Effect of anisotropy, angle, and critical tensile stress and confiningpressures on evaluation of shale brittleness index — Part 1:Methodology and laboratory study

Qing Wang1, Bo Zhang2, Shiguang Guo1, and Jianguang Han3

Abstract

Brittleness is an important evaluation parameter in shale fracturing. Current methods of brittleness evalu-ation can be classified into two categories: elastic parameter-based and mineral content-based methods. How-ever, both categories neglect the effect of anisotropy on the brittleness index (BI) computation of shaleresources. We have redefined a new BI by integrating failure criteria stress and anisotropy parameters estimated(BIac) from seismic waves. According to the new definition, the BI at one analysis point varies with the incidentangle of the seismic wave and confining pressures. We applied the BIac method to laboratory-measured shalesamples acquired from the Monterey Formation, Santa Maria Basin. We found that the delta parameter δ is moreresponsive to the BIac than the gamma γ and epsilon ε anisotropic parameters, and it indicates a good linear fitrelationship with the BIac at different angles. The slope of the linear is variable with the angles, thus delta can beused to predict the BIac in the Monterey Formation, Santa Maria Basin.

IntroductionUnconventional energy has become amajor field of oil

and gas exploitation resources worldwide (Wang andGale, 2009). Shale develops in 75% of clastic sedimentarybasins and covers most hydrocarbon reservoirs (Hornby,1998). As one of the clean energy resources, shale gashas gained increasing attention from industry and aca-demia (Zhu et al., 2011). Shale has very typical aniso-tropic characteristics. This anisotropy is manifested inits electrical and elastic properties. The anisotropic fea-tures are also exhibited in elastic waves at low and highfrequencies. The distribution of organic matter in clayminerals and the stratified developmental characteristicsof parallel layers lead to the anisotropic characteristics ofshale (Sone and Zoback, 2013a). The reason why shaleanisotropy research attracts so much attention is thatignoring it results in errors in seismic data processing,petrophysical analysis, seismic interpretation, and hy-draulic fracturing (Sone and Zoback, 2013b; Thomsen,2013). The porosity, permeability, fracture characteris-tics, total organic carbon (TOC) content, and brittlenessindex (BI) of shale reservoirs are key parameters in theevaluation of shale formations.

The TOC content is an important parameter for shalegeophysical evaluation, and it is directly reflected in thegeophysical response (Vernik and Milovac, 2011; Zhu

et al., 2012). The TOC content affects the porosity, elasticmodulus, elastic anisotropy, and geomechanical behav-ior of shale formations (Sayers, 2013a). The brittlenessestimation of resource plays is one of the most importanttasks in unconventional reservoir characterization. Theconcept of brittleness, which was proposed by Yaraliand Kahraman (2011), is defined as the capacity of a rockformation to undergo failure and retain fractures duringhydraulic fracturing. In conventional reservoirs, mea-surements of brittleness are mainly used to evaluatethe suitability of coal-bearing formations for drilling, cut-ting by sawing, and mechanical extraction (Jin et al.,2014). In unconventional shale reservoirs, brittlenessprovides key information to evaluate the capability ofthe formation on creating an effective fracture networkthat conducts the hydrocarbons to the borehole (Zhanget al., 2015).

There are nearly 20 methods to determine brittleness,and those methods can be mainly classified into threegroups: (1) direct measurements of stress and strainin the laboratory, (2) measurements of mineral content,and (3) empirical procedures that use elastic moduli. Thedirect brittleness determined using the stress and strainin the laboratory is a static measurement. The mineralcontent-based method uses the mineral analysis of shalesamples. However, direct and mineral-based brittleness

1Beijing Information Science and Technology University, School of Information and Communication Engineering, Beijing, China. E-mail: mr.wangqing@qq.com; beiergo@yahoo.com.

2The University of Alabama, Geological Sciences, Tuscaloosa, Alabama, USA. E-mail: bzhang33@ua.edu (corresponding author).3Chinese Academy of Geological Sciences, SinoProbe Center, Beijing, China. E-mail: hanjianguang613@163.com.Manuscript received by the Editor 16 August 2018; revised manuscript received 16 March 2019; published ahead of production 23 April 2019;

published online 03 July 2019. This paper appears in Interpretation, Vol. 7, No. 3 (August 2019); p. T637–T646, 16 FIGS.http://dx.doi.org/10.1190/INT-2018-0143.1. © 2019 Society of Exploration Geophysicists and American Association of Petroleum Geologists. All rights reserved.

t

Technical papers

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estimation cannot provide a 3D brittleness perdition dueto the limited amount of samples. Empirical methodsbased on elastic moduli are widely accepted and usedin industry. The most famous method was proposedby Rickman et al. (2008).

Rickman et al. (2008) suggest that the renormalizedPoisson’s ratio and Young’s modulus can be used to cal-culate the BI (Higgins et al., 2008). However, Rickman’smethod ignores intrinsic anisotropy shale formations.The well-aligned organic matters in shale and otherminerals make the shale formation illustrate anisotropycharacteristics (Sone and Zoback, 2013a). As a result,Poisson’s ratio and Young’s modulus should exhibitanisotropy features (Luan et al., 2014). Therefore, thebrittleness estimation of shale using elastic moduli is alsoanisotropic and varies with orientation along which wemeasure the capability of producing fractures. Luan et al.(2014) and Higgins et al. (2008) study the anisotropy fea-tures of BI for shale samples. These samples have differ-ent degrees of cementation. They compute the BI usingthe ratio between anisotropic Young’s modulus and Pois-son’s ratio obtained in the laboratory.

Another shortfall of current popular BI estimation isthat those methods ignore the rock physics behavior ofshale formations. The failure criterion of rock is one ofthe original concepts used to determine the stress con-ditions of fracture and fail for brittle rocks (Cho andPerez 2014). Wang et al. (2015) propose a method usingan anisotropic brittleness criterion index that takes intoaccount parameters of anisotropy, failure criteria, andthe angle between the axis of symmetry and the propa-gating direction (Cho and Perez, 2014). However, thismethod does not consider the S-wave splitting of verti-cally transversely isotropic (VTI) media that result fromchanges in the incidence angles of seismic waves. The S-wave can affect the comprehensive and accuracy of BIestimation. The comprehensive S-wave information canmore accurately reflect the elastic information and frac-turing information of the rock in the shear direction.

Ignoring the anisotropy characteristics and failure cri-terion of shale formations would result in errors in seis-mic data processing, petrophysical analysis, seismicinterpretation, and hydraulic fracturing design (Sone andZoback, 2013a; Thomsen, 2013). In this paper, we defineBI by considering parameters estimated from the slowand fast S-wave, the anisotropy along a different orien-tation, and the failure criterion stress. This method is ap-plied to shale samples in the formations of the SantaMaria Basin. We further discuss the relationship amongbrittleness, porosity, and TOC under isotropic, aniso-tropic, and anisotropic conditions with failure criterionstress. The relationship between confining pressure (Pc)and BI under different TOCwas also studied. These stud-ies help us better understand the effects of anisotropicparameters, the orientation along which we measure thebrittleness, incident angle, critical tensile stress, and Pcon shale BI evaluation.

Theory and methodShale anisotropy is currently the focus of academic

and industrial research (Chopra et al., 2012; Guo et al.,2013; Sone and Zoback, 2013b; Marfurt, 2014). The shaleis usually characterized by transversely isotropic (TI)media. Its axis of rotational symmetry is perpendicularto the bedding plane (Sayers, 2013b). Anisotropy has asignificant influence on the determination of the BIand fracture mechanics of shale formations, and neglect-ing the anisotropy of shale will lead to errors in the fol-lowing parameter calculations (Heidari et al., 2014).

Anisotropy background of shaleShale has obvious anisotropic characteristics. Due to

the horizontal distribution of its deposition and that itsaxis of rotational symmetry is perpendicular to the bed-ding plane, it has often been reduced to a VTI mediummodel for research by academia and industry (Figure 1).Its elastic stiffness matrix can be expressed as

C ¼

������������

C11 C12 C13 0 0 0C12 C11 C13 0 0 0C13 C13 C33 0 0 00 0 0 C44 0 00 0 0 0 C44 00 0 0 0 0 C66

������������: (1)

Here, Cij is the elastic stiffness constant. Three aniso-tropic parameters can be expressed by stiffness coeffi-cients:

ε ¼ C11 − C33

2C33;

γ ¼ C66 − C44

2C44;

δ ¼ ðC13 þ C44Þ2 − ðC33 − C44Þ22C33ðC33 − C44Þ

: (2)Figure 1. Layered shale schematic.

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The parameters ε and γ represent the intensities of theP-wave and the S-wave anisotropy, respectively. Theterm ε represents the difference between the verticaland horizontal directions of the longitudinal wave veloc-ity. The larger γ is, the greater the shear strength of thetransverse wave. The physical meaning of parameter δ isnot as clear as epsilon and gamma, but it also control thewave propagation, especially the wave propagation inthe vicinity of the vertical direction. For isotropic media,ε, γ, and δ are all zero.

To calculate the elastic stiffness constant Cij, thethree-plug method was proposed by Vernik et al. (1994),which involves measurement of the velocities of S- andP-waves in directions perpendicular (0°), parallel (90°),and diagonal (45°) to the axis of symmetry using coreplugs aligned in these three directions:

C11¼ρV2Pð90°Þ;

C12¼C11−2ρV2shð90°Þ;

C33¼ρV2ρð0°Þ;

C44¼ρV2Sð0°Þ;

C13¼−C44þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4ρ2V4

ρð45°Þ−2ρV2Pð45°ÞðC11þC33þ2C44þðC11þC44ÞðC33þC44Þ

q;

C66¼ρV2Sð90°Þ: (3)

Full-angle anisotropic brittleness indexThe actual shale is not an ideal VTI medium (Yang

and Jun, 2018). S-waves are split into Sv and Sh waveswhen the inclination of the phase angle is not equal tozero (Figure 2). The TI media elastic wave phase veloc-ity at the moment is

VPðθÞ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12ρ

�ðC11þC44Þsin2θþðC33þC44Þcos2θþ

ffiffiffiffiD

ps �

;

V shðθÞ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1ρðC66sin2θþC44cos2θÞ

s;

V svðθÞ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12ρ

�ðC11þC44Þsin2θþðC33þC44Þcos2θ−

ffiffiffiffiD

p �s;

D¼½ðC11−C44Þsin2θ−ðC33−C44Þcos2θ�2þ4ðC13þC44Þ2sin2θcos2θ; (4)

where VPðθÞ, V shðθÞ, V svðθÞ, and ρ are the velocity of theP-wave, Sh-wave, Sv-wave, and the density of shale in“θ” degree, respectively. The term θ is the angle be-tween the measuring direction and axis of symmetry ofthe TI medium. We substitute equation 4 into the for-mula for Young’s modulus and Poisson’s ratio proposedby Sharma and Chopra (2015) and Xiong et al. (2018)and equations 5 and 6 can be obtained

vðθÞsh ¼ V 2PðθÞ − 2V2

shðθÞ2V 2

PðθÞ − 2V2shðθÞ

;

EðθÞsh ¼ ρV 2sh

3V2PðθÞ − 4V 2

shðθÞV2

PðθÞ − V 2shðθÞ

: (5)

vðθÞsv ¼V 2

PðθÞ − 2V2svðθÞ

2V 2PðθÞ − 2V2

svðθÞ;

EðθÞsv ¼ ρV 2sv3V2

PðθÞ − 4V 2svðθÞ

V2PðθÞ − V 2

svðθÞ: (6)

Then, we put equations 5 and 6 into Rickman’s BI cal-culation formula (Rickman et al., 2008). The full-angle BIis derived (equations 7–9):

EðθÞshn ¼ EðθÞsh − EðθÞsh;min

EðθÞsh;max − EðθÞsh;min;

vðθÞshn ¼ vðθÞmax − vðθÞshvðθÞsh;max − vðθÞsh;min

;

BðθÞsh ¼ EðθÞsh;n þ vðθÞsh;n2

: (7)

Figure 2. Wavefield propagation diagram.

Figure 3. The full-angle BI diagram. Note that θ is the anglebetween the measuring direction and the axis of symmetryof the TI medium. The term BðθÞ indicates the full-angle BI.The dashed arrows indicate a direction opposite to the solidarrows.

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EðθÞsvn ¼ EðθÞsv − EðθÞsv;min

EðθÞsv;max − EðθÞsv;min;

vðθÞsvn ¼ vðθÞsv;max − vðθÞsvvðθÞsv;max − vðθÞsv;min

;

BðθÞsv ¼EðθÞsv;n þ vðθÞsv;n

2: (8)

BðθÞ ¼ BðθÞsh þ BðθÞsv2

; (9)

where BðθÞ is the full-angle anisotropic BI. It takes intoaccount the angles, anisotropy parameters, and full wave-field information. From equations 7 to 9 based on Rick-man’s method, we obtain the shale BI caused bydifferent angles and different fluctuation directions.

Figure 3 shows a BI from 0° to 360° in the clockwisedirection. The numbers 1, 2, 3, and 4 represent the first,second, third, and fourth quadrants, re-spectively. Due to the symmetry of thefour quadrants, the value of the BI justneeds to be computed in one quadrant.

Full-angle anisotropic BI with failurecriteria stress (BIac)

The theory of the Griffith energy bal-ance indicates that a high Poisson’s ratioand a low Young’s modulus mean morefavorable conditions for rock failure(Kartashov, 1978; Wang et al., 2015). Thedecrease in Young’s modulus would re-duce the critical tensile stress required togenerate crack growth (Kartashov, 1978;Cho and Perez, 2014). Therefore, the BIof shale under hydraulic fracturing can-not be accurately reflected only by theYoung’s modulus and Poisson’s ratio.To solve this problem, the critical tensilestress parameters are used to balance theanisotropic BI (equations 7–9):

TCðθÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

πγEðθÞ4ð1 − vðθÞ2Þc

s¼

ffiffiffiγ

c

r•

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiπEðθÞ

4ð1 − vðθÞ2Þ

s; (10)

where TCðθÞ is the critical tensile stress in θ degree, γ isthe surface energy per unit area, and c is the crack radius.We cannot accurately obtain the parameters γ and c ofeach crack. It is assumed that the crack is expanding tothe same size. Thus, we consider

ffiffiffiffiffiffiffiγ∕c

pto be a constant,

and we useffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiπEðθÞ∕4ð1 − vðθÞ2Þ

pto balance full angles

anisotropic BI with failure criteria stress:

TCðθÞsh ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

πγEðθÞsh4ð1 − vðθÞ2ÞC

s¼

ffiffiffiγ

c

r•

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiπEðθÞsh

4ð1 − vðθÞ2Þ

s;

TCðθÞsv ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

πγEðθÞsv4ð1 − vðθÞ2ÞC

s¼

ffiffiffiγ

c

r•

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiπEðθÞsv

4ð1 − vðθÞ2Þ

s: (11)

Then, TCðθÞ is normalized as

Figure 4. The BI of the shale samples. The radius represents the BI of the corresponding angle. The BI value is calculated byRickman’s formula.

Figure 5. The anisotropic BI of the shale samples. The radius represents the BIof the corresponding angle. The BI value is calculated in equation 7. Note that(d) indicates that the S-wave is the Sh-wave.

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TCðθÞshn ¼ TCðθÞsh − TCðθÞsh;min

TCðθÞsh;max − TCðθÞsh;min;

TCðθÞsvn ¼ TCðθÞsv − TCðθÞsv;min

TCðθÞsv;max − TCðθÞsv;min: (12)

The greater the Young’s modulus, the stronger thebrittleness. However, a greater Young’s modulus wouldincrease the critical tensile stress, which leads to theneed for greater force to create crack growth (Karta-shov, 1978; Cho and Perez 2014; Yin et al., 2017). To bal-ance this deficiency in equations 7 and 8, we have nowcombined it with stress failure criteria:

BCðθÞsh ¼ BðθÞshTCðθÞsh;n

;

BCðθÞsv ¼BðθÞsv

TCðθÞsv;n: (13)

To average calculation of normalized BI calculatedfrom two kinds of S-waves:

BCðθÞ ¼BCðθÞsh þ BCðθÞsv

2; (14)

where BCðθÞ is the full-angle anisotropic BI with failurecriteria stress (BIac).

Application and discussionFull-angle anisotropic brittleness index withfailure criteria stress

We apply the Rickman’s method, full-angle aniso-tropic BI, and full-angle anisotropic BI with failure crite-ria stress to laboratory measurements of the shalesample obtained from the Monterey Formation, SantaMaria Basin. Figure 4 shows the Rickman’s method BIat full angles (0°–360°) with different TOC. They indicatethat the values of the BI are independent of the bedding

plane angle, whereas they increase withdecreasing TOC.

Figures 5 and 6 show the anisotropicBI at full angle at the same point based onthe Sh and Sv waves (equations 7 and 8),which represent the degree of crack frac-turing in different directions. As a resultof the existence of anisotropy, the BIcalculated in the full wavefield changesfrom a scalar to a vector. In addition,under different TOC, changes in anglesallow a great diversity of the BI. How-ever, the maximum value of BI decreaseswith increasing TOC similar to themethod proposed by Rickman et al.(2008).

Figure 7 shows the final result of aniso-tropic brittleness in the full wavefield. Acomparison with Rickman’s method ofisotropic calculation found that shalebrittleness indices show variations withangles and that variations in the brittle-ness indices of shales with different TOCat angles are not the same. It also illus-trates that the brittleness of a shale is re-lated to the TOC. In Figures 8 and 9,

Figure 7. The full-wave anisotropic BI of the shale samples. The radius represents the BI of the corresponding angle. The BI valueis calculated in equation 9.

Figure 6. The anisotropic BI of the shale samples. The radius represents the BIof the corresponding angle. The BI value is calculated in equation 8. Note that(d) indicates that the S-wave is the Sv-wave.

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failure criteria stress is introduced to calculate the aniso-tropic brittleness at full angles. This method was devel-oped for computing the anisotropic BI at full angles inthe full wavefield, whereas we introduced the fracturepropagation parameters from engineering mechanics tomitigate the shortcomings of the previous method (Fig-ure 10). From Figures 4 to 10, it is shown in turn thatas the parameters increase, the value of the shale index

becomes more and more accurate as the angle changes.The consideration of the stress criteria stress factormakes the shale BI more in line with the engineering me-chanics evaluation criteria.

The effect of rock properties on full-angleanisotropic BI with failure criteria stress

Rock properties such as the shale’s TOC,mineral com-position, and porosity affect the elasticmechanics characteristics of the shale,and at the same time provide anisotropiccharacteristics. Figure 11a providesthe relationship among the shale BI,the porosity in the isotropic case, andthe TOC. In the case of isotropy, the shaleBI decreases with the increase in poros-ity and TOC. In the anisotropic case, theBI at full angle also exhibits the samechange characteristics (Figure 11b).The difference is that the shale BI ischaracterized by cyclic changes at anglesbecause of the existence of anisotropy.However, previous studies suggest thatthe porosity and TOC were not theunique reference factor for evaluationand the BI of not all shales decreasedwith the increase in porosity and TOC.Even if they were the same shales, differ-ent perforation angles have various levelsof difficulty in shale fractures. Figure 11cshows the relationship between the de-crease in the shale BI with the increaseof TOC and porosity.

It can be seen that the change in an-gles has an effect on the trends in Fig-ure 11b and 11c. Thus, selecting a rightangle is able to find the best BI positionfor a shale. This phenomenon gave a rea-sonable explanation for the integrationof shale BI evaluation. The limited earlysingle mineral calculation method andRickman’s method need to be corrected.

We chose several representative an-gles from anisotropic brittlenesses: 0°,30°, 45°, and 90° for performing an analy-sis between anisotropy parametersand BI. Figure 12 shows, respectively,the full-angle brittleness anisotropy andthe relationship among the anisotropicparameters ε, δ, and γ. There is no betterlinear relationship between ε and γ andthe brittleness anisotropy at full angles(Figure 12a and 12c). For ε, when ε isgreater than 0.12, the BI changes littlewith ε. For gamma, there is no changein consistency between the BI and γ.The term δ shows a good linear relation-ship (Figure 12c). Parameter δ can be

Figure 8. The full-angle anisotropic BI with failure criteria stress. The BI valueis calculated in equation 13. Note that (d) indicates that the S-wave is theSh-wave.

Figure 9. The full-angle anisotropic BI with failure criteria stress. The BI valueis calculated in equation 13. Note that (d) indicates that the S-wave is theSv-wave.

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used for modeling and forecasting the anisotropic BI.Previous studies have demonstrated that the aniso-

tropy of BI, so the shale at different angles should havea different linear relationship at different angles be-tween the anisotropic parameters and the anisotropybrittle index because it has different characteristicssuch as porosity, TOC, and mineral content. Figure 13shows, respectively, the relationships between the full-angle BI with the failure criteria stress and anisotropyparameters ε, δ, and γ. Among them, BIac has a linearrelationship with ε at 0° and 45°; at 0°, 30°, 45°, and 90°,the linear slope between δ and BIac showed a largerdifference, in which the slope value is larger at 30°and 45°, with the influence of anisotropy on BIac beingthe largest. The parameter γ has shown no obvious lin-ear relationship.

Figure 14 shows the relationship between the Pcand the isotropic BI under different TOC. In Figure 14a,TOC is 0.19%, the BI reaches its peak when the Pcincreases to 40 MPa. Also, the brittleness reaches itsmaximum when the TOC increases to 8.12% at 2 MPa(Figure 14c).

The anisotropic BI in the full wavefield (Figure 15)and BIac (Figure 16) conducted an in-depth researchon this issue. By using the MTS instrument for the tri-axial Pc test, we obtained the longitudinal and trans-verse wave velocities of the shale under different Pcconditions, and then we calculate its various brittlenessindices. In addition to the same characteristics as in Fig-ure 15, we found that at any TOC, the maximum BIvalue changes with PCs at angles; that is, a certain givenPc value cannot always maintain the maximum BI value

Figure 11. The relationship among the shale BI, the TOC, and porosity: (a) isotropic condition, (b)anisotropic condition, and(c) anisotropic condition with failure criteria stress.

Figure 12. The full-angle brittleness anisotropy and the relationship among the anisotropic parameters: (a) ε, (b) δ, and (c) γ.

Figure 10. The full-angle anisotropic BI with failure criteria stress. The BI value is calculated in equation 14.

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at full angle. The BI is a dynamic change process. Whenwe evaluate the BI distribution and service fracturingdesign, we must consider different pressures, angles,

TOC, and so on. Figure 16 displays the most compre-hensive characteristics of this multiparameter in theBI evaluation.

Figure 15. The relationship between the anisotropic BI in the full wavefield and Pc under different TOCs.

Figure 16. The relationship between BIac and Pc under different TOCs.

Figure 13. The full-angle anisotropic BI with failure criteria stress among the anisotropic parameters: (a) ε, (b) δ, and (c) γ.

Figure 14. The relationship between Pc and the isotropic BI under different TOCs.

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ConclusionWe demonstrate that the brittleness of the shale

formation is an orientation-dependent parameter. Themaximum BI is not always parallel or perpendicular tothe bedding plane of shale formations. Anisotropicparameter δ demonstrates a better linear relationshipwith the BI, and the slope of the linear relationship varieswith the orientation along which we measure the brittle-ness. As the δ increases, the BI shows a tendency to be-come smaller. The analysis between Pc and the BIindicates that Pc heavily affects the brittleness computa-tion. However, the BI does not proportionally vary withthe value of the Pc. The Pc required to reach the maxi-mum BI also varies with orientation along which wemea-sure the brittleness of the rock. The proposed brittlenesscomputation can be used to determine the optimum ori-entation along which we should perform hydrofracturingof the shale formations.

AcknowledgmentsThe authors would like to thank the Beijing Informa-

tion Science and Technology University Research Fundproject (grant no. 182501), the Scientific ResearchProject of Beijing Educational Committee (grant nos.KM201811232010 and KM201911232012), the BeijingScience and Technology Innovation Service CapacityBuilding Fundamental Research Fund (grant no.PXM2018_014224_000032), and Project for PromotingConnotative Development of Universities “Information+” for the funding. Our appreciation also goes to theanonymous reviewers for their constructive comments.

Data and materials availabilityData associated with this research are available and

can be obtained by contacting the correspondingauthor.

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