Elastic-Softening-Plasticity Around a Borehole: An Analytical and Experimental Study

  • Adelina Lv
  • Hossein Masoumi
  • Stuart D. C. Walsh
  • Hamid Roshan
Original Paper


Analytical solutions of near-borehole stress–strain responses are often based on simple elastic-perfectly plastic constitutive models. However, such models are incapable of capturing rock softening, and thus often disagree with observed failure patterns. Therefore, in this study, we present a set of closed-form solutions to estimate the stress–strain response around a borehole in a finite medium obeying elastic-softening-plasticity. Solutions are presented for material subject to either Mohr–Coulomb (M–C) or Hoek–Brown (H–B) failure criteria. The proposed analytical solutions define two failure surfaces to capture the peak and residual stress states, which are linked by a softening law. We demonstrate how to identify these two surfaces by first conducting a set of triaxial tests on a sandstone sample to differentiate the deformation stages. The proposed methodology provides a practical approach to characterize the strain-softening response with good accuracy. Finally, the proposed analytical solutions are validated through comparisons with (a) numerical simulations and (b) hydrostatic experiments on hollow-cylinder sandstone samples analysed using Digital Image Correlation. The strain development and interfaces calculated from the proposed model are in good agreement with the experimental and numerical data.


Analytical solution Hoek–Brown Mohr–Coulomb Constitutive model Elastoplasticity Digital Image Correlation (DIC) 

List of symbols

\({{{{\upsigma}}}_{\text{r}}}\),\({\sigma _\theta }\)

Radial and tangential stress


Radial and tangential strain


Internal radius of the borehole


External radius of the medium


The interface between the elastic and softening zones


The interface between softening and residual zones


The radial stress at Re


Radial displacement


Poisson’s ratio


Young’s modulus


The angle of internal friction


Dilation angle




Original cohesion


Residual value of cohesion


Hoek–Brown parameter


Hoek–Brown parameter of original state


Hoek–Brown parameter of residual state


Uniaxial compressive strength (UCS)


Borehole pressure

\({\sigma _{\text{o}}}\)

External pressure



Adelina Lv gratefully acknowledges the Australian Government Research Training Program for providing her scholarship.

Supplementary material

603_2018_1650_MOESM1_ESM.tif (7.3 mb)
Supplementary material 1 (TIF 7523 KB)
603_2018_1650_MOESM2_ESM.tif (7.3 mb)
Supplementary material 2 (TIF 7523 KB)
603_2018_1650_MOESM3_ESM.tif (1.3 mb)
Supplementary material 3 (TIF 1371 KB)


  1. Abdulhadi NO, Germaine JT, Whittle AJ (2010) Experimental study of wellbore instability in clays. J Geotech Geoenviron Eng 137(8):766–776CrossRefGoogle Scholar
  2. Alejano L, Rodriguez-Dono A, Alonso E, Manín GF (2009) Ground reaction curves for tunnels excavated in different quality rock masses showing several types of post-failure behaviour. Tunn Undergr Space Technol 24:689–705CrossRefGoogle Scholar
  3. Alonso E, Alejano L, Varas F, Fdez-Manin G, Carranza-Torres C (2003) Ground response curves for rock masses exhibiting strain-softening behaviour. Int J Numer Anal Methods Geomech 27:1153–1185CrossRefGoogle Scholar
  4. Alquwizani SA (2013) Three-dimensional elasto-plastic modeling of wellbore and perforation stability in poorly consolidated sands. Master thesis. The University of Texas at Austin, Austin, Texas, USAGoogle Scholar
  5. Alsayed M (2002) Utilising the Hoek triaxial cell for multiaxial testing of hollow rock cylinders. Int J Rock Mech Min Sci 39:355–366CrossRefGoogle Scholar
  6. Basarir H, Genis M, Ozarslan A (2010) The analysis of radial displacements occurring near the face of a circular opening in weak rock mass. Int J Rock Mech Min Sci 47:771–783CrossRefGoogle Scholar
  7. Bellarby J (2009) Well completion design, vol 56. Elsevier, Amsterdam, NetherlandsGoogle Scholar
  8. Blaber J, Adair B, Antoniou A (2015) Ncorr: open-source 2D digital image correlation matlab software. Exp Mech 55:1105–1122CrossRefGoogle Scholar
  9. Brady BHG, Brown ET (2006) Rock mechanics: for underground mining. Springer, BerlinGoogle Scholar
  10. Brown ET, Bray JW, Ladanyi B, Hoek E (1983) Ground response curves for rock tunnels. J Geotech Eng 109:15–39CrossRefGoogle Scholar
  11. Carranza-Torres C (1998) Self-similarity analysis of the elasto-plastic response of underground openings in rock and effects of practical variables. University of Minnesota, MinneapolisGoogle Scholar
  12. Carranza-Torres C (2004) Elasto-plastic solution of tunnel problems using the generalized form of the Hoek–Brown failure criterion. Int J Rock Mech Min Sci 41:629–639CrossRefGoogle Scholar
  13. Carranza-Torres C, Fairhurst C (1999) The elasto-plastic response of underground excavations in rock masses that satisfy the Hoek–Brown failure criterion. Int J Rock Mech Min Sci 36:777–809CrossRefGoogle Scholar
  14. Carter JP, Yeung SK (1985) Analysis of cylindrical cavity expansion in a strain weakening material. Comput Geotech 1(3):161–180CrossRefGoogle Scholar
  15. Chen X, Tan C, Haberfield C (1999) Solutions for the deformations and stability of elastoplastic hollow cylinders subjected to boundary pressures. Int J Numer Anal Methods Geomech 23:779–800CrossRefGoogle Scholar
  16. Chen S, Abousleiman Y, Abass H (2014) An analytical elasto-plastic analysis for stability of axisymmetric wellbore. In: ASME 2014 33rd international conference on ocean, offshore and arctic engineering. american society of mechanical engineers, pp V005T011A026–V005T011A026Google Scholar
  17. Cui L, Zheng JJ, Zhang RJ, Dong YK (2015) Elasto-plastic analysis of a circular opening in rock mass with confining stress-dependent strain-softening behaviour. Tunn Undergr Space Technol 50:94–108CrossRefGoogle Scholar
  18. Detournay E, Fairhurst C (1987) Two-dimensional elastoplastic analysis of a long, cylindrical cavity under non-hydrostatic loading. In: International journal of rock mechanics and mining sciences & geomechanics abstracts. Elsevier, Amsterdam, pp 197–211CrossRefGoogle Scholar
  19. Detournay E, John CMS (1988) Design charts for a deep circular tunnel under non-uniform loading. Rock Mech Rock Eng 21:119–137CrossRefGoogle Scholar
  20. Ewy R, Cook N (1990a) Deformation and fracture around cylindrical openings in rock—I. Observations and analysis of deformations. In: International journal of rock mechanics and mining sciences & geomechanics abstracts, vol 5. Elsevier, Amsterdam, pp 387–407CrossRefGoogle Scholar
  21. Ewy R, Cook N (1990b) Deformation and fracture around cylindrical openings in rock—II. Initiation, growth and interaction of fractures. In: International journal of rock mechanics and mining sciences & geomechanics abstracts, vol 5. Elsevier, Amsterdam, pp 409–427CrossRefGoogle Scholar
  22. Fairhurst C (2014) Analysis and design methods: comprehensive rock engineering: principles, practice and projects. Elsevier, AmsterdamGoogle Scholar
  23. Fakhimi A, Carvalho F, Ishida T, Labuz J (2002) Simulation of failure around a circular opening in rock. Int J Rock Mech Min Sci 39:507–515CrossRefGoogle Scholar
  24. Forbes M, Masoumi H, Saydam S, Hagan P (2015) Investigation into the effect of length to diameter ratio on the point load strength index of Gosford sandstone. In: Proceedings of the 49th US rock mechanics geomechanics symposium (ARMA), 28 June–1 July, San Francisco, US, vol 1, pp 478–488Google Scholar
  25. Gay N (1973) Fracture growth around openings in thick-walled cylinders of rock subjected to hydrostatic compression. In: International journal of rock mechanics and mining sciences & geomechanics abstracts, vol 3. Elsevier, Amsterdam, pp 209–233CrossRefGoogle Scholar
  26. Guan Z, Jiang Y, Tanabasi Y (2007) Ground reaction analyses in conventional tunnelling excavation. Tunn Undergr Space Technol 22:230–237CrossRefGoogle Scholar
  27. Hoek E, Brown ET (1997) Practical estimates of rock mass strength. Int J Rock Mech Min Sci 34:1165–1186CrossRefGoogle Scholar
  28. Hoek E, Diederichs MS (2006) Empirical estimation of rock mass modulus. Int J Rock Mech Min Sci 43:203–215CrossRefGoogle Scholar
  29. International Society for Rock Mechanics (2007) The complete ISRM suggested methods for rock characterization, testing and monitoring: 1974–2006. In: International society for rock mechanics, commission on testing methodsGoogle Scholar
  30. Jaeger JC, Cook NG, Zimmerman R (1977) Fundamentals of rock mechanics. Wiley, New YorkGoogle Scholar
  31. Jain SK (1980) Fundamental aspects of the normality rule and their role in deriving constitutive laws of soils (no. 1). Engineering Pubns, HardcoverGoogle Scholar
  32. Jourine S, Valko PP, Kronenberg AK (2004) Modelling poroelastic hollow cylinder experiments with realistic boundary conditions. Int J Numer Anal Methods Geomech 28(12):1189–1205CrossRefGoogle Scholar
  33. Lai WM (1974) Introduction to continuum mechanics (unified engineering). Butterworth-Heinemann, OxfordGoogle Scholar
  34. Lamé G (1866) Leçons sur la théorie mathématique de l’elasticité des corps solides par G. Lamé. Gauthier-VillarsGoogle Scholar
  35. Lee YK, Pietruszczak S (2008) A new numerical procedure for elasto-plastic analysis of a circular opening excavated in a strain-softening rock mass. Tunn Undergr Space Technol 23:588–599CrossRefGoogle Scholar
  36. Marsden J, Dennis J, Wu B (1996) Deformation and failure of thick-walled hollow cylinders of mudrock a study of wellbore instability in weak rock. In: ISRM international symposium-EUROCK 96. International Society for Rock MechanicsGoogle Scholar
  37. Masoumi H (2013) Investigation into the mechanical behaviour of intact rock at different sizes. Ph.D. thesis, University of New South Wales, SydneyGoogle Scholar
  38. Masoumi H, Bahaaddini M, Kim G, Hagan P (2014) Experimental investigation into the mechanical behaviour of Gosford sandstone at different sizes. In Proceedings of the 48th US rock mechanics geomechanics symposium (ARMA), 1–4 June, Minneapolis, US, vol 2, pp 1210–1215Google Scholar
  39. Masoumi H, Douglas KJ, Russell AR (2016a) A bounding surface plasticity model for intact rock exhibiting size-dependent behaviour. Rock Mech Rock Eng 49:47–62CrossRefGoogle Scholar
  40. Masoumi H, Roshan H, Hagan PC (2016b) Size-dependent Hoek–Brown failure criterion. Int J Geomech 17(2):04016048CrossRefGoogle Scholar
  41. Masoumi H, Horne J, Timms W (2017) Establishing relationships for the effects of water content on the mechanical behaviour of Gosford sandstone. Rock Mech Rock Eng 50(8):2235–2242CrossRefGoogle Scholar
  42. Masoumi H, Roshan H, Hedayat A, Hagan PC (2018) Scale-Size dependency of intact rock under point load and indirect tensile Brazilian testing. Int J Geomech 18(3):04018006CrossRefGoogle Scholar
  43. McLean M, Addis M (1990) Wellbore stability: the effect of strength criteria on mud weight recommendations. In: SPE annual technical conference and exhibition, 1990. Society of Petroleum EngineersGoogle Scholar
  44. Medhurst TP (1997) Estimation of the in situ strength and deformability of coal for engineering designGoogle Scholar
  45. Meng QX, Wang W (2014) A novel closed-form solution for circular openings in generalized Hoek–Brown media. Math Probl Eng 2014:870835Google Scholar
  46. Mitaim S, Detournay E (2004) Damage around a cylindrical opening in a brittle rock mass. Int J Rock Mech Min Sci 41:1447–1457CrossRefGoogle Scholar
  47. Monfared M, Delage P, Sulem J, Mohajerani M, Tang AM, De Laure E (2011) A new hollow cylinder triaxial cell to study the behavior of geo-materials with low permeability. Int J Rock Mech Min Sci 48:637–649CrossRefGoogle Scholar
  48. Roshan H, Fahad M (2012b) Chemo-poroplastic analysis of a borehole drilled in a naturally fractured chemically active formation. Int J Rock Mech Min Sci 52:82–91CrossRefGoogle Scholar
  49. Roshan H, Oeser M (2012a) A non-isothermal constitutive model for chemically active elastoplastic rocks. Rock Mech Rock Eng 45(3):361–374CrossRefGoogle Scholar
  50. Roshan H, Masoumi H, Hagan P (2016) On size-dependent uniaxial compressive strength of sedimentary rocks in reservoir geomechanics. In Proceedings of the 50th US rock mechanics geomechanics symposium (ARMA), 26–29 June, Houston, US, vol 3, pp 2322–2327Google Scholar
  51. Roshan H, Masoumi H, Regenaur-Lieb K (2017) Frictional behaviour of sandstone: a sample-size dependent triaxial investigation. J Struct Geol 94:154–165CrossRefGoogle Scholar
  52. Roshan H, Lv A, Xu Y, Masoumi H, Regenauer-Lieb K (2018a) New generation of hoek cells geotechnical testing journal. Geotech Test J. (Accepted 10 April 2018) CrossRefGoogle Scholar
  53. Roshan H, Masoumi H, Zhang Y, Al-Yaseri AZ, Iglauer S, Lebedev M, Sarmadivaleh M (2018b) Micro-structural effects on mechanical properties of Shaly-sandstone. J Geotech Geoenviron Eng 144(2):06017019CrossRefGoogle Scholar
  54. Santos JB, Barroso EV, Vargas Jr EA, Castro JT, Gonçalves C, Campos E (2007) Studies of mechanisms associated with sand production using X-ray CT scan in real time. In: 11th ISRM congress. International Society for Rock MechanicsGoogle Scholar
  55. Shah KR (1997) An elasto-plastic constitutive model for brittle-ductile transition in porous rocks. Int J Rock Mech Min Sci 34(3–4):283.e1–283.e13Google Scholar
  56. Sharan S (2003) Elastic–brittle–plastic analysis of circular openings in Hoek–Brown media. Int J Rock Mech Min Sci 40:817–824CrossRefGoogle Scholar
  57. Silvestri V, Diab R, Samra GA, Bravo-Jonard C (2010) Undrained response of clay in hollow cylinder expansion tests. Geotech Test J 34:76–88Google Scholar
  58. Varas F, Alonso E, Alejano L, Manín GF (2005) Study of bifurcation in the problem of unloading a circular excavation in a strain-softening material. Tunn Undergr Space Technol 20:311–322CrossRefGoogle Scholar
  59. Vardoulakis I, Sulem J, Guenot A (1988) Borehole instabilities as bifurcation phenomena. In: International journal of rock mechanics and mining sciences & geomechanics abstracts, vol 3, pp 159–170CrossRefGoogle Scholar
  60. Vermeer PA, De Borst R (1984) Non-associated plasticity for soils, concrete and rock. HERON 29(3)Google Scholar
  61. Wang Y (1996) Ground response of circular tunnel in poorly consolidated rock. J Geotech Eng 122:703–708CrossRefGoogle Scholar
  62. Wang Y, Dusseault M (1991) Borehole yield and hydraulic fracture initiation in poorly consolidated rock strata—Part II. Permeable media. In: International journal of rock mechanics and mining sciences & geomechanics abstracts, vol 4. Elsevier, Amsterdam, pp 247–260CrossRefGoogle Scholar
  63. Wang S, Sloan S, Sheng D, Tang C (2012a) Numerical analysis of the failure process around a circular opening in rock. Comput Geotech 39:8–16CrossRefGoogle Scholar
  64. Wang S, Sloan S, Tang C, Zhu W (2012b) Numerical simulation of the failure mechanism of circular tunnels in transversely isotropic rock masses. Tunn Undergr Space Technol 32:231–244CrossRefGoogle Scholar
  65. Weng MC, Jeng FS, Huang TH, Lin ML (2005) Characterizing the deformation behavior of Tertiary sandstone. Int J Rock Mech Min Sci 42:388–401CrossRefGoogle Scholar
  66. Weng MC, Tsai LS, Hsieh YM, Jeng FS (2010) An associated elastic–viscoplastic constitutive model for sandstone involving shear-induced volumetric deformation. Int J Rock Mech Min Sci 47(8):1263–1273CrossRefGoogle Scholar
  67. Younessi A, Rasouli V, Wu B (2012) Experimental sanding analysis: thick walled cylinder versus true triaxial tests. In: Proceedings of 2nd southern hemisphere international rock mechanics symposium (SHIRMS), Sun City, South AfricaGoogle Scholar
  68. Yuan SC, Harrison J (2004) An empirical dilatancy index for the dilatant deformation of rock. Int J Rock Mech Min Sci 41:679–686CrossRefGoogle Scholar
  69. Zervos A, Papanastasiou P, Vardoulakis I (2001) Modelling of localisation and scale effect in thick-walled cylinders with gradient elastoplasticity. Int J Solids Struct 38:5081–5095CrossRefGoogle Scholar
  70. Zhu H, Deng J, Jin X, Hu L, Luo B (2015) Hydraulic fracture initiation and propagation from wellbore with oriented perforation. Rock Mech Rock Eng 48:585–601CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  • Adelina Lv
    • 1
  • Hossein Masoumi
    • 2
  • Stuart D. C. Walsh
    • 2
  • Hamid Roshan
    • 1
  1. 1.School of Minerals and Energy Resources EngineeringUNSW AustraliaKensingtonAustralia
  2. 2.Department of Civil EngineeringMonash UniversityClaytonAustralia

Personalised recommendations