## 1 Introduction
It is generally believed that overburden damage caused by mining is zoned and can be divided into caving zones, fracture zones, and subsidence zones. [PERSON] et al. [9] based their theory on the \"three-zone\" model, subdividing a lower part of the subsidence zone into a separate delamination zone and proposed the \"four-zone\" model. This model served as the basis for the grouting construction design in the case of overburden delamination [10]. In addition, it was used to prevent and control water inrush hazards. According to the \"four-zone\" theory, the subsidence zone is primarily influenced by the movement and deformation of the nearest key layer at the bottom, and the subsidence trajectory follows a paraboloid with an elliptical shape [11]. [PERSON] et al. [12] applied the \"three-zone\" model for a numerical simulation of surface subsidence and flooding. A scheme illustrating the mechanism of water inrush is presented in Figure 1.  
Figure 1: Scheme of water inrush from aquifers in the result of overlying strata failure. [PERSON] and [PERSON] [13] proposed the subsidence theory for the Upper Silesia region conditions. According to [PERSON] and [PERSON] [13], the maximum predicted subsidence is equal to the thickness of excavating seam multiplied by exploitation parameter a (values between 0 and 1). The maximum area of exploitation influences contains the mining area and area in the influence range of exploitation. The depth of seam and the rock mass parameter \(\tan(\beta)\) (where \(\beta\) is the angle of main influences) should be known to compute the range of exploitation. With known values of maximal subsidence and the range of exploitation, one can compute the inclination in the distance from the edge of exploited seam. For standard geological conditions of Upper Silesia Coal Basin, the assumption of \(\tan(\beta)=2\) is usually made [14]. The [PERSON] and [PERSON] [13] subsidence scheme is a classical one. Similar subsidence schemes are used in different coal regions of the world. In particular, they form the basis of current regulatory documents of Ukraine [15]. Based on the aforementioned hypotheses, many modern researchers study the mechanism of water inrush resulting from overburden fracture [16; 17; 18]. The primary research method in these studies is one of numerical simulation or its combination with other approaches. [PERSON] et al. [19] predicted the height of the water-conducting fracture zone using empirical formulas, a BP neural network, and CDEM numerical simulation. The predicted height of the two zones in the overlying strata of the working face was verified by the underground water-conducting fracture zone height observation method and the borehole peeping method. [PERSON] et al. [20] calculated the height of the water-conducting fracture zone in the working face through theoretical analysis, FLAC3D numerical simulation, and field measurements. [PERSON] et al. [21] proposed a method for determining the height of the fracture zone based on thin-plate theory, utilizing theoretical analysis, numerical simulation, and field measurements. A mechanical model of the key layer as a thin plate was established. [PERSON] et al. [22] proposed, applied, and validated a method for predicting the height of the water-flowing fractured zone, explaining the reasons for differences between coalfields in Northwest and East China. Discrete element software (3 DEC) was used to simulate the progressive failure and collapse of the overlying strata. [PERSON] et al. [23] calculated the maximum height of the failure zone by assuming it to be equal to the height of the plastic zone obtained from FLAC3D simulation. Additionally, they analyzed the vertical displacement, vertical stress, and the evolution of the plastic zone in the overburden rock under different working face advancing distances.