## 2 Research Methods  
### Calculation Method for the Void Fraction  
The voids produced by coal mining come from the mined volume. The voids in goaf should be less than the extracted volume of the coal seam, and the bulking volume of the broken direct roof and the bending deformation of the overlying rock prevent the formation of voids. Therefore, the calculation principle of the method was proposed, which was that the volume of voids is equal to the volume of coal seam extraction less the volume of direct roof bulking less the volume of overlying deformation. The calculation model of a void ratio in goaf was established, as shown in Figure 2. In the figure, the upper part of the top plate is composed of a thin plate model, and the lower part is composed of a broken expansion model. The total void ratio of the goaf can be obtained by combining the upper  
Figure 1: Measurement of voids in goaf. and lower models. The voids reduced by the deformation of the overburdened rock in the goaf can be calculated by establishing a thin plate model. The voids formed by the broken roof in the goaf can be calculated by establishing a broken roof expansion model. 1. Volume of the overlying deformation  
Based on the O-ring theory [30; 31; 32], the rock strata above a roof are regarded as thin elliptical boundary-fixed plates under the action of a uniformly distributed load. The settlement amount (\(w_{ki}\)) of any point (\(x\), \(y\)) on a thin plate can be calculated using Formula (1). The coordinate system of the model is shown in Figure 2, and the calculation to determine settlement amount \(w_{ki}\) (\(x\), \(y\)) of any point (\(x\), \(y\)) of any rock strata overlying a stope is shown in Formula (2). In the process of the deformation of the overlying strata on a roof, the void ratio of the two adjacent strata can be obtained, according to the different deformations in each layer, as \(\varphi_{i,\,i+1}\) (see Formula (3)). \[w_{ki}(x,y)=\frac{\left(\rho_{i}S_{i}T_{i}+q_{0i}\right)\cos\alpha\left(\frac {4\alpha^{2}}{l_{x}^{2}}+\frac{4\alpha^{2}}{l_{y}^{2}}-1\right)^{2}}{8D_{i}( \frac{48}{l_{x}^{2}}+\frac{32}{l_{x}^{2}l_{y}^{2}}+\frac{48}{l_{y}^{2}})} \tag{1}\]  
where \(\rho_{i}\) is the rock density of the \(i\)-th layer; \(T_{i}\) is the thickness of the \(i\)-th layer; \(q_{0i}\) is the upper load of the \(i\)-th stratum; \(\alpha\) is the dip angle of the coal seam; \(l_{x}\) is the strike length of the goaf; \(l_{y}\) is the dip width of the goaf; \(D_{i}\) is the bending stiffness of the \(i\)-th stratum; and \(D_{i}=ET^{2}/(12\ (1-v^{2}))\), where \(E\) is the elastic modulus and \(v\) is Poisson's ratio. \[w_{ki}(x,y)=\frac{w_{0i}(1-e^{-\frac{y}{2l_{i}}})(1-e^{-\frac{|y|}{2l_{i}}})}{ 1-e^{-\frac{|y|}{2l_{i}}}} \tag{2}\]  
and  
\[\varphi_{i,i+1}=\frac{\Delta w_{ki}dxdy}{\Delta\sum h_{i}dxdy}=\frac{w_{ki}-w _{ki+1}}{\sum h_{i}-\sum h_{i+1}}, \tag{3}\]  
where \(w_{0i}\) is the subsidence of the \(i\)-th stratum after it moves and stabilizes; \(l_{i}\) is the rock-breaking length of the \(i\)-th stratum; \(l_{y}\) is the length of the working face; and \(h_{i}\) and \(h_{i+1}\) are the thicknesses of layer \(i\) and layer \(i+1\), respectively. 2. Volume of voids after direct roof bulking  
Figure 2: Calculation model of the void ratio in goaf. The void ratio of a broken rock mass below a roof can be expressed by the broken expansion coefficient, as shown in Formula (4):  
\[\varphi=1-\frac{1}{K_{p}}, \tag{4}\]  
where \(Kp\) is the coefficient of the crushing expansion of the broken rock and \(\varphi\) is the void ratio of the broken rock. In the \(x\)-direction, the coefficient of the void's ratio changes in the strike direction of the working face, which can be obtained as follows:  
\[K_{p}=\frac{h_{d}+H-w_{b}|(y=0)}{h_{d}}, \tag{5}\]  
where \(w_{b}\)! (\(y=0\))= (\(H\)-\(h_{d}(K_{pb}-1)\))(1-\(e^{x/2}\)).