## 4 Results and Discussion
### Gas Pressure Field and Production Rate around Single and Multiple Vertical Wells  
For the vertical well mining, this paper mainly considers the single well, double wells, three wells, and four wells. As shown in Figure 14, the reservoir area range considered in the simulation process is \(L_{x}=200\) m, \(L_{y}=200\) m. The distance between the wells is 50 m. Detailed reservoir parameters are listed in Table 5. The reservoir thickness is 10 m. The initial reservoir pressure is 10 MPa, and the well hole pressure is 1 MPa. The desorption and adsorption are also considered where the Langmuir pressure is 4.48 MPa and the Langmuir volume is 7.2. Figure 11: The pressure distribution in the fracturing reservoirs containing a horizontal well. (**a**) Case 1: lower-permeability reservoir induced by the continuous small-sized proppant packet. (**b**) Case 2: higher-permeability reservoir induced by discontinuous large-sized proppant pillar. The legend range varies from 1 MPa (blue) to 10 MPa (red). Figure 12: The evolution trend of the average pressure in two kinds of propped reservoirs. Case 1 is the lower-permeability reservoir. Case 2 is the higher-permeability reservoir. Similarly to the horizontal well, assuming that the permeability of the reservoir with small-size proppant is 0.001 mD, it can be deduced that the permeability of the reservoir with large-size proppant is 0.027 mD. The detailed permeability parameters are listed in Table 6.  
Figure 14: Layout form of the drainage well. (**a**) Double vertical wells. (**b**) Three vertical wells. (**c**) Four vertical wells. The reservoir region is \(L_{x}\) = 200 m, \(L_{y}\) = 200 m, and the well distance is 50 m.  
Figure 13: The evolution trend of the gas production rate in two kinds of propped reservoir. (**a**) Production rate variation during the first 30 days. (**b**) Production rate variation from the 30 th day to 300 th day. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline Reservoir length (m) & 200 & Initial gas pressure (MPa) & 10 \\ Reservoir width (m) & 200 & Well pressure (MPa) & 1 \\ Reservoir thickness (m) & 10 & Langmuir pressure (MPa) & 4.48 \\ Drainage length of well (m) & 10 & Langmuir volume (m\({}^{3}\)/m\({}^{3}\)) & 7.2 \\ Average matrix porosity & 0.01 & Well distance (m) & 50 \\ Average fracture porosity & 0.034 & Reservoir temperature (K) & 352 \\ of reservoir & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Simulation parameters of reservoir including vertical well. The results are shown in Figure 15. Case 1 represents the reservoir with small-sized proppant and case 2 represents the reservoir with large-sized proppant. It is seen that the average reservoir pressure decreases from 10 MPa to 9.8 MPa for case 1, where the decrease is only 2%. The average reservoir pressure decreases from 10 MPa to 6.7 MPa for case 2, where the decrease is up to 33%. The production curves indicate that the gas production rate is in the range of 50-100 m\({}^{3}\)/d for case 1. However, the gas production rate is only in the range of 3-6 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. Figure 16 clearly shows the variation in the reservoir pressure during the first 3000 days. Compared to the case 1, the continuous proppant pack model, case 2 provides a faster drop in gas pressure in the reservoir. The change in the pressure field corresponds to the gas production rate curves. For the reservoir with double wells, the simulation results are shown in Figure 17. Case 1 represents the reservoir with small-sized proppant and case 2 represents the reservoir with large-sized proppant. The results show that the average reservoir pressure decreases from 10 MPa to 9.55 MPa for case 1, where the decrease is only 4.5%. The average reservoir pressure decreases from 10 MPa to 5.3 MPa for case 2, where the decrease is up to 47%.