Rock Mechanics and Rock Engineering

, Volume 52, Issue 12, pp 5071–5084 | Cite as

Uncertainty in In Situ Stress Estimations: A Statistical Simulation to Study the Effect of Numbers of Stress Measurements

  • Yu FengEmail author
  • John P. Harrison
  • Nezam Bozorgzadeh
Original Paper


Obtaining reliable estimates of the mean in situ stress state is crucial for much rock mechanics and rock engineering analysis. However, due to the variability inherent to stress measurements, a common question is how many stress measurements are required to obtain mean stress estimates of acceptable reliability. This paper investigates the effect of the number of stress measurements used on the uncertainties in estimates of mean stress state. We use Monte Carlo simulation in conjunction with a recently developed multivariate statistical model to simulate stress data sets of different sizes, and then investigate the uncertainties in the mean stress state estimated from these simulated stresses. Three sets of actual stress measurements published in the literature are used to provide realistic values for stress population parameters. We show that the uncertainties depend not only on the number of stress measurements but also on the stress components being estimated and the overall variability of the stress field; hence, suggesting a fixed universal minimum number of stress measurements is inappropriate. The results also show that the number of stress measurements required for reliably estimating the complete stress state may be significantly greater than those required for any one or two individual components of the mean stress state, and are likely to exceed the number available in rock engineering practice. In addition, with the small numbers of measurements typically used in rock engineering, large uncertainties are likely to exist and these may yield misleading stress estimates. We end with suggestions for reducing these large uncertainties in stress estimations.


Uncertainty Stress variability Stress estimation Multivariate statistics Monte Carlo simulation 

List of Symbols


ith stress tensor, i = 1, 2, …, n


Mean stress tensor

\(\sigma_{x}\), \(\sigma_{y}\), \(\sigma_{z}\)

Normal components of stress tensor referred to xyz Cartesian coordinate system

\(\tau_{xy}\), \(\tau_{xz}\), \(\tau_{yz}\)

Shear components of stress tensor referred to xyz Cartesian coordinate system

\(\bar{\sigma }_{x}\), \(\bar{\sigma }_{y}\), \(\bar{\sigma }_{z}\)

Mean of \(\sigma_{x}\), \(\sigma_{y}\), \(\sigma_{z}\), respectively

\(\bar{\tau }_{xy}\), \(\bar{\tau }_{xz}\), \(\bar{\tau }_{yz}\)

Mean of \(\tau_{xy}\), \(\tau_{xz}\), \(\tau_{yz}\), respectively


Stress vector comprising six distinct tensor components


Population mean vector


Population covariance matrix


Maximum-likelihood estimator (MLE) of population mean vector

\({\boldsymbol{\hat{\Omega }}}\)

MLE of population covariance matrix


Vector of the ith measured stress tensor, i = 1, 2, …, n


Mean of \({\mathbf{s}}_{i}\)


Major principal stress


Intermediate principal stress


Minor principal stress


Covariance matrix of stress data set \({\mathbf{s}}_{i}\)



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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Civil and Mineral EngineeringUniversity of TorontoTorontoCanada
  2. 2.GeoEngineering Centre at Queen’s-RMC, Royal Military College of CanadaKingstonCanada

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