## 5 Conclusions  
In this study, the subsurface drainage of a PFC with various equivalent water flow paths, including length and slope, was observed via two-dimensional finite element analysis. A series of analyses was conducted for PFCs with equivalent water flow path length values of \(10\), \(15\), \(20\), and \(30\) m and slope values of \(0.5\%\), \(2\%\), \(4\%\), \(6\%\), and \(8\%\). The PFCs were subjected to a range of rainfall intensities, from \(10\) to \(120\) mm/h, with a step of \(10\) mm/h to extract the time for surface ponding. Based on the results and discussions, the following conclusions were drawn.  
PFCs subjected to various rainfall intensity values resulted in various times for surface ponding. In general, when the rainfall intensity increased, there was a downward trend in the time for surface ponding of the PFC. The geometric designs of PFCs had a remarkable impact on the subsurface drainage of the PFC. PFCs with different geometric designs provide a wide range of time for surface ponding values. Among the three factors of the  
Figure 7: Relationship between the time for surface ponding (\(t_{p}\)) of the PFC and rainfall intensity (\(I\)) with different thickness (\(T\)) values.  
geometric design of the PFC, thickness has the most influence on subsurface drainage. It is of note that at a high rainfall intensity, the geometric design of the PFC does not seem to affect significantly the subsurface drainage of the PFC.  
The observation of the subsurface drainage of the PFC according to the equivalent water flow path including the length \(L_{R}\) and the slope \(S_{R}\) showed that, with \(L_{R}\), it has an inversely proportional trend, while with \(S_{R}\), it has a similar trend. Another highlighted implication of this study is that the PFC with a shorter \(L_{R}\) appears to be more sensitive to rainfall intensity and slope values. The present study provided additional evidence with respect to the effect of PFC thickness. As the thickness \(T\) increased, the results of the time for surface ponding \(t_{p}\) increased gradually. The current study only examined the subsurface drainage of PFCs using two-dimensional finite element analysis based on FEniCS and implemented it entirely in the Python programing language. Further experimental investigations are needed to evaluate the subsurface drainage of PFCs, considering other factors such as permeability, rutting, and environmental factors.  
Conceptualization, T.H.N. and M.T.P.; methodology, T.H.N.; software, M.T.P.; validation, M.T.P. and T.H.N.; formal analysis, M.T.P.; investigation, T.H.N.; resources, M.T.P.; data curation, T.M.D.H.; writing--original draft preparation, T.H.N.; writing--review and editing, K.K.A.T. and T.H.N.; visualization, M.T.P.; supervision, T.H.N.; project administration, T.M.D.H.; funding acquisition, V.H.H. All authors have read and agreed to the published version of the manuscript.  
This research was fully supported by Tra Vinh University under grant contract number 247/2022/HD.HDKH&DT-DHTV.  
Not applicable.  
Data availability StatementData are contained within the article.  
The authors would like to thank the Tra Vinh University.  
Author [PERSON] was employed by the company Akselos S.A. 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.