## 1 Introduction  
As the \"14 th Five-Year Plan\" outline was proposed and the \"Belt and Road\" strategy advanced in China, in order to meet the needs of social development and economic growth, the number of water conservancy, transportation, and other tunnel projects in China has been increasing annually. The selection of tunnel location is increasingly constrained by a variety of factors such as geographic location and construction environment [1; 2; 3; 4]. Therefore, tunnel construction under complex geological conditions has received increasing attention. Water-bearing faults are a common adverse geological condition encountered in tunnel construction. Due to the typical presence of fault breccia, fractured rock, mylonite, and fault gouge within these faults, the rock mass is loose and broken, and often exhibits characteristics of high stress and high-water pressure. Combined with improper construction management, the construction process is highly susceptible to sudden water and mud intrush disasters, posing a significant threat to tunnel construction and personnel safety. Consequently, conducting research on sudden water intrush disasters at the tunnel working face in faulted zones is of urgent importance in order to provide theoretical guidance for the construction of tunnel projects in fault zones. The anti-inrush body is the rock mass between the tunnel face and the fault that can block the water flow from the fault into the tunnel. The instability criteria of the protective layer at the tunnel working face and the minimum safe thickness of the anti-inrush body are important aspects in the study of the mechanism of sudden water inrush in tunnels near faults. In recent years, some scholars have conducted beneficial explorations and achieved fruitful results. Based on the total potential energy principle, [PERSON] et al. [5] established a mutation model for the instability of the surrounding rock of a deeply buried tunnel passing through a fault fracture zone, and derived the mechanical criteria for instability. [PERSON] et al. [6] conceptualized the tunnel-karst system, analyzed the mechanism and mode of sudden water inrush in faults, and derived a formula for calculating the minimum safe thickness of the rock mass in the fault section of a karst tunnel, which was verified by engineering examples. Based on the theory of simple silos and the characteristics of the stress distribution of the strata, [PERSON] et al. [7] calculated the deformation characteristics of the tunnel under the action of longitudinal loads in the fractured zone, and analyzed the effects of burial depth, fracture zone width, and inclination on the longitudinal deformation and stress of the tunnel. Based on the theory of silo and the limit equilibrium method, [PERSON] et al. [8] derived a formula for calculating the ground stress received by the anti-burst rock mass, obtained the mechanical criteria for sudden water and mud inrush in the fractured zone of water-bearing faults, and analyzed the effect of the width of the fault fractured zone on the minimum safe thickness of the anti-burst rock mass. [PERSON] et al. [9] adopted the principle of slice method and the rock strength to realize a semi-quantitative analysis method for calculating the minimum safe thickness of rock against water inrush, and studied the semi-analytical solution of the minimum safe thickness of rock against water inrush in filled karst caves. [PERSON] et al. [10] used fracture mechanics and hydraulic theory to analyze the lag effect and expansion effect of sudden water inrush at the tunnel working face, believing that the sudden water inrush at the tunnel working face is caused by the decrease of the critical water pressure for hydraulic fracturing due to the disturbance of excavation. [PERSON] et al. [11] investigated clayy originated rock properties through machine learning. They established a formula for calculating the safe thickness of the rock wall at the tunnel working face based on the critical water pressure.