## 2 Calculation Model of Drainage Radius of Single-Layer Gas Well
### Calculation Model of Drainage Radius of Multi-Layered Gas Well  
For a multi-layered gas reservoir, the following assumptions are made: (1) the gas reservoir is circular and enclosed, and a vertical production well is located in the center and produces from multiple layers; (2) the gas reservoir is a uniformly thick, isothermal, and horizontal medium; (3) the single-phase compressible gas satisfies Darcy's Law; (4) the effects of gravity and capillary forces are neglected; (5) the development of the gas reservoir relies solely on elastic expansion energy; (6) inter-layer crossflow is not considered. The productivity equation of the gas well when the \(j\)-th layer stays in pseudo-steady flow is given by:  
\[\psi_{\rm{R}j}=\psi_{\rm{w}fj}+A_{j}q_{\rm{sc}j} \tag{9}\]  
where the coefficient of productivity equation, \(A_{j}\), is defined as:  
\[A_{j}=\frac{1.291\times 10^{-3}T_{\rm{ij}}(\ln\frac{0.472r_{\rm{ej}}}{r_{\rm{w}}}+S _{j})}{K_{j}h_{j}} \tag{10}\]  
where \(j\) is the layer number, \(j=1\), \(\ldots\), \(n\); \(n\) is the number of layers in the gas well, dimensionless; \(\Psi_{\rm{R}j}\) is the pseudo-reservoir pressure function of layer \(j\), MPa\({}^{2}\)/(mPa\(\cdot\)s); \(\Psi_{\rm{w}fj}\) is the pseudo-bottomole flowing pressure function of layer \(j\), MPa\({}^{2}\)/(mPa\(\cdot\)s); \(A_{j}\) is the productivity coefficient of layer \(j\), MPa\({}^{2}\)/(mPa\(\cdot\)s)/(m\({}^{3}\)/d); \(q_{\rm{sc}j}\) is the daily gas production of layer \(j\), m\({}^{3}\)/d; \(T_{j}\) is the temperature of layer \(j\), K; \(r_{ej}\) is the drainage radius of layer \(j\), m; \(r_{\rm{w}}\) is the wellbore radius, m; \(S_{j}\) is the skin factor of layer \(j\), dimensionless; \(K_{j}\) is the permeability of layer \(j\), \(10^{-3}\)um\({}^{2}\); \(h_{j}\) is the thickness of layer \(j\), m.  
Figure 4: Relationship of \(P/Z\) and \(G_{\rm{P}}\) for well X1. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Property**} & **Material** & \multicolumn{2}{c}{**Flowing Material Balance**} & \multicolumn{2}{c}{**Modified Flowing Material**} \\  & **Balance** & \multicolumn{2}{c}{**Method**} & \multicolumn{2}{c}{**Balance Method**} \\ \cline{3-6}  & **Method** & **Value** & **Error (\%)** & **Value** & **Error (\%)** \\ \hline \(G_{\rm{i}}\) (\(10^{8}\) m\({}^{3}\)) & 0.9180 & 0.8150 & 11.22 & 0.9047 & 1.45 \\ \hline \(r_{\rm{e}}\) (m) & 617.9 & 582.2 & 5.78 & 613.4 & 0.73 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of calculation results for different methods for well X1. Take the derivative of Equation (9) with respect to time \(t\), and then substitute Equation (2) into it, yielding the following equation:  
\[\frac{\partial(P_{\text{Rj}}/Z_{\text{Rj}})}{\mu_{\text{gRj}}C_{\text{gRj}} \partial t}=\frac{\partial(P_{\text{wfj}}/Z_{\text{wfj}})}{\mu_{\text{gwfj}}C_{ \text{gwfj}}\partial t}+A_{j}\frac{\partial(q_{\text{scj}})}{\partial t} \tag{11}\]  
where \(P_{\text{Rj}}\) is the reservoir pressure of layer \(j\), MPa; \(Z_{\text{Rj}}\) is the gas compressibility factor corresponding to \(P_{\text{Rj}}\), dimensionless; \(\mu_{\text{gRj}}\) is the gas viscosity corresponding to \(P_{\text{Rj}}\), mPa-s; \(C_{\text{gRj}}\) is the compressibility corresponding to \(P_{\text{Rj}}\), MPa\({}^{-1}\); \(P_{\text{wfj}}\) is the bottomhole flowing pressure of layer \(j\), MPa; \(Z_{\text{wfj}}\) is the gas compressibility factor corresponding to \(P_{\text{wfj}}\), dimensionless; \(\mu_{\text{gwfj}}\) is the gas viscosity corresponding to \(P_{\text{wfj}}\), mPa-s; \(C_{\text{gwfj}}\) is the compressibility corresponding to \(P_{\text{wfj}}\), MPa\({}^{-1}\). For a constant-production gas well, if the production ratio of each layer remains unchanged, the second term on the right side of Equation (11) is 0, and Equation (11) can be transformed into Equation (4) which represents the flowing material balance equation for a closed constant-volume gas reservoir.