# Regression Models and Fuzzy Logic Prediction of TBM Penetration Rate

Vu Trieu Minh, Dmitri Katushin, Maksim Antonov and Renno Veinthal
From the journal Open Engineering

# Abstract

This paper presents statistical analyses of rock engineering properties and the measured penetration rate of tunnel boring machine (TBM) based on the data of an actual project. The aim of this study is to analyze the influence of rock engineering properties including uniaxial compressive strength (UCS), Brazilian tensile strength (BTS), rock brittleness index (BI), the distance between planes of weakness (DPW), and the alpha angle (Alpha) between the tunnel axis and the planes of weakness on the TBM rate of penetration (ROP). Four (4) statistical regression models (two linear and two nonlinear) are built to predict the ROP of TBM. Finally a fuzzy logic model is developed as an alternative method and compared to the four statistical regression models. Results show that the fuzzy logic model provides better estimations and can be applied to predict the TBM performance. The R-squared value (R2) of the fuzzy logic model scores the highest value of 0.714 over the second runner-up of 0.667 from the multiple variables nonlinear regression model.

## 1 Introduction

As part of a research project on new technologies for tunneling and underground works, the research group in Tallinn University of Technology has conducted several researches on development of new materials for TBM cutting bits and testing devices for impact-abrasive carbide wear performances. In this study, we would like to present some new empirical models to predict the TBM penetration rate using the actual measurement data from a real TBM project. Four regressions (two linear and two nonlinear models) are built using the latest Matlab 2017 version. The regression results show that these models can be used precisely in regulating the TBM penetration rates since all their coefficients are highly statistically significant with the R-square (R2) greater than 62%. A new fuzzy logic model is also developed and compared to these conventional regression models. The fuzzy logic model shows the best fit to the empirical data, achieving the highest R-square (R2) at 71.4%. Thus, fuzzy logic model becomes the best alternative to predict the TBM penetration rate.

Prediction of tunnel boring machine (TBM) performance is always an important issue since the fast development of new TBMs with high penetration rates for all types of rocks, soils, clays, and in different geological conditions. This paper serves as a contribution providing fundamental knowledge and theoretical methodologies to setup realistic models for prediction of TBM performance based on the environmental conditions and rock engineering properties.

Environmental conditions and rock engineering properties have strong effects on the TBM performance. Accurate predictions of TBM rate of penetration (ROP) based on the rock engineering properties will determine the efficiency of the project, determine the success of the project and reduce both the delay and failure costs. Typical statistical analyses on rock engineering properties for TBM penetration rate are presented in . This reference showed that the main factors of rock engineering properties influencing the prediction of ROP are the compressive strength, the tensile strength of rocks, the frequency and the orientation of the rock joints.

Various regression methods for TBM performance prediction models are reviewed and discussed in . This reference discussed the development of recent regression models to improve the prediction of ROP. The primary factors of rock engineering properties used for the ROP prediction are the uniaxial compressive strength (UCS), the rock brittleness index (BI), the rock types, and the TBM cutting speeds. This reference recommended that the use of combination models could assure a better level of confidence in estimation of TBM performance.

A new statistical regression model combined the semi theoretical model by Colorado School of Mines and the empirical model by Norwegian University of Science and Technology in Trondheim is introduced in . The study analyzes the strong and weak points of the two approaches and introduces a more accurate model by modifying the two models. Analyses of the results show that there are strong relationships of UCS, BI, the distance between planes of weakness (DPW), and the alpha angle (Alpha) between the tunnel axis and the planes of weakness on the TBM performance.

Recent studies on prediction of TBM performance are related to advanced statistical methods. A new method for TBM penetration estimation used Monte-Carlo simulation is presented in . In this study, the Monte-Carlo simulation is used to estimate the TBM performance based on the rock engineering properties of UCS, Brazilian Tensile strength (BTS), BI, spacing and orientation of discontinuities, and measured TBM ROP. The Monte-Carlo simulation indicates that the hardness and density of rocks provide the most and the least influence on the ROP, respectively.

A review of various artificial intelligence methods for prediction of TBM performance is presented in . The paper briefly discusses modern methods on principal component analysis (PCA), artificial intelligence (AI) based methods including artificial neural networks (ANN), support vector regression (SVR), adaptive neuro fuzzy inference system (ANFIS) to obtain the better predictions for TBM performance. The study concludes that the SVR model is better than the ANN and ANFIS models and recommends the use of SVR model for estimation of TBM penetration rate. A new adaptive neuro fuzzy inference model based on fuzzy c-means clustering algorithm (ANFIS-FCM) is introduced in . This model uses the robust artificial intelligence algorithms of ANFIS-FCM for estimation of TBM performance. Another fuzzy logic model to predict the TBM performance is presented in . The paper is based on the experience, the existed database, and used fuzzy logic model to predict the specific energy requiring for the TBM performance.

As stated earlier that our paper will introduce fundamental statistical analyses on environmental factors that influence the TBM performance. Two linear and two nonlinear regression models are developed. And finally a new fuzzy logic model is established and compared to the four regression models. This study uses the empirical data in  with ROP, BTS, BI, DPW, Alpha angle, and the measured ROP at 153 separated samples inside the tunnel station. This completed data set is shown in appendix 1. Fuzzy logic algorithms and calculations are based on  and . The stochastic models and distributions are referred to as in  and . The contents of this paper are as follows: Section 2 introduces the database distribution analyses; Section 3 develops four statistical regression models; Section 4 presents the development of fuzzy logic model and the performances comparison; Section 5 is the conclusion.

## 2 Analyses of database distributions

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 Table 1. 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.

VariableStandard ErrorStatusp-value
UCS (MPa)0.011‘In’0.0012
BI (kN/mm)0.0028‘In’8.7015e-21
DPW (m)0.0286‘In’4.0284e-12
Alpha (degree)7.9897e-04‘In’2.8022e-10
BTS (MPa)0.0221‘Out’0.7636

Table 2 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.

UCS(Mpa)BI(kN/mm)DPW(m)Alpha(deg)ROP(m/h)
Mean149.885Mean34.64052Mean1.023203Mean44.56863Mean2.046928
Std Error1.785838Std Error0.680811Std Error0.051934Std Error1.876011Std Error0.029031
Median141.4Median31Median0.8Median45Median2.03
Mode136.2Mode30Mode1.6Mode21Mode1.87
StdDvt22.08959StdDvt8.421163StdDvt0.64239StdDvt23.20497StdDvt0.359095
Variance487.9501Variance70.91598Variance0.412665Variance538.4706Variance0.128949
Kurtosis−0.70096Kurtosis1.234204Kurtosis−1.43876Kurtosis−1.03636Kurtosis0.534826
Skewness0.656379Skewness1.424299Skewness0.164301Skewness0.026211Skewness0.473351
Range81.4Range33Range1.95Range87Range1.8
Minimum118.3Minimum25Minimum0.05Minimum2Minimum1.27
Maximum199.7Maximum58Maximum2Maximum89Maximum3.07
Sum22932.4Sum5300Sum156.55Sum6819Sum313.18
Count153Count153Count153Count153Count153
At (95%)3.528269At(95.0%)1.345073At(95.0%)0.102606At(95.0%)3.706423At(95.0%)0.057357

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.

## 3 Regression Models for ROC

In order to establish the exact relationship between the rock engineering properties (UCS, BI, DPW, Alpha) and the actual measured ROP, four different regression models are built using the latest Matlab 2017 version:

Model 1:

ROC(1)=β0(1)+β1(1)UCS+β2(1)BI+β3(1)DPW+β1(1)Alpha,

Model 2:

ROC(2)=β0(2)+β1(2)UCS+β2(2)BI+β3(2)DPW+β4(2)log(Alpha),

Model 3:

ROC(3)=β0(3)+β1(3)UCS+β2(3)BI+β3(3)DPW+β4(3)Alphaβ5(3),

Model 4:

ROC(4)=β0(4)+β1(4)UCS+β2(4)BI+β3(4)DPWβ4(4)+β5(4)Alphaβ6(4),

where βk(i) indicates the coefficient (k) in the model (i) calculated from the different regression models.

### 3.1 Linear regression model (LRM)

The linear regression model (Model 1) for ROC is run in Matlab and the results are shown in Table 3

### Table 3

Linear regression model.

 R Square 0.621472 Standard Error 0.252897 Observations 153 Coeflcients Intercept 1.47937 UCS (MPa) −0.00347519 BI (kN/mm) 0.0308452 DPW (m) −0.216151 Alpha (degree) 0.0054099

From results in Table 3, a linear equation for prediction of ROP is presented in (1).

ROC=1.479370.00347519UCS+0.0308452BI0.216151DPW+0.0054099Alpha(1)

And a graphic of measured ROC and estimation of ROC from (1) is shown in Figure 3. ### Figure 3

Linear regression model.

In this model, the R-square is 0.621472 and the Standard Error is 0.223897. Next, another linear regression model will be built for the logarithmic values of Alpha.

### 3.2 Linear regression model with log(Alpha)

In this model (Model 2), the values of Alpha (degree) are transformed in logarithmic form and the results of this model are shown in Table 4.

### Table 4

Linear regression model with log(Alpha).

 R Square 0.645865 Standard Error 0.225982 Observations 153 Coefficients Intercept 0.96712 UCS (MPa) −0.0036577 BI (kN/mm) 0.029169 DPW (m) −0.21697 Log(Alpha) 0.19086

The linear regression equation with log(Alpha) for estimation of ROC is indicated in (2):

ROC=0.967120.0036577UCS+0.029169BI0.21697DPW+0.19086log(Alpha)(2)

The graphic of measured ROC and estimation of ROC from (2) is shown in Figure 4. ### Figure 4

Linear regression model with log(Alpha).

This model has the R-square value of 0.645865 and really higher than the previous model.Next, a new nonlinear regression model will be built for the exponential values of Alpha.

### 3.3 Nonlinear regression model with exponential Alpha (NLRM1)

In this model (Model 3), Alpha (degree) will be transformed in nonlinear exponential form. Results of this model are shown in Table 5.

### Table 5

Nonlinear regression model with exponential Alpha.

 R Square 0.664865 Standard Error 0.212584 Observations 153 Coefficients Intercept −20.072 UCS (MPa) −0.0034431 BI (kN/mm) 0.029011 DPW (m) −0.21886 b5(b5*Alpha^b6) 21.178 b6(b5*Alpha^b6) 0.0087451

The nonlinear regression equation with exponential Alpha for ROC is expressed in (3):

ROC=20.0720.0034431UCS+0.029011BI0.21886DPW+21.178Alpha0.0087451(3)

The graphic of measured ROC and estimation of ROC from (3) is shown in Figure 5. ### Figure 5

Nonlinear regression model with exponential Alpha.

This model has the R-square value of 0,664865, a little bit higher than the previous linear model. Next, another new nonlinear regression model will be developed for both exponential values of DPW and Alpha.

### 3.4 Nonlinear regression model with both exponential DPW & Alpha (NLRM2)

In this model (Model 4), both DPW and Alpha will be transformed in exponential forms. Results of this model are shown in Table 6.

### Table 6

Nonlinear regression model with exponential DPW and Alpha.

 R Square 0.667456 Standard Error 0.212121 Observations 153 Coefficients Intercept −12.661 UCS (MPa) −0.0035507 BI (kN/mm) 0.029086 b4(b4*DPW^b5) −0.32839 b5(b4*DPW^b5) 0.64199 b6(b6*Alpha^b7) 13.878 b7(b6*Alpha^b7) 0.01312

The nonlinear regression equation with both exponential DPWand Alpha for estimation of ROC is presented in (4)

ROC=12.6610.0035507UCS+0.029086BI0.32839DPW0.64199+0.01312Alpha0.013121(4)

The graphic of measured ROC and estimation of ROC for this model is shown in Figure 6. ### Figure 6

Nonlinear regression model with DPW & Alpha.

Similarly, this model has the R-square value of 0.667456, a little bit higher than the previous nonlinear model. All four regression models provide very good fitness coefficients. Model 4 achieves the highest R-square value of 0.667456 and becomes the best regression model to predict ROC. Next, a fuzzy logic model will be built to predict the ROP performance and compared to the four statistical regression models.

## 4 Fuzzy Logic Model and Comparison

Motivation of building up a fuzzy logic model is the ability of an intelligent model that can deal with imprecise and uncertain inputs. Then, the use of fuzzy logic can avoid the complex of deterministic formulas and mathematic modellings. There are huge of successful applications of fuzzy logic in industries. For instance, most of anti-lock braking systems (ABS) in automotive engineering are used fuzzy logic algorithms. Algorithms of fuzzy logic model in this paper are referred to in  with an application of fuzzy logic for controlling clutch engagement and vibration reduction, and in  with the use of two fuzzy logic methods of Mamdani and Sugeno for a nonlinear and complicated system.

This fuzzy logic model is set up in Matlab version 2016 as shown in Figure 7. The inputs for this fuzzy logic model are the 153 discrete samples of UCS (MPa), BI (kN/mm), DPW (m), and Alpha (degree) in appendix 1. The output of this fuzzy logic model is the ROP prediction. ### Figure 7

Fuzzy logic model in Matlab.

Four discrete inputs are designed with membership functions of five (5) levels of very low (VL), low (L), medium (M), high (H) and very high (VH). The output of ROP is designed with seven (7) levels of very very low (VVL), very low (VL), low (L), medium (M), high (H), very high (VH) and very very high (VVH). The design of fuzzy logic model in Matlab Simulink is shown in figure 8. ### Figure 8

Design of fuzzy logic model.

Table 7 shows the first 20 fuzzy logic rules among the total of 321 fuzzy logic rules designed for this model.

### Table 7

Fuzzy logic rules.

 1 IF UCS is VL and BI is H and DPW is H and Alpha is M THEN ROP is M 2 IF UCS is VL and BI is H and DPW is M and Alpha is H THEN ROP is H 3 IF UCS is L and BI is VL and DPW is L and Alpha is VH THEN ROP is VH 4 IF UCS is L and BI is VL and DPW is H and Alpha is H THEN ROP is VL 5 IF UCS is M and BI is L and DPW is VH and Alpha is M THEN ROP is L 6 IF UCS is M and BI is L and DPW is VL and Alpha is L THEN ROP is M 7 IF UCS is H and BI is H and DPW is L and Alpha is VL THEN ROP is H 8 IF UCS is H and BI is M and DPW is M and Alpha is L THEN ROP is VH 9 IF UCS is VH and BI is H and DPW is H and Alpha is M THEN ROP is H 10 IF UCS is VH and BI is H and DPW is VH and Alpha is VH THEN ROP is M 11 IF UCS is H and BI is M and DPW is H and Alpha is H THEN ROP is L 12 IF UCS is H and BI is M and DPW is M and Alpha is M THEN ROP is VL 13 IF UCS is M and BI is H and DPW is L and Alpha is L THEN ROP is L 14 IF UCS is M and BI is H and DPW is VL and Alpha is VL THEN ROP is M 15 IF UCS is L and BI is VH and DPW is H and Alpha is L THEN ROP is H 16 IF UCS is M and BI is L and DPW is M and Alpha is H THEN ROP is M 17 IF UCS is M and BI is L and DPW is M and Alpha is H THEN ROP is L 18 IF UCS is M and BI is L and DPW is VL and Alpha is M THEN ROP is M 19 IF UCS is H and BI is L and DPW is VL and Alpha is VH THEN ROP is H 20 IF UCS is VH and BI is VL and DPW is M and Alpha is M THEN ROP is L

The viewer of fuzzy logic rules for four (4) inputs and output is shown in Figure 9. ### Figure 9

Fuzzy logic rules viewer.

The fuzzy logic surface is shown in Figure 10. ### Figure 10

Fuzzy logic surface.

Results of the fuzzy prediction of ROP are shown in Table 8.

### Table 8

Results of fuzzy logic ROP prediction.

 R Square 0.714365 Standard Error 0.204694 Observations 153

Graphic of fuzzy logic model prediction is shown in Figure 11. ### Figure 11

Fuzzy logic model estimation.

Performances comparison of four (4) stastitical regression models vs. the fuzzy logic model is shown in Table 9.

### Table 9

Comparison of regression models vs. fuzzy logic model.

VariablesLRMLRM lg(Alpha)NLRM1NLRM2Fuzzy Logic
R Square0.6214720.6458650.6648650.6674560.714365
Standard Error0.2528970.2259820.2125840.2121210.204694
Observations153153153153153

From Table 9, fuzzy logic model provides the best performance with R square value of 0.714 compared to the second runner-up of 0.667 in the nonlinear regression model with both exponential DPW & Alpha (NLRM2).

Finally a comparison of the best performances of the nonlinear regression model with both exponential DPW & Alpha (NLRM2) vs. fuzzy logic estimation is shown in Figure 12. It is clearly that the fuzzy logic model provides better prediction of ROP. ### Figure 12

Comparison of NLRM2 vs Fuzzy logic estimation.

## 5 Conclusions

The purpose of this study is to develop reliable models to predict the TBM penetration rate based on empirical measurement data. Four regression models are built including a linear model (model 1), a logarithmic model (model 2), an exponential model (model 3), and a two variables exponential model (model 4). The regression results prove that the exponential regression models provide better fitness to predict ROP.

Lately, a fuzzy logic model is also developed and compared to the above classical regression models. Then, the fuzzy logic model achieves the best fitness to predict ROP. It is suggested that the use of fuzzy logic as well as other artificial intelligences can be used as also very good alternatives to predict ROP. In this study, the fuzzy logic model has provided the best accurate output predictions dealing with the uncertain inputs. In the next stage of this study, some other artificial intelligence methods as well as the use of neural networks will be tried to discover better alternatives to predict ROP in the future.

# Acknowledgement

The research leading to these results has received funding via the project NeTTUN from the European Union’s Seventh Framework Programme for Research, Technological Development and Demonstration on (FP7 2007-2013) under Grant Agreement 280712 (VFP566).

1. Conflicts of interest: The authors would like to confirm that there is no conflict of interests associated with this publication and there is no financial fund for this work that can affect the research outcomes.

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