Risk Assessment of Water Inrush in Tunnels: A Case Study of a Tunnel in Guangdong Province, China

Abstract

Fractured tectonic zones with developed groundwater are one of the major causes of water inrush in the construction of igneous tunnels; thus, it is highly important to assess the risk of water inrush. In this study, a total of six evaluation attributes, groundwater level, amount of inrush water, permeability coefficient, strength of the surrounding rock, rock integrity, and width of the jointed and fault fracture zone, were selected for the risk assessment of water inrush, and fuzzy theory was applied to the treatment of the uncertainty in the evaluation attributes. On this basis, the MULTIMOORA (multiple multi-objective optimization by ration analysis) and the model of nearness degree of incidence were combined to obtain the new model of MULTIMOORA–nearness degree of incidence for the risk assessment of water inrush. A deep-underground, extra-long tunnel under construction in southern China was used as an example for validation. The six tunnel sections assessed on site were ZK91 + 195~236, K91 + 169~186.5, K91 + 203~238, ZK94 + 238~198, K94 + 112~82, and K94 + 076~034. K94 + 112~82 was assessed at risk Level III, while the remainder were assessed at risk Level IV. The site conditions were also Level III for section K94 + 112~82 and Level IV for the remainder. The assessment results for the above six tunnel sections are consistent with the field conditions, which verified the validity of the model of the MULTIMOORA–nearness degree of incidence.

1. Introduction

As an important part of national transport systems, tunnels have developed rapidly worldwide over the last decade or so [1]. At the same time, new records are being set for the depth of the tunnels, the span of excavation, etc. [2]. As a result, geological hazards such as water inrush are often encountered during construction [3,4,5]. Water-rich fracture zones are one of the main causes of water inrush in tunnels [6]. The occurrence of water hazards can be a serious threat to the safety of life and property of construction workers [7,8,9], in addition to involving groundwater drainage and environmental protection issues [10]. The risk assessment of the water inrush hazard is, therefore, of great importance for the next step in risk control [11].

A number of studies have been carried out by academics on the assessment of the risk of water inrush in tunnels [12,13,14]. Li et al. [15] constructed a risk evaluation system for water inrush in karst tunnels through fuzzy mathematics and AHP theory. Xue et al. [16] worked out the potential function for the risk of water or mud inrush and the index of water inrush for assessing and predicting the risk of sudden water inrush from faults, according to the cusp catastrophe model. Zhang et al. [17] established a risk assessment system for karst water inrush based on extension theory. Zheng et al. [18] derived the activation coefficient of the fault inrush water by means of numerical simulation and established an inrush water risk assessment system based on the acceptability of activation coefficient. On the basis of considering the objectivity of the field data and the subjectivity of expert knowledge, Sun et al. [19] proposed a new hybrid copula-based nonparametric Bayesian model for risk assessment of water inrush.

The water inrush disasters in karst areas are characterized by high flows, high velocities, etc. [20]. Hence, current water inrush hazard assessments are focused on karst tunnels. Due to the higher compressive and shear strength of igneous rocks [21,22], igneous rock areas are usually considered to be ideal areas for underground construction, although tunnels in igneous areas are less well researched. Under the effect of fracture formation, the destructive nature of water inrush in tunnels in igneous rock areas is more easily overlooked. In addition, many existing methods for risk assessment of water inrush are limited by differences in the classification [23] and errors in the calculation of evaluation attributes, which will directly cause deviations in the assessment results. More often than not, risk and uncertainty are not considered together in practical applications [24].

Multicriteria decision making (MCDM) approaches solve the problem of searching for the highest-priority sample in a group of samples based on the characteristics of multiple evaluation attributes, and they assign priority levels to individual samples within the sample group [25]. In 2010, Brauers and Zavadskas [26] improved MOORA to MULTIMOORA (multiple multi-objective optimization by ration analysis) [27], which is more robust. MULTIMOORA is a type of MCDM [28]. It has been found to be mathematically simple. In MULTIMOORA, there are three different methods for ranking target objects which are processed by the aggregation tool, resulting in a reliable slave ranking. This method has wide application prospects.

In this study, MULTIMOORA was used for the first time to assess the risk of water inrush in the tunnel, with the aim of avoiding errors arising from the discrepancy in the division of property evaluation indicators. Several segments of the tunnel to be evaluated were selected as evaluation objects. The risk data samples of the tunnel sections were extracted from the monitoring data on site and placed in the MULTIMOORA theoretical system together with the standard level of risk. Then, they were combined to obtain the respective risk ranking. In addition, the evaluation objects between the standard samples of water inrush risk were divided into risk levels, depending on the model of nearness degree of incidence. For the purpose of dealing with uncertainty in risk assessment, fuzzy theory was introduced into the MULTIMOORA theoretical system [29]. In risk ranking, the weighting of the evaluation attribute indicators needs to take into account the veracity of the objective data and the level of experience of the engineer. The decision is, therefore, taken by means of a combined weighting. The FAHP method [30], which maximizes the characterization of the decision maker’s experience, is used for subjective weights, and the CRITIC algorithm [31], which takes into account the amount of information and independence of the attributes, is used for objective weights. This article shows that the model of MULTIMOORA–nearness degree of incidence can effectively assess the risk of water inrush in tunnels affected by fault structures in igneous areas, using a deep buried extra-long tunnel under construction in southern China as an example.

2. Theory and Methods

2.1. Fuzzy Set Theory

Definition 1.

A fuzzy set FS in a limited universe of discourse U is characterized by a membership function${\mu }_{FS}\left(x\right)$. The membership of x to FS is reflected by${\mu }_{FS}\left(x\right)$, where$\forall x\in U$satisfies

$$FS=\left\{x,{\mu }_{FS}\left(x\right)|x\in U\right\}.$$

(1)

Definition 2.

The membership function${\mu }_{FS}\left(x\right)$of a trapezoidal fuzzy number$a^{*}=\left(a_1,a_2,a_3,a_4\right)$can be expressed as

$${\mu }_{FS}\left(x\right)\left\{\begin{array}{c}0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}x<a_1\\ \frac{x-a_1}{a_2-a_1}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{a}_1\le x\le a_2\\ 1\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{a}_2<x<a_3\\ \frac{x-a_4}{a_3-a_4}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{a}_3\le x\le a_4\\ 0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}x>a_4\end{array}\right..$$

(2)

when$a_2=a_3$, it is a special form of a*, known as a triangular fuzzy number.

Definition 3.

Let there be a set R of non-negative real numbers, and$R=\left(r_1,r_2,\dots ,r_n\right)$. According to the Pauta criterion [32], R represented by trapezoidal fuzzy number$a^{*}=\left(a_1,a_2,a_3,a_4\right)$, which can be expressed as

$$\left\{\begin{array}{c}{a}_1=0.9a_2\\ a_2=\frac{1}{n}{\displaystyle {\sum }_{z=1}^n{r}_n-\sigma }\\ a_3=\frac{1}{n}{\displaystyle {\sum }_{z=1}^n{r}_n+\sigma }\\ a_4=1.1a_3\end{array}\right.,$$

(3)

where σ is the standard deviation.

Definition 4.

Any of$a^{*}=\left(a_1,a_2,a_3,a_4\right)$and$b^{*}=\left(b_1,b_2,b_3,b_4\right)$obey the following basic rules of arithmetic:

$$a^{*}\oplus b^{*}=\left(a_1+b_1,a_2+b_2,a_3+b_3,a_4+b_4\right),$$

(4)

$$a^{*}\ominus b^{*}=\left(a_1-b_1,a_2-b_2,a_3-b_3,a_4-b_4\right),$$

(5)

$$a^{*}\otimes b^{*}=\left(a_1{b}_1,a_2{b}_2,a_3{b}_3,a_4{b}_4\right),$$

(6)

$$a^{*}\oslash b^{*}=\left(a_1/b_1,a_2/b_2,a_3/b_3,a_4/b_4\right),$$

(7)

$$a^{*}\otimes R=\left(a_{1}R,a_{2}R,a_{3}R,a_{4}R\right),$$

(8)

$$a^{*}{}^{-1}=\left(\frac{1}{a_4},\frac{1}{a_3},\frac{1}{a_2},\frac{1}{a_1}\right),$$

(9)

where R is a real number.

Definition 5.

The degree of probability$Pr\left(a^{*}>b^{*}\right)$between$a^{*}=\left(a_1,a_2,a_3,a_4\right)$and$b^{*}=\left(b_1,b_2,b_3,b_4\right)$is calculated as follows:

$$\begin{array}{l}Pr\left(a^{*}\ge b^{*}\right)=\frac{\phi }{2}max\left[1-max\left(\frac{a_2-b_1}{a_2-a_1+b_2-b_1},0\right),1\right]+\frac{1}{2}max\left[1-max\left(\frac{a_3-b_2}{a_3-a_2+b_3-b_2},0\right),1\right]\\ +\frac{1-\phi }{2}max\left[1-max\left(\frac{a_4-b_3}{a_4-a_3+b_4-b_3},0\right),1\right],\end{array}$$

(10)

where$\phi \in \left[0,1\right]$, and the decision maker’s attitude toward the target is represented by$\phi$. When$\phi >0.5$, experts are optimistic. Conversely, experts are conservative when$\phi <0.5$. If$\phi =0.5$, experts are neutral.

Definition 6.

The distance$d=\left(a^{*},b^{*}\right)$between two trapezoidal fuzzy numbers$a^{*}=\left(a_1,a_2,a_3,a_4\right)$and$b^{*}=\left(b_1,b_2,b_3,b_4\right)$can be calculated using the vertex method as follows [33]:

$$d\left(a^{*},b^{*}\right)=\sqrt{\frac{1}{6}\left[{\left(a_1-b_1\right)}^2+2{\left(a_2-b_2\right)}^2+2{\left(a_3-b_3\right)}^2+{\left(a_4-b_4\right)}^2\right]}.$$

(11)

Definition 7.

The trapezoidal fuzzy number needs to be defuzzified and transformed to a crisp value during the application. There exists a score function between the trapezoidal fuzzy number and the crisp value, which is expressed as follows [34]:

$$F\left(a^{*}\right)=\left\{\begin{array}{c}\sqrt[4]{a_1{a}_2{a}_3{a}_4}, a_1,a_2,a_3,a_4\succ 0\\ -\sqrt[4]{a_1{a}_2{a}_3{a}_4}, a_1,a_2,a_3,a_4\prec 0\end{array}\right..$$

(12)

2.2. Calculation of Weights for the Appraisal Attributes

2.2.1. Subjective Weights on the Basis of FAHP

In this study, the subjective weights of the evaluation attributes were determined using the FAHP method [30]. Let $T=\left\{t_1,t_2,t_3,\dots ,t_n\right\}$ be a collection of objects. Then, for the purpose of this paper, the evaluation attributes were classified as sets of objects. The FAHP method was calculated as described below.

Step 1. The experts appraised the degree of importance between two evaluation attributes on the basis of the fuzzy linguistic variables in

Table 1. Fuzzy linguistic variables for weights of evaluation of attributes.

Fuzzy Linguistic Variables	Trapezoidal Fuzzy Number
Very high	(0.8, 0.9, 1.0, 1.0)
High	(0.7, 0.8, 0.8, 0.9)
Medium high	(0.5, 0.6, 0.7, 0.8)
Medium	(0.4, 0.5, 0.5, 0.6)
Medium low	(0.2, 0.3, 0.4, 0.5)
Low	(0.1, 0.2, 0.2, 0.3)
Very low	(0, 0, 0.1, 0.2)

. There were h experts who performed a level of importance analysis between any two objects $t_i\left(i=1,2,\dots ,n\right)$ to obtain trapezoidal fuzzy number evaluation values $p_{ij}^{1\ast },p_{ij}^{1\ast },\dots ,p_{ij}^{h\ast }$, which formed the evaluation matrix $E{M}_1$. Then, the weighted average matrix $E{M}_2={\left[\overline{p_{ij}^{\ast }}\right]}_{n\times n}$ was obtained, which is the weighted average of $E{M}_1$.

Step 2. The fuzzy synthesis degree S_i of each evaluation attribute was calculated according to

$$S_i={{\displaystyle \sum }}_{j=1}^n\overline{p_{ij}^{\ast }}\otimes {\left[{{\displaystyle \sum }}_{i=1}^{\mathrm{n}}{{\displaystyle \sum }}_{j=1}^{\mathrm{n}}\overline{p_{ij}^{\ast }}\right]}^{-1},$$

(13)

where ${{\displaystyle \sum }}_{j=1}^n\overline{p_{ij}^{\ast }}$ was calculated using Equation (14).

$${{\displaystyle \sum }}_{j=1}^n\overline{p_{ij}^{\ast }}=\left({{\displaystyle \sum }}_{j=1}^n\overline{p_{ij,1}^{}},{{\displaystyle \sum }}_{j=1}^n\overline{p_{ij,2}^{}},{{\displaystyle \sum }}_{j=1}^n\overline{p_{ij,3}^{}},{{\displaystyle \sum }}_{j=1}^n\overline{p_{ij,4}^{}}\right).$$

(14)

${\left[{{\displaystyle \sum }}_{i=1}^n{{\displaystyle \sum }}_{j=1}^n\overline{p_{ij}^{\ast }}\right]}^{-1}$ is represented by Equation (15).

$${\left[{{\displaystyle \sum }}_{i=1}^n{{\displaystyle \sum }}_{j=1}^n\overline{p_{ij}^{\ast }}\right]}^{-1}={\left({{\displaystyle \sum }}_{i=1}^n{{\displaystyle \sum }}_{\mathrm{j}=1}^n\overline{p_{ij,4}^{}},{{\displaystyle \sum }}_{\mathrm{i}=1}^n{{\displaystyle \sum }}_{\mathrm{j}=1}^n\overline{p_{ij,3}^{}},{{\displaystyle \sum }}_{\mathrm{i}=1}^n{{\displaystyle \sum }}_{\mathrm{j}=1}^n\overline{p_{ij,2}^{}},{{\displaystyle \sum }}_{\mathrm{i}=1}^n{{\displaystyle \sum }}_{\mathrm{j}=1}^n\overline{p_{ij,1}^{}}\right)}^{-1}.$$

(15)

Step 3. The degree of probability between the fuzzy synthesis degree S_i and S_j is defined as $P{r}_{ij}=Pr\left(S_i^{\ast }\ge S_j^{\ast }\right)$. The matrix of possibility degree $P={\left[P{r}_{ij}\right]}_{n\times n}$ was calculated using Equation (10). Finally, subjective weights for the evaluation attributes were obtained by

$${\omega }_{s_j}=\frac{{{\displaystyle \sum }}_{j=1}^{n}P{r}_{ij}+\frac{n}{2}-1}{n\left(n-1\right)}.$$

(16)

The vector of subjective weights can be expressed as ${\omega }_s=\left({\omega }_{s_1},{\omega }_{s_2},\dots ,{\omega }_{s_n}\right)$.

2.2.2. Objective Weights on the Basis of CRITIC

In this section, the aim is to effectively retain the amount of information on the evaluation attributes and ensure their independence. Therefore, the CRITIC algorithm [31] was applied to derive the objective weights of the evaluation attributes in a fuzzy environment, which was performed as described below.

Step 1. The evaluation matrix $V={\left(v_{ij}^{*}\right)}_{m\times n}$ of the objects was set up, where the value of the j-th evaluation attribute of the i-th object was denoted as $v_{ij}^{*}=\left(v_{ij,1},v_{ij,2},v_{ij,3},v_{ij,4}\right)$.

Step 2. The evaluation matrix $V={\left(v_{ij}^{*}\right)}_{m\times n}$ was normalized. For the evaluation attributes of the benefit category, a greater value of the attribute indicated a higher risk of water inrush. Consequently, it was normalized using Equation (17).

$$\widetilde{v_{ij,k}}=\frac{v_{ij,k}-min\left(v_{ij,k}\right)}{max\left(v_{ij,k}\right)-min\left(v_{ij,k}\right)};\left(k=1,2,3,4\right).$$

(17)

In addition, a higher value of the attribute indicates a lower risk of water inrush for the cost category. It was normalized using Equation (18).

$$\widetilde{v_{ij,k}}=\frac{\max \left(v_{ij,k}\right)-v_{ij,c}}{max\left(v_{ij,k}\right)-min\left(v_{ij,k}\right)};\left(k=1,2,3,4\right).$$

(18)

The normalized trapezoidal fuzzy number can be expressed as

$$\widetilde{v_{ij}^{*}}=\left(\widetilde{v_{ij,1}},\widetilde{v_{ij21}},\widetilde{v_{ij,3}},\widetilde{v_{ij,4}}\right).$$

(19)

Step 3. $\widetilde{v_{ij}^{*}}$ was defuzzified to obtain the corresponding score value $f_{ij}$ using Equation (12). The scoring matrix $F={\left(f_{ij}\right)}_{m\times n}$ was composed of $f_{ij}$. When $F={\left(f_{ij}\right)}_{m\times n}$ was obtained, the average (Ave) and the standard deviation (Std) of the score function values for each evaluation attribute were calculated as follows:

$$Av{e}_j=\frac{1}{m}{\displaystyle \sum }_{i=1}^m{f}_{ij},$$

(20)

$$St{d}_j=\sqrt{\frac{1}{m-1}{\displaystyle \sum }_{i=1}^m{\left(f_{ij}-Av{e}_j\right)}^2}.$$

(21)

Step 4. The coefficient of variation (Cv), correlation coefficient (Cor), and conflict coefficient (Con) for each evaluation attribute were obtained using the following equation:

$$C{v}_j=\frac{St{d}_j}{Av{e}_j},$$

(22)

$$Co{r}_{lj}=\frac{{{\displaystyle \sum }}_{i=1}^m\left(f_{il}-Av{e}_l\right)\left(f_{ij}-Av{e}_j\right)}{\sqrt{{{\displaystyle \sum }}_{i=1}^m{\left(f_{il}-Av{e}_l\right)}^2}\sqrt{{{\displaystyle \sum }}_{i=1}^m{\left(f_{ij}-Av{e}_j\right)}^2}},$$

(23)

$$Co{n}_j={\displaystyle \sum }_{l=1}^n\left(1-Co{r}_{lj}\right),$$

(24)

where $C{v}_j$ represents the coefficient of variation of the j-th attribute, $Co{r}_{lj}$ represents the correlation coefficient between the 𝑙-th attribute and the j-th attribute, and $Co{n}_j$ denotes the conflict coefficient of the j-th attribute.

Step 5. The objective weights ${\omega }_{oj}$ for each evaluation attribute were obtained according to

$${\omega }_o{}_j=\frac{C{v}_{j}Co{n}_j}{{{\displaystyle \sum }}_{l=1}^n\left(C{v}_{j}Co{n}_j\right)}.$$

(25)

Ultimately, the vector of objective weights could be expressed as ${\omega }_o=\left({\omega }_o{}_1,{\omega }_o{}_2,\dots ,{\omega }_o{}_n\right)$.

2.2.3. Combined Weights of Evaluation Attributes

The principle of minimum discriminant information [35] is widely used to solve for the combined weights. Therefore, this paper applied this principle to find the composite weight ${\omega }_j$. Let the goal function be

$$\left\{\begin{array}{l}minJ\left(\omega \right)={\displaystyle \sum }_{j=1}^n\left({\omega }_j\ln \frac{{\omega }_j}{{\omega }_{s_j}}+{\omega }_i\ln \frac{{\omega }_j}{{\omega }_o{}_j}\right)\hfill \\ s.t.{\displaystyle \sum }_{j=1}^n{\omega }_j=1,{\omega }_j\ge 0,j=1,2,\dots ,n\hfill \end{array}\right..$$

(26)

The goal function was solved to obtain Equation (27) as follows:

$${\omega }_j=\frac{\sqrt{{\omega }_{s_j}{\omega }_{o_j}}}{{{\displaystyle \sum }}_{j=1}^n\sqrt{{\omega }_{s_j}{\omega }_{o_j}}}.$$

(27)

Then, the combined weight vector can be denoted as $\omega =\left({\omega }_1,{\omega }_2,\dots ,{\omega }_n\right)$.

2.3. Extended MULTIMOORA Theory with Weights

The weights of the evaluation attributes are not taken into account in the traditional MULTIMOORA method, resulting in a certain amount of error in the risk ranking. As a result, the evaluation attributes are given a combined weight in the extended MULTIMOORA, according to the different needs of the engineering reality for risk assessment [36,37]. The specific operational steps for the extended MULTIMOORA are given below.

Step 1. The evaluation matrix $V={\left(v_{ij}^{*}\right)}_{m\times n}$ is normalized using

$$x_{ij}^{*}=\left(x_{ij,1},x_{ij,2},x_{ij,3},x_{ij,4}\right)=\left(\frac{v_{ij,1}}{\overline{v_j}},\frac{v_{ij,2}}{\overline{v_j}},\frac{v_{ij,3}}{\overline{v_j}},\frac{v_{ij,4}}{\overline{v_j}}\right),$$

(28)

$$\overline{v_j}=\sqrt{{\displaystyle \sum }_{i=1}^h{\left(v_{ij,4}\right)}^2}.$$

(29)

Step 2. The first risk rank is derived using the ratio system based on trapezoidal fuzzy numbers, which is calculated as follows:

$$y_{i,c}={\displaystyle \sum }_{j=1}^g{w}_j{x}_{ij,c}-{\displaystyle \sum }_{j=g+1}^h{w}_j{x}_{ij,c};\left(c=1,2,3,4\right),$$

(30)

$$y_i^{*}=\left(y_{i,1},y_{i,2},y_{i,3},y_{i,4}\right).$$

(31)

There are g evaluation attributes of the benefit category and n–g evaluation attributes of the cost category, while ${\omega }_j$ denotes the combined weight of the j-th evaluation attribute. $y_i^{*}$ is defuzzified according to Equation (10) to obtain $Y_i$. A larger $Y_i$ indicates a higher risk factor for the sample to water inrush. $Y_i$ is ranked from largest to smallest to obtain the first risk rank, which is named Rank₁.

Step 3. The second risk rank is constructed using the reference point approach based on trapezoidal fuzzy numbers, which is defined as

$$\widetilde{x_j^{*}}=\left\{\begin{array}{c}max{x}_{ij,1},max{x}_{ij,2},max{x}_{ij,3},max{x}_{ij,4},j\le g\\ min{x}_{ij,1},min{x}_{ij,2},min{x}_{ij,3},min{x}_{ij,4},j>g\end{array}\right.,$$

(32)

$$d_i=\underset{j}{min}d\left({\omega }_j\widetilde{x_j^{*}},{\omega }_j{x}_{ij}^{*}\right),$$

(33)

where $d\left({\omega }_j\widetilde{x_j^{*}},{\omega }_j{x}_{ij}^{*}\right)$ is calculated using Equation (11). A smaller $d_i$ indicates a higher risk factor for that sample. Later, $d_i$ is sorted from smallest to largest to obtain the second risk ranking named Rank₂.

Step 4. The third risk rank is built using the full multiplicative form on the basis of trapezoidal fuzzy numbers. Similarly, it is expressed as follows:

$$u_{i,k}=\frac{{{\displaystyle \prod }}_{j=1}^g{x}_{ij,k}{}^{w_j}}{{{\displaystyle \prod }}_{j=g+1}^n{\widetilde{x_{ij}}}^{w_j}},$$

(34)

$$u_i^{*}=\left(u_{i,1},u_{i,2},u_{i,3},u_{i,4}\right),$$

(35)

$$U_i=\sqrt[4]{u_{i,1}{u}_{i,2}{u}_{i,3}{u}_{i,4}},$$

(36)

where a larger $U_i$ indicates a higher risk factor for that sample. U_i is ordered from largest to smallest to obtain the third risk rank, named Rank₃.

Step 5. According to the dominance theory, Rank₁, Rank₂, and Rank₃ above are summarized into a single rank named Sum-Rank. Next, Sum-Rank is sorted from smallest to largest to obtain the Final Rank. A smaller value of Final Rank indicates a higher risk level.

2.4. The Model of Nearness Degree of Incidence

The geometric features of the data series can be analyzed using the model of nearness degree of incidence (NDI) and classified by correlation. The traditional NDI model is somewhat deficient in its ability to handle the sequences of oscillating data, and it is prone to large errors [38]. Therefore, the improved NDI model proposed by Geng et al. [39] was used in this study.

Definition 8.

Assume that there are m n-dimensional sequences as follows:

$$\begin{array}{l}{\alpha }_1=\left({\alpha }_1\left(1\right),{\alpha }_1\left(2\right),\dots ,{\alpha }_1\left(n\right)\right)\\ {\alpha }_2=\left({\alpha }_2\left(1\right),{\alpha }_2\left(2\right),\dots ,{\alpha }_2\left(n\right)\right)\\ \hfill \dots \hfill \\ {\alpha }_{\mathrm{m}}=\left({\alpha }_m\left(1\right),{\alpha }_m\left(2\right),\dots ,{\alpha }_m\left(n\right)\right).\end{array}$$

(37)

Then, the direct geometric distance between ${\alpha }_i$and${\alpha }_j$can be expressed as

$$\Delta D_{ij}(k)=\left|{\alpha }_i\left(k\right)-{\alpha }_j\left(k\right)\right|.$$

(38)

Definition 9.

The relative distance of incidence of any two n-dimensional sequences$a_i$and$a_j$can be expressed as

$$\Delta R_i{}_j(k)=\frac{\Delta Dij(k)}{1+\left|\underset{1\le z\le m}{\max }{\alpha }_z\left(k\right)-\underset{1\le z\le m}{\min }{\alpha }_z\left(k\right)\right|}.$$

(39)

Definition 10.

The improved NDI model of any two n-dimensional sequences$a_i$and$a_j$can be given as

$${\gamma }_i{}_j={\omega }_k\Delta R_i{}_j(k),$$

(40)

where$1\le i, j\le m;1\le k\le n$;${\omega }_k$denotes the weight of the k-th column of the sequence. A larger${\gamma }_{ij}$indicates a higher correlation between$a_i$and$a_j$.

3. The Model of Risk Assessment for Water Inrush

3.1. Model Frameworks

In this study, the basic process of the risk assessment model for water inrush proposed is shown in Figure 1, and its key processes are shown below.

Step 1. With reference to the relevant literature and the existing research base, the risk evaluation attributes for water inrush are selected, and the risk standard levels of water inrush are also determined.

Step 2. Through geological survey data and construction site monitoring, the raw engineering data for the evaluation property values of the individual assessment objects are obtained. According to fuzzy theory, the original data are characterized by trapezoidal fuzzy numbers and form a sample dataset.

Step 3. The subjective weights obtained through FAHP and the objective weights obtained through the CRITIC method are treated to derive the combined weights of the evaluation attributes, according to the principle of minimum discriminant information.

Step 4. The risk priority of each sample of objects to be assessed and the standard samples of water inrush risk are assessed using the extended MULTIMOORA theory. Then, their respective risk ranking level is obtained.

Step 5. The evaluation objects between the standard samples of water inrush risk are divided into risk classes, depending on the model of nearness degree of incidence.

3.2. Selection of Evaluation Attributes and Determination of Standard Levels

Water inrush is influenced by a wider range of factors, and various factors that affect the ease of determination and subordination are not entirely consistent. In summary, the factors can be categorized into construction, engineering, geological, and hydrogeological factors. In igneous areas, groundwater, the quality of the surrounding rock, and faults play a decisive role. Therefore, in this study, the various types of evaluation attributes were selected from the above three aspects on the basis of their accessibility and relative independence. Nevertheless, the construction factors are difficult to quantify specifically by objective data, and the influence of this component was taken into account in the selection of other indicators. Hence, it was assumed in this study that the tunnel was constructed with construction methods that are effective and recognized by the industry. Construction factors were not considered separately.

According to the available research results [14,23,40], groundwater level (E1), amount of water inrush (E2), permeability coefficient (E3), strength of the surrounding rock (E4), rock integrity (E5), and width of the jointed and fault fracture zone (E6) were selected as indicators for the dynamic risk evaluation of water inrush in the tunnel. One of E1 and E2 describes the degree of groundwater development. At the same time, E3, E4, and E5 describe the condition of the surrounding rock. In addition, E6 characterizes the faults.

With reference to Li et al. [41], Wang et al. [42], and Wang et al. [43] on the classification of water inrush, the six attribute factors mentioned above were classified into four levels, namely IV, III, II, and I. Among these, Level IV is the highest risk level. Conversely, Level I is the lowest risk level, as delineated in

Table 2. Standard levels of risk for water inrush.

Risk Level	Ⅰ	Ⅱ	Ⅲ	Ⅳ
Groundwater level (MPa)	<0.2	0.2~0.5	0.5~1	≥1
Amount of water inrush (m³/h)	<90	90~180	180~360	≥360
permeability coefficient (m/d)	<0.01	0.01~0.1	0.1~1	≥1
Strength of the surrounding rock	≥450	350~450	250~350	<250
Rock integrity	≥0.75	0.5~0.75	0.25~0.5	<0.25
Width of jointed and fault fracture zone (m)	<2	2~5	5~10	≥10

. The corresponding disaster consequences for standard levels of risk are summarized in

Table 3. The corresponding disaster consequences of standard levels.

Risk Level	Ⅰ	Ⅱ	Ⅲ	Ⅳ
The total amount of water inrush (million m³)	≤0.3	0.3~0.6	0.6~0.9	≥0.9

.

4. Case Study

4.1. Background

One of the tunnels under construction in this study is situated in Guangdong Province in southern China. The tunnel is a double-tunnel split design with a total length of over 6300 m and a maximum depth of 740 m. The tunnel site area is located in a tectonically uplifted, eroded, and denuded area of low-to-medium mountainous terrain. The stratigraphic lithology of the area is mainly Jurassic felsic tuff, Jurassic andesitic porphyrites, and Yanshan-age granites. From the New to the Old Period, they are Quaternary Pleistocene Alluvium (Q₄^al+pl), Quaternary Deluvium and Eluvoal Soils (Q^dl+el), the Upper Jurassic Gaojiping Group (J₂₊₃gj^b), etc. As shown in Figure 2, the tunnel is affected by the regional Lotus Mountain deep and large fracture zone, from which the fault structure of the tunnel body is developed. The surrounding rock level of the tunnel is II to V, which is broken. Figure 3 shows a longitudinal section and a geological cross-section of the tunnel on the basis of the CSAMT technique. Low resistance indicates that the rock in the area is fractured and that faulted tectonic zones exist.

Table 4. Physical and mechanical parameters of the rocks.

Location	Dry Density (g/cm³)	Total Porosity (%)	Uniaxial Saturated Compressive Strength (MPa)	Rock Elastic Modulus (GPa)	Poisson’s Ratio
ZK91 + 240	2.67~2.69	3.2	90.1~113.5	2.2~11.4	0.02~0.16
ZK94 + 198	2.55~2.59	5.2~5.7	33.8~101.8	27~67	0.13~0.3

Note: all parameters are measured values, except for rock elastic modulus and Poisson’s ratio, which are empirical values in Guangdong, China. In addition, the minimum value in uniaxial saturated compressive strength of rock is the strength value of the sample with fractures.

shows the physical and mechanical parameters of the rocks sampled at ZK91 + 240 and ZK94 + 198. As a result, the rock at ZK91 + 240 is rhyolitic crystalline tuff, while the rock at ZK94 + 198 is fine-grained black mica granite. The common cross-section of the tunnel is a horseshoe shape, as shown in Figure 4. It has a maximum height of 13.31 m and a maximum width of 10.18 m. Groundwater types in the tunnel site area can be classified into loose rock pore water and bedrock fracture water according to their natural conditions, transport rules, and spatial characteristics. Among them, loose rock pore water is locally sporadic, while bedrock fracture water is widespread. This also led to a large amount of high-pressure water inrush at the time of construction, as illustrated in Figure 5. The safety of the people and property was seriously threatened.

4.2. Raw Engineering Data

In order to accurately and dynamically assess the risk level of water inrush in the construction sections of the tunnel, the MULTIMOORA–nearness degree of incidence model was applied. In this study, six tunnel sections affected by fault structures were selected for illustration, numbered ZK91 + 195~236 (Ex1), K91 + 169~186.5 (Ex2), K91 + 203~238 (Ex3), ZK94 + 238~198 (Ex4), K94 + 112~82 (Ex5), and K94 + 076~034 (Ex6). All of these sections are affected by faults. The envelope type indicators are obtained via sampling of the envelope on site and in-house testing. In the tunnel site area, the permeability coefficients of the surrounding rock are gained through pumping tests and pressure water tests. The width of the fault-faulted tectonic zone and the amount of groundwater surge are obtained by means of advanced geological drilling and advanced geological prediction. Subsequently, the levels of the groundwater pressure are measured through the closed borehole.

Four raw data were recorded for all evaluation attributes of the tunnel section to be assessed, as shown in Appendix A. The aim was to evaluate the risk levels of water inrush in the six tunnel sections mentioned above; hence, the standard levels of water inrush risk in Table 2 are referenced and given as a dataset in the form of four samples of standard levels of risk, as also shown in Appendix A. It is worth noting that the number of evaluation values for the evaluation attributes should be consistent across all samples. According to Equation (2), the data for all samples are represented by trapezoidal fuzzy numbers. This allows the uncertainty in the array to be preserved in the operation, the results of which are shown in Appendix B.

4.3. Determination of Weights for Evaluation Attributes

As can be seen from Figure 1, the combined weights of the evaluation attributes need to be obtained first according to the steps below.

Step 1. Four experienced and highly professional experts were invited to conduct a two-by-two evaluation of the six evaluation attributes by applying the fuzzy linguistic variables in Table 1, the results of which are shown in Appendix C. According to Equations (10) and (13)–(15), the probability matrix EM is obtained as

$$EM=\left[\begin{array}{cccccc}0.500& 0.000& 0.255& 0.114& 0.037& 0.350\\ 1.000& 0.500& 0.404& 0.983& 0.919& 1.000\\ 0.745& 0.000& 0.500& 0.330& 0.106& 0.596\\ 0.886& 0.017& 0.670& 0.500& 0.259& 0.762\\ 0.963& 0.081& 0.894& 0.741& 0.500& 0.924\\ 6500& 0.000& 0.404& 0.238& 0.076& 0.500\end{array}\right].$$

On the basis of Equation (16), the data in EM are extracted and substituted to obtain the final subjective weights of the evaluation attributes from E1 to E6, which are ${\omega }_s=\left(0.225,0.087,0.191,0.164,0.130,0.204\right)$.

Step 2. From Table 2, it is clear that groundwater level, amount of inrush water, permeability coefficient, and the width of jointed and fault fracture zone fell into the evaluation attributes of the benefit category. On the contrary, the strength of the surrounding rock and rock integrity belonged to evaluation attributes of the cost category. Thus, the data in Appendix B were normalized using Equations (17) and (18) and defuzzified using Equation (12), resulting in the score function matrix

$$F=\left[\begin{array}{cccccc}1.000& 0.594& 1.000& 0.502& 0.623& 0.766\\ 0.923& 0.978& 0.798& 0.353& 0.528& 0.302\\ 0.900& 1.000& 0.840& 0.313& 0.464& 0.519\\ 0.455& 0.483& 0.515& 0.155& 0.392& 0.485\\ 0.012& 0.102& 0.010& 0.036& 0.062& 0.405\\ 0.779& 0.544& 0.604& 0.105& 0.490& 0.617\\ 0.000& 0.000& 0.000& 0.000& 0.000& 0.000\\ 0.000& 0.075& 0.000& 0.222& 0.333& 0.174\\ 0.001& 0.149& 0.004& 0.444& 0.667& 0.436\\ 0.002& 0.299& 0.043& 1.000& 1.000& 0.872\end{array}\right].$$

The objective weights ${\omega }_s=\left(0.209,0.134,0.201,0.213,0.125,0.118\right)$ of the evaluation attributes from E1 to E6 were finally determined with reference to Equations (20)–(25).

Step 3. Ultimately, the combined weight $\omega =\left(0.219,0.109,0.198,0.189,0.126,0.157\right)$ of the evaluation attributes from E1 to E6 were obtained, referring to Equations (26) and (27).

The above calculations show that the combined weight values for the groundwater level, permeability coefficient, and strength of the surrounding rock exceeded the average. Under the influence of the igneous fracture tectonic zone, they were the key factors influencing the water inrush in the tunnel. In addition, the rock integrity and width of the jointed and fault fracture zone were factors reflecting the condition of the surrounding rock, but they reflected less information than the key factors, as reflected by the lower combined weighting. The last evaluation attribute, amount of inrush water, was used as a dynamic on-site correction indicator, and it accordingly received the smallest combined weighting.

4.4. Risk Ranking Based on Extended MULTIMOORA

In the previous section, the combined weights of the six evaluation attributes were obtained. Immediately following the process in Figure 1, the risk ranking of the samples of tunnel segments to be assessed was carried out.

Firstly, the data in Appendix B were normalized using Equations (28) and (29), where the normalization was applied in a different way to that in the CRITIC method. The results are shown in Appendix D.

Secondly, the risk ranking Rank1, Rank2, and Rank3 of the 10 samples was obtained according to Equations (30)–(36), as shown in Appendix E, Appendix F and Appendix G. The risk rankings in Rank1, Rank2, and Rank3 were combined to obtain the Sum-Rank according to the dominance theory described in Section 2.3. The Sum-Rank was sorted from smallest to largest and given a final ranking (Fin-Rank); the results are shown in

Table 5. Risk ranking based on extended MULTIMOORA.

Samples	Rank1	Rank2	Rank3	Sum-Rank	Fin-Rank
EX1	3	3	3	9	3
EX2	1	1	1	3	1
EX3	2	2	2	6	2
EX4	6	4	5	15	5
EX5	8	6	6	20	7
EX6	4	5	4	13	4
S1	10	10	10	30	10
S2	9	9	9	27	9
S3	7	8	8	23	8
S4	5	7	7	19	6

.

From Table 5, the standard samples S1 to S4 were all ranked in decreasing order in the risk ranking Rank1, Rank2, and Rank3, which led to the Fin-Rank being similarly ranked in decreasing order. Among them, the standard sample S4 was the riskiest, as observed in the initial setting condition. The results in Table 4 were proven to be valid. Accordingly, the Fin-Rank of the six samples to be assessed was, from largest to smallest, K91 + 169~186.5 (Ex2), K91 + 203~238 (Ex3), K91 + 203~238 (Ex1), K94 + 076~034 (Ex6), ZK94 + 238~198 (Ex4), and K94 + 112~82 (Ex5). Among them, Ex1, Ex2, Ex3, Ex4, and Ex6 had a risk ranking priority greater than the standard sample S4 (Level IV) of water inrush. The above five samples were set at Level IV. According to Table 3, the amount of water inrush would be greater than or equal to 0.9 million m³ at these five tunnel sections during the construction excavation phase. In addition, Ex5 had a risk ranking priority that fell between standard sample S3 (Level III) and S4 (Level IV). The risk ranking of Ex5 was further determined through the model of nearness degree of incidence described in Section 2.4.

4.5. Calculation of Nearness Degree of Incidence

In Appendix B, the data for Ex5, S3, and S4 are shown, which are characterized by trapezoidal fuzzy numbers. They were defuzzified according to Equation (12). In accordance with Equation (37), the data feature vectors were constructed as

$$\begin{array}{l}{\alpha }_1=\left(4.934,122.154,0.206,423.781,0.683,4.625\right)\\ {\alpha }_2=\left(0.499,179.548,0.100,249.373,0.249,4.987\right)\\ {\alpha }_3=\left(0.997,359.097,0.997,0.000,0.000,9.975\right).\end{array}$$

The above results were substituted into Equations (38)–(40) to find γ₁₂ = 0.742 and γ₁₃ = 0.500. It is clear that γ₁₂ ≻ γ₁₃. Although Ex5 had a higher risk ranking priority than S3 in MULTIMOORA, it was closer to standard sample S3 and far from S4. It was reasonable for Ex5 to be classified as Level III. According to Table 3, the amount of water inrush at Ex5 would be 0.6 to 0.9 million m³ during construction. Accordingly, the risk levels of the six tunnel sections were classified.

4.6. Field Validation for Feature Vector and Model Comparison

During the tunnel excavation period, the daily water inrush at the construction sites of the six tunnel sections was recorded, and the total amount of water inrush was summed up once every five days, with the water inrush recorded as shown in Figure 6. These data were used to verify the correctness of the predicted results of the model.

As shown in Figure 6, the cumulative total amount of water inrush during excavation from EX1 to EX6 was monitored at the tunnel inlet and outlet, ignoring the smaller amounts of seepage in the remaining areas of the tunnel. The total amount of water inrush sorted from largest to smallest was EX2 (3.1 million m³), EX3 (2.37 million m³), EX1 (1.78 million m³), EX4 (1.16 million m³), EX6 (1.09 million m³), and EX5 (0.77 million m³). The above ranking results are in line with the risk ranking results in Table 5. Moreover, the total amount of water inrush for the six samples mentioned above coincided with the corresponding disaster consequences in Table 3, indicating that the division of their risk level of water inrush was reasonable. It is a clear demonstration that the risk of water inrush in igneous zones affected by fracture structures could be effectively assessed by the MULTIMOORA–nearness degree of incidence model.

As a further validation of the advantages of the model proposed here, it was compared with the traditional risk matrix method and actual levels on site.

Table 6. Comparative analysis of assessment results.

Samples	The Model in This Study	Risk Matrix Method	Actual Levels on Site
EX1	IV	III	IV
EX2	IV	III	IV
EX3	IV	III	IV
EX4	IV	III	IV
EX5	III	III	III
EX6	IV	III	IV

shows the results of their comparison. The risk matrix method is the recommended method for assessing major risk sources according to the Ministry of Transport of China’s “Guidelines for Construction Safety Risk Assessment of Highway Bridge and Tunnel Projects (for Trial Implementation)”. The core part of the model here, MULTIMOORA, was, for the first time, used for the assessment of the risk aspect of water inrush in the tunnel. The errors caused by the variability in the division of the attribute evaluation indicators were minimized. At the same time, the importance of evaluation attributes should be inconsistent during practical application. The application of combined weights to measure the importance of evaluation attributes ensures the accuracy of the risk assessment. The uncertainty in risk assessment is dealt with using fuzzy theory to solve the problem of ignoring uncertainty in previous risk assessments. Finally, the samples between the standard levels in MULTIMOORA are classified using the model of nearness degree of incidence.

As shown in Table 6, the risk matrix method was more subjective in the risk rating process. It was limited in its treatment of uncertainty. Hence, the risk matrix method had a lower level compared to the actual situation on site. In contrast, the model in this study was more accurate and more advantageous in its assessment. It can be seen from Table 5 and Table 6 that limitations inevitably existed in a single ranking system and, thus, affected the final assessment results. In MULTIMOORA, the samples were given three different rankings and ultimately received their own final ranking through the dominance theory. A Fin-Rank that is more robust than a single ranking was obtained. With this in mind, standard samples were introduced. This could help to classify the level of the samples to be assessed. It is obvious that some of the samples were ranked between the two standard samples. Finally, the model of nearness degree of incidence was applied to solve the above problem. In Table 6, it can be concluded that the assessment results for EX1, EX2, EX3, EX4, EX5, and EX6 were consistent with the actual situation on site. In summary, the MULTIMOORA–nearness degree of incidence model was superior to traditional assessment methods in fractured tectonic zones in igneous rock areas. This is a clear demonstration that the risk of water inrush in igneous zones affected by fracture structures can be effectively assessed using the MULTIMOORA–nearness degree of incidence model.

5. Conclusions

• Multicriteria decision-making approaches can also form an effective system for assessing the risk of water inrush when combined with the model of nearness degree of incidence in a rational way. In this study, the MULTIMOORA–nearness degree of incidence model could effectively assess the risk of water inrush in tunnel sections in fractured tectonic zones in igneous rock areas. The six tunnel sections assessed on site were ZK91 + 195~236, K91 + 169~186.5, K91 + 203~238, ZK94 + 238~198, K94 + 112~82, and K94 + 076~034. K94 + 112~82 was assessed at risk Level III, while the remainder were assessed at risk Level IV. The site conditions were also Level III for section K94 + 112~82 and Level IV for the remainder.
• The new MULTIMOORA–nearness degree model is more robust and has a strong capability of data analysis, thus being better capturing the data features reflecting the quantitative differences between the data. As a result, the characteristics of the data were retained to the maximum extent possible, and data distortion was avoided. Hence, it was more advantageous and accurate in the assessment of water inrush risk in tunnels than the traditional methods.
• Fuzzy theory has certain advantages in dealing with uncertainty. Therefore, the model was mathematically analyzed in a fuzzy environment for a sample dataset. Meanwhile, the combined weighting of the evaluation attributes was established during the assessment process. Through the above steps, the degree of importance between the evaluation attributes was measured. The level of experience of the engineer and the objective veracity of the data were simultaneously taken into account. As a result, the risk assessment of water inrush in the tunnel could be successfully carried out.
• The high risk of water inrush should not be ignored when underground projects with large depths of burial are built across fractured tectonic zones in igneous rock areas. The risk of water inrush should be assessed in a timely manner to avoid the hazards threatening the safety of people and property, while protecting regional groundwater resources.