Closed-Form Solution to the Poromechanics of Deep Arbitrary-Shaped Openings Subjected to Rock Mass Alteration
Publication: International Journal of Geomechanics
Volume 20, Issue 12
Abstract
The aim of this paper is to propose a closed-form solution to the poromechanics problem of stress and pore pressure distribution around noncircular openings at great depth subjected to hydrostatic water pressure and far-field geostresses. The problem is solved by superposing the effects of fluid and solid skeleton obtained over simple circular geometries and generalizing the obtained expression to various geometries of mined ore bodies using complex variable functions and conformal mapping techniques. The principal stresses obtained analytically over the opening boundary and within the domain are compared with the results of finite-element analysis to verify the proposed approach. The comparison conducted for a representative noncircular opening indicates good agreement between the analytical and numerical methods. Hence, a parametric study is used to investigate in detail the stress variation under different opening dimensions, heterogeneous initial total stress conditions, and far-field pore pressure values. The proposed solution could be instrumental in the design of underground openings and deep mass alterations (that include local anthropogenic damage zones). It could also be used to provide reinforcement solutions where stresses can reach the mechanical stability thresholds.
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Acknowledgments
The first author would like to acknowledge financial support provided by the China Scholarship Council (CSC) under grant number 201606420056. The authors would like to thank the anonymous reviewers for their considerable effort in improving the paper.
Appendix. Detailed Derivation of the Coefficients an and bn
Substituting Eqs. (9), (12), and (13) into (7), the equation is derived aswhere: α = an arbitrary point on the unit circle boundary in the ξ plane. It can be expressed as α = eiθ, thus .
(28)
Using Cauchy’s integral operator and integrating Eq. (28), the following equation is obtained:where the function f0(α) can be expressed as
(29)
(30)
The ψ1(α) in Eq. (29) is the boundary value of ψ1(ξ), which is an analytic function exterior to the unit circle. Therefore, the first integral term on the left hand side of Eq. (29) by using Cauchy’s integral is obtained as:The complex constant is omitted here and in what follows, since it does not contribute to the stresses. The right hand side (RHS) of Eq. (29) can be expressed as:
(31)
(32)
Expanding the first term on the right-hand side of Eq. (32) yields
(33)
Note that the first integrand (c−1α + c0) is a boundary value of an analytic function (c−1ξ + c0) interior to the unit circle, whereas the other terms are analytical functions exterior to the unit circle. The denominator (α − ξ) is also an analytical function interior to the unit circle, given that |ξ| > 1. Therefore, the integral of the first term is zero and the rest can be obtained using the Cauchy-Goursat theory:
(34)
Similarly, the second term on the right-hand side of Eq. (32) gives
(35)
Combining Eqs. (34) and (35), the right-hand side of Eq. (29) can be expressed asThe analytic function of ψ1(ξ) is determined byBy conjugating Eq. (28) and following a similar procedure, the analytic function of χ1(ξ) can be rewritten as
(36)
(37)
(38)
The second term in Eq. (37) can be expressed asandAs shown previously, the ratio has (n) order pole at the origin. Therefore, it can be written in the following form:
(39)
(40)
(41)
The product of and has (n − 2) order pole at the origin and it takes the form
(42)
Hence, the Cauchy integral is solved
(43)
The coefficients dn and dj′ are known for a given conformal mapping function whereas kn and are unknown. Combining Eqs. (39), (41), and (42), the coefficient kn is expressed as
(44)
Another set of equations is then required to obtain these coefficients. Substituting Eq. (43) into (37) yieldstherefore, kn can also be expressed aswhere cn, B, and B′ are known. Combining Eqs. (44) and (46) the coefficient an is obtained. Therefore, the analytic function ψ1(ξ) is derived and the series number (n) is related to the conformal mapping terms.
(45)
(46)
Using the same procedure, the second term in Eq. (38) is written as
(47)
Therefore, the Cauchy integer of Eq. (48) reads
(50)
References
Ahmed, S., T. M. Müller, M. Madadi, and V. Calo. 2019. “Drained pore modulus and biot coefficient from pore-scale digital rock simulations.” Int. J. Rock Mech. Min. Sci. 114: 62–70. https://doi.org/10.1016/j.ijrmms.2018.12.019.
Biot, M., and D. Willis. 1957. “The elastic coeff cients of the theory of consolidation.” J. Appl. Mech. 24: 594–601.
Bobet, A. 2003. “Effect of pore water pressure on tunnel support during static and seismic loading.” Tunnelling Underground Space Technol. 18 (4): 377–393. https://doi.org/10.1016/S0886-7798(03)00008-7.
Cai, M. 2002. Rock mechanics and engineering. [In Chinese.] Beijing: Science Press.
Chauhan, M. M., and D. S. Sharma. 2015. “Stresses in finite anisotropic plate weakened by rectangular hole.” Int. J. Mech. Sci. 101: 272–279. https://doi.org/10.1016/j.ijmecsci.2015.08.007.
Dienstmann, G., F. de Almeida, A. Fayolle, F. Schnaid, and S. Maghous. 2018. “A simplified approach to transient flow effects induced by rigid cylinder rotation in a porous medium.” Comput. Geotech. 97: 134–154. https://doi.org/10.1016/j.compgeo.2017.11.014.
Do, D.-P., N.-H. Tran, D. Hoxha, and H.-L. Dang. 2017. “Assessment of the influence of hydraulic and mechanical anisotropy on the fracture initiation pressure in permeable rocks using a complex potential approach.” Int. J. Rock Mech. Min. Sci. 100: 108–123. https://doi.org/10.1016/j.ijrmms.2017.10.020.
Dong, X., A. Karrech, H. Basarir, and M. Elchalakani. 2018a. “Extended finite element modelling of fracture propagation during in-situ rock mass alteration.” In Proc., 52nd US Rock Mechanics/Geomechanics Symp., 1–8. Alexandria, VA: American Rock Mechanics Association.
Dong, X., A. Karrech, H. Basarir, M. Elchalakani, and C. Qi. 2018b. “Analytical solution of energy redistribution in rectangular openings upon in-situ rock mass alteration.” Int. J. Rock Mech. Min. Sci. 106: 74–83. https://doi.org/10.1016/j.ijrmms.2018.04.014.
Dong, X., A. Karrech, H. Basarir, M. Elchalakani, and A. Seibi. 2019a. “Energy dissipation and storage in underground mining operations.” Rock Mech. Rock Eng. 52 (1): 229–245. https://doi.org/10.1007/s00603-018-1534-x.
Dong, X., A. Karrech, C. Qi, M. Elchalakani, and H. Basarir. 2019b. “Analytical solution for stress distribution around deep lined pressure tunnels under the water table.” Int. J. Rock Mech. Min. Sci. 123: 104124. https://doi.org/10.1016/j.ijrmms.2019.104124.
Exadaktylos, G., and M. Stavropoulou. 2002. “A closed-form elastic solution for stresses and displacements around tunnels.” Int. J. Rock Mech. Min. Sci. 39 (7): 905–916. https://doi.org/10.1016/S1365-1609(02)00079-5.
Fang, Q., H. Song, and D. Zhang. 2015. “Complex variable analysis for stress distribution of an underwater tunnel in an elastic half plane.” Int. J. Numer. Anal. Methods Geomech. 39 (16): 1821–1835. https://doi.org/10.1002/nag.2375.
Fraldi, M., and F. Guarracino. 2010. “Analytical solutions for collapse mechanisms in tunnels with arbitrary cross sections.” Int. J. Solids Struct. 47 (2): 216–223. https://doi.org/10.1016/j.ijsolstr.2009.09.028.
Harr, M. E. 1962. Groundwater and seepage. New York: McGraw-Hill.
Hassani, A. N., H. Farhadian, and H. Katibeh. 2018. “A comparative study on evaluation of steady-state groundwater inflow into a circular shallow tunnel.” Tunnelling Underground Space Technol. 73: 15–25. https://doi.org/10.1016/j.tust.2017.11.019.
Huo, H., A. Bobet, G. Fernández, and J. Ramírez. 2006. “Analytical solution for deep rectangular structures subjected to far-field shear stresses.” Tunnelling Underground Space Technol. 21 (6): 613–625. https://doi.org/10.1016/j.tust.2005.12.135.
Jaeger, J. C., N. G. Cook, and R. Zimmerman. 2009. Fundamentals of rock mechanics. Hoboken, NJ: John Wiley & Sons.
Jafari, M., and E. Ardalani. 2016. “Stress concentration in finite metallic plates with regular holes.” Int. J. Mech. Sci. 106: 220–230. https://doi.org/10.1016/j.ijmecsci.2015.12.022.
Kargar, A., R. Rahmannejad, and M. Hajabasi. 2014. “A semi-analytical elastic solution for stress field of lined non-circular tunnels at great depth using complex variable method.” Int. J. Solids Struct. 51 (6): 1475–1482. https://doi.org/10.1016/j.ijsolstr.2013.12.038.
Karrech, A. 2013. “Non-equilibrium thermodynamics for fully coupled thermal hydraulic mechanical chemical processes.” J. Mech. Phys. Solids 61 (3): 819–837. https://doi.org/10.1016/j.jmps.2012.10.015.
Karrech, A., M. Attar, E. Oraby, J. Eksteen, M. Elchalakani, and A. Seibi. 2018. “Modelling of multicomponent reactive transport in finite columns—application to gold recovery using iodide ligands.” Hydrometallurgy 178: 43–53. https://doi.org/10.1016/j.hydromet.2018.03.020.
Karrech, A., O. Beltaief, R. Vincec, T. Poulet, and K. Regenauer-Lieb. 2015a. “Coupling of thermal-hydraulic-mechanical processes for geothermal reservoir modelling.” J. Earth Sci. 26 (1): 47–52. https://doi.org/10.1007/s12583-015-0518-y.
Karrech, A., C. Schrank, R. Freij-Ayoub, and K. Regenauer-Lieb. 2014. “A multi-scaling approach to predict hydraulic damage of poromaterials.” Int. J. Mech. Sci. 78: 1–7. https://doi.org/10.1016/j.ijmecsci.2013.10.010.
Karrech, A., C. Schrank, and K. Regenauer-Lieb. 2015b. “A parallel computing tool for large-scale simulation of massive fluid injection in thermo-poro-mechanical systems.” Philos. Mag. 95 (28–30): 3078–3102. https://doi.org/10.1080/14786435.2015.1067373.
Lekhnitskii, S. G. 1968. Anisotropic plates. New York: Gordon and Breach Science Publishers.
Lu, A., Z. Xu, and N. Zhang. 2017. “Stress analytical solution for an infinite plane containing two holes.” Int. J. Mech. Sci. 128: 224–234. https://doi.org/10.1016/j.ijmecsci.2017.04.025.
Lu, A., N. Zhang, and L. Kuang. 2014. “Analytic solutions of stress and displacement for a non-circular tunnel at great depth including support delay.” Int. J. Rock Mech. Min. Sci. 70: 69–81. https://doi.org/10.1016/j.ijrmms.2014.04.008.
Mahjoub, M., and A. Rouabhi. 2019. “A hydromechanical approach for anisotropic elasto-viscoplastic geomaterials: Application to underground excavations in sedimentary rocks.” Underground Space 4 (2): 109–120. https://doi.org/10.1016/j.undsp.2018.06.001.
Manh, H. T., J. Sulem, and D. Subrin. 2015. “A closed-form solution for tunnels with arbitrary cross section excavated in elastic anisotropic ground.” Rock Mech. Rock Eng. 48 (1): 277–288. https://doi.org/10.1007/s00603-013-0542-0.
Ming, H., M.-S. Wang, Z.-S. Tan, and X.-Y. Wang. 2010. “Analytical solutions for steady seepage into an underwater circular tunnel.” Tunnelling Underground Space Technol. 25 (4): 391–396. https://doi.org/10.1016/j.tust.2010.02.002.
Muskhelishvili, N. I. 1954. Some basic problems of the mathematical theory of elasticity. Berlin: Springer Science & Business Media.
Nieć, M., E. Sermet, J. Chećko, and J. Górecki. 2017. “Evaluation of coal resources for underground gasification in Poland. Selection of possible UCG sites.” Fuel 208: 193–202. https://doi.org/10.1016/j.fuel.2017.06.087.
Pan, Z., Y. Cheng, and J. Liu. 2013. “Stress analysis of a finite plate with a rectangular hole subjected to uniaxial tension using modified stress functions.” Int. J. Mech. Sci. 75: 265–277. https://doi.org/10.1016/j.ijmecsci.2013.06.014.
Potts, D. M., L. Zdravković, T. I. Addenbrooke, K. G. Higgins, and N. Kovačević. 2001. Vol. 2 of Finite element analysis in geotechnical engineering: Application. London: Thomas Telford.
Savin, G. N. 1961. Stress concentration around holes. New York: Pergamon Press.
Sharma, D. S. 2012. “Stress distribution around polygonal holes.” Int. J. Mech. Sci. 65 (1): 115–124. https://doi.org/10.1016/j.ijmecsci.2012.09.009.
Sharma, D. S. 2016. “Stresses around hypotrochoidal hole in infinite isotropic plate.” Int. J. Mech. Sci. 105: 32–40. https://doi.org/10.1016/j.ijmecsci.2015.10.018.
Shi, G.-P., J.-H. Zhu, B.-H. Li, and J.-H. Yang. 2014. “Elastic analysis of hole-edge stress of rectangular roadway.” Rock Soil Mech. 35 (9): 2587–2593.
Sokolnikoff, I. S. 1956. Mathematical theory of elasticity. New York: McGraw-Hill.
Terzaghi, K., and R. B. Peck. 1948. Soil mechanics in engineering practice. New York: John Wiley & Sons.
Tran, N.-H., D.-P. Do, and D. Hoxha. 2017. “A closed-form hydro-mechanical solution for deep tunnels in elastic anisotropic rock.” Eur. J. Environ. Civ. Eng. 22: 1–17. https://doi.org/10.1080/19648189.2017.1285253.
Wang, H., X. Chen, M. Jiang, F. Song, and L. Wu. 2018. “The analytical predictions on displacement and stress around shallow tunnels subjected to surcharge loadings.” Tunnelling Underground Space Technol. 71: 403–427. https://doi.org/10.1016/j.tust.2017.09.015.
Yang, Z., C.-B. Kim, H. G. Beom, and C. Cho. 2010. “The stress and strain concentrations of out-of-plane bending plate containing a circular hole.” Int. J. Mech. Sci. 52 (6): 836–846. https://doi.org/10.1016/j.ijmecsci.2010.02.001.
Zhang, Z., and Y. Sun. 2011. “Analytical solution for a deep tunnel with arbitrary cross section in a transversely isotropic rock mass.” Int. J. Rock Mech. Min. Sci. 48 (8): 1359–1363. https://doi.org/10.1016/j.ijrmms.2011.10.001.
Zhao, G., and S. Yang. 2015. “Analytical solutions for rock stress around square tunnels using complex variable theory.” Int. J. Rock Mech. Min. Sci. 80: 302–307. https://doi.org/10.1016/j.ijrmms.2015.09.018.
Zienkiewicz, O. C., R. L. Taylor, P. Nithiarasu, and J. Zhu. 1977. Vol. 3 of The finite element method. London: McGraw-Hill.
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Received: Nov 22, 2019
Accepted: Jul 31, 2020
Published online: Sep 21, 2020
Published in print: Dec 1, 2020
Discussion open until: Feb 21, 2021
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