○E

The Effects of Varying Injection Rates in OsageCounty, Oklahoma, on the 2016 Mw 5.8 PawneeEarthquakeby Andrew J. Barbour, Jack H. Norbeck, and Justin L. Rubinstein

ABSTRACT

The 2016 Mw 5.8 Pawnee earthquake occurred in a regionwith active wastewater injection into a basal formation group.Prior to the earthquake, fluid injection rates at most wells wererelatively steady, but newly collected data show significant in-creases in injection rate in the years leading up to earthquake.For the same time period, the total volumes of injected waste-water were roughly equivalent between variable-rate and con-stant-rate wells. To understand the possible influence of thesechanges in injection, we simulate the variable-rate injection his-tory and its constant-rate equivalent in a layered poroelastichalf-space to explore the interplay between pore-pressureeffects and poroelastic effects on the fault leading up to themainshock. In both cases, poroelastic stresses contribute a sig-nificant proportion of Coulomb failure stresses on the faultcompared to pore-pressure increases alone, but the resultingchanges in seismicity rate, calculated using a rate-and-state fric-tional model, are many times larger when poroelastic effects areincluded, owing to enhanced stressing rates. In particular, thevariable-rate simulation predicts more than an order of mag-nitude increase in seismicity rate above background rates com-pared to the constant-rate simulation with equivalent volume.The observed cumulative density of earthquakes prior to themainshock within 10 km of the injection source exhibits re-markable agreement with seismicity predicted by the variable-rate injection case.

Electronic Supplement: Animations of the evolution of pore pres-sure, vertical displacement, and cylindrical tensor strains in theinjection simulation domain, and a set of input commands forrunning simulation A using poel in ASCII format.

INTRODUCTION

Injection-induced earthquakes are well documented in Okla-homa (Ellsworth, 2013; Keranen et al., 2013; Walsh andZoback, 2015), where the rate of earthquakes has increasedsharply since 2009. Studies of the recent sequences in Okla-

homa explain the increased seismicity as a result of increasedpore pressures reactivating pre-existing faults (e.g., Keranenet al., 2014). Hubbert and Rubey (1959) demonstrate thisbehavior analytically, whereby increases in fluid pressure cancause slip on pre-existing structures. The influence of fluidpressure on the frictional stability of faults in the subsurface hasbeen argued to explain the earliest known cases of injection-induced seismicity at Rocky Mountain Arsenal and Rangely(Evans, 1966; Healy et al., 1968; Raleigh et al., 1976; Hsiehand Bredehoeft, 1981).

Although direct pore-pressure effects are typically believedto be the cause of injection-induced earthquakes, poroelasticeffects may be important to consider. For instance, recent stud-ies consider the possibility that aseismic processes resultingfrom injection can be responsible for changes in seismicity onfaults with frictional instability (Bourouis and Bernard, 2007;McClure and Horne, 2011; Segall and Lu, 2015; Deng et al.,2016; Fan et al., 2016; Norbeck and Horne, 2016a). In general,this has been difficult to establish because of a lack of obser-vations of pore-fluid pressure in proximity to earthquakesequences, but measured slip histories from controlled fluid-injection experiments (Guglielmi et al., 2015) demonstrate thiseffect clearly. Furthermore, numerical simulations (Zhang et al.,2013; Chang and Segall, 2016; Norbeck and Horne, 2016b)demonstrate that fault-related variations in permeability caninfluence the location and timing of induced seismicity; carefulanalyses of seismicity and injection patterns (McNamara et al.,2015; Yeck,Weingarten, et al., 2016) support this conclusion.

In this article, we explore the possibility that poroelasticeffects arising from fluid injection influenced the Mw 5.8Pawnee earthquake (Fig. 1). Notably, new injection data innearby Osage County show significant variability in the yearspreceding the earthquake, whereas other wells show relativelyconstant rates of injection (Fig. 2). This provides an opportu-nity to compare the influence of the direct pore-fluid pressureeffect and the poroelastic effect on the generation of seismicity,focusing on the difference between constant-rate and variable-rate wastewater disposal. We compare simulations of the time-varying poroelastic stresses and pore pressure resulting from

doi: 10.1785/0220170003 Seismological Research Letters Volume 88, Number 4 July/August 2017 1

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transient fluid injection with the equivalent constant-rate in-jection history based on an equivalent volume of fluid. Ourresults show clear differences in spatial and temporal patternsin Coulomb failure stress, depending on whether or not po-roelastic effects are considered. Our model predicts more thanan order of magnitude increase in the maximum seismicity rateprior to the mainshock, about a factor of 3 larger than the con-stant-rate result because of differences in the stressing ratehistory.

DATA AND METHODS

Injection DataThere are numerous wells within 15 km of the mainshock epi-center, but we restrict our study to the nine wells (Table 1) withinjection rates into the Arbuckle Group greater than0:3 kbbl=mo (Fig. 1), wells that could have had the largest in-fluence on the seismicity. Much of this injection activity hasbeen either fairly small compared with similar wells in Okla-homa or continued at a relatively constant rate (Fig. 2). Thisincludes the two closest wells to the mainshock epicenter,OLDHAM and SCROGGINS, ∼2:4 and ∼5:4 km from

the mainshock, respectively; these wells were injecting atrelatively constant rates between 146 and 211 kbbl=mo since2012. Because we are interested in looking at the interplaybetween the poroelastic effects and hydrologic effects of var-iable fluid-injection rates, we examine the two closest wellswith the largest changes in injection behavior: SOUTHBEND2A-11 (OS6301) and RICE 2A-9 (OS6379), located ∼6:5 and∼8:7 km from the epicenter of the mainshock, respectively(SOUTH BEND is ∼3:2 km from RICE). More specifically,injection rates at SOUTH BEND and RICE increased from 0to 288 and 404 kbbl=mo, respectively, over a period of ∼6months beginning in early 2013. This rapid change is followedby a steady reduction in rates to level the characteristic of theaverage in this period for all wells within 15 km of the main-shock, ∼92 kbbl=mo, and following the Mw 5.8 mainshockthese wells were shut-in. (All wells within a zone of regulatoryaction were shut-in after the mainshock.) These data are usedin our poroelastic simulations.

Poroelastic Reservoir ModelingOur model domain comprised four major lithologic unitsrepresenting the relatively uniform lithology in north-central

Regulatory Body

OCCEPA

0 20 40 60 km

N

KO

P

0 100 200 km

0

(a)

(b)

(c)

100 200 km

Injection Well

OCCEPA

Fault Stability

HighModerateLow

0 5 10 km

B

SHmax

▴ Figure 1. The 2016 Mw 5.8 Pawnee sequence. (a) Mapped faults from the Oklahoma Geological Survey and counties surrounding thesequence (K, Kay; N, Noble; P, Pawnee; O, Osage). (b) Locations of Mw ≥5 earthquakes, disposal wells by regulatory body, and regionalearthquakes. The relative size of a given disposal-well symbol shows the relative size of the total volume of fluid injected since 2015.(c) Map of the Pawnee sequence (Yeck, Hayes, et al., 2016) with mapped faults colored by their stability with respect to the maximumhorizontal stress direction (Holland, 2013; Alt and Zoback, 2016; Walsh and Zoback, 2016). The Sooner Lake fault (dashed line) is inferredfrom aftershock epicenters. Contours of radial distance are shown at 10 and 20 km (as in b). OCC, Oklahoma Corporation Commission; EPA,Environmental Protection Agency. The color version of this figure is available only in the electronic edition.

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Oklahoma (Luza and Lawson, 1980; Christenson et al., 2011).Injection is assumed to be entirely within the Arbuckle Group,a laterally extensive basal carbonate formation group which wetreat as a single layer between 1300 and 1900 m depth (below

ground surface [bgs]) based on information from well-comple-tion diagrams. The injection interval overlies semi-infinitecrystalline basement rock and is overlain by a confining layerof finite thickness. An unconfined layer representing sedimen-

1111111111111111111

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11111111111

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33333333333333333

3

3333333333333333333

3333333333333333333

444444

4444

4444444444444444444444444444444444444444444444

5555555555555

555

555555

55555

55555

555555555555555555555555

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77777777777777777777777777777777777777777777777777777777

88888888888888

8888888888

888888

88888888888888888888888888

99999999

999999

999999

9999

99999999999999999999999999999999

OLDHAM(2.4 km)

SCROGGINS(5.4 km)

SOUTH BEND(5.9 km)

STRICKER(6.5 km)

RICE(8 km)

RAINEY(8.3 km)

CARTMELL(9 km)

ROGERS(10.7 km)

CARTER(12.1 km)

OLDHAM(2.4 km)

SCROGGINS(5.4 km)

204

211

288

235

404 46

49

214

340

2012 2013 2014 2015 2016

325 kbbl/moper div

2012 2013 2014 2015 20160.0

0.5

1.0

1.5Total

OCC

EPA

Mbb

l/mo

204

211

288

235

404 46

49214

340

OCCEPA

(a)

(c)

(b)

▴ Figure 2. Injection histories of active Arbuckle disposal wells within 15 km of the epicenter (Table 1) from 2012 through the mainshockorigin time. (a) Time series for each well on the same scale (shifted for visualization purposes) with epicentral distances given in parenthesesbelow the name. Peak injection rates are labeled for each curve. Each division of the vertical scale represents 325 kbbl=mo. The injectionhistories used in the poroelastic simulations are marked with asterisks. (b) Aggregate injection time series for each regulatory body com-pared to total disposal volumes per month, in Mbbl/mo. (c) Map of the earthquake sequence (Yeck, Hayes, et al., 2016) with the locations ofinjection wells, regulatory classifications, and peak rates shown in (a). The gray lines are county boundaries, and the dotted circle shows aradial distance of 10 km; zero-volume, low-rate (see the Injection Data section), or non-Arbuckle wells are marked by the symbol ×. The colorversion of this figure is available only in the electronic edition.

Table 1Active Arbuckle Disposal Wells within 15 km of the Pawnee Mainshock Epicenter

ID

Name Number Longitude (°) Latitude (°) Distance (km)

Injection Interval (m bgs)

OCC (API) EPA (OS) Upper Lower3511701087 OLDHAM 6 −96.940230 36.449992 2.4 1283 16303511723382 SCROGGINS 1 SWD −96.994580 36.421030 5.4 1301 1459

6301 SOUTH BEND 2A-11 −96.949900 36.480900 6.5 1337 16103511722902 STRICKER 9D −96.865170 36.443755 6.5 1237 1463

6430 RAINEY 1D −96.877800 36.488100 8.4 1222 14776379 RICE 2A-9 −96.985900 36.488200 8.7 1336 1722

3511722558 CARTMELL 15-1D −96.855070 36.381214 8.9 1274 16403511723470 ROGERS 1-13D −96.833876 36.377607 10.7 1273 16153510324349 CARTER 1-5SWD −96.997119 36.332713 12.1 1467 1842

OCC, Oklahoma Corporation Commission; EPA, Environmental Protection Agency.

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tary units overlies the confining layer. Figure 3 shows the gen-eral model setup. We assume that all layers behave as homo-geneous linear poroelastic media with varying responses, whichwe describe in detail next.

We solve the general boundary value problem of fluid-mass flux into a system governed by the equations of linearporoelasticity (compare with, Wang, 2000); these are basedon isothermal equilibrium, Hooke’s law of elasticity, andDarcy’s flow law (conservation of mass). We use the spectralelement method of Wang and Kümpel (2003), which calcu-lates time-varying cylindrically symmetric solutions in a layeredhalf-space. We assume an unconfined free-surface boundarycondition (p � 0) and a volumetric source time functionfor injection based on real data (described above). The injec-tion simulation captures the injection-rate variability in OsageCounty from 2013 through late 2016, predominately from thewells RICE and SOUTH BEND (see Fig. 2).

The particular poroelastic parameters specified for eachlayer in the half-space include Skempton’s coefficient B, thedrained and undrained Poisson’s ratios ν and νu, the hydraulicdiffusivity D, and the elastic shear modulus μ. Poroelastic sol-utions are affected by four coupled parameters

EQ-TARGET;temp:intralink-;df1a;40;136α � 3�νu − ν�B�1 − 2ν��1� νu�

�1a�

EQ-TARGET;temp:intralink-;df1b;40;103β � 9�1 − 2νu��νu − ν�2μB2�1 − 2ν��1� νu�2

�1b�

EQ-TARGET;temp:intralink-;df1c;311;733χ � 9D�1 − νu��νu − ν�2μB2�1 − ν��1� νu�2

�1c�

EQ-TARGET;temp:intralink-;df1d;311;703λ � 2νμ1 − 2ν

�1d�

(Kümpel, 1991; Wang and Kümpel, 2003; Segall, 2010), inwhich α is Biot’s coefficient of effective stress, β is the bulkcompressibility, χ is the Darcy conductivity, and λ is Lamé’sfirst constant of elasticity. We treat these coupled parametersas homogeneous within each layer except in the basement, forwhich we use a variable hydraulic diffusivity that decreases log-arithmically from the top of the layer to 8 km depth (then isconstant at all depths below). We combine this depth-depen-dent diffusivity with a homogeneous shear modulus to re-present a typical depth-dependent permeability profile infractured crystalline rock (Ingebritsen and Manning, 2010).This type of permeability structure is known to control hy-draulic conductivity in critically stressed crystalline rock (Bar-ton et al., 1995; Townend and Zoback, 2000).

We perform two simulations of the same variable injectionrates with the goal of isolating the sensitivity to elastic param-eters in the four layers (see Table A1). The first, referenced assimulation A, uses elasticity constants representative of thegeneral rock lithology, adapted from Roeloffs (1996), Wang(2000), and Jaeger et al. (2007). The second, referenced as sim-ulation B, fixes the parameters in each layer to those for thebasement layer in simulation A. In both simulations, theSkempton’s coefficient is set to a single value across all layers,B � 0:75; this may be unrealistically high for some units in thesedimentary section (Lockner and Stanchits, 2002), but is gen-erally valid for granitic basement (Wang, 2000; Jaeger et al.,2007), and direct pore-pressure observations (Kroll et al.,2017) suggest a comparable value for the Arbuckle Group.

We set the drained Poisson’s ratio to ν � 0:25, and theundrained Poisson’s ratio to νu � 0:38. With the Skempton’scoefficient stated earlier, this gives an effective stress coefficientof α � 0:75. This value of α is higher than the value generallyassumed for bulk granite, α � 0:47, because the undrainedPoisson’s ratio is higher than expected for the same rock type(seeWang, 2000, their table C.1); however, published values ofα are generally calculated from laboratory measurements (e.g.,Rice and Cleary, 1976) and are thus subject to the influence ofmeasurement variability. We use statistical inference to quan-tify the conditional probability distributions of each parameterbecause uncertainties are not generally reported, to our knowl-edge. Based on 104 realizations in a Monte Carlo simulation ofα (equation 1a), with the values used in simulation A for thebasement layer (Table A1) and two assumed levels of relativeuncertainty (5% and 10%), we find broad distributions of α:standard deviations are 0.11 and 0.23 for 5% and 10% relativeuncertainty, respectively, and lower bounds are 0.53 and 0.27 at95% confidence. The details of these distributions demonstratethat, although the inferred value of α is perhaps unlikely forbulk material, it is not implausible statistically. Furthermore,the effects of pervasive hydraulically conductive (critically

Simulation time, years

10−2

m3

s

0 (2013) 1 2 3 3.840

1

2

3

4

(a)

(b)

(c)

▴ Figure 3. Details of the injection simulation domain. (a) Time-varying injection is modeled in a cylindrically symmetric layeredporoelastic half-space with four major layers (dimensions not toscale). (b) The injection source in map view (to scale), basedon the location of RICE (see Table 1), with the vertical receiverfault used to calculate Coulomb failure stresses; the seismic se-quence is shown for reference. (c) Injection rates used in the tran-sient-injection simulation; zero time represents 1 January 2013. Thecolor version of this figure is available only in the electronic edition.

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stressed) fractures in crystalline rock include elevated compli-ance and specific storage (Barbour, 2015; Xue et al., 2016);thus, although our simulations may represent a limiting caseof bulk material properties, they likely capture the enhancedsensitivity to external stresses (e.g., injection of wastewater) ex-pected for the Precambrian basement.

Seismicity Rate ModelingThe coupled geomechanical and hydrologic simulations pro-vide spatial and temporal distributions of stress and fluid pres-sure throughout the model domain. We assume that theorientation and location of the fault plane that hosted themainshock follows the distribution of relocated aftershocks(Yeck, Hayes, et al., 2016) and is vertical. The top of the faultplane begins at the intersection of the injection layer and base-ment. Then, we calculate the Coulomb stress changes along thefault (Pollard and Fletcher, 2005) caused by fluid injection

EQ-TARGET;temp:intralink-;df2;52;541Δτf � Δτ� f �Δσ � Δp�; �2�

in which Δτ is the change in shear stress, Δσ is the change innormal stress on the fault, Δp is the change in fluid pressurewithin the fault zone, and f is the static friction coefficient.Tensile normal stresses are taken as positive in this sign con-vention; hence, positive Δσ � Δp indicates unclamping of thefault. Furthermore, positive Δτ represents left-lateral shear;hence, positive Δτf represents failure-promoting shear stresschanges, assuming the maximum principal stress is compres-sional and oriented at N85°E (Walsh and Zoback, 2016).At each point along the fault, the changes in shear and normalstresses are obtained by rotating the principal stress tensor fromthe poroelastic simulation into the direction of the fault pole.The fault surface is assumed to have a uniform static coefficientof friction f � 0:65, as laboratory studies of basement rock inOklahoma indicate (Carpenter et al., 2016).

Following the empirical approach introduced originally byDieterich (1994) and expanded upon by Segall and Lu (2015)and Chang and Segall (2016), we use a rate-and-state frictionframework to model the evolution of seismicity rate along thefault in response to changes in the pre-existing state of stress. Inthis model, seismicity rate evolves according to

EQ-TARGET;temp:intralink-;df3;52;254

dRdt

� Rtc

�_τf_τ0− R

�; �3�

in which _τf is the Coulomb stressing rate, _τ0 is the backgroundstressing rate, tc � Aσ̄=_τ0 is a characteristic timescale overwhich the seismicity rate returns to background levels, A isthe rate-and-state direct-effect parameter, σ̄ � σ � p is theeffective normal stress, and R is defined as the ratio betweenthe current seismicity rate and a reference seismicity rate.When R � 1, the predicted seismicity rate is equal to the back-ground rate. We solve equation (3) numerically using anexplicit Runge–Kutta method (Griffiths and Smith, 2006).The rate-and-state frictional properties used in our model

are representative of granite at hydrothermal conditions (Blan-pied et al., 1995) and are listed in Table A2.

An advantage of this frictional model is that it is possibleto consider arbitrary loading conditions. In this study, we ex-amine the effects of a relatively rapid change of injection ratesover a period of roughly 3.7 yrs in two wells near the fault, inOsage County; seismicity rate calculations are based on thesimulated Coulomb stress changes (from the poroelasticitymodel) associated with the idealized disposal well operations.We repeat these calculations with the direct pore-pressureeffect isolated, wherein only pore-pressure changes are takenfrom the solution; we verified this approach with a traditionalhydrologic model using an equivalent layered structure(P. Hsieh, personal comm., 2016). It is important to recognizethat these idealizations are not a representation of the full his-tory of injection leading up to the mainshock; but they canhelp explain the effects of variations in injection rate. It is alsoimportant to recognize that the seismicity rate model used inthis study does not simulate the nucleation, rupture, or arrestprocesses of individual earthquake events in a deterministicmanner: changes in seismicity rate can only be interpretedin terms of their effect on earthquake statistics.

RESULTS

Stress Changes from InjectionThe poroelastic injection simulations show features consistentwith general hydrologic models. That is, the strongest changesare seen in pore-fluid pressure changes, which are strongly de-pendent on the hydraulic diffusivities of each rock layers. Thepresence of a confining layer promotes fluid diffusion radiallyin the Arbuckle-Simpson layer and, to a lesser extent, in thevertical direction into the basement, resulting in a pore-pressure distribution that decays both in depth and radial dis-tance from the injection source. Based on supplementarysimulations, the realistic depth-decreasing diffusivity in thesemi-infinite basement layer enhances the depth-decay in pore-pressure diffusion rates.

Despite the general consistency between our poroelasticsimulations and hydrologic models, the poroelastic model in-cludes significant variations in the stress field at all locationsthat cannot be replicated in a decoupled or partially coupledhydrologic model. Figure 4 shows snapshots of the evolution ofstresses and pore pressure for simulation A (Table A1). There isa nonlinear relationship between the change in excess porepressure Δp and the change in mean stress Δσm, defined bychanges in principal stress (Δσ1 ≥ Δσ2 ≥ Δσ3)

EQ-TARGET;temp:intralink-;df4;323;181Δσm � �Δσ1 � Δσ2 � Δσ3�=3; �4�indicating the influence of laminar fluid-volume flux (χ∇p)through the model domain. Figure 4 also shows elevated differ-ential stresses

EQ-TARGET;temp:intralink-;df5;323;116Δσd � Δσ1 − Δσ3; �5�that are initially concentrated near the injection source duringthe injection transient and eventually focus around the main-

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shock centroid depth. Although elastic stresses initially domi-nate pore-pressure changes, increases in Δσd are on the orderof the size of Δσm and Δp, indicating that poroelastic shearstresses exhibit considerable influence on the stress state of therock, even at distances much greater than the injection sourcedepth. The strongest changes in pore pressure and stresses areclose to the injection source, and the sharpest contrasts in pore-pressure and stress changes are related to contrasts in material

properties. In the sedimentary layer for example, significantstress changes are generated but very little pore-pressure diffu-sion occurs because of the relatively low diffusivity of the con-fining layer.

At depths of the mainshock centroid, the effect of materialcontrasts controls regions of high Δσd that are localized. Inparticular, the depth-dependent hydraulic diffusivity structurehas the strongest influence on localization, as the following

▴ Figure 4. Simulated changes in poroelastic stresses and pore pressure using a realistic source time function and variable elastic param-eters—Simulation A (see Table A1). Snapshots are at (a) 0.5 yrs, (b) 1.51 yrs, (c) 2.52 yrs, (d) 3.68 yrs, just after the mainshock (arrows point tothe centroid); and (e) 3.84 yrs, when seismicity had declined to a few earthquakes per week. (Left) The source time function with vertical linesshowing the time of the snapshot and the earthquake sequence. (Middle two columns) The mean stress (equation 4) and differential stress(equation 5). The circles show the locations of seismicity relative to the injection source, and are colored gray if they have not yet occurred atthe given snapshot. (Right) The pore pressure. Color scales are linear from 0 to 100 kPa, and contours are logarithmic for decadal factors of 1(solid lines), 2 (dashed lines), and 5 (short dashed lines). The color version of this figure is available only in the electronic edition.

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comparison between the two simulations demonstrates. In sim-ulation B, the elastic parameters are set equivalently in alllayers, based on the values for the basement fault (Table A1).Even though this fixed-elasticity equivalent of simulation A isunrealistic, especially in shallow sedimentary layers, it is an in-structive exercise for understanding the sensitivity of the simu-lated stress changes to variations in elastic parameters of therock. Figure 5 compares the evolution of stresses and pore pres-sure at the centroid of the mainshock for both simulations; thesimulations are nearly identical at each point in time, with theexception of the minimum principal stress at early times andthe pore pressure at later times. We find that elastic stressesdominate pore-pressure effects at the mainshock centroidwhen the principal stress aspect ratio

EQ-TARGET;temp:intralink-;df6;52;577r � �Δσ2 − Δσ3�=�Δσ1 − Δσ3� �6�(e.g., Bott, 1959) is less than 1/2. The transition from r ≤ 1=2to r > 1=2 occurs ∼1:7 yrs after the variable-rate injection be-gins, around when peak differential stresses occur, with peakpore-pressure changes occurring roughly 0.5 yrs later. Elevatedlevels of differential stress at centroid depths are apparent inboth the homogeneous elasticity and layered elasticity cases,but the maximum difference between the two solutions is lessthan 1 kPa; hence, even with homogeneous elastic structure,stress localization is controlled by the hydraulic diffusivities ineach layer. The pronounced dependence of both pore-pressureand stress changes on hydraulic diffusivity is a direct conse-quence of interrelated hydromechanical parameters (Kümpel,1991; Barbour and Wyatt, 2014).

In addition to the effects of elasticity, there is another dif-ference between the poroelastic and hydrologic models; thetiming of peak pore-pressure changes at centroid depths is dif-ferent. Although a hydrologic model predicts a pressure in-crease everywhere in the model domain assuming boundarycontinuity, the poroelastic simulation predicts a decrease inpore pressure in the basement caused by instantaneous elasticstrain in the rock at the initiation of injection that is roughly0.5 yrs long. This is a phenomenon sometimes observed inshallow water wells in crystalline aquifers (e.g., Wolff, 1970).Despite the small size of the pressure reduction relative to pres-sure increases at later times, the observed lag represents a tem-porary fault-stabilizing pressure reduction �Δp < 0� prior toimminent pressure rise �Δp > 0� from pressure diffusion. Theduration of this delay depends on distance to the source andexhibits sensitivities to the difference between the drained andundrained Poisson’s ratios (Rice and Cleary, 1976) ν and νuand the initial conditions of the simulation. The magnitudeof the initial change in injection rate is particularly important,but the opposite effect occurs in the transition to zero injec-tion; thus, in certain faulting regimes it is theoretically possibleto mitigate damaging effects of rapid shut-in by carefully taper-ing injection rates (Segall and Lu, 2015).

Seismicity Rate EvolutionIn the seismicity rate change model, the extent of the SoonerLake fault is represented as a 4 km (along depth) by 8 km

▴ Figure 5. Simulated temporal variation in stress changes atcentroid depths. (a) Simulated injection rates. The vertical lineshows the timing of the mainshock relative to the beginning ofthe simulation; wells were shut-in shortly after the mainshock.(b) Changes in pore pressure and principal stresses. Solid linesshow the quantities at the mainshock centroid; shaded regionsshow the variation from the centroid to where the fault intersectsthe injection interval at ∼2 km (at the same radial distance). Peakvalues occur more than a year after peak injection. (c) Principalstress aspect ratio (equation 6); elastic stress changes dominatepore-pressure changes when this is less than 1/2. (d) Changes inthe Coulomb failure stress Δτf , the differential stress Δσd , themean stress Δσm , and the maximum shear stress Δτmax with porepressure Δp shown for reference. Peak Δτf occurs at the time ofpeakΔσm , after peakΔτmax and slightly before peakΔp. The colorversion of this figure is available only in the electronic edition.

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(along strike) vertical plane with a strike of N65°E and an ori-entation of∼45° to the injection source. The fault tip is locatedat ∼4:2 km relative to the injection source (RICE) and 1.9 kmbelow the surface. The Coulomb stress change acting on thefault is used to inform the seismicity rate model, and thestressing rate is assumed to be constant between the weeklong

timesteps from the injection model, and is not sensitive tosmall variations in dip angle. In Figure 6, we show the distri-bution of Coulomb stresses resolved on the fault, and the as-sociated seismicity rate changes, at several snapshots in timeduring simulation A. There is a general correspondence be-tween the spatial pattern of stress and seismicity rate, with

▴ Figure 6. Simulated changes in (left) Coulomb failure stresses (CFS; equation 2; in kPa), and (right) seismicity rate (equation 3) on theSooner Lake fault associated with the variable injection-rate simulation. Snapshots are shown at (a) 0.5 yrs, (b) 1.51 yrs, (c) 2.52 yrs, andjust after the mainshock at (d) 3.68 yrs, as in Figure 4. The vertical axes are true depth, and the horizontal axes are distance along strike,both in kilometers. The color version of this figure is available only in the electronic edition.

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some aftershocks following isostress contours. The maximumCoulomb stress change is on the order of Δτf ≈ 40 kPa at themainshock centroid, and the maximum seismicity rate changeis R ≈ 16, suggesting more than an order of magnitude increaseabove the background rate.

In our simulations, the maximum predicted seismicity rateoccurs roughly one year following the period of peak injectionrate. During 2015, the predicted seismicity rate declinessteadily until early 2016, at which point the rate stabilizesat an average rate of R ≈ 5. The predicted seismicity rate tracksthe stress changes closely and resembles a time-shifted andsmoothed version of the injection rate, which appears to bea consequence of the sensitivity of equation (3) to the stressingrate for these loading conditions. We do not observe an im-mediate correlation between the predicted peak seismicity ratesand the timing of the seismic sequence. Comparing the seis-micity rate calculation using the full poroelastic solution withthe pore-pressure effect isolated—only fluid pressure changesaffect the state of stress along the fault—reduces the maximumrate to R ≈ 3. Furthermore, the value of R appears to be rel-atively insensitive to elastic structure, which again implies thathydraulic diffusivities exhibit the strongest influence on thefully coupled solution.

DISCUSSION

The frictional stability of the Sooner Lake fault prior to the2016Mw 5.8 sequence may have been reduced by active waste-water injection in close proximity to the fault. Although pres-sure diffusion is indeed the dominant mechanism for reducingthe effective stress on the fault, the magnitudes of shear andnormal stresses induced by coupling between elastic deforma-tion of the solid matrix and pressure diffusion are comparableto pore-pressure changes, to within a factor of 2. The distri-bution of Coulomb failure stresses predicted by our simula-tions between the centroid depth and the top of the faultat the same radial distance (shaded region in Fig. 7) impliesthat even though high-permeability pathways may be impor-tant to consider, they are not necessary to explain the relativetiming between this injection transient and the seismic se-quence; a stronger control on this timing is due to gross per-meability structure, which depends on both the hydraulic andelastic properties.

We have not yet considered the effects of steady injectionrates at wells closer to the rupture (e.g., wells in PawneeCounty). Appealing to the same physical mechanism usedto inform the seismicity rate changes, we assume that injectionat these wells must have influenced stressing rates on the fault;however, without strong changes in injection rate their influ-ence may have been subtle, albeit important. For example, theseismicity rate model (equation 3) is highly sensitive tostressing rates as we found in the variable-rate case; but, afternearly a decade of steady injection, as annual records of injec-tion volume extending back to 2006 indicate (e.g., OLD-HAM), stress and pore-pressure accumulation would berelatively slow. To test this, we simulated the effect of long-

term constant injection beginning in 2011, two years prior tothe previous simulations. To make the results comparable, weuse a constant-rate injection with the same total volume offluid as in the variable-rate case. We find that constant-rateinjection loads the fault slower than in the transient cases(Fig. 7). Additionally, this constant-rate simulation producesa much different temporal change in R, which roughly ap-proaches steady-state conditions with the square root of time,and decreases rapidly after shut-in (Fig. 8). This comparisonstrongly suggests that long-term injection may have beenresponsible for a gradual loading of the fault to the point whereit primed the fault for failure triggered by the short-term high-rate injection transient. In the absence of transient injectionactivities, however, frictional failure may still have occurredat a much later time.

In addition to computing the rate increase R, we examinethe integral of the rate increase to test our results against ob-served seismicity (Fig. 8). This quantity, referred to as ΣR,

▴ Figure 7. Variation in Coulomb failure stress and stressing rateat the mainshock centroid for the variable injection-rate simula-tion (Fig. 5) compared to the constant injection-rate simulation;the shaded region shows the range from the mainshock to thefault tip at 2 km depth at the same radial distance (Fig. 6). Notethat the total volumes injected in each simulation are equivalent,and that the horizontal axes are time normalized by the relativetime between the start of the simulation and the mainshock origintime. The color version of this figure is available only in the elec-tronic edition.

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represents the forecast increase in earthquake rate at an indi-vidual point in time. In some sense, it is analogous to an earth-quake probability density function, whereby the probability ofan earthquake is the highest at times for which R is highest.Accordingly, we could consider ΣR to be an empirical cumu-lative density function. Examining this function for simulationA (Fig. 8) shows that although the instantaneous probability ofan earthquake may be lower at later times in the simulation, the

overall probability for a given earthquake or a collection ofearthquakes to have occurred is still increasing.

Foreshock activity and temporal variations in magnitudedistributions (Walter et al., 2017) lend support to this predic-tion. We also note that seismicity was not observed within a10 km radius of the modeled injection source until late-2013,when a clear increase in rate occurred. Even though the ma-jority of these events are not considered foreshocks, their

10−2

m3

s Simulatedinjection rates

constant rate

variable

0123

(a)

(b)

(c)

R

Solid: includesporoelastic stresses

Dashed: neglectsporoelastic stresses

constant−rate

Modeled seismicityrate changes

5

10

15

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grou

ndra

te =

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ty

M > 2.7 within 10 km of inj. source

Noble County (n=23)Osage County (n=1)Pawnee County (n=13)

M 5

.8

2011 2012 2013 2014 2015 2016 2017

Cumulative seismicityabove background

2011 2012 2013 2014 2015 2016 2017

▴ Figure 8. Seismicity rate calculations for variable and constant-rate injection models (simulation A). (a) The simulated injection rates.(b) The time-varying ratios of R—the induced seismicity rate (equation 3)—when considering the full poroelastic solution for stresses onthe fault (solid lines) versus the direct pore-pressure effect (dashed lines). (c) Simulated cumulative seismicity densities above back-ground levels prior to theMw 5.8 mainshock. For comparison, the points show the cumulative density of observed earthquakes above themagnitude of completeness (∼2:7 statewide) inside a 10 km radius away from the injection source (see Fig. 3); no earthquakes weredetected prior to late 2013. The legend shows the number of earthquakes by county; the closest events to the injection source are 2.4, 6.3,and 8.8 km for Osage, Pawnee, and Noble, respectively. The color version of this figure is available only in the electronic edition.

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cumulative density function resembles the predicted R densitiesfor the variable injection-rate case. In general, detectable seis-micity occurs approximately one year after variable-rate injec-tion begins, and is nearly coincident with peak injection rates.If the poroelastic stresses modeled for the Sooner Lake fault areequivalent for similarly oriented fault planes around the injec-tion source, assuming cylindrical symmetry is justified (Segalland Lu, 2015), then these events may be related to changes ininjection rate. And if earthquake magnitudes are random intime (e.g., Kagan and Knopoff, 1987), the overall probabilityof a large earthquake owing to increases in injection is expectedto increase even though it may be instantaneously lower atlater times.

The apparent connection between injection rate, pre-dicted seismicity rate, and observed cumulative density isconsistent with the results of Weingarten et al. (2015) thatdemonstrate that, although large total disposal volumes arenot a necessary condition for elevated seismicity, high injectionrates do imply a higher probability of inducing seismicity. Onepossible explanation for the lack of seismicity in other parts ofOsage County, where moderately high rates of fluid injectionoccur (Murray, 2014), may be from regional variations pore-pressure changes that occur because of compartmentalizationof the sedimentary section, which includes the ArbuckleGroup. This is supported by myriad faults with appreciablethrow that show up in comparable hydrostratigraphy in south-central Oklahoma (Christenson et al., 2011; Mashburn et al.,2013) and a general deepening of basal formation depths to thesouthwest (Luza and Lawson, 1980).

Fluid pressures in a compartmentalized reservoir willremain high in response to injection mass if flow is inhibited.In a region with highly variable injection histories (Weingartenet al., 2015) and a set of mapped faults with both diverse ori-entations and preferential regional directions (Holland, 2013;Alt and Zoback, 2016; Walsh and Zoback, 2016), the effect ofcompartmentalization and its influence on heterogeneous pres-sure distributions would be especially pronounced. Further-more, injection in Osage County is predominantly on thewest side of a frictionally stable north–south-trending faultthat intersects the Sooner Lake fault (Fig. 1c), which suggeststhat pore-pressure changes may have localized until transientpostseismic flow (Sibson, 1994) caused pressure equalization(Manga et al., 2016). In any case, the effects of reservoir com-partmentalization in the Arbuckle can be investigated directlywith data from a network of downhole pore-pressure monitor-ing systems.

One complicating factor in these interpretations is thepossibility of anisotropic or temporal variations in permeabil-ity. Active crustal shear strain maintains shear dilatancy overhydrothermal sealing processes, diminishing the response toquasi-static deformation in favor of enhanced flow in highstrain-rate plate-boundary regions (Barbour, 2015). Eventhough strain rates in Oklahoma inferred from intraplate seis-micity (Anderson, 1986) and geodetic measurements (Calaiset al., 2006) are low compared to those at plate boundaries,for example, they may still outpace hydrothermal sealing

processes in the region (Nathenson and Guffanti, 1988). Spa-tial variations in permeability exist, especially near faults, butthese may be compensated by variations in specific storage,which implies that hydraulic diffusivity may be relatively uni-form over large regions (Xue et al., 2016).

A shortcoming of the seismicity rate model used in thisstudy is that it calculates changes relative to the backgroundseismicity rate. The question then becomes:What is the back-ground rate? In the Pawnee study area, it is difficult to infer areliable estimate of the background seismicity rate because ofthe extremely small sample size of documented earthquakesprior to 2009 and because all major seismic sequences inOklahoma have been on unmapped faults (Yeck, Hayes, et al.,2016). The model results we show must be interpreted withinthe context of a probabilistic analysis; however, it is notpossible to quantify the effect on the seismic hazard withoutestablishing a background seismicity rate, which is subject tocurrent debate. The background seismicity rate in Oklahomahas been estimated (Ellsworth, 2013; Langenbruch and Zo-back, 2016) to be roughly one Mw ≥3 earthquake per yearinside the areas of interest defined by the Oklahoma Corpo-ration Commission (∼26 × 103 km2). If the seismicity ratefor our model domain can be inferred from this regional es-timate by accounting for the relative area covered by themodel (∼314 km2), the local background seismicity rate isroughly one Mw ≥3 earthquake per century. Our modelingresults suggest an increase in seismicity from a backgroundrate of roughly 0.01 earthquakes per year to roughly 0.15earthquakes per year in mid-2014. As we mentioned above,seismicity rates did increase in the years prior to the Pawneeearthquake, around the time of peak-predicted seismicity rate(Fig. 8), but these observations are limited and possibly tied tochanges in detection thresholds (i.e., network changes).

The present set of poroelastic injection- and seismicity-rate models do not incorporate any detailed information aboutthe initial state of stress acting on the fault, the fault’s initialproximity to shear failure, stress transfer effects during earth-quake rupture, or variations in stress owing to either geometricvariation, frictional asperities, or aseismic slip. In future studies,it would be worthwhile to use deterministic earthquakemechanical and hydrological models to identify the conditionsthat would have influenced earthquake nucleation and ruptureduring the Pawnee sequence (e.g., McClure and Horne, 2011;Norbeck and Horne; 2016b).

CONCLUSIONS

Simulations of time-varying Coulomb failure stresses on thefault induced by injection activities indicate that the combinedeffect of stress changes associated with a high-rate fluid-injection transient and long-term injection may have influ-enced the timing and location of the Pawnee earthquake. Eventhough diffusion is the principal mechanism for transmittingpore-fluid pressure changes in basement rock, the effects ofstrain coupling between the rock and fluid are significantand should not be ignored. Seismicity rate calculations using

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a rate-and-state friction model are sensitive to the rate ofchange of Coulomb failure stresses and predict increases ofmore than an order of magnitude above background rates. Pre-dicted seismicity rate increases generally resemble a laggedsmoothed version of the injection-rate history, and the ob-served seismicity leading up to the mainshock follows the cu-mulative density function of rates predicted by the variable-rateinjection simulation. If seismicity rates are affected by poroe-lastic stress changes, our models suggest that they may remainelevated for at least a year following shut-in of injection wellsafter the earthquake, which emphasizes the importance ofincorporating accurate geomechanical properties of the subsur-face in understanding the hazards associated with injection-in-duced seismicity, especially in low strain-rate environments.

DATA AND RESOURCES

Injection data for Osage County are from the EnvironmentalProtection Agency (EPA) and are available inⒺ the electronicsupplement to this article; data for other counties are fromthe Oklahoma Corporation Commission (OCC) Oil andGas Database (http://www.occeweb.com/og/ogdatafiles2.htm).Mapped faults are from the Oklahoma Geological Survey(http://www.ou.edu/content/ogs/data/fault.html). We usedpoel (i.e.,Wang and Kümpel, 2003), version 2012, for numeri-cal simulations of injection. Regional earthquakes are from theAdvanced National Seismic System (ANSS) earthquake catalog(http://www.ncedc.org/anss/catalog-search.html). All websiteswere last accessed March 2017.

ACKNOWLEDGMENTS

We thank Nancy Dorsey at the Environmental ProtectionAgency (EPA) for help accessing injection records from OsageCounty and thank Xiaowei Chen and Paul Hsieh for helpfuldiscussions. Elizabeth Cochran and Paul Hsieh graciously pro-vided thorough preliminary reviews on short notice. We thanktwo anonymous reviewers for comments that led to improve-ments in the clarity and presentation of this work. JackNorbeck was supported by the Stanford Center for Inducedand Triggered Seismicity and a Mendenhall PostdoctoralFellowship.

Any use of trade, firm, or product names is for descriptivepurposes only and does not imply endorsement by the U.S.Government.

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APPENDIX

SIMULATION PARAMETERS

In Table A1, we give the layered poroelastic parameters used insimulations of injection. Simulation A represents the case ofvariable elasticity parameters within each layer based on repre-sentative bulk properties (Wang, 2000; Jaeger et al., 2007),whereas simulation B represents the limiting case in which elas-ticity parameters are fixed for the entire domain based onparameters for granitic basement. In both simulations, the hy-draulic diffusivities are variable, but equivalent; thus, the effec-tive permeabilities are different between the two simulations,except in the basement rock.

In Table A2, we give the parameters used in the seismicityrate calculations, including the state parameter, backgroundstressing rate, and effective normal stress.

Andrew J. BarbourJack H. Norbeck1

Justin L. RubinsteinEarthquake Science Center

U.S. Geological Survey345 Middlefield Road

Menlo Park, California 94025 U.S.A.abarbour@usgs.gov

Published Online 3 May 2017

1 Also at Department of Energy Resources Engineering, Stanford Uni-versity, Stanford, California U.S.A.

Table A2Rate-and-State Frictional Properties Used in Seismicity

Rate Calculations

Parameter ValueA 0.003_τ0 10−4 (MPa/yr)σ̄ 10 (MPa)

Table A1Layered Poroelastic Parameters Used in Simulations of Wastewater Injection

Layer Thickness (m) μ (GPa) ν νu B D (m2=s)Simulation A: Variable elasticity parametersSedimentary 1100 10 0.3 0.4 0.75 0.1Confining (post-Simpson, e.g., Sylvan shale) 200 15 0.3 0.4 0.75 0.005Injection (Arbuckle-Simpson) 600 15 0.25 0.4 0.75 1.0Basement (granitic) ∞ 30 0.25 0.38 0.75 (1.0, 0.002)*

Simulation B: Fixed elasticity parametersSedimentary 1100 30 0.25 0.38 0.75 0.1Confining 200 30 0.25 0.38 0.75 0.005Injection 600 30 0.25 0.38 0.75 1.0Basement ∞ 30 0.25 0.38 0.75 (1.0, 0.002)*

μ, elastic shear modulus; ν, νu , drained and undrained Poisson’s ratio, respectively; B, Skempton’s coefficient; D, hydraulicdiffusivity.*Range from top of basement unit to 8 km; the values decrease logarithmically with relative depth.

14 Seismological Research Letters Volume 88, Number 4 July/August 2017

SRL Early Edition