The in situ stress state is a fundamental parameter in rock mechanics and rock engineering, and is often summarized and represented as the point estimate of the principal stresses associated with the mean of a number of measured stresses. However, to account for the likely large uncertainties associated with stress estimations, it is important to also provide confidence intervals. In this paper, we propose a novel method for constructing confidence intervals for the in situ stress state using a multivariate distribution model and Monte Carlo sampling. We show that this method addresses a particular shortcoming of the customary method for constructing in situ stress confidence intervals, namely that of a misconception of confidence intervals. We also show that the customary method does not yield correct confidence intervals, and this may have a significant impact on designs and analyses in rock engineering that incorporate stress uncertainty.
The state of in situ stress is a fundamental parameter in rock mechanics and rock engineering (Amadei and Stephansson, 1997; Brady and Brown, 2004; Hudson and Harrison, 2000). Often in practice, the in situ stress state within a rock mass volume of interest is summarized and represented using the principal mean stress, which is customarily calculated from the mean of the measured stresses (Amadei and Stephansson, 1997; Gao and Harrison, 2017, 2018a, 2018b; Hudson et al., 2003; Martin and Christiansson, 1991a, b; Walker et al., 1990). Statistically speaking, the principal mean stress of measured stresses is a point estimate of the in situ stress state.
However, it is widely recognized that localized measurements of in situ stress often display significant variability in fractured rock masses (Ask, 2003, 2006; Day-Lewis, 2008; Gao, 2017; Gao and Harrison, 2014, 2015, 2016a, b, 2018a, b; Hudson and Feng, 2010; Harrison et al., 2010), and the estimation of the in situ stress state within a rock mass volume may involve large uncertainty, particularly when, as is often the case in rock engineering, only a limited number of measured stresses are available. Unfortunately, the point estimation provides no information on uncertainty associated with the estimation of the in situ stress state, and ignoring this uncertainty may yield misleading and even erroneous results in rock mechanics and rock engineering analyses. The need to evaluate uncertainty becomes particularly pressing when we consider the continuously increasing application of reliability-based design (RBD) methods in rock engineering, as these explicitly require the uncertainties in design parameters such as stress and material properties to be appropriately characterized and incorporated in analyses (Bozorgzadeh, 2017; Bozorgzadeh et al., 2017). Thus, it is important to also provide confidence interval (CI) as a quantitative measure of uncertainty in estimations of the in situ stress state.