A First Order Quantification of Effects of Uncertainties in Hydro-fracturing Parameters on Tunnel Ovalization Estimates

  • Shashank PathakEmail author
  • Gondu Venkat Ramana
Original Paper


In-situ stresses are always present in rock-masses due to gravitational and tectonic forces. Excavation of a tunnel in such a pre-stressed media causes deformation of tunnel cross-section. Ovalization of tunnels due to in-situ stresses in rock-mass is an important design parameter. For estimation of tunnel ovalization, state of in-situ stresses needs to be determined first. In-situ stresses determined through hydro-fracturing technique (HFT) are dependent upon the three HFT parameters: (a) shut-in pressure, (b) re-opening pressure, and (c) fracture orientation. A critical review of previous studies indicates that HFT parameters are subjected to uncertainties due to (1) limitations of testing procedures and equipment, (2) assumptions and subjective engineering judgment associated with interpretation of test results, and (3) inherent variability of geological formations. Therefore, tunnel deformation estimates based on in-situ stresses determined through HFT would also be affected by these uncertainties. In this paper, a framework based on the first-order second moment method is developed to evaluate the effects of uncertainties in hydro-fracturing test data on tunnel deformation. The analysis indicates that uncertainty in tunnel deformation depends upon the uncertainty levels as well as magnitude of the three HFT parameters along with Poisson’s ratio, height of overburden, and the angular location of the point on the tunnel periphery where deformation is being estimated. It is also found that among the three parameters, the shut-in pressure has the maximum relative contribution in the resulting uncertainties in tunnel ovalization with an average of 68% (± 8% standard deviation). The proposed methodology is explained through an example case-study from Bukit Timah Granite rock-mass of Singapore. It is found that for a range of coefficient of variation of shut-in and re-opening pressure from 0 to 50%, the maximum coefficient of variation of tunnel deformation varies between 63 and 332%. In view of such high uncertainties, it is recommended that uncertainties of HFT parameters must be taken into account in the design procedure to avoid unsound engineering judgments.


In-situ stress Rock-mass Hydro-fracturing test Tunnel deformation Uncertainties 



  1. Amadei B, Stephansson O (1997) Methods of in situ stress measurement. In: Rock stress and its measurement. Springer, DordrechtGoogle Scholar
  2. Ang AHS, Tang WH (1975) Probability concepts in engineering planning and design, vol 1. Wiley, HobokenGoogle Scholar
  3. Baecher GB, Christian JT (2005) Reliability and statistics in geotechnical engineering. Wiley, HobokenGoogle Scholar
  4. Baumgärtner J, Zoback MD (1989) Interpretation of hydraulic fracturing pressure-time records using interactive analysis methods. Int J Rock Mech Min Sci Geomech Abstr 26(6):461–469CrossRefGoogle Scholar
  5. Benjamin J, Cornell CA (1970) Probability, statistics, and decision for civil engineers. McGraw-Hill, New YorkGoogle Scholar
  6. Brady BHG, Brown ET (2004) Rock mechanics for underground mining. Springer, BerlinGoogle Scholar
  7. Bredehoeft JD, Wolff RG, Keys WS, Shuter E (1976) Hydraulic fracturing to determine the regional in situ stress field, Piceance Basin Colorado. Geol Soc Am Bull 87(2):250–258CrossRefGoogle Scholar
  8. Brown ET, Hoek E (1978) Trends in relationships between measured in situ stresses and depth. Int J Rock Mech Min Sci Geomech Abstr 15(4):211–215CrossRefGoogle Scholar
  9. Cai M (2011) Rock mass characterization and rock property variability considerations for tunnel and cavern design. Rock Mech Rock Eng 44(4):379–399CrossRefGoogle Scholar
  10. Choi SO (2012) Interpretation of shut-in pressure in hydrofracturing pressure-time records using numerical modeling. Int J Rock Mech Min Sci 50:29–37CrossRefGoogle Scholar
  11. Doe T, Hustrulid W, Leijon B, Ingevald K, Strindell L (1983) Determination of the state stress at the stripa mine, Sweden. In: Zoback M, Haimson B (eds) Hydraulic fracturing stress measurements. National Academy Press, Washington, DC, USA, pp 119–129Google Scholar
  12. Fairhurst C (2003) Stress estimation in rock: a brief history and review. Int J Rock Mech Min Sci 40(7–8):957–973CrossRefGoogle Scholar
  13. Gao K, Harrison JP (2018) Multivariate distribution model for stress variability characterisation. Int J Rock Mech Min Sci 102:144–154CrossRefGoogle Scholar
  14. Genter A, Castaing C, Dezayes C, Tenzer H, Traineau H, Villemin T (1997) Comparative analysis of direct (core) and indirect (borehole imaging tools) collection of fracture data in the Hot Dry Rock Soultz reservoir (France). J Geophys Res Solid Earth 102(B7):15419–15431CrossRefGoogle Scholar
  15. Gronseth JM, Kry PR (1983) Instantaneous shut-in pressure and its relationship to the minimum in situ stress. Hydraul Fract Stress Meas 139:142Google Scholar
  16. Haimson BC (1993) The hydraulic fracturing method of stress measurement: theory and practice. Compr Rock Eng 3:395–412Google Scholar
  17. Haimson BC, Cornet FH (2003) ISRM suggested methods for rock stress estimation—part 3: hydraulic fracturing (HF) and/or hydraulic testing of pre-existing fractures (HTPF). Int J Rock Mech Min Sci 40(7):1011–1020CrossRefGoogle Scholar
  18. Hardy MP, Asgian MI (1989) Fracture reopening during hydraulic fracturing stress determinations. Int J Rock Mech Min Sci Geomech Abstr 26(6):489–497CrossRefGoogle Scholar
  19. Hast N (1958) Measurement of rock pressure in mines: sveriges geol. Undersokning Ser. C Arsbok 52(3):1–183Google Scholar
  20. ISRM (2007) International society for rock mechanics (ISRM) suggested methods for rock characterization, testing and monitoring, 1974–2006: part 3: hydraulic fracturing (HF) and/or hydraulic testing of pre-existing fractures (HTPF). In: Ulusay R, Hudson JA (eds) Compilation arranged by the ISRM Turkish National Group, Ankara, Turkey, pp 397–408Google Scholar
  21. Ito T, Sato A, Hayashi K (1997) Two methods for hydraulic fracturing stress measurements needless the ambiguous reopening pressure. Int J Rock Mech Min Sci 34(3–4):143-e1Google Scholar
  22. Ito T, Evans K, Kawai K, Hayashi K (1999) Hydraulic fracture reopening pressure and the estimation of maximum horizontal stress. Int J Rock Mech Min Sci 36(6):811–826CrossRefGoogle Scholar
  23. Kim K, Franklin JA (1987) Suggested methods for rock stress determination. Int J Rock Mech Min Sci Geomech Abstr 24:53–73Google Scholar
  24. Lakirouhani A, Detournay E, Bunger AP (2016) A reassessment of in situ stress determination by hydraulic fracturing. Geophys J Int 205(3):1859–1873CrossRefGoogle Scholar
  25. Lee MY, Haimson BC (1989) Statistical evaluation of hydraulic fracturing stress measurement parameters. Int J Rock Mech Min Sci Geomech Abstr 26(6):447–456CrossRefGoogle Scholar
  26. Lee H, Ong SH (2018) Estimation of in situ stresses with hydro-fracturing tests and a statistical method. Rock Mech Rock Eng 51(3):779–799CrossRefGoogle Scholar
  27. Martin CD, Kaiser PK, Christiansson R (2003) Stress, instability and design of underground excavations. Int J Rock Mech Min Sci 40(7):1027–1047CrossRefGoogle Scholar
  28. Pender MJ (1980) Elastic solutions for a deep circular tunnel, technical notes. Geotechnique 30(2):216–222CrossRefGoogle Scholar
  29. Pitts J (1984) A review of geology and engineering geology in Singapore. Q J Eng Geol 17:93–101CrossRefGoogle Scholar
  30. Public Works Department (1976) The geology of the Republic of Singapore. Public Works Department, SingaporeGoogle Scholar
  31. Rutqvist J, Tsang CF, Stephansson O (2000) Uncertainty in the maximum principal stress estimated from hydraulic fracturing measurements due to the presence of the induced fracture. Int J Rock Mech Min Sci 37(1):107–120CrossRefGoogle Scholar
  32. Sharma J, Chu J, Zhao J (1999) An overview of the geological and geotechnical features of Singapore. Tunn Undergr Spce Technol 14:419–431CrossRefGoogle Scholar
  33. Terzaghi K, Richart FE Jr (1952) Stresses in rock about cavities. Geotechnique 3(2):57–90CrossRefGoogle Scholar
  34. Tunbridge LW (1989) Interpretation of the shut-in pressure from the rate of pressure decay. Int J Rock Mech Min Sci Geomech Abstr 26(6):457–459CrossRefGoogle Scholar
  35. Wu C, Hao H, Zhou Y (2000) Statistical properties of the Bukit Timah granite in Singapore. J Test Eval 28(1):36–43CrossRefGoogle Scholar
  36. Zhao J, Hefny AM, Zhou YX (2005) Hydrofracturing in situ stress measurements in Singapore granite. Int J Rock Mech Min Sci 42(4):577–583CrossRefGoogle Scholar
  37. Zhao XG, Wang J, Cai M, Ma LK, Zong ZH, Wang XY, An QM (2013) In-situ stress measurements and regional stress field assessment of the Beishan area, China. Eng Geol 163:26–40CrossRefGoogle Scholar
  38. Zhao H, Ru Z, Zhu C (2017) Determination of the geomechanical parameters and associated uncertainties in hydraulic fracturing by hybrid probabilistic inverse analysis. Int J Geomech 17(12):04017115CrossRefGoogle Scholar
  39. Zoback MD, Haimson BC (1982) Status of hydraulic fracturing method for in situ stress measurements. In: Proceedings of the 23rd US symposium on rock mechanics, Berkeley, SME/AIME, pp 143–56Google Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Central Soil and Materials Research Station (CSMRS)New DelhiIndia
  2. 2.Indian Institute of Technology (IIT) DelhiNew DelhiIndia
  3. 3.National Institute of Technology (NIT) WarangalWarangalIndia

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