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

jnorbeck@stanford.edu

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.

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,

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

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.

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.

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

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

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.

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)

Norbeck and Horne

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

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.

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 ]

Norbeck and Horne

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 ]

REFERENCES Baisch, S., Vörös, R., Rothert, E., Stang, H., Jung, R., and Schellschmidt, R. 2010. A numerical model for fluid injection induced

seismicity at Soultz-sous-Forêts. International Journal of Rock Mechanics and Mining Sciences, 47 (3), 405-413. doi:10.1016/j.ijrmms.2009.10.001.

Dempsey, D., Kelkar, S., Davatzes, N., Hickman, S., Moos, D., and Zemach, E. 2014. Evaluating the roles of thermoelastic and poroelastic stress changes in the Desert Peak EGS stimulation. Proc., Thirty-Ninth Workshop on Geothermal Reservoir Engineering, Stanford, California, USA.

Dunham, E.M., Belanger, D., Cong, L., and Kozdon, J.E. 2011. Earthquake ruptures with strongly rate-weakening friction and off-fault plasticity, Part 2: Nonplanar faults. Bulletin of the Seismological Society of America, 101 (5): 2308-2322. doi: 10.1785/0120100076.

Ellsworth, W.L. 2013. Injection-induced earthquakes. Science, 341 (July). doi: 10.1785/gssrl.83.2.250.

Fang, Z., and Dunham, E.M. 2013. Additional shear resistance from fault roughness and stress levels on geometrically complex faults. Journal of Geophysical Research: Solid Earth, 118: 3642-3654. doi: 10.1002/jgrb.50262.

Gibbs, J.F., Healy, J.H., Raleigh, C.B., and Coakley, J. 1973. Seismicity in the Rangely, Colorado, area: 1962-1970. Bulletin of the Seismological Society of America, 63 (5): 1557-1570.

Ghassemi, A., Nygren, A., and Cheng, A. 2008. Effects of heat extraction on fracture aperture: Aporothermoelastic analysis. Geothermics, 37 (5), 525-539. doi: 10.1016/j.geothermics.2008.06.001.

Häring, M.O., Schanz, U., Ladner, F., et al. 2008. Characterisation of the Basel 1 enhanced geothermal system. Geothermics, 37 (5): 469-495. doi: 10.1016/j.geothermics.2008.06.002.

Horton, S. 2012. Disposal of hydrofracking waste fluid by injection into subsurface aquifers triggers earthquake swarm in Central Arkansas with potential for damaging earthquake. Seismological Research Letters, 83 (2): 250-260. doi: 10.1785/gssrl.83.2.250.

Jaeger, J.C., Cook, N.G.W., and Zimmerman, R. 2007. Fundamentals of Rock Mechanics. Oxford: Blackwell Publishing Ltd., 4th ed.

Kim, J., Tchelepi, H., and Juanes, R. 2011. Stability, accuracy, and efficiency of sequential methods for coupled flow and geomechanics. SPE Journal (June), 249-262.

Li, L. and Lee, S.H. 2008. Efficient field-scale simulation of black oil in a naturally fractured reservoir through discrete fracture networks and homogenized media. SPE Reserv. Eval. Eng., (August): 750-758.

Maxwell, S., and Rutledge, J. 2013. Deformation investigations of induced seismicity during hydraulic fracturing. Paper SEG-2013-1415 presented at the SEG Annual Meeting, 22-27 September, Houston, Texas, USA.

McClure, M.W. and Horne, R.N. 2011. Investigation of injection-induced seismicity using a coupled fluid flow and rate/state friction model. Proc., Thirty-Sixth Workshop on Geothermal Reservoir Engineering, Stanford, California, USA.

McClure, M.W. 2012. Modeling and Characterization of Hydraulic Stimulation and Induced Seismicity in Geothermal and Shale Gas Reservoirs. PhD dissertation, Stanford University, Stanford, California, USA (December 2012).

McClure, M.W., and Horne, R.N. 2013. Discrete Fracture Network Modeling of Hydraulic Stimulation: Coupling Flow and Geomechanics. Springer. doi: 10.1007/978-3-319-00383-2.

Norbeck and Horne

14

Mossop, A.P. 2001. Seismicity, Subsidence and Strain at The Geysers Geothermal Field. PhD dissertation, Stanford University, Stanford, California, USA (March 2001).

Norbeck, J., Huang, H., Podgorney, R., and Horne, R. 2014. An integrated discrete fracture model for description of dynamic behavior in fractured reservoirs. Proc., Thirty-Ninth Workshop on Geothermal Reservoir Engineering, Stanford, California, USA.

Norbeck, J.H., and Horne, R.N. 2014. An embedded fracture modeling framework for coupled fluid flow, geomechanics, and fracture propagation. Proc., International Conference on Discrete Fracture Network Engineering, Vancouver, British Columbia, Canada.

Nowacki, W. 1986. Thermoelasticity. Oxford: Pergamon Press, 2nd Ed.

Segall, P. 2010. Earthquake and Volcano Deformation. Princeton, N.J.: Princeton University Press.

Shou, K.J., and Crouch, S.L. 1995. A higher order displacement discontinuity method for analysis of crack problems. Int. J. Rock Mech. Min. Sci. Geomech. Abstr., 32 (1), 49-55.

Swiss Seismologic Service (SED). 2013. Earthquake chronology geothermal energy project in St. Gallen. URL http://www.seismo.ethz.ch/sed/news/Chronologie SG/index EN (accessed 10 November, 2013).

van Poollen, H.K., and Hoover, D.B. 1970. Waste disposal and earthquakes at the Rocky Mountain Arsenal, Derby, Colorado. J. Pet. Technol. (August): 983-993. SPE-2558-PA. doi: 10.2118/2558-PA.

Zoback, M.D. 2007. Reservoir Geomechanics. Cambridge: Cambridge University Press.