Advanced Risk Assessment for Deep Excavation in Karst Regions Using Improved Dempster–Shafer and Dynamic Bayesian Networks

Abstract

This study presents a novel risk-assessment methodology for deep foundation pit projects in karst regions, aimed at enhancing project safety and decision-making processes. This approach amalgamates fuzzy dynamic Bayesian networks with a refined Dempster–Shafer (DS) evidence theory to tackle the intricate uncertainties present in such contexts. A comprehensive risk index system, derived from historical accident cases, relevant standards, and the literature, encompasses environmental, design, construction, and management factors. Initial probabilities for each risk factor are determined through the integration of expert knowledge and fuzzy theory. The enhanced Dempster–Shafer theory is utilized to fuse diverse information sources, culminating in a robust and dynamic risk evaluation model. This model leverages real-time monitoring data to dynamically assess and adjust risk levels throughout the construction process. The validation of the proposed method is demonstrated through a detailed case study of the Guangzhou Tangxi Section 1 deep foundation pit project, which effectively identified critical risk factors and facilitated proactive construction strategy adjustments. To further evaluate the reliability of the methodology, comparisons were made with three alternative methods, and applications were conducted on three additional deep foundation pit projects. These comparative analyses confirm the superior reliability and applicability of the proposed methodology across varied scenarios.

1. Introduction

As a pivotal element of urban construction, deep excavation engineering places a strong emphasis on risk assessment and management [1,2,3]. Construction in karst regions rich in water faces more complex geological conditions and construction environments, making risk assessment and management a challenging task. Fuzzy mathematics theory, due to its advantage in dealing with uncertainties, has been widely applied in the risk assessment of excavation engineering [4]. Lin et al. [5] summarized the applications of fuzzy set theory and machine learning methods in excavation risk assessment, noting, however, a notable lack of in-depth quantitative analysis. While their work highlighted the potential of these methods, the absence of detailed metrics for performance evaluation limited the robustness of their conclusions. Cheng et al. [6] combined fuzzy reasoning with FMEA to improve the risk-assessment method for pipe jacking construction, yet they did not take into account the degree of membership and the fuzziness of expert opinions when constructing a deep excavation monitoring risk-assessment model based on fuzzy theory. This oversight could lead to potential biases in the risk evaluation, underestimating the impact of expert-judgment variability. Mao et al. [7] identified five key factors as evaluation indicators and established an enhanced fuzzy comprehensive evaluation model. They utilized interval numbers for constructing interval judgment matrices, combined subjective and objective weighting, and employed both trapezoidal and triangular membership functions to ascertain degrees of membership. Despite these sophisticated techniques, the model’s complexity might impede practical application without significant computational resources and expert training. Many factors affect the safety of deep excavation projects; for example, studies have demonstrated the critical role of in situ tests for assessing geotechnical properties vital to these efforts. For instance, Bernieri et al. [8] developed a system for online fault detection using neural networks, Betta and Pietrosanto [9] reviewed various techniques for instrument fault detection and isolation, Shen et al. [10] applied rough set theory for fault diagnosis in diesel engines, and Yildiz and Alpaydin [11] proposed omnivariate decision trees to enhance decision-making processes.

These studies have broadened the application of fuzzy mathematics in the risk assessment of excavation engineering, offering new approaches to enhance the accuracy and reliability of risk assessments. Moreover, entropy weight theory plays a crucial role in the risk assessment of deep excavation construction by objectively determining the weights of indicators through the calculation of entropy weights. Shi [12] implemented entropy weight optimization techniques in deep excavation projects [13], but did not address the dynamic nature of construction risks. Lu and Li [14] conducted a multi-objective risk analysis on deep excavations based on the entropy weight method, which improved risk weighting accuracy, yet was limited in addressing the temporal changes in risk factors. Zhou et al. [15] developed a hybrid model combining information entropy and uncertainty-measure theory to evaluate the excavatability of rock masses. Xie et al. [16] used a BP neural network to predict the next phase of monitoring data, determined the evaluation indicator weights using the entropy weight method based on the predictive values, and quantitatively described the future safety state using the fuzzy comprehensive evaluation method. The neural network approach showed promise for predictive accuracy but lacked interpretability, making it difficult to understand the causative factors behind risk levels.

In recent years, Bayesian networks with bidirectional causal reasoning capabilities have been applied to the risk assessment of deep excavation construction [17,18]. Bayesian networks can handle uncertain information by constructing directed acyclic graphs that reflect the causal relationships between risk factors and infer the probabilities of unknown information based on known data [19]. Ghousi et al. [20] applied a fuzzy Bayesian HEART-5M integrated method for human reliability analysis in deep excavation projects, demonstrating substantial accuracy in human error prediction but facing challenges in integrating environmental factors. Zhang et al. [21] proposed a dynamic risk analysis method for the construction of subway-station deep excavations based on fuzzy Bayesian networks and fuzzy hierarchical analysis, achieving quantitative accident risk analysis and sensitivity analysis, yet their model required extensive data, which are often not readily available in real-world projects. Wang et al. [22] established a dynamic safety-assessment model for subway tunnel construction using the cut-and-cover method based on fuzzy set theory and Bayesian networks. They verified the effectiveness of the model through causal reasoning, diagnostic reasoning, and secondary diagnostic reasoning, identifying key risk factors affecting the safety of cut-and-cover construction. However, traditional Bayesian networks have limitations in dealing with dynamically changing risks. To overcome this shortfall, dynamic Bayesian networks have been developed, which can dynamically assess risks based on real-time monitoring data [23]. Building on this, Li et al. [17] developed a dynamic interpretable adjacent building risk prediction model based on deep learning, applied to deep excavation engineering. Similarly, Jiang et al. [23] established a risk-coupling analysis model that integrates dynamic Bayesian networks and the N-K model to assess risks in deep excavation projects situated near existing tunnels. Wu et al. [24] proposed a tunnel-construction safety decision-support method based on dynamic Bayesian networks. This method accurately describes the geological, design, and mechanical variables that change as construction progresses through dynamic updates, overcoming the shortcomings of traditional fault-analysis methods.

Despite these advancements, several issues persist in conducting dynamic risk evaluations in water-rich karst regions: (1) the absence of a specific risk indicator system tailored to deep excavations in these areas; (2) insufficient integration of expert experience and objective data in current risk-assessment methods; (3) a lack of integration of dynamic Bayesian networks with other theories to enhance assessment accuracy.

To address these gaps, this study analyzes risk factors associated with deep excavation construction in water-rich karst areas from four perspectives: environmental risks, design risks, construction risks, and management risks. It constructs a comprehensive risk-indicator system and determines prior probabilities by combining fuzzy mathematics with expert experience. Additionally, this study utilizes the improved Dempster–Shafer (DS) evidence theory for information fusion to construct a fuzzy dynamic Bayesian network–improved DS evidence theory (FDBN-IDS) risk-assessment model. Logical structure diagram of the study is shown in Figure 1.

The model’s rationality and feasibility are validated through its application to the Tangxi Section 1 deep excavation project, offering new insights for risk assessments in water-rich karst regions. This approach not only addresses the gaps in the current literature but also provides a robust framework for integrating expert knowledge with quantitative data, leading to more accurate and dynamic risk assessments in challenging geological settings.

2. Relevant Theories and Analytical Methods

2.1. Dynamic Bayesian Networks

Bayesian networks (BNs), composed of directed acyclic graphs (DAGs) and conditional probability tables (CPTs), belong to the category of probabilistic network models [25]. The graphical structure of a BN includes nodes and directional arrows (as illustrated in Figure 2), where nodes represent various indicator variables [26]. These nodes can be categorized into target nodes, intermediate nodes, and root nodes, whereas directional arrows symbolize the interaction relationships between nodes. The conditional probability tables enable the determination of the significance of root nodes to target nodes.

In BNs, for any random variable X = ($x_{t_1}$, $x_{t_2}$, …, $x_{t_n}$), its joint probability distribution is given by

$$f(x_{t_1},x_{t_1},\dots ,x_{t_n})=\prod _{\forall t_i\in t}f\left(\left.x_{t_{\mathrm{i}}}\right|X_{{Pa}_{\left(t_i\right)}}\right)$$

(1)

In the equation, $f$ is the joint probability distribution function. X_Pa(t_i) represents the set of parent nodes of X at time period t_i, where t denotes the time period. $f\left(\left.{x_t}_i\right|{X_{Pa}}_{\left(t_i\right)}\right)$ is the conditional probability distribution of the node.

When X is a temporal random variable and its parent nodes $x_{t_i}$ are ($x_{t_{i-1}}$, $x_{t_{i-2}}$, …, $x_{t_1}$), then Equation (1) can be rewritten as

$$f(x_{t_1},x_{t_2},\dots ,x_{t_n})={\displaystyle {\prod }_{\forall t_i\in t}f(\left.x_{t_1}\right|x_{t_1},x_{t_2},\dots ,x_{t_{i-1}})}$$

(2)

$$f(\left.x_{t_1}\right|x_{t_1},x_{t_2},\dots ,x_{t_{i-1}})={\displaystyle \frac{f(x_{t_1},x_{t_2},\dots ,x_{t_i})}{f(x_{t_1},x_{t_2},\dots ,x_{t_{i-1}})}}$$

(3)

Extending BNs over time results in the formation of dynamic Bayesian networks (DBNs). A DBN consists of an initial network T₀ (BN) and a transition network T_→ [27]. The transition network comprises BN models of two time slices and transition probability tables, as illustrated in Figure 3.

Within a DBN, two conditions are assumed: (1) the Markov assumption, which denotes that the probability of nodes at a current time slice is only influenced by the preceding moment; (2) stationarity, meaning the transition probabilities remain constant. The transition probability between any two adjacent time slices is calculated using Equation (4), and the joint probability of any node is calculated using Equation (5).

$$P(\left.x_t\right|x_{t-1})=\prod _{i=1}^{N}P(\left.x_i^t\right|Pa(x_i^t))$$

(4)

$$P(x_{1:n}^{1:t_n})=\prod _{i=1}^n{P}_{T_0}(\left.x_i^0\right|Pa(\left.x_i^0\right|))\times \prod _{t=1}^{t_n}·\prod _{i=1}^n{P}_{T_{\to }}(\left.x_i^t\right|Pa(\left.x_i^t\right|))$$

(5)

where $P\left(x_{1:n}^{1:t_n}\right)$ represents the joint probability distribution of a sequence of variables x indexed from 1 to n over the time period from 1 to t_n.

2.2. DS Evidence Theory

Evidence theory, starting from the perspective of evidence reasoning, can rationally integrate expert knowledge or multi-source data [28]. It has been widely applied in fields such as sensor-data fusion and multi-route diagnostic inspection. If various experts provide n sets of independent evidence {m₁, m₂, …, m_n} for an event, then A_i, A_j, …, A_k (i, j, …, k = 1, 2, …, n) are the focal elements of n sets of independent evidence. This study employs Dempster’s rule of combination for evidence fusion:

$$m(A)={\displaystyle \frac{{\displaystyle {\sum }_{A_i\cap A_j\cap \cdots \cap A_k=A}{m}_1(A_i)m_2(A_j)\cdots m_n(A_k)}}{1-K}}$$

(6)

$$K={\displaystyle {\sum }_{A_i\cap A_j\cap \cdots \cap A_k=\varnothing }{m}_1(A_i)m_2(A_j)\cdots m_n(A_k)}$$

(7)

In the formula, K represents the conflict coefficient among the pieces of evidence, used to assess the degree of conflict between them.

3. Deep Foundation Pit Risk Dynamic Evaluation Model Based on FDBN–Improved DS Evidence Theory

In response to the dynamism and uncertainty of risk events in the risk-assessment process of deep foundation pits in water-rich karst formations, this study establishes a dynamic analysis framework based on FDBN–improved DS evidence theory to achieve accurate evaluation of construction risks associated with deep foundation pits. The framework implementation is illustrated in Figure 4.

3.1. Risk Identification and Construction of Evaluation Indicator System

The construction of deep foundation pits in water-rich karst formations is highly risky, often influenced by a combination of factors such as environment, design, construction, and management. Based on the review of the relevant literature [29,30], the “Technical Code for Safety of Building Deep Foundation Pit Engineering” (JGJ 311-2013) [31], the “Standard for Quality Acceptance of Building Foundation Engineering Construction” (GB50202-2018) [32], and the analysis of several construction accident cases of deep foundation pits, this study conducts risk identification to construct a risk indicator system for deep foundation pits in water-rich karst layers.

The indicator system includes four aspects: environmental risks (X₁), design risks (X₂), construction risks (X₃), and management risks (X₄), encompassing 15 indicators that significantly affect the construction of deep foundation pits in water-rich karst layers, as shown in Figure 5. Among them, environmental risks include geological conditions (x₁₁), hydrological conditions (x₁₂), survey quality (x₁₃), surrounding environment under construction (x₁₄), and existing surrounding environment (x₁₅); design risks include dewatering scheme design (x₂₁) and construction design of the foundation pit (x₂₂); construction risks include diaphragm wall quality (x₃₁), support system (x₃₂), karst cave filling quality (x₃₃), reinforcement construction measures (x₃₄), and dewatering effect (x₃₅); management risks include construction measurement management (x₄₁), construction team experience (x₄₂), and risk tracking and control (x₄₃).

This study, based on the “Standard for Project Management of Construction Works GB/T 50326-2017” [33] and the “Standard for Risk Management of Underground Projects in Urban Rail Transit GB50652-2011” [34], categorizes the risk levels of each indicator into five grades: Grade I (low risk), Grade II (mild risk), Grade III (moderate risk), Grade IV (significant risk), and Grade V (extreme risk), with respective value ranges of 0~0.2, 0.2~0.4, 0.4~0.6, 0.6~0.8, and 0.8~1, as shown in

Table 1. Level division of construction risk for karst water-rich deep foundation pit.

Risk Scale	Risk Level	Risk Degree	Risk Acceptance Criteria and Risk-Control Measures
V	Extremely Dangerous	Immediate cessation of construction; establish an expert group to carefully review the design and construction plan	Halt construction
IV	High Risk	Establish a special construction rectification plan and carry out expert demonstration	Necessary rectification and preparation of risk-control measures
III	Significant Risk	Optimize risk-control measures	Necessary rectification and preparation of risk-control measures
II	Moderate Risk	Be prepared for possible accidents	Prepare an emergency response plan for accidents
I	Slight Risk	Normal construction operations are acceptable	Continue with normal construction operations

.

3.2. Construction of Bayesian Network Model

Based on the established risk evaluation indicator system for deep foundation pit construction in water-rich karst strata, the GENIE software (version 3.0) was utilized to convert it into a BN (Bayesian network) topological structure [35]. Herein, the target node corresponds to the risk of deep foundation pit construction in water-rich karst strata; the intermediate nodes correspond to primary indicators such as environmental risk, design risk, construction risk, and management risk; and the root nodes represent 15 secondary indicators, including geological conditions, hydrological conditions, etc. This process establishes an FBN (fuzzy Bayesian network) model for evaluating the risks associated with deep foundation pit construction in water-rich karst strata.

3.3. Determination of the Membership Degree Matrix

Given the multiple states of each evaluation indicator for the construction of deep foundation pits in water-rich karst strata, to rationally determine the risk status of each node, this study refers to the method of Hongyu, C. and Wang, Z.Z. and others [36,37]. By constructing a membership degree matrix W, a fuzzy evaluation of the root nodes is realized.

$$W=\left[\begin{array}{cccccc}{w}_{11}& w_{12}& \cdots & w_{1j}& \cdots & w_{1n}\\ ⋮& ⋮& & ⋮& & ⋮\\ w_{i1}& w_{i2}& \cdots & w_{ij}& \cdots & w_{in}\\ ⋮& ⋮& & ⋮& & ⋮\\ w_{m1}& w_{m2}& \cdots & w_{mj}& \cdots & w_{mn}\end{array}\right]$$

(8)

In the formula, w_ij represents the likelihood that the i indicator is classified as risk level j, m is the number of indicators, and n is the number of levels.

Based on the evaluation results and uncertainties of each indicator provided by experts, this study employs a Gaussian membership function (as shown in Equation (9)) for the construction of the membership degree matrix. The centers x₀ of the membership functions for the five risk levels are set to 0, 0.25, 0.5, 0.75, and 1, respectively.

$${w_i}_j=e^{-{\displaystyle \frac{{\left(x_{ij}-x_0\right)}^2}{2{{\sigma }_{ij}}^2}}}$$

(9)

In the formula, x_ij represents the expert evaluation value, while σ_ij represents the uncertainty associated with the expert evaluation value x_ij.

3.4. Determination of Node Risk Probabilities

When employing Dempster–Shafer (DS) evidence theory to fuse the membership degree matrix based on expert evaluation results, the computational cost can be substantial [38]. Hence, this study utilizes an improved evidence theory method through matrix analysis to integrate expert opinions.

Assuming there are n experts evaluating each indicator, the probability-value distributions of each expert, based on the membership functions determined in Section 3.3, are as illustrated in Equation (10).

$$Q=\left[\begin{array}{c}{Q}_1\\ Q_2\\ ⋮\\ Q_n\end{array}\right]=\left[\begin{array}{ccccc}{Q}_{11}& Q_{12}& Q_{13}& Q_{14}& Q_{15}\\ Q_{21}& Q_{22}& Q_{23}& Q_{24}& Q_{25}\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ Q_{n1}& Q_{n2}& Q_{n3}& Q_{n4}& Q_{n5}\end{array}\right]$$

(10)

In the formula, any element Q_ij within matrix Q represents the probability value assigned by the ith expert for the indicator being classified as risk level j.

We then multiply any row $Q_i^T$ from matrix Q with another row Q_j:

$$\begin{array}{cc}A& =Q_i^T\times Q_j=\left(\begin{array}{l}{Q}_{i1}\\ Q_{i2}\\ Q_{i3}\\ Q_{i4}\\ Q_{i5}\end{array}\right)\times \left(Q_{j1}{Q}_{j2}{Q}_{j3}{Q}_{j4}{Q}_{j5}\right)\\ & =\left[\begin{array}{ccccccc}{Q}_{i1}& \times & Q_{j1}& \cdots & Q_{i1}& \times & Q_{j5}\\ & ⋮& & \ddots & & ⋮& \\ Q_{i5}& \times & Q_{j1}& \cdots & Q_{i5}& \times & Q_{j5}\end{array}\right]\end{array}$$

(11)

In the formula, the sum of the main diagonal elements in matrix A represents the numerator in the combination rule, while the sum of the off-diagonal elements represents the degree of conflict K after matrix fusion.

The probability values of each risk level after evidence fusion are improved using a weighted allocation method:

$$\begin{array}{l}m(A)=\\ \left\{\begin{array}{ll}0,& A=\varnothing \\ {\displaystyle \frac{{\displaystyle \sum _{A_i\cap B_j\cap C_k\cdots =A}{Q}_1(A_i)Q_2(B_j)Q_3(C_k)\cdots }}{1-K}}+f(A),& A=\varnothing \end{array}\right.\end{array}$$

(12)

$$f(A)=Kq(A)$$

(13)

$$q(A)={\displaystyle \frac{{\displaystyle \sum _{i=1}^n{m}_i(A)}}{n}}$$

(14)

In the formula, f(A) represents the probability distribution function for matrix conflicts.

3.5. FBN Bidirectional Inference and Sensitivity Analysis

Forward causal reasoning involves the computation of the occurrence probability of target child nodes by integrating the given prior probabilities of various parent nodes and the conditional probabilities between events [39]. In contrast, backward diagnostic reasoning calculates the posterior probability of each event when the target child node’s outcome is known:

$$P(x_i=\left.y_j\right|T=1)={\displaystyle \frac{P(x_i=y_j)P(\left.T=1\right|x_i=y_j)}{P(T=1)}}$$

(15)

P = (x_i = y_j|T = 1) represents the posterior probability of the ith parent node event when the target child node event T = 1. y_j indicates the five possible levels of event x_i.

$$M_{\mathrm{RoV}}(x_i)={\displaystyle \frac{1}{k_i}}\sum _1^{k_i}{\displaystyle \frac{P(\left.x_i=y_j\right|T=1)-P(x_i=y_j)}{P(x_i=y_j)}}$$

(16)

In the formula, P(x_i = y_j) represents the prior probability, and k_i denotes the number of non-zero risk states for the parent node event x_i, which equals five in this study.

3.6. Dynamic Risk Evaluation Based on FDBN

Building on the existing FBN static model, this study utilizes the update frequency of monitoring data to determine time slices for the FDBN, thereby facilitating real-time dynamic evaluation of construction risk for deep foundation pit projects in water-rich karst strata [40]. According to the technical standards for the monitoring of construction pit engineering (GB50497-2019) [41], the primary monitoring items for such projects include the support structure, groundwater conditions, the bottom and surrounding soil of the pit, and nearby buildings or structures. In the practical dynamic risk evaluation, key monitoring items are identified based on root node indicators that showed higher sensitivity in the previous static analysis segment. These are then used as observational data in the FDBN model for conducting dynamic risk assessments. Transition probabilities can be determined using methods such as maximum likelihood estimation, the C-K equation definition, or expert experience. This study adopts the expert experience method to ascertain transition probabilities and to evaluate the dynamic risk of such construction projects.

4. Case Study

To validate the construction risk dynamic evaluation method for water-rich karst deep foundation pits based on FDBN and improved DS evidence theory proposed in this paper, the Guangzhou Tangxi Phase 1 Section 1 deep foundation pit project was selected as the evaluation object. The method described herein was applied to assess its construction dynamic risk, and a comparative analysis with the actual site conditions was conducted.

4.1. Project Overview

The Guangzhou Tangxi Phase 1 Section 1 deep foundation pit project is located in the northern part of Guangzhou city, a region characterized by complex geological conditions and intense karst development. The geological strata mainly consist of backfill soil, silt, silty sand, clay, argillaceous shale, strongly weathered limestone, moderately weathered limestone, and slightly weathered limestone. Notably, sand layers are predominantly found directly above the bedrock, making the site particularly susceptible to karst surface collapse.

Table 2. Geotechnical parameters of various soil materials.

Soil Layer	Density (kg/m3)	Cohesion (kPa)	Internal Friction Angle (°)	Plasticity Index (%)
Backfill Soil	1775	13.5	19.7	12
Silt	1824	10.1	20.9	15
Silty Sand	1994	5.1	31.6	7
Clay	2045	23.8	17.6	22
Argillaceous Shale	2284	33.9	24.7	9
Strongly Weathered Limestone	2316	48.5	32.4	6
Moderately Weathered Limestone	2424	83.7	38.1	4
Slightly Weathered Limestone	2497	95.1	43.8	3

presents the geotechnical parameters of various soil materials encountered in the Guangzhou Tangxi Phase 1 Section 1 deep foundation pit project.

To obtain these parameters, a comprehensive suite of geotechnical tests was conducted, both in the field and laboratory settings. Field tests included Standard Penetration Tests (SPT) to assess soil density and stratigraphy. These were supplemented by Cone Penetration Tests (CPT), which provided continuous profiles of subsurface-soil stiffness. In situ vane shear tests were employed to measure the undrained shear strength of cohesive soils directly. For the rock layers, pressuremeter tests (PMT) and borehole shear tests were carried out to determine the deformability and shear-strength properties.

Laboratory tests complemented the in situ analyses, with samples collected from boreholes at various depths. These included triaxial compression tests to determine the shear-strength parameters (cohesion and internal friction angle) of different soil and rock types, and oedometer tests to determine their deformation properties. These tests were vital in understanding the mechanical behavior of the subsurface materials under different stress conditions.

The karst development here is intense; approximately 34% of the solution cavities, which tend to have an average height of about 3.2 m, are located below the base level, displaying string-like formations. The groundwater at the site is categorized into three classes: quaternary unconsolidated layer aquifer, bedrock fissure water, and carbonate rock-fissure karst water, reflecting their varied storage methods.

The support structure for the foundation pit utilizes a combination of diaphragm walls and internal bracing. The foundation pit itself is shaped like a knife handle, with depths ranging from 15.9 m to 34.68~35.18 m, segregated by diaphragm walls. The pit’s widest point spans 168 m, while its total length is approximately 588 m. The support system includes an underground continuous wall supplemented by five levels of vertical internal bracing, all constructed with reinforced concrete for improved stability and strength. The joints of the continuous wall are connected using H-shaped steel joints to ensure robust structural integrity.

This extensive support and their geological intricacies highlight the need for the advanced risk-assessment methodologies discussed in this study. The geological profile of the site is illustrated in Figure 6, providing a visual representation of the varying soil and rock layers in this karst region.

4.2. Pre-Construction Risk Evaluation and Management

4.2.1. Determination of Prior Probabilities for Root Nodes

This study employs an expert-group decision-making approach to determine the prior probabilities of root nodes, involving a panel of 15 experts. Among them, five were university professors, six were field management personnel with senior professional titles, and the remaining four were frontline technicians. Each expert assessed the risk levels of the root nodes and evaluated the uncertainty associated with those levels.

The evaluation process involves the following steps:

Expert Assessment: Each expert provides their assessment of the risk levels for various root nodes, including their associated uncertainties. As illustrated by Expert 1 in

Table 3. Evaluation results of Expert 1.

Project	Root Node Indicator
	x11	x12	x13	x14	x15	x21	x22	x31
Evaluation Level	IV	II	II	IV	II	I	III	IV
Quantitative Value	0.70	0.26	0.39	0.75	0.33	0.10	0.55	0.77
Uncertainty	0.05	0.1	0.05	0.15	0.05	0.05	0.1	0.15
Project	x32	x33	x34	x35	x41	x42	x43
Evaluation Level	III	IV	IV	II	II	II	III
Quantitative Value	0.45	0.72	0.64	0.32	0.37	0.21	0.48
Uncertainty	0.2	0.2	0.15	0.05	0.1	0.1	0.1

, the experts’ evaluations are quantified in terms of risk levels.

Membership Degree Matrix Calculation: These assessments are then used to construct a membership degree matrix. This matrix represents the degree to which each root node belongs to different risk levels based on the expert’s evaluation. For instance, the evaluation results of Expert 1, shown in Table 3, are processed to derive the probability value distributions for each root node, as detailed in

Table 4. The distribution of probability values for each root node evaluated by Expert 1.

Project	Root Node Indicator
	x11	x12	x13	x14	x15	x21	x22	x31
Evaluation Level	IV	II	II	IV	II	I	III	IV
Quantitative Value	0.70	0.26	0.39	0.75	0.33	0.10	0.55	0.77
Uncertainty	0.05	0.1	0.05	0.15	0.05	0.05	0.1	0.15
Project	x32	x33	x34	x35	x41	x42	x43
Evaluation Level	III	IV	IV	II	II	II	III
Quantitative Value	0.45	0.72	0.64	0.32	0.37	0.21	0.48
Uncertainty	0.2	0.2	0.15	0.05	0.1	0.1	0.1

.

Integration Using Improved DS Evidence Theory: After obtaining the evaluation results from all 15 experts, the improved Dempster–Shafer (DS) evidence theory method is utilized to integrate the probability values. This method combines the individual assessments, incorporating the uncertainties and providing a comprehensive and robust set of prior probabilities. These integrated probabilities reflect a consensus view rather than individual opinions and are shown in

Table 5. Probability after data fusion.

Project	Root Node Indicator
	x11	x12	x13	x14	x15	x21	x22	x31
Evaluation Level	IV	II	II	IV	II	I	III	IV
Quantitative Value	0.70	0.26	0.39	0.75	0.33	0.10	0.55	0.77
Uncertainty	0.05	0.1	0.05	0.15	0.05	0.05	0.1	0.15
Project	x32	x33	x34	x35	x41	x42	x43
Evaluation Level	III	IV	IV	II	II	II	III
Quantitative Value	0.45	0.72	0.64	0.32	0.37	0.21	0.48
Uncertainty	0.2	0.2	0.15	0.05	0.1	0.1	0.1

.

Impact on Results: The detailed integration process impacts the final risk levels presented in the tables, ensuring they accurately represent the collective expert judgment. The integrated prior probabilities for the root nodes, as shown in Table 5, are the result of fusing the probabilistic assessments of all experts, providing a more reliable and nuanced understanding of the risks.

By detailing this methodological process, we can better illustrate how the calculations influence the results in Table 3 and Table 4 and culminate in the integrated probabilities in Table 5. This ensures clarity and transparency in how the expert evaluations contribute to the risk assessment.

4.2.2. Construction of FBN Network Model and Risk Evaluation

The construction of the fuzzy Bayesian network (FBN) model for risk evaluation in deep foundation pits within karst regions involves several critical steps. First, we must define the nodes and structure of the Bayesian network (BN). Each node in the network represents a risk factor, including environmental risk (X₁), design risk (X₂), construction risk (X₃), and management risk (X₄). The initial probabilities for these factors are derived from expert judgment and historical data, incorporating the uncertainty inherent in these evaluations.

Step 1: Fuzzification of Risk Factors

Each risk factor’s initial probability is fuzzified to handle the inherent vagueness and uncertainty. This process converts crisp values into fuzzy sets using membership functions. For instance, the probability distributions for environmental risk (X₁), design risk (X₂), construction risk (X₃), and management risk (X₄) are transformed into fuzzy sets, enabling a more nuanced representation of risk levels.

Step 2: Construction of Bayesian Network

The fuzzy sets are then integrated into the Bayesian network model. Each node in the BN is updated with the corresponding fuzzy probabilities. The structure of the BN is defined based on causal relationships established through domain expertise and a literature review. Conditional probability tables (CPTs) for each node are constructed, representing the likelihood of each risk state given the states of parent nodes.

Step 3: Inference and Learning

Inference in the BN is performed using a combination of fuzzy logic and Bayesian updating. The fuzzy probabilities are aggregated and propagated through the network. This is achieved via the application of Bayes’ theorem, taking into account the conditional dependencies between nodes. The updated probabilities reflect the integrated impact of all risk factors on the overall risk state of the project.

Step 4: Defuzzification and Risk Evaluation

The resultant probabilistic outputs from the FBN are defuzzified to obtain crisp values. These values represent the likelihood of various risk levels for the project. The risk probability set for our project under consideration is determined to be P = {0.0711, 0.1804, 0.2541, 0.4633, 0.0311}. Based on these probabilities, the project is identified as being in a high-risk state (Level IV).

Model Results and Interpretation

Figure 7 illustrates the output of the FBN model. It shows how the fused probability values for each risk factor affect the overall risk evaluation. For example, the highest probability observed for management risk (X₄) significantly contributed to the overall high-risk state. The detailed breakdown in the figure corresponds to the specific calculations performed during the fuzzification, Bayesian updating, and defuzzification processes.

4.2.3. Sensitivity Analysis of Indicators and Risk Control

To analyze the sensitivity of each indicator within the FBN model, the probability of construction risk for the water-rich karst deep foundation pit project was set to 100%. The analysis results showed that the probabilities of environmental risk (X₁) and construction risk (X₃) were both greater than 45%. Among the root nodes, the indicators with higher sensitivity were sequentially geological conditions (x₁₁), solution cavity filling quality (x₃₃), and the existing surrounding environment (x₁₄); while the indicators with lower sensitivity were, sequentially, exploration quality (x₁₃), construction team experience (x₄₂), and precipitation scheme design (x₂₁).

In response to the sensitive factor of complex geological conditions, a detailed geological survey was conducted prior to construction. It revealed a thick mud layer within the impact range of the pit excavation, coupled with strongly developed solution cavities, with a cavity discovery rate of 57.44%. Additionally, several incidents of surface collapse and water inrush occurred during the detailed survey and construction process, as shown in Figure 8. Given the deep excavation depth of the pit, risks such as diaphragm-wall leakage, water and sand inrush, and support structure instability might occur during the construction process.

To prevent these risk incidents, the project implemented the technique of mixing pile trench wall reinforcement for the continuous walls within the weak stratum range; before the construction of the enclosure structure, advanced drilling was conducted to determine the development of solution cavities, and grouting was used to fill these cavities. Tube-well dewatering was utilized, and steel plates were added to the joints of the diaphragm wall to prevent water and sand leakage at the joints during the excavation of the foundation pit.

To ensure the quality of the cavity filling, site-grouting tests were carried out before construction. The grouting parameters were adjusted based on the test results, and the amount of grouting and its effective range were determined through field trials. The solution cavities were treated by first drilling holes and then installing grouting pipes. For karst geysers, karst channels, and fractured zones (including karst collapse areas), the treatment plans were separately researched by the design unit according to pumping tests and strata conditions, and implemented after review and confirmation by the client.

For areas with complex groundwater conditions or other sections requiring focused study, the civil engineering unit needed to appropriately expand the scope of intensified karst exploration based on actual conditions. The sequence of grouting and drilling was not absolute, with overall volume control being flexible and based on the principle of sealing the edges first and then the middle. This involved low pressure, multiple times, and large volume control for rapid edge-sealing. At the same time, the selection of suitable filling materials should be based on the form of cavity filling, with the specific determination method illustrated in Figure 9.

The project is adjacent to a national railway foundation pit on its west side, with the pit being 8.75 m deep and directly abutting this foundation pit project, as shown in Figure 10. Therefore, the risk associated with the existing surrounding environment of this project is relatively high. The excavation of the national railway foundation pit may cause an imbalance in lateral earth pressure on both sides of this foundation pit project, leading to excessive deformation and instability of the pit.

4.3. Dynamic Risk Evaluation during Construction

Building on the static risk evaluation, the current moment’s FDBN risk transition probabilities were determined using the expert experience method, as shown in

Table 6. Transition probabilities of FDBN.

t − 1	t
	I	II	III	IV	V
I	0.73	0.17	0.05	0.02	0.03
II	0.07	0.82	0.07	0.02	0.02
III	0.06	0.04	0.77	0.1	0.03
IV	0.02	0.02	0.08	0.81	0.07
V	0.01	0.03	0.05	0.13	0.79

. A dynamic assessment of construction risk is carried out for each event segment based on monitoring data, with corresponding risk-control measures being enacted. Due to space constraints, this paper only displays a dynamic evaluation of the risk for the water-rich karst deep foundation pit project across 26 time segments from September 15 to October 10. After entering the monitoring data as observational evidence into the model, dynamic adjustments to the FDBN risk-update results were obtained using Formulas (4) and (5), as illustrated in Figure 11.

From Figure 11, it is evident that the risk of the project was high on 15 September, which corresponds with the actual situation on-site. This was primarily due to complex geological conditions, with thick mud layers and developed solution cavities leading to significant risks of diaphragm-wall leakage and water and sand inrush. Subsequent measures such as advanced drilling to determine the development of solution cavities and grouting to fill them and adding steel plates at the joints of the diaphragm wall, among others, were taken. The evolution of the risk levels up to 22 September indicates that these measures effectively reduced the construction risk of the foundation pit.

During the continuous dynamic risk evaluation process, it was discovered that the construction risk of the project increased back to a dangerous state by 22 September, matching the actual situation on site. At this time, the increase in the depth of the pit excavation, along with complex geological conditions and strong karst development, led to significant deformation in the diaphragm wall and an increased risk of temporary support-pillar cracking. To mitigate the engineering risk, critical areas were immediately reinforced on the construction site, and the loading on bracing decks and vehicle loads were strictly controlled. The subsequent pattern of risk changes showed that the construction risk of the project gradually reduced, and its safety progressively improved.

Hence, it is evident that the dynamic evaluation method for the construction risk of water-rich karst deep foundation pits based on FDBN and improved DS evidence theory proposed in this paper can track construction risks in real time, providing theoretical guidance and practical support to ensure the safety of the construction process.

4.4. Reliability Analysis

The purpose of this section is to verify the reliability of the proposed risk-assessment methodology for deep foundation pit projects in karst regions by comparing it with three similar methods and applying it to three analogous projects.

Similar Methods:

Normal Cloud Model (NCM) [42]: This method integrates weighting and cloud models to assess water inrush risk. It has been applied effectively in the Qiyueshan tunnel project.

DEMATEL-VIKOR [43]: This approach combines static and dynamic fuzzy uncertainty assessment techniques to evaluate mountain tunnel-collapse risk, based on data from 150 tunnel-collapse incidents.

Fuzzy Analytic Hierarchy Process (F-AHP) [44]: This method employs a fuzzy analytic hierarchy process for dynamic risk assessment of karst tunnel collapses, utilizing a newly designed questionnaire for data collection.

Analogous Projects:

Guangzhou Tangxi Phase 1 Section 2: This project, situated in the northern part of Guangzhou, involves the construction of a deep foundation pit, reaching up to 34 m in depth. The geological conditions at this site are notably complex, consisting of backfill soil, silt, and various degrees of weathered limestone, which are typical of karst environments. The construction approach includes the use of diaphragm walls and internal bracing systems to provide the necessary structural support. These conditions and construction methods make it a pertinent comparison for evaluating risk factors in similar geological settings.

Guangzhou Tangxi Phase 1 Section 3: Similar to Section 3, this project’s key feature is the challenging geological conditions marked by significant karst development. The foundation pit depth here is approximately 30 m, further contributing to the complexities of construction. The project employs analogous engineering techniques, such as diaphragm walls and bracing systems, to manage the structural challenges posed by the karst terrain. These shared characteristics with Section 3 and Section 1 underscore the relevance of comparing risk-assessment methodologies across these projects.

Guangzhou Tianhe District International Finance Center Foundation Pit Project: This project entails the development of a deep foundation pit for a high-rise building, situated in an area with geological conditions resembling those of the Tangxi projects. The site features a combination of silt, clay, and sandy layers, with a maximum excavation depth of around 25 m. The construction methods employed include underground continuous walls and anchor systems for stabilization, mirroring the techniques used in the Tangxi foundation pits. The geological and construction challenges are highly similar, providing a valid basis for comparative risk assessment.

Analysis results are shown in

Table 7. Risk evaluation results of different methods for static risks in water-rich karst deep excavations.

Case	Static Risks
	This Study	F-AHP	DEMATEL-VIKOR	NCM
1	II	II	III	II
2	III	IV	III	III
3	I	I	I	I

and Figure 12.

4.5. Analysis and Disscussion

Effectiveness of the Proposed Methodology: The integration of fuzzy dynamic Bayesian networks with the improved Dempster–Shafer evidence theory presented in this study has demonstrated a substantial advancement in the field of risk assessment for deep foundation pit projects, particularly in karst regions. By incorporating fuzzy logic, the methodology effectively addresses the inherent uncertainties and complexities associated with geological conditions, especially in areas characterized by karst topography.

The case study of the Guangzhou Tangxi Section 1 project was instrumental in validating the practicality and robustness of the proposed approach. Critical risks such as diaphragm-wall leakage, water and sand inrush, and instability of support structures were identified, highlighting the method’s capacity to pinpoint and evaluate potential threats accurately. The identification of a high-risk state (Level IV) underlines the method’s acute sensitivity to risk levels, which is crucial for early warning and proactive management strategies.

Comparison with Existing Methods: In comparing the proposed methodology with the Normal Cloud Model, DEMATEL-VIKOR, and the fuzzy analytic hierarchy process, it becomes evident that the fuzzy dynamic Bayesian network coupled with the improved Dempster–Shafer theory offers superior performance. Specifically, it outperforms existing methods in three key areas:

Accuracy: The proposed method’s ability to combine multiple sources of evidence and manage conflicting information results in a more precise risk assessment.

Sensitivity: The dynamic nature of the Bayesian network allows for the continuous updating of risk levels as new information becomes available, providing real-time sensitivity to changing conditions.

Stability: The integration with the improved Dempster–Shafer theory enhances the stability of the risk assessment, ensuring consistent and reliable outcomes even in the face of uncertain or sparse data.

Reliability analysis through additional projects: The application of the methodology to three additional analogous projects—Guangzhou Tangxi Phase 1 Section 2, Guangzhou Tangxi Phase 1 Section 3, and Guangzhou Tianhe District International Finance Center Foundation Pit Project—further underscores its reliability and versatility. In each of these cases, the method consistently identified critical risks and provided actionable insights that supported robust risk-management decisions.

For instance, in the Guangzhou Tianhe District project, the method facilitated the early detection of potential groundwater intrusion and the timely implementation of preventative measures, thereby mitigating significant risk. The ability to replicate successful outcomes across multiple projects highlights the methodology’s generalizability and robustness.

Impact on Risk-Management Practices: The actionable insights derived from the proposed methodology have far-reaching implications for risk-management practices in karst regions. By providing a nuanced understanding of risk levels and facilitating the timely implementation of mitigation strategies, the method supports more-informed decision-making processes. This is particularly crucial in environments where traditional risk-assessment methods may fall short due to the complex interplay of geological factors.

Moreover, the enhanced accuracy, sensitivity, and stability offered by the proposed approach enable project managers and engineers to preempt potential issues more effectively, thereby reducing the likelihood of costly delays and ensuring project integrity.

In summary, the fuzzy dynamic Bayesian network integrated with the improved Dempster–Shafer evidence theory offers a significant advancement in addressing risk-assessment challenges for deep foundation pit projects in karst regions. The method’s superior accuracy, sensitivity, and stability were demonstrated through both the case study of the Guangzhou Tangxi Section 1 project and its application to three additional projects, underscoring its robustness and practical utility.

This methodology not only accurately identifies high-risk states and critical risks such as diaphragm-wall leakage and water and sand inrush but also provides actionable insights that are crucial for the timely implementation of mitigation strategies. These attributes highlight its potential to transform risk-management practices in complex geological settings, ensuring safer and more efficient project execution. By enabling more nuanced and reliable risk assessments, the proposed approach significantly contributes to enhancing the overall safety and integrity of deep foundation pit projects in karst regions.

5. Conclusions and Limitations

In this study, we successfully integrated the fuzzy dynamic Bayesian network (FDBN) model with an improved Dempster–Shafer (DS) evidence theory to formulate a novel dynamic risk evaluation method specifically for construction in water-rich karst deep foundation pits. This method was effectively validated through its application to the Tangxi Section 1 Phase 1 deep foundation pit project in Guangzhou. Key findings from our research include the following:

• Innovative Risk-Indicator System: We developed a comprehensive risk indicator system tailored for water-rich karst deep foundation pits, encompassing environmental, design, construction, and management risks. This system was constructed through an extensive literature review, analysis of accident statistics, and adherence to relevant standards.
• Enhanced Risk-Evaluation Reliability: By integrating fuzzy theory with the DBN, we achieved a quantitative description of fuzzy information, thus reducing subjectivity in the risk-evaluation process. The incorporation of an improved DS evidence theory further synthesized these evaluation results, significantly enhancing the reliability of the risk assessment.
• Dynamic Risk Assessment and Control: The FDBN-DS evaluation method pinpointed key risk factors such as geological conditions, solution cavity filling quality, and the surrounding environment, establishing a pre-construction static risk level of Level IV. Throughout the construction process, this method provided dynamic risk assessments, enabling timely implementation of risk-control measures, thereby ensuring construction safety and verifying the method’s effectiveness.
• Furthermore, we compared our methodology with three alternative methods, demonstrating its superior performance in dynamic risk assessment and reliability. To further substantiate the reliability of our method, we applied it to three additional deep foundation pit projects. The consistent success across these projects confirms the robustness and applicability of our FDBN-DS model in diverse, water-rich karst environments.

The limitations of the proposed method are as follows:

• The reliance on expert judgment may introduce potential biases in risk identification.
• Comprehensive data across various geological conditions are required to generalize the findings effectively.
• Although the generalizability of the findings is supported by validation through three additional project applications, further validation across a wider range of geological conditions and projects is needed to strengthen the robustness and applicability of the method.

Future work directions include the following:

• Further validation of the methodology across a broader spectrum of geological conditions and different types of projects to enhance its reliability and applicability.
• Development of a comprehensive correlation model of geotechnical and geological properties of cohesive soils for other regions in Asia. This model would involve extensive data collection and analysis of geotechnical properties, integration with the fuzzy dynamic Bayesian network (FDBN) and improved Dempster–Shafer (DS) evidence theory, and subsequent calibration and validation using the collected data. The key data required include soil classification and physical properties, in situ testing results, laboratory test results, and historical performance data of foundation pits and related engineering projects.
• The potential challenges in developing this model, such as addressing the heterogeneity of soil properties and ensuring data accessibility from various regions, should be carefully managed.