# Numerical Study on the Hydrologic Characteristic of Permeable Friction Course Pavement

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{y}), cross slope (S

_{x}), length (L), width (W), and thickness (T) [9,10]. The intensity of rainfall (I) is also a critical factor that should be considered. Several researchers investigated the effects of these factors on the hydrologic characteristic of PFC pavement [9,10,11]. Ranieri [10] used a rainfall simulator to evaluate the hydrologic characteristic of a PFC pavement having dimensions of 1.45 × 0.75 × 0.07 m. It was tested for various longitudinal slopes (S

_{y}) and rainfall intensities (I). Piezometers were installed within the specimen for recording the head of water flow within the body of the PFC pavement. The results of the rainfall simulation experiments were in a good agreement with those from theoretical calculations. Based on the results of the experimental tests, they provided a chart that described the relationship between the ratio of the maximum head of water flow to the pavement length (H/L) and the ratio of the rainfall intensity to the permeability (4I/k). The thickness of the PFC pavement (T) that prevents surface ponding could be determined from the chart, and the chart can be used as a preliminary design tool. Charbeneau and Barrett [12] developed a mathematical model for estimating the steady-state phreatic line in PFC during rainfall. The results of their study showed that as the rainfall intensity (I) increased, the head of water flow within the body of the PFC pavement also increased. They concluded that the hydrologic characteristic of a PFC pavement is significantly dependent on rainfall intensity (I), as well as the permeability (k) and geometric design of the PFC pavement. Tan et al. [9] used a finite element program to investigate the effect of geometric design (S

_{x}, S

_{y}, T, and W) and rainfall intensity (I) on the hydrologic characteristic of PFC pavements. They assumed that the permeability (k) of porous asphalt was 20 mm/s. In each analysis case, they determined the maximum allowable rainfall intensity (I

_{a}) that prevented surface ponding over the PFC pavement. Based on the results of their study, a relationship between the ratio of the thickness to the width (T/W) of the pavement and the maximum allowable rainfall intensity (I

_{a}) was established. The results also showed that as the cross slope (S

_{x}) and the ratio of thickness to width (T/W) increased, the effect of the longitudinal slope (S

_{y}) on the drainage decreased. Recently, Liu et al. [13] proposed a computational model for predicting the water flow of PFC pavements during rainfall. They presented equations for calculating the fluid motion of the water flow. The water flow within the body was estimated as a two-dimensional flow using the diffusive wave equation, whereas the water flow over the surface was estimated as a three-dimensional flow using Richard’s equation. The results indicated that the thickness (T) of the PFC pavement has a significant effect on the time at which surface ponding is initiated, but a less significant effect on the phreatic line in PFC pavement. The time at which surface ponding is initiated (t

_{p}), known as the time taken for water flow over the PFC pavement surface, is an important factor that contributes to the calculation of the peak flow [14]. In low-impact development (LID) techniques, it is called the delay time of peak flow which is one of the typical performance measures for permeable pavements [15]. According to Liu et al. [13], the time taken for water flow over the PFC pavement surface is also critical for safety in transportation. It was also studied by Park et al. [16] in the case of using it for a permeable pavement (constructed with permeable block, subbase, and subgrade) and an impermeable pavement. They were tested with a rainfall simulator. In the test, rainfall intensities of 50 and 150 mm/h were used. The results showed that the time taken for water flow over the PFC pavement surface (t

_{p}) was in a range of 16–21 minutes for the permeable pavement. However, for the impermeable pavement, this value was only 10 minutes, regardless of rainfall intensity. It could be concluded that the permeable pavement results a higher time taken for water flow over the PFC pavement surface than the impermeable pavement. Another study of Huang et al. [17] evaluated the time taken for water flow over the PFC pavement surface for three different kinds of permeable pavement systems consisting of permeable interlocking pavers, porous asphalt, and porous concrete. The authors figured that the porous asphalt and permeable interlocking paver specimens provided the same time taken results. They concluded that the time taken for water flow over the PFC pavement has a strong relationship with the removal rate of pollutants.

## 2. Water Flow in a PFC Pavement

## 3. Modeling of Water Flow in a PFC Pavement and Analysis Parameters

#### 3.1. Transient Unsaturated/Saturated Seepage

_{x}and k

_{y}are the hydraulic conductivities in the horizontal and vertical directions, respectively, h is the total water head, k

_{vd}is the vapor conductivity, u

_{w}is the pore water pressure, γ

_{w}is the unit weight of water, ${m}_{2}^{w}$ is the coefficient of water storage obtained from the derivative of the soil–water characteristic curve, and t is the time.

_{s}is the saturated volumetric water content, θ

_{r}is the residual volumetric water content, Ψ is the soil suction, and a, n, and m are the material (fitting) parameters.

#### 3.2. Geometric Dimensions

_{R}of the cross slope S

_{x}and the longitudinal slope S

_{y}, as presented in Figure 2. The flow of water for a PFC pavement in a three-dimensional direction can be represented by the flow of water for one with a two-dimensional direction having an equivalent water flow path consisting of the length (L

_{R}) and slope (S

_{R}). These variables can be calculated from the width W, length L, cross slope S

_{x}, and longitudinal slope S

_{y}of the PFC pavement based on Equation (3) and Equation (4) [10]:

#### 3.3. Boundary Conditions

#### 3.4. Analysis Cases

_{R}and slope S

_{R}of the PFC pavement was determined according to Equation (3) and Equation (4). The equivalent length L

_{R}, ranging from 10 to 30 m, and the equivalent slope S

_{R}, ranging from 0.5% to 8%, as listed in Table 1, were selected for observation since the other values are exaggerated for a PFC pavement. For each geometric model, a rainfall intensity of 10 mm/h was simulated first, and this value was increased in steps of 10 mm/h up to a maximum value of 120 mm/h. For each iteration, the time taken for water flow over the PFC pavement surface (t

_{p}) was evaluated.

#### 3.5. Material Parameters

## 4. Results and Discussion

#### 4.1. Time Taken for Water Flow over the PFC Pavement Surface

_{R}and slopes S

_{R}) resulted in various times taken. Generally, as the rainfall intensity increased, the time taken for water flow over the PFC pavement surface decreased. This finding confirms that the hydrologic characteristic is dependent on rainfall intensity and geometric design.

#### 4.2. Effect of Geometric Design

_{p}decreases with the increase in the length of L

_{R}. For instance, for I = 40 mm/h and S

_{R}= 8%, as L

_{R}increased by 20 m (from 10 to 30 m), t

_{p}decreased by 9 min (from 26 to 17 min). These results indicate that surface ponding occurred earlier for the PFC pavement that had a longer length L

_{R}.

_{R}results in a corresponding increase in t

_{p}, as reported by Tan et al. [9]. For example, for L

_{R}= 10 m and I = 40 mm/h, t

_{p}increases by 12 min (from 14 to 26 min) when S

_{R}increases by 7.5% (from 0.5% to 8%).

_{R}= 30 m, different values of S

_{R}result in correspondingly similar values of t

_{p}. The finding highlights that the hydrologic characteristic of the PFC pavement is less sensitive to the slope S

_{R}when it has a longer length L

_{R}. Furthermore, it can be seen that the PFC pavements with different geometric designs have close results in the time taken for water flow over the PFC pavement surfaces at high rainfall intensities. It can be concluded that PFC pavements at high rainfall intensity have a similar hydrologic characteristic, regardless of the geometric design.

#### 4.3. Allowable Rainfall Intensity for the PFC Pavement

_{a}for the PFC pavement, which is the maximum rainfall intensity for which the water flow would remain only within the PFC pavement body, was estimated. In other words, this value of the rainfall intensity allows the head of water flow H inside the PFC pavement to be equal to its thickness T, regardless of the rainfall duration. The results are displayed in Table 4.

_{a}according to the slope S

_{R}and length L

_{R}are presented. It can be seen that when S

_{R}increases, there is an upward trend in I

_{a}. This observation is consistent with that of Tan et al. [9]. In addition, I

_{a}decreases when L

_{R}increases, which supports the results reported by Ranieri [29]. This observation implies that a PFC pavement with a shorter L

_{R}and longer S

_{R}exhibits higher drainage capacity.

_{R}is more sensitive to variations in the slope S

_{R}. For example, when the value of S

_{R}increases from 0.5% to 8%, for L

_{R}= 10 m, I

_{a}significantly increases by 27.5 mm/h (an increase from 2.5 to 30 mm/h), but for L

_{R}= 15 m, I¬¬

_{a}increases by only 19 mm/h. However, for L

_{R}= 20 m, I

_{a}slightly increases by 5 mm/h.

#### 4.4. Effects of Thickness and Porosity

_{R}= 10 m and S

_{R}= 2%. The results are displayed in Table 5 and Table 6.

_{p}doubles. According to Hou et al. [11], this could be attributed to the storage capacity of the PFC pavement, a PFC pavement with a higher porosity n has a higher water storage capacity.

#### 4.5. Design Charts for PFC Pavement

_{R}, and slope S

_{R}for the water flow path, the time taken for the water flow over the PFC pavement surface t

_{p}, and the permeability of porous asphalt k, were used to develop design charts in terms of the hydrologic characteristic for the PFC pavement. For each value of $\frac{{L}_{R}}{T}$, the values of $\frac{I}{k}$ are presented as a function of $\frac{{t}_{p}\times k}{{L}_{R}}$. Table 7 below gives the values of the five variables.

_{p}at the design rainfall intensity I using the design charts in Figure 8. The use of these charts is described in the following example. A PFC pavement with an equivalent water flow path has a length L

_{R}= 10 m, a thickness T = 0.05 m, and L

_{R}/T = 200. Assuming that the permeability k = 6.6 mm/s = 24,000 mm/h, at a design rainfall intensity I = 50 mm/h, the results of the time taken for the water flow over the PFC pavement surface t

_{p}for different slopes S

_{R}can be determined as follows (Equation 5):

_{p}were calculated and are listed in Table 8. The results from the design charts demonstrate that the time taken for the water flow over the PFC pavement surface (t

_{p}) can be determined efficiently.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Water flow in a PFC body [10].

**Figure 3.**Horizontal permeability test for porous asphalt specimens (after Ahn et al. [23]).

**Figure 4.**Vertical permeability test for porous asphalt specimen (after Ahn et al. [24]).

**Figure 6.**Relationship between the time taken for the water flow over the PFC pavement surface (t

_{p}) and rainfall intensity (I) with different slope (S

_{R}) values.

Equivalent Length, L_{R} (m) | Equivalent Slope, S_{R} (%) | Rainfall Intensity, I (mm/h) |
---|---|---|

10, 15, 20, 30 | 0.5, 2, 4, 6, 8 | 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120 |

**Table 2.**SWCC parameters for the PFC pavement (after Lim and Kim [25]).

Volumetric Water Content, θ_{s} (%) | Residual Volumetric Water Content, θ_{r} (%) | Material Parameters, | Soil Suction, Ψ (kPa) | |
---|---|---|---|---|

a | n | |||

20 | 0.001 | 2.23 | 1.63 | 0.01 |

L_{R} (m) | S_{R} (%) | Time Taken for Water Flow over PFC Pavement Surface, t_{p} (min) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

I = 10 (mm/h) | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 | 110 | 120 | ||

10 | 0.5 | 60 | 30 | 20 | 14 | 11 | 9 | 8 | 7 | 6 | 5 | 5 | 4 |

2 | 85 | 35 | 22 | 15.5 | 12.5 | 10 | 8.5 | 7.5 | 6.5 | 5.5 | 5.5 | 5 | |

4 | - | 49 | 26 | 18 | 14 | 11 | 9 | 8 | 7 | 6 | 5.5 | 5 | |

6 | - | - | 35 | 21 | 15 | 12 | 10 | 8 | 7 | 6 | 6 | 5 | |

8 | - | - | - | 26 | 18 | 13.5 | 11 | 9 | 8 | 7 | 6 | 5.5 | |

15 | 0.5 | 60 | 30 | 19 | 14 | 11 | 9 | 8 | 7 | 6 | 5 | 5 | 4 |

2 | 75 | 33.5 | 21 | 15.5 | 12 | 10 | 8.5 | 7.5 | 6 | 5.5 | 5 | 4.5 | |

4 | - | 41 | 24 | 17 | 13 | 10.5 | 9 | 7.5 | 6.5 | 6 | 5.5 | 4.5 | |

6 | - | 55.5 | 28 | 19 | 14 | 11.5 | 9.5 | 8 | 7 | 6 | 5.5 | 5 | |

8 | - | - | 34 | 21 | 15.5 | 12 | 10 | 8.5 | 7.5 | 6.5 | 6 | 5 | |

20 | 0.5 | 60 | 30 | 19 | 14 | 11 | 9 | 8 | 7 | 6 | 5 | 5 | 4 |

2 | 71 | 33 | 21 | 15 | 12 | 10 | 8 | 7 | 6 | 5 | 5 | 4 | |

4 | 104 | 38 | 23 | 16 | 12 | 10 | 8 | 7 | 6 | 5 | 5 | 4 | |

6 | - | 45 | 25 | 17 | 13 | 11 | 9 | 8 | 7 | 5 | 5 | 4 | |

8 | - | 59 | 29 | 19 | 14 | 11 | 9 | 8 | 7 | 6 | 5 | 5 | |

30 | 0.5 | 59 | 30 | 19 | 14 | 11 | 9 | 8 | 7 | 6 | 5 | 5 | 4 |

2 | 67 | 32 | 20 | 15 | 12 | 9 | 8 | 7 | 6 | 5 | 5 | 4 | |

4 | 84 | 35 | 21 | 15 | 12 | 10 | 8 | 7 | 6 | 5 | 5 | 4 | |

6 | 115 | 39 | 23 | 16 | 12 | 10 | 8 | 7 | 6 | 5 | 5 | 4 | |

8 | - | 45 | 25 | 17 | 13 | 10 | 9 | 7 | 6 | 5 | 5 | 4 |

Length, L_{R} (m) | Slope, S_{R} (%) | Allowable Rainfall Intensity, I_{a} (mm/h) |
---|---|---|

10 | 0.5 | 2.5 |

2 | 7.5 | |

4 | 15 | |

6 | 22.5 | |

8 | 30 | |

15 | 0.5 | 1.5 |

2 | 5 | |

4 | 10 | |

6 | 15 | |

8 | 20 | |

20 | 0.5 | 1 |

2 | 2.5 | |

4 | 5 | |

6 | 10 | |

8 | 15 |

**Table 5.**Time taken for water flow over the PFC pavement surface (t

_{p}) for I = 50 mm/h, L

_{R}= 10 m, and S

_{R}= 2%.

Thickness, T (m) | Porosity, n (%) | Time Taken for Water Flow over PFC Pavement Surface, t_{p} (min) |
---|---|---|

0.05 | 10 | 6 |

20 | 12.5 | |

0.1 | 10 | 13.5 |

20 | 27.5 |

**Table 6.**Time taken for water flow over the PFC pavement surface (t

_{p}) for I = 100 mm/h, L

_{R}= 10 m, and S

_{R}= 2%.

Thickness, T (m) | Porosity, n (%) | Time Taken for Water Flow over PFC Pavement Surface, t_{p} (min) |
---|---|---|

0.05 | 10 | 2.5 |

20 | 5.5 | |

0.1 | 10 | 6 |

20 | 12 |

$\frac{{L}_{R}}{T}$ | S_{R} (%) | $\frac{{t}_{p}\times k}{{L}_{R}}$ | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

I/k = 0.0003 | 0.0006 | 0.0008 | 0.0011 | 0.0014 | 0.0017 | 0.0019 | 0.0022 | 0.0025 | 0.0028 | 0.0031 | 0.0033 | ||

200 | 0.5 | 3.6 | 1.8 | 1.2 | 0.84 | 0.66 | 0.54 | 0.48 | 0.42 | 0.36 | 0.3 | 0.3 | 0.24 |

2 | 5.1 | 2.1 | 1.32 | 0.93 | 0.75 | 0.6 | 0.51 | 0.45 | 0.39 | 0.33 | 0.33 | 0.3 | |

4 | - | 2.94 | 1.56 | 1.08 | 0.84 | 0.66 | 0.54 | 0.48 | 0.42 | 0.36 | 0.33 | 0.3 | |

6 | - | - | 2.1 | 1.26 | 0.9 | 0.72 | 0.6 | 0.48 | 0.42 | 0.36 | 0.36 | 0.3 | |

8 | - | - | - | 1.56 | 1.08 | 0.81 | 0.66 | 0.54 | 0.48 | 0.42 | 0.36 | 0.33 | |

300 | 0.5 | 1.8 | 0.9 | 0.57 | 0.42 | 0.33 | 0.27 | 0.24 | 0.21 | 0.18 | 0.15 | 0.15 | 0.12 |

2 | 2.25 | 1.005 | 0.63 | 0.465 | 0.36 | 0.3 | 0.255 | 0.225 | 0.18 | 0.165 | 0.15 | 0.135 | |

4 | - | 1.23 | 0.72 | 0.51 | 0.39 | 0.315 | 0.27 | 0.225 | 0.195 | 0.18 | 0.165 | 0.135 | |

6 | - | 1.665 | 0.84 | 0.57 | 0.42 | 0.345 | 0.285 | 0.24 | 0.21 | 0.18 | 0.165 | 0.15 | |

8 | - | - | 1.02 | 0.63 | 0.465 | 0.36 | 0.3 | 0.255 | 0.225 | 0.195 | 0.18 | 0.15 | |

400 | 0.5 | 1.8 | 0.9 | 0.57 | 0.42 | 0.33 | 0.27 | 0.24 | 0.21 | 0.18 | 0.15 | 0.15 | 0.12 |

2 | 2.13 | 0.99 | 0.63 | 0.45 | 0.36 | 0.3 | 0.24 | 0.21 | 0.18 | 0.15 | 0.15 | 0.12 | |

4 | 3.12 | 1.14 | 0.69 | 0.48 | 0.36 | 0.3 | 0.24 | 0.21 | 0.18 | 0.15 | 0.15 | 0.12 | |

6 | - | 1.35 | 0.75 | 0.51 | 0.39 | 0.33 | 0.27 | 0.24 | 0.21 | 0.15 | 0.15 | 0.12 | |

8 | - | 1.77 | 0.87 | 0.57 | 0.42 | 0.33 | 0.27 | 0.24 | 0.21 | 0.18 | 0.15 | 0.15 | |

600 | 0.5 | 1.18 | 0.6 | 0.38 | 0.28 | 0.22 | 0.18 | 0.16 | 0.14 | 0.12 | 0.1 | 0.1 | 0.08 |

2 | 1.34 | 0.64 | 0.4 | 0.3 | 0.24 | 0.18 | 0.16 | 0.14 | 0.12 | 0.1 | 0.1 | 0.08 | |

4 | 1.68 | 0.7 | 0.42 | 0.3 | 0.24 | 0.2 | 0.16 | 0.14 | 0.12 | 0.1 | 0.1 | 0.08 | |

6 | 2.3 | 0.78 | 0.46 | 0.32 | 0.24 | 0.2 | 0.16 | 0.14 | 0.12 | 0.1 | 0.1 | 0.08 | |

8 | - | 0.9 | 0.5 | 0.34 | 0.26 | 0.2 | 0.18 | 0.14 | 0.12 | 0.1 | 0.1 | 0.08 |

Slope, S_{R} (%) | $\frac{{\mathit{t}}_{\mathit{p}}\times \mathit{k}}{{\mathit{L}}_{\mathit{R}}}$ | Time, t_{p} (h) | Time, t_{p} (min) |
---|---|---|---|

0.5 | 0.44 | 0.183 | 11 |

2 | 0.48 | 0.2 | 12 |

4 | 0.50 | 0.217 | 12.5 |

6 | 0.52 | 0.233 | 13 |

8 | 0.60 | 0.25 | 15 |

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**MDPI and ACS Style**

Nguyen, T.H.; Ahn, J.
Numerical Study on the Hydrologic Characteristic of Permeable Friction Course Pavement. *Water* **2021**, *13*, 843.
https://doi.org/10.3390/w13060843

**AMA Style**

Nguyen TH, Ahn J.
Numerical Study on the Hydrologic Characteristic of Permeable Friction Course Pavement. *Water*. 2021; 13(6):843.
https://doi.org/10.3390/w13060843

**Chicago/Turabian Style**

Nguyen, Tan Hung, and Jaehun Ahn.
2021. "Numerical Study on the Hydrologic Characteristic of Permeable Friction Course Pavement" *Water* 13, no. 6: 843.
https://doi.org/10.3390/w13060843