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Quantitative analysis of rock stress heterogeneity: Implicationsfor the seismogenesis of fluid-injection-induced seismicity
Cornelius Langenbruch1 and Serge A. Shapiro1
ABSTRACT
We compared elastic-rock heterogeneity measured by bore-hole logging to the occurrence of seismic events caused by hy-draulic fracturing of the corresponding rock sections. Ourobservations made from two hydraulic fracturing case studiessuggest that elastic-rock heterogeneity controls the occurrenceof fluid-injection-induced seismicity. The seismic events oc-curred preferentially in rock sections characterized by low Pois-son’s ratio and high Young’s modulus. Based on analyticsolutions of stress fluctuations in elastically heterogeneous me-dia in equilibrium to a homogeneous far-field stress, we quan-tified the relation between elastic-rock heterogeneity and stresschanges leading to fracture opening and reactivation in two endmember models of 1D and 3D heterogeneity. We found that sig-nificant fluctuations of rock stress originated from elastic rockheterogeneity. Moreover, we found that stress changes leading
to fracture opening and reactivation in rocks undergo scaleinvariance spatial fluctuations. This gave a possible physical ex-planation for the observed scale invariance of seismogenic proc-esses. We analyzed the physical meaning of a heterogeneityindex of rocks, which indicated rock sections of high Young’smodulus and low Poisson’s ratio. We found that this index canbe used to identify rock sections of high differential stress. Inaddition, our results suggest that it is related to the occurrenceprobability of brittle rock failure during hydraulic fracturing.Nevertheless, we found that fluctuations of Poisson’s ratioshowed an even stronger correlation to critical stress changeswhich resulted in opening and reactivation of fractures in rocks.We obtained an analytic solution of stress fluctuations in verticaltransverse isotropic layers such as shale deposits and applied itto a hydraulic fracturing case study of a shale gas reservoir. Alsoin this case, we observed strong relations between seismicityand fluctuations of stress in space.
INTRODUCTION
The injection of pressurized fluids through boreholes to stimulatehydrocarbon and geothermal reservoirs, is associated with the oc-currence of seismic events. This provides a unique opportunity tostudy different aspects of rock stress fluctuations during the processof earthquake nucleation and rupturing under well-known and evenadjustable conditions. The existence of boreholes at fluid injectionsites opens up the possibility to measure in situ physical rock prop-erties and their fluctuations over a broad range of scales. Moreover,borehole breakouts, drilling-induced tensile fractures, and hydraulicfracturing tests provide reliable in situ stress estimates (e.g., Zo-back, 2010). In addition, the driving forces leading to fluid-injec-tion-induced seismicity, that is, the mass and pressure of the injectedfluid, are known and controllable parameters. This allows us toquantitatively analyze the underlying processes using physics-based
models. For instance, a reconstruction of pore pressure changes trig-gering seismicity during fluid injection in sedimentary and crystal-line rocks by Rothert and Shapiro (2007) suggests that reduction ofeffective normal stress in the range of 103–106 Pa triggered the an-alyzed seismicity. This supports the concept of a critically stressedearth’s crust and suggests that significant heterogeneity of rockstress exists. In general, the hydraulic energy, added to the systemby the injection of pressurized fluids, results in an increase of porepressure in the connected, fluid-saturated pore, and fracture space ofrocks (e.g., Pearson, 1981; Shapiro et al., 1997; Zoback and Harjes,1997). In turn, this causes a decrease of the effective normal stressand a destabilization of the rock in consequence. If the frictionalstrength of preexisting fractures or the tensile strength of the intactrock is exceeded, brittle rock failure and associated seismic eventsoccur. Seismicity induced by fluid injections at any location will fall
Manuscript received by the Editor 29 January 2015; revised manuscript received 19 June 2015; published online 26 August 2015.1Freie Universität Berlin, Department of Geophysics, Berlin, Germany. E-mail: [email protected]; [email protected].© 2015 Society of Exploration Geophysicists. All rights reserved.
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GEOPHYSICS, VOL. 80, NO. 6 (NOVEMBER-DECEMBER 2015); P. WC73–WC88, 12 FIGS., 1 TABLE.10.1190/GEO2015-0061.1
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somewhere between these two end member causes, namely, shearreactivation and tensile opening (Shapiro and Dinske, 2009). Shearreactivation of preexisting fractures usually occurs due to fluid in-jections at geothermal locations, aiming to create enhanced geother-mal systems (EGS) in the crystalline basement. During this type ofhydraulic stimulation, the injection pressure stays below the mini-mum effective principal stress σ3 at the depth of injection. Becauseno new macroscopic fractures can be opened under compressivestress, the fluid-rock interaction is approximately linear (e.g., Lan-genbruch and Shapiro, 2010). During the hydraulic fracturing ofhydrocarbon reservoirs in low permeable sedimentary rocks, the in-jection pressure usually exceeds the large-scale minimum effectiveprincipal stress magnitude, and macroscopic tensile fractures areopened. This results in a significant increase of the rocks’ per-meability and for this reason in a highly nonlinear fluid-rock inter-action (e.g., Hummel and Shapiro, 2012). However because stresschanges leading to shear reactivation can be much lower than stresschanges leading to the opening of new fractures, the seismic eventcloud extends into the rock, beyond the opened hydraulic fracturelengths and widths (Evans et al., 1999). Studies of fluid-injection-induced seismicity from various case studies demonstrate that al-most all events show double-couple source mechanisms, corre-sponding to shear motion on planar faults. However, few tensileevents have been identified by the inversion of moment tensors(e.g., Zhao et al., 2014), which mathematically represent the move-ment on a fault during an earthquake. It is natural to assume that theoccurrence probability of seismic events at a given position in theunperturbed reservoir rock is controlled by the in situ state of stress,which defines the magnitude of critical stress changes necessary toopen new and reactivate preexisting fractures at this location. If rockstress heterogeneity in the unperturbed reservoir rock is importantfor the seismogenesis of fluid injection-induced seismicity, it willdetermine the spatial location and number of seismic events causedby injection of pressurized fluid. Consequently, rock stress hetero-geneity could explain the observation that fluid injection-inducedseismicity often is restricted to certain depth sections and showsstrong variability in the spatial occurrence. However, the underlyingcause and the strength of rock stress fluctuations, which eventuallycontrol the failure process, are still uncertain. The main purpose ofour study is to quantify critical stress changes in the unperturbedreservoir rock. We perform the quantification based on the only di-rectly accessible evidence of rock heterogeneity, which is given byfluctuations of physical-rock properties determined by boreholelogging. We analyze two hydraulic fracturing case studies. Basedon analytic solutions of stress fluctuations in layered linear elasticmedia (see Bourne, 2003), we quantify rock stress fluctuationsoriginating from the elastic-rock heterogeneity measured alongthe treatment wells. The end member model of 1D elastic hetero-geneity in a horizontally layered medium should be a good approxi-mation of sedimentary rocks, which are characterized by a muchsmaller degree of elastic heterogeneity in the horizontal than inthe vertical direction. We compute σ3 and the Coulomb failure stress(CFS) as indicators of fracture opening and reactivation probabilityand analyze if correlations to fluctuations of elastic properties mea-sured along the treatment well exist. In addition, we consider a sec-ond end member model of elastic heterogeneity, in which weassume that the degree of elastic heterogeneity in the verticaland horizontal directions is equivalent. This model is, for instance,representative for crystalline rocks, which are present at EGS and
deep scientific boreholes. Because all rocks show a certain degree ofhorizontal heterogeneity, stress fluctuations will fall somewhere be-tween the two analyzed end member models of 1D and 3D hetero-geneity. Recently, a brittleness index (BI) of rocks has beenproposed (Grieser and Bray, 2007; Rickman et al., 2008). Althoughthe physical meaning of the BI is unclear, it is frequently applied inthe hydrocarbon industry. We analyze if we can observe and physi-cally justify any relation between the BI and brittle rock failurecaused by hydraulic fracturing of sedimentary reservoir rocks. Eventhough the BI is based on isotropic elastic properties, it has beenoriginally introduced for application to unconventional shale reser-voirs, which show a high degree of elastic anisotropy. We extend theanalytic solutions of stress fluctuations in heterogeneous linear elas-tic media (Bourne, 2003) to the case of vertical transverse isotropic(VTI) layers. We apply the solutions to hydraulic fracturing of ashale gas reservoir located in the Horn River Basin, NortheasternBritish Columbia, Canada (e.g., Dunphy and Campagna, 2011).It has been observed that elastic rock heterogeneity determined fromanalysis of borehole logs is of a scale-invariant nature (e.g., Evanset al., 1999; Langenbruch and Shapiro, 2014). Moreover, analysisof borehole breakouts suggests that stress in rocks shows scale-invariant fluctuations (Shamir and Zoback, 1992; Day-Lewis et al.,2010). However, the physical origin of scale-invariant stress fluc-tuations and the relation to elastic-rock heterogeneity are unclear.If stress changes leading to fracture opening and reactivation inrocks undergo verifiable scale-invariant spatial fluctuations, itwould provide a physical explanation for scale invariance of seis-mogenic processes. Moreover, our results would be transferable toother scales of brittle rock failure. Our results will provide a basis tointegrate physics-based geomechanical models of rock stressheterogeneity into the research field of fluid-injection-inducedseismicity.
ROCK BRITTLENESS CONCEPTS
There are various definitions of material brittleness in literature(e.g., Perez Altamar, 2013). Most commonly, a material is definedas brittle if it breaks after experiencing only small deformation. Itmeans that, if stress is applied to a brittle material, it breaks withoutabsorbing much strain energy prior to failure. However, there arebrittleness concepts, which differ from this commonly acceptedphysical definition. Brittleness of rocks, for instance, has been de-fined by its mineral composition and total organic content (TOC)(Jarvie et al., 2007; Wang and Gale, 2009). Recently, a BI has beendefined as a combination of Young’s modulus E and Poisson’s ratioν determined from the P- and S-wave traveltime, as well as densitylogs along boreholes (Grieser and Bray, 2007; Rickman et al.,2008):
BI ¼�
E − Emin
Emax − Emin
þ ν − νmax
νmin − νmax
�×100
2; (1)
where Emin, νmin, Emax, and νmax are the minimum and maximumvalues of E and ν in the reservoir section of interest. The BI definedaccording to equation 1 is a relative quantity of a reservoir, and ac-cording to Rickman et al. (2008), it describes the rocks’ ability tofail under stress (ν) and to maintain a fracture (E), once the rockfractures. The authors state that zones of high BI in the reservoir,characterized by high E and low ν values, should be best suitable for
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hydraulic stimulation because they allow the development of a com-plex fracture network. Table 1 shows the BI of different materialscomputed according to equation 1. The Emin, νmin, and νmax are de-fined as maximum and minimum values in Table 1, respectively.The resulting brittleness values show that the index given by equa-tion 1 is obviously not an absolute measure for physical brittlenessof a material because, for instance, the BI of cork and steel is higherthan the BI of glass. The nomenclature of this heterogeneity indexof rocks seems to be meaningless. To clarify the physical meaningof the BI, we analyze its relation to brittle rock failure caused byhydraulic fracturing of hydrocarbon reservoirs.
OBSERVATIONS: CARTHAGE COTTON VALLEYGAS FIELD
Several hydraulic fracturing experiments were performed in theCotton Valley gas field in a tight gas sand stonereservoir in 1997. An overview of the operationalsetup is given in Rutledge et al. (2004). Due tothe hydraulic stimulation of the perforated zonesin the target formations, numerous seismicevents occurred. The locations of the eventsare shown in map and depth view in Figure 1a.Figure 1b shows the BI computed along treat-ment wells 21-10 and 21-09. The BI is computedaccording to equation 1. The values E and ν aredetermined from 1 m averaged sonic P- and S-waves traveltime, as well as density logs alongthe vertical treatment boreholes. We concentrateour analysis on the depth range (2600–2710 m)of stage-A (21-10) and stage-C (21-09) treat-ments. Seismic events induced by the hydraulicstimulation were registered and located with highprecision. Absolute depth errors, attributed tovelocity-model uncertainty, are as great as4 m, based on misalignment of the source loca-tions with the targeted (perforated) sand intervals(Rutledge and Phillips, 2003). The gray shadedareas in Figure 1b mark the perforated intervals.Brittle rock failure and the associated seismicityare triggered by pore pressure and stress changecaused by the injection of pressurized fluid. Be-cause these stress perturbations initially occuronly along the perforated zones, a comparisonbetween elastic rock heterogeneity and eventdensity with depth is only feasible along the per-forations. The event density (gray lines in Fig-ure 1) computed along well 21-10 (21-09)includes only events induced by stage-A(stage-C). As proposed by Rutledge and Phillips(2003) stage-A events of more than 2650 m wereshifted down 4 m, and events below 2680 m wereshifted up 2 m. The absolute depths of the clus-ters can be reasonably shifted up or down a fewmeters due to velocity uncertainties suggested bythe magnitude of station corrections (1 ms onaverage) Rutledge and Phillips (2003). A clearrelation between BI and event density is visiblealong the perforated intervals of the wells.Higher BI along a perforation results in higher
density of microseismic events. Observable deviations are in therange of location uncertainties with depth. Our observations suggestthat the probability of fracture opening and reactivation is increasingwith the BI. In the next section, we introduce a physics-based geo-mechanical model of rock stress heterogeneity to explain this ob-servation.
GEOMECHANICS: ROCK STRESS FLUCTUATIONSORIGINATING FROM ELASTIC ROCK
HETEROGENEITY
In this section, we compute rock stress fluctuations originatingfrom the elastic heterogeneity of the reservoir rock at Cotton Valleyand relate it to the occurrence probability of brittle rock failure dur-ing hydraulic reservoir stimulation. In general, brittle rock failureand the associated seismicity during hydraulic reservoir stimulation
Table 1. Young’s modulus, Poisson’s ratio, and resulting BI of differentmaterials computed according to equation 1. The resulting brittleness valuesshow that the index given by equation 1 is obviously not an absolute measurefor physical brittleness of a material because, for instance, cork and steel showhigher BI values than glass. The last column corresponds to mean valuescomputed from sonic and density logs of the Cotton Valley rock formation.
Cork Steel Glass Cotton Valley rock formation
Young’s modulus (GPa) 18.6 210 50–90 46.84
Poisson’s ratio 0.0 0.3 0.2–0.3 0.23
Brittleness index (%) 50 50 8–35 19
2600
2610
2620
2630
2640
2650
2660
2670
2680
2690
2700
27100 25
Events/m
2600
2610
2620
2630
2640
2650
2660
2670
2680
2690
2700
2710
25 50 75
BI[%]D
epth
[m]
0 25Events/m
25 50 75
BI[%]
21−09, stage−C21−10, stage−Aa) b)
Figure 1. (a) Source locations of microseismic events for treatments A and C. The topsand bottoms of the perforated intervals are marked with dashed lines in depth views(Figure taken from Rutledge et al., 2004). (b) Comparison of the density of inducedmicroseismic events with depth and the BI along wells 21-10 and 21-09 at the CottonValley gas field. The BI is plotted in black and the event density in gray. A comparisonbetween elastic heterogeneity and event density with depth is only feasible along theperforations. Higher BI along a perforation (gray shaded areas) results in a higher den-sity of microseismic events.
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occur due to tensile opening of new and shear reactivation of pre-existing fractures. Opening of new fractures can only occur if theminimum effective principal stress σ3 is becoming tensile due topore pressure and stress changes caused by the injection of pressur-ized fluids. The reactivation probability of preexisting fractures isgiven by the CFS (e.g., Jaeger et al., 2007)
CFS ¼ S0 − τ þ μðσn − ppÞ: (2)
The CFS is associated with the Mohr Coulomb failure criterion anddescribes the amount of stress change necessary to overcome thefrictional strength of a preexisting fracture of a given orientationin a given stress field. The τ and σn are shear and normal stressacting on the fault plane, and S0 is the cohesion, that is, the shearstress necessary to initiate failure in absence of normal stress. Thepp represents the pore pressure, and μ is the coefficient of internalfriction. A fault is in a stable state if CFS > 0. A fault is unstable ifCFS < 0 and failure occurs if CFS ¼ 0. We compute the CFS ac-cording to fractures optimally oriented for failure. The resulting val-ues of CFS correspond to the minimum change of stress from whichfracture reactivation is possible. We assume a friction coefficient ofμ ¼ 1.0 and cohesion S0 ¼ 5 MPa (see Fischer and Guest, 2011)for our computations. To solely analyze the result of measure-ment-based elastic rock heterogeneity on fracture strength, we keepfriction and cohesion constant. Moreover, this is done because no
direct measurements of heterogeneity of these parameters are avail-able. Figure 2a–2c shows Poisson’s ratio ν, Young’s modulus E,and BI computed from 1 m averaged P- and S-waves traveltimeand density logs along treatment borehole 21-10 at Cotton Valley.We averaged the raw data in 1-m intervals to account for the inher-ent averaging of the logging process over the active length of thetool (approximately 1 m). Because fluctuations of smaller scale thanthe active length of the tool cannot be resolved by the logging proc-ess, we eliminate these apparent small-scale fluctuations, which arephysically not justified. The distribution of ν and E with depth ishighly heterogeneous. The heterogeneity should be caused bychanges of mineral composition and structure (e.g., Watt et al.,1976), slip on crossing faults (e.g., Hickman and Zoback, 2004),damage zones (e.g., Faulkner et al., 2006), and other geologic struc-tures (e.g., Martin and Chandler, 1993). Most of these processes arenot restricted to certain lithological layers. Faulkner et al. (2006)analyze the relation between microfracture density and elastic prop-erties in fault zones and laboratory experiments on rock samples.Their results show that fluctuations of Young’s modulus and Pois-son’s ratio in rocks are governed by changes of the damage level. Inparticular, they find that increasing microfracture damage stronglyaffects the elastic properties of rocks, where Young’s modulus isdecreasing and Poisson’s ratio is increasing with the level of dam-age. It suggests that elastic rock heterogeneity measured in singlelithological units for which changes of the mineral composition canbe excluded could originate from changes of the damage level at
0.1 0.3
2550
2600
2650
2700
2750
2800
ν
Dep
th (
m)
20 40 60E(GPa)
20 4060 80BI(%)
20 40 60 800
0.02
0.04
0.06
0.08
E[GPa]
PD
F(E
)
Log dataBest fit
0.1 0.2 0.3 0.40
0.02
0.04
0.06
0.08
ν
PD
F(ν
)
log databest fit
−3 −2.5 −2 −1.5 −1 −0.5−7
−6
−5
−4
−3
−2
log10
(wavenumber) (1/m)
log 10
(PS
D(E
))
Power spectral densityLinear fit, slope=−1.09
−3 −2.5 −2 −1.5 −1 −0.5−7
−6
−5
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log10
(wavenumber) (1/m)
log 10
(PS
D(ν
))
Power spectral densityLinear fit, slope=−0.92
a) b) c)Pc
ν,E=−0.22
e) g)
d) f)
<E>=46.84 GPa
σE=10.40 GPa
<ν>=0.23
σν=0.05
Figure 2. Poisson’s ratio ν, Young’s modulus E, and BI computed from 1 m averaged P- and S-wave traveltime, and density logs along well21-10. The values ν and E show an inverse correlation characterized by a correlation coefficient of Pcν;E ¼ −0.22. The gray-shaded areascorrespond to perforated zones. (d and e) PDF of E and ν obtained from panels (a and b). The hEi, hνi, σE, and σν correspond to the mean valueand standard deviation of the best-fitting Gaussian distributions. (f and g) PSD of E and ν log data. The PSD characterizes the spatial corre-lation of ν and E.
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different scales. As is observed for most rocks, we find that fluc-tuations of ν and E at Cotton Valley are inversely interrelated (seeFigure 2a and 2b). We quantify this relation by calculating Pear-son’s correlation coefficient Pc given by the covariance of ν andE divided by the product of their standard deviations. The modulishow an inverse relation quantified by Pcν;E ¼ −0.22. The identi-fied inverse development of Young’s modulus and Poisson’s ratiowith increasing damage level (see Faulkner et al., 2006) may causethis inverse interrelation. In any case, measurements of elastic prop-erties along boreholes incorporate all processes, which finally re-sulted in elastic-rock heterogeneity existing in the present-daystress field. To compute the stress fluctuations originating from elas-tic rock heterogeneity, we consider a perfectly horizontally layeredlinear elastic medium in equilibrium to a homogeneous far-fieldstress (see Figure 3). The individual layers are characterized by dif-ferent values of E and ν. Analytic solutions of stress fluctuations inlayered media consisting of isotropic linear elastic layers are givenin Bourne (2003) (see Appendix A). We combine this model to insitu rock heterogeneity determined by borehole measurements forthe first time. Moreover, we test its applicability and verification bycomparing the resulting stress fluctuations with the occurrence offluid-injection-induced seismicity. These seismic events should bereliable indicators of in situ stress. We do not interpret the boreholemeasurement data or analyze core samples to introduce layers ofdifferent thickness into the model. Instead, we consider each meas-urement of Young’s modulus and Poisson’s ratio along well 21-10(see Figure 2a and 2b) as isotropic elastic properties of one elemen-tary layer of 1 m thickness, to account for as many scales as pos-sible. In agreement with Bourne (2003), we assume that the layerscharacterized by different elastic properties are perfectly coupled.This enforces all layers to undergo the same amount of strain inhorizontal direction and results in fluctuations of the horizontalprincipal stresses. The degree of elastic heterogeneity in the verticaldirection will control the magnitudes of fluctuations of the horizon-tal stresses. Because no elastic heterogeneity in the horizontal di-rection is present in a perfectly horizontally layered medium, thevertical stress will be constant throughout the model medium toachieve equilibrium to the far-field stress (see Appendix A andBourne, 2003). Moreover, we assume that thelayered medium is stress free before applicationof the external stress field. It means that residualstress, which should exist in rocks because lith-ification took place under certain nonzero stressconditions, is neglected. The stress fluctuationscomputed under this assumption will be theupper bound of possible stress fluctuations re-sulting from elastic rock heterogeneity. Thefar-field stress that we apply to the layeredmedium is determined from stress and pore pres-sure profiles at Cotton Valley (see Fischer andGuest, 2011). The stress regime corresponds tonormal faulting. Because we do not have reliableinformation on the magnitude of the maximumhorizontal principal stress σ2, we assume thatits magnitude is the mean value between σ1and σ3. The principal stress components of theexternally applied stress field are defined accord-ing to
σ1e ¼ 37.1 MPa ¼ Sv; σ2e ¼ 21.5 MPa ¼ SH;
σ3e ¼ 6.5 MPa ¼ Sh: (3)
These stress magnitudes correspond to effective principal stresscomponents in the depth of the considered rock section(2650 m). Compressive stress is always defined positive in the fol-lowing. We assume that the external stresses act normal to the boun-dary surfaces of the model. This means that no shear stresses exist atthe boundaries. As a result, the direction of principal stresses will beconstant in the layered medium. Figure 4 presents the stress fluc-tuations resulting in the layered medium corresponding to elasticrock heterogeneity characterized along well 21-10. The elasticheterogeneity at Cotton Valley results in significant fluctuationsof the horizontal principal stress magnitudes in the different elemen-tary layers. The Mohr circles between the minimum and maximumprincipal stress are color coded according to the BI (equation 1) ofthe corresponding rock section. A clear classification is visible.Rock sections characterized by high Young’s modulus and lowPoisson’s ratio show low magnitudes of minimum horizontal prin-cipal stress and correspondingly contain a high level of differentialstress (σ1 − σ3). This results in the highest ratio of shear to normalstress on fracture planes located in these layers. Thus, the stresschanges necessary to open new (σ3) and to reactivate preexistingfractures (CFS) are lowest in these layers. According to our model,brittle rock failure and the associated seismicity during hydraulicreservoir stimulation is most probable in rock sections characterizedby a high Young’s modulus and a low Poisson’s ratio. These rocksections are characterized by a high BI value. This is confirmed inthe histogram plot (Figure 4b and 4c), which represent the numberof layers in a given range of minimum principal stress σ3 and CFS.The individual bars are color coded according to the mean BI of thelayers in the corresponding stress level. As already discussed, σ3and CFS are indicator of fracture opening and reactivation proba-bility and are representing the occurrence probability of seismicevents during hydraulic reservoir stimulation. The lower σ3 andCFS are in a layer, the lower is the stress perturbation necessaryto trigger seismic events in the corresponding layer. The relation
Figure 3. Schematic illustration of a perfectly horizontally layered medium consistingof elastically homogeneous layers. We analyze stress fluctuations that result in the equi-librium state to an applied homogeneous far-field stress at the boundaries of the medium.The magnitudes of the far-field stress are given in the figure. We assume that the layerscharacterized by different elastic properties are perfectly coupled. This enforces alllayers to undergo the same amount of strain in the horizontal direction and resultsin fluctuations of the horizontal principal stresses in the individual layers.
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between elastic rock heterogeneity and the occurrence probabilityof seismicity resulting from our theoretical considerations are inagreement with the observation of a positive relation betweenthe BI and density of seismic events induced by hydraulic stimu-lation at the Cotton Valley gas field (see Figure 1). However, othermechanical properties might explain the observations of the corre-lation between event occurrence Sh and differential stress as well.For example, corresponding with this correlation is a tendency forthe stiffer layers to contain and sustain natural fracture develop-ment. Possibly there are more potential microseismic sourcesand associated permeability paths along which pore pressure canextend into the rock. To clarify this point, the controlling mecha-nisms of the site-specific elastic heterogeneity have to be analyzed.As mentioned above, different mechanisms like mineral composi-tion, structure, damage level, and other geologic structures contrib-ute to the elastic heterogeneity of rocks. Nevertheless, the obtainedrange of fracture opening and reactivation stress is in agreementwith critical stress changes reconstructed from microseismicity insedimentary and crystalline rocks (see Rothert and Shapiro,2007). Our results suggest that elastic rock heterogeneity controlsthe occurrence probability of brittle rock failure and associated seis-mic events during hydraulic reservoir stimulation. The high stresscontrast in the vertical direction restricts the occurrence of seismic-ity to certain depth ranges and explains the clear separation of eventclouds induced along the different perforations. To separately ana-lyze the influence of Young’s modulus and Poisson’s ratio fluctua-
tions on rock stress heterogeneity, we compare stress and elasticproperty fluctuations. The comparison is presented in Figure 5.In agreement with the theory discussed in Appendix A, we find thatrock stress fluctuations are controlled by deviations of in situ elasticmoduli from their mean values. We quantify the correlation strengthbetween stress and elastic-moduli heterogeneity, according to thecorrelation coefficient Pc shown at the bottom of the figure. Duringinterpretation of the comparison, we have to keep in mind that E andν are inversely interrelated in the Cotton Valley reservoir rock (seeFigure 2a and 2b). It means that, for instance, rock sections char-acterized by a high Young’s modulus will most probably be char-acterized by a low value of ν. Although the influence of ν and E onstress heterogeneity can be quantified using equation A-6, consid-ering the measured fluctuations of elastic properties is important tounderstand stress fluctuations in real rocks. However, this meansthat the obtained correlations between stress and elastic propertyfluctuations are only valid for the analyzed case study, which ischaracterized by a site-specific elastic rock heterogeneity andfar-field stress magnitudes. The mean values, standard deviations,and correlation strength (PC) of Young’s modulus and Poisson’sratio at other sites will be different. Moreover, equation A-6 showsthat not only the elastic rock heterogeneity but also the differentialstress magnitudes of the far-field principal stresses will influencethe resulting stress fluctuations. Rock failure indicators, whichare based solely on the elastic properties of rocks, cannot have auniversal physical meaning because the far-field differential stress
−5 0 5 10 15 20 25 30 35 400
5
10
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σn−P
p(MPa)
τ (M
Pa)
−10 0 10 200
10
20
30
CFS(MPa)
coun
ts
−10 0 10 200
10
20
30
σ3(MPa)
Cou
nts
Fracture opening: σ3 CFS=S
0−τ+μ(σ
n−P
p)
Fracture reactivation:
SV
SHS
h
b)
a)
c) 0510
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BI (%)
−3 −2 −1 0−6
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0
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(wavenumber) (1/m)
log 10
(pow
er s
pect
ral d
ensi
ty)
Power spectral density σ3
Linear fit, slope=−0.99
−3 −2 −1 0−6
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(wavenumber) (1/m)
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Power spectral density CFSLinear fit, slope=−0.99
d)
e)
Figure 4. Stress fluctuations resulting from elastic rock heterogeneity along treatment well 21-10 at Cotton Valley. (a) Mohr circle repre-sentation of stress fluctuations. The stars indicate shear and normal stress acting on optimally oriented fracture planes and color coding cor-responds to the BI. The externally applied homogeneous far-field stress is represented by the thick black circles. Close to critical stressed depthsections are characterized by high differential stress, resulting in high shear and normal stress acting on the fault planes. (b and c) Histogramplots of minimum principal stress σ3 and CFS. The bars are color coded according to the mean BI of the elementary layers in the correspondingstress level. (d and e) PSD of σ3 and CFS fluctuations with depth.
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magnitudes influences the strength of heterogeneous materials aswell. Comparison between rock stress and elastic heterogeneity(Figure 5) reveals that fluctuations of σ3 and CFS in the CottonValley reservoir rock are strongly correlated to fluctuations of Pois-son’s ratio (Pc ¼ 0.96). Moreover, the BI shows a strong inversecorrelation to CFS and σ3 (PC ¼ −0.92). This provides a physicalexplanation for the observed relation between event density withdepth and the BI (Figure 1). Nevertheless, our results suggest thatconsidering only fluctuations of Poisson’s ratio is the best indicatorfor the probability of fracture opening and reactivation and associ-ated seismic events at the Cotton Valley reservoir. Our model ofstress fluctuations in a perfectly horizontally layered medium inequilibrium to a homogeneous far-field stress seems to be a goodrepresentation of rock stress heterogeneity and resulting probabilityof fracture opening and reactivation. The obtained critical stresschanges leading to fracture opening and reactivation along well21-10 at Cotton Valley show low magnitudes. We note that the com-puted critical stress changes (CFS and σ3) represent the lowestthreshold. This is the case because we assumed that the layeredmedium is stress free before application of the external stress field.Moreover, the CFS is computed for optimally oriented preexistingfractures. Thus, the obtained values of CFS characterize the mini-mum stress changes from which reactivation of preexisting fracturesis possible. Higher stress changes are required to reactivate moreunfavorably oriented preexisting fractures. In the same sense, thevalue of σ3 represents the lowest threshold of stress changes causingthe opening of new fractures because we neglect the tensile strengthof the rocks. If the tensile strength of the rocks is known, the frac-ture opening stress can be computed according to the Griffith cri-terion (e.g., Jaeger et al., 2007). Nevertheless, the distribution ofCFS and σ3 computed from our model is representative for the spa-tial heterogeneity of the fracture reactivation andopening probability during hydraulic reservoirstimulation.
SCALE INVARIANCE OF CRITICALSTRESS CHANGES
It has been observed that heterogeneity inrocks is of a scale-invariant nature (e.g., Leary,1997; Day-Lewis et al., 2010; Langenbruchand Shapiro, 2014). It means that no character-istic scales exist above or below which the rockheterogeneity can be neglected. Figure 2f and 2gpresents the power spectral density (PSD) of Eand ν fluctuations measured along borehole21-10 at Cotton Valley (see Figure 2a and 2b).The power law dependence of the PSD functionson the wavenumber implies the scale-invariantfractal nature of the elastic rock heterogeneityat Cotton Valley. The power law exponent of β ≈1 is in agreement with the observation of univer-sality of the heterogeneous complexity of rocks(e.g., Leary, 1997; Langenbruch and Shapiro,2014). Our layered model thus shows scale-invariant fluctuations of Young’s modulus andPoisson’s ratio in the vertical direction. Figure 4dand 4e presents the PSD of σ3 and CFS fluctua-tions with depth, which we computed in the lay-ered medium. The PSD of the stress fluctuations
is characterized by a power law dependence on the wavenumber.Our results suggest that scale-invariant fluctuations of elastic prop-erties result in scale-invariant fluctuations of the fracture openingand reactivation probability. The power law exponent (β ¼−0.99), which is related to the ratio between small- to large-scalefluctuations of stress, is in agreement with the power law exponentof the PSD of elastic-moduli fluctuations (see Figure 2f and 2g).The existence of scale-invariant fluctuations of critical stresschanges in rocks provides a possible physical explanation for theobserved scale invariance of seismogenic processes. Scale invari-ance of seismogenic processes is, for instance, expressed in theGutenberg-Richter relation of earthquake magnitude scaling.Langenbruch and Shapiro (2014) show that the emergence of theGutenberg-Richter relation can be explained by the existence ofscale-invariant rock stress fluctuations. Scale invariance suggeststhe existence of elastic rock heterogeneity and corresponding rockstress fluctuations at all scales. However, our model only accountsfor heterogeneity down to the scale of the active length of the log-ging tool (approximately 1 m) and up to the length of the consideredrock section (300 m). Nevertheless, if scale invariance is fulfilled,the resulting stress fluctuations are valid on other scales. If, for in-stance, 300 measurements of stress or elastic properties are madealong well 21-10 at Cotton Valley on a constant but arbitrary scale,the result will be in agreement with the distributions shown in Fig-ures 2 and 4. The acquired data will be characterized by the samemean values, standard deviations, and power spectra of fluctuations.The maximum magnitude of measured fluctuation of stress andelastic properties, however, will increase with the number of mea-surements. This means that it is important to consider the scale onwhich elastic heterogeneity is obtained and the scale of the brittlefailure process for which critical stress changes should be quanti-
24 31 39σ
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3(MPa)
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27 37 47σ
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24 44 64
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−4 5 14CFS(MPa)
9 47 85
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−4 5 14CFS(MPa)
24 44 64
E(GPa)
29.0−=Pc69.0=Pc46.0−=Pc69.0=Pc Pc=−0.64 Pc=0.92 Pc=−0.96Pc=−0.92
Figure 5. Comparison of elastic rock heterogeneity measured along well 21-10 at Cot-ton Valley and stress fluctuations resulting in a perfectly horizontally layered medium inequilibrium to an applied far-field stress. The strength of the correlation between stressand elastic moduli fluctuations is quantified by the correlation coefficient Pc shown atthe bottom of the figure. Stress fluctuations are controlled by deviations of in situ modulifrom their mean values. Fluctuations of Poisson’s ratio show the strongest correlation tofluctuations of σ3 and CFS. This suggests that fluctuations of Poisson’s ratio are the bestindicator for the probability of fracture opening and reactivation during hydraulic stimu-lation at Cotton Valley.
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fied. The range of moment magnitudes (Mw ¼ −2.5 to −1.5) ofseismic events at Cotton Valley suggests that detectable eventsare characterized by source radii ranging from 0.35 to 1.1 m. Thisestimation is based on calculation of source radii according toBrune (1970) and Kanamori (1977) considering a constant stressdrop of 2.3 MPa (e.g., Goertz-Allmann et al., 2011). This scaleis comparable with the scale of the active length of the logging tool.However, it is unclear on which scale the rupture process is initiated.Moreover, scale invariance suggests that failure processes onsmaller scales exist, but they are not detectable by the seismic mon-itoring network at Cotton Valley. If scale invariance is fulfilled, fluc-tuations of CFS and σ3 on an arbitrary scale can be computed bydrawing the corresponding number of independent random samplesfrom the probability density function (PDF) of the CFS and σ3 dis-tribution shown in Figure 4b and 4c. This will be helpful for thefurther development of physics-based statistical models of fluid-in-jection-induced earthquakes (e.g., Rothert and Shapiro, 2003; Lan-genbruch and Shapiro, 2010; Goertz-Allmann and Wiemer, 2013).We use the best-fitting Gaussian distribution to compute the ex-pected magnitudes of CFS fluctuations on different scales rangingfrom 10−3 to 100 m (see Figure 6). We consider the 300-m-longborehole section shown in Figure 2. For instance, CFS fluctuationson the centimeter scale are determined by drawing 30,000 indepen-dent random samples from the Gaussian distribution shown in Fig-ure 4c. The scale-invariant nature of elastic rock heterogeneitymakes a prediction of absolute stress magnitudes at a given depthimplausible. The distributions of critical stress changes obtained inour study should be considered as probability density distributions.Absolute values of critical stress changes obtained by our and otherstudies should be viewed with caution. Measurements of stress inthe considered depth are the only reliable method to obtain absolutevalues of stress magnitudes. However, also in the case of stress mea-surements, the scale of the rock-failure process, which is used todraw conclusions about stress magnitudes, will be important.Mayerhofer et al. (2010) present some direct stress measurements
along well 21-09. The stress measurements are based on hydraulicfracturing stress tests. In the reservoir depth section considered inour study, they find stress fluctuations of approximately 1.33 MPa.The authors state that this stress contrast is too small to explain theconfinement of the hydraulic fracture in the targeted sand forma-tion. However, these results are based on only two direct stress mea-surements. Our results suggest that larger stress contrasts exist onsmaller scales. This may explain the confinement of the hydraulicfractures.
STRESS FLUCTUATIONS IN 3DHETEROGENEOUS MEDIA
We analyzed stress fluctuations in perfectly horizontally layeredmedia. However, it is clear that all rocks show elastic heterogeneityin the horizontal direction. Therefore, we consider a second endmember model of elastic heterogeneity. Here, we assume thatthe degree of elastic heterogeneity in the vertical and horizontal di-rections is equivalent. Our results will help to understand the influ-ence of horizontal heterogeneity. Measurements of elasticproperties are available along two vertical treatment wells (see Fig-ure 1) at Cotton Valley. We compute the correlation coefficients offluctuations of E and ν measured along the two wells. The highestcorrelation coefficients (PC;ν;ν ¼ 0.34 and PC;E;E ¼ 0.62) result fora layer tilt of 1.53° from horizontal. The tilt of the layers is suffi-ciently small to be neglected. The high values of the correlationcoefficients show that stress fluctuations in a perfectly horizontallylayered medium should be a good approximation for rock stressheterogeneity at Cotton Valley. Because no stress fluctuations occurwithin a single layer, but a strong contrast of critical stress changesexists between different layers, seismic event clouds in sedimentaryrocks are often restricted to certain depth ranges. However, our re-sults show that even the sedimentary rock at Cotton Valley showselastic heterogeneity in the horizontal direction. Figure 7 presents3D heterogeneous media, which we simulated according to the elas-tic heterogeneity along well 21-10 (see Figure 2). The media consist
−20 −15 −10 −5 0 5 10 15 20 25 3010
0
101
102
103
104
105
CFS (MPa)
Cou
nts
100 m scale
10−1 m scale
10−2 m scale
10−3 m scale
Figure 6. CFS fluctuations along the 300-m-long borehole sectionat Cotton Valley on different scales. The histogram of stress fluc-tuations along the considered rock section of 300 m on the differentscales is determined by drawing the corresponding number of in-dependent random samples from the PDF of CFS (see Figure 4c).Because larger stress fluctuations occur on smaller scales, moresmall magnitude events are triggered.
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y (m)
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E (GPa)
Figure 7. Simulated heterogeneous distributions of ν and E. Thedistributions are statistically equivalent to the log data along well21-10 (see Figure 2), that is, they show the same mean values, stan-dard deviations, and correlation coefficient between ν and E. More-over, the values along any 1D profile in the simulated media possessa power spectrum of fluctuations, given by k−1. Due to the appliedKriging procedure, the distribution of ν and E along the center (z-axis) of our model medium corresponds to measured values alongwell 21-10.
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of equally sized cubes characterized by different values of E and ν(for details on the simulation procedure, see Langenbruch and Sha-piro, 2014). Additionally, we applied Kriging to the simulated dis-tributions of ν and E to condition the media to measured valuesalong well 21-10. We applied a variation of the simple Krigingmethod according to Huang et al. (2011). The resulting distributionof ν and E is statistically equivalent to the elastic heterogeneityalong well 21-10. It means that ν and E show the same mean value,standard deviation, and correlation coefficient as the measured data(see Figure 2). Moreover, the fluctuations along any 1D profilethrough the simulated media possess a power spectrum of fluctua-tions given by k−1. Due to the applied Kriging procedure, the dis-tribution of ν and E along the center (z-axis) of our model mediumcorresponds to measured values along well 21-10. In Appendix B,we present new analytic solutions of the stress state in the individualcells of the 3D heterogeneous medium (equation B-2). In agreementwith the analysis of a layered medium, we assume that the elementsof the different elastic properties are perfectly bounded. This meansthat the strain inside the medium will be homogeneous, if a far-fieldstress is applied at its boundaries. We consider the normal faultingfar-field stress regime at Cotton Valley. Figure 8a and 8b shows thespatial distribution of σ3 and CFS in the complete 3D heterogeneousmedium. In Figure 8c–8e, the stress fluctuations along the z-axis ofthe medium are presented. Because heterogeneity in all three direc-tions in space is present, all three principal stress magnitudesundergo fluctuations. The comparison of stress fluctuations andelastic rock heterogeneity (see Figure 9) along well 21-10 revealsthat the maximum principal stress magnitude (SV ) is governed byfluctuations of Young’s modulus. In general, we find that the rela-tions between the fracture opening and reactivation probability andin situ elastic properties are in agreement with the results obtained ina layered medium. Nevertheless, the correlation strength betweenthe BI, σ3, and CFS is weaker than in the layered medium. Thisis so because the maximum principal stress undergoes significantfluctuations due to the existence of horizontal elastic heterogeneity.The color coding in Figure 8 and the correlation coefficient of PC ¼0.93 reveals that the BI is a very strong indicator of differentialstress. In agreement with the results obtained in the layeredmedium, fluctuations of ν seem to be the best probability indicatorof brittle rock failure and the associated seismicity during hydraulicreservoir stimulation. The results obtained for the isotropic 3Dheterogeneous medium will also be valid in a strike-slip and re-verse-faulting stress regime because interchanging the stress direc-tions has no effect on the correlations. The maximum principalstress is governed by fluctuations of E, and fluctuations of σ3are strongly correlated to ν. The magnitudes of stress fluctuationsin the three directions of space in the analyzed 3D heterogeneousmedium are equivalent. Because stress at a given depth will stronglyfluctuate in the horizontal direction, no restriction of seismicity to acertain depth section should occur in the 3D heterogeneous case.Crystalline rocks, which are present at EGS and deep scientificboreholes, such as the Continental Deep Drilling Site (KTB) (Ger-many) and the San Andreas Fault Observatory at Depth (SAFOD)(USA), are characterized by a comparable degree of heterogeneityin all three directions of space. As a result, stress magnitudes shouldfluctuate to the same degree in the horizontal and vertical directions.Natural and fluid-injection-induced earthquakes have been ob-served at the SAFOD and the KTB. Hypocenter locations of theseearthquakes and fluid-injection-induced earthquakes at EGS (e.g.,
Langenbruch and Shapiro, 2010; Langenbruch et al., 2011) show adiffuse spatial distribution in all three directions in space. This is inagreement with the consideration of similar fluctuations of stress inall three directions. A comparison of seismic event occurrence withdepth and elastic heterogeneity measured along injection boreholesin crystalline rocks does not seem to be feasible. In situ elastic prop-erties and resulting fluctuations of stress just a few meters awayfrom the borehole are significantly different than the fluctuationsmeasured along the borehole. Nevertheless, our results suggest thatrock stress heterogeneity has a significant influence on the seismo-genesis of fluid injection-induced seismicity in the 3D hetero-geneous case.
HORN RIVER SHALE GAS RESERVOIR: ROCKSTRESS FLUCTUATIONS IN LAYERED VERTICAL
TRANSVERSE ISOTROPIC MEDIA
Because the target sand stone formation at Cotton Valley is char-acterized by a minor degree of intrinsic anisotropy, the rock forma-tion can be considered as elastically isotropic. However, the BIoriginally has been introduced for application to unconventionalshale reservoirs, which show a high degree of elastic anisotropy.The anisotropy is mainly caused by alignment and lamination ofsofter and platy clay minerals and kerogen. Usually, this intrinsicanisotropy is described by VTI media. These media are character-ized by two different Young’s moduli and shear moduli (Ev, Eh, μv,and μh) in the vertical and horizontal directions, respectively. In ad-dition, three different Poisson’s ratios νvh, νhh, and νhv exist, wherethe first subscript defines the direction of applied stress and the sec-ond subscript gives the direction to which strain is compared. Onlyfive of these seven properties are independent (νvhEv
¼ νhvEh
andμh ¼ Eh
1þνhh) (e.g., Bower, 2014). The theoretical value range of
the elastic properties in the VTI case is given by (e.g., Amadei,1996)
Eh; Ev; μh; μv > 0; −1 < νhh < 1;
−Ev
Eh
ð1 − νhhÞ2
< νvh <Ev
Eh
ð1 − νhhÞ2
:(4)
In general, fluctuations of all five independent elastic properties in-fluence the state of stress in elastically heterogeneous VTI media.Thus, it is obvious that the BI, which is defined based on isotropicelastic properties, cannot represent the heterogeneity of shale res-ervoir rocks. However, unconventional shale reservoir rocks showstratification and strong relations between elastic properties, whichhave to be taken into account. We analyze a hydraulic fracturingcase study at a shale gas reservoir located in the Horn River Basin,Northeastern British Columbia, Canada. More than 3000 eventshave been recorded and located during the three-stage hydraulicfracturing treatment through a horizontal borehole. The event occur-rence is restricted to a certain depth range, which approximatelycorresponds to two lithologic layers (see Dunphy and Campagna,2011). These two organic-rich target shale formations (TOC ≈ 5%)are characterized by a high quartz content (approximately 65%) anda clay content of approximately 27%. The formations are overlainby a 800-m-thick clay-rich (approximately 70%) shale formation.Below the reservoir another clay-rich shale formation (approxi-mately 40%) and a thin carbonate layer are present (see Yu and Sha-piro, 2014). The clay-rich shale formations and the thin carbonate
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b)
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a)
0 10 20 30 40 50 600
5
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σn−P
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τ (M
Pa)
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10
20
30
σ3(MPa)
coun
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−10 0 10 200
10
20
30
CFS(MPa)
coun
ts
c)
d) e)
Figure 8. Stress fluctuations in the 3D heterogeneous medium (see Figure 7). (a and b) Spatial distribution of σ3 and CFS. (c) Mohr circlerepresentation of stress fluctuations along the z-axis (well 21-10). The stars indicate the shear and normal stress acting on optimally orientedfracture planes, and the color coding corresponds to the BI. The externally applied homogeneous far-field stress is represented by the thickblack circles. (d and e) Histogram plot of σ3 and CFS. The bars are color coded according to the mean BI of the elementary layers in thecorresponding stress level.
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VP(m/s)4000 6000
Dep
th (
m)
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1800
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VS(m/s)2000 3000 4000
ρ(Kg/m3)
2500 3000E
iso (GPa)
20 40 60 80ν
iso
0 0.2 0.4, γ , δ
0 0.2 0.4 0.6
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Ev,E
h (GPa)
20 60 100
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Eh
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h (GPa)
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,νhv
,νhh
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1700
1750
1800
1850
1900
νvh
νhv
νhh
BI (%)0 25 50 75
a) b) c) d) e) f) g) h) i) j)
Figure 10. Horn River shale gas reservoir: (a-c) VP, VS, and density logs along a vertical well in the study area. (d-f) Isotropic elastic propertiesand BI computed from panels (a-c). (g) Thomson parameter. (h-j) Anisotropic elastic properties of the VTI medium computed according topanels (a-c). (g) AVTI medium is characterized by five independent elastic properties. The anisotropic elastic properties satisfy the constraintsgiven in equation 4.
15 29 43σ
1−σ
3(MPa)
0.1 0.2 0.3
ν
15 29 43σ
1−σ
3(MPa)
9 47 85
BI(%)
−2 5 12σ
3(MPa)
2600
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Dep
th (
m)
σ3(MPa)
0.1 0.2 0.3
ν
−4 4 12CFS(MPa)
0.1 0.2 0.3
ν
−2 5 12σ
3(MPa)
24 44 64
E(GPa)9 47 85
BI(%)
24 34 44σ
1(MPa)
0.1 0.2 0.3
ν
24 34 44σ
1(MPa)
24 44 64
E(GPa)
−4 4 12CFS(MPa)
9 47 85
BI(%)
−4 4 12CFS(MPa)
24 44 64
E(GPa)
Pc=−0.08 Pc=0.93 Pc=−0.42 Pc=−0.53 Pc=−0.79 Pc=0.93Pc=0.96 Pc=−0.56Pc=0.89 Pc=−0.11
Figure 9. Relation between elastic properties and computed stress fluctuations along well 21-10 in the 3D heterogeneous medium shown inFigure 7. Correlation between stress and elastic moduli is quantified by the correlation coefficient Pc shown at the bottom of the figure. Stressfluctuations are controlled by deviation of the in situ moduli from their mean values. Fluctuations of Poisson’s ratio show the strongestcorrelation to the occurrence probability of brittle rock failure and associated seismic events.
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layer, seem to be fracture barriers. Density and P-and S-wave traveltime logs are measured along anearby vertical well in the study area (see Fig-ure 10a and 10c). Based on this information,we compute the isotropic Young’s modulus,Poisson’s ratio, and BI of the shale reservoir withdepth (see Figure 10d and 10f). In addition, in-formation about Thomson’s parameters (ϵ, γ, andδ) is available from analysis of a VSP survey,rock-physical constraints, and inversion of mi-croseismic data (Yu and Shapiro, 2014) (see Fig-ure 10g). The five independent elastic properties(Ev, Eh, νhh, νvh, and μv) in the VTI case can becomputed using VP, VS, ρ, ϵ, γ, and δ (e.g.,Grechka, 2009). The VP and VS correspond toP- and S-wave velocities along the symmetryaxis, respectively. Figure 10h–10j presents theresulting values of the anisotropic elastic proper-ties. The anisotropic elastic properties fall in thevalue range given by equation 4. In Appendix A,we extend the solutions of Bourne (2003) to thecase of elastically VTI layers. Based on thesenew analytic solutions (equation A-6), we areable to compute stress fluctuations in layeredVTI media. We consider the distribution of elas-tic properties corresponding to the Horn River
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14
16
σn−P
p(MPa)
τ (M
Pa)
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CFS(MPa)
coun
ts
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σ3(MPa)
Cou
nts
c)b)
a)
Sh
SH S
V
0510
20
30
40
50
60
70
80
BI (%)
Figure 11. Stress fluctuations resulting from the VTI elastic-rock heterogeneity of theHorn River shale gas reservoir rock in equilibrium to a homogeneous far-field stress.(a) Mohr circle representation of stress fluctuations. (b and c) Histogram plots of mini-mum principal stress σ3 and CFS. The bars are color coded according to the mean BI ofthe elementary layers in the corresponding stress level.
0 5 10 15
σ3(MPa)
0 50 100
BI(%)
0 5 10 15
CFS (MPa)
0 50 100
BI(%)
−5 0 5 10 15
1650
1700
1750
1800
1850
1900
σ3(MPa)
z (m
)
VTI
isotropic
20 40 60
events/m
15 20 25
σ2(MPa)
VTI
isotropic
Pc = −0.99 Pc = −0.99
lower fracture barrier
upper fracture barrier
Figure 12. Horn River Basin shale gas reservoir: Stress modeling results in a layered VTI medium created according to the data shown inFigure 10h–10j. From left to right: σ3 magnitude with depth. The black line shows the distribution resulting from the assumption of isotropicelastic properties (Figure 10d and 10e). The gray line shows the results according to a VTI medium (Figure 10h–10j). The second panel showsthe density of seismic events with depth. The event density follows the distribution of the minimum principal stress magnitude. An upper andlower fracture barrier corresponding to rock sections in a stable state of stress, restricts the depth range of seismic event occurrence. In the lasttwo panels to the right, we compare the magnitude of σ3 and CFS computed for the VTI medium to the BI computed according to isotropicelastic properties. BI, CFS, and σ3 show a strong inverse correlation.
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shale gas reservoir. The far-field stresses, which we apply to themedium are computed as follows: The vertical effective principalstress magnitude is determined from the lithostatic gradient andthe assumption of a hydrostatic pore pressure gradient. At the depthof 1775 m (the depth of the perforations), it is given by:SV ¼ 27.61 MPa. We assume that the frictional rock strength con-strains the magnitude of the minimum effective stress in the brittleearth’s crust. The minimum principal far-field stress can then bedetermined according to (see, Zoback, 2010)
SVSh
¼ ½ðμ2 þ 1Þ12 þ μ�2; (5)
where μ is the coefficient of friction. In agreement with the analysisin the previous section, we assume a friction coefficient of μ ¼ 1.This results in a minimum effective horizontal stress magnitude ofSh ¼ 4.74 MPa. The intermediate principal effective far-field stressmagnitude is assumed to be the mean value between SV and Sh(SH ¼ 16.17 MPa). Figure 11 shows the Mohr circle representationand the histogram of CFS and σ3 in the Horn River shale gas res-ervoir rock. The Mohr circles and histogram plots are color codedaccording to the value of the BI computed from isotropic elasticproperties (Figure 10d–10f). Even though the stress fluctuationsare computed according to anisotropic elastic properties, a strongcorrelation to the isotropic BI value is visible. In agreement withthe findings of the isotropic case at Cotton Valley, the probabilityof fracture opening (σ3) and reactivation (CFS) is increasing withthe BI. Moreover, the differential stress and the BI show a strongcorrelation. The bimodal distribution of the histograms of σ3 andCFS (Figure 11 bottom) results from the strong contrast of elasticproperties between the target shale formations and the overlayingclay-rich shale formation. The state of stress in the clay rich shaleformations is much less critical than the state of stress in the targetformations. The depth distribution of the seismic events induced bythe hydraulic fracturing is shown in Figure 12. The density of theevents with depth follows the distribution of the minimum principalstress magnitude resulting from our computation. Although theshale reservoir is characterized by a high degree of elasticanisotropy, the minimum principal stress magnitude computed ac-cording to VTI elastic properties and isotropic elastic properties(Figure 12 left) shows the same trend with depth. This explainsthe strong correlation between the BI and the computed probabilityof brittle rock failure in the shale reservoir rock (see Figure 12right). Moreover, we find that an upper and lower fracture barrierrestrict the depth range of seismic event occurrence. The fracturebarriers correspond to shale-rich rock sections in a stable state ofstress. Our results suggest that even though the analyzed shale res-ervoir rock is characterized by a high degree of intrinsic anisotropy,isotropic elastic properties will provide an estimate of the occur-rence probability of seismic events during hydraulic fracturing ofthe corresponding rock section.
CONCLUSIONS
We analyzed the relations between elastic-rock heterogeneity,rock-stress fluctuations, and occurrence of fluid injection-inducedseismicity. Our findings suggest that elastic-rock heterogeneity con-trols the occurrence probability of seismic events in the rock beinghydraulically fractured. The heterogeneity causes significant spatial
fluctuations of critical stress changes leading to opening of new andreactivation of preexisting fractures. Moreover, we demonstratedthat critical stress changes undergo scale-invariant fluctuations inrocks. The scale-invariant nature of rock-stress fluctuations iscaused by scale-invariant fluctuations of elastic-rock properties.This gives a possible physical explanation for the scale invarianceof seismogenic processes. We compare elastic-rock heterogeneityobtained from borehole logs with the spatial occurrence of seismicevents caused by hydraulic fracturing of the corresponding rocksection. Our comparison revealed that the seismic events occur pref-erentially in rock sections characterized by low Poisson’s ratio andhigh Young’s modulus. Based on analytic solutions of stress fluc-tuations in heterogeneous linear elastic media in equilibrium to ahomogeneous far-field stress, we quantify the relation between elas-tic-rock heterogeneity and critical stress changes. We analyzed twoend member models of 1D and 3D elastic heterogeneity. Because allrocks show a certain degree of horizontal heterogeneity, stress fluc-tuations will always fall somewhere between these two analyzedmodels. In both cases, stress fluctuations are governed by deviationsof in situ elastic properties from their mean values. We analyzed thephysical meaning of a heterogeneity index of rocks, which indicatesrock sections of high Young’s modulus and low Poisson’s ratio. Weshowed that this index can be used to identify rock sections of highdifferential stress. In addition, our results suggest that it is related tothe occurrence probability of brittle rock failure during hydraulicfracturing. Nevertheless, we find that fluctuations of Poisson’s ratioshow an even stronger correlation to critical stress changes, whichresult in opening and reactivation of fractures in rocks. We obtainedan analytic solution for stress fluctuations in VTI layers, such asshale deposits, and applied it to a hydraulic fracturing case studyof an unconventional shale reservoir in the Horn River basin. Alsoin this case, we observed strong relations between seismicity andfluctuation of stress in space. However, only if approximate mag-nitudes of the tectonic far-field stresses are known, it is possible toidentify proper rock sections for hydraulic stimulation by analysisof elastic-rock heterogeneity measured along boreholes. Our quan-titative study demonstrated that rock failure indicators such as theBI, which is based solely on elastic properties of rocks, cannot havea universal physical meaning. This is the case because the far-fielddifferential stress magnitudes influence the fracture strength ofheterogeneous materials as well.
ACKNOWLEDGMENTS
We thank the sponsors of the Physics and Application of SeismicEmission Consortium project and the Federal Ministry for the Envi-ronment, Nature Conservation and Nuclear Safety as a sponsor ofproject MeProRisk II for supporting the research presented in thispaper. We acknowledge J. Rutledge, two anonymous reviewers, andthe associate editor for helpful and constructive comments.
APPENDIX A
LAYERED TRANSVERSE ISOTROPIC MEDIUMUNDER A HOMOGENEOUS FAR-FIELD
STRESS
In this appendix, we present analytic solutions of stress fluctua-tions in heterogeneous linear elastic media consisting of elasticallyVTI layers in equilibrium to a homogeneous far-field stress. The
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solutions are a generalization of the solutions corresponding to elas-tically isotropic layers given in Bourne (2003). The consideration ofVTI layers allows application to instrinsically anisotropic rockssuch as shale deposits. This means that the elastic anisotropy ofthe individual layers considered in our model results from align-ment of platy clay minerals and kerogen particles and not fromthe layering of the complete medium. The effective macroscopicelastic properties of a layered medium, however, will be anisotropiceven in the case of isotropic elastic properties of the individuallayers. However, intrinsic rock anisotropy usually plays a more im-portant role than the anisotropy caused by the layering. The solu-tions of Bourne (2003) follow as a special case of our solutions forVTI layers. We assume the z-axis to be the symmetry axis. The elas-tic moduli of a VTI medium can be expressed by five independentquantities in terms of Young’s modulus (E), Poisson ratio (ν), andshear modulus (μ)
Eh ¼ Ex ¼ Ey; Ev ¼ Ez; νhh ¼ νxy ¼ νyx;
νvh ¼ νzx ¼ νzy; νhv ¼ νxz ¼ νyz;νvhEv
¼ νhvEh
;
μh ¼ μxy ¼ μyx ¼Eh
ð1þ νhhÞ;
μv ¼ μxz ¼ μzx ¼ μyz ¼ μzy; (A-1)
where h and v indicate the horizontal (x- and y-directions) and ver-tical (z-direction), respectively. The five independent elastic proper-ties can be expressed as Ev, Eh, νhh, νvh, and μv. These propertiescan be computed from VP, VS, ρ, ϵ, γ, and δ (e.g., Grechka, 2009).The VP and VS correspond to P- and S-wave velocities along thesymmetry axis, respectively. The ρ is the density and ϵ, γ, and δ arethe Thomson’s parameter. Hooke’s law for the VTI case reads
26666664
ϵxxϵyyϵzz2ϵyz2ϵzx2ϵxy
37777775¼
2666666664
1Eh
− νhhEh
− νvhEv
0 0 0
− νhhEh
1Eh
− νvhEv
0 0 0
− νhvEh
− νhvEh
1Ev
0 0 0
0 0 0 1μv
0 0
0 0 0 0 1μv
0
0 0 0 0 02ð1þνhhÞ
Eh
3777777775
26666664
σxxσyyσzzσyzσzxσxy
37777775:
In agreement with Bourne (2003), we consider a perfectly horizon-tally layered medium and perfectly coupled boundaries between thelayers. It means that no relative movement parallel to the layerboundaries can occur. Thus, the external application of a homo-geneous far-field stress (σij;e) will cause constant layer-parallelstrain. Moreover, we assume that the layered medium is stress freebefore application of the external stress field. This means thatresidual stress, which should exist in rocks because lithificationtook place under certain nonzero stress conditions, is neglected.The stress fluctuations computed under this assumption will bethe upper bound of possible stress fluctuations resulting from elasticrock heterogeneity. According to Hooke’s law, the layer parallelstrain is given by
ϵxx ¼1
Eh;iσxx;i −
νhh;iEh;i
σyy;i −νvh;iEv;i
σzz;i;
ϵyy ¼1
Eh;iσyy;i −
νhh;iEh;i
σxx;i −νvh;iEv;i
σzz;i;
ϵxy ¼1
2μh;iσxy;i;
(A-2)
where the index i indicates the elastic properties or stress magni-tudes in the ith layer. Now, we must find the horizontal strain thatis in equilibrium with the applied far-field stress. The sum of alllayer-parallel stresses has to be balanced by the horizontal compo-nents of the homogeneous far-field stress at the boundaries. Theapproach is comparable with effective medium descriptions offinely layered anisotropic media (Backus, 1962; Schoenberg andMuir, 1989), but with the distinction of investigating the internalstate of individual layers rather than the overall state of the ensemble(Bourne, 2003). The balance of forces between far-field stress andthe coupled layers requires
σxx;eXNi¼1
ti ¼XNi¼1
σxx;iti; σyy;eXNi¼1
ti ¼XNi¼1
σyy;iti;
σxy;eXNi¼1
ti ¼XNi¼1
σxy;iti: (A-3)
where ti is the thickness of the ith layer. The layer-parallel strain isthe same in each layer
σxx;e ¼ m1ϵxx þm2ϵyy þm4σzz;e;
σyy;e ¼ m1ϵyy þm2ϵxx þm4σzz;e; σxy;e ¼ m3ϵxy:
(A-4)
where m1, m2, and m4 are the thickness averaged quantities,where
m1;i ¼Eh;i
1 − ν2hh;i; m2;i ¼
Eh;iνhh;i1 − ν2hh;i
; m3;i ¼ 2μh;i;
m4;i ¼Eh;i
Ev;i
νvh;i1 − νhh;i
: (A-5)
Because we are considering a medium consisting of equally thicklayers, the thickness averaged coefficients m1, m2, m3, and m4 aregiven by the mean values of m1;i, m2;i, m3;i, and m4;i, respectively.In the case of varying layer thickness, the averaged coefficients aregiven by the thickness-weighted average quantities evaluated for allN layers (see Bourne, 2003). Eliminating the layer parallel strainsconsidering equation A-4 results in
σxx;i ¼ M1iσxx;e þM2;iσyy;e −M4;iσzz;e;
σyy;i ¼ M1iσyy;e þM2;iσxx;e −M4;iσzz;e;
σxy;i ¼ M3;iσxy;e:
(A-6)
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where σij;e are the components of the far-field stress and the cou-pling coefficients M1, M2, M3, and M4 are given by
M1;i ¼m1m1;i −m2m2;i
m21 −m2
2
; M2;i ¼m1m2;i −m2m1;i
m21 −m2
2
;
M3;i ¼m3;i
m3
; M4;i ¼ m4
m1;i þm2;i
m1 þm2
−m4;i:
(A-7)
All stresses involving the index z must be continuous to be in equi-librium to the applied far-field stress:
σzz;i ¼ σzz;e; σxz;i ¼ σxz;e; σyz;i ¼ σyz;e: (A-8)
The isotropic solutions of Bourne (2003) follow from settingEv ¼ Eh ¼ E, μv ¼ μh ¼ μ, and νvh ¼ νhh ¼ νhv ¼ ν in equa-tion A-5 according to
m1;i ¼Ei
1 − ν2i; m2;i ¼
Eiνi1 − ν2i
; m3;i ¼ 2μi;
m4;i ¼νi
1 − νi: (A-9)
APPENDIX B
3D HETEROGENEOUS MEDIUM UNDER AHOMOGENEOUS FAR-FIELD STRESS
If a homogeneous far-field stress is applied at the boundaries ofa 3D heterogeneous medium consisting of equally sized perfectlybounded cubes, which are characterized by different isotropicelastic properties (E and ν) (see Figure 7) not only the strain inthe horizontal, but also the strain in the vertical direction willbe homogeneous. This will result in fluctuations of all three prin-cipal stresses inside the medium. We use this model to obtainstress fluctuations in 3D elastically heterogeneous media. Thestrain resulting from application of a homogeneous far-field stressis given by
ϵxx ¼1
Eijk½σxx;ijk − νijkðσyy;ijk þ σzz;ijkÞ�;
ϵyy ¼1
Eijk½σyy;ijk − νijkðσxx;ijk þ σzz;ijkÞ�;
ϵzz ¼1
Eijk½σzz;ijk − νijkðσxx;ijk þ σyy;ijkÞ�;
(B-1)
where the subscripts (ijk) indicate a cube in the x-, y-, and z-di-rections. Stress fluctuations resulting in equilibrium to a homo-geneous far-field stress are given by
σxx;ijk ¼ Eijk
�½m1ðνijk − 1Þ þm2ð1 − 3νijkÞ�σxx;e
þ ½m2ðνijk − 1Þ −m1νijk�½σyy;e þ σzz;e�½ðνijk þ 1Þðm1 þm2Þ�½m1ð2νijk − 1Þ þm2ð1 − 4νijkÞ�
�;
σyy;ijk ¼ Eijk
�½m1ðνijk − 1Þ þm2ð1 − 3νijkÞ�σyy;e
þ ½m2ðνijk − 1Þ −m1νijk�½σxx;e þ σzz;e�½ðνijk þ 1Þðm1 þm2Þ�½m1ð2νijk − 1Þ þm2ð1 − 4νijkÞ�
�;
σzz;ijk ¼ Eijk
�½m1ðνijk − 1Þ þm2ð1 − 3νijkÞ�σzz;e
þ ½m2ðνijk − 1Þ −m1νijk�½σxx;e þ σyy;e�½ðνijk þ 1Þðm1 þm2Þ�½m1ð2νijk − 1Þ þm2ð1 − 4νijkÞ�
�; (B-2)
where m1 and m2 correspond to the mean values of
m1;ijk ¼Eijkðνijk − 1Þ2ν2ijk þ νijk − 1
; m2;ijk ¼EijkðνijkÞ
2ν2ijk þ νijk − 1;
(B-3)
respectively.
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