# Anisotropic Failure Strength of Shale with Increasing Confinement: Behaviors, Factors and Mechanism

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## Abstract

**:**

## 1. Introduction

## 2. Classifications on Anisotropic Failure Strength Behaviors of Shale by Experimental Results

#### 2.1. Degree of Anisotropic Failure Strength

_{2}and f also present a decreasing trend like the parameter $S{A}_{1}$, because they are all defined from the perspective of strength ratio. The parameter k

_{1}keeps almost constant because this parameter only considers the strength of the shale samples at the directions parallel and perpendicular to the weak planes, and the anisotropic strength behaviors induced by the structures cannot be reflected completely.

#### 2.2. Laboratory Experimental Database

#### 2.3. Different Types of Anisotropic Strength Behaviors Based on SA_{1}

_{1}: Significant decrease of $S{A}_{1}$ with increasing confinement.

_{3}increasing from 0 to 30 MPa (Figure 3a).

_{1}: Slight decrease of SA

_{1}with increasing confinement.

_{3}from 0 to 170 MPa (Figure 3b).

_{1}: Generally constant SA

_{1}with increasing confinement.

_{3}from 0 to 69 MPa (Figure 3c).

_{1}: Slight increase of SA

_{1}with increasing confinement.

_{3}= 40 MPa. Although there appears a little reduction to 1.27 and 1.21 when σ

_{3}= 60 and 100 MPa, they are still higher than the cases at σ

_{3}= 0 and 20 MPa (Figure 3d).

#### 2.4. Different Types of Anisotropic Strength Behaviors Based on SA_{2}

_{2}: Gradual decrease of SA

_{2}with increasing confinement.

_{2}: Slight increase of SA

_{2}with increasing confinement.

_{2}with the rise of confinement. Taking Greenriver Shale-2 as an example, $S{A}_{2}$ goes up gradually from 54.8 to 91.3 MPa with the confinement increases from 0 to 170 MPa (Figure 4b).

_{2}: Significant increase of SA

_{2}with increasing confinement.

#### 2.5. Discussions

## 3. Anisotropic Strength Behaviors Affected by Cohesion and Friction Angle of Weak Planes

#### 3.1. Bonded-Particle Discrete Element Modelling

#### 3.2. Modelling Analyses

#### 3.2.1. Influence of Cohesion of Weak Planes

- (1)
- In the case of a low to medium friction angle (${\varphi}_{\mathrm{w}}$ = 10° and 30°), the increase of cohesion ${c}_{\mathrm{w}}$ may transfer the $S{A}_{1}$ behaviors from significant decrease to slight decrease or even slight increase with the confinement going up;
- (2)
- For a high friction angle (${\varphi}_{\mathrm{w}}$ = 50°), increasing cohesion ${c}_{\mathrm{w}}$ can also change the $S{A}_{1}$ features from significant decrease to slight decrease, however, it is difficult to obtain the increasing trend of $S{A}_{1}$ with the rise of confinement;
- (3)
- Generally speaking, the lower cohesion ${c}_{\mathrm{w}}$ may be prone to lead to the significant decrease of $S{A}_{1}$ with the increasing confinement, while the higher cohesion ${c}_{\mathrm{w}}$ will weaken this trend, but whether it will be slight decrease or increase is dependent on the friction angle ${\varphi}_{\mathrm{w}}$ of the weak planes.

- (1)
- When friction angle ${\varphi}_{\mathrm{w}}$ = 10°, the increase of cohesion ${c}_{\mathrm{w}}$ may lower all $S{A}_{2}$ values under various confinements, and the increasing trend of $S{A}_{2}$ will be more significant with the increasing confinement;
- (2)
- When friction angle ${\varphi}_{\mathrm{w}}$ = 30°, the increase of cohesion ${c}_{\mathrm{w}}$ makes the slight decreasing trend of $S{A}_{2}$ transfer to a slight or significant increase with the increasing confinement;
- (3)
- When friction angle ${\varphi}_{\mathrm{w}}$ = 50°, the increase of cohesion ${c}_{\mathrm{w}}$ makes the significant decreasing trend of $S{A}_{2}$ transfer to a slight decrease as the confinement increases.

#### 3.2.2. Influence of Friction Angle of Weak Planes

- (1)
- For lower cohesion (${c}_{\mathrm{w}}$ = 10 MPa), the increasing friction angle ${\varphi}_{\mathrm{w}}$ can make the decreasing trend of $S{A}_{1}$ more and more significant;
- (2)
- For medium to higher cohesion (${c}_{\mathrm{w}}$ = 20 and 40 MPa), the increasing friction angle ${\varphi}_{\mathrm{w}}$ may transfer the slight increasing or almost constant trend of $S{A}_{1}$ to slight increasing behaviors;
- (3)
- As the cohesion ${c}_{\mathrm{w}}$ increases, the influence of friction angle ${\varphi}_{\mathrm{w}}$ on the degree of $S{A}_{1}$ changing behaviors is more and more limited.

- (1)
- For all cases of cohesion (${c}_{\mathrm{w}}$ = 10, 20 and 40 MPa), the increasing friction angle ${\varphi}_{\mathrm{w}}$ can induce the transferring of the $S{A}_{2}$ trend from going up to going down with the rise of confinement;
- (2)
- As the cohesion ${c}_{\mathrm{w}}$ increases, the influence of friction angle ${\varphi}_{\mathrm{w}}$ on the degree of $S{A}_{2}$ changing behaviors is more and more limited.

#### 3.2.3. Conjoint Analysis on Both Factors ${c}_{\mathrm{w}}$ and ${\varphi}_{\mathrm{w}}$

_{1}(significant decrease) to II

_{1}(slight decrease), III

_{1}(generally constant), or even IV

_{1}(slight increase). Meanwhile, the cases with lower friction angle ${\varphi}_{\mathrm{w}}$ are more prone to have weaker decreasing trend of $S{A}_{1}$ or even increase of $S{A}_{1}$. The phenomenon of increasing $S{A}_{1}$ with increasing confinements occurs for the cases with lower to medium friction angle (${\varphi}_{\mathrm{w}}$ = 10° and 30°) and higher cohesion (${c}_{\mathrm{w}}$ = 40 MPa).

_{2}(significant increase) to II

_{2}(slight decrease), or I

_{2}(significant decrease). What is more, the medium to higher cohesion ${c}_{\mathrm{w}}$ is more probable to induce significant increase of $S{A}_{2}$ with the increasing confinement.

_{1}(slight increase) for $S{A}_{1}$ behavior and Type III

_{2}(significant increase) for $S{A}_{2}$ behavior. The fracturing patterns are found to be closely related to the strength characteristics of the samples and the properties of the weak planes [40,41]. Figure 9 presents the typical failure patterns of Shale-5 samples with different inclination angles ($\beta $ = 30° and 90°) under different confinements (σ

_{3}= 0 and 60 MPa). For the case of $\beta $ = 30° (Figure 9a,b), the specimen mainly fails by vertical extension fractures in the shale matrix under uniaxial compression (σ

_{3}= 0 MPa), while shear failure planes can be observed crossing the weak planes under the confinement of σ

_{3}= 60 MPa. No obvious sliding can be observed along the weak planes, and the failure is mainly controlled by the strength of the shale matrix. For the case of $\beta $ = 90° (Figure 9c,d), the failure takes place by vertical extension along the weak planes under uniaxial compression (σ

_{3}= 0 MPa), and by shear fractures in the shale matrix under the confinement of σ

_{3}= 60 MPa. For both of these two cases, the strength of Shale-5 samples are not significantly weakened by the weak planes.

_{3}= 0 MPa; however, the failure is totally along the weak planes, and the fracture surface is very flat and quite smooth under higher confinement σ

_{3}= 20, 40, 60 and 100 MPa. These fracturing characteristics show that the weak planes of Shale-5 samples have high cohesion ${c}_{\mathrm{w}}$ but relatively low friction angle ${\varphi}_{\mathrm{w}}$. This estimation can be supported by the numerical results shown in Figure 11. For the numerical samples with $\beta $ = 60° and ${c}_{\mathrm{w}}$ = 40 MPa, different values of ${\varphi}_{\mathrm{w}}$ result in different failure characteristics under the confinement σ

_{3}= 30 MPa. For lower ${\varphi}_{\mathrm{w}}$ = 10°, the failure mainly slips along the weak planes. As ${\varphi}_{\mathrm{w}}$ increases to 30°, a few cracks can be observed in the shale matrix. When ${\varphi}_{\mathrm{w}}$ is as high as 50°, there are lots of fractures shown in the shale matrix. Although this numerical model is not exactly the same with the conditions of Shale-5 samples, it can demonstrate that lower ${\varphi}_{\mathrm{w}}$ may result in slip along the weak planes but higher ${\varphi}_{\mathrm{w}}$ may induce the fractures in the shale matrix for the samples with $\beta $ = 60° under high confinements.

_{n}and τ) on the weak planes by the stress transformation equations:

_{3}= 20, 40, 60 and 100 MPa) presented in Figure 12a, the normal and shear stresses on the weak planes can be calculated according to Equations (5) and (6), and they are plotted in Figure 12b. According to Coulomb’s criterion for structural planes:

_{1}(slight increase) for $S{A}_{1}$ behavior and Type III

_{2}(significant increase) for $S{A}_{2}$ behavior, it is consistent with the analyses by the numerical results that it is more prone to have slight increase of $S{A}_{1}$ and significant increase of $S{A}_{2}$ with increasing confinement for medium to higher cohesion ${c}_{\mathrm{w}}$ and lower to medium friction angle ${\varphi}_{\mathrm{w}}$.

## 4. Discussions

_{3}. Under lower confinement, friction angle ${\varphi}_{\mathrm{w}}$ has very limited influences on the strength, so cohesion ${c}_{\mathrm{w}}$ becomes more important here. As the confinement goes up, the role of friction angle ${\varphi}_{\mathrm{w}}$ with different values may have different degrees of enhancing, while the effect of cohesion ${c}_{\mathrm{w}}$ may not be improved significantly. Consequently, different combinations of ${c}_{\mathrm{w}}$ and ${\varphi}_{\mathrm{w}}$ may have various types of influences on the minimum strength ${\sigma}_{1,\mathrm{min}}$ with the increasing confinement σ

_{3}. Thereafter, different types of anisotropic strength behaviors can be shown for different shale samples with increasing confinement.

- (1)
- Case I: for lower cohesion (${c}_{\mathrm{w}}$ = 10 MPa) and lower friction angle (${\varphi}_{\mathrm{w}}$ = 10°), there is quite a large difference between ${\sigma}_{1,\mathrm{max}}$ and ${\sigma}_{1,\mathrm{min}}$ under lower confinement mainly resulted from the low value of ${c}_{\mathrm{w}}$, and the strength difference is also very considerable under higher confinement because the low value of ${\varphi}_{\mathrm{w}}$ cannot increase ${\sigma}_{1,\mathrm{min}}$ effectively with the increasing σ
_{3}. In this case, the anisotropic strength ratio $S{A}_{1}$ may be lowered with increasing confinement, while the anisotropic strength difference $S{A}_{2}$ may not increase or decrease significantly. - (2)
- Case II: for lower cohesion (${c}_{\mathrm{w}}$ = 10 MPa) and higher friction angle (${\varphi}_{\mathrm{w}}$ = 50°), the difference between ${\sigma}_{1,\mathrm{max}}$ and ${\sigma}_{1,\mathrm{min}}$ is again very large under lower confinement owing to the low ${c}_{\mathrm{w}}$, however, as the high value of ${\varphi}_{\mathrm{w}}$ can enhance ${\sigma}_{1,\mathrm{min}}$ significantly under higher confinement, the strength difference turns much smaller. In this case, both anisotropic strength ratio $S{A}_{1}$ and anisotropic strength difference $S{A}_{2}$ will decrease obviously with the increase of confinement.
- (3)
- Case III: for higher cohesion (${c}_{\mathrm{w}}$ = 40 MPa) and lower friction angle (${\varphi}_{\mathrm{w}}$ = 10°), the difference between ${\sigma}_{1,\mathrm{max}}$ and ${\sigma}_{1,\mathrm{min}}$ is much smaller than the first two cases as the cohesion ${c}_{\mathrm{w}}$ has quite a high value, while the strength difference becomes larger with the increasing confinement because the low value of ${\varphi}_{\mathrm{w}}$ leads to quite a low ${\sigma}_{1,\mathrm{min}}$. In this case, the anisotropic strength ratio $S{A}_{1}$ may remain almost constant or ever increase slightly with the increasing confinement, while the anisotropic strength difference $S{A}_{2}$ will increase significantly.
- (4)
- Case IV: for higher cohesion (${c}_{\mathrm{w}}$ = 40 MPa) and higher friction angle (${\varphi}_{\mathrm{w}}$ = 50°), there is quite a small difference between ${\sigma}_{1,\mathrm{max}}$ and ${\sigma}_{1,\mathrm{min}}$ under lower confinement attributed to the high value of ${c}_{\mathrm{w}}$, and the strength difference is also very limited under higher confinement because the high value of ${\varphi}_{\mathrm{w}}$ can increase ${\sigma}_{1,\mathrm{min}}$ effectively with the increasing σ
_{3}. Similar to the first case, the anisotropic strength ratio $S{A}_{1}$ may be lowered with increasing confinement, while the anisotropic strength difference $S{A}_{2}$ may not change significantly.

## 5. Conclusions

- (1)
- Two anisotropic strength parameters, $S{A}_{1}$ from the perspective of strength ratio and $S{A}_{2}$ from the perspective of strength difference, should both be researched for a comprehensive understanding of the anisotropic strength behaviors of shale under different confinements. $S{A}_{1}$ is better for comparing the anisotropic strength characteristics of different shale samples as a dimensionless coefficient, while $S{A}_{2}$ is easier to be applied to estimate the stability of a certain shale based on the strength criterion because it considers the specific values of strength differences;
- (2)
- Based on the laboratory experimental results of nine groups of different shale samples, it is found that there are four types of $S{A}_{1}$ behaviors (significant decrease, slight decrease, generally constant, and slight increase) and three types of $S{A}_{2}$ behaviors (gradual decrease, slight increase, and significant increase) with increasing confinement;
- (3)
- With the parallel bonded particle model simulating the rock material and smooth-joint model simulating the weak planes, the different types of anisotropic strength behaviors are well reproduced in the numerical models. By a series of systematic analyses, it is observed that cohesion ${c}_{\mathrm{w}}$ and friction angle ${\varphi}_{\mathrm{w}}$ of the weak planes are two dominant factors for the anisotropic strength behaviors;
- (4)
- The increase of cohesion ${c}_{\mathrm{w}}$ will change the $S{A}_{1}$ behaviors from significant decrease to slight decrease with increasing confinement, or even slight increase if the friction angle ${\varphi}_{\mathrm{w}}$ is medium to low. Meanwhile, the decrease of friction angle ${\varphi}_{\mathrm{w}}$ are more prone to transfer $S{A}_{2}$ behaviors from gradual decrease to slight increase with increasing confinement, or even significant increase if the cohesion ${c}_{\mathrm{w}}$ is medium to high;
- (5)
- The mechanism of the anisotropic strength behaviors have been analyzed based on the well-known Jaeger’s strength criterion, as well as the laboratory and numerical test results. Under lower confinement, cohesion ${c}_{\mathrm{w}}$ has more important roles as the friction angle ${\varphi}_{\mathrm{w}}$ has very limited influences on the strength. As the confinement goes up, the friction angle ${\varphi}_{\mathrm{w}}$ with different values may take different degrees of roles, while the effect of cohesion ${c}_{\mathrm{w}}$ is not easy to be improved significantly. Consequently, different combinations of ${c}_{\mathrm{w}}$ and ${\varphi}_{\mathrm{w}}$ may have various types of influences on the minimum failure strength with the increasing confinement, therefore different shale samples show different types of anisotropic behaviors with the increasing confinement.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Ewy, R.T.; Bovberg, C.A.; Stankovic, R.J. Strength anisotropy of mudstones and shales. In Proceedings of the 44th U.S. Rock Mechanics Symposium and 5th U.S.-Canada Rock Mechanics Symposium, Salt Lake City, UT, USA, 27–30 June 2010; American Rock Mechanics Association: Alexandria, VA, USA, 2010. [Google Scholar]
- Crawford, B.R.; Dedontney, N.L.; Alramahi, B.; Ottesen, S. Shear strength anisotropy in fine-grained rocks. In Proceedings of the 46th U.S. Rock Mechanics/Geomechanics Symposium, Chicago, IL, USA, 24–27 June 2012; American Rock Mechanics Association: Alexandria, VA, USA, 2012. [Google Scholar]
- Ambrose, J.; Zimmerman, R.W.; Suarez-Rivera, R. Failure of shales under triaxial compressive stress. In Proceedings of the 46th U.S. Rock Mechanics/Geomechanics Symposium, Minneapolis, MN, USA, 1–4 June 2014. [Google Scholar]
- Fjær, E.; Nes, O.-M. The impact of heterogeneity on the anisotropic strength of an outcrop shale. Rock Mech. Rock Eng.
**2014**, 47, 1603–1611. [Google Scholar] [CrossRef] - Zinszner, B.; Meynier, P.; Cabrera, J.; Volant, P. Ultrasonic, sonic ans seismic waves velocity in shale from tournemire tunnel. Impact of anisotropy and natural fractures. Oil Gas Sci. Technol.
**2002**, 57, 341–353. [Google Scholar] [CrossRef] - Huang, L.C.; Xu, Z.S.; Zhou, C.Y. Modeling and monitoring in a soft argillaceous shale tunnel. Acta Geotech.
**2009**, 4, 273–282. [Google Scholar] [CrossRef] - Niandou, H.; Shao, J.; Henry, J.; Fourmaintraux, D. Laboratory investigation of the mechanical behaviour of tournemire shale. Int. J. Rock Mech. Min. Sci.
**1997**, 34, 3–16. [Google Scholar] [CrossRef] - Hsu, S.-C.; Nelson, P.P. Mechanical properties and failure mechanisms for clay shale masses. In Proceedings of the DC Rocks 2001, the 38th U.S. Symposium on Rock Mechanics (USRMS), Washington, DC, USA, 7–10 July 2001; American Rock Mechanics Association: Alexandria, VA, USA, 2001. [Google Scholar]
- Mese, A.; Tutuncu, A. Mechanical, acoustic, and failure properties of shales. In Proceedings of the DC Rocks 2001, the 38th U.S. Symposium on Rock Mechanics (USRMS), Washington, DC, USA, 7–10 July 2001; American Rock Mechanics Association: Alexandria, VA, USA, 2001. [Google Scholar]
- Ciz, R.; Shapiro, S.A. Stress-dependent anisotropy in transversely isotropic rocks: Comparison between theory and laboratory experiment on shale. Geophysics
**2009**, 74, D7–D12. [Google Scholar] [CrossRef] - Cho, J.-W.; Kim, H.; Jeon, S.; Min, K.-B. Deformation and strength anisotropy of asan gneiss, boryeong shale, and yeoncheon schist. Int. J. Rock Mech. Min. Sci.
**2012**, 50, 158–169. [Google Scholar] [CrossRef] - Hathon, L.A.; Myers, M.T. Shale rock properties: Peak strength, acoustic anisotropy and rock fabric. In Proceedings of the 46th U.S. Rock Mechanics/Geomechanics Symposium, Chicago, IL, USA, 24–27 June 2012; American Rock Mechanics Association: Alexandria, VA, USA, 2012. [Google Scholar]
- Fjær, E.; Nes, O.M. Strength anisotropy of mancos shale. In Proceedings of the 47th US Rock Mechanics/Geomechanics Symposium, San Francisco, CA, USA, 23–26 June 2013; American Rock Mechanics Association: Alexandria, VA, USA, 2013. [Google Scholar]
- Park, B.; Min, K.-B. Bonded-particle discrete element modeling of mechanical behavior of transversely isotropic rock. Int. J. Rock Mech. Min. Sci.
**2015**, 76, 243–255. [Google Scholar] [CrossRef] - Wu, Y.; Li, X.; He, J.; Zheng, B. Mechanical properties of longmaxi black organic-rich shale samples from south china under uniaxial and triaxial compression states. Energies
**2016**, 9, 1088. [Google Scholar] [CrossRef] - McLamore, R.T.; Gray, K.E. A strength criterion for anisotropic rocks based upon experimental observations. In Proceedings of the Annual Meeting of the American Institute of Mining, Metallurgical and Petroleum Engineers, Los Angeles, CA, USA, 19–23 February 1967. [Google Scholar]
- Wu, Z.; Zuo, Y.; Wang, S.; Yi, T.; Chen, S.; Yu, Q.; Li, W.; Sunwen, J.; Xu, Y.; Wang, R.; et al. Numerical simulation and fractal analysis of mesoscopic scale failure in shale using digital images. J. Pet. Sci. Eng.
**2016**, 145, 592–599. [Google Scholar] [CrossRef] - He, J.; Afolagboye, L.O. Influence of layer orientation and interlayer bonding force on the mechanical behavior of shale under brazilian test conditions. Acta Mech. Sin.
**2017**, 1–10. [Google Scholar] [CrossRef] - Duan, K.; Kwok, C.Y.; Pierce, M. Discrete element method modeling of inherently anisotropic rocks under uniaxial compression loading. Int. J. Numer. Anal. Methods Geomech.
**2016**, 40, 1150–1183. [Google Scholar] [CrossRef] - Lisjak, A.; Tatone, B.S.A.; Grasselli, G.; Vietor, T. Numerical modelling of the anisotropic mechanical behaviour of opalinus clay at the laboratory-scale using fem/dem. Rock Mech. Rock Eng.
**2014**, 47, 187–206. [Google Scholar] [CrossRef] - Chen, H.; Jiao, Y.; Liu, Y. Investigating the microstructural effect on elastic and fracture behavior of polycrystals using a nonlocal lattice particle model. Mater. Sci. Eng. A
**2015**, 631, 173–180. [Google Scholar] [CrossRef] - Chen, H.J.; Jiao, Y.; Liu, Y. A nonlocal lattice particle model for fracture simulation of anisotropic materials. Compos. Part B Eng.
**2016**, 90, 141–151. [Google Scholar] [CrossRef] - Chen, T.; Feng, X.; Zhang, X.; Cao, W.; Changjian, F.U. Experimental study on mechanical and anisotropic properties of black shale. Chin. J. Rock Mech. Eng.
**2014**, 33, 1772–1779. [Google Scholar] - Sang, Y.; Yang, S.; Zhao, F.; Hou, B. Research on anisotropy and failure characteristics of southern marine shale rock. Drill. Prod. Technol.
**2015**, 38, 71–74. [Google Scholar] - Jia, C.G.; Chen, J.H.; Guo, Y.T.; Yang, C.H.; Xu, J.B.; Wang, L. Research on mechanical behaviors and failure modes of layer shale. Rock Soil Mech.
**2013**, 34, 57–61. [Google Scholar] - Yang, H.; Shen, R.; Fu, L. Composition and mechanical properties of gas shale. Pet. Drill. Tech.
**2013**, 31–35. [Google Scholar] - Saroglou, H.; Tsiambaos, G. A modified hoek–brown failure criterion for anisotropic intact rock. Int. J. Rock Mech. Min. Sci.
**2008**, 45, 223–234. [Google Scholar] [CrossRef] - Jaeger, J.C. Shear failure of anistropic rocks. Geol. Mag.
**1960**, 97, 65–72. [Google Scholar] [CrossRef] - Itasca Consulting Group Inc. PFC2D (Particle Flow Code in 2-Dimensions), version 4.0; Itasca Consulting Group Inc.: Minneapolis, MN, USA, 2008. [Google Scholar]
- Potyondy, D.O.; Cundall, P.A. Abonded-particle model for rock. Int. J. Rock Mech. Min. Sci.
**2004**, 41, 1329–1364. [Google Scholar] [CrossRef] - Potyondy, D.O. The bonded-particle model as a tool for rock mechanics research and application: Current trends and future directions. Geosyst. Eng.
**2015**, 18, 1–28. [Google Scholar] [CrossRef] - Zhang, X.-P.; Wong, L.N.Y. Cracking processes in rock-like material containing a single flaw under uniaxial compression: A numerical study based on parallel bonded-particle model approach. Rock Mech. Rock Eng.
**2012**, 45, 711–737. [Google Scholar] [CrossRef] - Zhang, X.-P.; Wong, L.N.Y. Crack initiation, propagation and coalescence in rock-like material containing two flaws: A numerical study based on bonded-particle model approach. Rock Mech. Rock Eng.
**2012**, 46, 1001–1021. [Google Scholar] [CrossRef] - Mas Ivars, D.; Pierce, M.E.; Darcel, C.; Reyes-Montes, J.; Potyondy, D.O.; Paul Young, R.; Cundall, P.A. The synthetic rock mass approach for jointed rock mass modelling. Int. J. Rock Mech. Min. Sci.
**2011**, 48, 219–244. [Google Scholar] [CrossRef] - Bahaaddini, M.; Hagan, P.C.; Mitra, R.; Hebblewhite, B.K. Parametric study of smooth joint parameters on the shear behaviour of rock joints. Rock Mech. Rock Eng.
**2014**, 48, 923–940. [Google Scholar] [CrossRef] - Bahrani, N.; Kaiser, P.K.; Valley, B. Distinct element method simulation of an analogue for a highly interlocked, non-persistently jointed rockmass. Int. J. Rock Mech. Min. Sci.
**2014**, 71, 117–130. [Google Scholar] [CrossRef] - Cheng, C.; Chen, X.; Zhang, S. Multi-peak deformation behavior of jointed rock mass under uniaxial compression: Insight from particle flow modeling. Eng. Geol.
**2016**, 213, 25–45. [Google Scholar] [CrossRef] - Peng, J.; Wong, L.N.Y.; Teh, C.I. Influence of grain size heterogeneity on strength and micro-cracking behavior of crystalline rocks. J. Geophys. Res. Solid Earth
**2017**, 122, 1054–1073. [Google Scholar] [CrossRef] - Han, Y.; Damjanac, B.; Nagel, N. A microscopic numerical system for modeling interaction between natural fractures and hydraulic fracturing. In Proceedings of the 46th US Rock Mechanics/Geomechanics Symposium, Chicago, IL, USA, 24–27 June 2012. [Google Scholar]
- Szwedzicki, T. A hypothesis on modes of failure of rock samples tested in uniaxial compression. Rock Mech. Rock Eng.
**2007**, 40, 97–104. [Google Scholar] [CrossRef] - Vervoort, A.; Min, K.-B.; Konietzky, H.; Cho, J.-W.; Debecker, B.; Dinh, Q.-D.; Frühwirt, T.; Tavallali, A. Failure of transversely isotropic rock under brazilian test conditions. Int. J. Rock Mech. Min. Sci.
**2014**, 70, 343–352. [Google Scholar] [CrossRef] - Jaeger, J.C.; Cook, N.G.W.; Zimmerman, R.W. Fundamentals of Rock Mechanics, 4th ed.; Blackwell Publishing: Hoboken, NJ, USA, 2007. [Google Scholar]

**Figure 1.**(

**a**) Anisotropic strength values of Greenriver Shale-2 samples under various confinements [16], and (

**b**) different changing trends of $S{A}_{1}$, $S{A}_{2}$ and some other anisotropic parameters with the increasing confinement.

**Figure 2.**Shale-5 samples with different oriented weak planes. (

**a**) β = 30°; (

**b**) β = 60°; and (

**c**) β = 90°.

**Figure 3.**Four types of anisotropic strength behaviors based on SA

_{1}with increasing confinement. (

**a**) Type I

_{1}; (

**b**) Type II

_{1}; (

**c**) Type III

_{1}; and (

**d**) Type IV

_{1}.

**Figure 4.**Three types of strength anisotropic behaviors based on SA

_{2}with increasing confinement. (

**a**) Type I

_{2}; (

**b**) Type II

_{2}; and (

**c**) Type III

_{2}.

**Figure 5.**Numerical models for the shale samples with different oriented weak planes. (

**a**) β = 0°; (

**b**) β = 30°; (

**c**) β = 60°; and (

**d**) β = 90°.

**Figure 6.**(

**a**) The peak strength values of Shale-1 samples at different loading directions under various confinements [25]; and (

**b**) the validated strength values of the numerical model.

**Figure 7.**Influence of cohesion of weak planes with the certain friction angle (

**a**) φ

_{w}= 10°; (

**b**) φ

_{w}= 30°; and (

**c**) φ

_{w}= 50° based on SA

_{1}(solid line) and SA

_{2}(dashed line).

**Figure 8.**Influence of friction angle of weak planes with the certain cohesion (

**a**) c

_{w}= 10 MPa; (

**b**) c

_{w}= 20 MPa; and (

**c**) c

_{w}= 40 MPa based on SA

_{1}(solid line) and SA

_{2}(dashed line).

**Figure 9.**Different failure patterns of Shale-5 samples with inclination angle $\beta $ = 30° under the confinements of (

**a**) σ

_{3}= 0 MPa; (

**b**) σ

_{3}= 60 MPa and $\beta $ = 90° under the confinements of (

**c**) σ

_{3}= 0 MPa; (

**d**) σ

_{3}= 60 MPa.

**Figure 10.**Different failure patterns of Shale-5 samples with inclination angle $\beta $ = 60° under the confinements of (

**a**) σ

_{3}= 0 MPa; (

**b**) σ

_{3}= 60 MPa and (

**c**) σ

_{3}= 100 MPa.

**Figure 11.**Different fracture characteristics of the samples with inclination angle $\beta $ = 60° under confinement of 30 MPa by numerical simulations. (

**a**) c

_{w}= 40 MPa, φ

_{w}= 10°; (

**b**) c

_{w}= 40 MPa, φ

_{w}= 30°; and (

**c**) c

_{w}= 40 MPa, φ

_{w}= 50°. Blue color shows the position of the weak planes; Red and magenta colors show the tensile and shear micro-cracks in the matrix; Cyan and green colors show the tensile and shear micro-cracks in the weak planes.

**Figure 12.**(

**a**) Peak strengths of Shale-5 sample ($\beta $ = 60°) under various confinements (σ

_{3}= 20, 40, 60, and 100 MPa); and (

**b**) normal and shear stresses on the weak planes based on the data in (

**a**) and the linearly fitted equation.

**Figure 13.**Four typical behaviors of maximum and minimum strengths with increasing confinement dominated by different combinations of cohesion ${c}_{\mathrm{w}}$ and friction angle ${\varphi}_{\mathrm{w}}$ of the weak planes based on numerical analyses. (

**a**) c

_{w}= 10 MPa, φ

_{w}= 10°; (

**b**) c

_{w}= 10 MPa, φ

_{w}= 50°; (

**c**) c

_{w}= 40 MPa, φ

_{w}= 10°; and (

**d**) c

_{w}= 40 MPa, φ

_{w}= 50°.

**Figure 14.**Four typical behaviors of maximum and minimum strengths with increasing confinement based on laboratory experimental results of (

**a**) Shale-1; (

**b**) Shale-4; (

**c**) Shale-5; and (

**d**) Shale-2.

Parameters | Descriptions | References |
---|---|---|

${k}_{1}=\text{}\frac{{\left({\sigma}_{1}-{\sigma}_{3}\right)}_{\parallel}}{{\left({\sigma}_{1}-{\sigma}_{3}\right)}_{\perp}}$ | Ratio between the failure stresses in the two principal directions parallel and perpendicular to the bedding planes, respectively | [7] |

${k}_{2}=\text{}\frac{{\left({\sigma}_{1}-{\sigma}_{3}\right)}_{\mathrm{max}}}{{\left({\sigma}_{1}-{\sigma}_{3}\right)}_{\mathrm{min}}}$ | Ratio of the maximum to minimum failure strengths | [7] |

${\mathsf{\sigma}}_{c}\left(\mathrm{max}\right)/{\mathsf{\sigma}}_{c}\left(\mathrm{min}\right)$ | Ratio of the maximum to minimum uniaxial compressive strength (UCS) | [11] |

$f=\text{}\frac{{\sigma}_{1,\mathrm{max}}-{\sigma}_{1,\mathrm{min}}}{{\sigma}_{1,\mathrm{max}}}$ | Ratio of the strength difference to the maximum strength | [15] |

${R}_{c}={\mathsf{\sigma}}_{ci\left(90\right)}/{\mathsf{\sigma}}_{ci\left(\mathrm{min}\right)}$ | Ratio between the UCS perpendicular to the beddings and the minimum UCS | [27] |

Samples | Description | Inclination Angle β (°) | Confinement (MPa) | Ref. |
---|---|---|---|---|

Greenriver shale-1 | Light brown to light gray; highly laminated, composed of fine grained calcite and dolomite particles inter-bedded with kerogen | 0, 15, 20, 30, 45, 60, 75, 90 | 7, 35, 70, 100, 170 | [16] |

Greenriver shale-2 | Much darker, with more oil; highly laminated, composed of fine grained calcite and dolomite particles inter-bedded with kerogen | 0, 10, 20, 30, 40, 60, 90 | 7, 35, 70, 100, 170 | [16] |

Outcrop shale-#8 | Gray to dark, with obvious plane of anisotropy shown in the photographs | 0, 15, 30, 45, 60, 75, 90 | 3, 21, 35, 48, 69 | [2] |

Top seal shale | - | 0, 15, 30, 45, 60, 75, 90 | 3, 7, 14, 21, 35 | [2] |

Shale-1 | Black shale from outcrop of Longmaxi Formation in China, with laminated structures from the SEM images | 0, 30, 60, 90 | 0, 10, 20, 30 | [25] |

Shale-2 | Cored black shale (3502.61~3508.63 m deep) of Longmaxi Formation in Sichuan, China, with planes of anisotropy | 0, 15, 30, 45, 60, 75, 90 | 0, 10, 20, 30, 40 | [24] |

Shale-3 | Black shale at the lower part of Longmaxi Formation in Guizhou, China, with laminated structures and micro-fissures from the SEM images | 0, 45, 90 | 0, 5, 10, 15, 20, 25 | [26] |

Shale-4 | Black shale of Niutitang Formation in China, showing obvious sedimentary rock feature from micrometer scale, with lamellar minerals | 0, 30, 45, 60, 90 | 0, 10, 20, 30, 40, 50 | [23] |

Shale-5 | Black shale from outcrop of Longmaxi Formation in Chongqing, China, with visible planes of anisotropy | 30, 60, 90 | 0, 20, 40, 60, 100 | This study |

Inclination Angle β (°) | Vp (m/s) | UCS (MPa) | E (GPa) |
---|---|---|---|

30 | 4370 | 191.3 | 29.8 |

60 | 4706 | 176.6 | 32.7 |

90 | 4964 | 200.2 | 34.5 |

Grain (Particles) | Cement (Parallel Bonds) | ||
---|---|---|---|

Ball density (kg/m^{3}) | 2700 | Bond modulus ${\overline{E}}_{\mathrm{c}}$ (GPa) | 21 |

Minimum ball radius (mm) | 0.36 | Normal bond strength (MPa) | 90 |

Ball radius ratio R_{max}/R_{min} | 1.66 | S.D. ^{1} normal bond strength (MPa) | 15 |

Contact modulus E_{c} (GPa) | 21 | Shearing bond strength (MPa) | 90 |

Coefficient of friction | 1.0 | S.D. ^{1} shearing bond strength (MPa) | 15 |

Normal to shearing stiffness ratio k_{n}/k_{s} | 2.5 | Normal to shearing bond stiffness ratio ${\overline{k}}_{\mathrm{n}}$/${\overline{k}}_{\mathrm{s}}$ | 2.5 |

^{1}S.D.: standard deviation.

Parameters | Values |
---|---|

Cohesion C_{sj} (MPa) | 20 |

Friction angle φ_{j} (°) | 50 |

Dilation angle ψ_{j} (°) | 0 |

Normal stiffness k_{n,sj} (GPa/m) | 1500 |

Shear stiffness k_{s,sj} (GPa/m) | 2500 |

Tensile strength σ_{n,sj} (MPa) | 5 |

SA_{1} | SA_{2} | ||||||
---|---|---|---|---|---|---|---|

${\varphi}_{\mathrm{w}}$ = 10° | ${\varphi}_{\mathrm{w}}$ = 30° | ${\varphi}_{\mathrm{w}}$ = 50° | ${\varphi}_{\mathrm{w}}$ = 10° | ${\varphi}_{\mathrm{w}}$ = 30° | ${\varphi}_{\mathrm{w}}$ = 50° | ||

${c}_{\mathrm{w}}$ = 10 MPa | I_{1} | I_{1} | I_{1} | ${c}_{\mathrm{w}}$ = 10 MPa | II_{2} | I_{2} | I_{2} |

${c}_{\mathrm{w}}$ = 20 MPa | III_{1} | II_{1} | II_{1} | ${c}_{\mathrm{w}}$ = 20 MPa | III_{2} | II_{2} | I_{2} |

${c}_{\mathrm{w}}$ = 40 MPa | IV_{1} | IV_{1} | II_{1} | ${c}_{\mathrm{w}}$ = 40 MPa | III_{2} | III_{2} | I_{2} |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Cheng, C.; Li, X.; Qian, H.
Anisotropic Failure Strength of Shale with Increasing Confinement: Behaviors, Factors and Mechanism. *Materials* **2017**, *10*, 1310.
https://doi.org/10.3390/ma10111310

**AMA Style**

Cheng C, Li X, Qian H.
Anisotropic Failure Strength of Shale with Increasing Confinement: Behaviors, Factors and Mechanism. *Materials*. 2017; 10(11):1310.
https://doi.org/10.3390/ma10111310

**Chicago/Turabian Style**

Cheng, Cheng, Xiao Li, and Haitao Qian.
2017. "Anisotropic Failure Strength of Shale with Increasing Confinement: Behaviors, Factors and Mechanism" *Materials* 10, no. 11: 1310.
https://doi.org/10.3390/ma10111310