## 2 Calculation Model of Drainage Radius of Single-Layer Gas Well
### Method Validation  
The drainage radius of multi-layer commingled gas production wells cannot be directly measured through reservoir well tests, so it is difficult to verify the accuracy of the proposed calculation model through the measured value of drainage radius. Therefore, in order to validate the accuracy of the calculation model of drainage radius, Eclipse numerical simulation software is applied to create a circular and enclosed gas reservoir model with a vertical production well in the center, which is used to simulate the entire production process of a multi-layer commingled gas production well. Specific properties of the gas reservoir model are shown in Table 4. Radial grids are adopted in the gas reservoir model and the grid block sizes in the radial direction, the theta direction and the vertical direction are 10 m, 11.25\({}^{\circ}\) and 1 m, respectively. Figures 6 and 7 show the gas-water relative permeability curve and gas PVT data for different layers in the model, respectively. The gas well is operated under a constant gas rate. As shown in Table 4, drainage radii of different layers in the gas reservoir model are given. Using the above data, reservoir simulation can be conducted to obtain production data (such as gas rate and bottomhole flowing pressure). Figure 8 presents the production data for three layers resulting from the numerical reservoir simulation. The calculated value of the drainage radius can be obtained by using the proposed calculation model and the obtained production data. And then, the derived calculation model can be verified by comparing the calculated values with the given values in the gas reservoir model.  
Figure 5: Flowchart of calculation model of drainage radius.  
Figure 6: Gas-water relative permeability curves.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Reservoir Property** & **Layer 1** & **Layer 2** & **Layer 3** \\ \hline \(K_{i}\) (mD) & 1 & 1.5 & 2 \\ \(h_{j}\) (m) & 6 & 8 & 10 \\ \(\phi_{j}\) (\%) & 7 & 7 & 7 \\ \(S_{\rm gij}\) & 0.8 & 0.8 & 0.8 \\ \(T_{\rm ij}\) (\({}^{\circ}\)C) & 88.66 & 90.68 & 92.91 \\ \(r_{\rm w}\) (m) & & 0.2 & \\ \(q_{\rm sc}\) (m\({}^{3}\)/d) & & 100,000 & \\ \(r_{\rm ej}\) (m) & 300 & 500 & 700 \\ \hline \end{tabular}
\end{table}
Table 4: Properties of gas reservoir model.  
Figure 7: Gas PVT for every layer.  
The aforementioned calculation procedures of drainage radius and the data of bottomhole flowing pressure and gas rate at 300 days and 600 days are applied to calculate drainage radius. Table 5 shows the calculated results for each layer. From Table 5, it can be observed that the drainage radius for layers 1 to 3 in gas reservoir are 295.7 m, 492.3 m and 716.3 m, respectively. The maximum relative error between the calculated values and the actual values of drainage radius from the Eclipse numerical gas reservoir model (Table 4) is 2.33%. These results suggest that the proposed calculation model of drainage radius of multi-layer commingled gas production well is significantly accurate. Also, it enables quick calculation of the drainage radius only using bottomhole flowing pressure and gas rate from two arbitrary time points. In summary, the proposed model is simple, efficient and highly accurate, meeting the engineering calculation requirements of on-site gas fields.