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Recent advances in mechanics of fracking and new results on 2D simulation of crack branching in anisotropic gas or oil shale

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Abstract

This article presents a comprehensive overview of several recent theoretical results at Northwestern University and demonstrates them by new numerical simulations of branching of hydraulic fractures. To model the inelastic behavior and fracturing of shale as an inherently anisotropic material, the recently developed spherocylindrical microplane model is described. Regarding the spread and branching of hydraulic cracks during the fracking process, it is emphasized that two kinds of water flow must be simulated: (1) the Poiseuille flow through the hydraulic fractures and natural cracks and (2) the Darcy diffusion flow of leak-off water through the pores of intact shale. The body forces due to gradient of Darcy flow pressure must be taken into account. The crack opening width is computed by means of the crack band model, in which each finite element is imagined to contain at the outset a potential cohesive crack, one in each of three spatial orientations, with the fracking water flowing through if the crack gets opened. The use of this model to suppress problems of mesh sensitivity due to localization of distributed fracturing is explained. Computer simulations of the growth of branched hydraulic system are preformed in two dimensions (2D) only. The results illustrate the effects of anisotropy and natural cracks on the evolution of 2D fracture patterns during the fracking process. These effects are not large, but much stronger effects are expected in future three-dimensional simulations.

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References

  1. Beckwith, R.: Hydraulic fracturing: the fuss, the facts, the future. J. Pet. Technol. 62(12), 34–40 (2010)

    Article  Google Scholar 

  2. Bažant, Z.P., Salviato, M., Chau, V.T., Viswanathan, H., Zubelewicz, A.: Why fracking works. J. Appl. Mech.-Trans. ASME. 81(10), 101010-1–101010-10 (2014)

    Google Scholar 

  3. Xie, H.P., Gao, F., Ju, Y., Xie, L.Z., Yang, Y.M., Wang, J.: Novel idea of the theory and application of 3D volume fracturing for stimulation of shale gas reservoirs. Chin. Sci. Bull. 61(1), 36–46 (2016)

    Google Scholar 

  4. Mokhtari, M., Bui, B.T., Tutuncu, A.N.: Tensile failure of shales: impacts of layering and natural fractures. In: SPE Western North American and Rocky Mountain Joint Meeting, Society of Petroleum Engineers (2014)

  5. Niandou, H., Shao, J.F., Henry, J.P., Fourmaintraux, D.: Laboratory investigation of the mechanical behaviour of Tournemire shale. Int. J. Rock Mech. Min. Sci. 34(1), 3–16 (1997)

    Article  Google Scholar 

  6. Masri, M., Sibai, M., Shao, J.F., Mainguy, M.: Experimental investigation of the effect of temperature on the mechanical behavior of tournemire shale. Int. J. Rock Mech. Min. Sci. 70, 185–191 (2014)

    Google Scholar 

  7. Heng, S., Guo, Y.T., Yang, C.H., Daemen, J.J.K., Li, Z.: Experimental and theoretical study of the anisotropic properties of shale. Int. J. Rock Mech. Min. Sci. 74, 58–68 (2015)

    Google Scholar 

  8. Sone, H.: Mechanical properties of shale gas reservoir rocks, and its relation to the in-situ stress variation observed in shale gas reservoirs. Ph.D. thesis, Stanford University (2012)

  9. Rassouli, F., Zoback, M.: A comparison of short-term and long-term creep experiments in unconventional reservoir formations. In: 50th US Rock Mechanics/Geomechanics Symposium, American Rock Mechanics Association (2016)

  10. Tsai, S.W., Wu, E.M.: A general theory of strength for anisotropic materials. J. Compos. Mater. 5(1), 58–80 (1971)

    Article  Google Scholar 

  11. Hill, R.: The Mathematical Theory of Plasticity, vol. 11. Oxford University Press, Oxford (1998)

    MATH  Google Scholar 

  12. Lee, Y.K., Pietruszczak, S.: Tensile failure criterion for transversely isotropic rocks. Int. J. Rock Mech. Min. Sci. 79, 205–215 (2015)

    Google Scholar 

  13. Rao, K.S., Rao, G.V., Ramamurthy, T.: A strength criterion for anisotropic rocks. Ind. Geotech. J. 16(4), 317–333 (1986)

    Google Scholar 

  14. Jaeger, J.C.: Shear failure of anistropic rocks. Geol. Mag. 97(1), 65–72 (1960)

    Article  Google Scholar 

  15. Hoek, E.: Strength of jointed rock masses. Geotechnique 33(3), 187–223 (1983)

    Article  Google Scholar 

  16. Duveau, G., Shao, J.F., Henry, J.: Assessment of some failure criteria for strongly anisotropic geomaterials. Mech. Cohesive-Frict. Mater. 3(1), 1–26 (1998)

    Article  Google Scholar 

  17. Jin, W.C., Xu, H., Arson, C., Busetti, S.: Computational model coupling mode ii discrete fracture propagation with continuum damage zone evolution. Int. J. Numer. Anal. Methods Geomech. 41(2), 223–250 (2016)

    Article  Google Scholar 

  18. Levasseur, S., Welemane, H., Kondo, D.: A microcracks-induced damage model for initially anisotropic rocks accounting for microcracks closure. Int. J. Rock Mech. Min. Sci. 77, 122–132 (2015)

    Google Scholar 

  19. Jin, W.C., Arson, C.: Discrete equivalent wing crack based damage model for brittle solids. Int. J. Solids Struct. 110, 279–293 (2017)

    Article  Google Scholar 

  20. Ravi-Chandar, K., Knauss, W.: An experimental investigation into dynamic fracture, III. On steady-state crack propagation and crack branching. Int. J. Fract. 26(2), 141–154 (1984)

    Article  Google Scholar 

  21. Garagash, D., Detournay, E.: The tip region of a fluid-driven fracture in an elastic medium. J. Appl. Mech.-Trans. ASME 67(1), 183–192 (2000)

    Article  MATH  Google Scholar 

  22. Bunger, A.P., Detournay, E., Garagash, D.I.: Toughness-dominated hydraulic fracture with leak-off. Int. J. Fract. 134(2), 175–190 (2005)

    Article  MATH  Google Scholar 

  23. Detournay, E.: Propagation regimes of fluid-driven fractures in impermeable rocks. Int. J. Geomech. 4(1), 35–45 (2004)

    Article  Google Scholar 

  24. Lecampion, B., Desroches, J.: Simultaneous initiation and growth of multiple radial hydraulic fractures from a horizontal wellbore. J. Mech. Phys. Solids. 82, 235–258 (2015)

    Article  MathSciNet  Google Scholar 

  25. Wang, H.: Poro-elasto-plastic modeling of complex hydraulic fracture propagation, simultaneous multi-fracturing and producing well interference. Acta Mech. 227(2), 507–525 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  26. Waters, G.A., Lewis, R.E., Bentley, D.: The effect of mechanical properties anisotropy in the generation of hydraulic fractures in organic shales. In: SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers (2011)

  27. Nagel, N.B., Sanchez-Nagel, M.A., Zhang, F., Garcia, X., Lee, B.: Coupled numerical evaluations of the geomechanical interactions between a hydraulic fracture stimulation and a natural fracture system in shale formations. Rock Mech. Rock Eng. 46(3), 581–609 (2013)

    Article  Google Scholar 

  28. Chau, V.T., Bažant, Z.P., Su, Y.W.: Growth model for large branched three-dimensional hydraulic crack system in gas or oil shale. Phil. Trans. R. Soc. A. 374(2078), 20150418 (2016)

    Article  Google Scholar 

  29. Chau, V.T., Li, C.B., Rahimi-Aghdam, S., Bažant, Z.P.: The enigma of large-scale permeability of gas shale: pre-existing or frac-induced? J. Appl. Mech.-Trans. ASME. 84(6), 061008-1–061008-11 (2017)

    Article  Google Scholar 

  30. Li, C.B., Caner, F.C., Chau, V.T., Bažant, Z.P.: Spherocylindrical microplane constitutive model for shale and other anisotropic rocks. J. Mech. Phys. Solids. 103, 155–178 (2017)

    Article  MathSciNet  Google Scholar 

  31. Bažant, Z.P.: Comment on orthotropic models for concrete and geomaterials. J. Eng. Mech. 109(3), 849–865 (1983)

    Article  Google Scholar 

  32. Brocca, M., Bažant, Z.P., Daniel, I.M.: Microplane model for stiff foams and finite element analysis of sandwich failure by core indentation. Int. J. Solids Struct. 38, 8111–8132 (2001)

    Article  MATH  Google Scholar 

  33. Cusatis, G., Beghini, H., Bažant, Z.P.: Spectral stiffness microplane model for quasi-brittle composite laminates. I. Theory. J. Appl. Mech.-Trans. ASME 75(1), 021009-1–021009-9 (2008)

    Google Scholar 

  34. Šmilauer, V., Hoover, C.G., Bažant, Z.P., Caner, F.C., Waas, K.W., Shahwan, K.: Multiscale simulation of fracture of braided composites via repetitive unit cell. Eng. Fract. Mech. 78(6), 901–918 (2011)

    Article  Google Scholar 

  35. Kirane, K., Salviato, M., Bažant, Z.P.: Microplane-triad model for elastic and fracturing behavior of woven composites. J. Appl. Mech.-Trans. ASME 83(4), 041006-1–041006-14 (2016)

    Article  Google Scholar 

  36. Bažant, Z.P., Oh, B.H.: Efficient numerical integration on the surface of a sphere. ZAMM-J. Appl. Math. Mech. 66(1), 37–49 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  37. Caner, F.C., Bažant, Z.P.: Microplane model M7 for plain concrete. I. Formulation. J. Eng. Mech. 139(12), 1714–1723 (2013)

    Article  Google Scholar 

  38. Bažant, Z.P., Xiang, Y.Y., Prat, P.C.: Microplane model for concrete. I: stress–strain boundaries and finite strain. J. Eng. Mech. 122(3), 245–254 (1996)

    Article  Google Scholar 

  39. Freeman, C., Moridis, G., Blasingame, T.: A numerical study of microscale flow behavior in tight gas and shale gas reservoir systems. Transp. Porous Media 90(1), 253–268 (2011)

    Article  Google Scholar 

  40. Ilgen, A.G., Heath, J.E., Akkutlu, I.Y., Bryndzia, L.T., Cole, D.R., Kharaka, Y.K., Kneafsey, T.J., Milliken, K.L., Pyrak-Nolte, L.J., Suarez-Rivera, R.: Shales at all scales: exploring coupled processes in mudrocks. Earth-Sci. Rev. 166, 132–151 (2017)

    Article  Google Scholar 

  41. Bažant, Z.P.: Crack band model for fracture of geomaterials. In: Eisenstein, Z. (eds.) Proceedings of 4th International Conference on Numerical Methods on Geomechanics, vol. 3. University of Alberta, Edmonton, pp 1137–1152 (1982)

  42. Bažant, Z.P., Oh, B.H.: Crack band theory for fracture of concrete. Mater. Struct. 16(3), 155–177 (1983)

    Google Scholar 

  43. Červenka, J., Bažant, Z.P., Wierer, M.: Equivalent localization element for crack band approach to mesh-sensitivity in microplane model. Int. J. Numer. Methods Eng. 62(5), 700–726 (2005)

    Article  MATH  Google Scholar 

  44. Bažant, Z.P., Planas, J.: Fracture and Size Effect in Concrete and Other Quasibrittle Materials. CRC Press, Boca Raton and London (1998)

    Google Scholar 

  45. Hull, K.L., Abousleiman, Y.N., Han, Y., Al-Muntasheri, G.A., Hosemann, P., Parker, S.S., Howard, C.B.: Nanomechanical characterization of the tensile modulus of rupture for kerogen-rich shale. SPE Journal. SPE-177628-PA (2017)

  46. Sneddon, I.N.: The distribution of stress in the neighborhood of a crack in an elastic solid. Proc. R. Soc. A: Math. Phys. Eng. Sci. 187(1009), 229–260 (1946)

    Article  Google Scholar 

  47. Dohmen, T., Zhang, J., Blangy, J.P.: Measurement and analysis of 3D stress shadowing related to the spacing of hydraulic fracturing in unconventional reservoirs. SPE Annual Technical Conference and Exhibition, Society of Petroleum Engineers (2014)

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Acknowledgements

Partial financial support from the US Department of Energy through Subcontract No. 37008 of Northwestern University with Los Alamos National Laboratory is gratefully acknowledged. The simulation of fracturing damage also received some support from ARO Grant W911NF-15-101240 to Northwestern University. The first author wishes to thank the Department of Science and Technology of Sichuan Province (Nos. 2015JY0280, 2012FZ0124) and CSC for supporting him as a Research Fellow at Northwestern University. Collaboration with Sichuan University which led to important understanding of Longmaxi shale is deeply appreciated.

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Authors and Affiliations

  1. Northwestern University, Evanston, IL, USA

    Cunbao Li & Viet T. Chau

  2. Key Laboratory of Energy Engineering Safety and Disaster Mechanics Ministry of Education, Sichuan University, Chengdu, China

    Cunbao Li & Heping Xie

  3. College of Architecture and Environment, Sichuan University, Chengdu, China

    Cunbao Li & Heping Xie

  4. McCormick Institute Professor and W.P. Murphy Professor of Civil and Mechanical Engineering and Materials Science, Northwestern University, 2145 Sheridan Road, CEE/A135, Evanston, IL, 60208, USA

    Zdeněk P. Bažant

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  1. Cunbao Li
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  2. Viet T. Chau
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  4. Zdeněk P. Bažant
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Correspondence to Zdeněk P. Bažant.

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This paper is dedicated to the memory of Franz Ziegler

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Li, C., Chau, V.T., Xie, H. et al. Recent advances in mechanics of fracking and new results on 2D simulation of crack branching in anisotropic gas or oil shale. Acta Mech 229, 975–992 (2018). https://doi.org/10.1007/s00707-017-2010-5

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  • DOI: https://doi.org/10.1007/s00707-017-2010-5

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