# Double Unloading Gas Control Technology for Fracturing Soft Coal Seams in Overlying Key Strata
## 2 Method
### Hydraulic Fracturing Radius Determination  
#### 2.3.1 Determination of Key Strata of Overlying Rock Strata  
1. Determine hard rock stratum according to load  
According to masonry beam theory, the \(n\)th layer of the overlying rock layer bears the load (\(q_{m}\)) of the \(n\)th layer to the \(m\)th layer to the \(m\)th layer (\(q_{m}\))\({}_{n}\):  
\[\left(q_{m}\right)_{n}=\frac{E_{n}h_{n}^{3}\sum_{i=n}^{m}\gamma_{i}h_{i}}{\sum _{i=n}^{m}E_{i}h_{i}^{3}}. \tag{1}\]  
In Formula (1), \(E_{i}\) represents the elastic modulus of the \(i\)th layer of the rock layer, GPa; \(\gamma_{i}\) is the unit weight of the \(i\)th layer, kg/m\({}^{3}\); and \(h_{i}\) is the thickness of layer \(i\), m.  
If layer \(m+1\) is a hard rock layer, it should have the following supporting characteristics:  
\[\left(q_{m}\right)_{n}>\left(q_{m+1}\right)_{n}. \tag{2}\]  
According to Equation (2), the load (\(q_{1}\)) of the first layer itself is 106.3 kPa, while the load (\(q_{2}\))\({}_{1}\) from the first layer to the second layer is 31.7 kPa. Based on Equation (2), it can be inferred that the second layer of overburden rock does not contribute to the loading of the first layer and can be considered as a hard rock stratum. The load (\(q_{2}\)) applied to the second layer itself is 208 kPa, the load (\(q_{3}\))\({}_{2}\) applied to the second layer up to the third layer is 275.1 kPa, the load (\(q_{4}\))\({}_{2}\) applied to the second layer up to the fourth layer is 281.4 kPa, the load (\(q_{5}\))\({}_{2}\) applied to the second layer up to the fifth layer is 314.7 kPa, and the load (\(q_{6}\)) applied to the second layer up to the sixth layer is 297.8 kPa. It can be seen that the sixth layer is a hard rock layer, and the load on the second layer is 314.7 kPa, which controls the upper three layers. Similarly, (\(q_{6}\)) = 188.5 kPa, (\(q_{7}\))\({}_{6}\) = 217.6 kPa, (\(q_{8}\))\({}_{6}\) = 255 kPa, (\(q_{9}\))\({}_{6}\) = 261.4 kPa, (\(q_{10}\))\({}_{6}\) = 296 kPa, (\(q_{11}\))\({}_{6}\) = 316.1 kPa, and (\(q_{12}\))\({}_{6}\) = 290.1 kPa; therefore, the twelfth layer is a hard rock layer, and the load that had been applied to the sixth layer is 316.1 kPa, controlling the upper five layers. Therefore, according to the theory related to the key strata [36], the second, sixth, and twelfth layers of the upper part of the coal seam were determined to be hard rock layers. 2. Calculate the breaking distance of the hard rock stratum \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Lithology** & **Unit Weight** & **Modulus of** & **Poisson’s** & **Compressive** & **Tensile Strength** & **Internal Friction** \\  & **kN/m\({}^{3}\)** & **Elasticity/GPa** & **Ratio/\(\mu\)** & **Strength/MPa** & **MPa** & **Angle**\({}^{\circ}\) \\ \hline Fine sandstone & 26 & 18 & 0.22 & 100 & 10 & 42 \\ Limestone & 26 & 40 & 0.20 & 120 & 18 & 45 \\ Sandy mudstone & 25 & 15 & 0.28 & 40 & 3 & 38 \\ Coal & 14 & 0.3 & 0.34 & 10 & 0.6 & 28 \\ \hline \end{tabular}
\end{table} 
Table 1: Physical and mechanical parameters of rock layers. \begin{table}
\begin{tabular}{c c c c} \hline
**Parameter Name** & **Numerical Value** & **Parameter Name** & **Numerical Value** \\ \hline Initial crack rate & 0.012 & Langmuir pressure constant (MPa) & 2 \\ Initial porosity & 0.049 & Maximum value of adsorbed gas (m\({}^{3}\)/t) & 28.8 \\ Initial permeability (mD) & 0.004 & Molar mass of coal gas (L/mol) & 22.4 \\ Initial gas pressure (MPa) & 1.6 & Density of coal (kg/m\({}^{3}\)) & 1380 \\ Dynamic viscosity of gas (Pa\(\cdot\)s) & \(1.08\times 10^{-5}\) & Negative pressure (kPa) & 21 \\ Elastic modulus of coal matrix (MPa) & 8139 & Poisson ratio (\(\upsilon\)) & 0.34 \\ Elastic modulus of coal (MPa) & 2713 & Single pore diffusion coefficient & \(5.599\times 10^{-12}\) \\ Klinkenberg factor (Pa) & 1.4 \(\times\) 105 & Constant amount of adsorbed gas b (MPa\({}^{-1}\)) & 0.494 \\ \hline \end{tabular}
\end{table} 
Table 2: Coal seam gas related parameters.