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Deformability modulus of jointed rocks, limitation of empirical methods, and introducing a new analytical approach
- 1. DEFORMABILITY MODULUS OF JOINTED ROCKS, LIMITATION OF EMPIRICAL METHODS, AND INTRODUCING A NEW ANALYTICAL APPROACH Mahdi Zoorabadi
- 3. Introduction The commission of Terminology, symbols and graphic representation of the International Society for Rock Mechanics ISRM ) ISRM, 1975 ) Modulus of elasticity or Young’s modulus (E) : The ratio of stress to corresponding strain below the proportionality limit of a material. Modulus of deformation of a rock mass (Em) : The ratio of stress (p) to corresponding strain during loading of a rock mass, including elastic and inelastic behavior Modulus of elasticity of a rock mass (Eem) : The ratio of stress (p) to corresponding strain during loading of a rock mass, including only the elastic behavior
- 4. Introduction Deformability Modulus Direct Methods (In situ Tests) Indirect Methods Empirical Equations (Based on rock mass classification systems) Back Analysis
- 5. Direct Methods: Borehole Expansion Tests Interfels Dilatometer Cambridge institute Dilatometer Goodman Jack (http://www.slopeindicator.com/pdf/goodman%20jack%20datasheet.pdf) 𝐸 = 𝐶. ∆𝜎 ∆𝑑 𝐶 = 1 + 𝜗 . 𝐷0
- 6. Direct Methods: Plate Loading Tests 𝐸 = 𝐶. ∆𝜎 ∆𝑑 𝐶 = 𝑎 2 [2 1 − 𝜗2 𝑐𝑜𝑡−1 𝑧 + 1 + 𝜗 𝑧 1 + 𝑧 ]
- 7. Direct Methods: Extra Large Flat Jack 𝐸 = 𝐶. ∆𝜎 ∆𝑑 𝐶 = (1 − 𝜗2)𝑘𝑖
- 9. Rock Mass Classification (Serafim and Pereira, 1983) (Hoek and Diederichs,2006)
- 10. Rock Mass Classification Parameter study on Hoek and Brown (1997) and Hoek and Diederichs (2006) equations (Zoorabadi 2010)
- 11. Stress dependency of deformability modulus which was not considered in empirical equation An applied normal stress on a rock fracture causes the fracture to close and decreases the aperture. Deformability of rock mass containing discontinuities would have different values at different depth or stress fields Stress Dependency of Deformability Modulus
- 12. Stress Dependency of Deformability Modulus New Procedure (Li, 2001) (Ebadi et al., 2011) 𝑘 𝑛 = 𝑘 𝑛𝑖 1 − 𝜎 𝑛 𝑉𝑚 𝑘 𝑛𝑖 + 𝜎 𝑛 −2 (Bandis et al.,1983)
- 13. Stress Dependency of Deformability Modulus New Procedure 𝑘 𝑛𝑖 = −7.15 + 1.75𝐽𝑅𝐶 + 0.02( 𝐽𝐶𝑆 𝑎𝑗 𝑎𝑗 = 𝐽𝑅𝐶 5 (0.2 𝜎𝑐 𝐽𝐶𝑆 − 0.1 𝑉𝑚 = 𝐴 + 𝐵 𝐽𝑅𝐶 + 𝐶 𝐽𝐶𝑆 𝑎𝑗 𝐷 (Bandis et al.,1983) (Barton and Choubey 1977) (Milne et al. 1991) (Barton 1982) (Barton and De Quadros 1997)
- 14. Case Study Joint set Dip Dip/Dir Spacing [m] JRC JCS A 85 113 2.03 13 30 B 64 41 1.77 13 30 C 80 331 3.83 13 30 Bedding plane 24 156 4 10 30 Elastic modulus of 16 GPa
- 15. Case Study From Measurement: • Maximum stress orientation: NW • Ratio between maximum horizontal stress and minimum horizontal stress is 𝜎 𝐻 𝜎ℎ = 1.5 (Nemcik et al. 2005) (Zoorabadi et al. 2015)
- 16. Case Study - Results • Deformability modulus at the ground surface (zero acting normal stress was assumed) was calculated to be 7.2 Gpa (around 0.45% of elastic modulus of intact rock) . • Deformability modulus increases significantly with depth increase: 0.78% of the elastic modulus of intact rock at depth of 50 m. • For depths deeper that 200 m, deformability modulus of a this rock mass would be more that 90% of the elastic modulus of intact rock.
- 17. Case Study - Results (Snomez and Ulusay 1999). 𝐽𝑣 = 𝑗=1 𝑛 1 𝑆𝑗 Joint surface condition of Fair/Good GSI value for this case would be between 60-70 with average of 65. Deformability modulus of rock mass would be around 10 GPa using Hoek and Diederichs (2006) and 15 GPa by Hoek and Brown (1997).
- 18. Conclusions • Deformability modulus is a stress dependent parameter and increases as applied stress increases. • All well-known empirical formulations do not consider this property of deformability modulus. • A new procedure is proposed to quantify the stress dependency of deformability modulus. • For this case study it was found that for depths higher than 200 m it approaches to the elastic modulus of intact rock.