Rock masses are discontinuous masses that present spatial anisotropy, being constituted by different rocks of different composition and mechanical properties. They can present different degrees and qualities of fractures, given their environment and geological history. Due to the complexity of working with realistic analytical models, in which all variables are included, historical problems of underground excavations have been solved with empirical models supported by the classification of rock mass, with great representation of the subdivision of classes for the type of reinforcement (use of mesh, rockbolts, shotcrete, etc.), since the diffusion of the New Austrian Tunnelling Method (NATM) [11]. A preliminary classification, usually in possession of data provided by drilling surveys and other methods of sampling and geotechnical characterization, allows the definition of the qualitative class of rock mass and definition of support types needed. As the excavation is carried out, with increased reliability of the information from the underground characterization, the class predicted should be revised and the appropriate support performed (Fig. 1).

Figure 1 Source: Stille and Palmström [13]

Fig. 1 shows the representation of the challenges faced in the field, where in the course of the excavation, the class of the rock mass must be reviewed to indicate the most appropriate support type to be performed.

Barton [12] proposed a diagram that indicates the choice and specification of support types of rock mass according to the classification by the Q-system. It uses the smallest excavation dimension (Excavation Span) or excavation height (the highest value among these) and the Excavation Support Ratio (ESR), a factor referring to the purpose of the excavation [3], which aims to consider the need for reliability, as previously mentioned.

For other rock mass classifications, there are empirical, graphical, and numerical relationships relating the classification of the rock mass with geotechnical properties and design parameters, such as the studies by Bieniawski [2,14], Unal [15], Palmström [16,17], and Mark [18], among others. This practice is common in the field of rock mechanics [19].

Risk analysis in underground works is highly recommended; from analysis by qualitative methods, analyzing generic risks to the project, as presented by Sturk et al. [1] and Domínguez et al. [30], as well as the use of methods such as risk matrix analysis. However, the applicability of probabilistic methods such as the Monte Carlo (Fig. 2), made possible by the computational capacity of processing and availability of robust databases of properties of interest, has been steering risk analyses towards the probabilistic field. Increasingly, non-managerial areas, such as engineering, have been using these tools, evolving the analysis of scenarios previously restricted to deterministic methodologies.

Figure 2 Source: Charbel [6]

In geotechnical engineering and excavations, a trend is identified in the use of probabilistic methods to determine Safety Factors and deformation parameters. The studies by Fattahi et al.[8], Lu et al. [9], Panthi [31], among others, start from the probabilistic distribution models of the parameters of RQD, Jn, Jr, Ja and Jw, for the simulation of the probability of the occurrence of empirical models of physical properties of rock mass, considering the Q-system and empirical relationships associated with it. Other studies have focused on the analysis of uncertainty of RMR parameters, such as that by Sari et al. [32].

The uncertainty associated with the occurrence of these and other parameters can be remedied by exhaustive field and laboratory tests. Watanabe [33] and Panthi [31] present extensive literature on statistical modeling of the input indexes in the RMR and Q-classifications, respectively. There is a tendency for the Normal, Lognormal, and Triangular distributions of the indices, although it must be considered that they tend to vary according to the geology (type of rock and location) under study and its genesis process (igneous, sedimentary or metamorphic). In his study, Charbel [6] presents, for the UCS (Uniaxial Compressive Strength) parameter, Lognormal and Normal distributions, varying according to the geological layer assessed. Table 1 presents a compilation of the distribution forms found by different authors for the Q-system and RMR parameters. It is noticeable that the largest number of studies on characterization and statistical analysis falls on the RQD, as highlighted by Zhang [5]. RQD is an extremely important parameter in the characterization of rock masses and can be related in isolation with other parameters related to the physical features of rock masses (such as modulus of elasticity and relationship between tensile strength of the rock mass and intact rock).

Table 1

Source: Authors

Considering that one can assume probability distributions for the parameters that compose the geomechanical classifications of RMR and Q, these can be simulated using the Monte Carlo method. Although the number and quality of information regarding the rock mass increase the extent to which an excavation is carried out, causing the dynamic review of a project and leading to the need for changes in its execution, the possibility of measuring the variability that a project may present is interesting for conceptual elaboration stages. In the case of geological environments, even with large ranges of data available, abrupt changes in the quality of the rock mass cannot be completely ruled out.

For conceptual cases or pre-feasibility studies, where data sources are limited, due to the execution of few physical surveys, the quantitative risk analysis helps to measure the potential scenarios that could occur. It is possible to visualize the amount of support elements to be used, execution times and, consequently, the costs of carrying out an excavation, be it a road, tunnel or an underground mine access. Although this is a distribution of expected values, and such distribution presents reliability proportional to the reliability of the input data, having a wide range of expected values allows a more realistic analysis than that of deterministic projects. Another possibility is the assessment of gain by densifying the available information (surveys and tests) in view of a possible decrease in uncertainty regarding the execution of the work in the cycle times, type and quantities of support elements and planned advances, among others. In the case of underground excavations, whether for mining or civilian purposes, we are dealing with works of around millions to easily billions of dollars.

Fig. 4 presents the histogram of simulated Q-values, based on the input parameters and their basic statistics. It is observed that simulated values ranged from 2.79 * 10^-4 to 49.96, presenting a distribution behavior of Lognormal type.

Figure 4 Source: Authors

There was a higher frequency of mass framing with Class 5, corresponding to Class D, depending on the Q-value. However, there were also other classifications such as Class C and E. Regarding the simulated support classes, there was a greater distribution among classes 4 to 6, but more frequently for classes 4 and 5 (Fig. 5).

Figure 5 Source: Authors

Fig. 6 shows the distribution in support classes, which considers, in addition to the Q interval, the ESR value and the height of the excavation.

Figure 6 Source: Authors

Considering the RMR self-support time graph and RMR values inferred from Q (eq. 2), the excavation self-support times, in hours, were simulated for excavation rates of 10, 5, and 3 meters. The automation of the calculation of the self-support times for these advances was performed by adjusting exponential equations, in this case, those with the highest correlation, to the graph of self-support times.

Figs. 7, 8, and 9 show the simulations of self-support times for excavation rates of 10, 5, and 3 meters, respectively. The simulated values start from time zero (referring to immediate collapse). There is an increase in the occurrence of scenarios with self-support time greater than 5 hours and an increase in averages of the simulations; with the decrease of the excavation span. The simulations indicate that if working with a 3-meter excavation rate, we are less likely to collapse in less than 5 hours, without carrying out support in the excavation. Such simulations are of great value for compatibility with the estimated times of operation cycles, with the time for cleaning the front and development of support. In this way, it is possible to identify situations with risk of collapse, when it is significant (project definition) the probability that the self-support time is less than the cycle time.

Figure 7 Source: Authors

Figure 9 Source: Authors