## 4 Results and Discussion
### Gas Pressure Field and Production Rate around Single and Multiple Vertical Wells
The production curves indicate that the gas production rate is in the range of 80-200 m\({}^{3}\)/d for case 1. However, the gas production rate is only in the range of 6-12 m\({}^{3}\)/d for case 2. These results show that the fracture propped by the large-sized proppant can remarkably increase the gas production rate by enhancing the reservoir permeability. \begin{table}
\begin{tabular}{c c c c} \hline
**Case 1: Small-Sized Proppant** & **Value** & **Case 2: Large-Sized Proppant** & **Value** \\ \hline Size \(d_{e}\) (mm) & 0.2 & Size \(D_{e}\) (mm) & 1 \\ Fracture permeability \(k\) (mD) & 8.53 \(\times\) 10\({}^{4}\) & Fracture permeability \(k\) (mD) & 2.34 \(\times\) 10\({}^{6}\) \\ Reservoir permeability \(k_{av}\) (mD) & 0.001 & Reservoir permeability \(k_{av}\) (mD) & 0.02743 \\ \hline \end{tabular} Where \(d_{e}=2r_{e}\).  
\end{table} 
Table 6: The proppant size and the corresponding permeability in two cases. Figure 15: Result comparison of two kinds of reservoirs with different permeability. (**a**) The average gas pressure variation in the reservoirs. Case 1 represents the low-permeability reservoir, where the fractures are filled by the small-size proppant pack. Case 2 represents the high-permeability reservoir, where the fractures are filled by the large-size discrete pillar. (**b**) The gas production rate evolution in the reservoirs. Figure 16: Comparison of the transient gas pressure fields when adopting single vertical extraction wells. (**a**) The left part is obtained based on the continuous model of the proppant pack. (**b**) The right part is obtained based on the discontinuous model of the proppant pillar. Compared to the single well, the double wells plan gives a faster pressure drop and higher production rate. Figure 18 clearly shows the variation in the reservoir pressure during the first 3000 days. Compared to case 1 of the continuous proppant pack model, case 2 provides a faster pressure drop in the reservoir. The change in the pressure field corresponds to the gas production rate curves. In conclusion, the large-sized proppant plan can provide a more efficient drainage rate. For the reservoir with three wells, the simulation results are exhibited in Figures 19 and 20. Case 1 represents the reservoir containing small-size proppant and case 2 represents the reservoir containing large-size proppant. The results show that during the first 3000 days, the average reservoir pressure decreases from 10 MPa to 9.3 MPa for case 1, where the decrease is only 7%. Meanwhile, the average reservoir pressure decreases from 10 MPa to 4.4 MPa for case 2, where the decrease is up to 56%. The production curves indicate that the gas production rate is in the range of 100-300 m\({}^{3}\)/d for case 1. However, the gas production rate is only in the range of 8-16 m\({}^{3}\)/d for case 2. These results further confirm that the fracture propped by the large-size proppant can remarkably increase the gas production rate by enhancing the reservoir permeability. Figure 17: Result comparison of two kinds of reservoirs with different permeability when adopting two vertical extraction wells. (**a**) The average gas pressure variation in the reservoirs. (**b**) The gas production rate evolution in the reservoirs. Figure 18: _Cont._  
Compared to the single well and double wells, the three wells plan provides a faster pressure drop and higher production rate. Figure 20 clearly shows the variation in the reservoir pressure during the first 3000 days. Compared to case 1 of the continuous prop  
Figure 19: Result comparison of two kinds of reservoirs with different permeability when adopting three vertical extraction wells. (**a**) The average gas pressure variation in the reservoirs. (**b**) The gas production rate evolution in the reservoirs. Figure 18: Comparison of the transient gas pressure fields when adopting two vertical extraction wells. (**a**) The left part is obtained based on the continuous model of the proppant pack. (**b**) The right part is obtained based on the discontinuous model of the proppant pillar.