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Tectonophysics

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Maximum magnitude of injection-induced earthquakes: A criterion to assessthe influence of pressure migration along faults

Jack H. Norbeck*,1, Roland N. HorneDepartment of Energy Resources Engineering, Stanford University, Stanford, CA 94305, USA

A R T I C L E I N F O

Keywords:Induced seismicityEarthquake mechanicsSeismic hazardRate and state friction

A B S T R A C T

The maximum expected earthquake magnitude is an important parameter in seismic hazard and risk analysisbecause of its strong influence on ground motion. In the context of injection-induced seismicity, the processesthat control how large an earthquake will grow may be influenced by operational factors under engineeringcontrol as well as natural tectonic factors. Determining the relative influence of these effects on maximummagnitude will impact the design and implementation of induced seismicity management strategies. In thiswork, we apply a numerical model that considers the coupled interactions of fluid flow in faulted porous mediaand quasidynamic elasticity to investigate the earthquake nucleation, rupture, and arrest processes for cases ofinduced seismicity. We find that under certain conditions, earthquake ruptures are confined to a pressurizedregion along the fault with a length-scale that is set by injection operations. However, earthquakes are some-times able to propagate as sustained ruptures outside of the zone that experienced a pressure perturbation. Wepropose a faulting criterion that depends primarily on the state of stress and the earthquake stress drop tocharacterize the transition between pressure-constrained and runaway rupture behavior.

1. Introduction

Disposal of wastewater associated with oil and gas operations byinjection into the subsurface is a common practice in the petroleumindustry. Changes in the state of stress at depth caused by fluid injectionhave reportedly generated significant levels of seismic activity nearUnderground Injection Control (UIC) class-II wells in several instances(Barbour et al., 2017; Frohlich, 2012; Frohlich et al., 2014, 2011; Healyet al., 1968; Hornbach et al., 2015; Horton, 2012; Hsieh andBredehoeft, 1981; Keranen et al., 2013; Kim, 2013; Rubinstein et al.,2014; Walsh and Zoback, 2015). In order to determine the seismichazard for a site, it is important to estimate parameters in probabilisticseismic hazard assessment models, such as the maximum expectedearthquake magnitude and the occurrence rate of a given-magnitudeearthquake (Ellsworth et al., 2015). Understanding how the interactionbetween injection well operational parameters and natural geologicsetting affects the behavior of induced earthquakes is difficult toquantify and has, so far, remained unresolved (Ellsworth, 2013; McGarret al., 2015).

Apart from ground motion estimates, the most influential para-meters in earthquake hazard analysis are the seismicity rate, theGutenberg-Richter (GR) frequency-magnitude scaling factor, and the

maximum earthquake magnitude (Petersen et al., 2014). If theseearthquake statistics can be quantified accurately, then the data can becombined to develop a probabilistic estimate of earthquake hazard for aparticular area. van der Elst et al. (2016) found that the maximummagnitude earthquakes observed in 21 separate cases of injection-in-duced seismicity were each as large as expected statistically based onthe local earthquake catalogs. Characterization of the hydromechanicalreservoir response to fluid injection must therefore be cast in terms ofunderstanding how these types of earthquake statistics can be expectedto change due to injection operations (Dempsey et al., 2016; Llenos andMichael, 2013).

Wastewater injection wells target injection horizons within natu-rally permeable brine aquifers, which are usually composed of sedi-mentary rocks. In most cases where relatively large earthquakes havebeen attributed to fluid injection, the earthquake hypocenters havebeen located beneath the target aquifers along faults that exist withinigneous basement rocks (Horton, 2012; Kim, 2013; Keranen et al.,2014; Hornbach et al., 2015). It has been suggested previously thatbasement faults may sometimes extend into overlying formations,providing a necessary hydraulic connection for pressure communica-tion (Ellsworth, 2013; Göbel, 2015; Göbel et al., 2016; Hornbach et al.,2015; McGarr, 2014). If the fluid pressure within a fault zone increases

https://doi.org/10.1016/j.tecto.2018.01.028Received 25 August 2017; Received in revised form 6 January 2018; Accepted 23 January 2018

* Corresponding author.

1 Now at Earthquake Science Center, U.S. Geological Survey, Menlo Park, CA 94025, USA.E-mail address: jnorbeck@stanford.edu (J.H. Norbeck).

Tectonophysics 733 (2018) 108–118

Available online 23 February 20180040-1951/ © 2018 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).

T

due to injection, the effective normal compressive stresses that provideresistance to shear slip are reduced, thereby bringing the state of stresson the fault closer to failure conditions (Ellsworth, 2013; Jaeger et al.,2007; Raleigh et al., 1976).

McGarr (2014) used an analytical poroelastic reservoir model andan assumption about the frequency-magnitude scaling of earthquakesequences to develop an expression for a theoretical upper bound onearthquake magnitude that was related linearly to the cumulative vo-lume of injected fluid. An implicit assumption was made that earth-quakes must be confined to regions that experience pressure change. Inthat study, it was concluded that data collected from 18 different casestudies of injection-induced seismicity supported the proposed re-lationship between maximum magnitude and injection volume. In thisperspective, the size of an earthquake is related closely to the injectionoperations. Göbel (2015) presented a comparison between inducedseismicity in Oklahoma and California based on regional-scale statisticsof earthquakes, injection rates, and injection pressures. In that study, itwas concluded that differences in the geologic setting likely played theprimary role in how injection-triggered seismicity has evolved in thosestudy areas over the past two decades, and the influence of injectionwell operations was of secondary importance. In this work, we exploredthe relationship between fluid injection, flow through porous media,and earthquake rupture along faults through numerical modeling ex-periments.

2. Faulting criterion

In reservoir engineering, the “distance of investigation ” has beenused to describe the location in the reservoir where pressure haschanged by a prescribed magnitude and is often interpreted as a pres-sure front (Horne, 1995). It is intuitive to understand that the likelihoodof interacting with hydraulic connections to basement faults increasesas injection continues and the pressure front migrates further from thewell. However, does this length-scale set a bound on the dimension ofan earthquake rupture and, ultimately, the earthquake magnitude?

We addressed this question by performing numerical simulationsthat modeled the coupled interactions between fluid flow in porousmedia, fluid flow in faults, and earthquake rupture physics (McClureand Horne, 2011; Norbeck and Horne, 2016). We modeled a scenariowhere fluid was injected into a permeable aquifer overlying imperme-able basement rock. A strike-slip fault zone in the vicinity of the wellwas located mostly within the basement rock, but a portion of the faultextended into the aquifer (see Fig. 1). The reservoir and fault geometryin our conceptual model were designed to be consistent with severalrecent instances of induced seismicity (Hornbach et al., 2015; Horton,2012; Keranen et al., 2014; Kim, 2013). In contrast to previous studies,for example see McGarr (2014) and Barbour et al. (2017), we modeledthe hydraulic interaction between the aquifer and the fault explicitly

and considered a rigorous treatment of the earthquake rupture processwithin the framework of rate-and-state friction theory.

We propose classifying faulting behavior into two separate cate-gories:

• Pressure-constrained ruptures (Type A) are limited by the extent of thepressure perturbation along the fault.

• Runaway ruptures (Type B) are controlled by traditional tectonicfactors such as fault geometry or stress heterogeneity.

This is a useful distinction because pressure-constrained behaviormight be considered more stable. For example, the maximum earth-quake magnitude might be expected to grow over time in a systematicmanner as larger patches of the fault are exposed to significant pressurechanges. For runaway rupture behavior, although fluid injection mayultimately be responsible for causing earthquakes to nucleate, the fac-tors controlling how large an earthquake will grow might depend moreclosely on characteristics of the natural geology, such as the size of thefault, geometric complexity, and stress heterogeneity.

We simulated sequences of injection-induced earthquake rupturesand found that the following faulting criterion, C, can be used to assessthe conditions that separate the two categories of behavior:

=Cff

,D

0

(1)

where fD is the dynamic friction coefficient and =f τ σ/0 0 0 is the ratio ofshear stress, τ0, to effective normal stress, σ0, acting on the fault beforeinjection begins (i.e., the prestress ratio). For C<1, pressure-con-strained behavior occurs within the pressure influenced region, and forC>1, runaway rupture behavior occurs. This faulting criterion de-scribes a subset of the transitional faulting behaviors investigated andquantified by Garagash and Germanovich (2012). As is discussed inSection 5.2, Eq. (1) can also be derived from an earthquake energybalance.

The parameter fD can be estimated from rate-and-state friction la-boratory experiments (Blanpied et al., 1991, 1995; Dieterich, 1992) orthrough controlled field experiments (Guglielmi et al., 2015). Theparameter f0 embodies the initial state of stress, the initial fluid pres-sure, and the orientation of the fault. In practical applications, theremay be considerable uncertainty in the state of stress and frictionalproperties of real faults. However, in the numerical experiments weperformed where the model properties were known with certainty, wefound that the value of the faulting criterion in Eq. (1) was good in-dicator of whether earthquake ruptures would arrest within the pres-sure-perturbed region or propagate in a sustained manner beyond thepressure front.

3. Hydromechanical coupling with a rate-and-state friction model

We performed our numerical experiments with a reservoir modelingsoftware called CFRAC (McClure and Horne, 2011, 2013; Norbeck,2016; Norbeck et al., 2016). The simulations involved a coupling be-tween fluid flow in an aquifer, fluid flow along a fault, and quasidy-namic earthquake rupture. Mass transfer between the aquifer domainand the fault was calculated using an embedded fracture modelingapproach (Karvounis and Jenny, 2016; Li and Lee, 2008; Norbeck et al.,2016; Ţene et al., 2017). Earthquake rupture, propagation, and arrestwere considered within the context of a rate-and-state friction con-stitutive framework.

3.1. Fluid flow in faulted porous media

In the embedded fracture modeling framework, the mass con-servation equations for the aquifer and fault domains are expressedseparately which allows for flexibility in the discretization strategy

Fig. 1. Conceptual reservoir model used to design the numerical modeling experiments. Apermeable basement fault extended slightly into a saline aquifer, allowing for pressurecommunication during fluid injection.

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(Norbeck et al., 2016). For a porous medium saturated with single-phase fluid, the mass balance equations can be written, for flow in theaquifer domain, as:

∇⋅ ⋅∇ + + = ∂∂

∼∼ρλk p mt

ρϕ( ) Ω ( ),a a wa fa a(2)

and, for flow in the fault domain, as:

∇⋅ ⋅∇ + + = ∂∂

∼∼ρλek p mt

ρE( ) Ω ( ).f f wf af

(3)

Here, p is the fluid pressure, ρ is the fluid density, λ is the inverse offluid viscosity, k is the permeability, ϕ is the porosity, e is the faulthydraulic aperture, E is the fault void aperture, and ∼m is a normalizedmass source term related to wells. The superscripts a, f, and w indicateproperties related to the aquifer, fault, and well domains, respectively.In addition to the usual terms related to flux, wells, and storage, theterms ∼Ω fa and ∼Ωaf are introduced to account for mass transfer betweenthe two domains. Upon integration over their respective control vo-lumes, these mass transfer terms take the following form (Hajibeygiet al., 2011):

= −p pΩ ϒ( ),fa f a (4)

where the parameter ϒ is a transmissibility called the fracture indexand is analogous to the Peaceman well index (Peaceman, 1978). Fluiddensity and matrix rock porosity are assumed to be slightly compres-sible and are calculated as ρ= ρ*exp [cw (p− p*ρ)] and

= −ϕ ϕ c p p*

exp [ ( * )]a aϕ ϕ . The fault is able to deform elastically, and the

hydraulic properties depend on the effective normal stress and shearslip as described in Section 3.3.

3.2. Quasidynamic earthquake rupture

Under the assumptions of quasidynamic linear elasticity, mechan-ical equilibrium along a fault can be described as (Ben-Zion and Rice,1997):

+ − = +τ τ ηV fσ sΔ ,0 (5)

where τ= τ0+ Δτ− ηV is the shear stress acting on the fault,= + −σ σ σ pΔ f

0 is the effective normal stress acting on the fault, f isthe fault friction coefficient, V= dδ/dt is the sliding velocity along thefault, Δτ and Δσ represent the quasistatic stress transfer caused by slipon the fault, η is the inertial damping parameter, and s is fault cohesion.In Eq. (5), a Mohr-Coulomb failure criterion is assumed. We modeled asingle planar fault, so slip on the fault did not affect the normal stress.In the case of a fault located along a bimaterial interface, it may bepossible that dynamic effects leading to a reduction in normal stressduring rupture propagation may promote sustained rupture even forcases where C<1 (Andrews and Ben-Zion, 1997). Bimaterial dynamicstress effects are not likely to be significant for injection-inducedearthquakes that occur along faults in basement rock, therefore weneglected the bimaterial effect in our simulations.

The friction coefficient is assumed to evolve according to a rate-and-state constitutive formulation (Dieterich, 1992; Rice et al., 2001):

= +f V a VV

( , Ψ) ln*

Ψ,(6)

where Ψ is the state variable. In this study, we used the aging law todescribe state evolution (Rojas et al., 2009):

∂∂

= − ⎧⎨⎩

− ⎡⎣⎢

−− ⎤

⎦⎥⎫⎬⎭t

bVδ

f V f Vb

Ψ 1 exp( , Ψ) ( )

,c

ss

(7)

where the steady-state friction coefficient is defined as:

= − −f V f b a VV

( ) * ( ) ln*

.ss (8)

In Eqs. (6) through (8), a is the direct-effect velocity-strengthening

parameter, b controls the magnitude of the state evolution effect, δc isthe characteristic slip distance over which friction evolves, and f* and V* are reference values that are introduced to define the steady-statefriction curve. The dynamic friction coefficient can be approximated asthe steady-state friction coefficient while sliding at seismic slip speeds(i.e., fD ≈ fss (V max)). Similarly, the static friction coefficient can beapproximated as the steady-state friction coefficient while sliding atlow slip speeds during the interseismic phase of the earthquake cycle(i.e., fS ≈ fss (V min)). In Eqs. (6) through (8), strongly rate-weakeningeffects (Di Toro et al., 2011; Nielsen, 2017; Spagnuolo et al., 2016) aswell as the frictional response to variable normal stress (Linker andDieterich, 1992) were neglected.

We considered a purely mode-II shear problem and assumed planestrain along the vertical dimension of the fault (i.e., the ruptures wereeffectively one-dimensional). Under these assumptions, fault properties(pf, V,Ψ, τ, σ ) are constant along the vertical dimension of the fault, butcan vary along the length of the fault. The seismic moment can becalculated as:

∫ ∫= =M μ δ dA μH δ x dxΔ Δ ( ) ,A

fL

00

f

(9)

where Δδ is the shear slip accumulated during an individual earthquakerupture, μ is the shear modulus, Hf is the height of the fault, Lf is thelength of the fault, and A is the surface area of the fault.

3.3. Stress and slip-dependent fault hydraulic properties

It is well understood that fault zone architecture can affect theability for faults to transmit fluids (Caine et al., 1996). We were in-terested in understanding how flow within the injection aquifer travelsalong the fault, therefore in the context of the classifications describedby Caine et al. (1996) our model fault can be interpreted as either adistributed or localized conduit that represents a fractured damagezone. In the fluid mass balance equation for flow along faults, Eq. (3),the flux term depends on the hydraulic aperture of the fault, e, and thestorage term depends on the void aperture, E. In our model, the faulttransmissivity and permeability were related to the hydraulic apertureof the set of fractures that make up the fault damage zone through thefollowing relationship:

= =T ek e12

.f f3

(10)

The fault porosity is the total fracture void volume relative to thephysical width of the fault damage zone:

=ϕ EW

.ff (11)

The fault is able to deform elastically which can influence its hy-draulic properties significantly. Following the models proposed byBarton et al. (1985) and Willis-Richards et al. (1996), we adopted thefollowing constitutive laws to relate the mechanical deformation of thefault to changes in hydraulic properties:

=+

++

e σ δ e δφ

( , ) *1 9

tan1 9

,

* *σ

σ

eσ

σe e1 2 (12)

=+

++

E σ δ E δφ

( , ) *1 9

tan1 9

.

* *σ

σ

Eσ

σE E1 2 (13)

The first term represents a nonlinear response to changes in effectivestress, and the second term accounts for shear-enhanced dilation thatoccurs as the fault slips. In Eqs. (12) and (13), e*, E*, σ*e1, σ*e2, σ*E1, andσ*E2 are constants that define the fault stiffness, while φe and φE areshear dilation angles. The effect of shear dilation during the earthquakerupture can cause transient pressure changes that occur relativelyquickly on the order of the rupture duration (McClure and Horne,

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2011).

3.4. Numerical discretization and solution strategy

Eqs. (2) and (3) were discretized using a forward-Euler finite vo-lume method. We used an embedded fracture modeling strategy tocouple flow in the fault and flow in the aquifer, so the aquifer dis-cretization did not conform to the fault discretization (Norbeck et al.,2016). A displacement discontinuity method was used to calculate thestatic stress change caused by slip on the fault, Δτ (Shou and Crouch,1995), and the hierarchical matrices algorithm described by Bradley(2014) was implemented to perform the slip-stress computations effi-ciently. During earthquake rupture, state and sliding velocity canchange rapidly causing fault strength to evolve in a highly nonlinearfashion. Therefore, to solve for sliding velocity, state, and stress in therate-and-state framework, we used an explicit third-order Runge-Kuttamethod (see Section 2.4.2 in Norbeck, 2016). To couple the fluid flowand fault mechanics governing equations (see Eqs. (2), (3), and (5)), weused a sequential coupling strategy in which the fluid flow equationswere solved first to obtain pa and pf, followed by the fault mechanicsequation to obtain V, Ψ, and Δτ. The coupling between flow and me-chanics arises because fluid pressure influences frictional strength (seeEq. (5)) and both slip and effective stress influence the fault's hydraulicproperties (see Eqs. (10) and (12)). The numerical solution strategy isdescribed in detail in Chapter 2 of Norbeck (2016).

4. Description of numerical experiments

We explored a wastewater disposal setting in which a basementfault was connected hydraulically to an overlying aquifer. In this study,it was essential to model the interaction between flow in the aquifer andflow in the fault realistically. Faults that are able to host inducedearthquakes of significant magnitude are not likely to exist entirelywithin the target injection aquifer, are on the scale of tens of kilometerslong, and have finite transmissivity. In our simulations, the transientnature of pressure diffusion along the fault influenced the earthquakenucleation, rupture, and arrest processes significantly.

We modeled three separate cases to test conditions spanning a broadrange of C values:

• Case 1: varied orientation of the fault

• Case 2: varied magnitude of the least principal stress

• Case 3: varied b in the rate-and-state model to influence dynamicfriction

In each case, 1 km, 5 km, and 10 km faults were considered. Tables 1through 4 list values of the model properties used in the simulations.Tables S1 through S3 in the Supporting Information provide detailedinformation on the stress and frictional conditions used to estimate Cfor Cases 1 through 3.

4.1. Model geometry and physical properties

The conceptual reservoir model used in this study was motivated byrecent case studies of injection-induced seismicity (Barbour et al., 2017;

Horton, 2012; Kim, 2013) and is illustrated schematically in Fig. 1. Inthe model, water was injected into a 4 km deep saline aquifer at aconstant rate of 10 kg/s (roughly 165,000 bbl/month) over a period ofthree years. This is representative of a relatively high-rate wastewaterdisposal well for the state of Oklahoma (Walsh and Zoback, 2015;Weingarten et al., 2015). Flow was two-dimensional in the aquifer andone-dimensional along the length of the fault. The state of stress andfrictional properties were homogeneous.

The aquifer was modeled as a wide channel. The aquifer was 100mthick, 1 km wide, and 25 km long. This situation could represent ascenario in which the injection well is bounded on two sides by im-permeable geologic structures or perhaps the effects of injection fromadjacent wells. The edge of the basement fault was 500m away fromthe injection well in the center of the channel aquifer. The orientationof the aquifer paralleled the strike of the fault. The fault was in directcontact with the aquifer for 500m, and was assumed to be surroundedcompletely by impermeable basement rock for the remaining extent ofthe fault. The aquifer-fault mass transfer terms (see Eq. (4)) were cal-culated for elements within the first 500m of the fault, assuming thatthe entire surface area of each fault element was embedded in theaquifer domain. The initial fluid pressure in the aquifer and fault wereboth p0= 40MPa, approximately equal to hydrostatic pressure at adepth of 4 km.

The permeability of the aquifer was ka=10-14 m2 (10md) and theporosity of the aquifer was ϕa=0.1. The aquifer overlaid impermeablebasement rock. The permeability of the fault was kf=10-12 m2. Rate-and-state friction properties used in the model were consistent withvalues obtained from laboratory experiments on granitic rocks(Blanpied et al., 1991, 1995). Additional important model parametersare listed in Tables 1 through 4. Despite our choice of a quite particularmodel geometry, our results are representative of other flow regimes

Table 1Model geometry.

Parameter Value Unit Description

Lf 1 or 5 or 10 km Fault lengthHf 50 m Fault heightWf 1 m Width of fault damage zoneLa 25 km Aquifer lengthHa 100 m Aquifer heightWa 1 km Aquifer width

Table 2Aquifer and fault hydraulic properties.

Parameter Value Unit Description

kf 1× 10-12 m2 Fault permeability

ϕ*f 0.01 – Porosity of fault damage zone

cϕf 0 Pa-1 Fault pore compressibility

ka 1× 10-14 m2 Aquifer permeabilityϕ*a 0.1 – Aquifer porosity

cϕa 4.4×10-10 Pa-1 Aquifer pore compressibility

p*ϕ 40 MPa Porosity reference pressure

Table 3Fluid properties.

Parameter Value Unit Description

λ-1 0.7× 10-3 Pa⋅s Water viscosityρ* 1000 kg⋅m-3 Reference water densitycw 4.4× 10-10 Pa-1 Water compressibilityp*ρ 0.1013 MPa Density reference pressure

Table 4Rate-and-state friction and elastic properties.

Parameter Value Unit Description

a 0.0100 – Direct-effect parameterb 0.0110 to 0.0175 – State evolution parameterδc 50×10-6 m Slip-weakening distancef* 0.6 – Reference friction coefficientV * 1× 10-9 m⋅s-1 Reference sliding velocityμ 30 GPa Shear modulusν 0.25 – Poisson's ratioη 3.15 MPa⋅s⋅m-1 Radiation damping parameter

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(e.g., two-dimensional or three-dimensional radial flow), which willonly affect the shape of the pressure profile along the fault and not therupture propagation or arrest processes on which the faulting criterionis based.

5. Results

Here, we present the results of our numerical experiments. InSection 5.1, we demonstrate the differences between pressure-con-strained and runaway rupture behavior. In Section 5.2, we elaborate onthe faulting criterion defined in Eq. (1) and show that it can also beused to define the approximate location for rupture arrest in the case ofpressure-constrained rupture behavior. In Section 5.3, we present theresults of a parametric study that validated our proposed faulting cri-terion for a broad range of conditions related to state of stress, faultgeometry, and fault frictional properties. Finally, the results of an ad-ditional simulation in which shear dilation effects affected the rupturepropagation process are presented in Section 5.4.

5.1. Injection-induced earthquake rupture behavior

Subsurface fluid injection can cause a change in reservoir pressure,Δp, which can be transmitted within a fault zone if there is a hydraulicconnection between a well and a fault. Assuming a Mohr-Coulombshear failure criterion, a fault can be expected to begin to fail if thefrictional resistance to slip is reduced to the level of shear stress actingon the fault, i.e., − =f σ p τ( Δ )S 0 0, where fS is the static friction coef-ficient of the fault (Jaeger et al., 2007). This failure criterion impliesthat there is a critical pressure perturbation that is required to initiateslip on a fault, Δpc,S (Garagash and Germanovich, 2012):

= −p σ τf

Δ .c SS

, 00

(14)

In rate-and-state friction theory, earthquake nucleation occurs whenshear stress exceeds the static strength over a sufficiently large coherentpatch of the fault that can be estimated as = − −L γμδ σ b a ν/[ ( )(1 )]c c ,where γ is a dimensionless prefactor that depends on geometrical effects(Ampuero and Rubin, 2008; Rice, 1980; Rice and Ruina, 1983; Riceet al., 2001). In the context of injection-induced seismicity, this requiresthat a pressure perturbation of at least Δpc,S must diffuse over thiscritical length-scale of the fault before an earthquake can nucleate,otherwise slip will be purely aseismic. Aseismic behavior over the entirethree-year injection duration was a common outcome in many scenarioswe modeled. This depended mostly on the initial proximity to failure ofthe fault.

For scenarios in which seismicity did occur, two general patterns ofearthquake rupture behavior emerged: a) earthquake ruptures thatwere confined within pressurized regions of the fault (Type A), and b)earthquake ruptures that propagated beyond the pressure front andwere limited by the size of the fault (Type B). Here, we provide ex-amples of model results to illustrate each type of behavior.

Figs. 2 and 3 show profiles of fault properties during typical pres-sure-constrained and runaway rupture events, respectively, on 5 kmlong faults. These two faults had the same orientation and stress state,but had different dynamic friction values. The earthquake events oc-curred at similar times (285 and 310 days after injection began, re-spectively). The pressure and effective stress distributions developedbecause the faults were relatively large and had finite permeability andstorativity. The location along the fault at which the pressure changedby 0.1MPa was taken as the pressure front. The pressure distributionwas effectively constant during each earthquake event because theearthquake ruptures occurred very quickly relative to pressure diffusiontime scales and fault permeability was assumed to be constant. In be-tween subsequent events, the pressure front migrated along the fault.

In Figs. 2 and 3, the earthquake rupture front can be identified asthe location along the fault where slip velocity, shear stress, and friction

are at their maximum. Ahead of the rupture front, the cumulative shearslip is zero. During an earthquake, the value of friction behind therupture front represents the dynamic friction, fD, and the value wellahead of the rupture front represents f0. The relative magnitude of thesetwo values is described by the faulting criterion in Eq. (1), and was themost influential factor governing earthquake behavior in these experi-ments.

For the pressure-constrained earthquake event, shown in Fig. 2, therupture died out before reaching the pressure front so that only a smallincremental accumulation of shear slip occurred during the earthquake.The magnitude of this earthquake was calculated as Mw=2.8. Therunaway rupture event shown in Fig. 3 displayed markedly differentbehavior. Friction behind the rupture front was below f0 which enableda stress drop to occur outside of the pressurized region, providing thenecessary energy to drive the rupture a significant distance beyond thepressure front. Slip was able to accumulate over the entire fault surfacearea which produced a correspondingly large seismic moment release.The magnitude of this earthquake was calculated as Mw=4.0.

5.2. Critical pressure perturbation for sustained rupture

Although the faulting criterion proposed in Eq. (1) was based onobservations from numerical experiments, it can also be interpretedwithin the theory of earthquake rupture dynamics as a limiting case ofan earthquake energy balance. The rupture and arrest processes aregoverned by a competition between fracture energy, Γ, and energy re-lease rate, G (Ampuero and Rubin, 2008; Rice, 1980). In the limit thatΓ→ 0, then G>0 will cause instability leading to earthquake rupture.The energy release rate scales with the stress intensity factor, K, as G ∼K2 (Rice, 1980). In turn, K ∼ΔτD depends on the stress drop behind therupture front (Rice, 1980):

= − +τ τ f σ f pΔ Δ .D D D0 0 (15)

In the case that − <τ f σ( ) 0D0 0 (or, equivalently, C<1), it is possiblefor a stress drop to occur only over the region that has been pressurized,so the rupture will be constrained by the pressure front.

If we relax Eq. (1) and now allow C to be a function of fluid pressureduring injection, then Eq. (15) can be used to determine the pressurechange at the transition point along the fault where C=1. This criticalpressure perturbation, Δpc,D, depends on the dynamic friction coeffi-cient:

= −p σ τf

Δ .c D oD

,0

(16)

For faults that exhibit pressure-constrained behavior, Δpc,D can be usedto define a nonarbitrary pressure front that represents approximatelythe distance at which earthquake ruptures will arrest. Portions of thefault that have experienced a pressure change of at least Δpc,D are ableto host sustained earthquake ruptures. The apparent similarity betweenEqs. (14) and (16) is encouraging, because it suggests that Eq. (16) canbe applied in a manner analogous to the Mohr-Coulomb failure cri-terion as a method for estimating the maximum extent of inducedearthquake ruptures.

We demonstrate this principle in Fig. 4. The location along the faultat which the faulting criterion transitions across C=1 is now taken asthe critical pressure front. In this case, this critical pressure front cor-responded with the rupture arrest location quite well. In general, theextent of the critical pressure front defined using Eq. (16) represents theapproximate location of rupture arrest.

5.3. Parametric study

We performed three sets of numerical experiments to assess thevalidity of the proposed faulting criterion and to isolate the effect ofdifferent parameters that influence the value of C. In Case 1, the effectof fault orientation was investigated. In Case 2, the principal stress ratio

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was varied. In Case 3, dynamic friction of the fault was varied. The stateof stress, fault orientation, and frictional properties of the fault for eachscenario are illustrated as Mohr circle representations in Fig. 5 and arealso summarized in Tables S1 through S3 in the Supporting

Information. For each case, we examined three different fault sizes:1 km, 5 km, and 10 km long.

In all scenarios, a sequence of earthquakes developed along the faultover the three year duration of injection. We compared different

Fig. 2. Earthquake rupture profiles during a typical pressure-constrained rupture event (Type A). The location of the pressurefront is indicated by the blue dashed line. The earthquake rupturewas confined to the pressurized region. The conditions for thisscenario are listed as Simulation 3-6 in Table S1. (For inter-pretation of the references to color in this figure legend, thereader is referred to the web version of this article.)

Fig. 3. Earthquake rupture profiles during a typical runawayrupture event (Type B). The location of the pressure front isindicated by the blue dashed line. The earthquake rupture pro-pagated beyond the pressure front and ultimately arrested afterreaching the fault boundary. The conditions for this scenario arelisted as Simulation 3-10 in Table S1. (For interpretation of thereferences to color in this figure legend, the reader is referred tothe web version of this article.)

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scenarios by normalizing seismic moment, M0, by the cumulative vo-lume of fluid injected up until the earthquake was induced, Q. We usedQ as a proxy for the distance of the pressure front because it is a tan-gible operational parameter. We are not attempting to demonstrate adirect relationship between M0 and Q. In Fig. 5, we show M0/Q fordifferent faults. The data presented represent the average value ofM0/Qfor earthquake sequences over the three year injection duration.

The results from the Case 1 experiments, shown in Fig. 5 (b), de-monstrate the marked contrast in behavior depending on the value ofthe faulting criterion, C, calculated using Eq. (1). For faults with C<1,the normalized seismic moment did not depend on the size of the faultbecause the dimension of the earthquake rupture was controlled by thepressure front. The earthquake magnitude of subsequent earthquakesincreased as the pressure perturbed larger zones of the fault. For faultswith C>1, the normalized seismic moment tended to be one to threeorders of magnitude larger and was dependent predominantly on thesize of the fault. In these types of earthquakes, once an event nucleatedit was able to propagate in a sustained manner far beyond the pressurefront, which resulted in relatively large earthquakes.

The results from the Case 2 and Case 3 experiments, shown in Fig. 5(d) and (f), further demonstrate the two distinct earthquake behaviorsover broader ranges of parameter space. It was observed that thetransition between pressure-constrained and runaway rupture behaviordid not occur strictly at C=1. For faults with C slightly > 1, pressure-constrained behavior tended to occur. In the models, the fault had afinite fracture energy. In the energy balance referenced in the discus-sion of Eq. (15), the effect of a finite fracture energy is to require G> Γ ,which explains why runaway rupture behavior required C slightly > 1in our numerical experiments.

5.4. Shear-enhanced dilation

In the simulations discussed previously, the fault hydraulic prop-erties were held constant to isolate the effect of the pressure profilealong the fault on the rupture propagation behavior. Under thoseconditions, the timescales for fluid pressure transients were much largerthan the speed of rupture propagation so there was effectively nocoupling between fluid flow and the earthquake rupture process. Here,we present the results of a simulation in which the effects of shear-enhanced dilation were considered. As opposed to the cases with

constant hydraulic properties, in this case the fluid pressure distributionalong the fault changed as the rupture propagated, slip occurred, andthe fault dilated. This hydromechanical coupling affected the rupturebehavior significantly.

In Eqs. (12) and (13), the fault hydraulic and void apertures dependon effective stress and shear slip. The constitutive properties used inthis simulation were similar to previous studies of induced seismicityand are provided in Table 5 (McClure and Horne, 2011; Norbeck andHorne, 2016). The reference hydraulic properties were chosen such thatthe permeability and porosity under the initial loading conditions wereequal to the fault properties used in the prior simulations. In this si-mulation, a constant injection pressure of 50MPa was specified at theleft boundary of the fault. All other model parameters were identical toSimulation 3-10 (see Table S1), which was an example of conditionsthat promoted runaway rupture (i.e., compare these results to Fig. 3).

The time evolution of various fault properties during a typicalrupture are shown in Fig. 6. This rupture was the second earthquake tooccur in the sequence, so both the cumulative slip from all ruptures, δ,and the additional slip accumulated during the rupture of interest, Δδ,are shown. Note that the pressure distribution before dynamic rupturenucleated shows a concave-down shape due to the strong contrast intransmissivity between the previously slipped patch and the unrupturedpatch, similar to the behavior observed at the Basel geothermal field(Mukuhira et al., 2017). The rupture began to nucleate near x=250mwithin the concentration of increased shear stress that resulted from theprior rupture. Slip speeds began to increase along the previously rup-tured patch where fluid pressures were largest, and once the criticalnucleation length of Lc was met the earthquake transitioned into adynamic rupture.

As the rupture nucleated and began to propagate, the accumulationof shear slip caused the fault to dilate, increasing the fault's pore vo-lume. In response, the fluid pressure along the slipping portion of thefault dropped nearly instantaneously which had the effect of strength-ening the slipping patch. A competition between the two terms in Eq.(13) caused a complex distribution of void aperture change to developas the rupture progressed. Ultimately, the rupture arrested after pro-pagating only a few tens of meters. As injection continued in betweenruptures, we observed several instances in which shear dilation en-couraged aseismic slip to occur by preventing nucleation from transi-tioning toward instability. Evidently, the shear dilation behavior was

Fig. 4. The faulting criterion (Eq. (1)) wasapplied to identify the maximum extent ofrupture propagation for pressure-con-strained ruptures. The critical pressure frontwhere pressure changed by at least Δpc,D isrepresented by the dashed blue line. Alter-natively, the critical pressure front locationcan be identified as the point along the faultwhere the faulting criterion transitionedacross C=1. This location corresponded tothe rupture arrest location. The conditionsfor this scenario are listed as Simulation 3-6in Table S1. (For interpretation of the re-ferences to color in this figure legend, thereader is referred to the web version of thisarticle.)

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strong enough in this particular case to prevent runaway ruptures fromoccurring under conditions that otherwise would have favored them. Itshould be emphasized that shear dilation effects may not always bestrong enough to prevent runaway rupture behavior.

6. Discussion

Many of the largest injection-induced earthquakes have occurredalong faults in basement rock beneath the target injection aquifers(Barbour et al., 2017; Hornbach et al., 2015; Horton, 2012; Keranenet al., 2014; Kim, 2013). Assessing the manner in which basement faultsare likely to respond to pressure perturbations caused by fluid injectionis important for developing management strategies and to characterizehazard for wastewater disposal wells. In general, seismicity is not aninherent outcome as a response to fluid injection, as evidenced by thevast number of currently operating UIC class-II wells that have not beenassociated with seismicity (Ellsworth, 2013). The results from the nu-merical experiments performed in this study indicate that when fluidinjection does induce seismicity, there exist two distinct types offaulting behavior that may emerge: a) earthquake ruptures that areconfined to the pressurized region of the fault and b) sustained earth-quake ruptures that propagate beyond the pressure front. Throughnumerical simulations, we observed that Eq. (1) can be used effectively

Fig. 5. (left) Mohr circle representations of the state ofstress (black semicircles), fault orientation (black dots), andfriction coefficients (colored dashed lines) for scenarioswith variable C values. (right) Seismic moment, M0, nor-malized by cumulative volume of fluid injected, Q. Theparameter M0/Q did not depend on fault size for C<1because the earthquake ruptures were limited by thepressure front (Type A behavior). In contrast, M0/Q de-pended strongly on fault size for C>1 because the rup-tures propagated beyond the pressure front and arrested atthe edge of the fault (Type B behavior). (For interpretationof the references to color in this figure legend, the reader isreferred to the web version of this article.)

Table 5Constitutive properties for simulation with shear dilation.

Parameter Value Unit Description

e* 2.1547×10-5 m Reference hydraulic apertureσ*e1 50 MPa Reference normal stressσ*e2 50 MPa Reference normal stressφe 2 deg. Shear dilation angleE* 0.0622 m Reference void apertureσ*E1 50 MPa Reference normal stressσ*E2 ∞ MPa Reference normal stressφE 25 deg. Shear dilation angle

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as a faulting criterion to evaluate whether or not the extent of thepressure front sets a limit on the earthquake rupture dimension for agiven fault.

Previous numerical modeling studies have also proposed a distinc-tion between pressure-constrained and runaway rupture behavior.Gischig (2015) modeled quasidynamic earthquake rupture on one-di-mensional fault planes with a homogeneous shear stress distribution.Dieterich et al. (2015) modeled quasidynamic earthquake rupture ontwo-dimensional fault planes with heterogeneous shear stress dis-tributions. In both of these studies, the distinction between the twobehaviors was observed, and it was reported that τ0 influenced thetransition between behaviors. By introducing Eq. (1), we demonstratedthat the transition between faulting behaviors is characterized moreappropriately by considering relative magnitudes of the prestress ratio,

=f τ σ/0 0 0, and the dynamic friction, fD.We modeled earthquake behavior along planar faults in a homo-

geneous state of stress and with homogeneous frictional properties. Thecoupled interaction between fluid flow and earthquake rupture pro-cesses for more realistic faults will have a strong influence on theevolution of earthquake sequences along faults. Fang and Dunham(2013) performed dynamic earthquake rupture simulations on roughfaults, and observed that nonplanar geometries tended to preventearthquakes from rupturing the entire fault. Dempsey and Suckale(2016) and Dempsey et al. (2016) performed semianalytical simulationsthat demonstrated the influence of heterogeneous patterns of shearstress on the rupture propagation process. In the presence of hetero-geneity, the pressure-constrained and surface-area-constrained rupturedimensions discussed in this study likely represent upper bounds forpressure-constrained and runaway ruptures, respectively.

Field observations consistent with pressure-constrained behavior,for example at the Basel (Mukuhira et al., 2017) and Soultz geothermalsites (Shapiro et al., 2011), have tended to occur when high-pressure

injection occurred directly into a fault zone. Mukuhira et al. (2017)presented evidence that large pressure gradients in the fault zone lim-ited the migration of seismicity at Basel suggesting that high-pressureinjection into a fault may promote pressure-constrained rupture beha-vior. In Oklahoma, the pressure changes activating basement faults arelikely to be relatively small which might suggest that rupture growthmay tend to be limited by tectonic factors. Whether much of the faultzone remains at ambient conditions, as was the case in our simulations,or the entire fault zone can become pressurized will influence the ex-pected faulting behavior (Garagash and Germanovich, 2012). The dis-tance of injection relative to a fault, the duration of injection, thelength-scale of the fault zone, and the hydraulic properties of thefaulted porous media will influence how pressure evolves along a fault,therefore is important to consider the operational context when at-tempting to assess the potential for pressure-constrained or runawaysruptures.

In practice, it may be difficult to assess the range of stress conditionsand frictional parameters with sufficient accuracy to apply the faultingcriterion with full confidence. Methods to constrain the state of stressand fluid pressure in the subsurface are well established (Jaeger et al.,2007), but it is not always possible to extrapolate stress measurementsmade in the injection aquifers to greater depths. Given that C dependson the current stress conditions on a fault, in future studies of Okla-homa seismicity it would be useful to address the impact of startinginjection at different points within the fault's earthquake cycle. Thelocations and orientations of basements faults are difficult to determinebefore injection begins, but could potentially be determined if sufficientmonitoring is performed. For example, at Guy, Arkansas a three-stationseismic array was installed after several relatively small earthquakeswere observed (Mw<3), which allowed for accurate determination ofevent hypocenters that defined a previously unknown fault (Horton,2012). Similarly, the fault structures shown in the earthquake

Fig. 6. Earthquake rupture profile during a typical rupture alonga fault that experienced the effects of shear dilation. The linesgrow progressively darker as time increases. As slip initiated,shear-enhanced dilation caused a near-instantaneous pressuredrop which increased the strength and ultimately caused therupture to arrest prematurely.

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relocation study performed by Schoenball and Ellsworth (2017) sug-gests that it may be possible to assume that potentially active basementfaults are distributed ubiquitously throughout the crust in the centraland eastern United States.

Uncertainty in the ability to determine the dynamic friction coeffi-cient accurately likely presents a significant barrier toward practicalapplication of our proposed faulting criterion. Moreover, recent la-boratory studies have shown evidence for strong dynamic weakeningalong faults while sliding at seismic slip speeds (Di Toro et al., 2011;Nielsen, 2017), an effect that was neglected in the classic rate-and-statefriction formulation used in this study. To overcome these limitations, itmay be useful to look toward field experiments. For example, the studyperformed by Guglielmi et al. (2015), where fluid was injected into anatural fault, demonstrated an ability to obtain direct in-situ mea-surements of the frictional properties of real faults. After obtainingappropriate information and acknowledging the uncertainty associatedwith the field data, one could use the criterion proposed in this study toassess the variability in expected faulting behavior near a wastewaterdisposal site.

7. Conclusions

In this work, we applied numerical modeling to investigate funda-mental physical processes that relate fluid flow through faulted porousmedia and earthquake rupture mechanics to identify controlling factorson maximum earthquake magnitude for induced earthquake sequences.We developed a faulting criterion that may be used to assess whetherthe maximum magnitude will be controlled predominantly by injectionoperations or by tectonic factors. The faulting criterion depended pri-marily on the state of stress, orientation of the fault, and the stress dropduring dynamic rupture. The faulting behavior was also influenced bythe nature in which pressure evolved along the fault zone. Our mod-eling results highlight a transition between conditions that allow forearthquake ruptures to be constrained by the pressurized region andconditions that promote runaway ruptures that are able to propagatewell beyond the pressure front.

The assumption that each individual earthquake has an equalprobability of growing into a large magnitude event is embedded inhazard analysis for natural seismicity. This study has implications forunderstanding the factors that influence whether the frequency-mag-nitude scaling assumption holds for cases of induced seismicity. Ourfaulting criterion suggests that for faults with low resolved shear stress,if an event is triggered by fluid injection then its maximum magnitudewill likely be bounded by the extent of the pressurized zone. On theother hand, the maximum magnitude of events triggered on critically-stressed faults will likely by influenced predominantly by tectonic fac-tors such as geometric or stress heterogeneity. Similarly, faults thatbehave with strongly rate-weakening friction during dynamic rupturewill tend to behave as runaway events no matter the prestress condi-tions or the extent of pressure perturbation. In practice, there is con-siderable uncertainty in many of the parameters that influence thefaulting criterion. Nonetheless, the criterion provides a basis for as-sessing which properties are most important to consider when per-forming site characterization and developing mitigation strategies forinduced seismicity.

Acknowledgements

The Stanford Center for Induced and Triggered Seismicity (oftheStanford School of Earth, Energy and Environmental Sciences) pro-vided funding for this study. The numerical simulations were performedat the Stanford Center for Computational Earth and EnvironmentalStudies. The authors thank two anonymous reviewers and S. Xu forthoughtful comments that improved this manuscript.

Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.tecto.2018.01.028.

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