Uncertainties in the estimation of in situ stresses: effects of heterogeneity and thermal perturbation

  • Hossein Agheshlui
  • Stephan Matthai
Original Article


In situ stress directions and magnitudes in subsurface are commonly estimated using two different methods: (1) borehole break-out—drilling-induced fracture interpretation complemented by extended leak-off tests; (2) stress estimation from acoustic measurements using advanced wireline tools. Both methods use stress perturbations around boreholes to estimate far field stresses employing Kirsch’s equations, which are only valid for boreholes aligned with one of the principal stresses, linear elastic deformation and homogeneous isotropic materials. Furthermore, the original stress state may have been altered by drilling-mud-circulation induced temperature changes. Using heuristic models including heterogeneity, this study investigates potential errors in the estimation of far-field stress due to (a) the generalisation of Kirsch’s equations to heterogeneous media and (b) plausible temperature perturbations. First, errors due to uncertainties in measuring wave slowness are analysed for an idealised homogeneous material. Second, errors due to application of Kirsch’s equations are investigated considering potential effects of frictional interfaces between layers, pore pressure in the rock matrix and thermal perturbations induced by drilling for example cases. To analyse stress in these complex scenarios finite element analysis was used, revealing strong effects of lithological and thermal variations. Stress magnitude was amplified by stiff layers and was attenuated by soft ones. At layer interfaces, substantial changes in stress orientation occurred. Kirsch’s equations for the considered cases resulted in errors in far field stresses as large as 44% in the magnitude and 90° in orientations. An uncertainty propagation analysis indicated a high accuracy of acoustic estimates for homogenous materials. However, a dramatic impact of small-scale heterogeneity may not be resolved by the logging process.


Stress perturbations around boreholes Kirsch equations Heterogeneity Thermal stress Acoustic logging Error propagation analysis 

List of symbols


Borehole diameter (M)


Young’s modulus (GPa)


Vector of body forces (N)


Shear modulus (GPa)


Thermal conductivity [W/(mK)]


Pore pressure (MPa)


Wellbore pressure (MPa)


Heat flux (W/m2)


Radial stress in a cylindrical coordinate system with its origin set to the wellbore centre


Hoop stress in a cylindrical coordinate system with its origin set to the wellbore centre


Temperature in bedrock


Temperature (K)


Compressive wave velocity (m/s)


Shear wave velocity (m/s)


Fast shear wave (m/s)


Slow shear wave (m/s)


Stoneley wave velocity (m/s)


Displacements in x, y, and z directions (m)


Biot coefficient


Thermal expansivity (1/K)


Maximum principal horizontal total far-field stress (MPa)


Minimum principal horizontal total far-field stress (MPa)


Vertical total stress (MPa)


Minimum hoop stress around wellbore (MPa)


Maximum hoop stress around wellbore (MPa)


Effective stress (MPa)




Angle measured from the major horizontal stress (°)


Coefficient of internal friction


Poisson’s ratio


Rock density (kg/m3)



Caroline Millotte (FEI Inc., Canberra, Australia) is thanked for the generation of the more complex facies model and mesh. Their original geometries were generated with SBED, Geomodelling Technology Corp., Alberta, Canada.

Compliance with ethical standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.


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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Peter Cook Centre for CCSThe University of MelbourneParkvilleAustralia

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