Probabilistic Framework for Ground Movement Induced by Shield Tunnelling in Soft Soil Based on Gap Parameter

Abstract

Numerical simulation and machine learning-based methods are frequently adopted when performing ground movement probabilistic analyses, considering the various uncertainties during shield tunnelling. However, numerical simulation takes time, while machine learning lacks interpretation somehow. New methods fully reflecting mechanisms and taking advantage of field data should be proposed and applied in probabilistic analysis. This study proposes a probabilistic framework from the mechanism and data aspect based on the GAP parameter. Solutions for three components of the GAP parameter are first improved through different methods. Coupling the uncertainty of the input parameters, a probabilistic framework estimating the risks from both mechanistic and data insights is then established. Furthermore, the spatial variability in soft soil is considered in the framework by calculating the equivalent parameters. Through an analysis of a practical case, the results show that the measured data can fall within the 95% confidence interval of the predicted displacement samples. The median of the predicted samples is highly consistent with the measured value, and by considering the spatial variability in soil, the results can be more accurate. As a result, the proposed probabilistic framework is verified as practically applicable when predicting ground movement while considering multiple uncertainties.

1. Introduction

Due to intricate engineering geological conditions and design optimizations influenced by humans, the factors relevant to shield tunnelling exhibit significant uncertainty. In this scenario, employing probabilistic approaches [1,2,3,4,5,6,7,8,9] to manage the construction and environmental risks associated with ground movement induced by shield tunnelling in soft soil is a trending and effective method.

Probabilistic analysis is usually conducted based on a specific mechanical model. With respect to ground movement induced by shield tunnelling, empirical solutions, theoretical solutions, and numerical models are generally adopted. Empirical solutions as well as theoretical solutions are simple and output results relatively quickly. Taking advantage of the simplicity and practical applicability of the Peck formula [10], the probability for the occurrence of a certain value of surface settlements [11,12,13,14] can be easily obtained. However, because of the oversimplification of these solutions, the statistics of ground loss (i.e., volume loss V_L) are often assumed to be fixed values according to experience. As a result, these methods are not able to capture the uncertainty induced by the complex shield tunnelling process [12], such as the posture adjustments of the shield machine and synchronous grouting at the tail void. In other words, valuable conclusions can be drawn through these solutions, but the experience-based statistics of ground loss are inadequate in accurately accounting for the uncertainty of ground loss. Furthermore, it is hard to recognize the main reason for the development of ground movement and to manage its risks.

A three-dimensional numerical model makes it possible to reproduce the details in tunnel construction. Construction details, such as the route of the tunnel machine, the force of pushing jacks, the chamber pressure, over-excavation, the formation of a shield tail void, and so on, can be well captured in a numerical model. Mollon et al. [6] noted that the natural variability in soil, the possible errors or omissions in the measurements of soil characteristics, and the imperfect control of a tunnel boring machine should be considered when conducting stochastic analysis. They established a numerical model for tunnel excavations and considered the uncertainty of cohesion, the friction angle, Young’s modulus of soil, Young’s modulus of the grouting material, the chamber pressure, the uniform grouting pressure, and the liquid length of the grouting. To cope with the characteristics of soft clay, Miro et al. [4,5] established a numerical model based on an advanced elasto-plastic constitutive model with isotropic hardening, namely the Hardening Soil model. Additionally, a Bayesian back analysis approach is adopted to reduce the uncertainties of soil properties during tunnel excavations. Franco et al. [2] considered the uncertainty of six geotechnical parameters and five variables corresponding to the layer’s bottom depths in a numerical model. However, as mentioned by the above researchers, performing probabilistic analysis based on a three-dimensional numerical model is time-consuming. This is the reason why proper methods such as collocation-based stochastic response surface methodology, quadratic polynomial regression, and the hybrid point estimation method are adopted to make a stochastic analysis more effective.

With the rapid development of computers and techniques, the construction and monitoring data of tunnelling can be collected in real time. The above-mentioned mechanical model is not able to make use of the data, leading to a waste of resources. Machine learning-based algorithms [7,15] give new insights into this problem. With the aid of machine learning-based algorithms, the relationship between the ground movement and multiple factors, including geometrical, stratigraphic, and construction parameters, can be figured out successfully. However, as this method is only driven by data, the relationship between the ground movement and multiple factors lacks interpretability without the support of mechanisms.

This study utilizes the GAP parameter [16] to account for the uncertainty in shield tunnelling and, thus, makes the mechanical mechanism and data-driven method work together to better explain the cause of ground loss. The GAP parameter is defined according to the ground loss induced at every stage of tunnel construction. Also, it is a basic parameter in many theoretical and empirical solutions that is used for solving ground displacement. Therefore, it is suitable for solving ground movement, accounting for both construction and geotechnical parameters. When the uncertainty of construction and geotechnical parameters is addressed, a stochastic analysis can be performed according to the equations of the GAP parameter and the corresponding ground movement prediction. However, improvement should be made in this method owing to the unrealistic assumptions in the existing solution. The improvement will be introduced in the following text, and a probabilistic framework is then established based on the improved solution. This study is organized into the three following parts. Firstly, the original solution for the GAP parameter is reviewed, and an improvement for the solution made in this study through different methods is clarified. Secondly, by characterizing the uncertainty of the input parameters, the probabilistic framework is proposed for predicting ground movement. Through the equivalent parameters, the spatial variability in soft soil is also considered in the framework. At last, the verification is carried out by comparing the measured data in a practical case with the analyzed results from the probabilistic framework.

2. Improved Solution for the GAP Parameter

2.1. GAP Parameter Method

To predict ground movement caused by tunnelling, a large number of measured data were collected, and the concept of ground loss was proposed as early as 1969 [10]. Assuming the soil was undrained, ground loss was the main cause of surface settlement. For decades, the Peck formula has been widely used. As the very important parameters in the formula, the ground loss ratio was always determined according to experience. Generally, an assumed ground loss ratio range is utilized in determining the value of surface settlement. Based on the closed-form solution of Verruiit and Booker [17], Loganathan and Poulos [18] proposed the equivalent ground loss parameter for the first time. Based on this equivalent ground loss parameter, Lee et al. [16] further classified the ground loss into three types according to shield tunnelling process by giving the definition of the GAP parameter: ground loss caused by three-dimensional deformation developed at tunnel face, the posture change in the shield machine, and at the shield tail in the duration of the lining erection. The three-proportion ground loss (1) ahead of the tunnel face, (2) over the shield, (3) upon the installation of the lining segments is shown in Figure 1. The equation for the GAP parameter is written as:

$$\mathrm{GAP}={u^{*}}_{3D}+\omega +G_p$$

(1)

where u^*_3D represents the ground loss induced by 3D deformation at the tunnel face; ω is the quality of workmanship regarding to the alignment problems of shield machine; G_p denotes the physical gap produced in the process of shield tail advance and lining segment erection. Also, the three components can be regarded as the ground losses ahead of the tunnel face u*_3D, over the shield ω, and upon the erection of the lining G_p. According to Lee et al. [16], detailed definitions of the components are shown as follows. For u*_3D, excavating a tunnel relieves in situ stresses, causing soil to move into the tunnel face and creating ground loss. According to 3D finite element analyses [19], this face loss during shield tunnelling can be approximated in 2D plane strain analysis by increasing the maximum allowable radial displacement at the tunnel crown. That is to say, u*_3D is an approximate value for equaling the 3D face losses with 2D plain volume loss. For ω, the ground loss over the shield is the displaced and excavated parts in excess of the diameter of the cutting shield. The reason for this kind of ground loss is the alignment problems encountered when steering the shield. The most common case is that the operator of the shield machine usually advances the shield machine at a slightly upward pitch relative to the actual design grade to avoid the diving tendency of the shield. This excessive pitch will cause overcutting of the ground near the crown of the tunnel. For G_p, the cause of ground losses upon the erection of the lining is that the shield diameter is larger than the lining diameter. Also, there is a space for erection of lining needed for protecting the lining. After this, the ground will deform into the void left by the combination of the thickness of tail skin and the clearance. However, when expanded lining and the ground subsidence reduction technique are adopted, the ground loss will be decreased.

Compared with the ground loss ratio mentioned in the Peck formula, the equivalent ground loss is more reliable. Therefore, many scholars followed the concept of it when solving ground movement. For example, Park [20] defined four different displacement modes in advance by using the ground loss determined by the GAP parameter and gave the calculation results under undrained conditions. Bobet [21] extended the solution of water-free soil by Einstein and Schwartz [22] to saturated soil and put forward another theoretical solution of ground movement considering the uniform convergence displacement mode, in which the GAP parameter is considered in the analytical formula. However, some scholars [23] pointed out that there are some unrealistic assumptions in the existing solution of the GAP parameter for simplicity, leading to empirical values being selected for some parameters related to soil properties and tunnel geometry, and, thus, the use of the solution may be accompanied by uncertainties. Therefore, the improved solution for the GAP parameters is proposed and introduced in the following text.

2.2. Improved Solution for u*_3D

Take a review for the estimation for u*_3D. The idea of the solution is to equate the ground loss in the longitudinal section shown in Figure 2 with the ground loss in the cross-section shown in Figure 3 according to the definition of the u*_3D.

The ground loss per unit length Δy in the longitudinal section is calculated as follows:

$$V_{\mathrm{f}}=(\pi {r_0}^2)k_1{\displaystyle \frac{{\delta }_{\mathrm{ymax}}}{\mathsf{\Delta }y}}$$

(2)

where r₀ is the design radius of the shield tunnel; δ_ymax is the maximum intrusive displacement on the tunnel face; k₁ is the parameter representing the uniformity degree of horizontal intrusive displacement on the tunnel face; Δy is considered to be equivalent to the longitudinal study range. In Equation (1), Δy is assumed to be 1.0D, indicating that the displacement only exists in front of the tunnel face about 1.0D; k₁ is assumed to be 1.0, denoting that the displacement occurring on the tunnel face is uniformly distributed. This is the main assumptions for the u*_3D solution. However, these assumptions deviate greatly from the realistic cases.

An analytical method [24] for predicting the three-dimensional equivalent gap u*_3D on the tunnel face is established to deal with the deviations caused by the above assumptions. The analytical method is combined with the empirical formula of three-dimensional deformation at the tunnel face in soft soil, and the formula is shown as follows:

$${\delta }_y={\delta }_{y\max }{{\delta }_y}^{\prime }=a_1{\displaystyle \frac{D}{{\lambda }^2}}\cdot {\displaystyle \frac{(1-\lambda )\cdot \gamma \cdot D\cdot H/D}{c_{\mathrm{u}}}}\cdot e^{-{\displaystyle \frac{x^2+{(z+0.25\cdot e^{-\frac{x^2+{(z+0.25)}^2}{2\cdot {({\omega }_1\cdot D)}^2}})}^2}{2\cdot {(\omega \cdot D)}^2}}}$$

(3)

The equivaling process of ground loss on the tunnel face is analyzed, and the solution of u*_3D is, thus, improved as follows:

$$u{*}_{3\mathrm{D}}=2\cdot {\displaystyle \frac{{\displaystyle \underset{S_0}{\iint }{\delta }_{\mathrm{y}}(x,z)-{\delta }_{\mathrm{y}1}(x,z)}dxdz}{\pi DdL}}$$

(4)

where S₀ represents the actual range of intrusive displacement; δ_y(x, z) is the displacement function of the tunnel face referring to Equation (3); δ_y1(x, z) is the displacement function at dL in front of the tunnel face. In combination with Sagaseta’s solution [25], δ_y1(x, z) can be deduced by solving the following integral,

$${\delta }_{\mathrm{y}1}(x,z)={{\displaystyle \underset{x_0=-D}{\overset{D}{\int }}{\displaystyle {\int }_{z_0=-D}^D\left.S_{y1}+S_{y2}+S_{y3}\right|}}}_{y=dL}d{x}_{0}d{z}_0$$

(5)

where S_yl, S_y2, S_y3 are the three displacement components of spherical ground loss located at (x₀, y₀, z₀). As the intrusive displacement δ_y(x, z) is at the tunnel face, y₀ = 0. For y = dL, δ_y1(x, z) can be solved as:

$${\delta }_{\mathrm{y}1}={{\delta }_{\mathrm{y}1}}^{\prime }+{{\delta }_{\mathrm{y}2}}^{\prime }$$

(6)

$${{\delta }_{\mathrm{y}1}}^{\prime }={\displaystyle \frac{1}{2\pi }}\cdot dL{\displaystyle \underset{x_0=-D}{\overset{D}{\int }}{\displaystyle {\int }_{z_0=-D}^D\begin{array}{l}(\frac{1}{{\left[{\left(x-x_0\right)}^2+d{L}^2+{\left(z-z_0\right)}^2\right]}^{\frac{3}{2}}}-\\ \frac{1}{{\left[{\left(x-x_0\right)}^2+d{L}^2+{\left(z+z_0\right)}^2\right]}^{\frac{3}{2}}}){\delta }_{\mathrm{y}}(x_0,z_0)d{x}_{0}d{z}_0\end{array}}}$$

(7)

$${{\delta }_{\mathrm{y}2}}^{\prime }=-{\displaystyle \frac{3}{2{\pi }^2}}\cdot {\displaystyle \underset{x_0=-D}{\overset{D}{\int }}{\displaystyle \underset{z_0=-D}{\overset{D}{\int }}{\delta }_{\mathrm{y}}(x_0,z_0)\cdot }}{S_{y3}}^{\prime }d{x}_{0}d{z}_0$$

(8)

$${S_{y3}}^{\prime }={\displaystyle {\int }_{u=-\infty }^{+\infty }{\displaystyle {\int }_{t=-\infty }^{+\infty }\begin{array}{l}\frac{1}{{\left[{\left(x-u-x_0\right)}^2+{\left(dL-t\right)}^2+{(z_0)}^2\right]}^{\frac{3}{2}}}\cdot \\ \frac{z_0(u+t)(x-u-x_0)(dL-t)}{{[u^2+t^2+{z_0}^2]}^{5/2}}\end{array}}}\cdot dudt$$

(9)

After discussion, the value of dL can be determined as 0.5D. Finally, compared with both the results from numerical simulation and field measurements, the error caused by the assumptions is verified to be successfully reduced. Furthermore, the analytical method shows better predictive performance than the existing method.

2.3. Updated Method for ω

ω is the ground loss above the shield machine, due to the calibration problem of shield during steering. The shield operator would, as a rule of thumb, lift the shield machine slightly to prevent it from bowing. This is the main cause of the overcut of the shield machine. In addition, the irregular up-and-down motion of the shield machine will also make the traveling route inconsistent with the designed route. Depending on the operator’s experience, it is difficult to predict ω very accurately in advance.

The existing methods for solving ω are reviewed as follows. Two extreme cases (denoted as case A and B in the following text) are assumed first. In case A, it is assumed that soil freely deforms towards the annular gap caused by overcut, without soil and lining contacting with each other. In case B, the final deformation of soil is hypothesized to exceed the tunnel periphery. However, due to the restriction of the lining, soil deformation equals the physical gap between soil and lining segments. Also, that means the development of deformation is not sufficient. Thus, cases A and B can be considered as unlined and lined cases in 3D numerical simulations, respectively. ω can be equated with 0.33u_i and 0.6G_p, respectively, according to the numerical simulation results. In summary, this solution gives a hypothetical range, which has not been verified in practice.

Therefore, Yang et al. [26] collected field data from Longqing Road to Baiyun Road stations of the Kunming Metro Line 5 project and analyzed the relationship between ω with construction and geological parameters. A multivariate probabilistic distribution for ω according to the definition of Johnson distribution family is established.

According to the definition of the Johnson distribution family, the original variables Y can be transformed into standard normal distribution X with the distribution parameters listed in

Table 1. The parameters of the multivariate probability distribution.

Y1–Y11	Marginal Distribution	The Parameters of the Multivariate Probability Distribution
		ax	bx	ay	by
The pressure of the pushing jacks (A group)	SB	3.336	0.4112	1.898	0.431
The pressure of the pushing jacks (C group)	SB	2.025	0.0978	1.283	0.236
The stroke of articulation jacks (D group)	SB	1.388	0.800	1.391	−0.119
The stroke of articulation jacks (C group)	SB	0.935	−0.413	0.628	0.066
The stroke of articulation jacks (A group)	SB	0.893	0.399	0.941	0.009
The stroke of articulation jacks (B group)	SB	1.518	−0.876	1.109	0.220
Total force	SU	2.525	0.201	0.283	0.533
Rotation speed	SB	1.662	−2.400	1.474	0.407
Chamber pressure	SB	3.352	1.340	1.530	−0.119
Unit weight	SB	1.446	−1.139	1.614	−0.547
ω	SB	2.118	−1.000	0.777	−0.186

. A correlation matrix of the multivariate probability distribution is shown in

Table 2. The correlation matrix.

C	X1	X2	X3		X4	X5	X6	X7	X8	X9	X10	X11
X1	1	0.23	0.26		0.33	−0.04	−0.04	0.33	0.06	0.15	0.16	−0.28
X2		1	−0.15		0.37	−0.18	0.26	0.38	0.15	0.36	0.12	0.02
X3			1		0.27	0.38	−0.44	−0.06	0.09	−0.15	−0.36	−0.60
X4					1	−0.19	0.28	0.29	0.01	0.14	−0.01	0.02
X5						1	0.31	−0.23	0.16	−0.08	−0.42	−0.27
X6							1	0.07	0.03	0.20	−0.01	0.38
X7								1	0.05	0.50	0.07	−0.06
X8									1	0.01	−0.25	−0.17
X9		Symmetry								1	−0.09	0.04
X10											1	0.23
X11												1

. The multivariate normal probability density function after transforming is defined uniquely by a mean vector μ and a covariance matrix C:

$$f(X)={\left|C\right|}^{-(1/2)}{(2\mathsf{\pi })}^{-(d/2)}{\mathrm{e}}^{-(1/2){(\mathit{X}-\mathsf{\mu })}^{\mathrm{T}}{\mathit{C}}^{-1}(\mathit{X}-\mathsf{\mu })}$$

(10)

The accuracy of the established multivariate probability distribution model is verified by simulating a large number of examples.

At the same time, with the field-acquired new data, the model can be updated showing good performance on dynamic prediction compared with the field data from three shield tunnels. The Bayesian updating process can be divided into two basic steps.

Step 1:

The parameters in the X space were first updated. X_i is standard normal with mean μ_i = 0 and standard deviation σ_i = 1 before the update. It can be shown that X_i is still normally distributed but with updated mean μ_i’ and updated standard deviation σ_i’ after the update (with the information of (X_j, X_k, ∙∙∙)). These two values are shown below:

$${{\mu }_i}^{\prime }=E(X_i\left|X_j,X_k,\cdots \right.)=\left[{\delta }_{ij}{\delta }_{jk}\cdots \right]{\left[\begin{array}{ccc}1& {\delta }_{jk}& \cdots \\ {\delta }_{jk}& 1& \\ ⋮& & \ddots \end{array}\right]}^{-1}\left[\begin{array}{c}{X}_j\\ X_k\\ ⋮\end{array}\right]$$

(11)

$${{\sigma }_i^{\prime }}^2=\mathrm{VAR}(X_i\left|X_j,X_k,\cdots \right.)=1-\left[{\delta }_{ij}{\delta }_{jk}\cdots \right]{\left[\begin{array}{ccc}1& {\delta }_{jk}& \cdots \\ {\delta }_{jk}& 1& \\ ⋮& & \ddots \end{array}\right]}^{-1}\left[\begin{array}{c}{\delta }_{ij}\\ {\delta }_{jk}\\ ⋮\end{array}\right]$$

(12)

where E and VAR denote mean and variance, respectively, and δ_ij refers to the product–moment correlation between X_i and X_j (seeing Table 2). For simplicity, the covariance matrix C and the updating database X were introduced as follows:

$$\mathit{C}=\left\{\left.\begin{array}{cccc}1& {\delta }_{ij}& {\delta }_{ik}& \dots \\ {\delta }_{ij}& 1& {\delta }_{jk}& \\ {\delta }_{ik}& {\delta }_{jk}& 1& \\ ⋮& & & \ddots \end{array}\right\}\right.=\left\{\left.\begin{array}{cc}{\mathit{C}}_{(11)}& {\mathit{C}}_{(12)}\\ {\mathit{C}}_{(21)}& {\mathit{C}}_{(22)}\end{array}\right\}\right.$$

(13)

$$X=\left\{\begin{array}{c}\frac{X_i}{X_j}\\ X_k\\ ⋮\end{array}\right\}=\left\{\begin{array}{c}{\mathit{X}}_{(1)}\\ {\mathit{X}}_{(2)}\end{array}\right\}$$

(14)

Then, Equations (11) and (12) can be rewritten as follows:

$${\mu }_i^{\prime }={\mathit{C}}_{(12)}{{\mathit{C}}_{(22)}}^{-1}{\mathit{X}}_{(2)}$$

(15)

$${{\sigma }_i^{\prime }}^2={\mathit{C}}_{(11)}-{\mathit{C}}_{(12)}{\mathit{C}}_{(22)}^{-1}{\mathit{C}}_{(21)}$$

(16)

Step 2:

The parameters in the Y space are updated. By substituting the updated mean μ’_i and the updated standard deviation σ’_i for the original mean μ_i and standard deviation σ_i in Equation (13), the following equation can be obtained:

$$\begin{array}{l}{\displaystyle \frac{X_i-b_{\mathrm{x}i}}{a_{\mathrm{x}i}}}={\displaystyle \frac{({\mu }_i^{\prime }+{\sigma }_i^{\prime }Z-b_{\mathrm{x}i})}{a_{\mathrm{x}i}}}\\ =[Z-{\displaystyle \frac{(b_{\mathrm{x}i}-{\mu }_i^{\prime })}{{\sigma }_i^{\prime }}}]/({\displaystyle \frac{a_{\mathrm{x}i}}{{\sigma }_i^{\prime }}})=\kappa (Y_{\mathrm{n}})\end{array}$$

(17)

where Z is the standard normal. As a result, the posterior distribution of Y_i is still a Johnson distribution of the same family type (SU, SB, or SL) with the following posterior parameters:

$$a_{\mathrm{x}i}^{\prime }=a_{\mathrm{x}i}/{\sigma }_i^{\prime }$$

(18)

$$b_{\mathrm{x}i}^{\prime }=(b_{\mathrm{x}i}-{\mu }_i^{\prime })/{\sigma }_i^{\prime }$$

(19)

$$b_{\mathrm{y}i}^{\prime }=b_{\mathrm{y}i}$$

(20)

$$a_{\mathrm{x}i}^{\prime }=a_{\mathrm{x}i}$$

(21)

As a result, this multivariate probability distribution model can be directly utilized in the following probabilistic analysis.

2.4. Uncertainty of G_p

Compared with u*_3D and ω, G_p is easier to determine, and its maximum equals the value of the tail void. When the lining segment is erected and the shield tail leaves the lining segments, an annular gap, named the tail void, is left between the lining segments and the actual excavation boundary. In the absence of support, soil deforms towards the lining segments to fill the tail void. The value of the tail void is roughly consistent with the difference between the external diameter of the shield machine and the external diameter of the lining segments, so that it is determined by the thickness of the shield shell and the operating space of the shield tail, which is generally 8–16 cm. If the tunnel diameter reaches more than 14 m, the tail void can reach 20 cm. Effective grouting can reduce the ground loss caused by the tail void, and the remaining gap after grouting is G_p. Namely, the relative value of G_p depends on the grouting effect. To ensure the grouting effect, the grouting rate is generally set to 120–250%. However, due to the liquidity of grouting material and the interaction between grouting material and soil, the effective grouting amount is inconsistent with the actual grouting amount. Some studies [27] have shown that when the grouting rate is set to 170%, the effective grouting ratio does not reach 100%. To conclude, the uncertainty of G_p depends on the effective grouting ratio.

The non-destructive geophysical exploration methods are widely adopted to identify the range of grouting diffusion. The ground-penetrating radar method is often used to test the grouting effect behind the shield shell due to its lightweight instrument, mature technology and high level of automation. The ground-penetrating radar can detect the distribution pattern of the grouting body along the circumferential and longitudinal directions, as well as whether there are gaps between the grouting body and the lining segments, and, thus, can directly and accurately determine the grouting effect. This is the basic principle of determining the physical gap by using ground-penetrating radar. On the basis of appropriate selection of detection parameters, proper data processing, sufficient simulation test comparisons and rich interpretation experience, it is reasonable to estimate the G_p by using the scientifically processed ground-penetrating radar results. Based on the ground-penetrating radar results in the five cases reported in the literature, five probability distribution models describing the effective grouting ratio were established [28]. The main parameters of the probability models are listed in

Table 3. The statistics of effective grouting ratio and probability models.

Model No.	Mean	Standard Deviation	COV	Kurtosis	Skewness	Distribution Type	Fitting Parameters	h	p
1	0.94	0.25	0.27	−0.11	0.34	Logarithmic logical	μ = −0.085; σ = 0.154	0	0.39
2	0.95	0.19	0.19	−0.28	−0.22	Normal	μ = 0.954; σ = 0.186	0	0.94
3	0.81	0.25	0.31	−0.75	−0.06	Normal	μ = 0.811; σ = 0.064	0	0.44
4	1.13	0.07	0.06	−0.63	0.001	Normal	μ = 1.128; σ = 0.070	0	0.99
5	1.40	0.21	0.15	2.18	0.86	Normal	μ = 1.404; σ = 0.209	0	0.64

. The effective grouting ratio inverted using the measured displacement data proves the practical applicability of the established models. Combined with the value of the tail void, the five established probability models can directly represent the uncertainty of G_p.

2.5. Spatial Variability in Soil

To account for the spatial variability in soil parameters, the finite element/difference method is usually employed, based on the theory of random fields, and the random finite element/difference method (RFEM/RFDM) is used within the framework of Monte Carlo simulation (referred to as MCS) to solve the problem. However, this method requires time-consuming computations, especially when the failure probability is very small. The equivalent method referring to [29,30] is adopted in the probabilistic framework to account for the spatial variability in soil. The equivalent method is based on the random field model, regression method, and MCS. The core of this method lies in solving the equivalent parameters that achieve the same effect as the random variable model for the random field model. Through the equivalent method, the statistical parameters of the equivalent parameters can be obtained. Using the regression method, an explicit response surface function of the random variable model is established. By inputting the equivalent parameters into the explicit response surface function for reliability analysis, the time cost of reliability analysis considering the spatial variability is reduced.

3. Development of the Probabilistic Framework

Based on the improved solution of the GAP parameter, the probabilistic framework can be developed. A flowchart is depicted in Figure 4, and the basic workflow is shown as follows.

Step 1:

Input the geometrical parameters of the shield tunnel excavated in soft soil. Obtain the buried depth of tunnel axis H, tunnel diameter D, tail void (D₁–D₂) of the given cross-section, where the external diameter of the shield machine is D1 and the external diameter of the lining segments is D₂.

Step 2:

Determine the basic parameters of probabilistic analysis related to u*_3D. In this study, referring to the improved solution for u*_3D, the chamber pressure, the unit weight and the undrain shear strength of soil are the key parameters. According to the project profile, the appropriate probability distribution model, mean value and COV can be selected for random sampling. Then, input the samples into the analytical Equation (4), which is mentioned in Section 2.2. In this way, the samples for u*_3D can be easily obtained with sample size equaling n.

Step 3:

Obtain the influencing parameters for ω at a certain cross-section of which the ground movement is to be calculated. The influencing parameters can be recognized as the pressure of pushing jacks and articulation jacks, total thrust, cutter speed, chamber pressure and unit weight based on the multivariate distribution mentioned in Section 2.3. Follow the Bayesian updating process and obtain the means and standard deviation of updated marginal distribution for ω. Also, conduct random sampling based on the updated marginal distribution and obtain the samples for ω. The sample size is also set as n.

Step 4:

If there are sufficient geological exploration results, then use them to deduce the probability distribution model of the effective grouting ratio. If there are no such results, the uncertainty can be characterized by assuming a proper probability distribution model, as mentioned in Section 2.4. With the tail void (D₁–D₂) calculated in step 1 known, the probability distribution for G_p is deduced. Continue sampling on the probability distribution of G_p. The sample size is the same as the other GAP components.

Step 5:

Summarize the samples u*_3D, ω and G^p, and, thus, obtain the samples of the GAP parameter. When inputting the GAP parameter in proper solution, the ground movement samples can be obtained. Then, the Monte Carlo strategy is adopted for the probability analysis.

Step 6:

Establish a finite difference/finite element model for the cross-section. Select appropriate autocorrelation function, probability distribution model and COV based on the spatial variability characteristics in the soil according to the statistics of soft soil collected by [31] and then carry out random finite difference/finite element analysis. Following the equivalent process, the equivalent parameters considering the spatial variability are obtained, and the probability analysis considering spatial variability can be carried out.

The number of Monte Carlo simulation times should be set in advance in the probability analysis. According to the regulations of the Ministry of Housing and Urban-Rural Development of the People’s Republic of China [32], the construction risk can be ranked as four levels considering the incident possibility and the corresponding loss, as shown in

Table 4. Construction risk level.

	Loss Level	A	B	C	D	E
Probability Level		Calamitous	Very Serious	Serious	Considerable	Ignorable
1	Frequent	I	I	I	II	III
2	Possible	I	I	II	III	III
3	Ocassional	I	II	III	III	IV
4	Infrequent	II	III	III	IV	IV
5	Impossible	III	III	IV	IV	IV

. The risk acceptance criteria stipulate that level I, II, III and IV correspond to unacceptable, unwilling, acceptable and negligible risks, respectively. In Table 4, when the incident possibility level is 3 to 5, no matter how high the risk loss level is, most of the construction risk levels are acceptable. In contrast, when the possibility level is 1 to 2, most of the project construction risk is unacceptable. Therefore, only the possibility levels 1–2 are considered in the subsequent analysis. As the possibility of level 2 in the regulation is 0.1~0.01, the number of Monte Carlo simulations is set to 10,000 times.

4. Application of the Proposed Framework

4.1. Project Information

This section introduces a real metro line project to explain the application of the proposed probabilistic framework. Kunming Metro Line 5 project starts from the Expo Park station in the north and ends at the Baofeng Village station in the south. The line from Longqing Road to Baiyun Road station is laid along Bailong Road. It first passes through the Jinshuihe River and then passes through the Shizha overpass and ramp pile foundation of the East 2nd Ring Road. Then, it turns west along Bailong Road to the intersection of Bailong Road and Baiyun Road and enters Baiyun Road station. The right line length is 975.658 m, and the left line length is 985.353 m.

The tunnel from Longqing Station to Baiyun Road Station mainly passes through plasticizing silty clay and peaty soil. The shield tunnel is buried 9.66 m–12.53 m in depth. There are many buildings on both sides of the construction site. The main environmental risk is that the shield tunnel crosses near the side of the pile foundation of the Shizha overpass and passes through the Jinzhi river, and so on. The minimum horizontal distance between the pile foundation of Shizha overpass and the right line is 0.83 m, and there may be a reinforcement area under the bridge of the Jinzhi river. This reinforcement area may invade the tunnel limit. The minimum horizontal distance between the left line and right line with the pile foundation is 3.29 m and 5.59 m, respectively. Near the construction area under Bailong Road, there are many underground pipelines, and the pipelines are densely distributed, which are mainly for tap water, sewage, electricity, communication and gas pipes. The buried depth of most pipelines is shallow, and only some drainage pipelines are buried relatively deep (about 2.3–3.9 m). Also, the surface settlement and soil displacement should be detected, to ensure the safety of the surrounding environment and underground pipelines.

4.2. Probabilistic Analysis

Field data from the Bailong tunnel interval introduced in Section 4.1 are used to illustrate the probabilistic framework proposed in this study.

The fifth ring of the left line is selected for discussion. On 11 July 2020, the shield machine reached the 9th ring, and the cumulative surface settlement of the 5th ring of the left line reached −26.54 mm. Following the workflow of the probabilistic framework mentioned in Section 3, a probabilistic analysis is shown as follows.

Step 1:

Input the geometry of the shield tunnel. The buried depth of the tunnel in the cross-section corresponding to the 5th ring is 10.24 m. The tunnel diameter is 6.31 m, and the tail void is 110 mm.

Step 2:

According to the project overview, the chamber pressure at this section is 1.32 bar. The unit weight and undrained shear strength at the tunnel axis are 16.56 kN/m³ and 15.87 kPa, respectively. These deterministic values are considered as the mean value in probabilistic analysis. As normal distribution is the most commonly used model, it is chosen to depict these three main parameters, and according to relevant engineering experience [28], COV for chamber pressure, unit weight and undrained shear strength are set 0.1, 0.05 and 0.4, respectively. Sampling is carried out according to the above-determined statistical characteristic. Input the samples into the theoretical formula of u*_3D, and obtain 10,000 sets of u*_3D samples.

Step 3:

When the shield machine passed through where the 5th ring is located, the pressure of the pushing jacks (A group) was 5.0 MPa; the pressure of the pushing jacks (C group) was 7.0 MPa; the strokes of articulation jacks of D, C, A, and B groups were 69, 102, 63, and 86 mm; the total force was 6400 kN; the rotation speed was 1.2 r/min; the chamber pressure was 0.132 MPa; and the average unit weight (which is different from the unit weight in step 2 in definition) was 18.5 kN/m³. After conducting Bayesian updating, the updated mean μ_x and the updated standard deviation σ_x were 0.758 and 0.730, respectively. The distribution parameters can also be updated as a_x = 2.902; b_x = −2.408; a_y = 0.777; b_y = −0.186. The updated multivariate distribution is sampled with a sample size of 10,000.

Step 4:

In this study, the effective grouting ratio is assumed to follow the normal distribution model 3 shown in Table 3. The statistical characteristic is determined by referring to an actual case in Shanghai, China [33].

Step 5:

Summarize the samples of the three components of the GAP parameter in the previous three steps, and then obtain the samples of the GAP parameter. Input the GAP parameter into the selected solution for ground movement. In this case, the selected solution is the analytical solution proposed by Loganathan and Poulos [18]. Obtain the displacement samples without considering the spatial variability, as shown in Figure 5. The statistical characteristics of u*_3D, G_p and ω are shown in Table 2.

Figure 5 shows the distribution for the three proportions of the GAP parameter and maximum of the surface settlement S_max. The distribution of S_max is relatively symmetrical, mainly concentrated in the range of −60~10 mm, in which some of the displacement is positive, that is, surface uplift. However, the frequency of surface uplift is much smaller than that of subsidence. The distribution of ω is consistent with the updated distribution pattern mentioned in step 3. The distribution of G_p is also symmetrical, consistent with the distribution pattern of the effective grouting ratio, and the distribution range is −30~60 mm. The distribution of u*_3D is obviously skewed to the left, ranging from 0 to 40 mm. The 95% confidence interval, median and measured results are also plotted in the figure, and it is not difficult to find that the measured S_max is in the 95% confidence interval and highly consistent with the median. The measured results of ω are also in the corresponding 95% confidence interval. This shows that the probabilistic framework without considering the spatial variability can predict the displacement in a practical way.

Step 6:

A numerical model of the tunnel is established according to the tunnel geometry obtained in step 1. As the elastic modulus E is the most relevant parameter in estimating ground movement, E is considered to be the spatial variability parameter.

For simplicity, Mohr–Coulomb model is chosen to be the soil constitutive model. The values of unit weight and E are determined as mentioned in step 2. The average values of cohesion and friction angle at the tunnel axis are 18.37 kPa and 6.2°, respectively. To simulate the non-uniform convergence mode at tunnel periphery, different stresses are released at the bottom and top of the tunnel to make the unbalanced forces around the tunnel consistent. When the stress release ratio at the top of the tunnel is 0.25, the GAP parameter and S_max are extracted, which are 44.38 mm and 26.56 mm, respectively. This is consistent with the median of the GAP parameter and S_max when not considering the spatial variability in Step 5, and this model is used to consider the spatial variability in E.

Then, set E as 1, 5, 10, 15 and 20 MPa, and the stress release ratio at the top of the tunnel is 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5 and 0.55, respectively, to establish an explicit response surface function. Figure 6 shows the change in S_max with the GAP parameter. It is not difficult to find that S_max changes almost linearly with the GAP parameter. No matter how the value of E changes, the slope of the S_max-GAP parameter curve stays almost unchanged, while the intercept decreases with the increase in E, showing an obvious inverse relationship, as shown in Figure 6. Therefore, an explicit response surface function can be established, as shown in Equation (10).

$$S_{\max }\left(E,g\right)=0.46\mathrm{GAP}+40.5/E$$

(22)

Thus, 3000 sets of random fields are generated to calculate the equivalent parameters of E_eq. The values of the spatial variability parameters are as follows: E is assumed to follow a lognormal distribution [8,9,15], and its mean is the elastic modulus value at the tunnel axis, which is 400c_u = 6.35 MPa. The autocorrelation function is a two-dimensional Gaussian function, and COV is set to 0.25; the horizontal and vertical correlation length is 20 m and 2 m, respectively [31,34,35].

Input 3000 groups of S_max and the GAP parameter into Equation (10), and obtain the equivalent parameter E_eq of E, as shown in Figure 7. Sample the fitting probability distribution of E_eq, and combine the GAP parameter samples generated in step 5 with Equation (10) to generate displacement samples considering spatial variability. The results are shown in Figure 7b. It is not difficult to find that the measured results of S_max are in the 95% confidence interval and are highly consistent with the median. Compared with Figure 5a, the 95% confidence interval can be further reduced after considering the spatial variability, and the median can be closer to the measured S_max value. This indicates that the proposed probabilistic framework shows better prediction performance when considering spatial variability, and it is necessary to consider the spatial variability in soil.

5. Conclusions

In this study, based on the improved solutions of the GAP parameter, the probabilistic framework for ground movement induced by shield tunnelling in soft soil is proposed, and the specific process of the method is described. This framework can fully consider the uncertainty during shield tunnelling from both mechanism and data aspects based on the GAP parameter. Combined with the field data of the fifth ring of the Bailong tunnel, referring to the uncertainty characterized both onsite and in the literature, the method is applied in practical projects. The results show that the method can fully consider various uncertainties in the shield tunnelling process through the improved solutions of the GAP parameter. From the perspective of a probabilistic point, the ground movement of the shield tunnel excavated in soft soil is predicted. The next step is when a certain standard is set, and the failure probability related to the ground movement of shield tunnelling can be efficiently calculated. Through the prediction for the ground movement of the fifth ring of the Bailong tunnel, it is found that the measured data can fall within the 95% confidence interval of the predicted samples, whether the spatial variability is considered or not, and the median of the samples is highly consistent with the measured values. By considering the spatial variability in soil, the results can be more accurate. As a result, the probabilistic framework can fully consider the uncertainties of the shield process and solve the existing dilemma of data waste and overlook of the mechanism.

However, the method proposed in this study has certain limitations. (1) The proposed framework can be more scientific and persuasive to incorporate the synergistic effects of each component of the GAP parameter with spatial variability into the solution of the GAP parameters. (2) The Monte Carlo simulation must be reduced by adopting more efficient methods in the probabilistic framework in future work. (3) The probabilistic framework can be extended to the application in other types of soil or more complex situations.