Geomechanical modeling of induced seismicity source parametersand implications for seismic hazard assessment

Bettina P. Goertz-Allmann1 and Stefan Wiemer2

ABSTRACT

We simulate induced seismicity within a geothermal re-servoir using pressure-driven stress changes and seismicitytriggering based on Coulomb friction. The result is a for-ward-modeled seismicity cloud with origin time, stress drop,and magnitude assigned to each individual event. Our modelincludes a realistic representation of repeating event clusters,and is able to explain in principle the observation of reducedstress drop and increased b-values near the injection pointwhere pore-pressure perturbations are highest. The higherthe pore-pressure perturbation, the less critical stress statesstill trigger an event, and hence the lower the differentialstress is before triggering an event. Less-critical stress statesresult in lower stress drops and higher b-values, if both arelinked to differential stress. We are therefore able to establisha link between the seismological observables and thegeomechanical properties of the source region and thus areservoir. Understanding the geomechanical properties isessential for estimating the probability of exceeding a certainmagnitude value in the induced seismicity and hence the as-sociated seismic hazard of the operation. By calibrating ourmodel to the observed seismicity data, we can estimate theprobability of exceeding a certain magnitude event in spaceand time and study the effect of injection depth and crustalstrength on the induced seismicity.

INTRODUCTION

Injecting fluid at high pressures into a reservoir rock formationreduces the effective stress and can lead to brittle failure (Terzaghi,1923; Healy et al., 1968; Hsieh and Bredehoeft, 1981; Zoback,2007). This creation or reactivation of fractures enhances accessto the reservoir fluids (oil, gas, or hot water) and is typically

associated with microearthquakes, which, if recorded and located,can give important clues about the properties and extent of the trea-ted reservoir volume, as well as about the effectiveness of the treat-ment (e.g., Pearson, 1981; Warpinski et al., 1995; Shapiro et al.,2002; Maxwell et al., 2009; Wessels et al., 2011). If the injectionpressure exceeds the minimum principal stress σ3 in the reservoir,tensile opening can occur and new fractures are created, which needto be kept open after the treatment with the help of a suitable “prop-pant” (typically sand, see Economides and Nolte, 2000). For fluidpressures lower than σ3, failure occurs predominantly throughshearing along preexisting fractures. In this case, permeability isenhanced mainly through the mismatch of rough fracture surfacesafter slipping (“self-propping,” see Brown and Bruhn, 1998). Inboth cases, the observed seismic emission is mainly from shear slipevents (e.g., Rutledge and Phillips, 2003), either by reactivation ofpreexisting fractures (Pearson, 1981), or through shear stress nearthe tip of newly created tensile fractures (Sneddon, 1946) that canextend into the host rock beyond the opened hydrofrac (Evans et al.,1999). Such seismically active shear fissures may be subject to ten-sile opening themselves as the pressure front advances, thus creat-ing a network of fractures and fluid pathways in the reservoir(Hopkins et al., 1998).Although the initiation of slip on a fracture is fairly well under-

stood, it is very unclear what properties and processes are limitingthe slip once it is initiated, and hence govern the size of a rupturearea. The size of the rupture area defines the magnitude (Kanamoriand Anderson, 1975), and hence the seismic energy release of amicroearthquake (Kanamori and Brodsky, 2004).On the one hand, the size of the rupture area can play a role in the

increase of reservoir permeability and/or drainage efficiency. On theother hand, there is a small, but sometimes non-negligible risk thatmicroseismic events of a noticeable strength are created that mightbe felt at the surface, or even cause damage (Deichmann and Giar-dini, 2009; Hitzman et al., 2012). It therefore needs to be ensuredthat the magnitude of induced seismic events does not exceed valueswhere shaking can affect surface infrastructure, and at the same time

Manuscript received by the Editor 19 March 2012; revised manuscript received 19 September 2012; published online 14 December 2012.1NORSAR, Kjeller, Norway; formerly ETH Zurich, Switzerland. E-mail: Bettina.Goertz-Allmann@norsar.no.2ETH Zurich, Swiss Seismological Service, Institute of Geophysics, Switzerland. E-mail: stefan.wiemer@sed.ethz.ch.

© 2012 Society of Exploration Geophysicists. All rights reserved.

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GEOPHYSICS, VOL. 78, NO. 1 (JANUARY-FEBRUARY 2013); P. KS25–KS39, 15 FIGS., 2 TABLES.10.1190/GEO2012-0102.1

ensuring that the economic objective of the stimulation is met. Wepresent in this paper an approach to forward model the spatio-temporal variation of microearthquake source properties. Themodel can explain observed radial variations of stress drop andearthquake size statistics. Through proper calibration to observedseismicity, the approach can provide estimates of the probabilityof exceeding a certain magnitude event. The latter capabilitymay be used in advanced traffic light systems to mitigate the seismicrisk of injection operations.Propagation of the microseismic event cloud through the medium

can be described by the diffusive process of pore-pressure relaxa-tion (Talwani and Acree, 1984). The hydraulic diffusivity of themedium can be estimated from the time-distance dependence ofthe seismicity triggering front (Shapiro et al., 1997, 2002). Apartfrom the spatio-temporal behavior of seismicity, source propertiesof induced seismicity appear to be also influenced by the hydraulicmedium properties. Using data from a geothermal operation inBasel, Goertz-Allmann et al. (2011) showed that the Brune stressdrop of induced events is significantly lower near the injection pointand inversely correlates with estimated pore-pressure values. Simi-larly, Michelet (2002), Bachmann (2012), and Bachmann et al.(2012) measured the spatial variation of Gutenberg’s b-value inseismicity clouds of geothermal injections and highlighted severalcases with b-values higher near the injection point where pore pres-sures are higher. The b-value describes the slope in the Gutenberg-Richter power law distribution, logðNÞ ¼ a-bM, where N is thecumulative number of events, M is the earthquake magnitude,and a denotes the a-value describing the productivity. Seismicsource property observations such as the latter contain informationabout the in situ stress regime: The b-value is linked to the differ-ential stress (Amitrano, 2003; Schorlemmer et al., 2005; Gulia andWiemer, 2010), and stress drop denotes the difference in shear stresson the fault plane before and after the earthquake (e.g., Kanamoriand Brodsky, 2004).A variety of approaches exist to forward-model the seismic re-

sponse of a stimulated reservoir. For example, Rothert and Shapiro(2003) developed a numerical model to trigger microseismicity dueto the process of pore-pressure relaxation and statistically prede-fined critical zones in the medium. Their model is able to describethe spatiotemporal distribution of induced seismicity reasonablywell, but it does not include any event size distribution or othersource parameters such as stress drop. A similar approach is usedby Langenbruch and Shapiro (2010) to investigate the behavior ofpostinjection seismicity and the approach by Hummel and Müller(2009) considers nonlinear pore fluid pressure diffusion. Schoenballet al. (2010) find that the influence of hydraulic properties and thecoupling of pore pressure to the stress field have an effect on theoccurrence of microseismicity, but again they do not concentrate onevent magnitudes. Kohl and Megel (2007) use a finite-element ap-proach to model the hydromechanical response of the medium.Their approach attempts to model the opening or shearing of dis-crete (but stochastically distributed) fractures in the medium. It cantake into account a Coulomb failure criterion to associate seismicitywith shearing, and hence model location and time of seismic events.However, it cannot predict seismic source properties such as mag-nitude and stress drop. Another study also includes the opening ofstochastically distributed fractures due to fluid-flow-induced stresschanges to obtain a microseismicity distribution (Bruel, 2007). Thisstudy also obtains a magnitude of each event from the slip and the

rupture area, but the magnitude distribution is basically predefinedby the fracture distribution where each fracture can only fail com-pletely. Baisch (2009) and Baisch et al. (2010) present an approachto physically model event sizes (magnitudes) by stress transferacross elementary slip patches. Their approach also considers a non-stationary permeability, but is restricted to individual fractureplanes. Assuming a single planar fracture (which may be the domi-nant fluid conduit) may be inconsistent with a pressure diffusionapproach (Jung, 1989). The restriction of the seismicity to a planeleads to an over-representation of modeled seismicity along the trig-gering front and an unrealistic representation of the seismicity backfront, the so-called Kaiser effect, which may be described by anabsence of seismicity due to stress relaxation (Kaiser, 1950). Withincreasing time, the isobar of the triggering front covers a largercircle on the fault, hence increasing the number of events along thatisobar. McClure and Horne (2011) propose a new stimulationmechanism based on coupling fluid-flow with a rate-and-state fric-tion model. This model could, in principle, also be used for hazardmitigation. However, its plausibility has not yet been fully testedagainst field observations. As an alternative physical modelingapproach, particle-based lattice solid models (e.g., Hazzard andYoung, 2002, 2004) would, in principle, be capable of simulatingthe effects that we seek to explain. However, to use them for theproblem at hand, they would have to be implemented in 3D atthe meso- or macroscale, which is computationally extremely chal-lenging. In addition, robust approaches are still lacking for the up-scaling of physical rock properties from the micro (i.e., grain) scaleto the meso (fracture) or macro (fault, stimulated volume) scale.Statistical approaches to model the seismicity aim at predicting

the total number and frequency-magnitude distribution of futureseismicity in a given time window. Based on the Gutenberg-Richterlaw of frequency-magnitude distribution and the Omori law of after-shock decay, the seismic hazard can be forecasted based on the pre-viously observed seismicity catalog (Ogata, 1988; Reasenberg andJones, 1989; Woessner et al., 2010). Such an approach was alsotested on the Basel induced sequence (Bachmann et al., 2011). Itis, in principle, independent of the physics of the triggering mechan-ism. However, the model is static in the sense that it cannot predictthe spatiotemporal variability of the b-value (as observed in Baselby Bachmann et al., 2012), which plays a major role in hazard as-sessment. It was shown by Barth et al. (2011) that time-decreasingb-values are required to explain the observed increase in seismichazard after shut-in.Observations such as the latter are a starting point for our study.

We develop a hybrid model in which we base the rupture initiationon pore pressure diffusion and Coulomb triggering. The size of theearthquake is modeled with a semiprobabilistic approach: We ran-domly assign event magnitudes from a given Gutenberg-Richterdistribution whose b-value is tied to differential stress. We cantherefore forward-model the spatiotemporal variation of b-valuesand event stress drop by way of physical modeling where b-valueand stress drop are tied to differential stress. The result is a hybridgeomechanical model of induced seismicity with probabilistic ele-ments that is able to explain the observed source property variationsto the first order.The main aim of this paper is twofold. First, we seek a possible

explanation for the observed spatio-temporal behavior of stressdrop and b-value (Goertz-Allmann et al., 2011; Bachmann,2012). We can reproduce these variations in principle with a simple,

KS26 Goertz-Allmann and Wiemer

hydraulically homogeneous and isotropic model. Second, weattempt to use the geomechanical model to predict aspects of thebehavior of an induced seismicity cloud. The modeled source prop-erty variations in space and time and the ability to include eventmagnitudes allow us to calculate the probability of exceeding a cer-tain magnitude event and therefore make a prediction of the seismichazard of an injection operation. This last step requires a calibrationof the model to observed data. In our case, we use the observedseismicity of the Basel geothermal stimulation (Häring et al.,2008; Deichmann and Giardini, 2009). Due to the calibration step,the obtained probabilities are strictly valid only for the Basel case.However, the main characteristics of the modeled probability curvesare independent of the calibration step and allow us to make somegeneralized statements. To illustrate the sensitivity to the underlyingstress regime, we present example calculations of event probabilityfor a few selected stress scenarios. If calibrated repeatedly using anevolving measured seismicity during a stimulation operation, thepresented method may be used in the future for a near-realtime pre-diction of induced seismicity. In addition to assigning magnitudesand stress drop to each event, our rigorous application of Mohr-Coulomb theory can also model repeating events at the same loca-tion. Repeating events are important to properly describe effectsclose to the injection point.We first present the details of the new hybrid modeling meth-

odology and the calibration to the Basel data before discussingthe features of the resulting seismicity cloud. This provides a geo-mechanical explanation for the variation of b-value and stress dropthat are observed in reality and successfully reproduced by ourmodel. We then present the calculation for the hazard estimation.At last, we discuss some principle features of the hazard estimatesthat are independent of the calibration step and may therefore beapplicable in different injection scenarios.

MODELING METHODOLOGY

We randomly distribute possible failure points (seeds) in a 3Dvolume with a constant background stress regime and constant fric-tion coefficient and cohesion. At each seed point, we randomly as-sign values for the minimum and maximum principal stress, σ3 andσ1, assuming a Gaussian perturbation to a given background stressregime. Focal mechanisms of the largest events in Basel suggestmostly strike-slip motion (Deichmann and Giardini, 2009). There-fore, in Basel, σ1 and σ3 are both horizontally oriented and σ2 wouldbe the vertical stress. However, for our model, the orientation of thebackground stress regime could be any of normal, thrust, or strikeslip, and is not relevant for the modeling. A Gaussian perturbationof the principle stresses could be interpreted as being due to thevariation of elastic parameters (Langenbruch and Shapiro, 2011).The assumed standard deviation of the stress fluctuation (10%)is consistent with the observed standard deviations of sonic velo-cities in the upper crust (Holliger, 1996). We impose a limit onσ1 such that it cannot exceed a maximum value as defined bythe strong crust limit (Zoback, 2007), and on σ3 such that it cannotbe smaller than the hydrostatic pore pressure ph. The stressdistribution for one example seed point is depicted in Figure 1a

as a Mohr diagram described by its radius R ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið12ðσ1 − σ3ÞÞ2

qand the center point ½1

2ðσ1 þ σ3Þ; 0�. The distribution of σ1 and

σ3 is indicated by the histograms for all triggered seeds (actual dis-tribution after application of limiting constraints) and dashed bellcurves (nominal perturbation of values). Table 1 lists values for the

assumed stress regime and perturbations. These cases are based onthe Basel geothermal examplewith numbers taken fromHäring et al.(2008) who investigated the depth-dependent stress regime in theBasel-1 borehole. At Basel, injection occurred at a depth of4.5 km within the Granite. The magnitudes of the principal stressesand their perturbations are also consistent with stress magnitudesmodeled from borehole observations by Sikaneta and Evans(2012) and a focal mechanism analysis by Terakawa et al.(2012). Because we require the medium to be in a stable equilibriumbefore the pore-pressure perturbation, the upper bound of σ1 iseffectively linked to the coefficient of friction. If we chose a smallercoefficient of friction, all stable systems (Mohr circle just below thefailure criterion) will have smaller average values of σ1. Note thatthe initial seed points are randomly distributed in the medium with-out spatial correlation. Many observations of the statistical proper-ties of elastic parameters in the upper crust suggest that structuretends to be spatially correlated with, for example, an exponentialor von Karman correlation function (e.g., Frankel and Clayton,1986; Holliger and Levander, 1992). However, it was shown indepth by Rothert (2004) that such a spatial correlation of seed points(the criticality criterion in his case) has little to no influence on thetriggered event cloud. Therefore, we decided not to introduce aspatial correlation.In an injection operation, the stress field is mainly modified by

the injection pressure. Increasing the pore pressure pp in the med-ium causes a reduction of the normal stress σn to an effective stress.If the stress is near the critical state (Mohr circle close to the

0 50 100 150 2000

20

40

60

80

τmax

σ1 − p

hσ

3 − p

h

τs= 0.85σ +7

n

τ s (M

Pa)

σn (MPa)

0 50 100 150 2000

20

40

60

80

p = 8.31 MPa

∆σ = 2.1 MPa

τ s (M

Pa)

σn (MPa)

a)

b)

040

80

120

160

200

Cou

nts

Figure 1. (a) Example stress distribution for an average crust at4.5 km depth at one point in the medium depicted by the Mohr dia-gram (shear stress τs versus normal stress σn). The bold dashed lineshows the failure envelope. Distribution of stress values around themean σ3 and σ1 are indicated by the dotted line (planned) and thehistogram (actual). The vertical dashed line shows the strong crustupper limit for σ1. Symbol ph denotes the hydrostatic pore pressure.(b) The black Mohr circle shows the original stress distribution atone point in the medium. The gray circle shows the effective stressof an induced event with the depicted critical pore pressure. Theblack dotted circle shows the postevent stress regime (decreaseof σ1 only) due to the indicated event stress drop Δσ.

Modeling induced seismicity hazard KS27

Coulomb envelope), the reduction of the normal stress may causethe shear stress τs to exceed the Coulomb failure envelope andhence trigger an event (Figure 1b)

τs ¼ μðσn − pÞ þ c; (1)

where μ denotes the coefficient of friction and c the cohesion. Faultplanes are assumed to be always optimally oriented; i.e., at half theangle prescribed by the point at which the Mohr circle is tangentialto the failure criterion. If p continues to increase, the failure envel-ope may be reached again at a later point in time due to the previousevent stress drop, thus triggering a repeating earthquake. We modelthe spatiotemporal evolution of the effective stress based on a lineardiffusion model in a hydraulically homogeneous and isotropic med-ium with a linearly increasing pressure source (Dinske et al., 2010),

pðtÞ ¼ 4πDa0ðp0 þ pttÞ; 0 < t < t0: (2)

For this case, Dinske et al. (2010) formulate an analytical solution tothe diffusion equation (Wang, 2000) that we will use in the follow-ing for our modeling. Values for the hydraulic diffusivity D, theeffective source radius a0, the starting pressure p0, the wellheadpressure gradient pt, and the injection time t0 are listed in Table 1,and are again chosen to closely follow the Basel case study. Withthe listed values, equation 2 denotes a simplified approximation tothe actual wellhead pressure in Basel. We assume an injection pointin the center of our volume. The assumption of a point source iscorroborated by the shape and evolution of the observed microseis-mic cloud in Basel which originates from a single point just belowthe casing shoe. An event is recorded once the Mohr circle hasreached the failure envelope.Note that the number of seed points can be adjusted to modify the

number of induced events. We chose the number of seeds (30,000)such that the modeled seismicity’s a-value, describing the overallevent productivity of the volume (Gutenberg and Richter, 1944) issimilar to the a-value measured from the Basel event cloud (4.3).The choice of seed points is bounded by considerations detailed in

Appendix A. We do not define a minimum and maximum triggeringpore pressure (criticality). Theoretically, a stable stress state can beinfinitesimally close to the failure envelope, and hence the mini-mum pore pressure required to trigger an event can be infinitesi-mally small. In practice, stable systems are systems with aminimum triggering pressure larger than the average backgroundpressure fluctuations imposed by, e.g., tidal variations, ambient seis-mic noise, etc.. Choosing a0 ¼ 70 m and 30,000 seeds results in atriggering front (envelope to all events) that roughly follows the2000 Pa pressure contour (Figure 2). 2000 Pa is the maximumtidal-induced pressure variation measured from borehole water levelgauges in Basel (K. Evans, personal communication, 2011), andtherefore a viable assumption for the minimum triggering pressure.With the analytical solution used in this study, we misuse the factora0 as a free parameter to fit realistic pressure values near the trig-gering front. This comes at the expense of incorrect pressure valuesvery close to the injection point (values beyond the 30 MPa contourin Figure 2). More realistic pressure estimates across the wholevolume might be achieved by including time-varying permeabilityenhancement. However, the unrealistically high values very close tothe borehole have no impact onto our triggered seismicity cloud andcan therefore be ignored. In general, it should be noted that theabsolute number of events triggered in the modeling is governedby four parameters: (1) the magnitude of the pressure distribution(controlled by the factor a0), (2) the average differential stress (con-trolled by our assumption of the background stress field), (3)the friction coefficient, and (4) the seed density. There exist manypossible combinations of these four parameters that result in verysimilar seismicity patterns and a-values. The large effective sourceradius a0 used in our model compared to Dinske et al. (2010) tradesoff with the number of seed points chosen in the model. The smallerthe source radius, the more seed points are required to induce anevent cloud with a similar a-value. We seek to constrain the realisticparameter range by reasonable a priori assumptions. However, theactual choice of these four parameters has little influence on theconclusion as long as the overall behavior of the seismicity cloudis matched.Schorlemmer et al. (2005) infer that the b-value acts as a

stress meter that depends inversely on differential stress σd. Thisinverse relationship has also been suggested by laboratory studies(Amitrano, 2003). To model the size of each induced event, we linkthe b-value to σd in an inversely proportional relationship, and thenrandomly draw a magnitude from a Gutenberg-Richter relation with

Table 1. Mean medium parameters.

Model Avg. crust Avg. crust Weak crust Strong crust

depth 4.5 km 2.5 km 4.5 km 4.5 km

σ̄3 (MPa) 75 42 75 75

σ̄1 (MPa) 185 105 147 232

max σ1 (MPa) 232 129 232 232

std. dev. (σ1, σ3) 10%

ph (MPa) 45 25 45 45

μ 0.85 0.85 0.6 1.0

No. of seeds 30,000

Cohesion (MPa) 7

a0 (m) 70

p0 (MPa) 9.5

pt (Pa / s) 48

t0 (s) 4.32 · 105

D ðm2∕sÞ 0.05

0 1 2 3 4 5 6 7 8

×105

0

200

400

600

800

1000

Time (s)

Dis

tanc

e (m

)

0.00

2

0.002

0.002

0.002

0.01

0.01

0.01

0.01

0.1

0.1

0.1

1

1 1

1010

# Repeats1 2 3 4 5+

3030

Figure 2. Time-distance plot of the simulated induced seismicity.The color denotes the number of repeats of an event. Note thatthe colorscale is clipped at five repeats. The dashed line showsthe shut-in time. The solid contour lines show isobars in MPa.

KS28 Goertz-Allmann and Wiemer

the corresponding b-value. It is this inverse relationship that pro-vides the coupling between the physical modeling and the probabil-istic magnitude assignment in our model. We seek to base thiscoupling term on empirical observations to the extent possible.For this purpose, we use the following working hypothesis:(1) the long-term average tectonic b-value is one or lower, and(2) the relation between differential stress and b-value is actuallylinear. The latter hypothesis is suggested by the laboratory resultsof Amitrano (2003), whereas the former assumption results from theobservation of b-values approaching one for the postinjection seis-micity at large distances and long after the injection (Bachmannet al., 2012), as well as mean b-values below one observed bySchorlemmer et al. (2004, 2005), and Gulia and Wiemer (2010).First, we choose σd ¼ 136 MPa as the upper differential stress lim-it, which corresponds to the mean differential stress includingrespective standard deviations of σ1 and σ3 for the average crustat 4.5 km depth (see Table 1). We fix the lower b-value limit(bmax) to one at σd ¼ 136 MPa, but we loop over possible upperb-value limits (bmax) at the hypothetical value of σd ¼ 0 (insetof Figure 3) and compute the L1 misfit between the mean b-valueof the observed data and the mean b-value from 50 model runs,each with a different realization of random seeds and stresses.The best fit between the Basel data and the modeled seismicityis observed at the minimum of the quadratic least-squares fit(0.058x2 − 0.46xþ 0.99), which corresponds to bmax ¼ 4� 0.24

(the uncertainty is estimated using Viegas et al. (2010)). We there-fore use a linear scaling relation of b ¼ −0.022σd þ 4. The result-ing relation between σd and b-value is suitable for the Basel data.Additional data points for coincident stress magnitude and seismi-city measurements will be required to investigate whether this rela-tion might be varying between locations or stress regimes.In the next step, we calibrate the modeled seismicity to the overall

number of events observed in the Basel data. This calibrationstep is necessary for the calculation of realistic a-values and henceabsolute values of forecasted magnitude probabilities. Instead ofmodifying the seed density, which also controls the overall numberof induced events but is computationally very expensive, we esti-mate a constant scaling factor s from 100 model runs withs ¼ ðP100

i¼1 Nobserved∕NmodelÞ∕100, where N is the overall numberof induced events. We obtain s ¼ 0.92, and therefore randomly pick92% of the modeled seismicity of each run for further analysis.Event stress drop Δσ is also assumed to be proportional to σd.

After an event is induced due to the increase in p, σ1 is reducedby 10%� 5% of σd. We define the difference in shear stress τs be-fore and after the reduction of σ1 to be the stress drop (Figure 1b).Due to the stress drop, the point in the medium becomes stableagain and a further increase in p at a later time may repeat the entireprocess at this point (Figure 1b). Note that the scaling between Δσand σd is arbitrarily defined. A higher percentage would lead tohigher absolute Δσ values and a larger standard deviation to a high-er scatter in individual Δσ. The scaling factor used results in a meanunsmoothed Δσ of 2.56 MPa and a variation over two orders ofmagnitudes of individual Δσ values. This is similar to stress-dropestimates obtained for the Basel induced seismicity (Goertz-Allmann et al., 2011).The result is a forward-modeled seismicity cloud with location,

origin time, stress drop, and magnitude assigned to each event loca-tion, and calibrated to match the overall number of events observedin Basel. About 1000 events are induced with about 50% of these

events being repeaters (seed points rupturing more than once,Figure 2). This is consistent with a multiplet analysis for the Baselmicroseismic events, which grouped 52% of all events into multi-plets (Kummerow et al., 2011). Most repeaters in our model occurclose to the injection point, where pore pressures are higher, and thenumber of repeats decreases with overall pore pressure perturba-tions, and hence with distance.

DISTANCE DEPENDENCE OF SOURCEPARAMETERS

Despite restriction to a hydraulically homogeneous and isotropicmedium, our model is able to explain the two seismological obser-vations of reduced stress drops (Goertz-Allmann et al., 2011) andincreased b-values (Bachmann, 2012; Bachmann et al., 2012) nearthe injection point where pore-pressure perturbations are highest. Italso realistically simulates the occurrence of repeating event clus-ters, consistent with observations (Kummerow et al., 2011). Color-coding b-value and stress drop (Figure 4a and 4b) for one model runshows the distance-dependence of both compared to the observedBasel seismicity (Figure 4c). To enable a fair comparison, weestimate the b-value from the synthetic seismicity cloud in the samemanner than for the real data: We compute the b-value from theevent magnitudes over a fixed number of 100 surrounding eventsto evaluate the lateral variation following Bachmann et al. (2012).Stress drop is spatially smoothed using a median filter over the clo-sest 20 events. The smoothing of Δσ is needed to investigate any

1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

| bob

serv

ed-b

mod

el |

bmax

1

136

2

5.5

0Diff. stress σd (MPa)

b-va

lue

Best fit: b = –0.022σd+4

bmin

}bmax

Figure 3. Data-driven definition of the relation between σD andb-value. Modulus of the overall b-value difference between theBasel data and the model versus possible upper b-value limits(bmax). The circles show the mean value from 50 model runs withstandard deviation. The solid line shows a quadratic fit with itsminimum determined at bmax ¼ 4 (dashed line). The respective bestfitting scaling relation is shown. The inset shows a sketch of theunderlying b-value versus differential stress σd scaling relationsusing different upper b-value limits. The lower b-value limit is fixedto one at σd ¼ 136.

Modeling induced seismicity hazard KS29

spatial variability and some spatial smoothing has also been appliedto the observed Basel data (Goertz-Allmann et al., 2011). We noticethat the variance of the b-values and especially of the stress drops isconsiderable, even though this is synthetic data and we based ourmodeling originally on a comparatively moderate variance of 10%for the principal stresses. To analyze the properties of the syntheticcloud in a statistically robust way, we run 100 model runs, each witha different realization of the random seeds and stresses. The result ofthe spatial b-value and stress drop variations is summarized inFigure 5. Although Figure 5a and 5b shows a 3D display and across-sectional view for one model run, Figure 5c shows a grid-stack of the respective b-value (top) and stress drop (bottom) over100 model runs. Mean b-value and mean stress drop show a stabledistance dependence from the injection point with high b-value andlow stress drop at the close distance. Figure 5d shows a correspond-ing plot for a constant b-value of 1.48 and a constant stress drop of2.5� 1.25 MPa, without any dependence on differential stress. In

this case, lateral changes in b-value or stress drop are random andno distance dependence is observed.If we plot the source parameters versus distance for the case of

stress-coupled b-values (Figure 6), we observe that the distance de-pendence is strongest up to 200–300 m and decreases at further dis-tances. This is especially visible if we compute the mean of themean values of each distance bin from 100 model runs (Figure 6cand 6d). The error bars in Figure 6a and 6b show the standard errorof the mean values and have been computed using bootstrap resam-pling over 1000 realizations. The error bars in Figure 6c and 6dshow the standard deviation.The higher the pore-pressure perturbation, the less critical stress

states still trigger an event and hence the lower the differential stresscan be before triggering an event (Figure 7). The inset in Figure 7ashows the decrease of differential stress with increasing pore pres-sure. This dependence results in lower differential stress close to theinjection point and an increase of differential stress at further dis-tances (Figure 7b). Therefore, less critical stress states result in low-er stress drops and higher b-values, if both are linked to differentialstress. We expect the lowest Δσ and highest b-value close to theinjection point where the pore pressure is the highest. In additionto the effect described above, the distance dependence of b-valueand stress drop is strengthened by repeating events. Each repeat atone location results in a smaller differential stress than its predeces-sor due to the previous event stress drop. This leads to increasinglyhigher b-value and decreasingly lower stress drop for each repeater.Because the number of repeaters increases close to the injectionpoint (Figure 2), the distance dependence of stress drop and b-valueis emphasized.

BREAK IN SELF-SIMILARITY OF SEISMICITY

Many source parameter studies indicate that Δσ is scale invariantfor natural seismicity (e.g., Abercrombie, 1995; Allmann andShearer, 2009), as well as for induced seismicity (e.g., Kwiatek et al.,2010; Goertz-Allmann et al., 2011). However, there are also variousstudies that found an increase of stress drop with magnitude (e.g.,Mayeda and Walter, 1996), and this topic remains a controversialquestion with important implications for seismic hazard analysis.Our model implicitly assumes a connection between b-value and

stress drop because both parameters are linked to differential stress.We therefore indirectly input a nonself-similar stress drop scalingwith magnitude in our model. Figure 8 shows the relation betweenstress drop and magnitude for different assumptions. No depen-dence of stress drop and magnitude is observed if the stress dropis linked to differential stress but includes a 5% scatter (Figure 8a),and we cannot distinguish the result from a constant stress dropmodel (Figure 8c). A slight increase of stress drop with magnitudeis observed for the raw stress drop values only if the additional5% scatter of the stress drop is omitted in the modeling (top ofFigure 8b). This can be better seen if we directly compare the meanvalues (Figure 8d). If a spatial smoothing is applied to the individualstress drops (bottom row of Figure 8), the expected dependencewith magnitude becomes indiscernible. This shows that even syn-thetic stress drop scaling is difficult to resolve. It appears to be anextremely challenging task to extract empirically sound scalingrelations from real data, where we have to deal with considerableadditional scatter due to noise and measurement errors. Further-more, our result suggests that a spatial smoothing of parametersshould not be applied if scaling relations are investigated.

# ev:964

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Basel observed seismicity

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tanc

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Simulated seismicity

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×105

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1

Time (sec)

Dis

tanc

e (k

m)

b-value1 1.5 2 2.5

Simulated seismicitya)

b)

c)

Figure 4. Time-distance plot of the induced seismicity. The colordenotes (a) b-value and (b) smoothed log stress drop for one modelrun. The dashed line denotes the shut-in time. Stars mark all eventswithM ≥ 2.5. (c) Observed Basel seismicity for all events above themagnitude of completeness (Mc ¼ 0.9). Color denotes the b-valuesestimated at a certain event location by Bachmann et al. (2012).Note that the colorbar in (c) is clipped at both ends for a better com-parison with the modeled seismicity.

KS30 Goertz-Allmann and Wiemer

TIME AND LOCATION OF LARGEMAGNITUDE EVENTS

Several conditions must be met for significant (damaging) earth-quakes to occur. There must be a fault large enough to allow sig-nificant slip, there must be stress present to cause the slip along thefault and this stress must exceed the strength of the fault. It is notclear today what is the upper possible magnitude limit (Mmax) forinduced events in a comparatively shallow reservoir setting (<5 km

depth). However, this is an important input parameter to probabil-istic seismic hazard assessment.Using our model, we investigate the probability of an event

exceeding a certain magnitude with respect to injection time anddistance from the injection point (Figure 9). We evaluate the eventsfor a- and b-values within specific time and distance bins. For eachbin, the probability P of a certain magnitude M event is defined as(Wiemer, 2000),

P ¼ 1 − e−ða−MbÞ: (3)

We choose time bins of 105 s (1.16 days), moving at 104 s (3.84 h)intervals, and distance bins of 100 m, moving across distance in10 m increments. To obtain more stable results, we analyze the in-duced seismicity from 100 model runs and stack individual prob-abilities. This allows us to compute a standard deviation to eachmean probability estimate. The probability of a large magnitudeevent is determined by two factors; (1) the overall number of eventsper bin (i.e., the a-value), and (2) the b-value in this bin. We find

that the mean probability of a large magnitude event increases withtime up to the shut-in time (upper row Figure 9). This can partiallybe explained by the steadily increasing number of induced events upto shut-in and therefore increasing a-values, which is reflected byincreasing probabilities up to the shut-in for a constant input b-valuemodel. A varying b-value causes a substantial increase of the meanprobability for the time period right after shut-in. The probability ofa large magnitude event also increases to much further distancesfrom the injection point for a varying b-value compared to a con-stant b-value (bottom row Figure 9). The increased probability atlater times and larger distances is especially prominent for largermagnitudes (Figure 9c). Note that the large difference in absoluteprobability before the shut-in time between the constant and thevarying b-value model mostly depends on the choice of a constantb-value of 1.48. A larger constant b-value would bring the probabil-ity levels of the two models closer at early times. The result of anincreased probability of larger magnitude events at later times justafter the shut-in and at larger distances to the injection point is con-sistent with observations of large magnitude events during hydrau-lic stimulations in geothermal systems not only in Basel(Deichmann and Giardini, 2009), but also elsewhere (Charlety et al.,2007; Evans et al., 2012).Note that the simple, hydraulically homogeneous, and isotropic

model is sufficient to explain the principle phenomenology. Theaccuracy of the prediction could certainly be improved by takinginto account more realistic hydraulic models. In Basel, for example,anisotropy of hydraulic diffusivity is probably significant judgingfrom the shape of the event cloud. This would have an effect on

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x (km)

1.44 1.48 1.52

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b-value

log10 stress drop (MPa)

d)

log10 stress drop (MPa)0.2 0.3 0.4 0.5

log10 stress drop (MPa)0.2 0.3 0.4 0.5

b-value1.2 1.6 2

b-value1.2 1.6 2

a) d)b) c)

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y (k

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−0.4 −0.2 0 0.2 0.4

−0.4

−0.2

0

0.2

0.4

x (km)

y (k

m)

Figure 5. Spatial b-value (upper panels) and smoothed stress drop (lower panels) distributions of induced events in (a) 3D view for one modelrun, (b) cross section for one model run, (c) grid-stack of all cross sections over 100 model runs, and (d) grid-stack of all cross sections over 100model runs using a constant input b-value and stress drop as described in the text. The cross sections include all events within 100 m to a planethrough the injection point.

Modeling induced seismicity hazard KS31

lateral variation of seismic hazard. On the other hand, the analyticsolution available for the simple homogeneous isotropic caseenables fast computation. A more complex hydraulic model wouldrequire a numerical modeling approach which may be incompatible

with the requirement of a real-time implementation in an advancedtraffic light system.

DEPENDENCE ON DEPTH ANDCRUSTAL STRENGTH

To test a more general applicability of the sim-ple geomechanical model for the purpose of for-ward-modeling expected seismic responses tofluid injections, we now investigate the influenceof some of the modeling parameters onto the re-sulting seismicity cloud and source parameters.First, we calculate one model with an injectionat a shallower depth of only 2.5 km, with corre-spondingly smaller overall stress magnitudes(Häring et al., 2008) and with only half the porepressure. In a second step, we vary the strength ofthe crust by adjusting σ1 and the coefficient offriction μ. Input parameters of the three addi-tional models are shown in Table 1.We keep the scaling relation between differen-

tial stress and b-value, respectively stress drop,exactly the same as before, even though this de-pendence was obtained heuristically on the basisof the Basel observations. Varying differentialstresses between the three different scenariostherefore lead to varying absolute values of aver-age stress drop and b-value. Our main interest isthe relative variation of the source parametersand probabilities with time and distance. Theserelative variations can be extracted reliably fromthe modeling runs, and some more general pos-tulates about induced seismicity can be deriveddespite variations in absolute values. It is unclearat this point whether any scaling between differ-ential stress and b-value or stress drop wouldbe varying in different geologic situations.However, for the time being we consider it

101

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(M

Pa)

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101

102

103

100.1

100.3

100.5

Smoo

thed

str

ess

drop

(M

Pa)

Distance (m)

c)

d)

Figure 6. (a) The b-value, and (b) stress drop versus distance from the injection point forone model run. The squares in (a) and (b) show the mean values over constant log dis-tance bins with respective standard errors from bootstrap resampling. (c) Mean b-value(crosses), and (d) mean stress drop (crosses) versus distance from the injection pointfrom 100 model runs. The squares in (c) and (d) show the mean of the mean valuesfrom 100 model runs with standard deviation.

0 20 40 60 80 100 120 140 1600

20

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τ s (M

Pa)

σd (MPa)

σ d (

MPa

)

50

100

150

0 10 20

50

100

150

Pore pressure (MPa)x-direction (km)

y-di

rect

ion

(km

)

−0.5 0 0.5

−0.4

−0.2

0

0.2

0.4

σd (MPa)70 80 90 100 110 120 130 140

a) b)

Figure 7. (a) Differential stresses of all induced events (colored dots) plotted at the minimum distance of the Mohr circle to the failure envelope(red dashed line). The maximum and minimum differential stresses are indicated by the black circles. The inset shows differential stress versuspore pressure. (b) Cross section through injection point of stacked differential stress from 100 model runs.

KS32 Goertz-Allmann and Wiemer

a realistic assumption because it explains the observation at Baselquite well.At shallow depth and for a weak crust, we observe overall much

higher b-values above two (stars and diamonds, respectively,Figure 10) compared to the original model at 4.5 km depth (circles),

for which we obtain a good fit to the observed Basel seismicity(squares). The strong crust model results in a much lower b-value(inverted triangles). The stronger the distance dependence of theb-value, the worse the fit of the data to a constant average b-value.This is particularly evident for the strong crust model (inverted

0

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ooth

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Magnitude1 1.5 2 2.5 3 3.5

Magnitude1 1.5 2 2.5 3 3.5

Magnitude

Varying stress dropNo additional scatterConstant stress drop

a) b) c) d)

Figure 8. Stress drop versus magnitude for (a) the original model where stress drop is linked to differential stress and a 5% scatter is included,and (b) the same as (a) but without the scatter. (c) The constant stress drop of 2.5� 1.25 MPa and (d) the comparison of the mean values of (a)to (c). Upper row shows raw stress drops and bottom row spatially smoothed stress drops. The bold symbols show the mean value within 0.2magnitude bins with respective standard error from bootstrap resampling using 1000 realizations.

0 1 2 3 4 5 6 7 8×105

0

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abili

ty

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M > 3

0 100 200 300 400 500 6000

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ty

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abili

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M > 4

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0.035

Prob

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ty

Distance (m)

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ty

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M > 5

0 100 200 300 400 500 6000

0.5

1

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3 ×10 −3

Prob

abili

ty

Distance (m)

Shut-in

Basel ML3.4

Varying bConstant b

Varying bConstant b

Varying bConstant b

a) b) c)

Figure 9. Probability of an event exceeding a magnitude (a) M3, (b) M4, and (c) M5 to occur at a certain time (top row) and distance from theinjection point (bottom row). Error bars show the standard deviation computed from 100 model runs. The dashed line marks the shut-in time intime, and the location of the largest observed Basel event in distance. The different colors denote the model where the b-value is linked todifferential stress (white), and the model with a constant input b-value (gray).

Modeling induced seismicity hazard KS33

triangles). The distance dependence of the b-value is greatly re-duced for the shallow and weak crust models due to the reducedrange of differential stresses (Figure 11). Interestingly, the stressdrop does not show a strong difference in the distance dependencebetween the different models.In the next step, we impose a depth-dependent gradient onto the

background stress field for the average crust case and an injectionsource at 4.5 km depth. Following Häring et al. (2008), the back-ground stress field changes as follows

phðzÞ ¼ 10MPa

kmz (4)

σ1ðzÞ ¼ 42MPa

kmz� 10% (5)

σ3ðzÞ ¼ 17MPa

kmz� 10%. (6)

Because we leave the relation between differential stress and b-va-lue the same, we would expect to observe a depth dependence of theb-value on top of the radial dependence from the injection point.Figure 12 shows that this is indeed observed. However, the depthdependence is a secondary effect to the radial dependence and there-fore difficult to observe in practice. Revealing the mean depth de-pendence in the synthetic data requires correcting each event first bythe radial dependence of b-value and stress drop as shown inFigure 6c and 6d, respectively. The result is a variation of b-valuesfrom 1.4 near the top of the seismicity cloud to 1.2 for the deepestevents (diamonds in Figure 12a). Similar to the b-value, aweak depth dependence is also imprinted onto the stress drops

(Figure 12b). Figure 12c and 12d shows the depth variation ofb-value and stress drop of the observed Basel data. As expected,the large scatter in the data does not allow us to observe such rathersubtle depth variations in reality. Although at least the meanb-values per depth bin (white squares in Figure 12c) are not incom-patible with a depth variation, the scatter is too large to reliably re-solve a depth dependence with statistical significance. The observedstress drops show no indication of a depth dependence.In summary, our results suggest that the probability of large

events (M ≥ 4) is reduced if we either stimulate less deeply, or drill

101

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ss d

rop

(MPa

)

Distance (m)

Average at 4.5 km

Shallow

Weak

Strong

101

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1

1.5

2

2.5

3

3.5

b-va

lue

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a)

b)

2

3

4

0.9

Figure 11. Here, (a) b-value, and (b) stress drop versus distancefrom the injection point for various model setups. The symbolsshow the mean over constant log distance bins of the mean valuesfrom 100 model runs with respective standard deviations for theaverage crust model at 4.5 km depth (circle), the shallow crust mod-el at 2.5 km depth (stars), the weak crust model (diamonds), and thestrong crust model (inverted triangles).

1 1.5 2 2.5 3 3.5 4

0

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1

1.5

2

2.5

3

3.5

Magnitude M

log(

N)

Observed: log(N) = 4.3-1.48 MAverage: log(N) = 4.36-1.55 MWeak crust: log(N) = 4.96-2.24 MStrong crust: log(N) = 3.88-0.91 MShallow crust: log(N) = 4.92-2.43 M

Mc

Figure 10. Gutenberg-Richter relations of one model run for differ-ent model setups compared to the observed Basel data (squares). Thecircles show the average crust model at 4.5 km depth. The diamondsshow the weak crust model. The inverted triangles show the strongcrust model, and the stars show the shallow crust model. See Table 1for input parameters. The magnitude of completenessMc is marked.

KS34 Goertz-Allmann and Wiemer

into softer formations such as sediments, insteadof granite (Figure 13a and 13b). For the strongcrust model, the probability of a M ≥ 4 eventis overall higher compared to the average crustmodel at 4.5 km depth (Figure 13c).However, because we do not know whether

our scaling between b-value and σd is universallyvalid, or only valid for Basel, we cannot interpretthe absolute probability levels anywhere otherthan Basel where we derived our scaling relation.Nevertheless, this does not prevent us from com-paring the relative probabilities within one sce-nario before or after shut-in or with distance.The common feature of most modeled scenariosis the observation of the highest probability of alarge magnitude event after well shut-in and atlarger distances to the injection point. To showthis more quantitatively, we calculate normalizedcumulative probabilities in time and radial dis-tance to the injection point (Figure 14) for thedifferent model set-ups. For each case, we com-pare the cumulative probability ratios before andafter the shut-in time, as well as closer andfurther away than 300 m distance to the injectionpoint. Individual values are listed in Table 2.Note that the 300 m distance level is arbitrarilychosen to separate the induced seismicity intotwo equal groups. For example, we find thatour model predicts a 61.5% probability for aM ≥4 event after the shut-in time, and a 53.2% prob-ability for an event beyond 300 m if b-valueis coupled to differential stress. These probabil-ities are higher than their respective counter-parts before the shut-in and closer than 300 m,

4 4.2 4.4 4.6 4.8 51

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With no distance correctionWith distance correction

4 4.2 4.4 4.6 4.8 52

2.5

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3.5

4

Smoo

thed

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ess

drop

(M

Pa)

Depth (km)

Figure 12. Here, (a) b-value and (b) smoothed stress drop versus focal depth for themodeled seismicity (average crust, injection at 4.5 km). Circles show the data (meanfrom 100 model runs with standard deviation) without correction for the distance tothe injection point. Diamonds show the same data after correction for the radial distancedependence. (c) Observed b-value versus focal depth for the Basel seismicity estimatedby Bachmann et al. (2012). (d) Observed stress drop versus focal depth for the Baselseismicity estimated by Goertz-Allmann et al. (2011). The bold squares in (c) and (d)show the mean value within 0.1 km depth bins with respective standard error from boot-strap resampling using 1000 realizations.

0 1 2 3 4 5 6 7 8×105

0

1

2×10−4

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abili

ty

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Weak crust

0 100 200 300 400 500 6000

1

2×10−4

Prob

abili

ty

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0 1 2 3 4 5 6 7 8×105

00.10.20.30.40.50.60.70.8

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0 100 200 300 400 500 6000

0.10.20.30.40.50.60.70.8

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abili

ty

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0 1 2 3 4 5 6 7 8×105

00.51

1.52

2.53

3.54 ×10−5

Prob

abili

ty

Time (s)

Shallow crust

0 100 200 300 400 500 6000

0.51

1.52

2.53

3.54 ×10−5

Prob

abili

ty

Distance (m)

a) b) c)

Shut-in

Basel ML3.4

Figure 13. Probability of an event exceeding a magnitude M4 for (a) a shallow crust model at 2.5 km depth, (b) a weak crust model, and for (c) astrong crust model to occur at a certain time (top row) and distance from the injection point (bottom row). Error bars show the standard deviationcomputed from 100 model runs. The dashed line marks the shut-in time, and the location of the largest observed Basel event in distance.

Modeling induced seismicity hazard KS35

respectively. A constant b-value always predicts higher probabil-ities before the shut-in time and closer than 300 m to the injectionpoint compared to the model where b-value is linearly linked todifferential stress. Whereas the distance dependence is reducedfor the shallow crust model, we also find that the weak and strongcrust models predict higher probabilities of a M ≥ 4 event aftershut-in and at larger distances. Two important consequencesarise from this observation: (1) it is insufficient to shut-in an injec-tion operation upon observation of a threshold magnitude event

(so-called traffic-light system), because the largest magnitudeprobability is yet to come after the shut-in, irrespective of the ap-plied threshold magnitude. (2) increased probabilities for largermagnitude events at larger distances from the well means that alarger area around the injection well is potentially affected by in-creased seismic hazard. Although risk mitigation is always possibleby limiting the injected volume, the decision process to alter theoperation for risk mitigation has to start at a much earlier timeand needs to take into account the time variability of the b-value.

0 1 2 3 4 5 6 7 8×105

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mal

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ulat

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M≥3M≥4 Varying b M≥5M≥3M≥4 Constant b M≥5

}}

Shut-in

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Shut-in

c)a)

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}}

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mal

ized

cum

ulat

ive

prob

abili

ty

Distance (m)

M≥4: Average M≥4: ShallowM≥4: WeakM≥4: Strong

d)b)

Figure 14. Normalized cumulative probability of an event exceeding a certain magnitude versus time (a and c) and distance to the injectionpoint (b and d). (a and b) Comparison of an average crust model at 4.5 km depth for varying b-value (black curves) and constant b-value (graycurves). (c and d) Comparison of an average crust at 4.5 km depth (solid black), shallow crust (dashed gray), weak crust (dotted gray), andstrong crust model (solid gray). The vertical lines marks the shut-in time (a and c) or the 300 m distance (b and d).

Table 2. Comparison of normalized cumulative probabilities pcum of exceeding a certain magnitude event in time and distance tothe injection point for various model set-ups.

Magnitude M ≥3 ≥4 ≥5 ≥3 ≥4 ≥5 ≥4Cumulative probability [%] Varying b Constant b 2.5 km Weak Strong

Before shut-in 55.3 38.5 23.2 75.5 73 68.4 65.3 30.8 45.2

After shut-in 44.7 61.5 76.8 24.5 27 31.6 34.7 69.2 54.8

≤300 m 54.7 46.8 39.4 72.5 70 64.3 60.1 41.6 50.1

≤300 m 45.3 53.2 60.6 27.5 30 35.7 39.9 58.4 49.9

KS36 Goertz-Allmann and Wiemer

Note that, for a constant b-value, the injected volume is proportionalto the total seismic moment (McGarr, 1976), whereas for a varyingb-value the injected volume is proportional to the overall number ofevents (Shapiro et al., 2010). We speculate that the correlationbetween differential stress and pore pressure perturbation (insetin Figure 7a) is the main driver in a hydraulic stimulation setting.This is corroborated not only by observations of Bachmann et al.(2012), but also by b-value observations in hydrocarbon reservoirsettings (Maxwell et al., 2009; Wessels et al., 2011): High b-valueswhen new (tensile) fractures open (previously stable points withcorresponding low differential stress) due to high pore pressureclose to the injection and low b-values when preexisting fractures(with larger differential stress, i.e., higher criticality) are reactivated.Our study suggests that a priori information about the stresses in

the area of interest are important for some initial estimation of theexpected seismic hazard. Furthermore, the implementation of anear-realtime probabilistic seismic hazard prediction model seemsfeasible and can be used to replace previously used traffic light sys-tems. Our model can also be used to investigate the effect on theinduced seismicity and the respective seismic hazard using varyingstimulation approaches such as a different injection flow rate or adifferent wellhead pressure.

CONCLUSIONS

We propose a simple geomechanical model for induced seismi-city that links differential stress to b-value and stress drop. With thismodel, we can forward-model the induced seismicity response to ahydraulic injection in space and time. In addition, we can forward-model source parameters such as stress drop and magnitude ofevents in a stochastic way. The latter allows us to estimate the seis-mic hazard associated with an injection operation by calculating theprobability of exceeding a certain magnitude event. Our model canexplain the observation of increasing stress drops and decreasingb-values as a function of radial distance from the injection well.The probability of exceeding a certain magnitude event is largerafter well shut-in, and reaches further out from the injection point.The overall procedure can be coupled with incoming seismicity datafrom an ongoing stimulation operation, and thus allows to estimatethe seismic hazard of an injection operation in real time. In caseswhere the in-situ stress field and reservoir rock properties are knownsufficiently well beforehand, the model can be used to estimate theseismic response to fluid injection before the stimulation such thatcritical parameters (e.g., injection volume, time, and pressure) canbe planned and optimized for particular projects.

ACKNOWLEDGMENTS

We are grateful to GeoPower Basel for permission topublish these results. This research is supported by the projectGEOTHERM that is funded by the ETH Competence Center forEnvironment and Sustainability. This work is also supported bythe European Commission through funds provided within theFP7 project GEISER, grant agreement no. 241321-2. We wish tothank Alexander Goertz, Valentin Gischig, and Corinne Bachmannfor fruitful discussions that helped to improve the manuscript. Wethank three anonymous reviewers and the associate editor ShawnMaxwell for their thorough comments which helped to significantlyimprove the manuscript.

APPENDIX A

CONSTRAINTS ON THE SEED DENSITY

We assume that every seed point in our volume represents a disk-shaped fracture of random orientation with a radius that depends onthe assigned magnitude. Magnitudes are randomly assigned, butfollow a Gutenberg-Richter distribution with a prescribed b-value.The higher the seed density (seed points per volume), the more arethe fractures intersecting each other, up to the point where intersect-ing fractures are completely separating individual parts of the rockframe from each other. The fracture porosity at which this is reachedis termed critical porosity ϕc (Nur et al., 1998); it is the point be-yond which the fractured material would fall apart. Obviously, wehave to choose the seed density such that the fracture porosity staysbelow this critical value.Assuming circular ruptures, we can define a moment-dependent

fault radius (Eshelby, 1957),

r ¼�7

16

M0

Δσ

�1∕3

; (A-1)

withM0 the seismic moment, and Δσ the stress drop, which we willassume constant for simplicity in this exercise. We know thatlog10M0 ∝ 1.5Mw, and hence log10r ∝ 0.5Mw, with the Mw distri-bution following a Gutenberg-Richter statistic above the magnitudeof completeness. Consequently, the radius r will follow a similardistribution, and we have to find a meaningful average radius r̄ overthis distribution.For disk-shaped fractures, the crack density can be given as

(Kachanov, 1992)

ρ ¼ NV0

r̄3; (A-2)

where V0 denotes a reference volume, r̄ is the average radius of acircular rupture for a given event distribution, and N is the overallnumber of cracks (seeds) in the volume. To define a porosity to amedium cracked in such a manner, we need to assign an openingwidth or aperture d to these cracks. For an ellipsoidal shape of thecrack, the porosity can then be stated as (Saenger et al., 2004),

ϕ ¼ 4N3V0

πr̄2d: (A-3)

To define a lower critical porosity limit for such a medium, we con-sider a simplistic thought experiment with uniformly oriented andregularly distributed fracture sets (Figure A-1). We attempt to fill acube of side length n · a with a network of nonintersecting fractureswith diameter 2r, spaced a distance a from each other. In this man-ner, we can fill the cube with 1∕2 n cracks in the direction of the twolong axes of the crack, and n cracks in the direction of the apertured ≪ r, such that to fill the whole cube, it requires 1∕4 n3 cracks.Assume now that the medium is intersected by three mutually per-pendicular fracture networks with a total of 3∕4 n3 cracks. In orderfor these fracture networks to fully intersect each other, the crackradius r needs to be such that it describes the circumscribed circle ofa square of side length a,

a ¼ rffiffiffi2

p: (A-4)

Modeling induced seismicity hazard KS37

Using equation A-3, we can now state the porosity for this case as

ϕc ¼1

2ffiffiffi2

p πα; (A-5)

where α denotes the crack aspect ratio d∕r. We assume that for morecomplicated situations, e.g., randomly oriented fractures, the criticalporosity would be higher. Combining equations A-2 and A-3, wecan define an upper bound for the crack density ρ, which we equalto the seed density in our modeling,

ρ ≤3

8ffiffiffi2

p ≈ 0.27; (A-6)

and with that define an upper bound for the number of seeds in ourmodeling as

N ≤0.27V0

r3: (A-7)

To give an example, for an average rupture radius r̄ of 20 m, whichcorresponds very roughly to assuming a magnitude of completeness(mode of the magnitude distribution) of one, we could fill our 1 km3

modeling volume with a maximum of 33,125 seeds. Using 30,000seeds, as we have done in our modeling, fulfills criterion A-6, andstays below this simple critical porosity limit.

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