## 2 Calculation Model of Drainage Radius of Single-Layer Gas Well
### Modified Flowing Material Balance Method
Substituting Equations (2) and (3) into Equation (1) yields the flowing material balance equation for a closed constant-volume gas reservoir:  
\[\frac{\partial(P_{\rm R}/Z_{\rm R})}{\mu_{\rm gR}C_{\rm gR}\partial G_{\rm P}} =\frac{\partial(P_{\rm wf}/Z_{\rm wf})}{\mu_{\rm gwf}C_{\rm gwf}\partial G_{ \rm P}} \tag{4}\]  
where \(P_{\rm R}\) is the reservoir pressure, MPa; \(Z_{\rm R}\) is the gas compressibility factor corresponding to \(P_{\rm R}\); \(\mu_{\rm gR}\) is the gas viscosity corresponding to \(P_{\rm R}\), mPa\(\cdot\)s; \(C_{\rm gR}\) is the compressibility corresponding to \(P_{\rm R}\), MPa\({}^{-1}\); \(P_{\rm wf}\) is the bottomhole flowing pressure, MPa; \(Z_{\rm wf}\) is the gas compressibility factor corresponding to \(P_{\rm wf}\); \(\mu_{\rm gwf}\) is the gas viscosity corresponding to \(P_{\rm wf}\); mPa\(\cdot\)s; \(C_{\rm gwf}\) is the compressibility corresponding to \(P_{\rm wf}\), MPa\({}^{-1}\). Assuming a correction factor \(\lambda\) and \(\lambda=\frac{\mu_{\rm gR}C_{\rm gR}}{\mu_{\rm gwf}C_{\rm gwf}}\), then:  
\[\frac{\partial(P_{\rm R}/Z_{\rm R})}{\partial G_{\rm P}}=\lambda\frac{\partial (P_{\rm wf}/Z_{\rm wf})}{\partial G_{\rm P}} \tag{5}\]  
where \(\lambda\) is the correction factor for flowing material balance method. As \(P_{\rm R}\) is unknown, the value of \(\mu_{\rm gR}C_{\rm gR}\) is difficult to obtain. \(P_{\rm R}\) can be derived by converting the production pressure drop and bottomhole flowing pressure during the pseudo-steady stage, so:  
\[\lambda=\frac{\overline{\mu_{\rm g}C_{\rm g}}}{\mu_{\rm gwf}C_{\rm gwf}} \approx\frac{\left(\mu_{\rm g}C_{\rm g}\right)\left|{}_{P_{\rm wf}+\Delta P} \right.}{\left(\mu_{\rm g}C_{\rm g}\right)\left|{}_{P_{\rm wf}}\right.} \tag{6}\]  
where \(\Delta P\) is the production pressure drop. From Equation (5) and the physical properties of natural gas at high pressure (as shown in Figure 1), we can obtain \(|\lambda|<1\), so \(|\partial(P_{\rm R}/Z_{\rm R})/\partial G_{\rm P}|<|\partial(P_{\rm wf}/Z_{\rm wf })/\partial G_{\rm P}|\). Also, the following characteristics can be obtained: (1) at the same production pressure drop, the lower the reservoir pressure, the greater the difference between \(|\partial(P_{\rm R}/Z_{\rm R})/\partial G_{\rm P}|\) and \(|\partial(P_{\rm wf}/Z_{\rm wf})/\partial G_{\rm P}|\); (2) at the same reservoir pressure, the larger the production pressure drop, the greater the difference between them. The modified flowing material balance method is as follows: (1) plot the relationship curve between \(P_{\rm wf}/Z_{\rm wf}\) and \(G_{\rm P}\) using known bottomhole flowing pressure and cumulative gas production data, as shown in Figure 2; (2) conduct linear fitting for the data points showing a linear trend to determine the slope of the fitted line (\(-\)m); (3) calculate the value of \(\lambda\) by Equation (6); (4) calculate the value of \(-\lambda m\), and then use this value as the slope to draw a straight line through the point \(P_{\rm i}/Z_{\rm i}\), the intercept of this straight line on the horizontal coordinate is the original gas in place (\(G_{\rm i}\)) calculated using the modified material balance method (Equation (7) is the formula for this straight line); (5) substitute the value of \(G_{\rm i}\) into the volumetric method formula (Equation (8)) [19] to obtain the gas well's drainage radius (\(r_{\rm e}\)):  
\[\frac{P}{Z}=-\lambda mG_{\rm P}+\frac{P_{\rm i}}{Z_{\rm i}} \tag{7}\]where \(m\) is the slope of the straight line section of \(P_{\rm wf}/Z_{\rm wf}\sim G_{\rm p}\), \(\rm MPa/10^{8}\ m^{3}\); \(P_{\rm i}\) is the original formation pressure, MPa; \(Z_{\rm i}\) is the gas compressibility factor corresponding to \(P_{\rm i}\).