Prediction of TBM penetration rate for different surrounding rocks and cutter head diameters

Abstract

The penetration rate (PR) of tunnel boring machines (TBMs) is often used to evaluate tunneling performance and predict construction period costs. However, most penetration rate (PR) prediction models are based on a single specific project, which leads to poor universality of the models.Furthermore, the value of the cutter head speed set for the prediction of the construction period in the survey and planning stages depends on manual experience and lacks theoretical guidance. Therefore, based on a set of engineering data from TBM of different surrounding rocks and diameters, this study statistically analyzed the distribution law of the cutter head speed (N) and the relationship between the field penetration index (FPI), geological parameters (uniaxial compressive strength (UCS) and the integrity coefficient of the rock mass (Kv)), and penetration (P). The results show that the FPI is strongly correlated with the geological parameters and the P. The geological and tunnelling parameters are the main factors affecting the penetration rate (PR). On the basis of this, prediction models of the FPI and P and a calculation model of the cutter head speed were developed, and a prediction model of the PR was obtained. The accuracy and reliability of this model were verified and analyzed for the EHe (EH) project. The average prediction error was 15.15 %.

1. Introduction

Tunnel boring machines (TBMs) have been widely used in many tunnel construction projects, such as railways, highways, water conveyances, and hydropower development, due to their advantages in construction efficiency, tunnelling safety, and ecological protection. Recently, the analysis and prediction of tunnelling performance has become a hot topic in the TBM industry [1]. The penetration rate (PR) is one of the main indicators to evaluate the tunnelling performance of TBMs. Research on PR prediction can provide theoretical support for the evaluation of the performance of TBM tunnelling and the prediction of the construction period. Currently, research on TBM PR prediction focuses mainly on two aspects: on the one hand, theoretical analysis of indoor tests or numerical simulations; on the other hand, analysis of engineering data based on tunnelling parameters. Concerning theoretical analysis, Boyd [2] made extensive use of numerical simulations to perform a dimensional analysis of tunnel performance and proposed the concept of specific energy. PR can be obtained by calculating the driving power required by the cutter head. However, the specific energy cannot be calculated and analyzed at present; thus, prediction models have limited practical application. The CSM model primarily calculates the penetration of the TBM on the bar using force balance method through many linear cutting tests. However, the data obtained from an indoor cutting test ignore the characteristics of the rock, leading to relatively conservative prediction results. The NTNU model [3] is based on a large number of geological parameters and the realization of many laboratory tests; it considers many factors that affect the tunnelling parameters and uses various mechanical variables to predict the PR. However, this model is heavily based on previous engineering data. Therefore, it is necessary to reintroduce the revised data for a new tunnel project. Therefore, the NTNU model has significant limitations. Regarding engineering data analysis, Alber [4] proposed the concept of specific penetration based on field excavation data and established a relationship model between specific penetration and uniaxial compressive strength (UCS). However, this model only considers UCS and ignores the influence of other factors; therefore, its prediction accuracy is low. Du et al. [5] studied the field data of a super-long tunnel in Northeast China, and selected factors such as UCS and integrity coefficient (Kv) to establish a field penetration index (FPI) prediction model using a multivariate nonlinear fitting method. Zhou et al. [6] used a data mining clustering method to select geological and excavation parameters to build a prediction model for excavation speed. Gholaimi [7] selected UCS, structural plane conditions, and other influencing factors to develop a PR prediction model using a neural network. Mahdevari et al. [8] selected UCS, Brazilian tensile strength (BTS) and brittleness index (BI), among other parameters, to develop a PR prediction model using a support vector machine. Yagiz [9] and Armaghani et al. [10] used a particle swarm optimization algorithm to predict PR. Luo et al. [11] analyzed the relationship between penetration, cutter head thrust and torque, and developed a prediction model for PR with UCS, Kv, and cutter head thrust as input. Wu et al. [12] analyzed the engineering data from several tunnels and established a prediction formula for the FPI based on factors such as UCS, Kv, and tunnel diameter (D). Xu et al. [13] established penetration prediction formulas using curve fitting and multiple linear regression based on engineering data analysis. Feng et al. [14] predicted the cutter head thrust, rotating speed, and FPI of a TBM based on data from the Jilinyinsong project, and their results showed that the prediction effect was good. Sahinoglu, UK et al. [15] have emphasized that before starting a TBM project, performance analysis is a crucial stage necessary to inform the initial investment decision and project cost estimation. PR is also the main index of tunneling performance analysis. However, bad geology (rockburst, fault, sudden water inrush, etc.) will have a certain impact on it [16,17]. As a summary, we can conclude that most of the proposed PR prediction models are based on real tunnelling data from TBMs in a single specific project. Therefore, these models feature are poor in universality. Some real-time PR prediction models are based on data after excavation and cannot be used in the planning stage of the construction period. On the basis of the field data of several different TBM projects, this paper fully considers the influence of geology and TBM diameter on tunneling speed (PR), and establishes a more universal PR prediction model. The construction period can be predicted scientifically based on geological parameters, thus solving the problem that the prediction of construction period in the stage of engineering investigation and planning lacks a theoretical basis. Therefore, this research topic has an important engineering significance.

2. Factors affecting the PR of TBMs

Based on previous studies and a large number of engineering practices, it has been shown that UCS, Kv, F, and N are the main factors affecting the PR of TBMs. In particular, Kv can be obtained by testing the rock mass and rock wave velocity during exploration or construction. It can also be calculated using the number of joints of the surrounding rock exposed after excavation for subsequent verification of the measured Kv values. The Code for Geological Exploration of Water Resources and Hydropower Engineering (GB 50487-2008) states that the integrity of rock mass is closely related to the degree and stability of rock mass fragmentation; the degree of correlation is shown in Table 1. With an increase in Kv, the surrounding rock becomes more complete, and the stability of the rock mass improves.

Table 1.

Relationship between Kv and degree of rock integrity.

Kv	>0.75	0.75–0.55	0.55–0.35	0.35–0.15	<0.15
Integrity degree	Complete	Relatively complete	Relatively broken	Broken	Extremely broken

The magnitude of P is determined by the F and surrounding rock conditions. N is primarily regulated by TBM drivers according to geological conditions in combination with subjective experience. Under the condition of poor surrounding rock, the TBM driver actively controls the thrust and speed of the cutter head, thereby limiting penetration. Therefore, this study introduces an FPI to establish the relationship between tunnelling and geological parameters. This FPI fully reflects the difficulty of TBM tunnelling in a rock mass. The calculation formula is as follows:

$$FPI=\frac{F}{nP}$$ (1)

where n represents the number of disc cutters.

3. Prediction model of FPI based on geological parameters

3.1. Correlation analysis between FPI and geological parameters

Using data in the range of 82 + 606-77 + 548 in the Dahuofang (DHF) project, the influence of UCS and Kv on the FPI distribution was analyzed, and a three-dimensional scatter and colour map of the data was rendered. The results are shown in Fig. 1.

Fig. 1.

Three-dimensional scatter and map of FPI, UCS, and Kv.

Fig. 1 clearly shows the distribution law of the FPI based on UCS and Kv. In the case where both UCS and Kv are small, or UCS is small, but Kv is large, the FPI is low, indicating that the TBM can easily penetrate. However, in the range where UCS and Kv are both large, the distribution of the FPI is also evident. When UCS is in the range of 60–70 MPa and Kv is in the range of 0.6–0.7, FPI is in the range of 38–44 kN·r·mm⁻¹. The distribution of the FPI is annular when UCS is in the range of 60–80 MPa and Kv is in the range of 0.5–0.7. Interestingly, the FPI is strongly correlated with UCS and Kv, and its distribution law is clear.

3.2. FPI prediction model

The engineering data of the Zhuxishuiku (ZXSK), DHF and ErHe (EH) projects (including TBM7 and TBM8) were selected. The pile numbers were 21 + 891-17 + 731, 82 + 620-77 + 546, 172 + 578-169 + 816 and 173 + 863-177 + 410, respectively. The correlation between FPI, UCS, and Kv was analyzed. Their scatter plot was drawn and a fitting analysis was carried out. The results are shown in Fig. 2.

Fig. 2.

Correlation between FPI and UCS and Kv.

As shown in Fig. 2, the FPI of the four TBMs is highly correlated with UCS and Kv. The correlation coefficient R² is close to 0.9, showing an exponential relationship and a significant positive correlation. With an increase in UCS and Kv, the FPI also increases. As the rock becomes harder and more complete, it becomes more difficult for the TBM to penetrate, which is consistent with real working conditions.

According to this analysis, the FPI is clearly correlated with UCS and Kv. However, the correlation between other geological parameters and the FPI is not significant, or there is a collinear superposition effect with UCS and Kv, and its data are difficult to obtain in the exploration stage. Therefore, UCS and Kv were assumed to be independent variables and FPI was taken as the dependent variable for prediction and analysis. Furthermore, the FPI has a nonlinear relationship with UCS and Kv, which exhibited an exponential distribution. Therefore, FPI, UCS, and Kv were regressed using multiple nonlinear regressions.

We selected 400 groups of data from four TBMs from the ZXSK, EH (including TBM7 and TBM8) and DHF projects for multivariate non-linear regression fitting. We initially used the following optimal mathematical expression:

$$FPI=\mathrm{a}\cdot \exp \left(b\cdot UCS+c\cdot Kv+d\right)+e$$ (2)

where a, b, c, d and e are all mathematical regression parameters with initial values a = 0.328, b = 0.017, c = 2, d = 1, and e = 0.15; additional constraints were established as follows: a≥0, b ≥ 0, and c ≥ 0.

After 19 iterations, the sum of the squares of the residuals remained unchanged, and the resulting value was too small. The estimated values of the optimal regression parameters were a = 0.497, b = 0.014, c = 0.683, d = 2.190 and e = 0.836. The correlation coefficient R² of the fitting formula was 0.801. All in all, the empirical formula for multivariate nonlinear regression of the FPI is as follows:

$$FPI=0.497\cdot \exp \left(0.014UCS+0.683K\mathrm{v}+219\right)+8.367$$ (3)

which exhibits a high overall degree of fitting. The corresponding FPI value can be obtained according to the geological data in the preliminary investigation stage of a project.

4. Prediction model of P based on FPI

Previous studies [5] showed that P is strongly correlated with FPI. However, previous studies were based on fitting single engineering data, and prediction models lacked universality. Therefore, this study introduced TBM cluster data into the relationship model between P and FPI.

The pile number ranges of the DHF, EH (including TBM1, TBM2, and TBM3), Shenzhen Metro Line 6 (SZM-6), and Gaoligongshan (GLGS) projects are 82 + 620-77 + 546, 14 + 133-17 + 922, 57 + 797.40–54 + 415.19, 62 + 973-67 + 570, 0 + 242-2+757, and 226 + 126–225 + 784, respectively. Based on these tunnelling data, the relationship between P and FPI was determined by a fitting regression method, as shown in Fig. 3.

Fig. 3.

Correlation between P and FPI in different projects.

As shown in Fig. 3, there is a strong correlation between P and FPI: their correlation coefficient R² is close to or greater than 0.9. The decrease in P with an increase in the FPI indicates that the drivability of the rocks worsens. When the FPI is small, the degree of dispersion of P is large. With an increase in the FPI, the degree of fluctuation of P decreases. This is mainly because when the FPI is small, the rock can be better excavated, and the main driver can cause a large fluctuation in P by slightly adjusting F. However, when the FPI is large, the rock mass is difficult to tunnel, and the change in P is small if the main driver increases the thrust. Furthermore, it can be seen that the fitting curves of the seven projects are very close. To some extent, the proposed approach can effectively avoid the problem of poor universality of previous prediction models caused by the use of a single project. To this end, a unified P and FPI fitting formula was established using TBM data from multiple projects. Accordingly, 800 groups of excavation data from the seven projects were fitted and regressed as a whole, and a prediction model of P suitable for multiple projects was established, as shown in Fig. 4.

Fig. 4.

Fitting relationship between P and FPI based on multi-engineering.

Fig. 4 shows that by fitting and regressing the data of the aforementioned seven projects as a whole, the relationship between P and FPI is exponential and exhibits a strong correlation; furthermore, the correlation coefficient R² reaches 0.883. The data in Fig. 4 show the same trend as those in Fig. 3. The prediction model of P is expressed as follows:

$$P=15.202\exp \left(-0.023\text{FPI}\right)$$ (4)

5. Theoretical calculation of the cutter head speed for different surrounding rocks and diameters

The cutter head speed is related to the surrounding rock and the diameter of the cutter head. However, in the stages of survey, design or construction planning, there is no theoretical calculation method to set the cutter head speed for different surrounding rocks and diameters; thus, the cutter head speed is always determined by experience. Therefore, based on TBM engineering data for different diameters, this study proposes a theoretical calculation formula for the cutter head speed using a statistical method.

5.1. Theoretical calculation formula for the cutter head speed

When designing a TBM, it must be considered that a too high cutter head speed leads to high temperatures. This affects the performance and life of bearings and seals. Simultaneously, the side cutters are quickly damaged. Therefore, the allowable peripheral linear speed (v) of the cutter head is generally set to 2.5 m/s. Based on this principle, the cutter head speed can be determined as follows.

$$\mathrm{N}=\raisebox{1ex}{$v$}\!\left/ \!\raisebox{-1ex}{$\pi D$}\right.=\raisebox{1ex}{$2.5\times 60$}\!\left/ \!\raisebox{-1ex}{$\pi D$}\right.=\raisebox{1ex}{$47.7$}\!\left/ \!\raisebox{-1ex}{$D$}\right.$$ (5)

Furthermore, the relationship between the speed of the cutter head and its diameter can be expressed as follows:

$$\mathrm{N}=\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$D$}\right.$$ (6)

where x is the speed factor of the cutter head, generally set in the range 45–50, and the corresponding circumferential linear speed of the cutter head is 2.4–2.6 m/s.

In addition, rotation of the cutter head during TBM tunnelling causes a certain amount of centrifugal force on the rock slag. The slag tapping effect of the TBM is affected when the centrifugal force is excessively large. Hence, in the design of the rotational speed, it must be considered that the centrifugal acceleration generated by the rock debris should be smaller than its gravitational acceleration g. That is, the following expression must be satisfied:

$$\raisebox{1ex}{$v^2$}\!\left/ \!\raisebox{-1ex}{$r$}\right.\le g$$ (7)

where r is the radius of the cutter head.

The following expression can be derived from Equations (5), (7):

$$v=\raisebox{1ex}{$N\pi D$}\!\left/ \!\raisebox{-1ex}{$60$}\right.\le \sqrt{\raisebox{1ex}{$gD$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$$ (8)

The allowable cutter head speed that meets the design requirements is then obtained:

$$N\le 42.6D^{-0.5}$$ (9)

Equation (8) shows that the cutter head speed is often calculated according to the maximum cutter head speed in TBM designs. However, in practical engineering, the TBM operator adjusts the cutter head speed according to his personal experience and changes in the surrounding rock conditions; thus, it significantly departs from the maximum cutter head speed.

To obtain a reasonable formula for calculating the cutter head speed, the cutter head diameter and rated speed of some representative TBM projects in China, such as ZXSK, Nabang (NB), Chongqing Metro (CQM) (including Lines 5 and 6), Xinjiang EH (including Ⅶ and Ⅳ sections), Jilinyinsong (JLYS), DHF, Shaanxi Yinhanjiwei (YHJW), Gaoligongshan (GLGS) and Jinping (JP) Ⅱ hydropower station, were statistically analyzed. The distribution law of the rotation speed of the TBM cutter heads was studied for different diameters and the results are listed in Table 2.

Table 2.

Statistics of cutter head diameter and rotation speed of some representative TBM projects in China.

Project	TBM type	Diameter/(m)	Cutter head speed/(r/min)
ZXSK	open-type	4.03	0–12
NB	open-type	4.53	0–12
CQM-6	open-type	6.36	0–10.53
ABH	open-type	6.53	0–9.8
CQM-5	Single shield	6.83	0-0.54-11.9
EH-Ⅶ	open-type	7	0–10.9
EH-Ⅳ	open-type	7	0–10.9
JLYS	open-type	7.93	0–6.7
DHF	open-type	8.03	0-4.63-6.93
YHJW	open-type	8.05	0–6.87
GLGS	open-type	9.03	0-3.4-6.5
JP	open-type	12.4	0–4.5

Table 2 shows that with an increase in the diameter of the cutter head, the rated speed of the cutter head decreases. This indicates that the diameter has a certain influence on the speed of the TBM cutter head. Moreover, the influence of the surrounding rock conditions on the speed of the TBM can be evaluated using the speed distribution law of different surrounding rock types in different projects. The statistics of the regular distribution are shown in Fig. 5.

Fig. 5.

Speed distribution laws of TBM cutter head for different diameters in different projects.

Fig. 5 shows that the speed of the cutter head decreases with an increase in the diameter of the cutter head under the same surrounding rock conditions. The overall trend of the cutter head speed for different types of surrounding rock is that it also decreases with the deterioration of the surrounding rock. For the same diameter of the TBM and under the same surrounding rock grade, the cutter head rotation speed is similar, showing only a small change, as in EH-TBM1, EH-TBM2, EH-TBM10, and EH-TBM5. The average rotation speed of the cutter head varies in the range of 0∼1.4 r/min for the same surrounding rock grade. However, the rotating speed distribution of TBM cutter heads featuring different diameters varies considerably for the same surrounding rock grade. For example, in the NB project and EH project, the average rotating speed of the TBM cutter head changes by 41.15 % for the Class II surrounding rock. Note that the diameter and surrounding rock are the main factors that influence the speed of the cutter head. The average and maximum values of the rotating speeds of the eight TBM cutter heads for various surrounding rock grades resulting from statistical analysis are listed in Table 3.

Table 3.

Distribution statistics of the rotation speed of eight TBM cutter heads for different types of surrounding rock.

Project Class	Ⅱ		Ⅲa		Ⅲb		Ⅳ		Ⅴ
	Mean	Max	Mean	Max	Mean	Max	Mean	Max	Mean	Max
NB	11.3	11.2	10.6	10.6	10.5	10.5	9.1	9.1	8.5	8.6
ABH	5.9	5.7	5.2	5.2	5.2	5.2	4.6	4.5	–	–
EH-TBM1	6.5	6.6	6.0	6.0	5.6	5.6	3.8	3.7	–	–
EH-TBM2	6.5	6.5	6.0	6.0	5.5	5.5	4.1	4.0	–	–
EH-TBM10	7.0	6.8	6.3	6.4	5.9	5.7	3.9	4	2.8	2.7
EH-TBM5	6.6	6.6	6.3	6.2	5.8	5.7	5.4	5.2	3.5	3.5
DHF	–	–	5.9	6.0	5.9	6.1	5.4	5.4	3.9	3.7
GLGS	–	–	5.1	5.2	4.8	4.7	4.6	4.5	3.0	3.0

In summary, the diameter and surrounding rock are the main factors influencing the speed of the cutter head. Therefore, these factors are included in the calculation of the cutter head speed, which is reflected in the form of the correction coefficient. The diameter factor is represented by $k_1$, while the surrounding rock factor is represented by $k_2$. In this study, a TBM with D = 7.03 m was used as a reference. Therefore, the calculation formula for the cutter head speed is as follows:

$$N=k_1{k}_2{N}_0$$ (10)

where $N$ denotes the speed of the cutter head (rpm), $k_1$ is the diameter coefficient, $k_2$ denotes the surrounding rock coefficient, and $N_0$ is the speed of the cutter head for the reference diameter, expressed in r/min.

5.2. Determination of correction coefficient of cutter head diameter

According to research conducted by Du Yanliang et al., the influence of the diameter of the TBM on the cutter head speed is primarily related to the linear speed of the edge cutter. The relationship between the cutter head speed and the diameter of the TBM in the TBM design is as follows.

$$N=\frac{X}{D}$$ (11)

where X is the velocity coefficient, generally 45 or 50, and D is the diameter of the cutter head.

The calculation formula of the reference speed $N_0$ is as follows:

$$N_0=\frac{X}{D_0}$$ (12)

The influence of diameter on the speed of the TBM cutter head can be measured by calculating the ratio of the cutter head speed $N$ to the reference cutter head speed $N_0$, that is, the diameter coefficient $k_1$. The formula for calculating the diameter coefficient $k_1$, expressed in terms of the ratio of diameter $D_0$ to design diameter $D$, is as follows:

$$k_1=\frac{N}{N_0}=\frac{D_0}{D}$$ (13)

5.3. Selection of correction coefficient of surrounding rock

To obtain the cutter head rotation speed of the TBM reference diameter, the average rotation speeds of six 7.03 m diameter TBM cutter heads were statistically compared and analyzed in the Xinjiang EH project; the results are shown in Fig. 6.

Fig. 6.

Statistical analysis of cutter head speed for the TBM reference diameter (D = 7.03 m) in the EH project.

A comparative analysis of the average cutter head speeds shown in Fig. 6 reveals that the distribution of the TBM cutter head speed remains within a small range for the same surrounding rock grade. The maximum and minimum values of the cutter head speed are only 0.5 r/min for Class II and Class IIIb, 0.6 r/min for Class IIIa, 1.6 r/min for Class IV, and 1 r/min for Class V. The overall fluctuation range is small. It can be observed that different surrounding rocks have different effects on the cutter head speed. The reference rotational speed $N_0$ of the reference diameter $D_0$ (7.03 m) for different surrounding rock grades can be obtained by averaging the rotational speeds of six sets of 7.03-m-diameter TBM for the same surrounding rock grade. The results are summarized in Table 4.

Table 4.

Reference cutter head speed$N_0$for different surrounding rock grades.

$D_0$ (m)	$N_0$ (r/min)
	Ⅱ	Ⅲa	Ⅲb	Ⅳ	Ⅴ
7.03	6.6	6.0	5.5	4.5	3.0

It is necessary to study and analyse the law of cutter head speed for the same TBM diameter in different projects considering the influence of surrounding rock factors. Therefore, the tunnelling data of the TBM in the NB (D = 4.53 m) and Beijiangyinshui (BJYS) (D = 4.53 m) projects were selected, and the cutter head speed was statistically compared and analyzed. The results are shown in Fig. 7.

Fig. 7.

Statistical analysis of cutter head speed for the same TBM diameter in different projects.

As shown in Fig. 7, the diameters of the TBMs in the BJYS and NB projects were the same; however, the cutter head speeds were clearly different for the same surrounding rock grade. The main reason for this is the influence of surrounding rock factors. The physical and mechanical properties of the two projects differed from those of the surrounding rock lithology for the same surrounding rock grade. The UCS of the BJYS project is 60.15–203 MPa, the quartz content is 20%–45 % and the lithology is medium-coarse-grained biotite granite. The UCS of the Nabang project is 100–220 MPa, the quartz content is 25%–35 %, and the lithology is mixed gneiss. Without the influence of the diameter factor, the cutter head rotation speed of the TBM in the BJYS project for Class II surrounding rock is approximately 0.68 that of the NB project, and approximately 0.67 for Class III surrounding rock. The value of $k_2$ can be obtained from Equation (10) without the influence of the diameter factor, as expressed in Equation (14). The cutter head rotation speed of the TBM in the BJYS project for Class II surrounding rock is approximately 0.68 that of the NB project and is approximately 0.67 for Class III surrounding rock. That is, the value of the correction coefficient $k_2$ for the surrounding rock is 0.67 and 0.68, respectively.

$$k_2=\frac{N}{k_1{N}_0}$$ (14)

To cover more specifications of the TBM, the cutter head speeds of the TBM in the ABH (D = 6.53 m), DHF (D = 8.03 m) and GLGS (D = 9.03 m) projects were statistically analyzed, as shown in Table 2. First, $k_1$ was calculated according to Equation (7). Then, the distribution range of the surrounding rock correction coefficient $k_2$ was obtained according to · cutter head speed n and the cutter head speed N₀ for the reference diameter, as shown in Table 5.

Table 5.

Range of the correction coefficient $k_2$ of surrounding rock.

Class	$k_2$
Ⅱ	0.65–0.75
Ⅲa	0.75–0.85
Ⅲb	0.85–0.95
Ⅳ	0.95–1.05
Ⅴ	1.05–1.15

In summary, according to Equations (10), (13), and Table 4, Table 5, the rotation speed of the TBM cutter head can be calculated for different diameters and surrounding rock grades.

6. PR prediction model

PR represents the tunnelling distance of the TBM per unit of time, which can be expressed as the product of N and P:

$$\mathit{PR}=N\cdot P$$ (15)

Based on the above prediction model, the PR prediction model can be expressed as follows:

$$\mathit{PR}=15.202{\mathrm{k}}_1{k}_2{N}_0\cdot e^{-0.023FPI}$$ (16)

The application flow of the PR prediction model is shown in Fig. 8.

Fig. 8.

Application flow of the PR prediction model.

Based on geological parameters (UCS, Kv, etc.) obtained in the geological exploration or construction stages, the data are first cleaned to eliminate abnormal data. Second, the FPI value is obtained by entering the FPI model. Third, the FPI is substituted into the P model to obtain the value of P. Fourth, considering the influence of the diameter of the TBM and the surrounding rock, the diameter correction coefficients ($k_1$) and the surrounding rock ($k_2$) are set and substituted into the theoretical calculation formula for N, and the value of n is obtained. Finally, based on the above model, a prediction model for the tunnelling speed is obtained.

7. Model verification and analysis

7.1. Project verification of FPI prediction model

Based on geological data from the investigation stage, the FPI prediction model can obtain the corresponding FPI value. The prediction results for the FPI were compared and tested by selecting 30 datasets from the DHF, EH-TBM7 and EH-TBM8 projects. A comparison between the predictions of the FPI and the corresponding real values is shown in Fig. 9.

Fig. 9.

Comparison between predicted and real values of the FPI.

As shown in Fig. 9, the predicted value of the FPI is generally close to the real value, with a maximum prediction error of 23.5 % and an average prediction error of 7.84 %. A verification of project site data showed that the prediction accuracy of this model is good.

7.2. Project verification of penetration prediction model

Thirty groups of data from the EH-TBM7 and EH-TBM8 projects were selected to verify the prediction effect of the P model. A comparison of the predicted and the real values of p is shown in Fig. 10.

Fig. 10.

Comparison between predicted and real values of P.

As shown in Fig. 10, the real value of p is close to the predicted value, with a maximum prediction error of 25.5 % and an average prediction error of 11.91 %. The prediction errors of some data points in Fig. 10 are relatively large. Note that the FPI itself has certain prediction errors; these errors may be caused by artificial regulation of TBM drivers. However, in general, the prediction effect of P is good, which can better reflect the relationship between P and FPI, thus setting the foundation for subsequent prediction of PR.

7.3. Project verification of PR prediction model

The universality and precision of the PR prediction method were verified by selecting 40 data sets from the EH-TBM7 and EH-TBM8 projects in Xinjiang. The predicted results are listed in Table 6.

Table 6.

Statistics of PR prediction.

UCS	Kv	Predicted				Actual	RE (%)
		FPI	P	N	PR	PR
66	0.6	25.22	8.5	5.8	49.4	50	1.28
76	0.69	28.98	7.8	6.8	53.1	49	8.32
80	0.63	29.30	7.7	7.1	55.0	44	25.1
69	0.63	26.31	8.3	6.6	54.8	47	16.6
…	…	…	…	…	…	…	…
81	0.7	30.63	7.5	6.9	51.9	50	3.71
65	0.56	24.54	8.6	5.2	45.0	48	6.34
82	0.73	31.41	7.4	6.8	50.2	47	6.79
76	0.59	27.62	8.1	6.5	52.3	48	9.06

The data in Table 6 are plotted as a point chart in Fig. 11 to compare the predicted and real PR values.

Fig. 11.

Comparison between predicted and real values of PR.

As shown in Fig. 11 and Table 6, the predicted PR values are consistent with the corresponding real values. Nevertheless, the predicted values are slightly larger than the actual values overall. The main reason for this is that in real tunnelling processes of TBMs, the main driver takes the initiative to reduce the cutter head speed or thrust according to the influence of geological conditions and tools. The predicted value is based on previous tunneling parameters, leading to a predicted value that is slightly larger than the real value. However, through validation analysis, we found that the average prediction error was 15.15 %, the maximum prediction error was 28.7 %, and the minimum prediction error was only 0.14 %. Overall, this indicates that the PR prediction effect is good. Moreover, it can not only meet the cost prediction of the construction period in the early stage of TBM project, but also provide theoretical guidance for tool consumption prediction and tunneling performance evaluation.

8. Discussion

The above research systematically and comprehensively analyzes the influencing factors of TBM driving speed, the correlation analysis of geological parameters, penetration and FPI, the theoretical calculation of cutter head speed and the determination method of surrounding rock coefficient and diameter coefficient, but there are four points that need special explanation.

• (1)This paper is based on a number of TBM projects, with different working conditions of the test data sources. It can meet the conditions of considering the influence of various factors on the driving speed. Therefore, the relevant analysis of the field construction data under the conditions of different diameters and surrounding rocks in this paper makes the conclusion more practical and credible.
• (2)This paper studies and analyzes the correlation between geological parameters and FPI in view of the lack of theoretical guidance in the period planning of engineering survey and design. The influence of diameter and geological factors is considered at the same time, and the prediction model of PR is established, which is of great significance to time limit planning and cost prediction.
• (3)The influence of TBM diameter and surrounding rock grade on the prediction model of PR is considered in the study, but the factors such as groundwater, rockburst or large deformation have not been considered. The following research will bring more influencing factors into the model, so that the model can be continuously improved and adapted to different working conditions and has stronger universality.
• (4)The collected engineering data has some limitations due to the uncertainty of field working conditions. The errors in model prediction and verification are inevitable, but they are all within the allowable range, which shows that the technical route studied is feasible. As the data continues to improve, the prediction accuracy of the model will be guaranteed.

9. Conclusions

Based on data from several engineering sites, this study analyzed the main factors that influence the excavation speed to accurately estimate the cost of the construction period in the engineering investigation stage. Prediction models for FPI and P were constructed. A theoretical calculation method for the cutter head speed was developed for different diameters and surrounding rock grades. Furthermore, a relationship model was established between geological parameters and PR. The following results were obtained. (1) The cutter head speed is mainly related to the diameter of the TBM and surrounding rock. With an increase in the diameter of the TBM or deterioration of the surrounding rock, the rotational speed also decreases. Therefore, the diameter and surrounding rock correction coefficients were introduced and the value of the correction coefficient was determined. A theoretical calculation method was established for a cutter head speed suitable for different diameters and surrounding rock grades. (2) A fitting relationship was established between FPI and UCS and Kv, which were exponentially distributed. Through multivariate nonlinear regression fitting, an empirical model of the FPI was obtained and verified. The average prediction error was 7.84 %. The P and FPI data of seven projects were studied using regression analysis and a more universal prediction model of P was constructed. The average prediction error was 11.91 %, as verified by field data. (3) A prediction method for PR was developed and a prediction model for PR was established based on geological parameters. According to the geological parameters, the PR of TBM can be predicted, and then the construction period can be estimated. Through engineering verification and analysis, the maximum prediction error was found to be 28.7 %, the minimum prediction error was only 0.14 %, and the average PR prediction error was 15.15 %.

In this study a more universal PR prediction model was established based on field data from several projects and considering the influence of geology and TBM diameter on PR. In addition to providing a cost prediction of the construction period in the early stage of the TBM project, the proposed approach also provides theoretical guidance for tool consumption prediction and evaluation of tunnel performance.

Data availability statement

Data will be made available on request.

CRediT authorship contribution statement

Yalei Yang: Writing – original draft. Du Lijie: Writing – review & editing. Tang Rong: Data curation. Wei Fei: Resources, Conceptualization. Zhang Huilan: Methodology, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.