All empirical data used to build regression models and fuzzy logic model are indicated appendix 1. They are the real measurements of a TBM project with 153 rock samples randomly taken along a TBM boring tunnel of 7.5 km. The data consists of the measured rate of penetration (ROP) through rock measured properties of uniaxial compressive strength (UCS), Brazilian tensile strength (BTS), rock brittleness index (BI), distance between planes of weakness (DPW) and the alpha angle (Alpha) between the tunnel axis and the plane of weakness. The statistical description for all measurement data with their shape, mean, standard deviation, Kurtosis and skewness can provide some fundamental properties of their distribution and possible correlation to the regression model for ROP.

The statistical distributions of all variables are shown in Figure 1 for all 153 sampled points. The UCS (MPa) has a distribution curve with Mean of 153.6836, Standard Deviation of 22.08959, Kurtosis of −0.70096, and Skewness of 0.656379. The BTS (MPa) has a distribution curve with Mean of 9.545098, Standard Deviation of 0.864652, Kurtosis of 0.113056, and Skewness of −0.51696. The BI (kN/mm) has a distribution curve with Mean of 34.64052, Standard Deviation of 8.421163, Kurtosis of 1.234204, and Skewness of 1.424299. The DPW (m) has a distribution curve with Mean of 1.023203, Standard Deviation of 0.64239, Kurtosis of −1.43876, and Skewness of 0.164301. The Alpha (degree) has a distribution curve with Mean of 44.56863, Standard Deviation of 23.20497, Kurtosis of −1.03636, and Skewness of 0.026211.

Figure 1 Distributions of database vs. their normal curves.

From the distributions of database vs their normal curves in Figure 1, it is assumed that the ROP is a dependent variable and can be estimated from the other five independent variables of UCS, BI, DPW, and Alpha. Before taking any regression process, a stepwise test is conducted for all five independent variables of UCS, BTS, DPW, and Alpha on ROC to see the significance of their p-values supporting the assumption that they are independent variables and influenced the ROC performance.

Results of the stepwise test are shown in . The stepwise analyses accept UCS (MPa), BI (kN/mm), DPW (m), and Alpha (degree) in the regression model for prediction of ROC, all of their p-values are well significantly below 5% (p-value <0.05). The stepwise test rejects BTS (MPa) as an independent variable influencing ROC with a very large p-value of 0.7636. Conclusion of this stepwise test is that only four engineering rock properties (UCS, BI, DPW, and Alpha) have affected the ROP in significant levels. BTS has to be removed from the regression since it has no effect on the ROP prediction.

Table 1 Results of stepwise test.

shows the summary of main statistical values for all four (4) variables that will be used to build the regression models for ROP.

Table 2 Statistical summary of variables.

A graphic that show the relationship of ROP to all four rock engineering properties is shown in Figure 2.

Figure 2 Rock engineering properties on ROP.

The rock engineering properties show that the UCS and the BI will be the most reliable parameters for predicting the ROP since their statistical correlation coefficients to ROP are higher than 0.6. The DPW and the alpha angle are also important parameters to estimate ROC with their correlation coefficients to ROP are higher than 0.5. Therefore, in the next section, four (4) rock engineering properties of UCS, BI, DPW, and Alpha will be used to build regression models for ROC.

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