## 2 Calculation Model of Drainage Radius of Single-Layer Gas Well
### Calculation Model of Drainage Radius of Multi-Layered Gas Well
The material balance principle for a closed constant-volume gas reservoir [20; 21; 22] is given by:  
\[\frac{P_{\text{Rj}}}{Z_{\text{Rj}}}=\frac{P_{\text{ij}}}{Z_{\text{ij}}}(1- \frac{C_{\text{pj}}}{G_{\text{ij}}})=\frac{P_{\text{ij}}}{Z_{\text{ij}}}- \frac{C_{\text{pj}}}{2696\times 10^{8}\pi r_{\text{ej}}^{2}h_{\text{j}} \phi_{j}S_{\text{gij}}/T_{\text{ij}}} \tag{12}\]  
where \(P_{\text{ij}}\) is the original reservoir pressure of layer \(j\), MPa; \(Z_{\text{ij}}\) is the deviation factor corresponding to \(P_{\text{ij}}\); \(T_{\text{ij}}\) is the temperature of the reservoir, K; \(G_{\text{pj}}\) is the cumulative gas production of layer \(j\), \(10^{8}\) m\({}^{3}\); \(G_{\text{ij}}\) is the controlled reserves of layer \(j\), \(10^{8}\) m\({}^{3}\); \(S_{\text{gij}}\) is the initial gas saturation of layer \(j\); \(\phi_{j}\) is the porosity of layer \(j\). Substitute Equation (12) into Equation (11):  
\[-\frac{\partial(P_{\text{wfj}}/Z_{\text{wfj}})}{\partial t}=\frac{\mu_{\text{ gwfj}}C_{\text{gwfj}}\overline{\eta}_{\text{scj}}}{\mu_{\text{gRj}}C_{\text{gRj}}*2696 \pi r_{\text{ej}}^{2}h_{\text{j}}\phi_{j}S_{\text{gij}}/T_{\text{ij}}}+A_{j} \mu_{\text{gwfj}}C_{\text{gwfj}}\frac{\partial(q_{\text{scj}})}{\partial t} \tag{13}\]  
where \(\overline{\eta}_{\text{scj}}\) is the average daily gas production of layer \(j\), m\({}^{3}\)/d. Assuming \(\alpha_{j}=\frac{\partial(P_{\text{wfj}}/Z_{\text{wfj}})}{\partial t}\) and \(\beta_{j}=\frac{\partial(q_{\text{scj}})}{\partial t}\), Equation (13) can be simplified as:  
\[-\alpha_{j}=\frac{\mu_{\text{gwfj}}C_{\text{gwfj}}\overline{\eta}_{\text{scj} }}{\mu_{\text{gRj}}C_{\text{gRj}}*2696\pi r_{\text{ej}}^{2}h_{\text{j}}\phi_{ j}S_{\text{gij}}/T_{\text{ij}}}+A_{j}\mu_{\text{gwfj}}C_{\text{gwfj}}\beta_{j} \tag{14}\]  
where \(\alpha_{j}\) is the change rate of the pseudo-bottomhole flowing pressure in layer \(j\), MPa/d; \(\beta_{j}\) is the change rate of the gas rate in layer \(j\), m\({}^{3}\)/d\({}^{2}\). From Equations (10) and (14), it can be observed that the two terms on the right side of Equation (14) are related to the drainage radius. Moreover, if the change rate of the gas rate in layer \(j\) is smaller, the second term on the right side of Equation (14) is smaller, and the impact of \(A_{j}\) on the calculation results of the drainage radius is smaller. The calculation procedures for the drainage radius of a multi-layer commingled gas production well are as follows: (1) obtain bottomhole flowing pressures \(P_{\text{wfj}}(t_{1})\) and \(P_{\text{wfj}}(t_{2})\) and gas rates \(q_{\text{scj}}(t_{1})\) and \(q_{\text{scj}}(t_{2})\) at two arbitrary times \(t_{1}\) and \(t_{2}\) through gas production profile tests; (2) use the data from step (1) and the definition formulas of \(\alpha_{j}\) and \(\beta_{j}\) to calculate their values; (3) assume an initial value for the drainage radius \(r_{\text{ej}}\) and substitute it into Equation (14), and then calculate the value of the right side of Equation (14); (4) compare the values of the left and right sides of Equation (14), if the error between them is less than \(1\times 10^{-4}\), the initial value of \(r_{\text{ej}}\) is the drainage radius of multi-layer commingled gas production well; if the error is larger than \(1\times 10^{-4}\), re-assume an initial value of \(r_{\text{ej}}\) and repeat steps (3) and (4) until the error between the values on the left and right sides of Equation (14) meets the setting requirements; then, the obtained \(r_{\text{ej}}\) is the drainage radius of multi-layer commingled gas production well. The procedure used in the model calculations is shown in the flowchart of Figure 5.