## 1 Introduction  
Tunnel construction in karst areas [1] could suffer from the risk of geological disasters such as karst water and mud inrush [2] and landslides. Regarding the tunnelling, the specific spatial dimensions of the karst cave, the specific orientation of the karst cave, the presence or absence of a filling medium inside the karst cave, and the pressure of the filling medium inside the karst cave will all have an impact on the ability of the tunnel to pass through the karst area [3], which is fraught with safety hazards. In comparison with the continuous homogeneous single-phase medium stratum, the stratum of a karst region is subject to the combined effect of the gas phase, liquid phase and solid phase, which will easily lead to the expansion and complication of the security risk of tunnel construction [4] if it is constructed blindly or handled inappropriately. Therefore, it is necessary to assess the safety risk of tunnel construction in karst areas and lay a theoretical foundation for preventing and resolving the major risks of tunnel construction in karst areas. In recent years, a number of scholars at home and abroad have established a safety evaluation system and method for tunnel construction in karst areas by applying theoretical methods, such as fuzzy comprehensive evaluation method [5], analytic hierarchy process [6], attribute recognition theoretical model [7] and other theoretical methods, to predict and analyse the possibility of geological hazards occurring during the construction of the tunnels and to judge the safety risk level of karst tunnel construction. However, the fuzzy comprehensive evaluation method [8] and the attribute recognition theoretical [9] model require the high reliability of data, and there are certain subjective considerations. And the hierarchical analysis method requires a large amount of data statistics, the weightsare difficult to determine, and the human subjective factors have a greater impact on the whole comprehensive evaluation process. The analytic hierarchy process [10] requires a large amount of data statistics, the weights are difficult to determine, and human subjective considerations have a significant impact on the overall comprehensive evaluation process. For the purpose of objectively identifying the disaster risk of water and mud intrusion during tunnel construction in karst areas, several scholars at home and abroad have taken elastic-plastic mechanics [11], damage and fracture mechanics [12], fluid mechanics [13,14] and other theoretical bases, and revealed and analysed the disaster mechanism of karst water and mud inrush under the action of high-pressure karst water in tunnel construction through researching and analysing the minimum safe thickness of the rock and soil bodies against breakout between the karst tunnel space and the space containing high-pressure medium. Although the method for calculating the minimum safe thickness [15,16] of the rock and soil masses for preventing the breakout could effectively determine the interaction mechanism between the tunnel and the karst water, for complex geological conditions such as large buried depth and the randomness of geotechnical parameters, the risk level of water surge in deeply buried tunnels in karst regions under complex circumstances could not be determined in a satisfactory way. In view of the large influence of human subjective considerations in the evaluation of the tunnel risk level in the above research status, firstly, based on the upper bound theorem of limit analysis [17] and [PERSON]'s theory, the first damage model for deep-buried tunnel construction should be proposed. And based on the upper bound theorem of limit analysis and energy dissipation theory, the second damage model for deeply buried tunnel construction should be proposed.