Abstract

This study developed a numerical model to observe the water flow in permeable friction course pavement (PFCP) during rainfall events. The model was established based on FeniCS, which is widely known as an open-source computing platform for solving partial differential equations. The results showed that rainfall intensity significantly affected time for surface ponding in the PFCP. A higher rainfall intensity resulted in a lower time for surface ponding of the PFCP. The evaluation for the effect of suction pressure on the PFCP showed that the suction pressure in the PFCP had a remarkable effect on the time for surface ponding. As the suction pressure in the PFCP increased, the time for surface ponding increased. The results in this study are based on numerical modeling. In the future, further experimental studies in the laboratory and in the field are needed to validate the water flow in PFCP during rainfall events considering other factors such as permeability, rutting, and environmental factors.

1 Introduction

Industrialization and urbanization processes have quickly increased. As a consequence, construction including roads and buildings have increased (Dai et al. 2021). This progress has changed natural permeable surfaces to artificial impermeable ones and brought detrimental impacts to water hydrology in cities such as flooding and pollution (Chu et al. 2018; Wu et al. 2020; Zhang et al. 2020). To date, green solutions have been introduced to resist the above detrimental impacts. Among them, one solution that is used universally is permeable pavement systems (PPS). The PPS itself is constructed with a high porosity; hence, the rainwater could go through and drain out, productively (Eisenberg et al. 2015). Since water could infiltrate the PPS, the peak flow and pollutants in the rainwater could be scaled down. Therefore, PPS has become a prospective solution for water management in urban areas (Eck et al. 2012; Berbee et al. 1999; Roseen et al. 2009).

Permeable friction course pavement (PFCP) is an element of PPS. Its structure includes a layer of porous asphalt laid over conventional asphalt pavement (Manrique-Sanchez and Caro 2019). The operation of PFCP in water management for urban areas depends on several aspects. They could be categorized into three types: material property such as suction pressure, porosity, and permeability; design geometric such as longitudinal slope, cross slope, and pavement length; and environmental factors such as rainfall intensity, and freeze thaw cycles (Tan et al. 2004; Ranieri 2002; Pal and Das 2019). To observe the water flow in the PFCP during rainfall, several studies have been adopted. Typically, the water flow in the PFCP could be assessed by using experiments and numerical modeling. For evaluating the water flow in the PFCP, numerical modeling has generally been used because of its benefits, such as saving cost, saving time, and high accuracy results (Ali 2018).

A number of studies have used numerical modeling to detect the water flow in PFCP. Tan et al. (2004) utilized the numerical model program SEEP3D to extract the water flow in the PFCP. In that study, the water flow in the body of the PFCP (subsurface flow) and the allowable rainfall intensity for the PFCP, which was the maximum rainfall intensity that PFCP dealt with no surface ponding, were evaluated. The PFCPs with various of geometric designs, including cross slope, longitudinal slope, thickness, and width (S_x, S_y, T, and W), were simulated with varying rainfall intensity (I). It was assumed that the permeability (k) of the PFC was 20 mm/s. The numerical results of the study agreed with those from the experiment. Moreover, PFCP with different k values had different allowable I. As the k of the PFCP increased, the allowable I increased. In this study, the effect of suction pressure on the water flow in the PFCP was not mentioned. The water flow in the PFCP has also been researched by (Elkhateeb et al. 2022). In their study, the effect of porosity on the water flow of PFCP was carried out. To meet the scope, PFCPs with different porosity, 19%, 22%, 24% and 27% were simulated. According to the results, the porosity had a high impact on the k of PFCP, and therefore, it also had a high impact on the water flow in the PFCP. Furthermore, the study determined that when the porosity increased, the speed of water flow decreased. This study also did not deal with the suction pressure when modeling the water flow in the PFCP. Recently, Nguyen and Ahn (2021) have introduced a study of PFCP considering the effect of geometry (slope, length, and thickness). In that study, the authors conducted numerical simulations to evaluate the time for surface ponding (t_p) of PFCP in different rainfall intensities based on the SVFlux 2D program. The results highlighted that at a high rainfall intensity, the water flow of PFCP was not sensitive to the PFCP geometry. Among the geometries, the thickness T of PFCP was the most important factor for the water flow in the PFCP. Once again, the suction pressure was not considered in the study.

The literature has reported that numerical models could effectively assess the water flow in PFCP. The water flow in the PFCP significantly depends on material characteristics such as suction pressure, porosity, permeability, and rainfall intensity (Muttuvelu and Kjems 2021; Kuruppu et al. 2019;  Santhanam and Majumdar 2020). To evaluate the water flow in PFCP, the phreatic line of water head in the body of PFCP, the time for surface ponding and the water outflow could be used. To date, there has been a research gap for the effect of rainfall intensity and suction pressure on water flow in PFCP. Thus, this study was adopted to fill the gap and to better understand the impact of rainfall intensity and suction pressure on water flow in PFCP. To evaluate the water flow of the PFCP in this study, the time for surface ponding (t_p) was used.

2 Water Flow in the Body of PFCP

Theoretically, in initially dry conditions, the water from rainfall flows in the PFCP body includes subsurface flow and surface flow. While the subsurface flow immediately occurred, the surface flow happened later, whenever the PFCP became fully saturated. The subsurface flow in the PFCP body had a water shape that was laminar, or between laminar and turbulent (Ranieri et al. 2011). It has been reported that this shape was not consistent and significantly depended on the rainfall intensity, duration, material property, and the geometry of the PFCP. While the shape of water flow in a PFCP is displayed in Figure 1, the PFCP domain in the current study is presented in Figure 2 below. It was constructed with 7591 elements. The element had a triangular shape and was 0.1 cm.

Figure 1 Water shape in a PFC body (Nguyen and Ahn 2021; Ranieri 2002).

Figure 2 PFCP domain in the current study.

2.1 PFCP model and boundary conditions

To achieve its scope, this study modeled PFCP using FEniCS, which was implemented entirely using Python programming language. The PFCP model was modeled in a two-dimensional network. To simulate the water flow in PFCP, the model utilized three types of boundary conditions: "natural", "review", and "zero-flux". While the “natural” boundary condition described the rainfall, the “review” boundary condition simulated the drain of water. The “zero-flux” boundary condition indicated that there was no water draining out. The positions that the boundary conditions used in the PFCP are displayed in Figure 1, above. The AB line, where the rainfall intensity infiltrates into the PFCP body, was a “natural” boundary condition. The BC line, where the rainwater in the PFCP drains out laterally, was a “review” boundary condition. Finally, CD and DA, where the water cannot drain out, was a “zero-flux” boundary condition.

2.2 Analysis cases

This study assumed that PFCP had isotropic permeability and the saturation permeability of the PFCP was 10 mm/s. Since this permeability result was extracted from a laboratory experiment by Yoo et al. (2020), it was represented by the PFCP. The length and the slope of the PFCP were 10 cm and 4%, respectively. It is well-known that the rainfall intensity and suction pressure in the PFCP had a significant effect on its subsurface flow. Therefore, this study evaluated how rainfall intensity and suction pressure influence its subsurface flow. In the analysis program, the time for surface ponding results were recorded. Tables 1 and 2, below, present the details of the analysis cases in the study.

Table 1 Analysis cases to observe the effect of rainfall intensity and suction pressure for PFCP with slope of 4% and length of 10 cm.

3 Governing equations and variational formulation

In this study, time-dependent flow was used to simulate the subsurface drainage of PFCP.  In the simulation, it was assumed that the water was incompressible and had a constant volume. Based on these assumptions, the governing equation for water flow in the PFCP is described in Equation 1: 

(1)

Where:

kx and ky	=	horizontal and vertical directions of the hydraulic conductivities,
h	=	total water head,
kvd	=	vapor conductivity,
uw	=	pressure of the pore water,
γw	=	water's unit weight,
	=	water storage coefficient obtained from the soil water characteristic curve (SWCC's) derivative, and
t	=	time.

Notedly, this study ignored vapor flow and treated the PFCP as having uniform permeability. Therefore, Equation 1 could be rewritten as a partial differential equation:

(2)

The process of subsurface drainage in the PFCP is considered as a time-dependent issue. It is governed by the partial differential equation (PDE), presented in Equation 2. This equation was solved using an implicit Euler method. Equation 3, below, is the result, see below:

(3)

Where:

This equation can also be solved using the Finite Element Method (FEM). The weak form of Equation 3 is shown in Equation 4:

(4)

Where:

For all v belonging to a suitable function space:

The SWCC for the material indicates the non-linear relationship between the volumetric water content and the suction pressure in the PFCP. In this study, the SWCC parameters were determined based on Van Genuchten’s equation (Fredlund and Xing 1994), which is presented in Equation 5: 

(5)

Where:

4 Results and Discussion

The model in this study could be used to describe the water flow in the body of PFCP. Based on the water head in the PFCP body, the time for surface ponding results t_p, (the time that the water head in the body of PFC reach the top its surface), were extracted and reported. Figures 3, 4, and 5 show how the water flow in the PFCP had the suction pressure of 40 kPa and dealt with a rainfall intensity of 80 mm/h after 10 minutes, 15 minutes, and 20 minutes. Apparently, the water head in the body of PFCP increased according to the time of rain event.

Figure 3 Water flow in the PFCP after 10 minutes.

Figure 4 Water flow in the PFCP after 15 minutes.

Figure 5 Water flow in the PFCP after 20 minutes.

The results showing the time for surface ponding in the PFCP according to rainfall intensity and suction pressure are displayed in Table 2. The results in the table show that different rainfall intensities and suction pressures of PFPC resulted in a wide range of time for surface ponding for water flow in the PFCP.

Table 2 Results of time for surface ponding t_p with rainfall intensity and suction pressure.

4.1 Effect of rainfall intensity on the time for surface ponding

The results of time for surface ponding for PFCP in terms of initial suction pressure of 10 kPa, 20 kPa, 30 kPa, and 40 kPa are presented in Figure 6, below. Generally, the time for surface ponding for the PFCP decreases according to an increase in rainfall intensity and suction pressure.

Figure 6 Time for surface ponding of PFCP at different rainfall intensities.

The results showed that rainfall intensity remarkably impacted the time for surface ponding of the PFCP. As the rainfall intensity increased, the time for surface ponding decreased. For example, for the PFCP with a suction pressure of 10 kPa, at the rainfall intensity of 10 mm/h, the time for surface ponding was 44 minutes. When the rainfall intensity increased to 20 mm/h, the result was 36 minutes. This means that when the rainfall intensity increases, the effectiveness of PFCP in water management decreases. This behavior implied that at a high rainfall intensity, the water flow in PFCP provided a similar water flow. In contrast, the study by Hsieh and Chen (2013) implied that at a higher rainfall intensity, the PFCP results show a wide difference in water flow behavior. This idea needs to be investigated in a future study. The dependence of on rainfall intensity on the time for surface ponding of PFCP is consistent with that in studies by Nguyen and Ahn (2021), and Huynh et al. (2023).

The curves in Figure 6 clearly show that the reduction of time for surface ponding is higher at a low rainfall intensity. Nevertheless, this reduction is low at high rainfall intensity. For instance, when then suction pressure as 10 kPa, the rise of rainfall intensity by 10 mm/h (from 10 mm/h to 20 mm/h) caused a large drop in the time for surface ponding by 12 minutes (from 44 minutes to 36 minutes). However, in the case of the rainfall intensity increased by 80 mm/h (from 80 mm/h to 160 mm/h), a small drop was found for the time for surface ponding, only by 4 minutes (from 25 minutes to 21 minutes). It indicated that the level of rainfall intensity impacts strongly on the time for surface ponding of PFCP. It is noted that as the rainfall intensity increases, the results of time for surface ponding become closer together. It could be concluded that at the high rainfall intensity, the behavior of the PFCP is similar. Based on the discussion, the conclusion was made that the rainfall intensity significantly impacts the subsurface water drainage of the PFCP. This impact is clearer at the low rainfall intensity. At the high rainfall intensity, the subsurface water drainage of PFCPs is quite similar. It means the subsurface drainage of PFCPs is similar at high rainfall intensity. This behavior mostly related with that in the study of Nguyen and Ahn (2021) and Huynh et al. (2023), who also found that at the high rainfall intensity, the geometry and material properties of PFCP showed a low impact to the water flow.

4.2 Effect of suction pressure on the time for surface ponding

The results of time for surface ponding for PFCP according to different suction pressures are displayed in Figure 7 below. The curves in Figure 7 clearly illustrate that there is a difference in the time for surface ponding for the PFCP in terms of varying suction pressure. Apparently, the results showed that as the suction pressure in the PFCP increased, the time for surface ponding increased. For example, at rainfall intensity of 10 mm/h, suction pressure of 10 kPa, the time for surface ponding was 44 minutes. When the suction pressure was 40 kPa, the time for surface ponding was 73 minutes. The explanation could be that the PFCP with higher suction pressure had more ability to store rainwater, and therefore it resulted in a higher time for surface ponding. In other words, the PFCP with higher suction pressure provided a higher resistance to water flooding in urban areas.

Figure 7 Time for surface ponding for the PFCP according to different suction pressures.

It is noted that a steep increase in the suction pressure resulted in a steady increase of time for surface ponding. For instance, at a rainfall intensity of 10 mm/h, when the suction pressure increased four times, from 10 to 40 kPa, the time for surface ponding increased by 1.66 times, from 44 minutes to 73 minutes. At a rainfall intensity of 80 mm/h, the time for surface ponding increased by 1.4 times, from 25 minutes to 35 minutes. Hence, it could be concluded that the increase of suction pressure had a low impact on the subsurface drainage of PFCP.

It was noticed that at a low suction pressure, PFCPs at different rainfall intensities have a small range of time for surface ponding. Nevertheless, at a high suction pressure, the times for surface ponding do not vary much. For instance, with the suction pressure of 10 kPa, the times for surface ponding were 21, 25, 30, 36, and 44 minutes for rainfall intensities of 10, 20, 40, 80 and 160 mm/h. The range is 13 minutes; however, with a suction pressure of 40 kPa, the range is 44 minutes. This behaviour indicated that the subsurface drainage of PFCPs is not sensitive to its suction pressure at a high rainfall intensity. In other words, at high rainfall intensity, the behaviour of the PFCPs is similar. This implication is in agreement with the study of Nguyen and Ahn (2021) and Huynh et al. (2023). In the future, further studies for the effect of suction pressure should be adopted to enhance this idea of the current study.

5 Conclusion

The current study established a numerical model to observe the water flow in PFCPs during rainfall events considering rainfall intensity and suction pressure based on the FeniCS open-source platform, which is implemented entirely in Python programing language. Based on the model, the time for surface ponding results in this study were extracted. The results showed that the times for surface ponding of the PFCPs are strongly dependent on rainfall intensity and the suction pressure. As the rainfall intensity increased, the time for surface ponding decreased. Observation of the effect of suction pressure showed a similar trend; as the suction pressure increased, the time for surface ponding increased. This study highlighted that at a high rainfall intensity, the behavior of PFCPs seemed to be similar, regardless of the suction pressure. The outcome of this study could be used as a reference for pavement engineering in the design of PFCP. The innovation of this study is that it develops a numerical model based on open-source program to extract the water flow of PFCP. It is noted that the results of this study were based on the numerical modeling. In the future, further experimental investigation in the laboratory and in the field should be adopted.

Acknowledgements

We acknowledge Ho Chi Minh City University of Technology (HCMUT), and VNU HCM for supporting this study. This paper is a part of the SCNSU 2024.

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