## 2 Calculation Model of Drainage Radius of Single-Layer Gas Well  
### Modified Flowing Material Balance Method  
[PERSON] and [PERSON] [18] firstly proposed the flowing material balance equation, stating that when a gas reservoir is in a pseudo-steady state, the bottomhole flowing pressure and the average reservoir pressure decrease at the same rate. Also, they further concluded that the straight line of \(p_{\mathrm{wf}}/Z_{\mathrm{wf}}\sim G_{\mathrm{P}}\) is parallel to that of \(p/Z\sim G_{\mathrm{P}}\). However, this method did not consider the pressure's impact on natural gas viscosity and its compressibility, assuming these properties remain constant. Figure 1 shows the relationship curve between the product of natural gas viscosity and its compressibility and reservoir pressure in the Yanan gas field in Shaanxi Province, China. It can be seen that the product of natural gas viscosity and its compressibility increases rapidly as pressure decreases, which is contrary to the assumption of previous research. Therefore, we propose the modified flowing material balance method, and the specific method is as follows. For a single-layer 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; (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. When the flow phase of gas reservoir satisfies the pseudo-steady state:  
\[\frac{\partial\psi_{\mathrm{R}}}{\partial t}=\frac{\partial\psi_{\mathrm{wf} }}{\partial t} \tag{1}\]  
where \(\Psi_{\mathrm{R}}\) is the pseudo-pressure function of reservoir pressure (\(P_{\mathrm{R}}\)), \(\mathrm{MPa^{2}}/(\mathrm{mPa\cdot s})\); \(\Psi_{\mathrm{wf}}\) is the pseudo-pressure function of bottomhole flowing pressure (\(P_{\mathrm{wf}}\)), \(\mathrm{MPa^{2}}/(\mathrm{mPa\cdot s})\); \(t\) is the production time, day. According to the definition of the compressibility of natural gas, the relationship of pseudo-pressure function (\(\Psi\)) and pseudo-pressure (\(P/Z\)) can be determined as:  
\[\frac{\partial\psi}{\partial t}=\frac{2P}{\mu_{\mathrm{g}}Z}\left(\frac{PC_{ \mathrm{g}}}{Z}\right)^{-1}\frac{\partial(P/Z)}{\partial t}=\frac{2}{\mu_{ \mathrm{g}}C_{\mathrm{g}}}\frac{\partial(P/Z)}{\partial t} \tag{2}\]  
Figure 1: Relationship curve between the product of natural gas viscosity and its compressibility and reservoir pressure. where \(\Psi\) is the pseudo-pressure function of pressure \(P\), MPa\({}^{2}\)/(mPa\(\cdot\)s); \(P\) is the pressure at any point in the reservoir, MPa; \(Z\) is gas compressibility factor at pressure \(P\); \(P\)/\(Z\) is the pseudo-pressure, MPa; \(\mu_{\rm g}\) is the gas viscosity, mPa\(\cdot\)s; \(C_{\rm g}\) is the compressibility of natural gas, MPa\({}^{-1}\). When the gas well maintains constant production, the daily gas production remains constant, so:  
\[\partial G_{\rm P}=10^{-8}q_{\rm sc}\partial t \tag{3}\]  
where \(G_{\rm P}\) is the cumulative gas production, \(10^{8}\) m\({}^{3}\); \(q_{\rm sc}\) is the daily gas production, m\({}^{3}\)/d.