## 3 Validation
### Case Validation  
In this paper, based on the above model equations and numerical calculation methods, the one-dimensional, two-dimensional, and three-dimensional calculation programs are developed. The reliability of the developed program has been examined and verified by comparing the simulation results with the field production data; for example, by comparing with the Barnett Basin gas production data. The detailed validation can be found in previous research [13]. Here, we give an example to further validate the reliability of the present computation program. The detailed reservoir parameters are listed in Table 1. The main reservoir parameters include the reservoir length, width, thickness, averaged matrix porosity, fracture porosity, initial gas pressure, and so on. Furthermore, the adsorption and desorption are considered in the present simulation. The adsorption and desorption parameters are listed in Table 1. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline Reservoir half length (m) & 550 & Initial gas pressure (MPa) & 20.34 \\ Reservoir half width (m) & 145 & Well pressure (MPa) & 3.69 \\ Reservoir thickness (m) & 90 & Langmuir pressure (MPa) & 4.48 \\ Artificial fracture half length (m) & 70 & Langmuir volume & 7.2 \\ Artificial fracture width (m) & 0.003 & Matrix porosity (inner and outer zones) & 0.04 \\ Artificial fracture thickness (m) & 90 & Fracture porosity (inner and outer zones) & 0.04 \\ Artificial fracture distance (m) & 30.5 & Artificial fracture porosity & 0.5 \\ Artificial fracture number & 28 & Permeability in outer zone (mD) & \(4\times 10^{-5}\) \\ Well half length (m) & 426.7 & Permeability in inner zone (mD) & \(1\times 10^{-4}\) \\ Reservoir temperature (K) & 352 & Permeability in artificial fracture (mD) & 1800 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Barnett shale reservoir and well parameters. Figure 3: Schematic diagram of traditional trilinear seepage model. It is important to conduct the grid independence test. In order to accurately calculate the flow field information, the grid near the artificial fracture should be fine, and the grid in the area far away from the artificial fracture can be sparse, as shown in Figure 4. To ensure the accurate calculation of the gas production rate, it should be ensured that the first grid point near the well should fall in the linear area of pressure distribution; that is, it must be ensured that \(\mathrm{d}p/\mathrm{d}y\) does not depend on the grid selected for calculation. The grid distribution is shown in Figure 4. The \(X\) direction grid dependency test shows that the calculation results are consistent between \(N_{x}\times N_{y}\) = 581 \(\times\) 41 and \(N_{x}\times N_{y}\) = 871 \(\times\) 41. There is a slight difference (Figure 5) between \(N_{x}\times N_{y}\) = 581 \(\times\) 41 and \(N_{x}\times N_{y}\) = 291 \(\times\) 41. Therefore, to ensure that the calculation results do not depend on the number of grids, the number of grids used in the \(X\) direction is \(Nx\) = 581. Similarly, the grid dependency test is conducted in the \(Y\) direction, and it is found that the calculation results are consistent between \(N_{x}\times N_{y}\) = 581 \(\times\) 41 and \(N_{x}\times N_{y}\) = 581 \(\times\) 61. There is a slight difference (Figure 5) between \(N_{x}\times N_{y}\) = 581 \(\times\) 41 and \(N_{x}\times N_{y}\) = 581 \(\times\) 21. Therefore, we finally select \(Nx\times Ny\) = 581 \(\times\) 41 as the computational grid. The minimum grid spacing corresponding to the \(X\) direction is \(dx_{\mathrm{min}}\) = 0.003 m, and the minimum grid spacing corresponding to the \(Y\) direction is \(dy_{\mathrm{min}}\) = 0.5 m. In fact, we did not solve the region of the artificial fracture. The vicinity of the artificial fracture was defined as the boundary with constant well pressure. As a result, there is not any grid in the vicinity of the artificial fracture. For the selection of time step, in this paper, the implicit scheme is used to solve the problem.