## 5 Conclusions  
In this study, the calculation methods of drainage radius of single-layer/multi-layer commingled gas production wells in a closed constant-volume gas reservoir are investigated. Based on the results of analytical studies, the following conclusions can be drawn:  
1. The novel calculation model of the drainage radius of a single-layer gas well is proposed and proven to be effective and accurate. Results from the case study indicate that the error of calculated drainage radius is only 0.73%, significantly lower than that of the flowing material balance method (5.78%).  
Figure 11: Comparison of measured and simulated gas rate splitting coefficients of each layer in well X6.  
2. The newly derived calculation model of the drainage radius of multi-layer commingled gas production well is established by coupling a production capacity equation with the principle of mass balance in the closed constant-volume gas reservoir. This model is significantly concise and efficient, requiring only two gas production profile tests to swiftly calculate the drainage radius for each layer within the gas reservoir.
3. The proposed calculation model of the drainage radius of the multi-layer commingled gas production well is validated using reservoir numerical simulation. The results indicate that a maximum relative error of this model is only 2.33%, demonstrating that it has good simplicity, significant efficiency and high precision and meets the engineering calculation demands in gas fields.
4. Sensitivity analysis reveals that the calculation model of drainage radius is most sensitive to the physical properties of natural gas, while the influences of formation coefficient, skin factor and change rate of the gas rate are comparatively small.
5. The case study of a closed constant-volume gas reservoir with multi-layer commingled gas production well demonstrates the proposed calculation model of drainage radius has satisfactory accuracy and reliability. It can achieve a good history-match of the actual measured bottomhole flowing pressure and gas rate splitting coefficients for each layer with the simulated data.  
Conceptualization, C.X., W.Z., L.Z. and X.Q.; methodology, C.X., W.Z., X.Q. and Y.W.; software, L.Z. and Y.X.; validation, C.X., W.Z., L.Z., X.Q. and Y.X.; formal analysis, C.X. and W.Z.; investigation, X.Q. and Y.W.; resources, L.Z.; data curation, L.Z. and Y.X.; writing--original draft preparation, C.X. and W.Z.; writing--review and editing, W.Z., L.Z., X.Q. and Y.W.; visualization, Y.W. and W.Z.; supervision, X.Q. and W.Z.; project administration, Y.W. and X.Q.; funding acquisition, W.Z, X.Q. and Y.W. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the West Light Foundation of the Chinese Academy of Sciences (Grant No. XAB2021 YW03), the Natural Science Foundation of Shaanxi Province of China (Grant No. 2023-JC-YB-368), the National Natural Science Foundation of China (Grant No. 51804036), the Key Research and Development Program of Shaanxi Province (Grant No. 2023-YBGY-308) and Fundamental Research Funds for the Central Universities (Grant No. 300102271205).  
Not applicable.  
The data that support the findings of this study are available from the corresponding author, [PERSON], upon reasonable request.  
The authors would like to thank the Research Institute of Shaanxi Yanchang Petroleum (Group) Co., Ltd. for its support.  
Authors [PERSON], [PERSON], [PERSON] and [PERSON] were employed by the company Research Institute of Shaanxi Yanchang Petroleum (Group) Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.