Uncertainty Analysis in Tunnel Deformation Using Monte Carlo Simulation

Abstract

Calculating in situ rock stresses is increasingly imperative due to the complexities and limitations associated with testing techniques, equipment, assumptions, and the subjective interpretation of test results. Estimating in situ stress from hydraulic fracturing tests (HFT) is one of the most effective methods. HFT data rely on three parameters: (1) shut-in pressure, (2) reopening pressure, and (3) fracture orientation. The previously available literature developed a first-order approximation for estimating the effect of these uncertainties on tunnel deformation estimates using Pender’s solution. This study conducts a detailed Monte Carlo analysis across four cases with results compared with the first-order approximate analysis. The cases are defined by the least horizontal principal stress, maximum horizontal principal stress, the direction of the fracture, and angular coordinates concerning the tunnel center. Case b represents maximum tunnel deformation (percentage of tunnel diameter) when the tunnel alignment is along the direction of the maximum horizontal principal stress. Cases c and d represent minimal tunnel deformation (percentage of tunnel diameter) when the tunnel alignment is along the direction of the least horizontal principal stress and the most horizontal principal stress, respectively. It is observed that the first-order analysis provides reasonable results for Cases b, c, and d. However, significant errors exist in Case a, which involves the largest tunnel deformation (percentage of tunnel diameter) when the tunnel alignment is along the direction with the least horizontal principal stress. This case is the most critical and exhibits less fluctuation in the Monte Carlo simulation (MCS) results compared with the first-order approximation (FOA). This study demonstrates that MCS offers a more precise approach to analyzing uncertainties and estimating their impact on tunnel deformation. Specifically, for Case a, a detailed Monte Carlo analysis outperforms the approximate analysis, making MCS a more reliable method for assessing the effects of uncertainty in hydrofracturing parameters on tunnel ovalization estimates. Numerical models are predominantly used in designing rock engineering projects. In this study, we used both first-order approximations and Monte Carlo simulations to estimate the coefficients of variation in tunnel deformation, comparing four different cases. Our findings emphasize the importance of accounting for uncertainties in hydrofracturing test method parameters during the design process to ensure reliable engineering decisions. The analysis is divided in to four cases. The analysis showed that Case a is particularly critical in the design process, exhibiting less variability and fluctuation in Monte Carlo simulation results compared with first-order approximations. Additionally, we visualized tunnel cross-section deformations under different horizontal stresses while keeping another parameters constant. The results indicate that maximum deformation occurs at angles θ=0° or 90°, highlighting the significant impact of stress orientation on tunnel behavior. Monte Carlo simulations proved more accurate in predicting tunnel deformation under large parameter uncertainties, presenting a robust method for incorporating hydrofracturing test uncertainties into design methodologies. This approach reduces the risk of poor decisions and enhances the reliability of engineering outcomes. Practicing engineers can use first-order analysis for initial assessments, but should employ Monte Carlo simulations for more accurate and comprehensive insights.