## 1 Introduction  
With the rapid development of China's economy, the demand for transportation has increased dramatically. Consequently, infrastructure construction represented by expressways and high-speed railways has developed rapidly, and the proportion of tunnels in such projects has become increasingly significant. The southwest region, as an important coal resource gathering area and major production region in southern China, has many goafs formed by long-term mining activities. During the construction of the tunnel, it is easy to be affected by various factors around the tunnel, which will cause the deformation and displacement of the tunnel [1; 2; 3; 4]. The impact of this disturbance is more serious when crossing the goaf. During the process of tunnels crossing through coal seam goafs, disturbances to the strata will occur, disrupting the original equilibrium state of the goaf, leading to activation and collapse of the goaf rock layers. This poses threats to tunnel construction and operation, resulting in uneven settlement of the tunnel, significant deformation of the surrounding rock, intrusion of lining structures beyond the tunnel's architectural limits, and cracking of the lining structure [5; 6; 7; 8; 9; 10; 11]. For example, the Yudushan Tunnel of the Yanqing-Chongli Expressway, a major traffic project for the Beijing Winter Olympics, encounters a goaf at its entrance. The goaf lies above the tunnel along the slope body, and tunnel excavation disrupts the mechanical balance of the goaf, potentially causing large-scale landslides that directly endanger the tunnel itself and the construction and operational safety of the bridge outside the tunnel entrance [12]. Similarly, during the construction of the Heijadi Tunnel's exit section on the Shanghai-Kumming Railway Passenger Line, the influence of the goaf led to deformation and cracking of the secondary lining and the formation of long, thorough surface cracks, severely impacting the normal construction and operation of the tunnel [13]. The Taoping Tunnel has experienced continuous issues such as lining cracking and bulging due to goafs since its construction. Over a period of more than 50 consecutive days, the maximum longitudinal crack at the vault extended up to 8.1 m, with a maximum width increase of 32 mm and multiple instances of lining detachment and spalling, severely threatening traffic safety [14]. In order to reduce the influence of the goaf on the stability of the surrounding rock of the tunnel, one of the methods of minimizing the deformation of overlying workings is filling the workings with rock backfill, which has good compressibility [15; 16]. Another method to ensure the stable excavation and operation of the tunnel is to carry out advanced reinforcement during tunnel excavation. This scheme usually uses bolt and pipe shed combined with corresponding position grouting to achieve the effect of reinforcement of the corresponding position of the tunnel [17; 18; 19; 20; 21; 22], and some supporting devices are also used to support the tunnel. In order to effectively guide the reinforcement of the goaf, it is necessary to design a targeted reinforcement scheme according to the location and characteristics of different goafs, which will save money and construction time. The above method requires the influence law of different goafs on the surrounding rock of the tunnel as the premise guidance. Therefore, given the high-risk construction and operational safety challenges posed by tunnels near goafs, it is of great significance to explore the deformation characteristics of the surrounding rock and the stress conditions of the initial support in tunnels undercrossing multi-situational goafs and to analyze the factors affecting the stability of these tunnels. In recent years, many scholars have conducted extensive research on the impact of goafs on the stability of tunnel surrounding rocks and initial support.