# An Experimental Study of Industrial Site and Shaft Pillar Mining at Jinggezhuang Coal Mine
## 2 Probability Integral Method Based on Stochastic Model  
The PIM (also referred to as the geometry integral method) is a mine prediction function based on the stochastic medium model [18]. It was first used in China by [PERSON] and [PERSON] [1] in the 1960s. This method requires only eight parameters based on site surveys and it provides reliable deformation results. In addition, the method is widely used by engineers and technical personnel. The method is mostly used in China and it is a government-recommended method for mine subsidence and damage prediction [19]. In this method, it is assumed that a rock is formed with a small stochastic element, as shown in Figure 2, and that deformation (or subsidence) can be calculated based on the deformation of small elements. It is believed that the mining surface subsidence is caused by the relative movement of the particle model as shown in Figure 2. The particles completely lose contact with each other and can move relatively. When the small particles below are removed, a small cell of the adjacent cell of the upper layer will fall into the removed cell grid due to gravity. According to this stochastic process, the probability curve of surface movement \(P(x,z)\) similar to the bell function can be derived. To obtain the surface movement curve after a large number of unit bodies have been removed, only use integration with \(P(x,z)\). The subsidence of a small-element mining can be calculated using Equation (1) [1]. The deformation for underground mining can be obtained using Equations (2)-(6) [20]. Figure 1: The location of JGZ and brief map of its industrial sqare. \[W_{e}(x)=\frac{1}{r}e^{-\pi\frac{x^{2}}{r^{2}}} \tag{1}\]  
\[W(x,y)=\sum\limits_{j=1}^{n}\iint\limits_{D_{j}}W_{0}W_{e}(x,y)dsdt=\sum\limits _{j=1}^{n}\iint\limits_{D_{j}}\frac{W_{0}}{r_{j}^{2}}e^{-\pi\frac{(x-s)^{2}+(y- t)^{2}}{r_{j}^{2}}}dsdt \tag{2}\]  
\[ \begin{split} i(x,y,\varphi)&=\frac{\partial W(x,y) }{\partial x}\cos\varphi+\frac{\partial W(x,y)}{\partial y}\sin\varphi\\ &=\sum\limits_{j=1}^{n}\iint\limits_{D_{j}}\frac{-2\pi W_{0}}{r_ {j}^{4}}\{[(x-s)\cos\varphi+(y-t)\sin\varphi]e^{-\pi\frac{(x-s)^{2}+(y-t)^{2}} {r_{j}^{2}}}\}dsdt\end{split}  \tag{3}\]  
\[ \begin{split} K(x,y,\varphi)&=\frac{\partial i(x,y, \varphi)}{\partial x}\cos\varphi+\frac{\partial i(x,y,\varphi)}{\partial y} \sin\varphi\\ &=\sum\limits_{j=1}^{n}\iint\limits_{D_{j}}\left[\frac{-2\pi W_{0 }}{r_{j}^{4}}\{1-\frac{2\pi}{r_{j}^{2}}[(x-s)\cos\varphi\right.\\ &\left.+(y-t)\sin\varphi]^{2}\}e^{-\pi\frac{(x-s)^{2}+(y-t)^{2}} {r_{j}^{2}}}\right]dsdt\end{split}  \tag{4}\]  
\[ \begin{split} U(x,y,\varphi)&=\sum\limits_{j=1}^{n} \iint\limits_{D_{j}}\frac{-2\pi bW_{0}}{r_{j}^{3}}\{[(x-s)\cos\varphi\\ &\left.+(y-t)\sin\varphi]e^{-\pi\frac{(x-s)^{2}+(y-t)^{2}}{r_{j} ^{2}}}\}dsdt\end{split}  \tag{5}\]  
Figure 2: Theoretical model for calculating mining subsidence using probability integral method. \[ \begin{split}&\varepsilon(x,y,\varphi)=\sum_{j=1}^{n}\iint\limits_{D_{j}} \left[\frac{-2\pi bW_{0}}{r_{j}^{3}}\{1-\frac{2\pi}{r_{j}^{2}}[(x-s)\cos\varphi \right.\\ &\left.+(y-t)\sin\varphi]^{2}\right\}e^{-\frac{\pi(x-s)^{2}+(y-t) ^{2}}{r_{j}^{2}}}\right]dsdt\end{split}  \tag{6}\]  
where (\(x\),\(y\)) are the coordinates of a surface point; \(\varphi\) is the direction angle of surface movement; \(W_{0}\) is the maximum ground subsidence, provided by \(W_{0}=m\alpha cosa;W_{e}(x,y)\) is the subsidence of a small element mining; \(r\) is the major influence radius, which is provided by \(r=H/tan\beta\); \(H\) is the mining depth; \(q\) is the subsidence factor; \(b\) is the displacement factor; \(m\) is the mining thickness; \(tan\beta\) is the tangent of the major influence angle; \(D\) is the calculated mining area, which is obtained by subtracting the deviation of an inflection point (\(S\)) from the actual mining area; \(n\) is the number of mining panels; and \(\alpha\) is the dip angle of the coal seam.