Risk Assessment of Tunnel Construction Deformation Under Spatial Variation in Hydraulic Parameters

Abstract

Tunnel construction in soft soil environments involves significant geological and hydraulic uncertainty, particularly where permeable sandy interlayers within soft clay are prone to seepage-induced instability and excessive settlement. Although hydraulic–mechanical coupling is widely recognized, the spatial variability of key soil parameters (e.g., permeability and elastic modulus) is often inadequately represented, limiting quantitative evaluation of heterogeneous ground effects on construction-induced deformation. In this study, statistical analyses of site investigation and monitoring data are conducted to characterize parameter distributions and transverse settlement trough morphology, supporting model validation. A fluid–solid hydro-mechanical coupled numerical model in ABAQUS demonstrates that groundwater flow increases maximum surface settlement from 3.18 cm to 3.58 cm, confirming the significance of hydraulic coupling. To quantify spatial variability effects, a stochastic finite element framework based on random field theory is developed, showing that variations in vertical correlation length influence both the mean and dispersion of maximum settlement. Specifically, under a settlement control threshold of 40 mm, the failure probability decreases from 24.21% to 1.01% as the vertical correlation length increases from 1.5 m to 6 m. Finally, an engineering-oriented risk assessment framework is established using settlement trough area as the core loss indicator; its lognormal distribution is verified, and failure probability and reliability indices are integrated with code-based thresholds to evaluate construction risk under different scenarios, with the resulting risk levels ranging from Relatively High (Level III) to Moderate (Level II).

1. Introduction

Spatial variability of geotechnical parameters is an intrinsic characteristic of natural soils formed through complex depositional and post-depositional processes. Early research recognized that soil properties exhibit inherent randomness rather than deterministic uniformity [1]. Subsequent developments in probabilistic geotechnics established formal frameworks for uncertainty quantification, including first-order reliability approaches [2] and random field theory for spatially correlated media [3]. Comprehensive statistical characterization of geotechnical variability was later summarized by Phoon and Kulhawy [4]. Compared with slopes and shallow foundations, tunnel–soil interaction systems are mechanically more complex and more sensitive to spatial heterogeneity in both strength and hydraulic parameters [5]. Insufficient representation of subsurface variability may therefore lead to substantial discrepancies between assumed and actual ground conditions, potentially resulting in excessive settlement during construction.

Hydro-mechanical coupling plays a decisive role in tunnel excavation [6], particularly in saturated soft soils. Excavation induces unloading and stress redistribution while simultaneously altering groundwater flow paths, leading to pore pressure evolution and effective stress changes. Numerical investigations have demonstrated that inclusion of seepage significantly increases predicted displacement and stress levels around tunnels [7]. Analytical solutions have further clarified pore pressure distributions in underwater tunnelling scenarios [8]. More recent coupled analyses confirm that neglecting seepage may underestimate deformation and consolidation settlements in soft ground [9]. Despite these advances, most hydro-mechanical simulations adopt deterministic ground models, typically assuming homogeneous or simplified layered soils, without explicitly incorporating spatial randomness in permeability and stiffness.

Random field theory has been extensively applied in slope and foundation reliability analyses to represent spatially correlated soil parameters. For example, the influence of coefficient of variation and correlation length on foundation reliability has been quantified using two-dimensional random fields [10], and the impact of autocorrelation structure on probabilistic risk assessment has been systematically examined [11,12]. In tunnel engineering, spatial variability has been introduced into face stability and deformation analyses [13,14,15]. These studies demonstrate that parameter variability can significantly affect predicted mechanical response and failure probability. However, applications in soft soil tunnelling remain limited, especially regarding hydraulic parameters such as permeability, which often exhibit large dispersion and strong spatial correlation. The quantitative relationship between hydraulic parameter spatial variability and construction-induced surface settlement therefore remains insufficiently clarified.

From a risk assessment perspective, probabilistic concepts were formally introduced into tunnelling by Einstein [16], who emphasized systematic risk analysis in underground construction. Monte Carlo-based probabilistic frameworks were subsequently employed in major infrastructure projects to classify and manage construction risks. Integrated evaluation approaches combining fault tree analysis and multi-criteria decision methods have also been proposed for TBM tunnelling [17]. Furthermore, probabilistic assessment of groundwater-related hazards has highlighted the importance of hydrogeological uncertainty in underground works [18]. Multi-level frameworks incorporating both quantitative indicators and expert judgment have been developed to evaluate tunnelling-induced risks to adjacent structures [19,20]. Although these contributions have advanced tunnel risk management methodologies, several limitations remain. Many existing approaches emphasize qualitative or semi-quantitative evaluation, without directly linking spatial parameter randomness to probabilistic deformation outcomes. In addition, loss quantification is often decoupled from reliability analysis, limiting the consistency of risk metrics. Most importantly, the combined influence of hydraulic parameter spatial variability and hydro-mechanical coupling on settlement probability distributions has rarely been integrated into a unified, engineering-oriented risk assessment framework.

To address the limitations in previous studies, this research investigates tunnel construction deformation in soft soil considering the spatial variability of hydraulic parameters and their influence on construction risk. Existing studies have generally recognized the importance of hydro-mechanical coupling in soft ground tunnelling, but the heterogeneous distribution of key parameters such as permeability and elastic modulus has often been simplified or ignored, leading to insufficient understanding of uncertainty in deformation response and risk evolution [21,22,23]. In particular, the combined effects of seepage, parameter randomness, and spatial correlation on settlement behavior have not been systematically quantified.

Accordingly, this study conducts statistical analyses of site investigation and monitoring data to characterize soil parameter distributions and settlement trough features, and develops a fluid–solid coupled numerical model to evaluate the effect of groundwater seepage on tunnel-induced deformation. Based on random field theory, a stochastic finite element framework is established to assess the influence of spatial variability and correlation structure on settlement uncertainty. Furthermore, an engineering-oriented risk assessment framework is proposed by integrating settlement-related loss indicators, failure probability, and reliability indices to evaluate construction risk under different scenarios. The study provides a quantitative basis for risk-informed tunnel design and construction management in soft soil environments.

2. Statistical Analysis of Hydraulic Parameters

Based on extensive geotechnical investigation data from completed projects in Shanghai, a database of soil parameters was established and representative layers were selected for statistical analysis. Particular focus was placed on the permeability coefficient and compression modulus, which govern groundwater flow and soil mechanical response, respectively, in hydro-mechanical coupling. Tunnel excavation disturbs the initial equilibrium state of the ground, triggering interaction between the seepage and stress fields. Changes in groundwater level and pore water pressure alter the effective stress, leading to soil deformation and settlement. Therefore, permeability and compressibility are the key parameters controlling the coupled hydraulic–mechanical response.

The probability distributions of permeability coefficients for typical Shanghai soil layers are shown in Figure 1. After log₁₀ transformation, the data generally limited outliers likely caused by stratigraphic heterogeneity. The coefficients of variation (COV) exceed 0.5 for all layers, indicating significant spatial variability and underscoring the necessity of probabilistic characterization in coupled analyses of tunnel excavation.

Similarly, a statistical analysis was conducted on the compression modulus of each soil layer to characterize its distribution pattern and variability. The results indicate noticeable differences among the soil layers, with mean compression modulus values ranging approximately from 3.84 to 12.47 MPa and coefficients of variation between 0.15 and 0.30, suggesting a moderate level of dispersion overall. The corresponding mean values and coefficients of variation for each soil layer are summarized in

Table 1. Statistical characteristics of compression modulus for each soil layer.

Soil Layer	Sample	Modulus Mean/MPa	Modulus COV	Permeability Mean/m/s	Permeability COV
②3	96	9.46	0.21	1.19 × 10−6	0.65
⑤1	102	3.84	0.30	1.15 × 10−9	0.71
⑤4	118	7.04	0.15	1.53 × 10−8	0.84
⑦2	99	12.47	0.23	4.68 × 10−8	0.48

, providing a basis for subsequent deformation analysis and parameter selection.

3. Methodology

Based on the previously established permeability database for the Shanghai soft soil region, the statistical characteristics of key hydraulic and mechanical parameters, including mean values, coefficients of variation, and probability distribution types, have been obtained. These statistical descriptors provide the quantitative foundation for subsequent deterministic and stochastic modelling. To systematically evaluate the influence of hydraulic parameter variability on tunnel excavation deformation, a hydro-mechanical probabilistic analysis framework is developed, as illustrated in Figure 2. The workflow integrates deterministic finite element modelling, spatial random field generation, and batch stochastic simulations to capture both coupled seepage–deformation mechanisms and parameter uncertainty.

The procedure consists of the following stages:

(1)

A deterministic hydro-mechanical finite element model is first established in ABAQUS and validated against monitoring data.

(2)

Spatial correlation lengths and statistical distributions of hydraulic parameters are defined based on site investigation results.

(3)

Correlated random fields are generated using the Karhunen–Loève (KL) expansion method and then mapped onto the finite element mesh.

(4)

Monte Carlo simulations are then performed by repeatedly assigning random fields, running ABAQUS analyses, and extracting settlement responses until the target number of samples is reached.

(5)

Finally, all simulation outputs are compiled for statistical evaluation and probabilistic risk assessment.

3.1. Hydro-Mechanical Coupled Numerical Model

Tunnel excavation in saturated soft soils triggers a coupled response between deformation of the soil skeleton and groundwater seepage. Excavation-induced unloading modifies the total stress field and changes drainage conditions, which in turn drives pore pressure redistribution and alters effective stress. As a result, ground deformation is controlled not only by mechanical stress release but also by the transient evolution of pore water pressure and seepage flow. To capture these interactions, a fully coupled hydro-mechanical model is adopted, in which stress equilibrium of the porous medium and seepage continuity are solved simultaneously, because deformation and pore pressure dissipation occur concurrently during tunnel excavation and cannot be accurately captured using uncoupled mechanical or seepage analyses. This model therefore provides a deterministic basis for evaluating seepage-amplified settlement and subsequent stochastic analyses with spatially variable hydraulic parameters.

The equilibrium equation of the porous medium is expressed as

$$\nabla •\sigma +\rho \mathrm{g}=0$$

(1)

where σ is the total stress tensor, ρ is the bulk density of the saturated soil–fluid mixture, and g is the gravitational acceleration vector. According to the effective stress principle,

$$\sigma ={\sigma }^{\prime }-uI$$

(2)

where σ′ is the effective stress tensor carried by the soil skeleton, u is the pore water pressure, and I is the identity tensor. The seepage behaviour follows Darcy’s law,

$$q=-k\nabla h$$

(3)

where q is the Darcy seepage velocity vector (discharge per unit area), k is the hydraulic conductivity (permeability coefficient), and h is the hydraulic head. Darcy’s law is selected because groundwater flow in soft soils and sandy interlayers generally remains under laminar flow conditions, for which this relationship is widely validated. By combining Darcy’s law with mass conservation, the seepage continuity equation is written as

$$\nabla •q+{\displaystyle \frac{\partial {\epsilon }_v}{\partial t}}=0$$

(4)

where ε_v is the volumetric strain of the soil skeleton and t is time. This equation is required to satisfy fluid mass conservation while accounting for volume change in the deformable porous medium during consolidation. Equations (1)–(4) form the governing system for hydro-mechanical coupling.

In practical seepage processes, hydro-mechanical coupling manifests as a bidirectional interaction. Variations in pore water pressure modify the effective stress state, thereby influencing deformation and settlement. Conversely, deformation alters pore structure and hydraulic conductivity, leading to redistribution of pore pressure and seepage velocity. This nonlinear feedback can be viewed as direct coupling (pore pressure acting on the soil skeleton) and indirect coupling (deformation-induced hydraulic evolution). The governing equations are solved simultaneously using a pore pressure–displacement finite element formulation, where displacement and pore pressure are primary variables. This unified framework captures excavation-induced stress release, pore pressure redistribution, and consolidation settlement, providing a realistic deterministic basis for subsequent stochastic analyses considering spatial variability of hydraulic parameters.

3.2. Finite Element Model

A two-dimensional plane strain finite element model was developed to simulate the transverse response of shield tunnel excavation based on Abaqus 2022. The model dimensions are 100 m × 80 m to minimize boundary effects. The tunnel has an outer diameter of 15 m, lining thickness of 0.65 m, and burial depth of 24.5 m, as shown in Figure 3.

The soil layers from top to bottom include silty sand (②₃), clay (⑤₁), silty clay (⑤₄), fine sand (⑦₂), clay (⑧₁₁), and interbedded sandy clay and silty sand (⑧₂₂), as shown in Figure 3. Soil behavior was modeled using the Mohr–Coulomb constitutive model. The compression modulus obtained from site investigation was converted into elastic modulus for numerical implementation. Material parameters of soils, lining, and grouting are summarized in

Table 2. Model soil parameters and material parameters.

Material	Layers/m	γ/(kN/m3)	E/MPa	μ	c/kPa	φ/°
②3	16.50	18.20	40.00	0.30	13.00	34.50
⑤1	19.10	17.70	15.00	0.35	28.00	23.50
⑤4	2.10	19.50	20.00	0.35	53.00	22.00
⑦2	3.90	18.80	75.00	0.30	7.00	36.00
⑧11	22.30	17.70	75.00	0.35	34.00	22.50
⑧22	16.10	18.20	80.00	0.30	34.00	26.00
Lining	/	27.00	28,800.00	0.20	/	/
Grouting	/	20.00	200.00	0.30	/	/

. γ denotes unit weight, E denotes elastic modulus, μ denotes Poisson’s ratio, c denotes cohesion, and φ denotes internal friction angle.

The model consists of 32,632 nodes and 32,116 elements, using coupled pore pressure–displacement elements (CPE4P) and reduced-integration elements (CPS4R). The CPE4P element is a four-node bilinear plane strain element with coupled displacement and pore pressure degrees of freedom, where each node includes two displacement components and one pore pressure variable, enabling simulation of hydro-mechanical coupling behavior. The CPS4R element is a four-node bilinear plane stress element with reduced integration, primarily used to model structural components such as tunnel lining, featuring two displacement degrees of freedom per node and improved computational efficiency. Lateral boundaries were constrained in the horizontal direction, the bottom boundary was fully fixed, and the ground surface was defined as a free boundary. Tunnel excavation was simulated using element deactivation and lining activation to represent stress release and structural installation. The initial seepage field was assumed to be hydrostatic. After excavation, pore pressure redistribution and consolidation were simulated over a two-year period to obtain stabilized settlement.

3.3. Random Field Modelling of Hydraulic Parameters

Spatial variability of hydraulic parameters is an intrinsic characteristic of natural soils and must be incorporated to realistically represent heterogeneous ground conditions. In this study, permeability coefficient and compression modulus is modelled as a spatially correlated random field. The spatial correlation between two arbitrary points is described by an exponential autocorrelation function:

$$\rho \left({\tau }_1,{\tau }_2\right)=\exp \left[-2\left({\displaystyle \frac{{\tau }_1}{{\delta }_x}}+{\displaystyle \frac{{\tau }_2}{{\delta }_y}}\right)\right]$$

(5)

where $\rho \left({\tau }_1,{\tau }_2\right)$ is the correlation coefficient, ${\tau }_1$ and ${\tau }_2$ are the horizontal and vertical separation distances between two spatial locations, and d_x and d_y denote the horizontal and vertical correlation lengths, respectively. The correlation length controls the spatial persistence of parameter fluctuations.

Given a vector of independent standard normal random variables $\mathit{\xi }$, the correlated Gaussian random field $H^D(x,y)$ can be expressed as

$$H^D(x,y)=\mathit{L}\mathit{\xi }$$

(6)

where $H^D(x,y)$ represents the spatially correlated standard normal field.

Since hydraulic conductivity typically follows a lognormal distribution in soft soils, a transformation is applied to obtain the physical random field:

$$H_i(x,y)=\exp \left[{\mu }_{\ln i}+{\sigma }_{\ln i}{H}_i^D\left(x,y\right)\right]$$

(7)

where $H^D(x,y)$ is the hydraulic conductivity at spatial location (x,y), and ${\mu }_{\ln i}$ and ${\sigma }_{\ln i}$ are the mean and standard deviation of the logarithmic permeability.

The generated random field is mapped onto the finite element mesh, assigning spatially varying hydraulic properties to each element. Through this procedure, both the statistical characteristics and spatial correlation structure of hydraulic parameters are preserved. The random field model therefore provides a quantitative representation of parameter heterogeneity and forms the basis for stochastic hydro-mechanical simulations.

4. Probabilistic Analysis of Tunnel Considering Spatial Variability

4.1. Influence of Hydro-Mechanical Coupling

The primary focus of the hydro-mechanical analysis is the evolution of the seepage field during tunnel excavation. When hydro-mechanical coupling is not considered, the post-excavation pore water pressure remains stratified and approximately follows the initial hydrostatic distribution, as shown in Figure 4a. In contrast, when coupling effects are included, the pore pressure distribution around the tunnel changes significantly, as illustrated in Figure 4b. Negative excess pore water pressure develops in the vicinity of the tunnel, accompanied by steep hydraulic gradients and densely distributed pore pressure contours. This indicates intensified seepage activity around the excavation boundary, which may increase the risk of water inflow during construction.

Vertical displacement of ground surface nodes was extracted from the numerical simulations in ABAQUS to evaluate settlement response. A fully coupled pore pressure–displacement analysis was conducted using the soil consolidation procedure, in which nodal displacement and pore water pressure were simultaneously solved at each excavation stage through incremental finite element iterations. Tunnel excavation was simulated by stepwise stress release and corresponding boundary updates, while the final surface settlement profile was obtained from the vertical displacement output (U2) of nodes along the ground surface. The maximum settlement was identified as the peak downward displacement from the computed settlement trough. As shown in Figure 5, the final maximum surface settlement reaches 3.58 cm when seepage is considered, compared with 3.18 cm in the uncoupled case. The larger deformation under coupled conditions demonstrates that seepage effects amplify excavation-induced settlement. This discrepancy highlights that neglecting hydro-mechanical coupling may lead to underestimation of ground deformation. Therefore, under similar geological and hydrogeological conditions, incorporating seepage effects into settlement prediction models is essential for improving accuracy and ensuring engineering reliability.

4.2. Analysis Scenarios

Soft soil parameters exhibit pronounced directional dependence in their spatial correlation structure, with vertical correlation lengths generally much smaller than horizontal ones. The vertical correlation length is commonly within 1.0–3.0 m and, for Shanghai clays, typically ranges from 1.5 m to 25 m. In contrast, the horizontal correlation length is considerably larger and may reach several tens of meters in relatively stable depositional environments. In this study, both horizontal and vertical correlation lengths were selected based on regional statistics and model dimensions.

Five random field scenarios were defined using different combinations of horizontal and vertical correlation lengths (first value: horizontal; second value: vertical). Comparisons among Cases 4, 5, and 2 assess the influence of horizontal correlation length, while Cases 1–3 isolate the effect of vertical correlation length. A control variable approach was adopted, independently varying the correlation lengths of elastic modulus and permeability while keeping other parameters constant. The detailed scenarios are summarized in

Table 3. Random field parameter setting cases.

Case	$\mathbf{Horizontal}\mathbf{Distance}{\mathit{\delta }}_{\mathit{h}} \left(\mathbf{m}\right)$	$\mathbf{Vertical}\mathbf{Distance}{\mathit{\delta }}_{\mathit{v}} \left(\mathbf{m}\right)$	${\mathit{\delta }}_{\mathit{h}}/{\mathit{\delta }}_{\mathit{v}}$
1	15.00	1.50	10.00
2	15.00	3.00	5.00
3	15.00	6.00	2.50
4	6.00	3.00	2.00
5	9.00	3.00	3.00

, and the corresponding statistical parameters are listed in

Table 4. Mean values and coefficients of variation of parameters for each soil layer.

Type	Mean of k (m/s)	COV of k	Mean of $\mathit{E}$ (MPa)	COV of $\mathit{E}$
②3	1.19 × 10−6	0.65	20.00	0.21
⑤1	1.15 × 10−9	0.71	10.00	0.30
⑤4	1.53 × 10−8	0.84	8.00	0.15
⑦2	4.68 × 10−8	0.48	30.00	0.23

.

As shown in Figure 6, the color contours represent different parameter magnitudes, with the scale bar located on the right-hand side. Blue regions indicate lower values, whereas yellow regions correspond to higher values. By fixing the mean value of the random field and varying the correlation length, the spatial distribution characteristics can be clearly observed. With increasing correlation length, the spatial influence range of a given value expands in both horizontal and vertical directions, and the overall dispersion of parameter values becomes more pronounced.

4.3. Convergence Analysis

During the stochastic finite element simulations, it is necessary to determine an appropriate number of Monte Carlo realizations $N$ to ensure stable probabilistic characteristics of tunnel surface deformation, including the mean value and coefficient of variation. Figure 7 illustrates the relationship between the number of Monte Carlo simulations and the probabilistic characteristics of tunnel settlement. It can be observed that when the number of simulations exceeds 200, both the mean and coefficient of variation of settlement exhibit a clear convergence trend. Quantitative verification further indicates that the variation in these two probabilistic indicators remains within 2% under this condition. Accordingly, the number of Monte Carlo simulations in this study is set to N = 200.

4.4. Effect of Correlation Length on Surface Settlement

Based on the hydro-mechanical coupling numerical model established in Section 3.1, ground surface deformation under various working conditions was computed. The results indicate that as the correlation length increases, the dispersion of settlement responses becomes more pronounced and the distribution range broadens, reflecting enhanced uncertainty. In addition, the majority of stochastic simulation results, together with their mean values, exceed the deterministic prediction, implying that neglecting spatial variability of soil may lead to an underestimation of ground surface settlement.

As illustrated in Figure 8a, when the horizontal correlation length remains constant, increasing the vertical correlation length causes a gradual expansion of the fluctuation range of the settlement curve cluster, although the magnitude of this change is relatively limited. Conversely, Figure 8b shows that with a constant vertical correlation length, increasing the horizontal correlation length results in a slight decreasing trend in the fluctuation range, and its influence is comparatively weaker than that of the vertical correlation length. Notably, when spatial variability is taken into account, the maximum settlement induced by tunnel excavation may significantly exceed the deterministic results. Therefore, incorporating correlation length in the analysis of settlement is essential for accurately quantifying the extent and intensity of variability effects.

As shown in Figure 9, when the vertical correlation length increases from 1.5 m to 6 m, the mean value of the maximum ground surface settlement exhibits a decreasing trend, while the coefficient of variation increases, indicating enhanced dispersion. Under this condition, the results tend to approach the deterministic prediction, and the calculated mean settlement value becomes relatively conservative. In contrast, the horizontal correlation length has a limited influence on the mean settlement. However, as it increases, the settlement curves become more concentrated and the degree of dispersion decreases.

4.5. Failure Probability Assessment

Given the significant influence of spatial variability of soil parameters on ground deformation, Gong et al. [24] proposed evaluating the allowable ground deformation using the exceedance probability. Accordingly, under a given allowable deformation criterion, the performance function for ground deformation can be established, as expressed in Equation (8):

$$Z=S_{\max }-S_{\lim }$$

(8)

where $S_{\max }$ denotes the maximum ground deformation obtained from a single stochastic simulation, and $S_{\lim }$ represents the allowable ground deformation.

The exceedance probability is defined as the probability that the maximum ground deformation $S_{\max }$ the allowable value $S_{\lim }$, as shown in Equation (9):

$$P_f={\displaystyle \frac{N_f}{N}}\times 100\%$$

(9)

where $N$ is the total number of stochastic simulations under a specific working condition, and $N_f$ is the number of simulations in which the maximum ground deformation exceeds the allowable value.

The reliability index $\beta$ can be expressed as Equation (10):

$$\beta ={\displaystyle \frac{{\mu }_Z}{{\sigma }_Z}}={\displaystyle \frac{\left({\mu }_{S_{\max }}-S_{\lim }\right)}{{\sigma }_{S_{\max }}}}$$

(10)

where ${\mu }_{S_{\max }},{\sigma }_{S_{\max }}$ are the mean value and standard deviation, respectively, of the maximum deformation obtained from the $N$ stochastic simulations.

Referring to the Technical Code for Monitoring of Urban Rail Transit Engineering [25], the specified control value for ground surface settlement induced by shield tunneling in the Beijing area is −30 mm. International recommendations generally assess allowable deformation based on structural response rather than a single settlement threshold. For example, the ITA/AITES report indicates that total settlements of approximately 50 mm and differential settlements of 20 mm are often acceptable for standard structures, provided that relative rotation remains within tolerable limits [26]. Statistical data from field measurements indicate that surface settlement caused by shield tunneling is generally more pronounced in medium-soft to soft soils, where approximately 90.2% of monitoring points record settlement values within −45 mm. In hard to medium-hard soils, about 94.1% of monitoring points show settlement values within −40 mm, and uplift values are mostly within +10 mm. The project investigated in this study is located in medium-soft to soft soil. Considering both the domestic code requirement and the range of international settlement control criteria, three control thresholds—30 mm, 35 mm, and 40 mm—are selected for failure probability analysis.

As shown in Figure 10, when the settlement control value is 40 mm, the failure probabilities for Cases 1–3 (constant horizontal correlation length, with vertical correlation lengths of 1.5 m, 3 m, and 6 m) are 24.21%, 4.05%, and 1.01%, respectively. When the settlement control value is 35 mm, the failure probabilities for Cases 1–3 are 91.58%, 61.68%, and 15.15%, respectively. When the settlement control value is 30 mm, the failure probabilities for Cases 1–3 are 100%, 99.38%, and 84.85%, respectively. As the vertical correlation length of the soil permeability coefficient and elastic modulus increases, the vertical continuity of soil parameters is enhanced and overall homogeneity improves. On the one hand, this mitigates the concentrated compression effect caused by locally low elastic modulus (high compressibility) or high permeability (strong drainage consolidation) zones, thereby reducing extreme settlement induced by local parameter anomalies. On the other hand, it causes the mean settlement within the excavation influence zone to converge more closely to the overall statistical level, lowering the exceedance risk arising from parameter dispersion. Consequently, the failure probability of ground surface settlement decreases.

Under different settlement control values, for Cases 4 and 5 (constant vertical correlation length, with horizontal correlation lengths of 6 m, 9 m, and 15 m), the failure probability increases as the horizontal correlation length increases. Although a larger horizontal correlation length also enhances parameter uniformity in the horizontal direction, its influence on the evolution mechanism of the surface settlement trough differs. More uniform horizontal parameters reduce the “inhibitory effect” of local high elastic modulus (low compressibility) or low permeability (weak drainage consolidation) zones on settlement, leading to lateral expansion of the high-settlement region centered along the tunnel axis. As a result, excessive settlement develops from isolated discrete points into continuous zones, increasing the number of failure samples in statistical terms and ultimately leading to a higher failure probability of ground surface settlement.

5. Risk Assessment

5.1. Risk Assessment Procedure

To reasonably assess engineering risk and support risk-informed decision-making, it is necessary to classify risk events into different levels. The criteria for risk classification should comprehensively consider the project construction stage, actual scale, importance level, and specific risk management objectives. In general, the risk level is jointly determined by the probability of occurrence and the associated consequences (loss). Therefore, it is first necessary to establish grading standards for both probability and loss, and then determine the overall risk level of an event accordingly.

(1)

Grading standard for risk occurrence probability:

The grading standard for risk occurrence probability during metro shield tunneling construction may refer to the classification method specified in Code for Risk Management of Underground Construction in Urban Rail Transit (GB 50652-2011) [27], as shown in

Table 5. Grading standard for risk occurrence probability in metro construction.

Level	Frequency	Probability of Occurrence	Score
1	Rare	P < 0.01%	1
2	Unlikely	0.01% ≤ P < 0.1%	2
3	Occasional	0.1% ≤ P < 1%	3
4	Possible	1% ≤ P < 10%	4
5	Frequent	10% ≤ P	5

and

Table 6. Risk probability classification standard [27].

Level	I	II	III	IV	V
Description	Improbable	Rare	Occasional	Possible	Frequent
Probability	$P<0.01\%$	$0.01\%\le P<0.1\%$	$0.1\%\le P<1\%$	$1\%\le P<10\%$	$P\ge 10\%$

.

(2)

Risk loss levels and evaluation criteria:

According to the probability of risk occurrence and the associated loss, the risk level for engineering construction should be classified into four grades, in accordance with the provisions shown in

Table 7. Risk level standard.

Level		A	B	C	D	E
Likelihood		Catastrophic	Extremely Severe	Severe	Considerable	Negligible
1	Frequent	I	I	I	II	III
2	Likely	I	I	II	III	III
3	Occasional	I	II	III	III	IV
4	Rare	II	III	III	IV	IV
5	Impossible	III	III	IV	IV	IV

.

In addition, the assessment of risk uncontrollability should be incorporated into the risk evaluation process, and its classification criteria are presented in

Table 8. Grading standard for risk uncontrollability.

Uncontrollability Level	I	II	III	IV
Description	Easily controllable	Controllable	Difficultly control	Hard control
Score	1	2	3	4

.

The risk magnitude is defined as the product of the probability of occurrence and the potential consequences. The calculation formula is expressed as follows:

$$Risk=Probability\times Consequences$$

(11)

which can be simplified as

$$R=P\times C$$

(12)

When incorporating the uncontrollability of risk, the improved expression is

$$R=f(P,C,U)=P\times C\times U$$

(13)

where

R—risk magnitude (risk level);

P—estimated score of risk occurrence probability (see Table 5);

C—estimated score of risk-induced loss (see Table 6);

U—estimated score of risk uncontrollability (see Table 7).

Using the above formula, the magnitude of a specific risk can be calculated. Furthermore, a fuzzy evaluation method is adopted to classify the calculated risk values according to the concept of fuzzy intervals. The grading results and corresponding estimated value ranges are presented in

Table 9. Risk Classification criteria.

Level	Risk Degree	Risk Description	Estimated Value
I	Low	The risk loss level (including probability and consequence) is low and no control measures are required	$\left(0,12\right]$
II	Moderate	The risk loss level is moderate and can be easily controlled	$\left(12,48\right]$
III	Relatively High	The risk loss level is relatively high but controllable	$\left(48,75\right]$
IV	High	The risk loss level is very high and difficult to control	$\left(75,100\right]$

.

5.2. Loss Assessment

In the previous section, the maximum ground surface settlement was selected to calculate the failure probability because it serves as a clear indicator for evaluating engineering safety. It directly corresponds to the critical “failure” state and determines whether the most sensitive and critical buildings or pipelines within the affected area may be damaged. Once the settlement exceeds the prescribed limit, functional or structural failures—such as pipeline rupture or building cracking—may occur. Therefore, the maximum settlement is a direct criterion for identifying whether the project is in a failure state.

In contrast, the settlement trough area is an index that comprehensively reflects both the extent and severity of deformation. It provides a better quantification of the economic loss and consequence severity once failure occurs. Unlike the maximum settlement, which represents a single point value, the settlement trough area simultaneously considers the influence width of settlement and the magnitude of settlement at different locations. Accordingly, the maximum settlement is used to determine the probability of risk occurrence, while the settlement trough area is adopted as a loss indicator to quantify the potential consequences if risk occurs. Combining these two indices enables a more comprehensive risk assessment.

Based on the stochastic analysis results of the five working conditions presented in the previous section, the ground surface settlement trough area under each condition was calculated, and its statistical characteristics were analyzed. The corresponding frequency histograms are shown in Figure 11. As the vertical correlation length increases (δ_v), the mean settlement trough area decreases from 1.87 m² to 1.28 m², while the coefficient of variation increases. Conversely, as the horizontal correlation length (δ_h) increases, the mean settlement trough area rises from 1.40 m² to 1.66 m², whereas the coefficient of variation gradually decreases.

A fitting analysis was conducted on the distribution of ground surface settlement trough area under Cases 1–5. The results indicate that both the normal and lognormal distributions provide satisfactory fits, while the lognormal distribution performs better overall, yielding more accurate results, more stable residuals, and no systematic bias. It also demonstrates stronger adaptability to data with moderate dispersion. In particular, at the tail of the data (high-probability regions), the deviation is smaller, and the simulated data more closely match the actual distribution characteristics. A comparison of fitting performance indices is presented in

Table 10. Comparison of fitting indices.

Case	${\mathit{R}}^2$		RMSE		MAE		p Value (α = 0.05)
	Normal	Lognormal	Normal	Lognormal	Normal	Lognormal	Normal	Lognormal
1	0.87	0.88	1.61	1.53	1.28	1.19	0.62	0.83
2	0.89	0.90	1.40	1.35	0.90	0.89	0.93	1.00
3	0.93	0.94	1.62	1.55	1.03	0.98	0.87	0.97
4	0.83	0.84	1.93	1.98	1.46	1.39	0.97	0.91
5	0.84	0.85	1.54	1.49	1.20	1.19	0.82	0.99

.

All evaluation indices show that the lognormal distribution outperforms the normal distribution. Specifically, the coefficient of determination ($R^2$) approaches 1 when the explanatory power for data variability is stronger (with moderate dispersion and higher absolute values). The root mean square error (RMSE) approaches 0 when the average deviation between fitted and observed values is smaller. The mean absolute error (MAE) approaches 0 when the absolute difference between fitted and actual values is lower. For the Kolmogorov–Smirnov (KS) test, when $p>0.05$, the distribution hypothesis is accepted, and a larger p-value indicates higher reliability.

Therefore, the data for Case 1 approximately follow a lognormal distribution with a mean of 1.87 and a standard deviation of 0.14, exhibiting no significant skewness, as shown in Figure 12. The corresponding probability density function (PDF) and cumulative distribution function (CDF) are given in Equations (14) and (15), respectively.

$$f\left(x\right)={\displaystyle \frac{1}{x\sqrt{2\pi }\times 1.87}}\times e^{-{\displaystyle \frac{{\left(\ln x-1.87\right)}^2}{2\times {0.14}^2}}}$$

(14)

$$F\left(x\right)=\Phi \left({\displaystyle \frac{\ln x-1.87}{0.14}}\right)$$

(15)

Similarly, the data for Case 2 approximately follow a lognormal distribution with a mean of 1.66 and a standard deviation of 0.14; Case 3 follows a lognormal distribution with a mean of 1.33 and a standard deviation of 0.15; Case 4 follows a lognormal distribution with a mean of 1.40 and a standard deviation of 0.22; and Case 5 follows a lognormal distribution with a mean of 1.29 and a standard deviation of 0.14. Based on the fitted distribution curves, the quantile value can be determined using Equation (16),

$$x_p=e^{(\mu +\sigma \cdot z_p)}$$

(16)

where $z_p$ denotes the p-quantile of the standard normal distribution.

In accordance with risk probability classification standards and drawing on the concept of graded risk control, deformation induced by shield tunnel construction is categorized into different control levels. The loss control values (i.e., settlement trough area) corresponding to different risk levels under six working conditions are calculated. The loss values associated with each failure level adopted in this study for graded construction deformation control are presented in

Table 11. Loss values corresponding to failure levels for each case.

Level	V	IV	III	II	I
Description	A	B	C	D	E
Probability	$P<0.01\%$	$0.01\%\le P<0.1\%$	$0.1\%\le P<1\%$	$1\%\le P<10\%$	$P\ge 10\%$
Case 1	$S\ge 2.178$	$2.153\le S<2.178$	$2.112\le S<2.153$	$2.034\le S<2.112$	$S<2.034$
Case 2	$S\ge 2.074$	$2.034\le S<2.074$	$1.974\le S<2.034$	$1.833\le S<1.974$	$S<1.833$
Case 3	$S\ge 1.705$	$1.669\le S<1.705$	$1.609\le S<1.669$	$1.502\le S<1.609$	$S<1.502$
Case 4	$S\ge 2.048$	$1.968\le S<2.048$	$1.858\le S<1.968$	$1.679\le S<1.858$	$S<1.679$
Case 5	$S\ge 1.596$	$1.566\le S<1.596$	$1.516\le S<1.566$	$1.435\le S<1.516$	$S<1.435$

S denotes the transverse ground surface settlement trough area (m²).

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5.3. Result Analysis

Using the risk level classification method established in this section, and based on the calculated failure probabilities of ground surface deformation from the previous section, together with the calculated ground loss values and the proposed control thresholds presented herein, the safety risk of ground surface deformation under the six working conditions was evaluated. The assessment results are shown in

Table 12. Risk levels corresponding to each case.

Case	1	2	3	4	5
Estimation	75	60	45	45	45
Level	III	III	II	II	II
Degree	Relatively High	Relatively High	Moderate	Moderate	Moderate

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Under conditions of relatively small vertical correlation length, the risk level is higher, which is consistent with general engineering experience. Risk control can be implemented throughout the entire process of “pre-construction prediction—in-construction control—post-construction remediation.” Through accurate prediction, graded control, and dynamic adjustment, ground surface deformation can be maintained within allowable limits, thereby effectively managing safety risks associated with tunnel-induced surface deformation.

Prior to construction, potential high-risk zones should be identified based on geological investigation data and numerical simulation results. During construction, differentiated deformation control measures should be adopted according to real-time monitoring data (e.g., the development trend of the settlement trough) and corresponding risk levels (such as Level III or Level II). For example, pre-construction ground improvement methods, including deep mixing piles and high-pressure jet grouting piles, may be applied to reinforce weak strata; precise site investigation should be conducted to obtain reliable soil mechanical parameters; and adjacent pipelines and structures should be pre-strengthened and protected. During shield tunneling, excavation parameters should be optimized by controlling advance rate, cutterhead rotation speed, synchronous grouting pressure and volume, maintaining earth chamber pressure balance, and strictly regulating muck discharge. In addition, segment installation quality and tail void grouting effectiveness must be ensured to prevent groundwater seepage. After construction, secondary grouting reinforcement may be implemented, and multidimensional monitoring points—covering ground surface, tunnel structure, and adjacent pipelines—should be arranged for dynamic observation. Timely repair of segment cracking and water leakage should also be carried out. For special conditions such as river crossings, soft soil lenses, or tunneling adjacent to existing tunnels, targeted measures including composite cutterheads, intensified grouting holes, and isolation piles should be adopted. By implementing a quantitative risk-based dynamic control strategy throughout the entire construction process, the management effectiveness of ground deformation risk can be significantly enhanced, ensuring the safety of tunnel construction and the surrounding environment.

6. Conclusions and Outlook

This study establishes a probabilistic risk assessment framework for tunnel construction deformation in soft soils by explicitly incorporating spatial variability of hydraulic parameters within a hydro-mechanically coupled finite element model. The proposed framework integrates three components: (i) statistical characterization of permeability and compression modulus based on site investigation data to define parameter distributions and correlation structures; (ii) development of a stochastic hydro-mechanical finite element model using random field theory and Monte Carlo simulation to quantify settlement uncertainty; and (iii) construction of an engineering-oriented risk evaluation system that combines failure probability, reliability index, and settlement trough area as a loss indicator to classify construction risk levels under different correlation scenarios.

Although the proposed framework provides useful engineering insights, its applicability is constrained by the project-specific database, simplified constitutive assumptions, and the limited consideration of uncertain parameters. Future studies are therefore encouraged to incorporate three-dimensional excavation effects, additional geotechnical uncertainties, and real-time monitoring data for model updating and broader validation. The main conclusions are summarized as follows:

(1)

Hydro-mechanical coupling significantly amplifies excavation-induced ground deformation. Compared with the uncoupled condition, inclusion of seepage effects increases the maximum surface settlement from 3.18 cm to 3.58 cm, corresponding to an increase of 12.6%, confirming that neglecting groundwater–soil interaction may lead to systematic underestimation of deformation. Moreover, stochastic simulations demonstrate that most settlement realizations and their mean values exceed deterministic predictions, highlighting the necessity of incorporating spatial variability in soft soil tunnelling analysis.

(2)

The spatial correlation structure of hydraulic parameters plays a decisive role in deformation uncertainty and failure probability. As the vertical correlation length increases from 1.5 m to 6 m, the mean settlement trough area decreases from 1.87 m² to 1.28 m², while the failure probability under a 40 mm settlement threshold decreases markedly from 24.21% to 1.01%. Under a 35 mm threshold, the corresponding failure probabilities decrease from 91.58% to 15.15%. Increasing the vertical correlation length reduces the mean maximum settlement but increases its coefficient of variation, resulting in a significant decrease in exceedance probability under prescribed settlement control thresholds. In contrast, increasing the horizontal correlation length slightly reduces dispersion but enlarges the lateral continuity of high-settlement zones, thereby increasing failure probability. These results indicate that vertical correlation exerts stronger control over deformation magnitude, whereas horizontal correlation primarily influences the spatial extent of risk.

(3)

The settlement trough area follows a lognormal distribution and provides a rational quantitative indicator for consequence assessment. Compared with the normal distribution, the lognormal model yields higher goodness-of-fit indices and more stable residual behavior. By integrating failure probability, settlement-related loss, and uncontrollability factors into a unified risk matrix, the evaluated risk levels range from Moderate (Level II) to Relatively High (Level III), depending on correlation scenarios. Smaller vertical correlation lengths correspond to higher risk levels, which is consistent with engineering observations in heterogeneous soft strata.

Overall, the proposed framework establishes a systematic pathway from spatial parameter characterization to probabilistic deformation prediction and quantitative risk classification. It provides methodological support for risk-informed design and dynamic construction control of shield tunnels in saturated soft soil environments.