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


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.


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


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


Wellbore axis


Radial distance from the borehole axis


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


Poisson’s ratio of the rock


Differential pressure between a wellbore and adjacent formation pores


Uniaxial compressive strength


P-wave velocity


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


Parameter of an exponential density function



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).


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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

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