
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

73025
10.1038/s41598-024-73025-3
Article
Enhancing ground loss rate prediction in soft-soil shield tunneling: a synergistic approach of peck back analysis and eXtreme Gradient Boosting and bayesian optimization
Ding Zhi 13
Tu Wenrong Twy8718@163.com

123
Li Xinjia 1
1 https://ror.org/01wck0s05 Department of Civil Engineering, Hangzhou City University, Huzhou Street 51#, Hangzhou, 310015 Zhejiang China
2 https://ror.org/00a2xv884 grid.13402.34 0000 0004 1759 700X College of Civil Engineering and Architecture, Zhejiang University, Hangzhou, 310058 China
3 Key Laboratory of Safe Construction and Intelligent Maintenance for Urban Shield Tunnels of Zhejiang Province, Hangzhou, 310015 Zhejiang China
20 9 2024
20 9 2024
2024
14 219355 7 2024
12 9 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
This study investigates the prediction of ground loss rate during soft-soil shield tunneling using Peck’s back analysis method and XGBoost model. Bayesian optimization is employed to determine optimal hyperparameters, ensuring comprehensive and efficient model tuning. The XGBoost model is compared with Random Forest (RF) and Support Vector Machine (SVM) models to benchmark its performance. The results demonstrate the superior accuracy and robustness of the XGBoost model. Also, the results show that the soil properties and the grouting factors of the excavation face affect the duration of the instantaneous settlement of the ground surface. There is a specific correlation between the depth-to-diameter ratio, the coefficient of variation in the advancing speed of the shield machine, the maximum surface subsidence, and the ground loss rate. The prediction model of the ground loss rate based on the combined approach of Peck back analysis and eXtreme Gradient Boosting and Bayesian optimization has high reliability in soft-soil layers, and this method can provide a specific reference for predicting construction risk in related projects.

Keywords

Shield tunnel
Ground loss
Soft soil
Peck back analysis method
eXtreme Gradient Boosting
Bayesian optimization
Subject terms

Civil engineering
Structural geology
Key Research and Development Program of Zhejiang2023C03182 issue-copyright-statement© Springer Nature Limited 2024
==== Body
pmcIntroduction

Since 1980, shield tunneling has been widely used in China1. Especially in recent years, shield tunneling has played an important role in vigorously promoting urban rail transit construction, highway construction, and urban drainage system renovation. As of now, China has become one of the main countries in the construction of shield tunnels worldwide, and most domestic subways are constructed using shield tunneling methods2. Among them, Hangzhou Metro, as a new first tier city with developed economy in Southeast China, has witnessed particularly rapid subway construction. By the end of 2022, the operating mileage of Hangzhou’s subway has exceeded 500 km, the scale of construction in progress has continued to grow. Data shows that in 2022, Hangzhou’s urban rail transit completed construction investment of over 30 billion yuan, ranking fourth in the country, and the mileage of subway under construction reached 220 km. As shown in Fig. 1, Mileage Statistics of China’s Subway Operation Lines.

Fig. 1 Mileage statistics of China’s subway operation lines.

Behind the high-speed construction of the subway, there are also huge engineering risks hidden. In the process of shield tunneling, due to urban planning restrictions, new tunnels inevitably cross the existing subway tunnels, bridges, underground pipelines, and urban building foundations, thereby posing significant potential safety hazards3. Therefore, shield tunneling must ensure the safety and stability of surrounding buildings. The impact of shield tunneling on adjacent buildings is primarily reflected in the formation disturbance caused during construction, resulting in surface settlement. Ground loss is one of the crucial factors causing surface subsidence and is often expressed by the ground loss rate in engineering. The ground loss rate refers to the ratio of the unit soil loss volume to the actual excavated soil volume, which can comprehensively reflect the impacts of the construction method, formation conditions, support time, construction management quality, and the technical level on the ground settlement4. Therefore, for shield tunneling projects, conducting comprehensive analysis and prediction of ground loss rate can help the construction party better understand geological conditions and existing risks, thus taking practical and effective measures to avoid accidents and ensure the smooth implementation of the project. For example, Huo et al.5 and Zhou et al.6 have conducted the numerical simulation study on surface settlement caused by tunnel construction. Wang et al.7 have conducted research on theoretical and empirical formulas and calculation methods for surface settlement caused by tunnel excavation.

Peck’s formula is a commonly used ground loss rate prediction method in traditional engineering8. Some scholars have further studied the composition of ground loss and summarized the corresponding calculation formula9–11. Zhu et al.12,13 and Wu et al.14,15 collected the measured settlement data during the construction of domestic subway tunnels, obtained the ground loss rate caused by different construction methods using the back analysis method, and summarized the change rules of parameters such as the maximum surface settlement, the width of the settlement trough, and the ground loss rate at various relative depth-to-diameter ratios (H/D), formation conditions, and construction methods. The practical value of the ground loss rate derived from Peck’s back analysis method based on measured data is accurate and widely used15. However, this method needs sufficient samples and field-measured data, and data processing has particular complexity for different strata and engineering conditions. As such, the rise of machine learning has offered new ideas for predicting the ground loss rate. Like empirical formulas, machine learning does not need a rigorous theoretical basis and only requires establishing the relationship between inputs and outputs through an algorithm based on a large amount of data. Shi et al.16 used the backpropagation neural network (BPNN) to predict the surface deformation caused by excavating a shallow, buried soft soil tunnel in Brasilia and found that the error was reduced by half compared to the general model at the time. Since then, many scholars have introduced machine learning algorithms to predict surface deformation caused by tunneling17–21. Zhang et al.22 established a model for predicting the settlement trough using construction parameters, geological parameters, and geometric parameters of the tunnel as inputs based on random forest (RF) and long short-term memory algorithms. Dindarloo et al.23 employed the decision tree model to estimate the range of ground settlement caused by shield tunneling based on parameters such as the burial depth of the tunnel and the excavation diameter.

Based on the existing research, this paper collects data on the construction of several shielded soft-soil sections in the Hangzhou area, calculates the ground loss rate caused by shield tunneling using Peck’s back analysis method, and obtains the influences of geological parameters and the geometric parameters of the tunnel on the ground loss rate through data analysis. An innovative method to estimate the ground loss rate, combining the surface settlement value, detailed geological parameters, the geometric parameters of the tunnel, and construction parameters, is developed based on eXtreme Gradient Boosting (XGBoost).

Methodology

In theoretical research, ground loss rate is generally used to characterize the geological losses caused by shield tunneling construction, and geological losses are the main factor causing ground settlement. Therefore, the study of ground loss rate is of great significance for controlling surface subsidence deformation. The existing research on ground loss rate mainly focuses on summarizing empirical formulas for deformation. Most empirical formulas have specific applicable conditions, and some parameter values in the formulas still depend on empirical values. For soft soil, relying on the PECK back analysis method based on measured data to obtain empirical values of ground loss rate is a relatively accurate and widely used method. However, in practical applications, the PECK formula requires the collection of settlement data for the entire cross-section, and the calculation process requires linear fitting and other operations. If ground subsidence cannot be well fitted with PECK Gaussian curve, it is often difficult to apply.

On the basis of existing research, this paper collected shield tunneling construction data for several soft soil shield tunneling sections in Hangzhou area, and calculated the ground loss rate caused by shield tunneling using the PECK back analysis method. By analyzing the data, the influence of geological parameters, tunnel geometry parameters, etc. on the ground loss rate can be obtained. In order to predict the ground loss rate more efficiently, an innovative approach combining surface settlement values, detailed geological parameters, tunnel geometric parameters, and construction parameters was proposed. This approach involved developing an XGBoost-based prediction model for ground loss rate. The flowchart outlining the proposed analytical method can be found in Fig. 2.

Fig. 2 A flowchart of the proposed analytical method.

The three key components of the model in this research are data preprocessing, XGBOOST model training, and XGBOOST model testing. First, the initial dataset is cleaned completely in preprocessing, filling in missing values, and handling outliers. Next, the processed data are divided into initial training and testing sets in a 4:1 ratio. As shown in the Fig. 3. In order to improve the XGBoost model parameters, the training set is split into a training set and validation set after a set of hyperparameters is chosen by a random grid search. Finally, the end of the training phase is determined by metrics such as mean squared error, root mean squared error (RMSE), and mean absolute error (MAE), etc., until the search is completed.

Fig. 3 Structure of XGBOOST model.

In urban subway construction, the magnitude of ground loss rate caused by shield tunneling is often difficult to accurately predict due to numerous factors. In response to this issue, this study proposed a Bayesian optimized eXtreme Gradient Boosting (XGBoost) prediction model. Based on engineering examples, the model input took into account geological parameters, construction parameters, and tunnel geometric parameters (influencing factors). The importance of each feature was analyzed through feature selection, while reducing the dimensionality of the input parameters. The optimal hyperparameters were determined using Bayesian optimization algorithm, and the predicted results were compared with manually tuned XGBoost, Random Forest (RF), and Support Vector Machines (SVM) models. The methodological framework is shown in Fig. 4.

Fig. 4 Bayesian optimized eXtreme Gradient Boosting.

Statistical analysis of ground loss rate in Hangzhou soft-soil area

Soft soil, composed mainly of clay and silt, is characterized by high compressibility, low shear strength, high moisture content, time-dependent behavior, and sensitivity to stress changes. These properties make soft soil particularly prone to ground settlement during shield tunnel construction. The excavation process alters stress distribution in the soil, leading to deformation and settlement. Maintaining face stability is challenging due to the soil’s low shear strength and compressibility, often resulting in face deformation. Understanding these factors is crucial for accurately predicting and managing ground settlement in such conditions.

Back analysis method based on Peck’s formula

As early as 1969, Peck analyzed several engineering cases of tunnel construction based on actual projects and found that the surface settlement curve above the tunnel cross section could be well fitted with a Gaussian curve, thereby proposing Peck’s famous formula as follows:1 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S\left( x \right)={S_{{\text{max}}}}\mathcal{exp}\left( { - \frac{{{x^2}}}{{2{i^2}}}} \right)$$\end{document}

2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${S_{\mathcal{max}}}=\frac{{{V_{\mathcal{loss}}}}}{{i\sqrt {2\mathcal{\pi }} }}$$\end{document}

3 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V_{\mathcal{loss}}}=\mathcal{\pi }{R^2}\eta$$\end{document}

where S(x) represents the ground settlement at point x; x is the horizontal distance from the calculation point to the tunnel center; Smax indicates the maximum ground settlement above the center point of the tunnel; i denotes the horizontal distance from the reverse bend point of the Gaussian settlement curve to the center of the tunnel, generally referred to as the width of the ground settlement trough; Vloss is the loss of soil mass per unit length; η stands for the ground loss rate.

The following measures are usually taken to obtain the deformation in Eq. (1) in order to evaluate whether the settlement of the surface cross section conforms to the Gaussian curve:4 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{{S\left( x \right)}}{{{S_{{\text{max}}}}}}={\text{exp}}\left( { - \frac{{{x^2}}}{{2{i^2}}}} \right)$$\end{document}

Simultaneously taking the logarithm of both sides of Eq. (4) yields the following:5 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{ln}}\frac{{S\left( x \right)}}{{{S_{{\text{max}}}}}}= - \frac{1}{{2{i^2}}}{x^2}$$\end{document}

Equation (5) demonstrates that there should be a linear relationship between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{ln}}\frac{{S\left( x \right)}}{{{S_{{\text{max}}}}}}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${x^2}$$\end{document}, with a slope of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$- \frac{1}{{2{i^2}}}$$\end{document}. Therefore, drawing the regression curve of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{ln}}\frac{{S\left( x \right)}}{{{S_{{\text{max}}}}}}$$\end{document} versus \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${x^2}$$\end{document} on the Cartesian coordinate system can determine i.

Combining Eqs. (2) and (3) yields η as follows:6 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta =\frac{{{S_{{\text{max}}}}i\sqrt {2{\text{\varvec{\uppi}}}} }}{{{\text{\varvec{\uppi}}}{R^2}}}$$\end{document}

The above method for calculating the ground loss rate is called the back analysis method based on Peck’s formula. The back analysis method enjoys the advantage of high accuracy and is widely used in practical engineering; however, there is still a relatively complex problem when calculating the ground loss rate on a large scale, that is, Peck’s formula is applied on the premise that the conditions are undrained. Nevertheless, in the process of soft-soil tunnel excavation, it is always difficult to define the turning point between the “drainage stage” and “undrained stage”. At present, many experts and scholars have conducted much research to distinguish the drainage stage from the undrained stage, that is, the soil consolidation settlement stage from the construction settlement stage24–27. In combination with the research results of Wei et al.26, the relatively fuzzy period is 3–8 days after the shield tail passes in the actual calculation process. There are often distinct differences between various projects and even different sections. The error in calculating the ground loss rate by taking a fixed surface settlement after the shield tail passes is often significant. The ground settlement at the turning point of the ground settlement rate is used to calculate the ground loss rate. Although its accuracy is high according to the principles, this method is feasible for several sections. Many sections cause considerable calculations, thereby making the method complex.

Factors affecting turning point of settlement rate

Formation factors

Given the above problems, this section collects data on the surface settlement of several shielded soft-soil sections in the Hangzhou area. The typical geological profile of the project area is shown in the Fig. 5. Table 1 lists the geological conditions of the sections.

Fig. 5 Geological profile map (Muddy silty clay, silty clay).

Table 1 The geological parameters at various intervals.

Tunnel crossing stratum	Shielded section	c (kPa)	ø (°)	ES (MPa)	
Muddy silty clay	a. Liming Road Station to Jianqiao Old Street Station	8	6.5	2.5	
b. Xiangjisi Station to Daguan Station	9.7	13.5	1.3	
c. 7#air shaft to Xiaoshan Airport Station	9	8	2	
d. 5#air shaft to West Lake Cultural Square Station	13	10	2.5	
e. Tiaoxi Station to Cangqian Vehicle Base	6	4.5	0.9	
Silty clay	f. Gouyangdu Station to Haoyun Street Station	40	16.5	6.2	
g. Chuyun Road Station to Hangxing Road Station	51.9	16.8	8.7	
h. Xintiandi Street Station to Gaotingba Station	46	20	11	
i. Pingan Bridge Station to Chuyun Road Station	48	18	6.5	
j. West Railway Station to Chuangjing Road Station	44	21	9	

The surface settlement of the central measuring point of each section was calculated to study the influence of geological parameters on the turning point of the surface settlement rate after the shield tail passes.

Figure 6 plots the longitudinal surface settlement above the section axis of some muddy, silty clay layers. As shown in Fig. 6a, the settlement curve at the section of ring 765 between Liming Road Station and Jianqiao Old Street Station shows that when the shield machine cut is about 8 m away from the measuring point, the surface has evident settlement, and then the surface deformation gradually stabilizes. When the shield tail of the shield machine leaves the monitoring section, the soil enters the transient settlement stage, manifested by the rapid increase in the surface settlement above the axis. This process lasts about 8 days, that is, the surface settlement begins to stabilize after 8 days. As shown in Fig. 6b, the settlement curve at the section of ring 200 between 7# air shaft and Xiaoshan Airport Station shows that when the shield machine cut is about 6 m away from the measuring point, the surface has apparent settlement deformation; afterward, the trend of change in the surface settlement is similar to that in Fig. 6a. The rapid settlement stage lasts about 6.5 days, slightly less than 8 days.

Fig. 6 The longitudinal surface settlement of the muddy, silty clay layer: (a) the section of ring 765 between Liming Road Station and Jianqiao Old Street Station; (b) the section of ring 200 between 7# air shaft and Xiaoshan Airport Station.

Figure 7 delineates the longitudinal surface settlement above the section axis of some silty clay layers. As shown in Fig. 7a, the settlement curve at the section of ring 480 between Xintiandi Street Station and Gaotingba Station is similar to that in Fig. 6. When the shield machine cut is about 10 m away from the measuring point, the surface has evident settlement; then the surface deformation gradually stabilizes. When the shield tail of the shield machine leaves the monitoring section, the soil enters the transient settlement stage, lasting about 3 days. As shown in Fig. 7b, the settlement curve at the section of ring 810 between Chuyun Road Station and Hangxing Road Station demonstrates that when the shield machine cut is about 8 m away from the measuring point, the surface has obvious uplift deformation; afterward, the trend of change in the surface settlement is similar to that in Fig. 7a. The rapid settlement stage lasts about 3.5 days.

Fig. 7 The longitudinal surface settlement of the silty clay layer: (a) the section of ring 480 between Xintiandi Street Station and Gaotingba Station; (b) the section of ring 810 between Chuyun Road Station and Hangxing Road Station.

Comparing Fig. 6 with Fig. 7 shows that the ground settlement above the tunnel axis can be divided into multiple stages during the whole process of shield tunneling.

The first stage: the shield machine is far from the settlement measuring point. At this time, the surface does not settle, or the settlement is not apparent. It can be understood that the soil at the measuring point is located outside the scope of the shield machine disturbance or in the micro disturbance area.

The second stage: when the horizontal distance between the shield machine and the measuring point is about 6–12 m, the ground surface has a significant deformation, which is generally settlement deformation and, in some cases, uplift deformation; it should be noted that Liang et al.24 reported that the horizontal distance between the shield machine and the measuring point was about one to two times the diameter of the shield tunnel. At this time, the cut earth bin pressure of the shield machine primarily affects the surface deformation, which mainly plays a role in maintaining the stability of the excavation surface. Ideally, the pressure of the earth bin should always be equal to the static lateral earth pressure of the excavation face. However, shield tunneling is a dynamic process. In actual operations, the pressure of the earth bin can only approximate to the lateral earth pressure. Therefore, as the earth pressure in the earth bin is higher than the lateral earth pressure, the soil mass in front of the shield machine has uplift deformation. On the contrary, as the earth pressure in the earth bin is less than the lateral earth pressure, the soil mass in front of the shield machine has settlement deformation.

The third stage: when the shield tail is separated from the pipe segment, it enters the transient settlement development stage, and the surface settlement above the axis develops rapidly. This is because a specific overbreak during the shield construction results in a particular gap between the excavated soil and the laid pipe segment. Once the shield tail is separated, the surrounding disturbed soil rushes to the shield tail gap. Although the shield tail grouting can effectively alleviate this process, the unfixed slurry cannot provide sufficient support. Therefore, the surface settlement develops rapidly when the upper soil gradually moves toward the tunnel.

The fourth stage: after the development of the instantaneous settlement stabilizes, the soil enters the consolidation settlement stage, manifested by the evident, slowing surface settlement rate. The soil settlement in this stage is mainly caused by the dissipation of the excess static pore water pressure and the occurrence of drainage consolidation. For soft soils such as silty clay, the consolidation settlement period is often long, and the settlement accounts for a significant proportion of the total settlement value. Therefore, it is necessary to distinguish the settlement value from the total settlement value in the undrained stage, that is, to find the dividing point between the instantaneous settlement stage and the consolidation settlement stage.

The instantaneous settlement duration of the two sections in the muddy, silty clay layer in Fig. 6 is 6.5 and 8 days, respectively, while that of the two sections in the silty clay layer in Fig. 7 is 3 and 3.5 days, respectively, which is significantly shorter than that in the muddy, silty clay layer. In order to analyze the influence of soil strength on the boundary of instantaneous settlement and consolidation settlement, we calculate the instantaneous settlement duration of some sections in the section (see Table 1). According to Table 2, in the muddy, silty clay layer, the shortest average duration of instantaneous settlement is 6.7 days in the section between the 5# air shaft and the West Lake Cultural Square Station, and the longest average is 9.7 days in the section between Liming Road Station and Jianqiao Old Street Station. In the silty clay layer, the shortest average duration of instantaneous settlement is 3.9 days between Xintiandi Street Station and Gaotingba Station, and the longest average is 5.1 days between West Railway Station and Chuangjing Road Station. Figure 8 depicts the box diagram of the instantaneous settlement duration of each section. The instantaneous settlement duration of the surface above the axis is significantly higher when the shield passes through the muddy, silty clay layer than the silty clay layer. The nature of the soil layer of the excavation face significantly impacts the division of the instantaneous settlement and consolidation settlement of the stratum. We believe this is because the compressive modulus and strength of the muddy, silty clay are lower than those of the silty clay, giving rise to its higher compressibility and proper rheology. After being disturbed, it produces excessive settlement more easily than silty clay, resulting in its longer stability time.

Table 2 Time dividing point of the instantaneous settlement and consolidation settlement of each section.

Tunnel crossing stratum	Shield section	Sample size	Instantaneous settlement duration (days)	The average duration of instantaneous settlement (days)	
Muddy, silty clay	a. Liming Road Station to Jianqiao Old Street Station	51	6–12.5	9.7	
b. Xiangjisi Station to Daguan Station	52	5.5–14	8.9	
c. 7#air shaft to Xiaoshan Airport Station	57	3.5–11	8.1	
d. 5#air shaft to West Lake Cultural Square Station	60	4–9	6.7	
e. Tiaoxi Station to Cangqian Vehicle Base	47	5.5–15	9.6	
Silty clay	f. Gouyangdu Station to Haoyun Street Station	42	2.5–8	4.6	
 g. Chuyun Road Station to Hangxing Road Station	46	3–9.5	4.2	
h. Xintiandi Street Station to Gaotingba Station	52	2.5–7.5	3.9	
i. Pingan Bridge Station to Chuyun Road Station	53	2.5–9	4.4	
j. West Railway Station to Chuangjing Road Station	55	3–8	5.1	

Fig. 8 The box diagram of the instantaneous settlement duration of each section.

Grouting factors

Grouting is generally divided into synchronous grouting and secondary or multiple grouting. Synchronous grouting is to fill mortar into the gap between the segment and the stratum through the synchronous grouting system at the end of the shield machine during the shield construction so as to quickly stabilize the soil and reduce the settlement. Excessive grouting pressure not only offsets the original settlement but also leads to surface uplift and deformation. Liang et al.24 researched the impact of synchronous grouting on the division of instantaneous settlement and consolidation settlement and believed that the time required for instantaneous settlement due to grouting was longer than that in the case of no uplift; it was delayed by 2–3.5 days more than that in the case of no uplift, and the greater the amount of ground uplift, the higher the instantaneous settlement rate.

In order to further study the impact of grouting, we collected some data on the grouting-induced uplift section settlement. We selected the section of ring 320 between Xintiandi Street Station and Gaotingba Station as an example because the section was relatively special. When the shield tail leaves the monitoring section, the shield machine stops immediately, so it is unnecessary to consider the impact of subsequent construction. As shown in Fig. 9, the surface uplift stage caused by grouting compensation is classified into the instantaneous settlement stage. Although the grouting compensation causes uplift deformation on the surface, in the final analysis, this stage is still in the rapid development stage of settlement compared with the section without uplift; however, the former offsets the excessive settlement, resulting in uplift deformation. According to Fig. 9, after the shield tail is detached, the surface settlement above the axis of the section of ring 320 develops rapidly, with a settlement rate of about 2.6 mm/day. After the settlement lasts for 2 days, the shield tail is grouted and compensated, and the surface is rapidly raised by about 7 mm to stabilize the segment and soil. With the gradual disappearance of grouting pressure, the surface settlement develops rapidly again, lasting for about 4.5 days with a settlement rate of about 3.4 mm/day; it then enters the consolidation settlement stage.

Comparing Fig. 7a with Fig. 9 reveals that the instantaneous settlement rate of the section of ring 480 between Xintiandi Street Station and Gaotingba Station is about 2.3 mm per day, which is similar to that of the section of ring 320 before grouting compensation. Nevertheless, the instantaneous settlement rate of the section of ring 320 after grouting compensation is significantly higher than before, consistent with the research results of Liang et al.24. Compared with the section of ring 480, that of ring 320 is affected by grouting to uplift, and the duration of instantaneous settlement is prolonged. Because the duration of instantaneous settlement without grouting to uplift is unknown, the specific impact of grouting factors is difficult to quantify. Considering the collected data and the average duration of instantaneous settlement in the case of no uplift in the interval, grouting-induced uplift delays the time for the surface above the axis to enter the consolidation settlement by about 4 days. This specific time may also be affected by the amount of grouting and grouting pressure, and the impact of secondary or multiple groutings is not considered.

Fig. 9 The analysis of the influence of grouting factors on section settlement (section of ring 320 between Xintiandi Street Station and Gaotingba Station).

Statistics for ground loss rate and analysis of influencing factors

Considering “Statistical analysis of ground loss rate in Hangzhou soft-soil area”, Peck’s back analysis method is employed to use the surface settlement monitoring data at the boundary point of the instantaneous settlement and consolidation settlement stages of each section for Gaussian curve fitting. In combination with the research method of Wu et al.14, the data with a goodness of fit of more than 0.80 are utilized to calculate the ground loss rate of each monitoring section. Table 3 presents the specific collected data and back analysis results.

Table 3 The back analysis results of the measured data on the shield tunnel.

Tunnel crossing stratum	Shield section	Sample size	η (%)	\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar {\eta }$$\end{document} (%)	H/D	Smax (mm)	
Muddy, silty clay	a. Liming Road Station to Jianqiao Old Street Station	35	0.36–3.10	1.20	1.43–2.62	9.52–45.21	
b. Xiangjisi Station to Daguan Station	40	0.01–1.21	0.40	1.85–3.15	0.61–16.55	
c. 7#air shaft to Xiaoshan Airport Station	43	0.02–2.98	0.41	2.30–3.65	1.37–71.28	
d. 5#air shaft to West Lake Cultural Square Station	56	0.06–0.81	0.18	2.79–4.38	1.08–19.64	
e. Tiaoxi Station to Cangqian Vehicle Base	36	0.44–2.22	0.86	1.25–2.23	3.94–36.85	
Silty clay	f. Gouyangdu Station to Haoyun Street Station	40	0.03–0.78	0.31	1.64–4.22	0.26–33.09	
 g. Chuyun Road Station to Hangxing Road Station	45	0.01–1.28	0.23	2.52–2.72	0.38–30.46	
h. Xintiandi Street Station to Gaotingba Station	35	0.03–1.53	0.54	1.49–3.24	1.98–33.09	
i. Pingan Bridge Station to Chuyun Road Station	42	0.04–2.97	0.75	2.01–3.40	0.98–34.79	
j. West Railway Station to Chuangjing Road Station	33	0.15–1.76	0.52	2.64–3.48	1.25–29.89	

η is the ground loss rate, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar {\eta }$$\end{document} indicates the average value of the ground loss rate, H represents the depth of burial of the tunnel, D denotes the tunnel excavation diameter, and Smax stands for the maximum settlement of the ground.

According to the collected ground loss rate and relevant data, this section primarily studies the correlation between the geological conditions, the depth-to-diameter ratio (the ratio of the depth of burial of the tunnel to the excavation diameter), the advancing speed, the variation coefficient of the advancing speed, and the maximum surface settlement and ground loss rate.

Correlation between geological conditions and ground loss rate

Figure 10 delineates the cumulative probability of the ground loss rate. There is no significant difference in the probability of the ground loss rate when the shield passes through the muddy, silty clay layer and the silty clay layer. The ground loss rate is between 0% and 3%, and the ground loss rate of most sections is less than 0.5%. In addition, the distribution box diagram of the ground loss rate in different sections in Fig. 11 demonstrates that, even for the same geology, the ground loss rates in various sections still differ remarkably. Taking the muddy, silty clay layer as an example, the average ground loss rate between Liming Road Station and Jianqiao Old Street Station is 1.2%, while that in the section between 5# air shaft to West Lake Cultural Plaza Station is only 0.18%. In the soft soil layer, the influence of the physical and mechanical properties of the formation on the ground loss rate is insignificant.

Fig. 10 The cumulative probability of the ground loss rate.

Fig. 11 The distribution box diagram of the ground loss rate in different sections.

Correlation between depth-to-diameter ratio and ground loss rate

Figure 12 plots the relationship between the depth-to-diameter ratio and the ground loss rate. An H/D less than 1.6 indicates ultrashallow tunnel sections, and an H/D between 1.6 and 2.5 represents shallow tunnel sections; a depth-to-diameter ratio higher than 2.5 denotes deep tunnel sections. The data on the ultrashallow tunnel section are insufficient, so they are not analyzed. In the section of the shallowly buried tunnel, the ground loss rate decreases as the depth-to-diameter ratio increases, and at the same depth-to-diameter ratio, the ground loss rate of the muddy silty clay layer is slightly higher than that of the silty clay layer. In the section of the deeply buried tunnel, when H/D is less than 3.75, the distribution of the ground loss rate is relatively scattered, and it is difficult to find the rule. According to the analysis of Wu et al.14, the ground loss rate of the shield tunnel with a small- and medium-sized diameter generally presents an increasing and then decreasing trend as the depth-to-diameter ratio enlarges. This phenomenon is pronounced in the silty clay layer. When H/D is higher than 3.75, the ground loss rate is generally low, that is, raising the depth-to-diameter ratio has almost no effect on the ground loss rate.

Fig. 12 The relationship between the depth-to-diameter ratio and the ground loss rate.

Correlation between advancing speed of shield machine and ground loss rate

Figure 13 depicts the relationship between the advancing speed and the ground loss rate. No apparent relationship can be fitted between the advancing speed of the shield machine and the ground loss rate, whether in the muddy, silty clay layer or the silty clay layer. On the one hand, we believe that the shield system is very complex, and the propulsion speed is only one of the more critical parameters, which is affected by other tunneling parameters or geological conditions and consequently affects other tunneling parameters. Of course, the most direct impact originates from the subjective judgment of the shield operator. However, the shield operator is relatively experienced, so there is often no case that the driving speed is too high or too slow, leading to the imbalance of the face, thereby causing the ground loss rate to be too high. The set driving speed is basically within a reasonable range. On the other hand, the average propulsion speed contains limited content, which cannot fully represent the impact of construction factors and cannot consider the changes in the construction factors in a single ring. Therefore, it is difficult to judge the ground loss rate only from the measured data on the advancing speed of the shield machine.

Fig. 13 The relationship between the advancing speed of the shield machine and the ground loss rate.

Correlation between coefficient of variation in advancing speed of shield machine and ground loss rate

Figure 14 delineates the correlation between the coefficient of variation in the advancing speed of the shield machine and the ground loss rate. The coefficient of variation is the ratio of the standard deviation to the average value. The larger the coefficient of variation, the higher the degree of dispersion. This section uses the data on each stable section of the shield and does not take into account the long-term shutdown section. The ground loss rate rises with an increase in the coefficient of variation on the trend. The relationship between the coefficient of variation in the advancing speed of the shield machine and the ground loss rate is similar to a power function, and the index values in the power function regression equation are between zero and one. In other words, the increase in the ground loss gradually slows down as the coefficient of variation in the advancing speed of the shield machine rises. The variation coefficient of the shield tunneling parameters reflects the discreteness of the tunneling parameters during the construction process. The discreteness primarily reflects the difficulties of shield construction, which is more likely to cause significant ground losses. In practical engineering, the variation coefficient of the shield tunneling parameters cannot be too large. Therefore, within the normal range, it can be considered that the variation coefficient of the shield tunneling parameters, such as the ground loss rate and the advancing speed of the shield machine, approximately fulfill a power-law relationship.

Fig. 14 The relationship between the coefficient of variation in the advancing speed of the shield machine and the ground loss rate.

Correlation between maximum surface subsidence and ground loss rate

Figure 15 plots the relationship between the maximum surface settlement and the ground loss rate during the construction stage. There is a specific linear correlation between the ground loss rate and the maximum settlement of the ground surface. The correlation between the maximum settlement value of the ground surface and the ground loss rate can be used to estimate the ground loss rate caused by shield tunneling, and its formula is expressed by the following:7 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta =\left\{ \begin{gathered} 0.085*{S_{{\text{max}}}}{\text{ max}} \hfill \\ 0.038*{S_{{\text{max}}}}{\text{ min}} \hfill \\ \end{gathered} \right.$$\end{document}

There is a particular linear relationship between the ground loss rate and the maximum surface settlement because, on the one hand, the ground loss is the direct cause of the surface settlement, and the surface settlement is the most direct manifestation of the ground loss. On the other hand, according to Peck’s formula and Eq. (6), the ground loss rate is directly proportional to the maximum surface settlement during the construction stage, and the slope is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{{i\sqrt {2{\text{\varvec{\uppi}}}} }}{{{\text{\varvec{\uppi}}}{R^2}}}$$\end{document}.

Fig. 15 The relationship between the maximum surface settlement and the ground loss rate.

The above analysis presents the correlation results between the ground loss rate and various factors, including geological conditions, burial diameter ratio (the ratio of tunnel burial depth to excavation diameter), advancing speed, coefficient of variation of advancing speed, and maximum surface settlement value. The correlation order, in terms of impact on the ground loss rate, is as follows: maximum settlement value > coefficient of variation of advancing speed > burial diameter ratio > advancing speed > geological conditions.

To further investigate this relationship, shield construction data from multiple soft soil shield sections in the Hangzhou area were collected. The PECK back analysis method was employed to calculate the ground loss rate caused by shield tunneling. Through thorough analysis of the collected data, the influence of geological parameters, tunnel geometric parameters, and other factors on the ground loss rate was determined.

Prediction of ground loss rate based on XGBoost

“Statistics for ground loss rate and analysis of influencing factors” demonstrates that the ground loss rate is related to the geological conditions, the depth-to-diameter ratio, the coefficient of variation in the advancing speed of the shield machine, and the maximum surface settlement during construction. Therefore, this section aims to obtain a more accurate ground loss rate by establishing a prediction model of the XGBoost ground loss rate based on Bayesian optimization, which is based on the maximum settlement value of the surface during the construction stage, combined with the detailed geological parameters, geometric parameters of the tunnel, and construction parameters.

eXtreme Gradient Boosting

XGBoost is an integrated algorithm improved and optimized based on the gradient lifting decision tree model. Its essence is a gradient lifting algorithm based on classification and regression tree (CRAT), and its mathematical model can be understood as an additive model composed of K classification regression trees:8 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat {y}_i}=\sum\limits_{{k=1}}^{K} {{f_k}({x_i})} {\text{ }}{f_k} \in F$$\end{document}

where k is the number of trees, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f_k}$$\end{document} indicates a function in function space F, and F represents the set of all possible CARTs.

Assume that the final XGBoost model has t trees, i.e., the prediction model with t rounds. The model of round t is composed of retaining the prediction models of the previous t – 1 rounds and adding a new function as follows:9 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{gathered} \hat {y}_{i}^{{(0)}}=0 \hfill \\ \hat {y}_{i}^{{(1)}}={f_1}({x_i})=\hat {y}_{i}^{{(0)}}+{f_1}({x_i}) \hfill \\ \hat {y}_{i}^{{(2)}}={f_1}({x_i})+{f_2}({x_i})=\hat {y}_{i}^{{(1)}}+{f_2}({x_i}) \hfill \\ \ldots \hfill \\ \hat {y}_{i}^{{(t)}}=\sum\limits_{{k=1}}^{K} {{f_k}({x_i})} =\hat {y}_{i}^{{(t - 1)}}+{f_t}({x_i}) \hfill \\ \end{gathered}$$\end{document}

To ensure that the final expression effect of the model improves after the new function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f_t}({x_i})$$\end{document}is added, we need to introduce the total objective function as follows:10 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Ob{j^{(t)}}=\sum\limits_{{i=1}}^{n} {l({y_i},\hat {y}_{i}^{{(t)}})} +\sum\limits_{{i=1}}^{t} {\Omega ({f_i})}$$\end{document}

11 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l({y_i},{\hat {y}_i})={({y_i},{\hat {y}_i})^2}$$\end{document}

12 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega ({f_t})=\gamma T+\frac{1}{2}\lambda \sum\limits_{{j=1}}^{T} {{\omega ^2}}$$\end{document}

In Eq. (10), the first part is the loss function of the model itself, that is, the mean squared error (MSE); the second part is the regularization penalty term; \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document} are the penalty coefficients of the model; T is the number of leaves; \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega$$\end{document} indicates the fraction of the leaf.

Equation (10) is rearranged to obtain Eq. (13):13 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Ob{j^{(t)}}=\sum\limits_{{i=1}}^{n} {l({y_i},\hat {y}_{i}^{{(t - 1)}}+{f_t}({x_i}))} +\Omega ({f_t})+\varvec{c}$$\end{document}

where c is a constant term.

Taylor expansion is performed on Eq. (13) to obtain the following:14 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Ob{j^{(t)}}=\sum\limits_{{i=1}}^{n} {\left[ {{g_i}{f_t}({x_i})+\frac{1}{2}{h_i}f_{t}^{2}\left( {{x_i}} \right)} \right]} +\Omega ({f_t})+\varvec{c}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${g_i}=\frac{{\partial \left( {l\left( {{y_i},{{\hat {y}}^{\left( {t - 1} \right)}}} \right)} \right)}}{{\partial {{\hat {y}}^{\left( {t - 1} \right)}}}}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${h_i}=\frac{{{\partial ^2}\left( {l\left( {{y_i},{{\hat {y}}^{\left( {t - 1} \right)}}} \right)} \right)}}{{\partial {{\hat {y}}^{\left( {t - 1} \right)}}}}$$\end{document}.

Converting Eq. (14) from traversing on the sample to traversing on the leaf node and eliminating the constant term, we may express the final objective function as follows:15 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Ob{j^{(t)}}=\sum\limits_{{j=1}}^{T} {\left[ {{G_j}{\omega _j}+\frac{1}{2}\left( {{H_j}+\lambda } \right)\omega _{j}^{2}} \right]} +\gamma T$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${G_j}=\sum\nolimits_{{i \in {I_j}}} {{g_i}}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H_j}=\sum\nolimits_{{i \in {I_j}}} {{h_i}}$$\end{document}.

The final objective function is a quadratic equation of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\omega _j}$$\end{document}, and Eqs. (16) and (17) express its minimum point and minimum value, respectively:16 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\omega _j}= - \frac{{{G_j}}}{{{H_j}+\lambda }}$$\end{document}

17 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Obj= - \frac{1}{2}\sum\limits_{{j=1}}^{T} {\frac{{G_{j}^{2}}}{{{H_j}+\lambda }}} +\gamma T$$\end{document}

Bayesian optimization

Bayesian optimization is an influential global optimization algorithm and one of the most advanced and promising technologies in probabilistic machine learning and artificial intelligence28. It primarily includes two core parts: the probabilistic agent model and the acquisition function. Its framework is an iterative process, that is, it selects the next most potential evaluation point (xi) by maximizing the acquisition function and then evaluates the target function value (yi) according to the selected evaluation point; afterward, it adds the newly obtained xi and yi to the historical evaluation point set, updates the probability proxy model, and iterates to obtain the optimal solution in turn. The process for establishing the ground loss rate prediction model is depicted in Fig. 16.

Fig. 16 Establishment process of ground loss rate prediction model.

Model development

Based on the measured data in “Statistical analysis of ground loss rate in Hangzhou soft-soil area”, 370 sets of complete data are screened and processed to form the ground loss rate prediction data set, 80% of which are randomly selected as the training set, and the rest 20% are the testing set. Table 4 displays the parameters and samples.

The input parameters include the depth-to-diameter ratio in the geometric parameters of the tunnel; the cohesion, the internal friction angle, and the compressive modulus in the geological parameters; and the propelling speed and the coefficient of variation in the propelling speed in the construction parameters. It should be noted that the maximum surface settlement in the construction stage is introduced into the model as the input parameter. The maximum surface settlement in the construction stage is initially affected by the tunnel geometry, construction parameters, and geological factors. This value contains some information about these influencing factors. Taking it as a feature can significantly reduce the dimensions of the input parameters.

Table 4 Raw parameters and samples.

Parameter	1	2	3	4	5	…	370	
Geometric parameters	Depth-to-diameter ratio	1.5648	1.6296	1.6898	1.7376	1.8441	.	2.9012	
Geological parameters	Cohesion	8	8	8	8	8		48	
Internal friction angle	6.5	6.5	6.5	6.5	6.5		18	
Compressive modulus	2.5	2.5	2.5	2.5	2.5		6.5	
Construction parameters	Propelling speed	45.2	50.1	40.2	42.3	39.5		42.8	
Coefficient of variation in the propelling speed	46	63	67	136	131		59	
Maximum surface settlement	11.51	11.43	12.65	11.54	15.92	…	9.39	

Invoking command feature_importances obtains the characteristic importance score of the prediction model of the ground loss rate and draws a pie chart according to the characteristic importance score, as shown in Fig. 17. The maximum settlement value of the ground surface during the construction stage is most crucial in the prediction model, reaching 48.16%; in other words, the prediction of the ground loss rate by the model is primarily based on the characteristics of the maximum settlement value of the ground surface during the construction stage. Secondly, the importance of the coefficient of variation in the propelling speed reaches 19.05%. The driving parameters in the shielding process fluctuate too much, that is, it is challenging to control the ground loss rate when the shield is blocked.

Among geological parameters, the relative importance of the internal friction angle of soil is higher than that of the cohesion and compression modulus. The importance of the advancing speed of the shield machine and the depth-to-diameter ratio in the prediction model can be ignored. The maximum ground settlement value during the construction stage is most affected by the diameter and depth of burial of the tunnel. It can be understood that this value contains preliminary information on the depth-to-diameter ratio. Therefore, in the same way, the prediction model of the ground loss rate does not consider the grouting volume, grouting pressure, and other factors that have been considered in the surface deformation prediction model.

To sum up, all the input parameters are ranked according to their significance: the maximum surface settlement in the construction stage > the coefficient of variation in the propulsion speed > the internal friction angle > cohesion > the compressive modulus > the propulsion speed > the depth-to-diameter ratio. We omit the characteristics with a negligible impact on the prediction results and select the maximum surface settlement value, the coefficient of variation in the propulsion speed, the internal friction angle, cohesion, and the compressive modulus in the construction stage as the input parameters. Detailed parameters are shown in Table 5.

Fig. 17 Evaluating the importance of the model features.

Table 5 Final selected parameters and samples.

Parameter	1	2	3	4	5	…	370	
Geological parameters	Cohesion	8	8	8	8	8		48	
Internal friction angle	6.5	6.5	6.5	6.5	6.5		18	
Compressive modulus	2.5	2.5	2.5	2.5	2.5		6.5	
Construction parameters	Coefficient of variation in the propelling speed	46	63	67	136	131		59	
Maximum surface settlement	11.51	11.43	12.65	11.54	15.92	…	9.39	

The Bayesian optimization method is used to determine the optimal hyperparameters, and the four-fold cross-validation iterative training model is employed to prevent the overfitting of the model. According to the performance of the testing data set, the number of iterations is determined to be 100. Table 6 presents the optimal values of the Bayesian optimization hyperparameters of the XGBoost model set in this paper, and the other hyperparameters are taken as default values.

Table 6 The optimal values of the bayesian optimization hyperparameters of the XGBoost model.

Hyperparameter	Optimal value	
Max_depth	3	
Learning_rate	0.24	
Min_child_weight	3.82	
Subsample	1	
Colsample_bytree	0.32	
Reg_alpha	0.001	
Gamma	0.001	

Model predictions

This work compares the Bayesian optimization XGBoost model with RF and support vector machine (SVM) models and conducts model training and prediction using the same data set to verify its performance. We select mean squared error, root mean squared error (RMSE), mean absolute error (MAE), and coefficient of determination (R2) as the evaluation indices. Figure 18 compares the test results.

According to Fig. 18, the prediction model of the ground loss rate based on XGBoost has high reliability in the soft-soil layer, and its prediction accuracy is the highest among the three models: R2 is over 99%, MSE is 0.0005, RMSE equals 0.0222, and MAE is 0.0168. In order to analyze the prediction effect of the model in the case of a significant ground loss rate, we conduct statistical analysis on the measured and predicted data of nine sections with a ground loss rate higher than 1.0% in the testing set, as presented in Table 7. The mean absolute prediction error in the ground loss rate is 0.0175%, which is relatively small, proving the high prediction accuracy of the model and thereby indicating its particular application value.

Fig. 18 Comparing the predictions of various models: (a) XGBoost; (b) SVM; (c) RF.

Table 7 The prediction error of the section with a high ground loss rate.

The actual ground loss rate (%)	The predicted ground loss rate (%)	Absolute error (%)	
2.2154	2.1932	0.0222	
2.2131	2.2146	0.0014	
2.1188	2.1031	0.0157	
1.9119	1.8630	0.0489	
1.8310	1.8215	0.0095	
1.7109	1.7259	0.0150	
1.5276	1.5258	0.0019	
1.2003	1.1644	0.0358	
1.1752	1.1679	0.0073	
Mean absolute error (%)	0.0175	

Conclusions

This study collected data on the surface settlement of 10 shield sections in the Hangzhou area and calculated the ground loss rate caused by shield tunneling based on Peck’s back analysis method. The influence of the geological parameters on the turning point of the surface subsidence rate after the shield tail passes through was studied, and various factors affecting the ground loss rate were analyzed. By establishing a prediction model of the ground loss rate based on XGBoost, we innovatively proposed an intelligent method to predict the ground loss rate, combining the surface subsidence value and detailed geological parameters. The following conclusions could be drawn from the above findings:

The nature of the soil layer on the excavation surface markedly impacted the dividing point of the instantaneous settlement and consolidation settlement of the stratum. When the shield passed through the muddy, silty clay layer, the duration of the instantaneous settlement of the surface above the axis was significantly longer than that of the silty clay layer. Grouting delayed the entrance of the surface above the axis into consolidation settlement by about 4 days, which might also be affected by the grouting volume and pressure.

The influences of the physical and mechanical properties of the soft-soil strata on the ground loss rate were insignificant. In the shallowly buried tunnel section, the ground loss rate decreased with an increase in the depth-to-diameter ratio. At the same depth-to-diameter ratio, the ground loss rate of the muddy, silty clay layer was slightly higher than that of the silty clay, deeply buried tunnel section. The ground loss rate of the shield tunnel with a small- and medium-sized diameter generally increased first and then dropped as the depth-to-diameter ratio enlarged, which was particularly obvious in the silty clay layer when H/D was higher than 3.75. The ground loss rate was generally low, that is, raising the depth-to-diameter ratio had almost no effect on the ground loss rate.

The ground loss rate increased with the coefficient of variation in the advancing speed of the shield machine, following a power-law relationship. The indices in the power function regression equation were between zero and one. In other words, the ground loss rate increased slowly with the coefficient of variation in the advancing speed of the shield machine. There was also a specific linear correlation between the ground loss rate and the maximum surface subsidence.

The prediction model of the ground loss rate based on XGBoost had high reliability in the soft-soil layer: R3 was over 99%, MSE equaled 0.0005, RMSE was 0.0222, and MAE was 0.0168. The absolute prediction error of the model was small at high ground loss rates, that is, it had high prediction accuracy, indicating its high application value.

Combining this method with advanced automation technologies, such as autonomous shield machines and robotic systems, could lead to more precise control and adjustment of tunneling parameters. This integration would enhance efficiency and reduce the risk of ground loss. Extending the method to address a wider range of geological conditions through additional data sources and predictive models will increase its versatility. This adaptation will make the method more applicable to complex and variable ground conditions encountered in modern tunneling projects. Integrating the method with predictive maintenance systems can help foresee and mitigate potential issues before they affect construction. Analyzing patterns and trends from real-time data can enable proactive problem-solving and risk management.

Author contributions

W.T.: Conceptualization; methodology; writing-original draft preparation; validation. Z.D.: Funding acquisition; project administration; resources; supervision; writing-reviewing and editing. X.L.: Methodology; writing-reviewing and editing.

Funding

This work was supported by Key Research and Development Program of Zhejiang [Grant No. 2023C03182], the National Natural Science Foundation of China [Grant No. 52278418] and Key Project of Zhejiang Provincial Natural Science Foundation [Grant No. LHZ20E080001].

Data availability

The data that support the findings of this study are available from Hangzhou Metro Group Co., Ltd but restrictions apply to the availability of these data, which were used under license for the current study, and so are not publicly available. Data are however available from the corresponding authors upon reasonable request and with permission of Hangzhou Metro Group Co., Ltd.

Declarations

Competing interests

The authors declare no competing interests.

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
==== Refs
References

1. Li, C., Li, S., Gao, X. & Zhao, Y. A comparative study of inappropriate behaviors of Chinese and Japanese contractors in construction disputes under FIDIC conditions of contract. J. Legal Affairs Dispute Resol. Eng. Constr. (2), 04518003 (2018).
2. Li H Zhang D Chen Y Analysis and suggestion on shield tunneling technology for metro construction J. Constr. Eng. Manag. 2020 146 9 04020070 10.1061/(asce)co.1943-7862.0001895
Li, H., Zhang, D. & Chen, Y. Analysis and suggestion on shield tunneling technology for metro construction. J. Constr. Eng. Manag. 146 (9), 04020070. 10.1061/(asce)co.1943-7862.0001895 (2020).
3. Ding Z Wei XJ Zhang X Yin XS 2019 Analysis of the field monitoring data on soil movements and adjacent building settlement due to shield tunneling 10.1108/EC-07-2018-0314
Ding, Z., Wei, X. J., Zhang, X. & Yin, X. S. Analysis of the field monitoring data on soil movements and adjacent building settlement due to shield tunneling. Eng. Comput. 36(4), 1219–1237. 10.1108/EC-07-2018-0314 (2019).
4. Qian L Huang H Li J Evaluation of ground loss rate for soft soil tunneling in urban area J. Perform. Constr. Facil. 2021 35 1 04020101 10.1061/(asce)cf.1943-5509.0001562
Qian, L., Huang, H. & Li, J. Evaluation of ground loss rate for soft soil tunneling in urban area. J. Perform. Constr. Facil. 35 (1), 04020101. 10.1061/(asce)cf.1943-5509.0001562 (2021).
5. Huo R Study on the settlement of large-span metro station’s baseplate caused by the tunnels newly built beneath it Adv. Mech. Eng. 2019 11 2 1687814018825161 10.1177/1687814018825161
Huo, R. et al. Study on the settlement of large-span metro station’s baseplate caused by the tunnels newly built beneath it. Adv. Mech. Eng. 11 (2), 1687814018825161 (2019).
6. Han S Ma G A case study on safety management for metro construction based on Sydenham accident Adv. Civil Eng. 2020 2020 1 13 10.1155/2020/8891971
Han, S. & Ma, G. A case study on safety management for metro construction based on Sydenham accident. Adv. Civil Eng. 2020, 1–13. 10.1155/2020/8891971 (2020).
7. Wang J Zhou P Song Z Li S Zhang Q A new calculation method for tunneling-caused stratum settlement KSCE J. Civ. Eng. 2022 26 6 2624 2640 10.1007/s12205-022-1258-z
Wang, J., Zhou, P., Song, Z., Li, S. & Zhang, Q. A new calculation method for tunneling-caused stratum settlement. KSCE J. Civ. Eng. 26 (6), 2624–2640 (2022).
8. Peck, R. B. Deep excavations and tunneling in soft ground. In Proceedings of 7th International Conference on Soil Mechanic and Foundation Engineering Mexico City 225–290 (1969).
9. Klar A Klein B Energy-based volume loss prediction for tunnel face advancement Géotechnique 2014 64 10 776 786 10.1680/geot.14.P.024
Klar, A. & Klein, B. Energy-based volume loss prediction for tunnel face advancement. Géotechnique. 64 (10), 776–786. 10.1680/geot.14.P.024 (2014).
10. Mair, R. & Taylor, R. Theme lecture: Bored tunnel in the urban environment. In Proceeding of the Fourteenth International Conference on Soil Mechanics and Foundation Engineering 2353–238 (Balkema, 1997).
11. Vu MN Broere W Bosch JW Volume loss in shallow tunneling Tunn. Undergr. Space Technol. 2016 59 10 77 90 10.1016/j.tust.2016.06.011
Vu, M. N., Broere, W. & Bosch, J. W. Volume loss in shallow tunneling. Tunn. Undergr. Space Technol. 59 (10), 77–90. 10.1016/j.tust.2016.06.011 (2016).
12. Zhu CH Li N Estimation method and laws analysis of surface settlement due to tunneling Rock. Soil. Mech. 2016 37 s2 533 542 10.16285/j.rsm.2016.S2.068
Zhu, C. H. & Li, N. Estimation method and laws analysis of surface settlement due to tunneling. Rock. Soil. Mech. 37 (s2), 533–542. 10.16285/j.rsm.2016.S2.068 (2016).
13. Zhu CH Li N Estimation and regularity analysis of maximal surface settlement induced by subway construction Chin. J. Rock Mechan. Eng. 2017 36 s1 3543 3560
Zhu, C. H. & Li, N. Estimation and regularity analysis of maximal surface settlement induced by subway construction. Chin. J. Rock Mechan. Eng. 36 (s1), 3543–3560 (2017) (in Chinese).
14. Wu CS Zhu ZD Comparative study on ground loss ratio due to shield tunnel with different diameters Chin. J. Geotech. Eng. 2018 40 12 2257 2265 10.11779/CJGE201812013
Wu, C. S. & Zhu, Z. D. Comparative study on ground loss ratio due to shield tunnel with different diameters. Chin. J. Geotech. Eng. 40 (12), 2257–2265. 10.11779/CJGE201812013 (2018).
15. Wu CS Zhu ZD Statistical analysis of ground loss ratio caused by different tunnel construction methods in China J. Zhejiang Univ. (Eng. Sci.) 2019 53 1 19 30 10.3785/j.issn.1008-973X.2019.01.003
Wu, C. S. & Zhu, Z. D. Statistical analysis of ground loss ratio caused by different tunnel construction methods in China. J. Zhejiang Univ. (Eng. Sci.). 53 (1), 19–30. 10.3785/j.issn.1008-973X.2019.01.003 (2019).
16. Shi CH Cao CY Lei MF An analysis of the ground deformation caused by shield tunnel construction combining an elastic half-space model and stochastic medium theory KSCE J. Civ. Eng. 2017 21 5 1933 1944 10.1007/s12205-016-0804-y
Shi, C. H., Cao, C. Y. & Lei, M. F. An analysis of the ground deformation caused by shield tunnel construction combining an elastic half-space model and stochastic medium theory. KSCE J. Civ. Eng. 21 (5), 1933–1944. 10.1007/s12205-016-0804-y (2017).
17. Dong X Zhang J Chen Y Machine learning-based model for ground settlement prediction during shield tunneling: a case study in Nanjing, China Tunn. Undergr. Space Technol. 2021 109 103733 10.1016/j.tust.2021.103733
Dong, X., Zhang, J. & Chen, Y. Machine learning-based model for ground settlement prediction during shield tunneling: a case study in Nanjing, China. Tunn. Undergr. Space Technol. 109, 103733. 10.1016/j.tust.2021.103733 (2021).
18. Feng G Fan H Han L Machine learning approach for predicting surface subsidence caused by underground excavation J. Geotech. GeoEnviron. Eng. 2021 147 6 04021012 10.1061/(asce)gt.1943-5606.0002628
Feng, G., Fan, H. & Han, L. Machine learning approach for predicting surface subsidence caused by underground excavation. J. Geotech. GeoEnviron. Eng. 147 (6), 04021012. 10.1061/(asce)gt.1943-5606.0002628 (2021).
19. Pourtaghi A Lotfollahi-yaghin MA Wavenet ability assessment in comparison to ANN for predicting the maximum surface settlement caused by tunneling Tunn. Undergr. Space Technol. Incorp. Trenchless Technol. Res. 2012 28 257 271 10.1016/j.tust.2011.11.008
Pourtaghi, A. & Lotfollahi-yaghin, M. A. Wavenet ability assessment in comparison to ANN for predicting the maximum surface settlement caused by tunneling. Tunn. Undergr. Space Technol. Incorp. Trenchless Technol. Res. 28, 257–271. 10.1016/j.tust.2011.11.008 (2012).
20. Wang J Li W Zhang D A novel method for predicting surface subsidence induced by tunneling using a machine learning algorithm Comput. Geotech. 2020 119 103358 10.1016/j.compgeo.2019.103358
Wang, J., Li, W. & Zhang, D. A novel method for predicting surface subsidence induced by tunneling using a machine learning algorithm. Comput. Geotech. 119, 103358. 10.1016/j.compgeo.2019.103358 (2020).
21. Zhou J Feasibility of Random-Forest approach for prediction of ground settlements induced by the construction of a shield-driven tunnel Int. J. Geomech. 2016 17 6 04016129 10.1061/(ASCE)GM.1943-5622.0000817
Zhou, J. et al. Feasibility of Random-Forest approach for prediction of ground settlements induced by the construction of a shield-driven tunnel. Int. J. Geomech. 17 (6), 04016129. 10.1061/(ASCE)GM.1943-5622.0000817 (2016).
22. Zhang P A critical evaluation of machine learning and deep learning in shield-ground interaction prediction Tunneling Undergr. Space Technol. 2020 106 103593 10.1016/j.tust.2020.103593
Zhang, P. et al. A critical evaluation of machine learning and deep learning in shield-ground interaction prediction. Tunneling Undergr. Space Technol. 106, 103593. 10.1016/j.tust.2020.103593 (2020).
23. Dindarloo SR Siami-Irdemoosa E Maximum surface settlement based classification of shallow tunnels in soft ground Tunn. Undergr. Space Technol. 2015 49 320 327 10.1016/j.tust.2015.04.021
Dindarloo, S. R. & Siami-Irdemoosa, E. Maximum surface settlement based classification of shallow tunnels in soft ground. Tunn. Undergr. Space Technol. 49, 320–327. 10.1016/j.tust.2015.04.021 (2015).
24. Liang RZ Pan JL Lin CG Shan HF Sun LW Settlement boundary induced by shield tunneling in soft ground J. Zhejiang Univ. (Engineering Science) 2014 48 7 1148 1154 10.3785/j.issn.1008-973X.2014.07.002
Liang, R. Z., Pan, J. L., Lin, C. G., Shan, H. F. & Sun, L. W. Settlement boundary induced by shield tunneling in soft ground. J. Zhejiang Univ. (Engineering Science). 48 (7), 1148–1154. 10.3785/j.issn.1008-973X.2014.07.002 (2014).
25. Ng RMC Lo KY Rowe RK Analysis of field performance-the thunder-bay tunnel Can. Geotech. J. 1986 23 1 30 50 10.1139/t86-005
Ng, R. M. C., Lo, K. Y. & Rowe, R. K. Analysis of field performance-the thunder-bay tunnel. Can. Geotech. J. 23 (1), 30–50. 10.1139/t86-005 (1986).
26. Wei G Selection and distribution of ground loss ratio induced by shield tunnel construction Chin. J. Geotech. Eng. 2010 32 9 1354 1360
Wei, G. Selection and distribution of ground loss ratio induced by shield tunnel construction. Chin. J. Geotech. Eng. 32 (9), 1354–1360 (2010) (in Chinese).
27. Yi X Rowe RK Lee KM Observed and calculated pore pressures and deformations induced by an earth balance shield Can. Geotech. J. 1993 30 3 476 490 10.1139/t93-041
Yi, X., Rowe, R. K. & Lee, K. M. Observed and calculated pore pressures and deformations induced by an earth balance shield. Can. Geotech. J. 30 (3), 476–490. 10.1139/t93-041 (1993).
28. Ghahramani Z Probabilistic machine learning and artificial intelligence Nature 2015 521 7553 452 459 10.1038/nature14541 26017444
Ghahramani, Z. Probabilistic machine learning and artificial intelligence. Nature. 521 (7553), 452–459. 10.1038/nature14541 (2015).26017444
