## 1 Introduction  
The West to East Gas Pipeline Project acts as a crucial strategic energy transmission conduit, linking natural gas resources in Western China with the consumer markets in the East [1]. However, as the pipeline traverses extensive areas, the issue of pipeline damage due to subsidence in the mining areas increases, particularly within the Central China Plain. This region, notable for its extensive history of mining, large scale operations, shallow coal seams, and early adoption of unconventional, disorganized, and intensive mining practices, is highly susceptible to surface subsidence and ground cracks. These disturbances cause significant risks to the safety and stability of the buried pipelines [2, 3]. Consequently,researching the deformation characteristics and mechanical behavior of buried pipelines in mining areas is of paramount theoretical and practical importance. The deformation mechanism of the overlying rock in these mining areas crucially determines the laws of surface subsidence, which in turn precipitate the destruction of buried pipelines. This phenomenon has captured the interest of numerous scholars and has been the subject of extensive physical experimental research across various mining processes and geological conditions [4; 5; 6; 7; 8; 9; 10]. [PERSON] et al. [11] have conducted simulations indicating that surface subsidence was initially minimal prior to the fracturing of the hard and thick key layer. Following the fracturing, there was a dramatic increase in the subsidence of the overburden rock layer, leading to the rapid compaction of the off-layer and triggering disasters such as gas missions, water bursts, and swift ground subsidence. [PERSON] et al. [12] utilized physical experiments to investigate the dynamic boundaries of the rock layers above the coal mines, categorizing the overlying rock into three regions based on the direction of motion vectors and discovering that the movement boundary of the overlying rock forms an S-shaped curve. [PERSON] et al. [13] analyzed the transport dynamics of the key magmatic rock layers, observing that the basal offshoots of these layers undergo transitional phases characterized by a'single valley', progressing to an 'inverted trapezoid', and culminating in a 'double valley' pattern. [PERSON] et al. [14] investigates hard roof failure caused by edge cracks, emphasizing its impact on overburden movement and providing insights for preventing instability-related disasters in mining. [PERSON] et al. [15] examines water jet efficiency in rock breaking, focusing on its role in controlling overburden movement and mitigating gas disasters in mining operations. Many studies have focused on the movement of overlying rock, while others have examined the mechanical responses of pipelines under various geological conditions, such as seismic activity, faulting, and soil deformation. [PERSON] et al. [16] developed a methodology to analyze lateral and longitudinal motion responses of buried pipelines under seismic influences, demonstrating that the maximum downward force on a pipe is only about one-third of that predicted by the traditional bearing capacity equation. [PERSON] et al. [17] explored pipeline-soil interactions during mining activities in two distinct phases: a coordinated deformation phase and a partial suspension stage. They developed a mechanical model using segmented elastic beams for each phase. [PERSON] et al. [18] investigated the interactions between buried pipelines and reverse fault ruptures through numerical simulations, finding that an increase in sediment thickness leads to a reduction in the normalized fault offset at the onset of buckling. [PERSON] et al. [19] conducted small-scale indoor tests and physical experiments to examine the effects of traffic loading on buried pipelines in soft soil, exploring the deformation mechanisms and the applicability of the modified Spangler-Iowa calculation method.