Tunnel collapse risk assessment based on improved quantitative theory III and EW-AHP coupling weight

Abstract

It is a multi-criteria decision issue to conduct a risk assessment of the tunnel. In this paper, a tunnel collapse risk assessment model based on the improved theory of quantification III and the fuzzy comprehensive evaluation method is proposed. According to the geological conditions and the construction disturbance classification method, the evaluation factors are selected, and the tunnel collapse risk level is divided into 5 levels according to the principle of maximum membership degree. The three groups of scores with the largest correlation ratio are calculated by the theory of quantification III to form the X, Y, and Z axes of the spatial coordinate system, The spatial distance of each evaluation factor is optimized by the Kendall correlation coefficient combined with the empirical formula, so that it can be used to judge the probability of the occurrence of the evaluation factor; taking the coupling of the objective entropy weight method (EW) and the subjective analytic hierarchy process (AHP) as the weight. Finally, the fuzzy comprehensive evaluation method is used to determine the possibility classification of tunnel collapse. Taking the Ka-Shuang water diversion tunnel as a case study, the comparison between the evaluation results of 10 tunnel samples and the status quo of the actual engineering area verifies the reliability of the method.

Introduction

Due to the intricacy, concealment, and uncertainty of tunnel construction, safety accidents frequently occur during the procedure. Through statistical analysis of geological hazard events recorded in the tunnel system between 2002 and 2018, it was concluded that collapse was the main geological hazard during tunnel construction¹. A tunnel collapse will not only increase construction difficulty, cost and the possibility of secondary disaster, but also endanger the safety of construction workers. Therefore, the assessment of tunnel collapse risk is a necessary measure to ensure the safe construction of tunnels.

Because a number of reasons can lead to tunnel collapse, the risk assessment of tunnels should be regarded as a multi-criteria issue². Kim et al.³ deduced 14 influencing factors that lead to tunnel collapse during the process of construction from five aspects: geotechnical engineering characteristics, tunnel geometric conditions, groundwater conditions, excavation conditions, as well as support and reinforcement conditions. When establishing a comprehensive risk assessment system for tunnel collapse, large deformation of surrounding rock and mud scouring, Li et al.⁴ selected 12 influencing factors as comprehensive evaluation indicators from three aspects: engineering geological conditions, hydrological conditions, and construction methods. After analyzing typical tunnel collapse cases, Ou et al.⁵ selected 11 influencing factors as risk assessment system indicators from five aspects: engineering geological conditions, natural environment, design and construction, construction organization and management, and advanced geological forecasting. In this paper, considering the influence of geological conditions and construction disturbance, 11 influencing factors are selected as the evaluation factors for the collapse risk of the northern water diversion tunnel with reference to the method of Zhai et al.⁹.

For multi-criteria problems, the determination of weight in traditional methods is very important. Chu⁶ and Dai⁷ used the AHP method to reasonably distribute the weights of the factors affecting the safety of tunnel construction, and used the membership function and the exponential weight to gradually calculate the corresponding risk level. The weight determination process of this method relies on expert experience, and the membership function classification is subjective. Gao et al.⁸ established a comprehensive risk assessment model for tunnel collapse based on entropy weight and grey correlation degree. Zhai et al.⁹ used the entropy weight method and the analytic hierarchy process combined with the undetermined measure theory to determine the multi-index comprehensive evaluation vector, and evaluated the tunnel collapse risk according to the principle of maximum membership degree. Both Gao and Zhai combine objective and subjective methods to determine weights, which are further improved compared to the former's evaluation reliability. With the development of theories about artificial intelligence, the Artificial Neural Networks (ANNs) are used to predict the stability conditions of roadways¹⁰. He et al.¹¹ combined the Interpreted Structure Modeling (ISM) and the Fuzzy Bayesian Networks (FBN), where the Fuzzy Bayesian Networks (FBN) obtain the prior probability and conditional probability of the node by aggregating the opinions of experts, using the Similarity Aggregation Method (SAM), respectively, to determine the hierarchical relationship and the interaction strength of each risk factor for risk analysis. Mahdevari et al.¹² found that the Particle Swarm Optimization (PSO) algorithm can significantly improve the performance of the Adaptive Neuro-Fuzzy Inference System (ANFIS), so the PSO-ANFIS model was proposed to predict unstable areas of underground roads. Zhou et al.¹³ optimized the Support Vector Machine (SVM) of the machine learning model through the Whale Optimization Algorithm (WOA), and established a WOA-SVM model to classify the extrusion behavior of the tunnel surrounding rock. In addition to the evaluation methods in the above section, Cloud model^14,15, interval risk assessment¹⁶, event tree analysis¹⁷, fault tree analysis¹⁸, BP neural network¹⁹ are also used in tunnel and underground engineering risk assessment. To sum up, the current risk assessment of tunnels is usually a combination of subjective and objective methods. Although some studies have shown that grading standards are also ambiguous, the ambiguity of evaluation factors is the main factor²⁰.

Therefore, this paper selects a branch of multivariate analysis "The Theory of Quantification III" as the basic theory for determining the weight of evaluation factors. In contrast to the qualitative or quantitative benchmark variables of Theory of Quantification I and II, the Theory of quantification III is a method that combines multiple qualitative and quantitative data to establish a comprehensive evaluation model^21,22,23, with the advantages of applicability and objectivity, that is currently rarely used in the natural sciences. After calculating the weights of the evaluation factors, combined with the "normal distribution" membership function of the Fuzzy Comprehensive Evaluation Method, the tunnel collapse risk is classified. Section “Introduction” introduces the process of constructing the evaluation model by combining the Improved Theory of Quantification III with the Fuzzy Comprehensive Evaluation Method; Section “Build the model” applies the model to engineering examples and analyzes the evaluation results; Section “Example verification” discusses the rationale for model construction and its mathematical implications.

Build the model

The improved theory of quantification III

General calculation method

The Theory of Quantification III is based on a reflection matrix constructed by dividing qualitative or quantitative data into several disjoint intervals, and assigning an appropriate value b_j (j = 1,2,…,m) to each category, which is called the category score, so that categories with similar response situations have similar scores; At the same time, a corresponding value y_i (1,2,…,n) is also assigned to each sample, which is called the sample score, so that samples with similar reaction conditions have similar scores. In this way, the score b_j (or y_i) has inherent meaning as a quantitative representation of categories (or samples), so it can comprehensively express the relationship between categories (or samples), and analyze the dominant factors in the variables.

In the Theory of Quantitative, qualitative variables are called items, and the different intervals that each item is divided into are called categories. Suppose there are s variables in total, of which there are m classified items, the j-th item has r_j categories, and a total of $r = \sum\limits_{{}}^{{}} {r_{j} \left( {1 \le j \le m} \right)}$ categories. From this, n sample data can be constructed into a reflection matrix X with n rows and r + s columns:

$$ X = \left( {\begin{array}{*{20}c} {\delta _{{1\left( {1,1} \right)}} } & \cdots & {\delta _{{1\left( {1,r_{1} } \right)}} } & \cdots & {\delta _{{1\left( {m,1} \right)}} } & \cdots & {\delta _{{1\left( {m,r_{m} } \right)}} } & {u_{{11}} } & \cdots & {u_{{1s}} } \\ {\delta _{{2\left( {1,1} \right)}} } & {} & {\delta _{{2\left( {1,r_{1} } \right)}} } & {} & {\delta _{{2\left( {m,1} \right)}} } & {} & {\delta _{{2\left( {m,r_{m} } \right)}} } & {u_{{21}} } & {} & {u_{{2s}} } \\ \vdots & \cdots & \vdots & \cdots & \vdots & \cdots & \vdots & \vdots & \cdots & \vdots \\ {\delta _{{n\left( {1,1} \right)}} } & {} & {\delta _{{n\left( {1,r_{1} } \right)}} } & {} & {\delta _{{n\left( {m,1} \right)}} } & {} & {\delta _{{n\left( {m,r_{m} } \right)}} } & {u_{{n1}} } & {} & {u_{{ns}} } \\ \end{array} } \right) $$

(1)

In the Eq. (1):

$$ \delta_{i} \left( {j,r_{j} } \right) = \left\{ \begin{gathered} 1\left( {{\text{When\;the\;qualitative\;data\;of\;item\;}}j\;{\text{in\;the\; }}i{\text{th\;sample\;is\;}}r_{j}\;{\text{\;category}}} \right) \hfill \\ 0\left({{\text{When\;the\;qualitative\;data\;of\;item\;}}j\;{\text{in\;the\;}}i{\text{th\;sample\;is\;not\;the\;}}r_{j}\;{\text{\;category}}} \right) \hfill \\ \end{gathered} \right. $$

(2)

In the Eq. (2): $\delta_{i} \left( {j,r_{j} } \right)$ represents the response on the r_j category of the i-th sample j item, and $u\left( {i,k} \right)$ represents the k-th quantitative variable’s response value in the i-th sample.

In the analysis method of Theory of Quantification III, the total r + s dimension category is assigned a score in the form of:

$$ b = \left\{ {b_{{11}} , \ldots b_{{1r}} , \ldots ,b_{{m1,}} , \ldots ,b_{{mr_{m} }} , \ldots ,a_{1} , \ldots ,a_{s} } \right\}^{T} $$

(3)

Average score for the i-th sample:

$$ y_{i} = \frac{1}{m + s}\left( {\sum\limits_{j = 1}^{m} {\sum\limits_{k = 1}^{{r_{j} }} {b_{jk} \delta \left( {j,k} \right) + \sum\limits_{k = 1}^{s} {a_{i} u_{ik} } } } } \right) $$

(4)

Therefore, the main problem of the Theory of Quantification III is transformed into solving vector b_j (relationship between categories) and vector y_i (relationship between samples). The specific solution process is as follows:

• (1) Record the sample score as $Y = \left\{ {y_{1} ,y_{2} , \cdots ,y_{n} } \right\}^{T}$, the sum of each sample’s responses on the j item k category is $g_{jk} = \sum\limits_{i = 1}^{n} {\delta_{i} \left( {j,k} \right)}$, sample score $Y = \left\{ {y_{1} ,y_{2} , \cdots ,y_{n} } \right\}^{T} = \frac{1}{m + s}Xb$, then the overall mean of the n sample scores is:

  $$ \overline{y} = \frac{1}{{n\left( {m + s} \right)}}g^{T} b $$

  (5)

• (2) Considering each sample as a group, the between-group variance is:

  $$ \sigma_{b}^{2} = \frac{1}{n}\sum\limits_{i = 1}^{n} {\left( {y_{i} - \overline{y} } \right)^{2} } = \frac{1}{{n(m + s)^{2} }}b^{T} Hb $$

  (6)

  In the Eq. (6):

  $$ H = X^{T} X - \frac{1}{n}gg^{T} $$

  From Eq. (6), we can get:

  $$ n\left( {m + s} \right)^{2} \sigma_{b}^{2} = b^{T} Hb $$

• (3) The total variance of the sample is:

  $$ \sigma^{2} = \frac{1}{{n\left( {m + s} \right)}}\left( {\sum\limits_{j = 1}^{m} {\sum\limits_{k = 1}^{{r_{j} }} {b_{jk}^{2} g_{jk} + n\sum\limits_{i = 1}^{s} {a_{i}^{2} } } } } \right) - \overline{y}^{2} = \frac{1}{{n\left( {m + s} \right)}}b^{T} Lb $$

  (7)

  Among them, G is a diagonal matrix of order r + s:

  $$ L = G - \frac{1}{{n\left( {m + s} \right)}}gg^{T} $$

• (4) The correlation ratio between sample group variance and total variance is:

  $$ \eta^{2} = \frac{{\sigma_{b}^{2} }}{{\sigma^{2} }} = \frac{{b^{T} Hb}}{{\left( {m + s} \right)b^{T} Lb}} $$

  (8)

To maximize the correlation ratio and satisfy the constraints $b^{T} Lb = 1$, $g^{T} b = 0$, the expression for solving the vector b is:

$$ Hb = \lambda \left( {m + s} \right)Lb $$

(9)

where λ represents the eigenvalue of the equation.

Both the category score b and the sample score yi obtained from this are one-dimensional, and its geometric meaning refers to that the feature vector b is regarded as a factor axis, and the sample score vector yi is regarded as the projection on this axis. The corresponding largest eigenvalue indicates the direction in which this axis gives the projection the greatest degree of dispersion. Therefore, when the one-dimensional representation effect is not ideal, the eigenvector b corresponding to the largest top k eigenvalues $\lambda_{1} \ge \lambda_{2} \ge \cdots \ge \lambda_{k} > 0$ can be selected to classify the categories²¹.

Improvement steps

The weights calculated by the quantitative theory mainly consider the frequency of occurrence of each factor and the internal meaning of the scores y_i of each sample. However, in practical engineering, the influence of the correlation between the categories b_i of each evaluation factor cannot be ignored^{24,25,26,27,28}. Considering that the quantitative theory has more detailed classification scores for various categories, we use the Kendall algorithm to calculate the correlation between the factors. The core idea is to calculate the number of different pairs between two ordered sets. The calculation process is as follows:

Suppose that there are N objects in a group of φ:

$$ \varphi = \left\{ {a,b, \ldots ,x,y} \right\} $$

(10)

An ordered set of N objects can be decomposed into an ordered pair of $1/2N\left( {N - 1} \right)$, for example $\varphi = \left\{ {a,b,c,d} \right\}$ then:

$$ \varphi_{1} = \left\{ {\left[ {a,c} \right],\left[ {a,b} \right],\left[ {a,d} \right],\left[ {c,b} \right],\left[ {c,d} \right],\left[ {b,d} \right]} \right\} $$

(11)

The difference distance between the two ordered pairs φ₁ and φ₂ is denoted as $d_{\Delta } \left( {\varphi_{1} ,\varphi_{2} } \right)$.

Hence the following Eq. (12) for the Kendall correlation coefficient²⁹. It is given:

$$ \tau = 1 - \frac{{2 \times \left[ {d_{\Delta } \left( {\varphi_{q} ,\varphi_{2} } \right)} \right]}}{{N\left( {N - 1} \right)}} $$

(12)

i.e.$\tau = P(same) - P(diffreent)$.where τ represents the difference between the probability that a pair of randomly obtained objects are in the same order and the probability that they are in a different order.

In this paper, a correlation (τ ≥ 0.5) is assumed between the two influencing factors whose correlation degree is greater than 0.5. To this end, the correlation feedback is used to adjust the spatial distance (i.e. the weight), and according to the Rocchio equation³⁰, we propose the following update function:

$$ \kappa_{f}^{\prime } = \kappa_{f} + \frac{\beta }{{\left| {cases_{REL} } \right|}}\sum\limits_{{b_{i} \in cases_{REL} }}^{{}} {b_{i} } - \frac{\gamma }{{\left| {cases_{NR} } \right|}}\sum\limits_{{b_{j} \in cases_{NR} }}^{{}} {b_{j} } $$

(13)

Among them, casesREL represents a group that is correlated with a certain evaluation factor U_i, and casesNR is a group that is not correlated with a certain evaluation factor Ui. b_i and b_j take values in the set of scores with the largest correlation ratio in the quantitative theoretical calculation results. In practice, γ is often taken as 0.25, and β is often taken as 0.75³⁰.

Coupling weights

The Entropy Weight Method (EW) and the Analytic Hierarchy Process (AHP) are commonly used objective and subjective weighting methods in Multiple Criteria Decision Making^31,32,33. In this paper, the multiplicative synthesis normalization method is used to determine the EW-AHP coupling weight as the initial weight.

The Entropy Weight Method (EW)³⁴ uses the entropy value g_ij to measure the amount of information. Assuming that the evaluation index Ui is equivalent to the importance of other indicators, it is represented by $\omega \left( {0 \le \omega_{ij} \le 1,\sum\limits_{j = 1}^{m} {\omega_{ij} = 1} } \right)$, then $\omega_{ij}$ is the weight of the evaluation factor Ui. The specific calculation of Eq is as follows:

• (1) Perform a dimensionless processing on x_ij to get $x_{ij}^{^{\prime}}$:

  $$ \begin{gathered} {\text{Positive indicator}}:x_{ij}^{\prime } = \frac{{x_{ij} - \min \left( {x_{ij} } \right)}}{{\max \left( {x_{ij} } \right) - \min \left( {x_{ij} } \right)}} + \alpha \hfill \\ {\text{Inverse indicator}}:x_{ij}^{\prime } = \frac{{\min \left( {x_{ij} } \right) - x_{ij} }}{{\max \left( {x_{ij} } \right) - \min \left( {x_{ij} } \right)}} + \alpha \hfill \\ \end{gathered} $$

  (14)

  In the equation, $\min \left( {x_{ij} } \right)$ is the minimum value and $\max \left( {x_{ij} } \right)$ is the maximum value. In order to eliminate the influence of 0 value, add a minimum value close to the value of $x_{ij}^{^{\prime}}$ after the dimensionless processing to translate. In this paper, $\alpha$ is taken as 0.001.

• (2) Perform a standardized processing, P_ij refers to the proportion of the j-th index of the i-th sample in the overall data:

  $$ P_{ij} = {{x_{ij}^{*} } \mathord{\left/ {\vphantom {{x_{ij}^{*} } {\sum\limits_{i = 1}^{n} {x_{ij}^{*} } }}} \right. \kern-\nulldelimiterspace} {\sum\limits_{i = 1}^{n} {x_{ij}^{*} } }} $$

  (15)

• (3) Calculate the difference coefficient g_i of the j-th index:

  $$ g_{j} = 1 + \frac{1}{\ln n}\sum\limits_{i = 1}^{n} {p_{ij} \ln p_{ij} } $$

  (16)

• (4) Calculate the weight $\omega_{j}$ of the j-th index:

  $$ \omega_{j} = {{g_{j} } \mathord{\left/ {\vphantom {{g_{j} } {\sum\limits_{j = 1}^{m} {g_{j} } }}} \right. \kern-\nulldelimiterspace} {\sum\limits_{j = 1}^{m} {g_{j} } }} $$

  (17)

The Analytic Hierarchy Process (AHP) method determines the subjective weight of the evaluation index. Firstly, the identified risk factors causing tunnel collapse are compared in pairs through expert judgment according to the 9-level evaluation method³⁵ to form a judgment matrix S, and the consistency test is carried out. Then the judgment matrix is calculated by the square root method to obtain the weight of each factor³³.

• (1) Constructing the judgment matrix:

  $$ M = \left( {\begin{array}{*{20}c} {b_{1} } & \ldots & {b_{j} } \\ \vdots & \ddots & \vdots \\ {b_{j} } & \cdots & {b_{j} } \\ \end{array} } \right)_{j \times j} $$

  (18)

  b_j represents the importance level determined by the 9-level evaluation method⁴⁰. By calculating the maximum eigenvalue of the judgment matrix M, the normalized eigenvector is obtained, and the the weight value of each index is obtained too.

• (2) In order to verify whether the importance level assigned to risk indicators is reasonable, the consistency test method of Wang et al.³⁶ is used.

  $$ CI = \frac{{\lambda_{\max } - n}}{n - 1} $$

  (19)
  $$ CR = CI/RI $$

  (20)

  In the equation, CI is the consistency index, RI is the random consistency index³⁶, and CR is the consistency ratio.

The composite weight is the coupling of objective data and subjective experience. In this paper, the multiplicative synthesis normalization method is used to calculate the coupling weight of the evaluation factor, and the EW-AHP coupling weight is used as the initial weight. After calculating the initial weight, combine the previous TQ-III weight into the following Eq. (21) to calculate the final weight:

$$ \omega_{j} = \left( {\alpha_{j} .\beta_{j} } \right)/\sum\limits_{j = 1}^{m} {\left( {\alpha_{j} .\beta_{j} } \right)} $$

(21)

In the Eq. (21): $\omega_{j}$ is the comprehensive weight of the j-th evaluation factor; $\alpha_{j}$ and $\beta_{j}$ are the TQ-III and EW-AHP weights of the j-th evaluation factor, and m is the number of evaluation factors.

Fuzzy comprehensive evaluation

Considering that the evaluation of tunnel collapse possibility is affected by the uncertainty of many factors, the application of the Fuzzy Comprehensive Evaluation Method can often achieve better practical results in tunnel collapse risk evaluation. In this method, the membership degree is calculated through the mapping of the factor set to the comment set, and then combined with the weight calculated by the Theory of Quantification III, a relatively objective fuzzy comprehensive evaluation result can be obtained.

Common membership function forms include "trapezoid", "semi-trapezoid", "normal", "k-th parabolic", "Cauchy", "Γ", "ridge" and so on³⁷. Considering that the evaluation factors in this paper are scattered, the "normal" type membership function is selected to calculate the membership degree of the factor set U to the comment set V³⁸. In the risk level classification of this paper, level I belongs to a small fuzzy distribution, levels II, III, and IV belong to an intermediate fuzzy distribution, and level V belongs to a large fuzzy distribution. The formula of the smaller membership function designed according to the normal distribution is:

$$ A_{{\text{I}}} \left[ {x_{j} \left( i \right)} \right] = \left\{ \begin{gathered} 1,x_{j} \left( i \right) \le a_{1} \hfill \\ \exp \left[ { - \left( {\frac{{x_{j} \left( i \right) - a}}{\sigma }} \right)^{2} } \right] \hfill \\ \end{gathered} \right.,x_{j} \left( i \right) > a_{1} $$

(22)

The three intermediate membership functions are:

$$ A_{{{\text{II}}}} \left[ {x_{j} \left( i \right)} \right] = \exp \left[ { - \left( {\frac{{x_{j} \left( i \right) - a_{2} }}{\sigma }} \right)^{2} } \right] $$

(23)

$$ A_{{{\text{III}}}} \left[ {x_{j} \left( i \right)} \right] = \exp \left[ { - \left( {\frac{{x_{j} \left( i \right) - a_{3} }}{\sigma }} \right)^{2} } \right] $$

(24)

$$ A_{{{\text{IV}}}} \left[ {x_{j} \left( i \right)} \right] = \exp \left[ { - \left( {\frac{{x_{j} \left( i \right) - a_{4} }}{\sigma }} \right)^{2} } \right] $$

(25)

The larger membership function is:

$$ A_{{\text{V}}} \left[ {x_{j} \left( i \right)} \right] = \left\{ \begin{gathered} 0,\;x_{j} \left( i \right) \le a_{5} \hfill \\ 1 - \exp \left[ { - \left( {\frac{{x_{j} \left( i \right) - a}}{\sigma }} \right)^{2} } \right],\;x_{j} \left( i \right) > a_{5} \hfill \\ \end{gathered} \right. $$

(26)

In the formula, A represents the membership degree of the i-th factor of the j-th sample at a certain level, $x_{j} \left( i \right)$ represents the measured data of the i-th factor of the j-th sample. where a_i is the location parameter of the normal distribution and σ is the shape parameter describing the degree of dispersion of the normal distribution. The formula is as follows.

$$ a_{i} = \frac{1}{{2\left( {e_{i} + e_{i + 1} } \right)}} $$

(27)

$$ \sigma = 0.6\left( {e_{i + 1} - e_{i} } \right) $$

(28)

e_i and e_i+1 are the upper and lower limits of a certain risk level evaluation factor.

The fuzzy judgment matrix calculated by the membership function is:

$$ B = \left( {\begin{array}{*{20}c} {b_{11} } & \ldots & {b_{1v} } \\ \vdots & \ddots & \vdots \\ {b_{u1} } & \cdots & {b_{uv} } \\ \end{array} } \right)_{u \times v} $$

(29)

In the Eq. (29): u is the evaluation factor; v is the risk level; b_uv represents the membership degree of the u-th evaluation factor to the risk level v.

After obtaining the comment set, weight set, and single-factor judgment matrix, make a fuzzy linear transformation on the fuzzy judgment matrix and turn the weight set into a fuzzy subset of the comment set:

$$ A = W \times B $$

(30)

In the Eq. (30): W is the coupling weight of Sect. “Coupling weights” (Fig. 1).

Figure 1

Model building flowchart.

As shown in Fig. 1, the first step is to construct a reflection matrix from the Theory of Quantification III, and calculate the scores of various objects. Then select the three groups of scores with the largest correlation ratio to form the X, Y, and Z axes, calculate the spatial distance κ of each category and, the Kendall correlation coefficient based on κ, then optimize the coefficient with the Rocchio equation to obtain the weight of each influencing factor (ie TQ-III weight); The second step is to calculate the multiplicative synthesis normalized weight of EW-AHP as the initial weight, which is coupled with the TQ-III weight so that the weight can be dynamically adjusted with the characteristics of the project area; The third step uses the Fuzzy Comprehensive Evaluation Method of the normal distribution membership function to classify the possibility of tunnel collapse.

Example verification

Overview of research object’sproject and geology

Overview of the project

The second phase of the Beijiang Water Supply Project consists of three diversion tunnels: West-Second, Ka-Shuang, and Double-Third. The total length of the tunnels is 516.2 km, of which the Ka-Shuang tunnel is 283.3 km long, making it the longest water delivery tunnel in the world. The average burial depth is 428 m, and the maximum burial depth is 774 m, which is a no-pressure diversion tunnel. The diameter of the cavern excavated by drilling and blasting is 6.64–7.4 m, and the diameter of the cavern excavated by TBM is 7.1 m. The surrounding rock of the cavern is mainly grade II and grade III (According to the relevant standards of Chinese tunnel construction⁴⁴, the quality of surrounding rock is divided into 6 grades from good to poor, and grades II and III are hard rocks with better comprehensive quality), accounting for 86.2%, and the saturated compressive strength is mostly between 50 and 140 MPa³⁹.

Topography and physiognomy

The project is located in the hilly area and low mountain area between the southern slope of the Altai Mountains and the northern slope of the East Tianshan Mountains, with an altitude of 750–1300 m, the terrain is undulating, the slope of the mountain is gentle, and the bedrock is mostly exposed, the physiognomy is mainly desert³⁹.

Stratigraphic lithology

The strata in this area are dominated by the ancient strata of the Devonian and Carboniferous, followed by granite, and very few areas are Permian and Triassic strata. Among them, the Devonian and Carboniferous tuffs, tuffaceous sandstones and calcareous sandstones strata have a total length of 209.1 km, accounting for 73.8% of the tunnel length; the total length of the biotite granite and granodiorite strata intruded in the late Hercynian is 59.6 km, accounting for 21% of the tunnel length; the mudstone, sandstone and glutenite of the Permian and Triassic have a total length of 12.1 km, accounting for 4.3% of the tunnel length³⁹.

Geological structure

The project area is located in the two major tectonic unit intervals of the Altai fold system and the Junggar-North Tianshan fold system. Due to the influence of the fold structure on the topography and geological structure, a series of compressive faults and compressive torsional faults have developed in this area. There are 5 regional fault zones with obvious structural traces on the surface, 72 secondary faults, and the general width of the fracture zone is 10–30 m. Through the comprehensive analysis of drilling core exposure, downhole TV snooping, geophysical sound wave testing and other methods. It is found that the faults and fissures near the tunnel do not developed, and the fissures are dominated by medium-steep dips. The surface water in the project area is lacking, and the groundwater is mainly composed of a small amount of bedrock fissure water. The water quality is poor, and it is corrosive to concrete and steel bars in concrete structures³⁹. In this paper, 10 typical sections are selected as the evaluation objects in the research area (Fig. 2).

Figure 2

Location of the Ka-Shuang Tunnel (Arcgis10.8 https://www.esri.com/en-us/arcgis/products/mapping/overview).

By summarizing the existing research results, the factors that cause tunnel collapse are mainly divided into geological condition factors, construction factors and design factors⁴⁰. Refering to the statistical analysis results of Zhu Jie et al.², on 242 road tunnels, 104 railway tunnels, 35 hydraulic tunnels, a total of 381 effective collapse cases, the main influencing factors are the grade of surrounding rock (18.35%), groundwater (11.15%), rainfall (10.19%), supporting method (7.29%), fractured and broken zone (6.90%), the integrity of rock mass (6.13%), tunnel Buried depth (4.39%). Among them, the rainfall factor is generally considered in the shallow buried section or the opening, so it is not considered in this paper.

In summary, combined with the literature induction in the introduction chapter. According to the geological conditions and the classification method of construction disturbance, this paper selects the area of equivalent cross-section U₁, depth ratio U₂ (the ratio of tunnel buried depth to diameter), the width of the fracture zone U₃ (the width of a section of strongly fragmented rock caused by a fault or a dense zone of fissures²), the strength of uniaxial compression U₄, RQD U₅ (rock quality index, reflecting the geological conditions of the stratum where the tunnel is located), the percentage of over excavation U₆, the grade of surrounding rock U₇ (the tunnel surrounding rock grade determined by the longitudinal wave), the integrity of rock mass U₈ (the ratio of the rock mass elastic longitudinal wave to the rock elastic longitudinal wave), the groundwater condition U₉, the degree of weathering U₁₀ and the method of support U₁₁ are used as tunnel collapse risk assessment factors (Fig. 3).

Figure 3

Evaluation index hierarchy diagram.

The tunnel-related survey data⁹ in Table 1 are used as the basic data of Quantitative Theory III and the basic data of the decision matrix of the fuzzy comprehensive evaluation method (Table 1).

Table 1 Tunnel monitoring data.

In this paper, referring to Zhai et al.⁹ and Hyu et al.⁴³ classification standard of tunnel collapse risk factors, a comment set V = {v₁, v₂, v₃, v₄, v₅} is formulated with five risk levels, which are grade I (no risk) and grade II (slight risk), grade III (high risk), grade IV (slight collapse), and grade V (severe collapse) (Table 2).

Table 2 Tunnel classification.

The improved theory of quantification III

According to the grading standard in Table 2, the Theory of Quantification III is used to calculate and determine the main controlling factors affecting tunnel collapse. The 11 projects have 5 sub-categories respectively, and the total number of categories is 55. After calculation, the first three largest eigenvalues λ₁ = 0.124, λ₂ = 0.090, λ₃ = 0.078 and their corresponding correlation ratios are 41, 18, and 28%, the sum has reached 87%, which can largely represent the information of influencing factors. The maximum eigenvalue and corresponding category scores are shown in Table 3.

Table 3 Influencing factor category scores.

It can be seen from Table 3 that the correlation ratio of λ₁ eigenvalue is 41%, and its trend change is the main macroscopic reflection of the influencing factors of tunnel collapse, and can be used as the main factor axis for screening sensitive factors. In order to more comprehensively reflect the information of the influencing factors, according to the previous restriction of g^Tb = 0, the category scores b₁, b₂, b₃ corresponding to λ₁, λ₂, λ₃ are regarded as the spatial X, Y, and Z factor axes, and the zero point is establish as the origin of the spatial coordinate system, based on the spatial distance from the origin to perform category screening and classification (Fig. 4) (Table 4).

Figure 4

The distance of each category compared to the origin.

Table 4 Spatial distance statistics of influencing factor categories.

Since each item is a subset of each item, the final spatial distance of each item is obtained by the sum of the corresponding category distances (Table 5):

$$ \kappa_{ui} = \sum\limits_{{r_{1} = 1}}^{m} {\kappa_{{r_{i} }} } $$

Table 5 Spatial distance of each project.

Since the theory mainly considers the frequency of occurrence of each factor and the inherent meaning of each sample, the correlation between the categories of evaluation factors in practical engineering cannot be ignored^{24,25,26,27,28}. Considering that the quantitative theory has more detailed classification scores for various categories, we use the Kendall algorithm to calculate the correlation between each factor. The core idea is to calculate the number of different pairs between two ordered sets.

Convert the original monitoring data of the tunnel in Table 1 into the spatial distance calculated in Sect. “Improvement steps”, as shown in Table 6 below.

Table 6 Spatial distance of influencing factors of each sample.

Equation (12) The Kendall algorithm is used to calculate the correlation between the evaluation factors as shown in the following Fig. 5:

Figure 5

Correlation analysis between various influencing factors.

The results are shown in Fig. 5: the correlation between the U₁ (the area of equivalent cross-section) and the U₁₁ (the method of support) is as high as 0.8, the correlation between the U₁ (the area of equivalent cross-section) and the U₂ (depth ratio) is also 0.76, and the correlation between U₂ (depth ratio) and U₁₁ (the method of support) also reached 0.61, and the comprehensive analysis was in line with the actual situation of tunnel construction.

To this end, the correlation feedback is used to adjust the spatial distance (i.e. the weight), and according to the Rocchio Eq. (13), Fig. 6:

Figure 6

Comparison of spatial distance κ before and after optimization.

It can be seen that after optimization and adjustment, based on the characteristics of the case problem, several evaluation factors with strong correlation with other evaluation factors fluctuate greatly, indicating that reasonable weights should be allocated according to the strength of correlation within the evaluation factor group.

The spatial distance of each item is normalized to obtain the theory of quantification III weight of each evaluation factor (Table 7):

$$ \alpha_{j} = {{\kappa_{uj} } \mathord{\left/ {\vphantom {{\kappa_{uj} } {\sum\limits_{j = 1}^{11} {\kappa_{uj} } }}} \right. \kern-\nulldelimiterspace} {\sum\limits_{j = 1}^{11} {\kappa_{uj} } }} $$

Table 7 TQ-III weights.

Coupling weights

According to the calculation process given by the Entropy Weight method in Sect. “Coupling weights”, the original data is dimensionless by Eq. (14), as shown in Table 8:

Table 8 Dimensionless data.

The difference coefficient is calculated by Eqs. (15) and (16), and then brought into Eq. (17) to obtain its weight value under the Entropy Weight method, as shown in Table 9:

Table 9 EW weights.

According to the calculation process given by the Analytic Hierarchy Process Sect. “Coupling weights”, the judgment matrix M₁, M₂ is constructed for the two levels of geological condition and construction disturbance by Eq. (18):

$$ M_{1} = \left( {\begin{array}{*{20}c} 1 & 1 & 1 & {1/2} & {1/2} & {1/3} & {1/3} \\ 1 & 1 & 1 & 1 & 1 & {1/3} & {1/3} \\ 1 & 1 & 1 & {1/2} & {1/2} & {1/2} & {1/2} \\ 2 & 1 & 2 & 1 & 1 & 1 & 1 \\ 2 & 1 & 2 & 1 & 1 & 1 & 1 \\ 3 & 3 & 2 & 1 & 1 & 1 & 1 \\ 3 & 3 & 2 & 1 & 1 & 1 & 1 \\ \end{array} } \right) $$

$$ M_{2} = \left( {\begin{array}{*{20}c} 1 & 1 & {1/2} & {1/2} \\ 1 & 1 & 1 & {1/2} \\ 2 & 1 & 1 & 1 \\ 2 & 2 & 1 & 1 \\ \end{array} } \right) $$

According to the judgment matrix, the largest eigenvalue and the eigenvector of the matrix are calculated, and then bring them into the Eqs. (19) and (20) to pass the consistency test (CI =  < 0.1), so as to obtain the AHP weight value of each evaluation index. as shown in Table 10.

Table 10 AHP weights.

After calculating the weight value of each hierarchy, the AHP weights are obtained by coupling the two hierarchies in a ratio of 7/4.

After calculating the weight of the entropy weight method (EW) and the weight of the analytic hierarchy process (AHP), the EW-AHP coupling weight is obtained using the multiplicative synthesis normalization Eq. (21), which is used as the initial weight (as shown in Fig. 7).

Figure 7

EW-AHP coupling weights.

Continue to use the multiplication synthesis normalization Eq. (21) to couple the EW-AHP initial weight and the TQ-III weight calculated earlier to obtain the final synthesis weight (as shown in Fig. 8).

Figure 8

TQ-III and EW-AHP coupling weights.

Fuzzy comprehensive evaluation

In this paper, referring to the method of Cao et al.⁴², a fuzzy comprehensive evaluation model of tunnel collapse risk is established, and it is applied to the assessment of the collapse risk of the Ka-shuang diversion tunnel. The 11 evaluation factors selected above are taken as the factor set in the fuzzy comprehensive evaluation, and recorded as U = {U₁, U₂, U₃, U₄, U₅, U₆, U₇, U₈, U₉, U₁₀, U₁₁} (Table 1 in Sect. “Overview of research object’ sproject and geology”). And a comment set with 5 risk levels V = {v₁, v₂, v₃, v₄, v₅} is formulated at the same time (Table 2 in Sect. “Overview of research object’ sproject and geology”) .

After determining the evaluation original data and the risk level limit of each factor, the position parameter a_i and the shape parameter σ of the normal distribution are calculated by Eqs. (27) and (28), as shown in Table 11:

Table 11 Normal distribution constant.

Among them, since U₉, U₁₀, and U₁₁ are qualitative variables, they are divided into equal proportions from 0 to 1 when calculating the membership degree.

After calculating the position parameter a_i and the shape parameter σ, the membership matrix B of each tunnel sample is calculated by Eqs. (22)–(26). Combined with the coupling weight W calculated in Sect. “Coupling weights”, the final risk rating of the 10 tunnel samples is calculated by Eq. (30).

Due to the large number of tunnel samples and the complicated calculation, this paper takes the tunnel sample S1 as an example, and calculates the fuzzy judgment matrix B₁ of S1 by Eq. (22)–(26) as:

$$ B_{1} = \left( {\begin{array}{*{20}c} 0 & {0.35} & {0.66} & {0.21} & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 1 & {0.3} & 0 & 0 & 0 \\ 0 & {0.06} & 1 & 0 & 0 \\ 0 & 1 & {0.22} & 0 & 0 \\ {0.5} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {0.89} & 1 \\ 0 & {0.93} & {0.73} & 0 & 0 \\ 1 & {0.21} & 0 & 0 & 0 \\ 1 & {0.21} & 0 & 0 & 0 \\ 0 & 0 & {0.06} & 1 & 0 \\ \end{array} } \right) $$

After calculating the membership matrix B₁ of S1, combined with the coupling weight W obtained above, the fuzzy comprehensive vector $A_{1} = \left( {0.4436 \, 0.2719 \, 0.1783 \, 0.2295 \, 0.1562} \right)$ of S1 is obtained by Eq. (30). After normalization, the membership degree $A_{1}^{\prime } = \left( {0.3467 \, 0.2125 \, 0.1394 \, 0.1793 \, 0.1221} \right)$ corresponding to each risk level is obtained.

According to the principle of maximum membership degree, the maximum membership degree in $A_{1}^{^{\prime}}$ is 0.3467, the corresponding risk level is level I (no risk), and the degree of membership in level II (slight risk) also reaches 0.2125, therefore, the possibility of S1 collapsing is very small. The remaining 9 tunnel samples were evaluated by the same method, and the final results are shown in Table 12.

Table 12 Comparison of evaluation results with field investigation.

It can be seen from Table 12 that:

• (1) There is one sample with the risk possibility of tunnel collapse level I (no risk), which is S1.The results field investigation shows that the surrounding rocks there are of high quality, with few fissures, complete and stable as a whole, and the walls of the caves are relatively smooth after excavation^9,41.

• (2) There are four samples with the risk possibility of tunnel collapse level II (slight risk), namely S5, S6, S7 and S9. The results of field investigation show that the surrounding rock at S5 is of high quality grade, with few cracks, the whole is complete and stable, and the wall of the cave is relatively smooth after excavation; The surrounding rock at S6 has relatively good integrity and stability, and there is a slight local rockfall. The surrounding rock fractures at S7 and S9 are not well developed, and the overall stability and integrity are good^9,41.

• (3) It is worth noting that S4 is classified as level IV according to the maximum membership degree, but the membership degrees of level II (slight risk) and level IV (slight collapse) are very close and the overall trend is biased towards the low risk area. Therefore, the risk possibility of the S4 is considered to be level III (high risk) as a compromise. The actual investigation on the site shows that the surrounding rock fissures are not well developed, the overall stability and integrity are good, and local rock falls slightly too^9,41.

• (4) There are three samples with the risk possibility of tunnel collapse level IV (slight collapse), which are S2, S3, and S10. The actual investigation on the site shows that the surrounding rock at S2 has poor integrity and stability, showing a block-cracked structure. The vault collapsed along the structural plane, and the blocks fell off seriously, but no large-scale collapse occurred. The surrounding rock fissures at S3 were strongly cut, the fissure plane is smooth, and the vault collapses seriously along the structural plane. The f 51 torsional fault is developed at the bottom of the arch at S10. The rock mass in the fault is mylonite and fault gouge, showing a fractured structure and serious rockfall^9,41.

• (5) The tunnel collapse risk possibility level of V (severe collapse) is S8. The actual investigation on site shows: The surrounding rock of the tunnel vault at S8 slumped severely, and the length of the collapse with the development direction of the fissure was larger, and the scale of the collapse at the tuff with smaller thickness was larger, and the height reached 0.5 m^9,41.

By analyzing the evaluation results of 10 sample tunnels, the results made by this method are in line with the actual situation of the project, and are basically consistent with the evaluation results made by other methods. The evaluation results not only answer the question of whether the tunnel collapses, but also describe the level of each risk possibility in detail.

Discussion

Combined with project examples, from the comparison of evaluation results, we found that the results of this paper are slightly different from other literature evaluation results⁹ (using EW-AHP weights), and analyzed the reasons:

• (1) In this paper, the weights of U₃ (the width of the fracture zone) and U₄ (the strength of uniaxial compression) calculated by TQ-III are 0.1142 and 0.1181, respectively, which are greatly improved compared to EW-AHP weights of 0.0848 and 0.0527. Analysis of the reasons shows that there are 5 regional fault zones and 72 secondary faults with obvious structural traces on the surface of the project. The fracture zone is generally 10–30 m wide, and the overall fault zone has a high degree of development⁴¹. The calculation process in this paper is based on the actual geological structure characteristics of the tunnel. Affected by this, the corresponding increase in the weight is more suitable for the actual geological situation.

• (2) The TQ-III weights of U₁ (the area of equivalent cross-section) and U₁₁ (the method of support) are 0.0381 and 0.0596, respectively, which are significantly reduced compared to EW-AHP weights of 0.0631 and 0.1160. Analysis of the reasons shows that the project is a water diversion tunnel with a smaller cross-sectional area than traditional highway tunnels, and the disturbance to surrounding rock during excavation is correspondingly reduced. In addition, small-sized tunnels have lower requirements for support methods. Therefore, the importance of reducing U₁ and U₁₁ is in line with the actual tunnel construction status, reflecting the adaptability of this method.

• (3) The TQ-III weight of U₇ (the grade of surrounding rock) is 0.0788, which is significantly lower than the EW-AHP weight of 0.1772. Affected by the correlation analysis, as shown in Fig. 4, the correlation between U₇ and U₁₁ is − 0.6, so there is a corresponding decreasing trend. And the overall stability of the rock mass in the cave is good, the uniaxial saturated compressive strength is 15 ~ 140 MPa, and the surrounding rock grades are mostly II and III⁴¹. Therefore, the less weight assigned to U₇ is in line with the geological characteristics of the project area.

• (4) The weight of U₉ (the groundwater condition) TQ-III is 0.0883, which is significantly lower than that of EW-AHP, which is 0.1709. Combined with the general situation of the project area, because the research object is located on the edge of the desert, the surface water is very poor, and the bedrock fissure water is the main type of groundwater, and the water volume is weak. The results of drilling water pumping test by Deng et al.³⁹ showed that the surrounding rock of the tunnel belongs to the micro-level micro-permeable layer. Therefore, assigning a lower weight to U₉ is in line with the actual geological conditions of the project area.

To sum up, the mathematical meaning of the weights calculated based on TQ-III is that less weights are assigned to evaluation factors that appear less frequently, and more weights are assigned to frequently appearing factors. In practical projects, some influencing factors may have high weights, but they appear less or not even in the project. Therefore, reducing their weights and assigning them to other influencing factors can improve the accuracy of the evaluation.

In view of the fact that the test data source of this case is a water diversion tunnel with a cross-section of 7.1 m, the application effect on other sizes or types of tunnels needs to be verified. Through the analysis of the selected 11 evaluation factors, the evaluation process considers U₁ (the area of equivalent cross-section) and other 10 evaluation factors commonly used in various types of tunnels, so the theoretically analysis, this method is also applicable to other tunnels of different sizes and types. Therefore, in the following research, the author tries to continue to optimize the theory and expand the scope of application of the method.

Conclusion

Considering the complexity and subjectivity of multiple decision-making problems in tunnel risk assessment, this paper proposes a tunnel collapse risk classification method based on the Improved Theory of Quantification III coupling weight and the Fuzzy Comprehensive Evaluation Method. The new method calculates the weights by uniformly converting the quantitative and qualitative data of tunnel monitoring into quantitative scores using the Improved Theory of Quantification III, and improves the accuracy by coupling commonly used subjective and objective weights. The membership degree of each evaluation factor and the tunnel risk level is calculated according to the normal distribution function, so as to comprehensively judge the collapse risk level of each tunnel sample. The new method has obtained the following conclusions in the application of the Ka-shuang diversion tunnel.

• (1) This paper uses the Theory of Quantification III to establish a reflection matrix for tunnel monitoring data, thereby converting qualitative data into quantitative data, avoiding the subjectivity of the assignment method. And after the improvement, the correlation between the evaluation factors is considered, and the accuracy of the evaluation results is improved.

• (2) The case vertification shows that the weights determined by this method are based on the actual project monitoring data, change accordingly with the structural characteristics of the project area, and have the characteristics of practicability and flexibility.

• (3) The reliability of the method is verified by comparing the evaluation results of 10 tunnel samples with the status quo of project area.