## 4 Results and Discussion  
### Gas Pressure Field and Production Rate around a Horizontal Well  
According to the reservoir characteristics, the drilling and exploitation forms widely used at present include the vertical exploitation well and horizontal exploitation well. This paper mainly discusses these two forms. In addition, there will be multi-well production. Different drainage layouts have an important influence on the reservoir pressure field and production rate. This section first discusses the problem of the natural gas production rate under the condition of a single horizontal production well. The main reservoir parameters include the reservoir length, width, thickness, averaged matrix porosity, fracture porosity, initial gas pressure, and so on. Furthermore, the well length and well pressure are another two key parameters that affect the production rate of gas. The above parameters are given in Table 3. Most of these parameters are chosen upon referring to the previous references [13; 30; 31; 32]. The three-dimensional case is conducted based on the above numerical method. After the hydraulic fracturing, the reservoir forms a fracture network, as shown in Figure 8. In the fracture network, proppants exist, and these proppants support the fracture to avoid fracture closure under the action of ground stress. Without losing generality, it is assumed that the average fracture aperture is 1 mm and the fracture porosity is 0.5. For case 1, the fracture is continuously filled by the small-sized proppants, and its permeability can be assessed using [PERSON]'s model [20]. The results show that the fracture permeability is 8.53 \(\times\) 10\({}^{4}\) mD for case 1. For case 2, the fracture is discontinuously filled by the large-sized proppants, and its permeability can be assessed using [PERSON]'s model [13]. The results show that the fracture permeability is 2.34 \(\times\) 10\({}^{6}\) mD for case 2. The fracture permeability \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline Reservoir length (m) & 100 & Initial gas pressure (MPa) & 10 \\ Reservoir width (m) & 100 & Well pressure (MPa) & 1 \\ Reservoir thickness (m) & 100 & Langmuir pressure (MPa) & 4.48 \\ Drainage length of well (m) & 50 & Langmuir volume (m\({}^{3}\)/m\({}^{3}\)) & 7.2 \\ Average matrix porosity & 0.01 & Reservoir temperature (K) & 352 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Simulation parameters of reservoir including horizontal well. of case 2 is 27.43 times that of case 1. Assuming that the average permeability of reservoir 1 is 0.001 mD, it can be considered that the average permeability of reservoir 2 is 0.02743 mD, as shown in Table 4.  
Before the discussion, it is vital to assess the grid independency. Four kinds of grids are tested, including the coarse grid, \(N_{x}\times N_{y}\) = 41 \(\times\) 41, medium grid, \(N_{x}\times N_{y}\) = 81 \(\times\) 81, fine grid, \(N_{x}\times N_{y}\) = 121 \(\times\) 121, and the finest grid, \(N_{x}\times N_{y}\) = 161 \(\times\) 161. The grid dependency test shows that the calculation results are consistent between the medium grid and the fine grid. There are only slight differences (Figure 9) between the coarse grid and the medium grid. Therefore, to ensure that the calculation results do not depend on the number of grids, the medium grid is adopted here for the following simulation and discussion. Another important issue is the choice of time step. If the selection is too small, the calculation efficiency will be reduced. If it is too large, the convergence of the calculation results is difficult to ensure. In this paper, we choose two steps to test, including \(dt\) = 0.1 d and \(dt\) = 0.01 d. As shown in Figure 10, the results show that the calculation results given by the two different time steps are consistent, which indicates that the convergence solutions can be given by the two kinds of time steps mentioned above.