## 1 Introduction  
The extraction of coal mines is known to precipitate surface subsidence, which can inflict substantial damage on surface infrastructures and aquatic systems [1; 2; 3]. Reports indicate that approximately 14 billion tons of coal lie beneath structures such as buildings, railways, and water bodies within China [4]. Traditional mining methods exacerbate ground subsidence, posing a significant threat to the structural integrity of surface constructions and railways [5]. Furthermore, during the process of longwall mining, it is common for horizontal separation to manifest between the key strata and the underlying rock layers of the working face [6]. This separation often leads to the accumulation of substantial quantities of groundwater or gas, adversely impacting mining safety [7]. In addressing these pervasive challenges, researchers have explored and refined various filling techniques. Notably, the method of grouting into overburden separation has emerged as a favored solution due to its simplicity, cost-effectiveness, and minimal interference with ongoing mining operations [8]. Grouting into overburden separation represents an innovative technique for controlling surface subsidence. The fundamental principle involves pumping a slurry, prepared at the surface, into the overburden separation via a grouting borehole. This filling slurry then supports the key strata, effectively mitigating the transmission of overburden movement to the surface. [PERSON] et al. [9] introduced a method for identifying the location of overburden separation and calculating the aperture size needed for effective separation. The practicality of these methods was substantiated through the use of a multi-point extensometer. Furthermore, [PERSON] et al. [10] developed the concept of isolated overburden grout injection, which constructs a supportive structure consisting of a sufficiently compacted area, an isolated coal pillar, and key strata. This structure is designed to significantly diminish surface subsidence. [PERSON] et al. [11] developed a method to calculate the optimal injection-extraction ratio, taking into account the characteristics of the main injection horizon and the geological features of the surrounding strata. This method aids in refining the design of the injection process. [PERSON] et al. [12] conducted a case study on the 7221 working face in Huaibei, where isolated overburden grout injection was utilized. They classified and analyzed the influence of various grouting parameters on controlling surface subsidence. Furthermore, [PERSON] et al. [13] undertook experimental research on the compression properties of slurry used for grouting into overburden separation and introduced a formula to determine the maximum permissible use of fly ash in the grouting mixture. As a technique to stabilize key strata, the structural integrity and movement of these strata are critical in influencing surface subsidence. To address the deflection of key strata, three primary mechanical models are considered: a cantilever or fixed support [14; 15], an elastic thin plate [16; 17; 18], or an elastic foundation beam [19; 20]. Of these, the elastic foundation beam model, capable of simulating complex boundary conditions, is most frequently employed to analyze the deflection of rock layers under filling conditions [21]. For predicting subsidence, the probability integration method is commonly used [22]. However, traditional applications of this method are limited to mining rectangular working faces with uniform mining heights. By incorporating the concept of equivalent mining height, it becomes possible to apply the probability integration method to predict surface subsidence in solid filling conditions within the goat, adapting it to varying working face configurations [23]. Numerous engineering practices have demonstrated the effective control of surface subsidence through grouting into overburden separation projects, and the subsidence control of the key strata is critical to the effect of the grouting.