much broader parametric study
must be performed in order to gain a more thorough understanding of
the range of conditions over which each process dominates.
ACKNOLEDGEMENTS The authors would like to thank the Stanford
Center for Induced and Triggered Seismicity for providing the
financial support for this work.
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Norbeck and Horne
12
NOTATION
Parameter Description and Typical Unit Parameter Description and
Typical Unit
a Rate-and-state direct effect parameter, [-] V Slip velocity,
[m ⋅s-1 ]
b Rate-and-state evolution effect parameter, [-] V0
Rate-and-state initial slip velocity, [m ⋅s-1 ]
cfP
Compressibility of fluid, [MPa-1 ] α Biot’s coefficient, [-]
crP
Compressibility of matrix rock porosity, [MPa-1 ] β Linear
thermal expansion coefficient, [
C-1 ]
cfT
Specific heat capacity of fluid, [ J ⋅kg-1 ⋅o C-1 ] δ
Cumulative shear slip discontinuity, [m ]
crT
Specific heat capacity of rock, [ J ⋅kg-1 ⋅o C-1 ] δij Kronecker
delta function, [-]
d Rate-and-state characteristic length, [m ] εij Components of
the strain tensor, [-]
e0 Fault hydraulic aperture at reference state, [m ] η Radiation
damping parameter, [MPa ⋅m-1 ⋅s ]
E0 Fault void aperture at reference state, [m ] θ Orientation of
the fault relative to the x-axis, [ deg ]
f Coefficient of friction, [-] Θ Quasistatic elastic stress
transfer, [MPa ]
f0 Rate-and-state coefficient of friction at reference state,
[-]
κ f Thermal conductivity of fluid, [W ⋅m-1 ⋅o C-1 ]
G Shear modulus, [MPa ] κ r Thermal conductivity of rock, [W
⋅m-1 ⋅o C-1 ]
H Fault height, [m ] Λ Lame’s modulus, [MPa ]
k Matrix rock permeability, [m2 ] µ Fluid viscosity, [ Pa ⋅s
]
K Bulk modulus, [MPa ] ν Poisson’s ratio, [-]
L Fault length, [m ] ρ f ,0 Density of fluid at reference state,
[ kg ⋅m−3 ]
Lc Rate-and-state critical perturbation length, [m ] ρr Density
of rock, [ kg ⋅m−3 ]
mw Mass rate of injection well, [ kg ⋅s-1 ] σ e,ref Reference
effective stress for hydraulic aperture normal stiffness, [MPa
]
p Fluid pressure, [MPa ] σ E,ref Reference effective stress for
void aperture normal stiffness, [MPa ]
p0 Initial fluid pressure, [MPa ] σ ij Components of the stress
tensor, [MPa ]
S Fault cohesion, [MPa ] σ ijM Mechanical stress, [MPa ]
T Temperature, [ C ] σ ijP Poroelastic stress, [MPa ]
Tw Temperature of injection fluid, [ C ] σ ijR Remote tectonic
stress, [MPa ]
T0 Initial temperature, [ C ] σ ijT Thermoelastic stress, [MPa
]
ui Components of the displacement vector, [m ] σ n Effective
normal stress, [MPa ]
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13
σ nM Mechanically-induced normal stress, [MPa ] ϕe,dil Hydraulic
aperture dilation angle, [ deg ]
σ nP
Poroelastically-induced normal stress, [MPa ] ϕE,dil Void
aperture dilation angle, [ deg ]
σ nR
Remote normal stress, [MPa ] φ0 Matrix rock porosity at
reference state, [-]
σ nT
Thermoelastically-induced normal stress, [MPa ] ΦP
Poroelastic potential, [m2 ]
τ Shear stress, [MPa ] ΦT
Thermoelastic potential, [m2 ]
τ s Shear strength, [MPa ] Ψ State, [ s ]
τ 0 Initial shear stress, [MPa ] Ψ0 Initial state, [ s ]
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