## 5 Conclusions  
The Beijing-Shanghai high-speed railway test section project used a fluid-solid coupling calculation model in FLAC\({}^{\text{3D}}\) software to establish four modes of negative pore water pressure distribution along the sand drain: concave parabola, convex parabola, linear, and uniform distribution. The soil consolidation process was simulated and calculated, and the results were compared with the measured values of the project. The following conclusions were drawn.  
The settlement of the soft foundation developed rapidly in the first 30 days, slowed down after 30 days, and eventually tended to stabilize. The settlement was most significant near the surface and gradually decreased with depth. The settlement-time curve at a depth of 22 m was nearly horizontal, indicating a weak vacuum effect at that depth.  
Within the first 5 days of vacuum preloading, the settlement calculation curves for all four distribution modes were highly consistent with the measured settlement curves. Between days 5 and 30, there was a significant gap between the calculated and measured values. After 30 days, the settlement curves for the concave parabolic and linear distribution modes coincided closely with the measured settlement curves, with differences ranging from 0.724 to 2.4 cm and from 0.01 to 1.65 cm, respectively. However, the differences between the calculated and measured settlement values for the convex parabolic and uniform distribution modes were consistent with the differences observed around day 30.  
The calculation results for the concave parabolic and linear pore water pressure modes were better than those for the convex parabolic and uniform distribution modes. The linear setting mode was slightly better than the concave parabolic setting mode.  
During the first 20 days, the pore water pressure dropped most significantly within the depth range of 0-18 m. Between days 20 and 30, the rate of the pore water pressure decrease was significantly reduced and eventually became stable. After 30 days, the pore water pressure at each depth was essentially constant.  
The influence depth of vacuum preloading for the concave parabolic and linear distribution modes was approximately 16 m, whereas the influence depth for the convex parabolic and uniform distribution modes was approximately 18 m. A comparison of the calculated settlement and the measured settlement showed that the influence depth of vacuum preloading was approximately 16 m.  
This article did not consider the impact of group sand drains. Therefore, we will consider this impact in future research.  
M.L.: Conceptualization and writing--original draft. J.C.: Data curation and formal analysis. R.Z.: Software. J.J.: Validation. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the National Natural Science Foundation of China (42107166), Hunan Provincial Natural Science Foundation (No. 2021 JJ40632, 2021 JJ30758, and 2022 JJ40521), Key project of Teaching Reform Research in Hunan Province (HNJG-2021-0209), Changsha Municipal Natural Science Foundation (No. kq2202065 and kq2202063), and Hunan Provincial Department of Education Project (No. 22B08253).  
Not applicable.  
All data, models, and code generated or used during the study are available from the corresponding author by request.  
The authors declare no conflict of interest.