استفاده از طراحی فاکتوریل دوسطحی برای تجزیه و تحلیل حساسیت آمار بلوک کلیدی از هندسه شکستگی

A stochastically modeled fracture network offers potential for more realistic assessment of stability status in underground excavations than predictions based entirely on deterministic features. The reliability of probabilistic models, however, depends strongly on an accurate estimation of the model's variables, i.e., the fracture network properties from the field and laboratory observations. In this study, predictions of keyblocks by implementing stochastically generated fractures in the Central Storage Facility for Spent Nuclear Fuel (CLAB 2 Centralt Lager Använt Bränsle) located in southeast Sweden are presented. The fracture network model is built by using fracture mapping in the floor of the facility and incorporates fracture size, shape, orientations, termination mode, spatial arrangement and fracture mechanical properties. The predicted volume of individual keyblocks is best-fitted with the Pareto probability distribution function. Subsequently, a statistical two-level factorial analysis is performed to examine the impact of both single fracture properties and their interactions on the predictions made. In the factorial experiments, the block predictions are made for eight different fracture models where three factors: fracture radii, orientation and termination are each assigned two levels intentionally departing from the best estimates found for the CLAB 2 site. This allows us to express the experiment results as the degree to which each of the eight computed block statistics deviated from the most likely prediction. It is found that fracture orientation is the only statistically significant factor influencing the keyblock statistics while the input from other variables/fracture properties and their interactions is less significant. The results of our study yield a prospective approach for improving the effectiveness of the stochastic model variable estimation and for more optimal field mapping strategies.

The volume, shape and amount of unstable rock blocks formed by intersections between joints/fractures and contours of underground chambers/tunnels depend on both the dimensions and the geometry of the excavation itself as well as on the geometry and other properties of fractures/joints intersecting the excavation. The commonly applied Block Theory of Goodman and Shi [1] focuses on the potentially largest keyblocks and orientations of major joint systems in relation to the orientation of the underground object. Several authors [2], [3], [4] and [5] attempted to employ a probabilistic approach for keyblock predictions based on the stochastic representation of fracture geometries and their locations. Many other researchers addressed the predictions of unstable rock blocks both for underground facilities and for rock slopes, and the numbers of the numerical codes for keyblock predictions and rock support design have been developed; SATIRN [6], UNWEDGE [7] and [8], DRKBA [9], MSB [10], KBTUNNEL [11], to mention only a few. These codes are all based on similar principles but differ in terms of the type of input data, model assumptions and potential outcome from the simulation analysis. Other well-known codes/methodologies offer less potential for modeling more complex fracture networks but on the other hand permit the examination of interactions between block triggering, displacement, deformation and stress field in relation to the time factor: for example, 3DEC [12] and DDA [13]. One important aspect of keyblock analysis refers to the estimation process of fracture geometries and mechanical properties based on field and/or laboratory measurements. Depending on the type of estimated fracture variable/property, its uncertainty will have a different impact on the final block predictions. Further, in some circumstances the interactions between several variables may influence the predictions to a higher degree than the individual variables do. Knowing which fracture properties significantly affect the amount of unstable blocks, their size or total removable volume along an underground chamber and which properties are of less importance could result in the optimization of field sampling strategy and of more cost-effective tunnel support design. The majority of these kinds of studies have been focused on how predicted block statistics depends on varying one property at the time e.g., by varying fracture size, fracture intensity or tunnel dimensions [14], [15] and [16]. Therefore, a deeper analysis of the possible interactions among properties of the fracture network and their quantitative bearing on block predictions could help us make more reliable geological risk estimates. This paper presents: (i) probabilistic three-dimensional keyblock predictions based on two-dimensional fracture mapping from the floor of the CLAB 2 underground chamber hosted in crystalline rock unit, southeast Sweden, and (ii) sensitivity analysis of keyblock predictions to fracture size, orientations and terminations based on statistical two-level factorial design.

Stochastic fracture network built from 2D fracture mapping in the CLAB 2 facility floor was derived. Orientation of fractures was approximated with two major clusters each following Fisher's pdf, fracture size, i.e., the representative radius was best fitted with lognormal distribution and fracture locations followed Poisson process. The proposed model was used for simulation of unstable blocks along the full length of the facility. The simulation included 10 Monte Carlo realizations and resulted in the average of 38 unstable blocks where their volume was best-fitted with the Pareto distribution. As the distribution of the block volume and the average block amount are of less practical value the estimation of the total reinforcement required should rely on other predicted parameters such as the total unstable volume Vt=57 m3 or the largest unstable block Vmax=12 m3. Since the predicted largest block volume showed by far more spread around the average than the total unstable volume, the last parameter was considered to be a more correct measure of instability of the cavern. The presented factorial analysis illustrated that incorrect estimation of fracture orientation pattern can have most severe consequences in terms of correct prediction of the stability conditions. The prediction of total unstable block volume based on models with three orientation clusters approximated with Fisher's pdf, i.e., plus level in factorial design was clearly closer to the prediction based on Most Likely Model than prediction derived from models where orientations were represented with non-parametric bootstrap distribution (minus level) as Fig. 5 indicates. Fig. 5 shows that also OR∗T∗SZ,OR∗T and SZ depart from the normal probability plot, however, these deviation is considerably smaller than OR and rather product of chance. The impact of three factor-interactions on the block prediction is rather a complex case, nonetheless, the quick look at Table 8 leads to the conclusion that Model 2 with OR+, T− and SZ- gives the closest match to the MLM. The obtained results are case specific, however, we conclude that in general, the accuracy in the prediction of total unstable volume is most sensitive to correct estimation of fracture orientations. Full-size image (3 K) Fig. 5. The impact of fracture orientations OR on measured quantity Y (deviation of the predicted total unstable volume from the most likely model for the CLAB 2 facility). Figure options Factorial analysis has a potential for more stringent sensitivity analysis and may be used for an optimal variable estimation for stochastic fracture models.