1 Introduction

At present, China has become one of the countries with the most serious tunnel collapse disasters in the world. The main factors leading to tunnel collapse of the tunnel were the poor geological condition and the ground water, especially the rich water, weak rock mass, broken formation and so on.

In the process of tunnel construction of Hurongxi highway, Chengdu-Chongqing highway, Zhuyong highway, Xinjian railway and Lanxin railway etc., serious collapse geological disasters were occurred. Most of the collapses were occurred in the regions of water rich and weak or broken zones. Taking Jinyunshan Tunnel of Chengdu-Chongqing highway as an example, five large-scale collapses were occurred during the construction. Especially the four largest collapses with 22 m long, 10–14 m wide and 18–25 m height caused the collapse amount of rock mass reaching to 4000–5000 m3. In the construction process of Longtan Tunnel of Hurongxi highway, three serious collapses were occurred with 9000 m3 collapse amount, which caused the schedule of tunnel delayed more than 1 year.

The tunnels which occurred the serious collapse disaster were mostly crossed the poor geological stratums, such as water rich, weak and broken strata, etc. The characteristics of collapse disasters mainly include high-frequency, large-scale and strong-disaster. It can not only cause heavy casualties and economic losses, but also cause the non-repairable damage and serious hidden danger to the water resources and ecological environment of the earth surface (Jiang et al. 2014).

Many scholars have done a lot of research on the risk assessment of tunnel collapse and water inrush (Bukowski 2001; Kong 2011; Li et al. 2015; Li and Li 2014; Meng et al. 2012; Yazdani-Chamzini 2014). Zhang et al. (2009) developed the theory and method of water inrush risk assessment of high risk karst tunnel, and then established the quantitative evaluation method of water inrush risk and the mechanism of the four color warning. Zhou et al. (2013) established a fuzzy analytic hierarchy model for the risk evaluation of collapse in shallow tunnel, and carried out the dynamic risk assessment in the construction process of the tunnel. Based on the attribute mathematical theory, Li et al. (2013a) established the attribute recognition model which used to evaluate the tunnel collapse risk.

Grey system theory was a kind of systematic scientific theory which initiated by Deng (2014). In the field of geotechnical engineering of disaster prevention and reduction, grey correlation analysis can be used to determine the degree of correlation between the factors of disaster based on the degree of similarity of the changing curve. Through the quantitative analysis of dynamic process development situation, this method completed the comparison of the geometric relation of the time series in the system. Finally, the grey correlation degree between the reference sequence and the comparison sequence was obtained. Gray correlation analysis method requires a minimum of 4 sample capacities, in the case of irregular data can also be applied, quantitative results can not appear to be inconsistent with the qualitative analysis of the case. The basic idea of the method was to carry out the non- dimensional treatment of the original observation data of the evaluation indexes, and to calculate the correlation coefficients, the correlation degrees and the size of the correlation degrees.

The risk evaluation index weights reflect the important degree of the evaluation goal, so it’s very important to ensure the rationality of the index (Zhang et al. 2015). Entropy weight method was a kind of objective weighting method. According to the variation degree of each index, the entropy weight of each index was calculated by the information entropy, and then the weight of each index was modified through the entropy weight. Finally, more objective index weights have been obtained.

At present, the grey system theory has been widely applied in the field of geotechnical engineering, but all of the researches are concentrated on the deformation monitoring of surrounding rock of tunnels (Hernandez et al. 2015; Yang and Pan 2015; Yin 2013). There is no find that the grey system theory was applied in tunnel collapse disasters. The tunnel collapse risk evaluation method which based on the grey system and entropy method is valuable to be established.

Although the traditional grey system theory has the advantages of few samples and no law, it was also because of the simple arithmetic mean method to determine the weight which leads to a large error. Based on the grey system theory and the entropy weight method, the risk assessment method was used to map the weight by the evaluation index information. This can avoid the subjectivity of weight calculation and provides an effective way for karst tunnel water inrush risk assessment. And which has a certain referential significance for the similar engineering.

2 Risk Assessment Theory Based on Grey System Theory and Entropy Weight Method

Risk assessment of tunnel collapse can be summed up as a measure of qualitative description. The comprehensive risk evaluation theory based on grey system theory and the entropy weight method can be simply summarized as: grey relational analysis and entropy method were used to determine the weight and correlation calculation.

2.1 Grey Correlation Analysis

Firstly, the analysis sequence was determined according to the evaluation object and the evaluation index. Then the decision matrix was constructed by the non-dimensional treatment. At last, the correlation coefficient was calculated, and the correlation coefficient matrix was established.

2.1.1 The Number of Analysis Sequence

Based on the evaluation object and evaluation purpose, the evaluation index system was determined, and the data was collected and evaluated. Set n data sequences to form the following matrix:

$$X = \left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} {x_{11} } & {x_{12} } \\ {x_{21} } & {x_{22} } \\ \end{array} } & {\begin{array}{*{20}c} \cdots & {x_{1n} } \\ \cdots & {x_{2n} } \\ \end{array} } \\ {\begin{array}{*{20}c} \vdots & \vdots \\ {x_{11} } & {x_{11} } \\ \end{array} } & {\begin{array}{*{20}c} \vdots & \vdots \\ \cdots & {x_{nn} } \\ \end{array} } \\ \end{array} } \right)$$
(1)

Risk assessment of tunnel collapse is a comprehensive evaluation system. This system analyzed n kinds of risk evaluation objects, and the evaluation objects were considered to have m evaluation indexes. According to the above conditions to establish the evaluation object range, X was a number of evaluation object intervals.

The reference data column was determined. Reference data columns should be an ideal comparison standard, the optimal value of each index (or the worst value) constitute the reference data column, also can choose other reference value according to the evaluation purpose. Record as

$$X_{0} = \left( {x_{0} \left( 1 \right),x_{0} \left( 2 \right), \ldots ,x_{0} \left( m \right)} \right).$$
(2)

2.1.2 Non-dimensional Variables

Because of the different dimensions of different factors in the system, it was necessary to carry out the non-dimensional treatment to the data when the grey relational grade analysis was carried out.

$$\left( {X_{0} ,X_{1} , \ldots ,X_{n} } \right) = \left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} {x_{0} \left( 1 \right)} & {x_{1} \left( 1 \right)} \\ {x_{0} \left( 2 \right)} & {x_{1} \left( 2 \right)} \\ \end{array} } & {\begin{array}{*{20}c} \cdots & {x_{n} \left( 1 \right)} \\ \cdots & {x_{n} \left( 2 \right)} \\ \end{array} } \\ {\begin{array}{*{20}c} \vdots & \vdots \\ {x_{0} \left( m \right)} & {x_{1} \left( m \right)} \\ \end{array} } & {\begin{array}{*{20}c} \vdots & \vdots \\ \cdots & {x_{n} \left( m \right)} \\ \end{array} } \\ \end{array} } \right)$$
(3)

The most dangerous situation was \(X_{0j}\) shown as

$$X_{0j} = { \hbox{max} }\left( {X_{1j} ,X_{2j} , \ldots ,X_{nj} } \right)$$
(4)
$$X_{ij}^{{\prime }} = X_{ij} /X_{0j}$$
(5)

The dimensionless matrix was obtained as follows

$$X_{ij}^{{\prime }} = \left[ {X_{i1}^{{\prime }} ,X_{i2}^{{\prime }} , \ldots ,X_{in}^{{\prime }} } \right].$$
(6)

2.1.3 Correlation Coefficient

The correlation coefficients of each comparison sequence and reference sequence were calculated respectively as follows

$$\xi_{i} \left( k \right) = \frac{{\mathop { \hbox{min} }\limits_{i} \mathop { \hbox{min} }\limits_{j} \left| {x_{0} \left( k \right) - x_{i} \left( k \right)} \right| + \rho \cdot \mathop { \hbox{min} }\limits_{i} \mathop { \hbox{min} }\limits_{k} |x_{0} (k) - x_{i} (k)|}}{{\left| {x_{0} (k) - x_{i} (k)} \right| + \rho \cdot \mathop { \hbox{min} }\limits_{i} \mathop { \hbox{min} }\limits_{k} |x_{0} (k) - x_{i} (k)|}}$$
(7)

where \(k = 1,2, \ldots ,m.\)

In the equation, ρ was the resolution factor, in the value of 0–1. The smaller of the ρ, the bigger of the difference between the correlation coefficient, and the stronger of the ability to distinguish. Usually, ρ was take as 0.5.

2.2 Weight Coefficient of Entropy Weight Method

Entropy weight method is a kind of objective weighting method. According to the variation degree of each index, entropy method took advantage of the information entropy to calculate the index’s entropy weight. Then entropy weight was used to modify the indexes. At last, the objective index weight was obtained.

According to the basic principle of information theory, the information was a measure of the system’s order degree, and the entropy was a measure of the system’s disorder degree.

First, the scheme M and the evaluation indicators N were constructed. Then the matrix B m×n was obtained through normalizing the matrix R.

figure a

In the equation, i = 1, 2,…, m, j = 1, 2,…, n. With the larger of the x ij, the collapse was more likely to occur, and the Eq. 8(a) was used to calculate b ij . Similarly, with the smaller of the x ij, the collapse was more difficultly to occur, and the Eq. 8(b) was used to calculate b ij .

The system were in a variety of different state. The probability of each state appeared as P i (i = 1, 2, …, m), and the entropy of the system could be defined as

$$e = - \mathop \sum \limits_{i = 1}^{m} P_{i} \cdot \ln P_{i}$$
(9)

Obviously, when P i  = 1/m (i = 1, 2, …, m), the probability of the occurrence of various states was consistent, and the maximum value of entropy were

$$e_{\hbox{max} } = \ln m$$
(10)

Existing m items to be evaluated, n evaluation indicators, the formation of the original evaluation matrix \(R = \left( {r_{ij} } \right)_{m \times n}\) or a certain indicators have the information entropy.

$$e_{j} = - \mathop \sum \limits_{i = 1}^{m} P_{ij} \cdot \ln P_{ij}$$
(11)

The final weight of each index was determined by the importance of the comprehensive index and the amount of information provided by the index.

Existing m items to be evaluated, n evaluation indicators, the formation of the original data matrix R = (r ij ) m×n :

$$R = \left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} {r_{11} } & {r_{12} } \\ {r_{21} } & {r_{22} } \\ \end{array} } & {\begin{array}{*{20}c} \cdots & {r_{1n} } \\ \cdots & {r_{2n} } \\ \end{array} } \\ {\begin{array}{*{20}c} \vdots & \vdots \\ {r_{m1} } & {r_{m2} } \\ \end{array} } & {\begin{array}{*{20}c} \vdots & \vdots \\ \cdots & {r_{mn} } \\ \end{array} } \\ \end{array} } \right)_{m \times n}$$
(12)

where the r ij was the evaluation value of the first i project for the first j index.

The process of seeking each index’s weight is shown as follows:

  1. 1.

    Count the proportion of the index value of the i item of the first j index

    $$P_{ij} = r_{ij} /\mathop \sum \limits_{i = 1}^{m} r_{ij}$$
    (13)
  2. 2.

    Calculate the j index of entropy

    $$e_{j} = - k\mathop \sum \limits_{i = 1}^{m} P_{ij} \cdot \ln P_{ij}$$
    (14)

    where k = 1/ln m

  3. 3.

    Calculate the entropy weight w j of the first j index

    $$W_{j} = \left( {1 - e_{j} } \right)/\mathop \sum \limits_{j = 1}^{n} \left( {1 - e_{j} } \right).$$
    (15)

2.3 Correlation Degree

Correlation degree E i was the degree of approximation of the evaluation scheme and the standard scheme. The large of the value Ei, the more similar of the two schemes. According to the size of the correlation degree of each program, it was easy to sort the pros and cons of values. And it also could be classified according to the correlation degree of the standard values.

E i was expressed as

$$E_{i} = \left[ {\begin{array}{*{20}l} {} \hfill & {A_{1} } \hfill & {A_{2} } \hfill & \cdots \hfill & {A_{m} } \hfill \\ {E_{i} } \hfill & {A_{2} } \hfill & {E_{2} } \hfill & \cdots \hfill & {E_{m} } \hfill \\ \end{array} } \right]$$
(16)

where \(E_{i} = \mathop \sum \limits_{j = 1}^{n} \omega_{j} \xi_{ij}\quad \left( {j = 1,2, \ldots ,n} \right)\).

3 Engineering Application Analysis

3.1 The Index of the Risk Evaluation

The different tunnel segment mileage was selected as the evaluation object. The statistical analysis of 76 large and medium landslides is carried out and the main factors of the tunnel collapse were obtained based on the hydrogeological environment of the tunnel before the collapse. Then the main evaluation indexes which caused the collapse were selected (Chen et al. 2009; Wang et al. 2009; Li et al. 2013b).

The unfavorable geology mainly include the fault fracture zone, poor geological stratums (loosely, water-rich and expansive). When the tunnel construction through the unfavorable geology, collapse was easier to happen, even cause serious consequences. Especially the fault fracture zone was more likely to lead to the collapse for the features of low strength, easy deformation, strong permeability, water resistance and so on. The fault fracture zone played a major role in the 36 collapses. In addition, the fault fracture zone played a more important role the small scale collapses. According to the collapse disaster intensity, the unfavored geology can be divided into four levels, which were strong disaster, medium disaster, weak disaster and micro disaster or no disaster (Table 1).

Table 1 Condition and score of unfavourable geology
Full size table

The buried depth of the tunnel. Tunnel buried depth was related to the initial stress field of surrounding rock and many factors. The greater the depth, the greater the initial stress and the less stable of the surrounding rock. Arch collapses were easier to occur in large buried deep tunnel, and the penetrating collapses were easier to occur in small buried deep tunnel.

The tunnel was located at the lower part of the mountain slope, and the ground elevation was between 1045 and 1120 m.

Construction measures and engineering disturbance were the direct factors of collapse. Improper excavation method, not timely support, monitoring measurement, nonstandard advance geological forecast and other factors can easily lead to the occurrence of collapses. The construction factor was restricted by the level of the construction unit, in order to simplify the analysis, the construction unit technology and management as one of the factors of fuzzy evaluation of construction factors, the construction measures to implement expert evaluation scoring system. The construction factor was restricted by the level of the construction unit, which was one of the factors that make the construction unit technology and management as one of the factors of fuzzy evaluation, and the construction measures were implemented by the expert evaluation system.

The integrity degree of rock mass was closely related to its structural features. The existence of structural plane in rock mass was a controlling factor which influences the stability of surrounding rock of tunnel. It can reduce the whole strength of rock mass and increase the deformation and rheology of the rock mass. It lead to the heterogeneity and discontinuity of rock mass. Rock integrity coefficient can reflect the integrity of rock mass, and its classification level was shown in Table 2.

Table 2 Grading criteria of evaluation indexes and expert scores for fatalness assessment of water inrush in karst tunnels
Full size table

The water flow of attribute recognition analysis is shown in Fig. 1.

Fig. 1
figure 1

Water flow of attribute recognition analysis

Full size image

Rock strength, usually with the strength of the rock was lower, the stability of the surrounding rock was poorer, easier to collapse, and the scale of collapses was larger.

Rock stratum was one of important factors which influence the stability of surrounding rock and groundwater flow. Groundwater recharge, runoff and discharge, seepage conditions and karst development degree was also subject to terrane influence, the stability of surrounding rock decreases with increase of the dip angle of strata, for supporting the supporting strength of the structure requirements higher.

The geological structure of tunnel was a monocline strata structure. The strata occurrence was between 294° \(\angle\) 43° and 290° \(\angle\) 25°. According to the survey of ground surface, the joints of rock mass mainly were L1, L2 and L3 three groups. L1 joint: occurrence was 85° \(\angle\) 68°. The density was generally 2–3 m, a micro crack opening shape, width was 2–4 mm, full of calcite, horizontal extension length of 1–3 m. The vertical cutting depth was 0.3–0.8 m. L3 joint: occurrence was 327° \(\angle\) 42°, density of 1–3 m, with micro open type, fracture width 1–3 mm, no filler, horizontal extension length of 1–2 m, the vertical shear deep 0.4–0.6 m.

The main factors of underground water which cause collapse were groundwater enrichment section, the softening of groundwater, erosion, immersion and so on. It can lead to lower rock strength. The higher the ground water level, the greater the water pressure, the higher risk of collapses. According to the research and statistical results. The bottom plates of the tunnel and underground water level elevation difference h were selected as the water inrush risk evaluation index. The underground water level was divided into four level: h < 10 m, 10 m ≤ h < 30 m, 30 m ≤ h ≤ 60 m, h > 60 m grade.

There was no flowing water in the tunnel area. The water in the tunnel was mainly rainfall. The catchment area of the slope near the tunnel was small. Surface runoff dissipated quickly during rainfall. Combined with the analysis of the hydrogeological boundary conditions, the groundwater in the tunnel was mainly caused by rainfall infiltration, and the yield of water was not large.

Combined with existing research results, seven main influence factors of the adverse geological, buried depth of tunnel, construction measures, rock mass integrity degree, rock strength, rock attitude and underground water level were selected as the evaluation indicators. Water inrush risk was classified as grade I (high risk), grade II (secondary risk), grade III (low risk), and grade IV (micro risk or no risk) four grades. Comprehensive Qiyueshan Tunnel actual hydrogeology condition, the evaluation index of quantitative grading standard was given. Each level were graded by experts. The higher the score, the greater the risk of water inrush, shown in Table 2.

According to the surrounding rock level, the tunnel was divided into seven different mileage segment which are shown in Table 3. Every 10 m an evaluation object. Each segment used the same the evaluation index.

Table 3 The length, number of measuring point and mileage number
Full size table

Taking a tunnel as an example, the tunnel was divided into seven sections. Each of which was provided with a measuring point. Each measurement point was assessed by the above 6 indicators. The measured data and grading standards of each segment were shown in Table 4.

Table 4 The different range of the measured data and the classification standard
Full size table

The decision matrix was established, and the above two forms were combined to carry out non dimensional treatment. According to the characteristics of the evaluation index, the judgment matrix was formed by MATLAB.\(X\,=\,\left| {\begin{array}{*{20}c} {} & {I_{1} } & {I_{2} } & {I_{3} } & {I_{4} } & {I_{5} } & {I_{6} } & {I_{7} } \\ {M_{1} } & {83} & {10} & {85} & {0.35} & {28.9} & 8 & {30} \\ {M_{2} } & {70} & {15} & {85} & {0.48} & {32} & 8 & {30} \\ {M_{3} } & {54} & {30} & {78} & {0.73} & {26.8} & {10} & {20} \\ {M_{4} } & {62} & {45} & {75} & {0.67} & {34.7} & {16} & {22} \\ {M_{5} } & {76} & {55} & {85} & {0.62} & {47.8} & {25} & {22} \\ {M_{6} } & {88} & {30} & {75} & {0.42} & {45} & {25} & {30} \\ {M_{7} } & {79} & {15} & {85} & {0.38} & {42.5} & {25} & {30} \\ {\text{I}} & {60} & {10} & {70} & {0.15} & {20} & {20} & {10} \\ {\text{II}} & {70} & {25} & {80} & {0.35} & {30} & {30} & {20} \\ {\text{III}} & {85} & {50} & {90} & {0.55} & {40} & {40} & {30} \\ {\text{IV}} & {100} & {80} & {100} & {0.8} & {50} & {45} & {40} \\ \end{array} } \right| .\)

The decision matrix was constructed,

$$X = \left| {\begin{array}{*{20}l} {} \hfill & {I_{1} } \hfill & {I_{2} } \hfill & {I_{3} } \hfill & {I_{4} } \hfill & {I_{5} } \hfill & {I_{6} } \hfill & {I_{7} } \hfill \\ {M_{1} } \hfill & {0.8300} \hfill & {0.1250} \hfill & {0.8500} \hfill & {0.4375} \hfill & {0.5780} \hfill & {0.1778} \hfill & {1.0000} \hfill \\ {M_{2} } \hfill & {0.7000} \hfill & {0.1875} \hfill & {0.8500} \hfill & {0.6000} \hfill & {0.6400} \hfill & {0.1778} \hfill & {1.0000} \hfill \\ {M_{3} } \hfill & {0.5400} \hfill & {0.3700} \hfill & {0.7500} \hfill & {0.9125} \hfill & {0.5220} \hfill & {0.2222} \hfill & {0.6667} \hfill \\ {M_{4} } \hfill & {0.6200} \hfill & {0.5625} \hfill & {0.7500} \hfill & {0.8375} \hfill & {0.6832} \hfill & {0.3556} \hfill & {0.7333} \hfill \\ {M_{5} } \hfill & {0.7600} \hfill & {0.6875} \hfill & {0.8500} \hfill & {0.7750} \hfill & {0.9450} \hfill & {0.6667} \hfill & {0.7333} \hfill \\ {M_{6} } \hfill & {0.8800} \hfill & {0.3750} \hfill & {0.7500} \hfill & {0.5250} \hfill & {0.9000} \hfill & {0.5556} \hfill & {1.0000} \hfill \\ {M_{7} } \hfill & {0.7900} \hfill & {0.1875} \hfill & {0.8500} \hfill & {0.4750} \hfill & {0.8500} \hfill & {0.5556} \hfill & {1.0000} \hfill \\ {\text{I}} \hfill & {0.6000} \hfill & {0.1250} \hfill & {0.7000} \hfill & {0.1875} \hfill & {0.4000} \hfill & {0.4444} \hfill & {0.2500} \hfill \\ {\text{II}} \hfill & {0.7000} \hfill & {0.3125} \hfill & {0.8000} \hfill & {0.4375} \hfill & {0.6000} \hfill & {0.6667} \hfill & {0.5000} \hfill \\ {\text{III}} \hfill & {0.8500} \hfill & {0.6250} \hfill & {0.9000} \hfill & {0.6875} \hfill & {0.8000} \hfill & {0.8889} \hfill & {0.7500} \hfill \\ {\text{IV}} \hfill & {1.0000} \hfill & {1.0000} \hfill & {1.0000} \hfill & {1.0000} \hfill & {1.0000} \hfill & {1.0000} \hfill & {1.0000} \hfill \\ \end{array} } \right|$$

The coefficients of entropy weight method were calculated

$$\xi_{ij} = \left| {\begin{array}{*{20}l} {} \hfill & {I_{1} } \hfill & {I_{2} } \hfill & {I_{3} } \hfill & {I_{4} } \hfill & {I_{5} } \hfill & {I_{6} } \hfill & {I_{7} } \hfill \\ {M_{1} } \hfill & {0.7202} \hfill & {0.3333} \hfill & {0.7447} \hfill & {0.4375} \hfill & {0.5090} \hfill & {0.3473} \hfill & {0.6364} \hfill \\ {M_{2} } \hfill & {0.5932} \hfill & {0.3500} \hfill & {0.7447} \hfill & {0.5224} \hfill & {0.5486} \hfill & {0.3473} \hfill & {0.6364} \hfill \\ {M_{3} } \hfill & {0.4875} \hfill & {0.4118} \hfill & {0.6364} \hfill & {0.8333} \hfill & {0.4779} \hfill & {0.3600} \hfill & {0.4667} \hfill \\ {M_{4} } \hfill & {0.5352} \hfill & {0.5000} \hfill & {0.6364} \hfill & {0.7292} \hfill & {0.5800} \hfill & {0.4044} \hfill & {0.4930} \hfill \\ {M_{5} } \hfill & {0.6458} \hfill & {0.5833} \hfill & {0.7447} \hfill & {0.6604} \hfill & {0.8883} \hfill & {0.5676} \hfill & {0.4930} \hfill \\ {M_{6} } \hfill & {0.7848} \hfill & {0.4118} \hfill & {0.6364} \hfill & {0.4795} \hfill & {0.8140} \hfill & {0.4961} \hfill & {0.6364} \hfill \\ {M_{7} } \hfill & {0.6757} \hfill & {0.3500} \hfill & {0.7447} \hfill & {0.4545} \hfill & {0.7447} \hfill & {0.4961} \hfill & {0.6364} \hfill \\ {\text{I}} \hfill & {0.5224} \hfill & {0.3333} \hfill & {0.5932} \hfill & {0.3500} \hfill & {0.4217} \hfill & {0.4406} \hfill & {0.3684} \hfill \\ {\text{II}} \hfill & {0.5932} \hfill & {0.3889} \hfill & {0.6832} \hfill & {0.4375} \hfill & {0.5224} \hfill & {0.5676} \hfill & {0.4667} \hfill \\ {\text{III}} \hfill & {0.7447} \hfill & {0.5385} \hfill & {0.8140} \hfill & {0.5833} \hfill & {0.6863} \hfill & {0.7975} \hfill & {0.6364} \hfill \\ {\text{IV}} \hfill & {1.0000} \hfill & {1.0000} \hfill & {1.0000} \hfill & {1.0000} \hfill & {1.0000} \hfill & {1.0000} \hfill & {1.0000} \hfill \\ \end{array} } \right|$$

A normalized judgment matrix was established. The entropy was calculated. The entropy and entropy weight were obtained as follows:

$$H = 0.9883, \, 0.9850, \, 0.9732, \, 0.9836, \, 0.9846, \, 0.9811, \, 0.9809.$$

3.2 The Calculation of the Weight of Each Evaluation Index

The weights of each evaluation index were calculated by Eq. (14).

The greater of the value of the evaluation indicators, the tunnel collapse is more likely to occur. Therefore, when the judgment matrix R was normalized, the b ij was calculated by the Eqs. (8). In the above equation, i = 1,2, …, 8, j = 1,2, …, 340.

The weight values of the evaluation index were calculate by the MATLAB software.

$$\omega = 0.0949, \, 0.1219, \, 0.2174, \, 0.1333, \, 0.1246, \, 0.1532, \, 0.1547.$$

3.2.1 Calculation of the Scheme Correlation Degree

The correlation degree of each scheme was obtained by Eq. (9).

$$E = \, 0.5443, \, 0.5505, \, 0.5328, \, 0.5578, \, 0.6562, \, 0.6028, \, 0.5965, \, 0.4429, \, 0.5354, \, 0.6971, \, 1.0000.$$

3.2.2 The Correction Factor of the Rainfall

The influence of rainfall on the stability of surrounding rock mass was mainly affected by precipitation, rainfall intensity, etc. Long and small rain could improve the amount of groundwater infiltration effectively. It could enhance the rock instability and weaken the strength of surrounding rock. At last, the bearing capacity of the surrounding rock was reduced and the collapse was caused. Short time and heavy rain increased the load of surrounding rock quickly, which was one of the most important inducing factors of the collapse.

Based on a large number of case statistics, it can be see that the risk coefficient of precipitation was 0.8 to 1.0 by the experts according to the maximum and total rainfall (shown in Table 5).

Table 5 Danger coefficient and grade of the different rainfall conditions
Full size table

At present, the rainfall of the tunnel was as follows: the rainfall was less than 2 mm/h with the total time of 3 h. By expert evaluation, the risk coefficient was 0.98. And then the risk grade of the tunnel collapse was modified by the risk coefficient. The corrected results were shown as follows (shown on Fig. 2)

$$E = \, 0.5334, \, 0.5395, \, 0.5221, \, 0.5466, \, 0.6431, \, 0.5907, \, 0.5846, \, 0.4430, \, 0.5367, \, 0.6937, \, 1.0000.$$
Fig. 2
figure 2

The risk assessment results of tunnel collapse

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Based on the assessment results, it is observed that the risk grades of tunnel collapse in every segment were: II, III, II, III, III, III, III.

3.2.3 Excavation Verification

The purpose of risk analysis and assessment was to develop risk response measures and strengthen the monitoring, thus to avoid or reduce the loss of risk. The risk of loss was avoided or reduced. By calculating the risk level of each segment, first and third of the collapse risk level of the segment was moderate to disaster. The tunnel entrance was usually reinforced advanced. After assessment, it was found that the third miles segment was a dangerous zone. The tunnel collapse in the section of the YK52+965 was shown in Fig. 3. The excavation results showed that it was consistent with the risk assessment. The collapse risk level was II.

Fig. 3
figure 3

Verification by excavation results

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Facts proved that the results of risk assessment were in good agreement with the field excavation. Therefore, the rationality and feasibility of the risk assessment model was verified.

4 Conclusions

  1. 1.

    Grey relational analysis was used to assess the risk level of tunnel collapse based on the grey theory and entropy weight theory was used to analyze the weight of the evaluation index. Thus, a comprehensive risk evaluation system of tunnel collapse was established. The evaluation result was objective and impartial, and it was more consistent with the reality. It provided an effective method and means for risk assessment of tunnel collapse.

  2. 2.

    Engineering application, comprehensive tunnel site hydrogeological conditions and a large number of collapse case statistics, select the unfavorable geology, tunnel buried depth, groundwater, rock mass integrity degree, rock strength, and the dip angle of strata, construction measures seven gushing main influence factors as evaluation index. The tunnel was divided by seven segment mileage. In all, 70 evaluation objects were selected, and the information of each evaluation object was quantified and graded. Then the entropy weight judgment matrix was constructed, and the weight values were calculated. Thus, the correlation degrees of the risk factors of tunnel collapse were obtained. At last, risk assessment of tunnel collapse was carried out, which proved that the risk assessment model for tunnel collapse was scientific and reasonable.

  3. 3.

    According to the actual rainfall conditions during the tunnel construction, the risk coefficients were put forward by experts. Which was used to modify the risk assessment results of tunnel collapse.

  4. 4.

    According to the results of risk assessment, real time risk monitoring of the tunnel was carried out in the process of tunnel construction. According to the feedback results, the corresponding risk management measures were put forward, which verified the rationality of the risk assessment plan.