Evaluation of Hydraulic Tunnel Lining Durability Based on Entropy–G2 and Gray Correlation–TOPSIS Methods

Abstract

Under long-term water flow, the physical and chemical properties of hydraulic tunnel linings are more likely to deteriorate than those of road tunnels, thus affecting the normal operation of tunnels. Evaluating the durability of hydraulic tunnel linings can help to grasp the durability of tunnels in a timely and accurate manner and provide a basis for the routine maintenance of tunnels. This paper proposes new methods for evaluating the durability of hydraulic tunnel linings. Firstly, the types of tunnel defects are divided, the durability indices corresponding to the defects are selected scientifically, and a hydraulic tunnel durability evaluation index system is established. Then, the G2 method is modified by the entropy value method to make it a subjective and objective weighting method, which can make the weights fit the reality while the calculation is easy, and the TOPSIS method is modified by the gray correlation degree to optimize the judgment criteria between the evaluation scheme and the ideal solution. Finally, the practicality and accuracy of this method are verified by the calculation of the five sections of a tunnel with lining durability grades of A, B, C, B, and C, respectively, which matched the calculation results of the RAGA-PP method in the related literature.

1. Introduction

China’s uneven distribution of water resources and significant differences in annual precipitation have seriously affected the economic development and improvement of living standards in many regions. To solve the uneven spatial and temporal distribution of water resources and to meet the production, living, and ecological requirements of water-scarce areas, China has invested in a series of water diversion and transfer projects, such as the Water Diversion Project from Datong River to Qinwangchuan District, the Water Diversion Project from Luanhe River to Tianjin City, and the South-to-North Water Diversion [1]. The durability of hydraulic tunnels is more likely to deteriorate than that of highway tunnels due to long-term exposure to water, external loads, and cyclical changes in the temperature field. Therefore, it is necessary to establish a scientific and practical evaluation method for the durability of hydraulic tunnel lining.

Currently, in tunnel defect research, Zhao, D.P. et al. [2] classified tunnel defects on the basis of the existing tunnel defect investigation and analysis, analyzed the causes of tunnel defects and the interaction between defects, and provided a reliable basis for tunnel defect management; Zhang, J.W. et al. [3] analyzed the temperature distribution characteristics of the lining along the tunnel axis under the influence of tunnel depth, tunnel radius, tunnel cavity wind speed, and thermal conductivity of the insulation layer through numerical simulation and numerical calculation, which provides a reference for the design of the reinforcement length of the tunnel against freezing defects. Xu, S.S. et al. [4] used radar scanning, borehole coring, and laser scanning to study the damage characteristics and damage causes of highway tunnel defects in water-rich strata, and innovatively proposed a complete set of solutions for defects such as cracks, water seepage, and counter-rising, which were applied in actual projects.

In the study of durability, Shi, X.M. [5] reviewed the role of mineral admixtures in concrete durability, methods for measuring chloride ingress into concrete, challenges in assessing concrete durability from chloride diffusivity, and service life models for reinforced concrete in chloride ion environments, suggesting future research problems to be addressed. Afroughsabet, V. [6] studied the effect of adding steel and polypropylene fibers on the mechanical properties and durability of high-strength concrete (HSC), tested the compressive strength, splitting tensile strength, flexural strength, electrical resistivity, and water absorption of concrete and found that the best performance was obtained with a mixture containing 0.85% steel and 0.15% polypropylene fibers. Thomas, J. [7] studied the strength and durability of recycled concrete aggregates (RCAs) and found that up to 25% of the natural aggregates in concrete can be replaced by RCAs without significantly affecting the strength and durability of the concrete, suggesting that natural aggregates can be partially replaced by RCA under moderate exposure conditions.

In the context of tunnel evaluation, Arends, B.J. [8] proposed a method for tunnel safety evaluation using probabilistic risk assessment based on three aspects: personal, social, and economic risks, and applied it in a Dutch tunnel project. Akula, P. [9] studied the effect of lime on the repair of hydraulic buildings and evaluated the long-term durability of the lime-treated broken concrete lining of the Friant-Kern canal. Li, Q.F. [10] applied the ANP and clouded matter element theory to the durability evaluation of concrete structures in hydraulic tunnels and verified the practicality of the evaluation method by example calculations.

From the above research status, it can be seen that, for the evaluation of concrete tunnel lining research, the current research object is mainly highway and railroad tunnels, and for the long-term erosion of hydraulic tunnels less research is conducted; the research content is mainly focused on the safety and risk evaluation of tunnels, while the durability evaluation of the attention is insufficient; the research method [11] makes it difficult to comprehensively take into account the actual situation, and the practicality of the research method needs to be further improved.

This paper establishes a relatively perfect index system for evaluating the durability of hydraulic tunnels on the basis of organizing the common defects of hydraulic tunnel linings. Then, a set of new durability evaluation methods are proposed: the G2 method is modified by the entropy value method to make it a subjective–objective empowerment method, and the TOPSIS method is improved by gray correlation degree. Finally, through the application of examples, the lining durability grades of five sections of Pandawling Tunnel were calculated, which verified the accuracy of the evaluation method in this paper.

2. Establishment of Evaluation Index System

To establish the durability evaluation index system of hydraulic tunnel lining, the following principles should be followed [12]: (1) Scientific principle. The selected evaluation indices have clear definitions, can be clearly distinguished from other indices, and can reflect the durability state of hydraulic tunnel lining. (2) Principle of comprehensiveness. The evaluation index system can reflect the durability status of the lining structure during the operation period of the hydraulic tunnel in a comprehensive and effective manner, and include the important characteristics and main influencing factors that reflect the durability status of the lining structure during the operation period of the hydraulic tunnel as far as possible, so that the evaluation results of the durability of the lining structure during the operation period of the hydraulic tunnel are accurate and reliable. (3) Principle of conciseness. The evaluation indices should be clearly divided into the evaluation level and hierarchical relationship, and the main and representative evaluation indices should be selected as far as possible, so as to reduce the number of evaluation indices and facilitate calculation and analysis. (4) Operability principle. The selected evaluation indicators should be easy to obtain and operable, and can be measured by the existing means and tools [13]. (5) Relevance principle. In order to make the evaluation index system of safety status of lining structure during the operation of hydraulic tunnels holistic, the selected evaluation indices should be related to each other and have certain relevance. (6) Principle of hierarchy. The evaluation index system should reflect the inner structure and key issues of different levels of indicators and build the evaluation index system into multiple levels to form a multi-level ladder analysis system containing multiple subsystems, with each level being independent of each other and the levels being interconnected.

Compared with road tunnels, hydraulic tunnels are not only affected by the external environment, but also suffer from long-term hydraulic erosion inside, which makes them more prone to lining damage. Hydraulic tunnels in cold and arid areas of northwest China often need to cross fault zones or large fracture zones, which are prone to cause tunnel sidewall and top arch cracking during operation, and tunnel fracture at fault boundaries, forming transverse cracks, water gushing from the fracture affected zone, sidewall cracking, sidewall projection deformation into the cave, and bottom slab bulge cracking. Due to the geomorphological characteristics of western China, influenced by landslides or floods, landslides and flash floods are easily formed during the rainy season, which will affect the structure of hydraulic tunnels to a certain extent [14]. The interior of the tunnel is in a water environment for a long time, and the durability state of the lining decreases as its structure and materials continue to deteriorate.

The following summarizes the lining structure defects during the operation of hydraulic tunnels, analyzes the causes and mechanisms of lining structure defects, and selects suitable evaluation indices according to the above-mentioned evaluation system establishment principles, and then establishes the hydraulic tunnel lining durability evaluation index system.

Lining cracks [15,16] are a direct manifestation of tunnel breakage, reflecting its bearing stress beyond its own strength, containing the lining deformation characteristics of the structure stressed to develop from a lower state to rupture, and is a specific indicator of the durability state of the lining structure during the operation of a hydraulic tunnel. Lining cracks can be divided into deformation cracks and stress cracks according to the cause, and can be divided into circumferential cracks, longitudinal cracks, and oblique cracks according to the direction. For the convenience of operation, the crack width and crack depth ratio (crack depth/liner thickness) are selected as the evaluation index of lining cracks.

Water leakage [4,17] is a common defect in hydraulic tunnels and is one of the causes of accelerated deterioration of the lining material, especially when water leakage shows strong acidity and the concrete is at risk of serious deterioration. Water leakage will lead to the reduction in lining strength, triggering defects such as dissolution, erosion, freezing and thawing, steel corrosion, and foundation frost swelling, accelerating the aging of concrete lining structure and shortening the service life of hydraulic tunnels. The leakage state and pH value of leaking water are selected as the evaluation indices of water leakage.

In hydraulic tunnel engineering, the high ground stress, high surrounding rock pressure, and high permeability pressure that the concrete lining is subjected to, the corrosion of various aggressive ions on the concrete lining structure and the carbonation of the concrete of the lining structure cause the problem of material deterioration of the concrete. At the same time, with the occurrence of reinforcement corrosion inside the lining, the cross-sectional area of reinforcement decreases, the joint bearing capacity with concrete decreases, and the tensile strength and ultimate tensile rate decrease significantly, thus reducing the bearing capacity and stability of the lining structure. In addition, concrete structures where reinforcement corrosion occurs are generally accompanied by cracks and spalling. The lining strength ratio (actual strength/design strength), lining thickness ratio (actual thickness/design thickness), and reinforcement interface loss rate (lost cross-section/design cross-sectional area) were selected as indicators for the evaluation of lining material deterioration [18].

Tunnel excavation will change the groundwater seepage channel, hydrostatic pressure as an additional load on the lining, dynamic water pressure on the fissure, and structural surface scouring; through the destruction of the rock body affecting the lining structure, on the other hand, the fissure water flow takes away the filling between the rock body, reducing the stability of the rock body, accelerating the disintegration of the rock body, rapidly reducing the bearing capacity of the rock body, and resulting in the collapse of the rock body and lining cracking deformation [19]. The deformation speed and deformation ratio (deformation volume/internal limit distance) are selected as the evaluation indices of lining deformation.

For lining spalling [20], on the one hand, hydraulic tunnels in cold regions are often threatened by drift ice, and a large amount of drift ice enters the hydraulic tunnels in winter and spring seasons, and drift ice with different velocities and different collision angles collide with the tunnel sidewalls, causing local damage to the tunnel lining, and long-term collision will cause concrete spalling of the hydraulic tunnel lining; on the other hand, influenced by the cold climate, when the surrounding rock behind the hydraulic tunnel lining structure freezes, it will generate frost swelling force. In the frost swelling surrounding rock, the water volume increases, which will easily cause lining frost swelling cracking near the vault, or cause concrete aggregate swelling out, mortar and concrete spalling, etc. Lining spalling in hydraulic tunnels during operation will cause reduction in the effective bearing section of concrete lining structure and corrosion of steel reinforcement, which will reduce its service life. The spalling diameter and spalling depth are selected as the evaluation indices of lining spalling.

Due to various reasons such as construction methods, construction quality, concrete shrinkage, etc., hydraulic tunnels can have more serious cavities behind the lining problems. The cavity behind the lining affects the stress state of the lining structure, thus causing cracking and deformation of the lining, and water leakage will also enter the lining interior along the cavity and cracks, causing lining concrete carbonation, reinforcement corrosion, frost damage, and other defect problems [21]. The continuous length of the cavity and the area of the cavity are selected as the evaluation indices of the cavity behind the lining.

In summary, the durability evaluation index system of hydraulic tunnel lining was established, as shown in Figure 1.

3. Evaluation Methodology

For ease of reading, the parameters used in this paper and their meanings are summarized in

Table 1. Meaning of each parameter.

Parameters	Meaning	Parameters	Meaning
X	Evaluation indicator set	U	Weighted criteria matrix
$X_j$	Evaluation of indicator j	$U^{+}$	Positive ideal solution
$X_{\min }$	Least important indicator	$U^{-}$	Negative ideal solution
$w_1$	Subjective weighting	$u_j^{+}$	Positive ideal solution element
$r_j$	Ratio of importance	$u_j^{-}$	Negative ideal solution element
$a_j$	Value of an assignment	$d_i^{+}$	Euclidean distance between scheme i and the positive ideal solution
$w_{1j}$	Subjective weight of indicator j	$d_i^{-}$	Euclidean distance between scheme i and the negative ideal solution
$D_j$	Assignment interval	$g_{ij}^{+}$	Gray correlation between indicator j in program i and the positive ideal solution
$d_{j1},d_{j2}$	Boundaries of the assignment interval	$g_{ij}^{-}$	Gray correlation between indicator j in program i and the negative ideal solution
$n(D_j)$	Midpoint of the interval $D_j$	$\lambda$	Resolution factor
$e(D_j)$	Width of the interval $D_j$	$g_i^{+}$	Gray correlation of program i with positive ideal solutions
$\epsilon$	Risk attitude factor	$g_i^{-}$	Gray correlation of program i with negative ideal solutions
$v_{ij}$	Measured value	$D_i^{+}$	Dimensionless value of $d_i^{+}$
$v_{j\min }$	Minimum measured value	$D_i^{-}$	Dimensionless value of $d_i^{-}$
$v_{j\max }$	Maximum measured value	$G_i^{+}$	Dimensionless value of $g_i^{+}$
${\overline{v}}_{\mathrm{i}j}$	Standard value corresponding to the measured value	$G_i^{-}$	Dimensionless value of $g_i^{-}$
$e_j$	Entropy	$S_i^{+}$	Positive composite index
$w_2$	Objective weighting	$S_i^{-}$	Negative composite index
$w_j$	Combined weights	$\alpha ,\beta$	Degree of preference for location, shape of evaluation program
$V$	Expansion of the decision matrix	$C_i^{+}$	Relative fit
$B$	Standard matrix

in the order in which they appear.

3.1. Entropy–G2 Method

The traditional G2 method is a subjective weight calculation method, and the introduction of entropy value into the G2 method makes it a subjective and objective weight calculation method, which can reduce the subjective factors of weight calculation and make the index weights more accurate [22,23,24].

The G2 assignment method does not require the construction of judgment matrices, and there is no need for consistency testing; the computational effort is greatly reduced compared with the hierarchical analysis method and the set-value iteration method; the number of indicators is unlimited, and indicators can be added or deleted at a later stage; the method is simple, intuitive, and easy to apply. However, the G2 assignment method, as a subjective assignment method, is subject to the subjective influence of experts and cannot accurately reflect the indicator weights. For this reason, this paper adopts the entropy value method to transform the G2 assignment method and determines the indicator weights jointly by expert scoring and indicator entropy value, making it a subjective and objective assignment method. The details are as follows:

The expert selects one and only one indicator that he considers to be the least important in the set of evaluation indicators $X=\{X_1,X_2,\dots ,X_m\}$ as a reference, which is noted as $X_{\min }$, and the expert gives the ratio of the importance of the evaluation indicator $X_j$ to the indicator $X_{\min }$ about a criterion based on his own experience, after which the subjective weight $w_1=(w_{11},w_{12},\mathrm{\dots },w_{1m})$ of each indicator is calculated based on the given value. The method is divided into the following two cases:

(1)

Point assignment

The ratio of the degree of importance of the indicator $X_j$ to the indicator $X_{\min }$ is a numerical value, let

$$r_j=a_j$$

(1)

where $j=1,2,..,m$, $a_j>1$, let $a_m=1$.

When the assignment of $a_j$ is accurate, the weight coefficient $w_{1j}$ of the evaluation indicator $X_j$ is calculated as follows:

$$w_{1j}=\frac{a_j}{{\displaystyle \sum _{j=1}^m{a}_j}},j=1,2,\mathrm{\dots },m$$

(2)

The values of $a_j$ can be assigned according to

Table 2. Assignment reference table.

${\mathit{a}}_{\mathit{j}}$	Meaning
1.0	The indicator $X_j$ has the same importance as indicator $X_{\min }$
1.2	The indicator $X_j$ is slightly more important than the indicator $X_{\min }$
1.4	The indicator $X_j$ is significantly more important than the indicator $X_{\min }$
1.6	The indicator $X_j$ is significantly more important than the indicator $X_{\min }$
1.8	The indicator $X_j$ is extremely more important than the indicator $X_{\min }$
1.1, 1.3, 1.5, 1.7	In between the above two degrees

below.

(2)

Interval assignment

In some cases, due to information asymmetry and other conditions, experts are not sure to assign an exact value to $a_j$ when they make a subjective assignment to $a_j$, but are able to give a range of values for $a_j$, at this point

$$r_j=a_j\in [d_{j1},d_{j2}]=D_j$$

(3)

where $d_{j2}>d_{j1}$, let $d_{m1}=d_{m2}=1$, $D_j$ are the importance intervals corresponding to $X_j$.

If the assignment of $D_j$ is accurate, the weighting system of the indicator $X_j$ is

$$w_{1j}=\frac{n(D_j)+\epsilon e(D_j)}{{\displaystyle \sum _{j=1}^m[n(D_j)+\epsilon e(D_j)]}}$$

(4)

where $n(D_j)$ is the midpoint of the interval $D_j$, i.e., $D_j=\frac{d_{j1}+d_{j2}}{2}$; $\epsilon$ is the risk attitude factor of the expert, which is taken as $\epsilon \in [-\frac{1}{2},0)$ for conservative experts, $\epsilon =0$ for neutral experts, and $\epsilon \in (0,\frac{1}{2}]$ for risky experts; $e(D_j)$ is the width of the interval $D_j$, i.e., $e_j=d_{j2}-d_{j1}$.

The entropy method is introduced into the G2 assignment method below. In natural science, thermodynamic entropy in physics is a measure of the disorderly state of a system. In the application of information system, information entropy is equivalent to thermodynamic entropy in the mathematical sense, but the meaning is mainly a measure of the degree of uncertainty of the system state. It is generally believed that the higher the value of information entropy, the more balanced the system structure is, the smaller the differences, or the slower the changes; conversely, the lower the information entropy, the more unbalanced the system structure is, the larger the differences, or the faster the changes [25]. Therefore, the weights can be calculated based on the entropy value magnitude, i.e., the degree of variation of each index value.

Due to the differences in the scale, order of magnitude, and positive and negative orientation of the indicators, it is difficult to compare them directly, and the initial data need to be standardized [26].

When the higher value of the indicator is more favorable to the evaluation system, it is calculated as a positive indicator, as shown in Equation (5).

$${\overline{v}}_{ij}=\frac{v_{ij}-v_{j\min }}{v_{j\max }-v_{j\min }}$$

(5)

The smaller the value of the indicator, the better it is for the development of the evaluation system, and it is calculated as a negative indicator, as shown in Equation (6).

$${\overline{v}}_{ij}=\frac{v_{j\max }-v_{ij}}{v_{j\max }-v_{j\min }}$$

(6)

where $i=1,2,\mathrm{\dots },n$, n is the number of evaluation programs, $v_{ij}$ is the measured value of the indicator $X_j$ in program i, $v_{j\min }$ and $v_{j\max }$ are the minimum and maximum value of the measured value of each program corresponding to $X_j$, respectively, and ${\overline{v}}_{\mathrm{i}j}$ is the standard value corresponding to the measured value of $X_j$.

The entropy of the indicator is then calculated according to Equations (7) and (8).

$$p_{ij}=\frac{\overline{v_{ij}}}{{\displaystyle \sum _{i=1}^n\overline{v_{ij}}}}$$

(7)

$$e_j=-\frac{1}{\ln n}{\displaystyle \sum _{i=1}^n{p}_{ij}\ln p_{ij}}$$

(8)

where $p_{ij}$ is the proportion of the whole accounted for by ${\overline{v}}_{ij}$ as an intermediate variable, and $e_j$ is the entropy value of the indicator $X_j$. The logarithm is used in Equation (8), and the normalized values will have zeros present and cannot be used directly, so a small translation of the normalized values is required.

If $e_j<e_m$, then the difference between the evaluation objects on the indicator $X_j$ is greater than that on the indicator $X_m$, so that $X_j$ is more useful for comparing the evaluation objects. Conversely, if $e_j>e_m$, then it means that indicator $X_j$ is less useful for the object of evaluation.

In summary, the entropy value is introduced into the G2 method, and let

$$r_j=a_j=\left\{\begin{array}{cc}\frac{e_m}{e_j}& \begin{array}{cc}& e_j<e_m\end{array}\\ 1& \begin{array}{cc}& e_j>e_m\end{array}\end{array}\right.$$

(9)

Then, the objective weight $w_2=(w_{21},w_{22},\mathrm{\dots },w_{2m})$ of each evaluation index can be calculated according to Equation (2).

Combining the subjective weight $w_1$ and objective weight $w_2$ calculated by the G2 method and the entropy-improved G2 method, the combined weights of the indicators $X_j$ are calculated by Equation (10).

$$w_j=\frac{{(w_{1_j}{w}_{2_j})}^{\frac{1}{2}}}{{\displaystyle \sum _{j=1}^m{(w_{1_j}{w}_{2_j})}^{\frac{1}{2}}}}$$

(10)

3.2. Gray Correlation–TOPSIS

The TOPSIS method [27] is a commonly used multi-objective decision-making method, the main feature of which is that the decision-maker defines the ideal solution and a kind of measured distance, and the degree of deviation of the solution from the ideal solution is measured according to the magnitude of the measured distance between each evaluated solution and the positive and negative ideal solutions of this decision problem [28], so the method is also often referred to as the ideal solution method. However, the traditional TOPSIS method only considers the closeness in distance between the solution and the positive and negative ideal solutions, and does not consider the similarity in shape [29]. Moreover, the TOPSIS method is based on raw data, which implies a certain objectivity, and if it is used with less data information, the TOPSIS method may not obtain the correct decision results. However, the gray correlation analysis method mines information from within the data and determines the superiority of the solution by analyzing the magnitude of the similarity of the geometric curves between each group of data and the reference data [30,31]; it does not require a large amount of data, which is exactly complementary to the TOPSIS method. Introducing the gray correlation into the traditional TOPSIS method, which is able to consider both the distance and the degree of shape similarity between the solution and the ideal solution, makes the durability evaluation more scientific.

The gray correlation was used to adapt the TOPSIS method, as detailed below:

Firstly, the measured values of each scheme to be evaluated and the cut-off values of the durability class form an incremental decision matrix $V={(v_{ij})}_{l\times n}$, and the standard matrix $B={(b_{ij})}_{l\times n}$ is obtained by normalizing the following Equation (11):

$$b_{ij}=\frac{v_{ij}}{({\displaystyle \sum _{i=1}^m{v}_{ij}^2{)}^{\frac{1}{2}}}}$$

(11)

where $i=1,2,\mathrm{\dots },l$, l is the sum of the number of evaluation schemes and the cut-off number of durability classes, $j=1,2,\mathrm{\dots },n$.

The weighted standard matrix U is obtained by multiplying the obtained standard matrix B with the derived integrated weight vector w, as shown in Equation (12).

$$U={(b_{ij}{w}_j)}_{l\times n}={(u_{ij})}_{l\times n}$$

(12)

Then, determine the positive ideal solution $U^{+}={(u_j^{+})}_{1\times n}$ and the negative ideal solution $U^{-}={(u_j^{-})}_{1\times n}$ of the weighted standard matrix U.

For the positive indicator, the positive ideal solution element $u_j^{+}$ and the negative ideal solution element $u_j^{-}$ are shown in Equation (13).

$$\begin{array}{c}{u}_j^{+}=\max \{u_{1j}^{+},u_{2j}^{+},\mathrm{\dots },u_{mj}^{+}\}\\ u_j^{-}=\min \{u_{1j}^{-},u_{2j}^{-},\mathrm{\dots },u_{mj}^{-}\}\end{array}$$

(13)

For the negative indicator, the positive ideal solution element $u_j^{+}$ and the negative ideal solution element $u_j^{-}$ are shown in Equation (14).

$$\begin{array}{c}{u}_j^{+}=\min \{u_{1j}^{+},u_{2j}^{+},\mathrm{\dots },u_{mj}^{+}\}\\ u_j^{-}=\max \{u_{1j}^{-},u_{2j}^{-},\mathrm{\dots },u_{mj}^{-}\}\end{array}$$

(14)

Calculate the Euclidean distance of scheme i for positive ideal solution $U^{+}$ and negative ideal solution $U^{-}$, respectively, as shown in Equation (15).

$$\begin{array}{c}{d}_i^{+}=({\displaystyle \sum _{j=1}^n{(u_{ij}-u_j^{+})}^2{)}^{\frac{1}{2}}}\\ d_i^{-}=({\displaystyle \sum _{j=1}^n{(u_{ij}-u_j^{-})}^2{)}^{\frac{1}{2}}}\end{array}$$

(15)

Calculate the gray correlation of scheme i regarding the jth indicator with positive and negative ideal solutions, as shown in Equation (16).

$$\begin{array}{c}{g}_{ij}^{+}=\frac{\underset{i}{\min }\underset{j}{\min }|u_j^{+}-u_{ij}|+\lambda \underset{i}{\max }\underset{j}{\max }|u_j^{+}-u_{ij}|}{|u_j^{+}-u_{ij}|+\lambda \underset{i}{\max }\underset{j}{\max }|u_j^{+}-u_{ij}|}\\ g_{ij}^{-}=\frac{\underset{i}{\min }\underset{j}{\min }|u_j^{-}-u_{ij}|+\lambda \underset{i}{\max }\underset{j}{\max }|u_j^{-}-u_{ij}|}{|u_j^{-}-u_{ij}|+\lambda \underset{i}{\max }\underset{j}{\max }|u_j^{-}-u_{ij}|}\end{array}$$

(16)

where $\lambda$ is the resolution factor, which is usually taken as 0.5.

Calculate the gray correlation of the ith solution with positive and negative ideal solutions, as shown in Equation (17).

$$\begin{array}{c}{g}_i^{+}=\frac{1}{n}{\displaystyle \sum _{j=1}^n{g}_{ij}^{+}}\\ g_i^{-}=\frac{1}{n}{\displaystyle \sum _{j=1}^n{g}_{ij}^{-}}\end{array}$$

(17)

In order to consider the degree of similarity in shape between scheme i and the positive ideal solution $U^{+}$ and the negative ideal solution $U^{-}$ under the action of Euclidean distance and gray correlation, respectively, $d_i^{+},d_i^{-},g_i^{+},g_i^{-}$ is dimensionless according to Equation (18).

$$\begin{array}{cc}{D}_i^{+}=\frac{d_i^{+}}{\underset{i}{\max d_i^{+}}},& D_i^{-}=\frac{d_i^{-}}{\underset{i}{\max d_i^{-}}}\\ G_i^{+}=\frac{g_i^{+}}{\underset{i}{\max g_i^{+}}},& G_i^{-}=\frac{g_i^{-}}{\underset{i}{\max g_i^{-}}}\end{array}$$

(18)

Since the magnitude of the values of $D_i^{-}$ and $G_i^{+}$ are both positively correlated with the positive ideal solution of the evaluation scheme, while $D_i^{+}$ and $G_i^{-}$ are negatively correlated with the positive ideal solution of the evaluation scheme, these four indicators are combined in Equation (19) as the composite indices $S_i^{+}$ and $S_i^{-}$.

$$\begin{array}{c}{S}_i^{+}=\alpha D_i^{-}+\beta G_i^{+}\\ S_i^{-}=\alpha D_i^{+}+\beta G_i^{-}\end{array}$$

(19)

where $\alpha ,\beta$ is the evaluator’s preference for location and shape, satisfying $\alpha ,\beta \in [0,1]$ and $\alpha +\beta =1$, the evaluator can set the value of $\alpha ,\beta$ according to his personal preference. The value of $S_i^{+}$ combines the proximity in distance and similarity in shape of solution i to the positive ideal solution, and the higher value indicates the better evaluation solution, and on the contrary, the higher value of $S_i^{-}$ indicates the worse evaluation solution.

For comparison purposes, the relative fit of scheme i is constructed by considering both the composite indices $S_i^{+}$ and $S_i^{-}$, as shown in Equation (20).

$$C_i^{+}=\frac{S_i^{+}}{S_i^{+}+S_i^{-}}$$

(20)

The larger the relative closeness indicating that the solution is closer to the positive ideal solution, the better the solution is, and vice versa. By comparing the relative closeness of the solution with the rank boundary, the corresponding evaluation rank of the solution can be obtained.

4. Example Applications

4.1. Project Overview

The Pandaoling tunnel is a key part of The Water Diversion Project from Datong River to Qinwangchuan District; it is the control project of the main canal and the longest pressureless hydraulic tunnel was completed in 1992. The tunnel has been in operation for 28 years since its opening in 1994, with a length of 15.723 km, a straight walled circular arch section, a net width of 4.2 m, a net height of 4.4m, a design flow of 29 m³/s, a design longitudinal slope of 1/1000, and a lining structure type of reinforced concrete.

Due to the poor geological conditions of the tunnel, there were problems such as lining concrete cracks, collapse of surrounding rock deformation and groundwater exposure during the construction period. During the operation of the Pandaoling Tunnel, several repairs and reinforcements were carried out on the sections affecting the durability of the tunnel lining structure, providing an effective guarantee for the safe operation of the hydraulic tunnel. However, many durability problems still exist in the lining structure of the Pandaoling Tunnel at this stage [32], the main reasons being: (1) The secondary lining concrete in most sections of the Pandaoling Tunnel was only C15 plain concrete at the early stage of construction, which makes it very easy to produce lining cracks, deformation, and other defect problems; (2) some sections of the Pandaoling Tunnel have serious groundwater exposure and high groundwater mineralization, and after long-term corrosion and washing by groundwater, the secondary lining concrete has insufficient ability to resist the deformation of the surrounding rock, resulting in the softening of the surrounding rock, which reduces the strength of the surrounding rock, which has a greater impact on the operation of the tunnel; (3) as the tunnel crosses a large fault zone, the geological conditions are poor and the burial depth of some sections is large, which means that the cracks in the secondary lining concrete have a tendency to increase slowly.

4.2. Weighting Calculation

In order to unify the experts’ scoring criteria, the least important indicator among the secondary indicators was determined in advance by the experts, i.e., the only reference for each expert was the same. After the experts’ discussion, the pH value was determined as the least important indicator $X_{\min }$ among the secondary indicators.

First, the subjective weights were calculated using the G2 assignment method.

Here, the values were assigned according to the interval method to reduce subjectivity, and the expert scoring data are shown in

Table 3. The score assigned by each expert.

Evaluation Indicators	$\mathbf{Importance} \mathbf{Interval} {\mathit{D}}_{\mathit{j}}$
	Expert 1	Expert 2	Expert 3
Crack width $X_1$	[1.7, 1.8]	[1.6, 1.8]	[1.7, 1.8]
Crack depth ratio $X_2$	[1.6, 1.7]	[1.6, 1.7]	[1.5, 1.7]
Leakage state $X_3$	[1.3, 1.4]	[1.2, 1.4]	[1.3, 1.4]
pH value $X_4$	1	1	1
Lining strength ratio $X_5$	[1.5, 1.6]	[1.6, 1.7]	[1.4, 1.6]
Lining thickness ratio $X_6$	[1.4, 1.6]	[1.5, 1.6]	[1.4, 1.5]
Reinforcement interface loss rate $X_7$	[1.1, 1.2]	[1.1, 1.2]	[1.2, 1.3]
Deformation speed $X_8$	[1.4, 1.6]	[1.5, 1.6]	[1.5, 1.7]
Deformation ratio $X_9$	[1.3, 1.4]	[1.4, 1.5]	[1.4, 1.5]
Spalling diameter $X_{10}$	[1.2, 1.4]	[1.3, 1.4]	[1.3, 1.4]
Spalling depth $X_{11}$	[1.2, 1.3]	[1.2, 1.3]	[1.3, 1.4]
Continuous length of the cavity $X_{12}$	[1.3, 1.5]	[1.4, 1.5]	[1.4, 1.5]
Area of the cavity $X_{13}$	[1.2, 1.3]	[1.3, 1.4]	[1.3, 1.4]

.

The indicator weights of each expert were calculated by Equation (4) and the average value was calculated to obtain the subjective weights $w_1$, as seen in

Table 4. Weight of each expert index.

Evaluation Indicators	Weights			$\mathbf{Average} \mathbf{Weight} {\mathit{w}}_1$
	Expert 1	Expert 2	Expert 3
$X_1$	0.0979	0.0912	0.0951	0.0947
$X_2$	0.0922	0.0901	0.0870	0.0898
$X_3$	0.0752	0.0690	0.0734	0.0725
$X_4$	0.0566	0.0556	0.0543	0.0555
$X_5$	0.0865	0.0901	0.0815	0.0861
$X_6$	0.0826	0.0845	0.0788	0.0820
$X_7$	0.0639	0.0623	0.0679	0.0647
$X_8$	0.0826	0.0845	0.0870	0.0847
$X_9$	0.0752	0.0790	0.0788	0.0777
$X_{10}$	0.0713	0.0734	0.0734	0.0727
$X_{11}$	0.0696	0.0679	0.0707	0.0694
$X_{12}$	0.0769	0.0790	0.0788	0.0782
$X_{13}$	0.0696	0.0734	0.0734	0.0721

, where Expert 1 and Expert 2 are conservative experts, so that ${\epsilon }_1=-0.2,{\epsilon }_2=-0.3$, and Expert 3 is a neutral expert, so that ${\epsilon }_3=0$.

Then, the G2 method with improved entropy value was used to calculate the objective weights of the evaluation indicators.

The durability evaluation indices of five tunnel sections (76 + 253 − 76 + 988, 76 + 988 − 77 + 833, 77 + 833 − 79 + 097, 79 + 097 − 79 + 435, 79 + 435 − 79 + 475) were measured, and the measured values are shown in Table 5.

Table 5. Measured values of durability evaluation index for each tunnel section.

Evaluation Indicators	Zone 1	Zone 2	Zone 3	Zone 4	Zone 5
$X_1$ (mm)	0.12	0.53	5.08	0.89	2.94
$X_2$	0.09	0.22	0.39	0.26	0.62
$X_3$ *	10	3	4	7	5
$X_4$	8.11	7.83	7.66	7.92	6.22
$X_5$	0.83	0.69	0.59	0.59	0.52
$X_6$	0.76	0.78	0.67	0.76	0.68
$X_7$ (%)	6	12	19	23	46
$X_8$ (mm/a)	2.62	2.82	3.22	3.01	2.65
$X_9$	0.43	0.32	0.31	0.29	0.18
$X_{10}$ (mm)	76.23	88.96	89.55	79.86	90.26
$X_{11}$ (mm)	6.33	8.68	8.19	9.02	8.27
$X_{12}$ (m)	0.85	1.27	0.87	1.68	1.41
$X_{13}$ (m2)	0.66	0.92	2.89	3.81	3.29

* Leakage state $X_3$ is a qualitative indicator, its grade classification for the use of the meaning of: 8–10 indicates that the existence of lining surface infiltration, no impact on the operation of water transmission; 6–8 indicates drips in the liner arch, small surges in the sidewalls, soaking seepage in the baseboards but no ponding, small amounts of hanging ice in the arch and sidewalls due to seepage, and ice buildup at the foot of the sidewalls, which may soon affect water conveyance operations; 4–6 indicates that there are surges in the arch of the lining, jet streams in the sidewalls, sand outflows, large hanging ice and swelling cracks in the arch lining due to water seepage, affecting the operation of water conveyance; 0–4 indicates that there is jet water flow in the arch, and there is surge water in the sidewalls that seriously affects the safety of water conveyance, accompanied by serious sand outflow and lining hanging ice, which seriously affects the operation of water conveyance. Next, its measured data were normalized by Equations (5) and (6), as shown in

Table 6. Measured standard values of durability evaluation index for each tunnel section.

Evaluation Indicators	Indicator Direction	Zone 1	Zone 2	Zone 3	Zone 4	Zone 5
$X_1$	− 1	1.0000	0.9173	0.0000	0.8448	0.4315
$X_2$	−	1.0000	0.7547	0.4340	0.6792	0.0000
$X_3$	+ 2	0.0000	1.0000	0.8571	0.4286	0.7143
$X_4$	+	1.0000	0.8519	0.7619	0.8995	0.0000
$X_5$	+	1.0000	0.5484	0.2258	0.2258	0.0000
$X_6$	+	0.8182	1.0000	0.0000	0.8182	0.0909
$X_7$	−	1.0000	0.8500	0.6750	0.5750	0.0000
$X_8$	−	1.0000	0.6667	0.0000	0.3500	0.9500
$X_9$	−	0.0000	0.4400	0.4800	0.5600	1.0000
$X_{10}$	−	1.0000	0.0927	0.0506	0.7413	0.0000
$X_{11}$	−	1.0000	0.1264	0.3086	0.0000	0.2788
$X_{12}$	−	1.0000	0.4940	0.9759	0.0000	0.3253
$X_{13}$	−	1.0000	0.9175	0.2921	0.0000	0.1651

¹ − indicates negative indicator. ² + indicates positive indicator.

.

Table 5. Measured values of durability evaluation index for each tunnel section.

Evaluation Indicators	Zone 1	Zone 2	Zone 3	Zone 4	Zone 5
$X_1$ (mm)	0.12	0.53	5.08	0.89	2.94
$X_2$	0.09	0.22	0.39	0.26	0.62
$X_3$ *	10	3	4	7	5
$X_4$	8.11	7.83	7.66	7.92	6.22
$X_5$	0.83	0.69	0.59	0.59	0.52
$X_6$	0.76	0.78	0.67	0.76	0.68
$X_7$ (%)	6	12	19	23	46
$X_8$ (mm/a)	2.62	2.82	3.22	3.01	2.65
$X_9$	0.43	0.32	0.31	0.29	0.18
$X_{10}$ (mm)	76.23	88.96	89.55	79.86	90.26
$X_{11}$ (mm)	6.33	8.68	8.19	9.02	8.27
$X_{12}$ (m)	0.85	1.27	0.87	1.68	1.41
$X_{13}$ (m2)	0.66	0.92	2.89	3.81	3.29

* Leakage state $X_3$ is a qualitative indicator, its grade classification for the use of the meaning of: 8–10 indicates that the existence of lining surface infiltration, no impact on the operation of water transmission; 6–8 indicates drips in the liner arch, small surges in the sidewalls, soaking seepage in the baseboards but no ponding, small amounts of hanging ice in the arch and sidewalls due to seepage, and ice buildup at the foot of the sidewalls, which may soon affect water conveyance operations; 4–6 indicates that there are surges in the arch of the lining, jet streams in the sidewalls, sand outflows, large hanging ice and swelling cracks in the arch lining due to water seepage, affecting the operation of water conveyance; 0–4 indicates that there is jet water flow in the arch, and there is surge water in the sidewalls that seriously affects the safety of water conveyance, accompanied by serious sand outflow and lining hanging ice, which seriously affects the operation of water conveyance. Next, its measured data were normalized by Equations (5) and (6), as shown in Table 6.

The entropy value of each indicator was calculated by Equations (7) and (8) and shifted to the right by a small amount of 0.01 with reference to the size of the standard value. The importance ratio of each indicator was calculated by Equation (9), and finally the objective weight $w_2$ was calculated by Equation (2), and the comprehensive weights w were obtained by combining the subjective and objective weights through Equation (10) and are shown in

Table 7. Calculation of comprehensive weights.

Evaluation Indicators	Entropy Value	Importance Ratio	$\mathbf{Objective} \mathbf{Weights} {\mathit{w}}_2$	Comprehensive Weights w	Serial Number
$X_1$	0.8405	1.0254	0.0718	0.0828	3
$X_2$	0.8414	1.0245	0.0717	0.0805	5
$X_3$	0.7601	1.0257	0.0718	0.0724	11
$X_4$	0.8636	1.0000	0.0700	0.0626	13
$X_5$	0.7476	1.1427	0.0800	0.0833	2
$X_6$	0.7539	1.1304	0.0792	0.0809	4
$X_7$	0.8529	1.0117	0.0709	0.0680	12
$X_8$	0.8249	1.0433	0.0731	0.0790	6
$X_9$	0.8526	1.0379	0.0727	0.0754	10
$X_{10}$	0.5962	1.3954	0.0977	0.0846	1
$X_{11}$	0.6961	1.2201	0.0854	0.0773	7
$X_{12}$	0.8081	1.0634	0.0745	0.0767	9
$X_{13}$	0.7358	1.1584	0.0811	0.0768	8

.

According to Table 7, the final comprehensive weights w of the indicators of the hydraulic tunnel lining durability evaluation system were obtained, of which the first three are spalling diameter $X_{10}$ (0.0847), lining strength ratio $X_5$ (0.0834), and crack width $X_1$ (0.0829). In this paper, it is considered that the spalling diameter $X_{10}$ is the most sensitive parameter to the tunnel lining durability.

For different tunnel types, the evaluation systems established by scholars are different, so it is difficult to compare the weights of these indicators one by one. The evaluation system established in literature [33] contains 13 evaluation indices, of which the top three weighted indices are crack length and width, crack depth and spalling depth, and the reason for this difference is that the ANP adopted in the literature [33] is a subjective method, compared with the subjective and objective weighting method in this paper, which is more comprehensive and objective.

4.3. Grade Evaluation

In reference to “Criteria for Deterioration Assessment of Railway Bridge and Tunnel Buildings” (Q/CR 405.2-2019) [34] and the related literature [35,36,37,38,39], the determination criteria for different grades of hydraulic tunnel lining durability index were established, as shown in

Table 8. Classification of durability evaluation index level.

Evaluation Indicators	A	B	C	D
$X_1$	(0, 0.2]	(0.2, 0.3]	(0.3, 0.4]	>0.4
$X_2$	(0, 0.3]	(0.3, 0.5]	(0.5, 0.7]	>0.7
$X_3$	(8, 10]	(6, 8]	(4, 6]	(0, 4]
$X_4$	(6, 14]	(5, 6]	(4, 5]	(0, 4]
$X_5$	>0.8	(0.5, 0.8]	(0.3, 0.5]	(0, 0.3]
$X_6$	>0.9	(0.75, 0.9]	(0.6, 0.75]	(0, 0.6]
$X_7$	(0, 3]	(3, 10]	(10, 25]	(25, 100]
$X_8$	(0, 1]	(1, 3]	(3, 10]	>10
$X_9$	(0, 0.25]	(0.25, 0.5]	(0.5, 0.75)	>0.75
$X_{10}$	(0, 50]	(50, 75]	(75, 150]	>150
$X_{11}$	(0, 6]	(6, 12]	(12, 25]	>25
$X_{12}$	(0, 3]	(3, 5]	(5, 10]	>10
$X_{13}$	(0, 1]	(1, 3]	(3, 5]	>5

.

The combination of the measured tunnel scenarios and the cut-off values of the durability grading criteria constitute the incremental decision matrix V, as shown in

Table 9. Incremental decision matrix.

	${\mathit{X}}_1$	${\mathit{X}}_2$	${\mathit{X}}_3$	${\mathit{X}}_4$	${\mathit{X}}_5$	${\mathit{X}}_6$	${\mathit{X}}_7$	${\mathit{X}}_8$	${\mathit{X}}_9$	${\mathit{X}}_{10}$	${\mathit{X}}_{11}$	${\mathit{X}}_{12}$	${\mathit{X}}_{13}$
AB *	0.2	0.3	8	6	0.8	0.9	3	1	0.25	50	6	3	1
BC	0.3	0.5	6	5	0.5	0.75	10	3	0.5	75	12	5	3
CD	0.4	0.7	4	4	0.3	0.6	25	10	0.75	150	25	10	5
1	0.12	0.09	0.02	8.11	0.83	0.76	0.06	2.62	0.43	76.23	6.33	0.85	0.66
2	0.53	0.22	0.31	7.83	0.69	0.78	0.12	2.82	0.32	88.96	8.68	1.27	0.92
3	5.08	0.39	0.48	7.66	0.59	0.67	0.19	3.22	0.31	89.55	8.19	0.87	2.89
4	0.89	0.26	0.29	7.92	0.59	0.76	0.23	3.01	0.29	79.86	9.02	1.68	3.81
5	2.94	0.62	0.39	6.22	0.52	0.68	0.46	2.65	0.18	90.26	8.27	1.41	3.29

* AB refers to the value of the demarcation scheme between grade A and grade B, BC refers to the value of the demarcation scheme between grade B and grade C, and CD refers to the value of the demarcation scheme between grade C and grade D. AB, BC, CD have the same meaning in Table 10, Table 11, Table 12, Table 13, Table 14.

.

The standard matrix B was obtained by normalizing according to Equation (11) and is shown in

Table 10. Standard matrix.

	${\mathit{X}}_1$	${\mathit{X}}_2$	${\mathit{X}}_3$	${\mathit{X}}_4$	${\mathit{X}}_5$	${\mathit{X}}_6$	${\mathit{X}}_7$	${\mathit{X}}_8$	${\mathit{X}}_9$	${\mathit{X}}_{10}$	${\mathit{X}}_{11}$	${\mathit{X}}_{12}$	${\mathit{X}}_{13}$
AB	0.0334	0.2457	0.4507	0.3144	0.4535	0.4287	0.0479	0.0813	0.2132	0.1933	0.1779	0.2518	0.1195
BC	0.0501	0.4095	0.3381	0.2620	0.2835	0.3572	0.1597	0.2439	0.4264	0.2900	0.3558	0.4197	0.3586
CD	0.0668	0.5733	0.2254	0.2096	0.1701	0.2858	0.3993	0.8131	0.6396	0.5800	0.7413	0.8395	0.5977
1	0.0200	0.0737	0.5634	0.4250	0.4705	0.3620	0.0958	0.2130	0.3667	0.2947	0.1877	0.0714	0.0789
2	0.0885	0.1802	0.1690	0.4103	0.3912	0.3715	0.1917	0.2293	0.2729	0.3440	0.2574	0.1066	0.1100
3	0.8487	0.3194	0.2254	0.4014	0.3345	0.3191	0.3035	0.2618	0.2644	0.3462	0.2429	0.0730	0.3455
4	0.1487	0.2130	0.3944	0.4150	0.3345	0.3620	0.3674	0.2447	0.2473	0.3088	0.2675	0.1410	0.4555
5	0.4912	0.5078	0.2817	0.3259	0.2948	0.3239	0.7347	0.2155	0.1535	0.3490	0.2452	0.1184	0.3933

.

According to the standard matrix B and the comprehensive weight w, the weighted standard matrix U was calculated according to Equation (12), and the positive and negative ideal solutions of each indicator were found according to Equations (13) and (14), as shown in

Table 11. Weighted standard matrix and positive and negative ideal solutions.

	${\mathit{X}}_1$	${\mathit{X}}_2$	${\mathit{X}}_3$	${\mathit{X}}_4$	${\mathit{X}}_5$	${\mathit{X}}_6$	${\mathit{X}}_7$	${\mathit{X}}_8$	${\mathit{X}}_9$	${\mathit{X}}_{10}$	${\mathit{X}}_{11}$	${\mathit{X}}_{12}$	${\mathit{X}}_{13}$
AB	0.0027	0.0196	0.0324	0.0195	0.0375	0.0344	0.0032	0.0064	0.0175	0.0162	0.0136	0.0191	0.0091
BC	0.0041	0.0327	0.0243	0.0163	0.0234	0.0287	0.0108	0.0191	0.0351	0.0243	0.0273	0.0319	0.0273
CD	0.0055	0.0458	0.0162	0.0130	0.0141	0.0229	0.0269	0.0637	0.0526	0.0487	0.0569	0.0638	0.0455
1	0.0016	0.0059	0.0405	0.0264	0.0389	0.0291	0.0065	0.0167	0.0302	0.0247	0.0144	0.0054	0.0060
2	0.0073	0.0144	0.0122	0.0255	0.0323	0.0298	0.0129	0.0180	0.0225	0.0289	0.0197	0.0081	0.0084
3	0.0697	0.0255	0.0162	0.0249	0.0276	0.0256	0.0205	0.0205	0.0218	0.0291	0.0186	0.0056	0.0263
4	0.0122	0.0170	0.0284	0.0258	0.0276	0.0291	0.0248	0.0192	0.0204	0.0259	0.0205	0.0107	0.0347
5	0.0404	0.0406	0.0203	0.0202	0.0244	0.0260	0.0496	0.0169	0.0126	0.0293	0.0188	0.0090	0.0300
$U^{+}$	0.0016	0.0059	0.0405	0.0264	0.0389	0.0344	0.0032	0.0064	0.0126	0.0162	0.0136	0.0054	0.0060
$U^{-}$	0.0697	0.0458	0.0122	0.0130	0.0141	0.0229	0.0496	0.0637	0.0526	0.0487	0.0569	0.0638	0.0455

.

According to Table 11, the gray correlations of different indicators corresponding to positive and negative ideal solutions for each scenario were calculated according to Equation (16), as shown in

Table 12. Positive gray correlation of program indicators.

	${\mathit{X}}_1$	${\mathit{X}}_2$	${\mathit{X}}_3$	${\mathit{X}}_4$	${\mathit{X}}_5$	${\mathit{X}}_6$	${\mathit{X}}_7$	${\mathit{X}}_8$	${\mathit{X}}_9$	${\mathit{X}}_{10}$	${\mathit{X}}_{11}$	${\mathit{X}}_{12}$	${\mathit{X}}_{13}$
AB	0.9688	0.7123	0.8078	0.8321	0.9603	1.0000	1.0000	1.0000	0.8739	1.0000	1.0000	0.7127	0.9166
BC	0.9323	0.5591	0.6775	0.7708	0.6876	0.8558	0.8186	0.7276	0.6025	0.8075	0.7139	0.5624	0.6150
CD	0.8986	0.4601	0.5834	0.7179	0.5782	0.7480	0.5895	0.3725	0.4597	0.5119	0.4407	0.3682	0.4627
1	1.0000	1.0000	1.0000	1.0000	1.0000	0.8641	0.9133	0.7673	0.6599	0.7999	0.9784	1.0000	1.0000
2	0.8581	0.8000	0.5455	0.9739	0.8384	0.8813	0.7783	0.7459	0.7760	0.7291	0.8482	0.9270	0.9350
3	0.3333	0.6341	0.5834	0.9587	0.7517	0.7947	0.6638	0.7065	0.7886	0.7261	0.8724	0.9963	0.6263
4	0.7631	0.7536	0.7369	0.9822	0.7517	0.8641	0.6124	0.7266	0.8151	0.7784	0.8321	0.8653	0.5426
5	0.4679	0.4952	0.6270	0.8470	0.7009	0.8019	0.4236	0.7640	1.0000	0.7226	0.8683	0.9050	0.5869

and

Table 13. Negative gray correlation of program indicators.

	${\mathit{X}}_1$	${\mathit{X}}_2$	${\mathit{X}}_3$	${\mathit{X}}_4$	${\mathit{X}}_5$	${\mathit{X}}_6$	${\mathit{X}}_7$	${\mathit{X}}_8$	${\mathit{X}}_9$	${\mathit{X}}_{10}$	${\mathit{X}}_{11}$	${\mathit{X}}_{12}$	${\mathit{X}}_{13}$
AB	0.3370	0.5652	0.6270	0.8395	0.5923	0.7480	0.4236	0.3725	0.4924	0.5119	0.4407	0.4324	0.4830
BC	0.3416	0.7222	0.7369	0.9127	0.7841	0.8558	0.4674	0.4329	0.6599	0.5830	0.5352	0.5161	0.6514
CD	0.3464	1.0000	0.8937	1.0000	1.0000	1.0000	0.6007	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
1	0.3333	0.4601	0.5455	0.7179	0.5782	0.8477	0.4413	0.4199	0.6025	0.5870	0.4450	0.3682	0.4627
2	0.3528	0.5200	1.0000	0.7320	0.6507	0.8319	0.4817	0.4266	0.5301	0.6321	0.4784	0.3792	0.4781
3	1.0000	0.6265	0.8937	0.7408	0.7147	0.9271	0.5392	0.4407	0.5243	0.6343	0.4710	0.3687	0.6391
4	0.3718	0.5416	0.6775	0.7274	0.7147	0.8477	0.5787	0.4332	0.5132	0.5992	0.4837	0.3906	0.7585
5	0.5368	0.8666	0.8078	0.8249	0.7675	0.9176	1.0000	0.4209	0.4597	0.6371	0.4722	0.3830	0.6861

.

Finally, the evaluation key parameters were calculated according to Equations (15) and (17)–(20), and the Euclidean distances $d_i^{+}$ and $d_i^{-}$, gray correlations $g_i^{+}$ and $g_i^{-}$, dimensionless values $D_i^{+}$, $D_i^{-}$, $G_i^{+}$, and $G_i^{-}$, composite indices $S_i^{+}$ and $S_i^{-}$, and relative fit $C_i$ are listed in

Table 14. Evaluation key parameters.

	${\mathit{d}}_{\mathit{i}}^{+}$	${\mathit{d}}_{\mathit{i}}^{-}$	${\mathit{D}}_{\mathit{i}}^{+}$	${\mathit{D}}_{\mathit{i}}^{-}$	${\mathit{g}}_{\mathit{i}}^{+}$	${\mathit{g}}_{\mathit{i}}^{-}$	${\mathit{G}}_{\mathit{i}}^{+}$	${\mathit{G}}_{\mathit{i}}^{-}$	${\mathit{S}}_{\mathit{i}}^{+}$ *	${\mathit{S}}_{\mathit{i}}^{-}$	${\mathit{C}}_{\mathit{i}}$
AB	0.0230	0.1386	0.1789	0.9854	0.9065	0.5281	0.9834	0.5798	0.9844	0.3793	0.7218
BC	0.0591	0.1066	0.4606	0.7581	0.7177	0.6307	0.7787	0.6925	0.7684	0.5766	0.5713
CD	0.1283	0.0682	1.0000	0.4850	0.5532	0.9108	0.6001	1.0000	0.5426	1.0000	0.3517
1	0.0229	0.1407	0.1788	1.0000	0.9218	0.5238	1.0000	0.5751	1.0000	0.3769	0.7262
2	0.0388	0.1266	0.3026	0.8999	0.8182	0.5764	0.8876	0.6328	0.8938	0.4677	0.6565
3	0.0837	0.1002	0.6519	0.7123	0.7258	0.6554	0.7874	0.7196	0.7499	0.6857	0.5223
4	0.0471	0.1149	0.3670	0.8168	0.7711	0.5875	0.8365	0.6450	0.8266	0.5060	0.6203
5	0.0805	0.0999	0.6269	0.7105	0.7085	0.6754	0.7686	0.7415	0.7395	0.6842	0.5194

* Since there is no preference for both location and shape in this paper, $\alpha ,\beta$ in Equation (19) was taken as 0.5 when calculating the composite indices $S_i^{+}$ and $S_i^{-}$.

below.

The relative fit $C_i$ in Table 14 not only allows for a visual comparison of the durability of the lining of each tunnel section, but also allows for the determination of the durability class of each tunnel lining by comparing it to the AB, BC, and CD boundary scenarios. The durability class of section 1 is A, that of sections 2 and 4 is B, and that of sections 3 and 5 is C.

RAGA-PP (Real Coded Accelerating Genetic Algorithm-Projection Pursuit model) is a multidimensional optimization model that organically combines the accelerating genetic algorithm and PP model [26], and its basic idea is to project high-dimensional data to the low-dimensional subspace by setting constraints based on the PP model. The basic idea is to project the high-dimensional data into the low-dimensional subspace based on the PP model by setting the constraints, optimizing the projection objective function, and optimizing the multi-dimensional projection direction by using RAGA to search for the projection direction vector that can reflect the characteristics of the high-dimensional data structure, so as to realize the comprehensive evaluation. This method can effectively avoid the subjective bias of the human-assigned values and the overlapping of the information between the index variables and does not need to make any assumption on the distribution pattern of the data. The literature [26] used this method to evaluate the durability of these five tunnel sections and obtained exactly the same results, which verified the simplicity and accuracy of the method in this paper.

5. Conclusions and Outlook

To ensure the high quality and efficient operation of hydraulic tunnels, a regular durability evaluation of tunnel linings is necessary. In this paper, the entropy–G2 method and the gray correlation–TOPSIS method were proposed to evaluate the durability of hydraulic tunnel lining, and the main conclusions are as follows:

(1)

According to the damage characteristics and causes, the hydraulic tunnel lining defects could be classified as lining cracks, water leakage, lining material deterioration, lining deformation, lining spalling, and cavity behind the lining, and the operable durability indices were selected to establish a scientific hydraulic tunnel lining durability evaluation index system.

(2)

The G2 method was modified with the entropy value method to form a unified form of subjective and objective assignment method, which ensures the simplicity of calculation while making the weights relevant to reality. Through calculation, the spalling diameter is the most sensitive parameter in this paper’s hydraulic tunnel lining durability evaluation system, with a weight of 0.0847. The TOPSIS method was modified with the gray correlation degree so that the evaluation scheme can consider both the distance and shape from the ideal solution, while solving the problem that the TOPSIS method is prone to errors when the amount of data is small.

(3)

Based on the formed hydraulic tunnel lining durability evaluation index system, the lining durability grades of A, B, C, B, and C for the five sections of the Pandoling tunnel were obtained using the method of this paper, and the results match with those of the RAGA-PP method, verifying its accuracy and scientificity.

This paper proposes a new method for evaluating the durability of hydraulic tunnel linings, but due to the limitations of data collection and the author’s knowledge, there are still some problems that need further research and improvement:

(1)

The materials and structural forms of hydraulic tunnel lining are diverse, and the existing defects are not the same, so a suitable durability evaluation index system needs to be established according to the actual situation in the actual evaluation process.

(2)

There is mutual influence among the defects of hydraulic tunnels, and the durability evaluation indices may be more or less related to each other, which is ignored in the evaluation index system and weight calculation in this paper. Therefore, considering the relationship between indicators is the direction of future research on durability evaluation of hydraulic tunnels.