Advertisement

Rock Mechanics and Rock Engineering

, Volume 51, Issue 9, pp 2761–2776 | Cite as

Stochastic Optimization of In Situ Horizontal Stress Magnitudes Using Probabilistic Model of Rock Failure at Wellbore Breakout Margin

  • Insun Song
  • Chandong Chang
Original Paper
  • 240 Downloads

Abstract

Constraining the relationship between the borehole breakout width and rock compressive strength in a vertical borehole is fundamental for determining the magnitudes of the principal in situ horizontal stresses. However, the deterministic relationship yields indeterminate solutions for the two unknown stresses. This paper describes a new method incorporating probabilistic distributions of rock strength and breakout width in a vertical wellbore section for the simultaneous determination of the magnitudes of both stresses. This method optimizes the complete set of in situ principal stresses by minimizing the misfit between a probabilistic model and the measured data of wellbore breakouts. The breakout model is established based on the Weibull distribution of rock strength at the margins of the breakout for a uniform set of far-field stresses. The inverse problem is solved by choosing the best-fit set of far-field stresses in a stress polygon using a grid search algorithm. This process also enables one to evaluate the statistical reliability in terms of sensitivity and uncertainty. The stochastic optimization process is demonstrated using borehole images and sonic logging data obtained from the Integrated Ocean Drilling Program (IODP) Hole C0002A, a vertical hole near the seaward margin of the Kumano basin offshore from the Kii Peninsula, southwest Japan.

Keywords

In situ stress Borehole breakout Rock failure Stochastic optimization Uncertainty Sensitivity 

List of symbols

\({\sigma _{ij}}\)

Stress tensor

\({\sigma _{ij}}'\)

Effective stress tensor

\({\delta _{ij}}\)

The Kronecker delta

Pf

Formation pore pressure

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

Maximum horizontal in situ stress

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

Minimum horizontal in situ stress

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

Vertical in situ (overburden) stress

\({\sigma _{\text{H}}}'\)

Maximum horizontal in situ effective stress

\({\sigma _{\text{h}}}'\)

Minimum horizontal in situ effective stress

\({\sigma _{\text{v}}}'\)

Vertical in situ effective stress

z

Wellbore axis

r

Radial distance from the borehole axis

θ

Angle of a radial direction measured from σH direction

\({\sigma _{\theta \theta }}\)

Tangential stress at wellbore wall

\({\sigma _{zz}}\)

Axial stress at wellbore wall

\({\sigma _{rr}}\)

Radial stress at wellbore wall

\({\sigma _{\theta \theta }}'\)

Tangential effective stress at wellbore wall

\({\sigma _{zz}}'\)

Axial effective stress at wellbore wall

\({\sigma _{rr}}'\)

Radial effective stress at wellbore wall

\({\theta _{\text{B}}}\)

Radial angle of breakout margin measured from σH direction

ν

Poisson’s ratio of the rock

ΔP

Differential pressure between a wellbore and adjacent formation pores

UCS

Uniaxial compressive strength

Vp

P-wave velocity

µ

Internal friction coefficient of intact rock

\({\sigma _1}\)

Maximum principal stress

\({\sigma _2}\)

Intermediate principal stress

\({\sigma _3}\)

Minimum principal stress

m, x0 and xu

Parameters of the Weibull probability density function

λ

Parameter of an exponential density function

Notes

Acknowledgements

This work was supported by the Basic Research Program of the Korea Institute of Geoscience and Mineral Resources (KIGAM) and in part by the project titled ‘International Ocean Discovery Program’, funded by the Ministry of Oceans and Fisheries, Korea. This work is also funded by the Energy Efficiency and Resources of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) Grant funded by the Korea Government Ministry of Trade, Industry and Energy (no. 20162010201980).

References

  1. Barton CA, Zoback MD, Burns KL (1988) In-situ stress orientation and magnitude at the Fenton geothermal site, New Mexico, determined from wellbore breakouts. Geophys Res Lett 15:467–470CrossRefGoogle Scholar
  2. Bell JS, Gough DI (1979) Northeast–southwest compressive stress in Alberta: evidence from oil wells. Earth Planet Sci Lett 45:475–482CrossRefGoogle Scholar
  3. Benz T, Schwab R (2008) A quantitative comparison of six rock failure criteria. Int J Rock Mech Min Sci 45:1176–1186.  https://doi.org/10.1016/j.ijrmms.2008.01.007 CrossRefGoogle Scholar
  4. Brudy M, Zoback MD (1999) Drilling-induced tensile wall-fractures: implications for determination of in-situ stress orientation and magnitude. Int J Mech Min Sci 36:191–215CrossRefGoogle Scholar
  5. Chang C, Haimson B (2000) True triaxial strength and deformability of the German Continental Deep Drilling Program (KTB) deep hole amphibolites. J Geophys Res 105:18,999–19,013.  https://doi.org/10.1029/2000JB900184 CrossRefGoogle Scholar
  6. Chang C, Zoback MD, Khaksar A (2006) Empirical relations between rock strength and physical properties in sedimentary rocks. J Petrol Sci Eng 51:223–237CrossRefGoogle Scholar
  7. Chang C, McNeill LC, Moore JC, Lin W, Conin M, Yamada Y (2010) In situ stress state in the Nankai accretionary wedge estimated from borehole wall failures. Geochem Geophys Geosyst 11:Q0AD04CrossRefGoogle Scholar
  8. Colmenares LB, Zoback MD (2002) A statistical evaluation of intact rock failure criteria constrained by polyaxial test data for five different rocks. Int J Rock Mech Min Sci 39:695–729.  https://doi.org/10.1016/S1365-1609(02)00048-5 CrossRefGoogle Scholar
  9. Devore JL (1991) Probability and statistics for engineering and the sciences, 3rd edn. Duxbury Press, Belmont, p 716Google Scholar
  10. Fairhurst C (1968) Methods of determining in-situ rock stresses at great depth. Tech. Report, I-86. Missouri River Div., U.S. Army Corps of EngineersGoogle Scholar
  11. Griffith AA (1924) Theory of rupture. In: Proceedings of the first international congress of applied mechanics, Delft, pp 55–63Google Scholar
  12. Gulick SPS, Bangs NL, Moore GF, Ashi J, Martin KM, Sawyer DS, Tobin HJ, Kuramoto S, Taira A (2010) Rapid forearc basin uplift and megasplay fault development from 3D seismic images of Nankai margin off Kii Peninsula, Japan. Earth Planet Sci Lett 300:55–62.  https://doi.org/10.1016/j.epsl.2010.09.034 CrossRefGoogle Scholar
  13. Haimson BC, Herrick CG (1985) In situ stress evaluation borehole breakouts experimental studies. In: Research and engineering applications in rock masses, Proceedings of 26th US Symposium on Rock Mechanics, Rapid City. AA Balkema, Rotterdam, pp 1207–1218Google Scholar
  14. Haimson BC, Herrick CG (1986) Borehole breakouts—a new tool for estimating in situ stress? In: Stephansson O (ed) Rock stress and rock stress measurement, Proceedings of international symposium, Stockholm. Centek Pub., Lulea, pp 271–281Google Scholar
  15. Haimson BC, Herrick CG (1989) Borehole breakouts and in situ stress. In: Rowley JC (ed) Drilling symposium 1989, 12th annual energy-sources technology conference and exhibition, Houston. The American Society of Mechanical Engineers, New York, pp 17–22Google Scholar
  16. Haimson BC, Song I (1993) Laboratory study of borehole breakouts in Cordova Cream limestone: a case of shear failure mechanism. Int J Mech Min Sci Geomech Abstr 30:1047–1056.  https://doi.org/10.1016/0148-9062(93)90070-T CrossRefGoogle Scholar
  17. Haimson B, Lin W, Oku H, Hung J-H, Song SR (2010) Integrating borehole-breakout dimensions, strength criteria, and leak-off test results, to constrain the state of stress across the Chelungpu Fault, Taiwan. Tectonophysics 482:65–72.  https://doi.org/10.1016/j.tecto.2009.05.016 CrossRefGoogle Scholar
  18. Huffman KA, Saffer DM (2016) In situ stress magnitudes at the toe of the Nankai Trough Accretionary Prism, offshore Shikoku Island, Japan. J Geophys Res Solid Earth 121:1202–1217.  https://doi.org/10.1002/2015JB012415 CrossRefGoogle Scholar
  19. Jaeger JC, Cook NGW, Zimmerman RW (2007) Fundamentals of rock mechanics. Blackwell Publishing, Oxford, p 475Google Scholar
  20. Kim K, Gao H (1995) Probabilistic approaches to estimating variation in the mechanical properties of rock masses. Int J Mech Min Sci Geomech Abstr 32:111–120CrossRefGoogle Scholar
  21. Kinoshita M, Tobin H, Moe KT, The Expedition 314 Scientists (2008) NanTroSEIZE Stage 1A: NanTroSEIZE LWD transect. IODP Prel. Rept., p 314.  https://doi.org/10.2204/iodp.pr.314.2008
  22. Lee H, Chang C, Ong SH, Song I (2013) Effect of anisotropic borehole wall failures when estimating in situ stresses: a case study in the Nankai accretionary wedge. Mar Petrol Geol 48:411–422.  https://doi.org/10.1016/j.marpetgeo.2013.09.004 CrossRefGoogle Scholar
  23. Mastin LG (1984) Development of borehole breakouts in sandstone. MS Thesis, Stanford University, Palo Alto, USAGoogle Scholar
  24. Mogi K (1971) Fracture and flow of rocks under high triaxial compression. J Geophys Res 76:1255–1269CrossRefGoogle Scholar
  25. Moore GF, Boston BB, Strasser M, Underwood MB, Ratliff RA (2015) Evolution of tectono-sedimentary systems in the Kumano Basin, Nankai Trough forearc. Mar Petrol Geol 67:604e616.  https://doi.org/10.1016/j.marpetgeo.2015.05.032 CrossRefGoogle Scholar
  26. Moos D, Zoback MD (1990) Utilization of observations of well bore failure to constrain the orientation and magnitude of crustal stresses: application to continental deep sea drilling project and ocean drilling program boreholes. J Geophys Res 95:9305–9325CrossRefGoogle Scholar
  27. Plumb RA, Hickman SH (1985) Stress-induced borehole elongation: a comparison between the four arm dipmeter and the borehole televiewer in the Auburn geothermal well. J Geophys Res 90:5513–5521CrossRefGoogle Scholar
  28. Sacks A, Saffer DM, Fisher D (2013) Analysis of normal fault populations in the Kumano forearc basin, Nankai Trough, Japan: 2. Principal axes of stress and strain from inversion of fault orientations. Geochem Geophys Geosyst 14:1973–1988.  https://doi.org/10.1002/ggge.20118 CrossRefGoogle Scholar
  29. Shih TT (1980) An evaluation of the probabilistic approach to brittle design. Eng Fract Mech 13:257–271CrossRefGoogle Scholar
  30. Song I (1998) Borehole breakouts and core disking in westerly granite: mechanisms of formation and relationship to in situ stress. Ph.D. Thesis, University of Wisconsin-MadisonGoogle Scholar
  31. Song I, Chang C (2017) In situ stress conditions at IODP Site C0002 reflecting the tectonic evolution of the sedimentary system near the seaward edge of the Kumano basin, offshore from SW Japan. J Geophys Res Solid Earth 122:4033–4052.  https://doi.org/10.1002/2016JB013440 CrossRefGoogle Scholar
  32. Song I, Haimson BC (1997) Polyaxial strength criteria and their use in estimating in situ stress magnitudes from borehole breakout dimensions. Int J Rock Mech Min Sci 34 (3–4):CD-ROM, Paper no. 116Google Scholar
  33. Song I, Saffer D, Flemings P (2011) Mechanical characterization of slope sediments: Constraints on in situ stress and pore pressure near the tip of the megasplay fault in the Nankai accretionary complex. Geochem Geophys Geosyst 12(5):Q0AD17.  https://doi.org/10.1029/2011GC003556 Google Scholar
  34. Song I, Rathbun AP, Saffer DM (2013a) Uncertainty analysis for the determination of permeability and specific storage from the pulse-transient technique. Int J Rock Mech Min Sci 64:105–111.  https://doi.org/10.1016/j.ijrmms.2013.08.032 CrossRefGoogle Scholar
  35. Song I, Huepers A, Olcott K, Saffer D, Dugan B, Strasser M (2013b) Interpretation of a leak-off test conducted near the bottom of the Kumano Forearc Basin strata at IODP Site C0002 in the Nankai accretionary complex, SW Japan. AGU Fall Meeting Abstracts, p 06Google Scholar
  36. Song I, Chang C, Lee H (2015) A stochastic prediction of in situ stress magnitudes from the predictions of rock strength and breakout width at IODP Hole C0002A in Nankai accretionary prism, SW Japan. EGU General Assembly 2015Google Scholar
  37. Tang CA, Liub H, Lee PKK, Tsui Y, Tham LG (2000) Numerical studies of the influence of microstructure on rock failure in uniaxial compression—Part I: effect of heterogeneity. Int J Rock Mech Min Sci 37:555–569CrossRefGoogle Scholar
  38. Terzaghi K (1954) Theoretical soil mechanics. Wiley, New YorkGoogle Scholar
  39. Tobin H, Kinoshita M, Ashi J, Lallemant S, Kimura G, Screaton E, Moe KT, Masago H, Curewitz D, The Expedition 314/315/316 Scientists (2009) NanTroSEIZE Stage 1 expeditions: introduction and synthesis of key results, in NanTroSEIZE Stage 1: Investigations of Seismogenesis, Nankai Trough, Japan. Proc. Integr. Ocean Drill. Program 314/315/316.  https://doi.org/10.2204/iodp.proc.314315316.101.2009
  40. Weibull W (1951) A statistical distribution function of wide applicability. J Appl Mech 18:293–297Google Scholar
  41. Wiebols GA, Cook NGW (1968) An energy criterion for the strength of rock in polyaxial compression. Int J Rock Mech Min Sci 5:529–549CrossRefGoogle Scholar
  42. Wong RHC, Chau KT, Wang P (1996) Microcracking and grain size effect in Yuen Long Marbles. Int J Rock Mech Min Sci Geomech Abstr 33:479–485CrossRefGoogle Scholar
  43. Zhou S (1994) A program to model the initial shape and extent of borehole breakout. Comput Geosci 20:1143–1160.  https://doi.org/10.1016/0098-3004(94)90068-X CrossRefGoogle Scholar
  44. Zoback MD (2007) Reservoir geomechanics. Cambridge University Press, New York, p 449.  https://doi.org/10.1017/CBO9780511586477 CrossRefGoogle Scholar
  45. Zoback MD, Moos D, Mastin L, Anderson RN (1985) Well-bore breakouts and in situ stress. J Geophys Res 90:5523–5538CrossRefGoogle Scholar
  46. Zoback MD, Barton CA, Brudy M, Castillo DA, Finkbeiner T, Grollimund BR, Moos DB, Peska P, Ward CD, Wiprut DJ (2003) Determination of stress orientation and magnitude in Deep wells. Int J Rock Mech Min Sci 40:1049–1076.  https://doi.org/10.1016/j.ijrmms.2003.07.001 CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Climate Change Mitigation and Sustainability DivisionKorea Institute of Geoscience and Mineral ResourcesDaejeonRepublic of Korea
  2. 2.Petroleum Resources TechnologyUniversity of Science and TechnologyDaejeonRepublic of Korea
  3. 3.Department of GeologyChungnam National UniversityDaejeonRepublic of Korea

Personalised recommendations