water

Article Numerical Study on the Hydrologic Characteristic of Permeable Course Pavement

Tan Hung Nguyen 1 and Jaehun Ahn 2,*

1 Faculty of Architectural, Civil and Environmental Engineering, Nam Can Tho University, Can Tho 900000, Vietnam; [email protected] 2 Department of Civil and Environmental Engineering, Pusan National University, Busan 46241, Korea * Correspondence: [email protected]; Tel.: +82-51-510-7627

Abstract: The hydrologic characteristic of a permeable friction course (PFC) pavement is dependent on the rainfall intensity, pavement geometric design, and porous asphalt properties. Herein, the hydrologic characteristic of PFC pavements of various lengths and slopes was determined via numerical analysis. A series of analyses was conducted using length values of 10, 15, 20, and 30 m and slope values of 0.5%, 2%, 4%, 6%, and 8% for the equivalent water flow path. The PFC pavements were simulated for various values of rainfall intensity, which ranged from 10 to 120 mm/h, to determine the time taken for water to flow over the PFC pavement surface. The results show that the time for water overflow decreased when the pavement length or rainfall intensity increased, and it increased when the slope increased. Finally, a series of design charts was developed to determine the time taken for water to flow over the PFC pavement surface for given rainfall intensities. Since this study was conducted based on numerical analysis, further studies are recommended to verify  experimentally the results presented. 

Citation: Nguyen, T.H.; Ahn, J. Keywords: hydrologic characteristic; permeable friction course pavement; geometric design Numerical Study on the Hydrologic Characteristic of Permeable Friction Course Pavement. Water 2021, 13, 843. https://doi.org/10.3390/ 1. Introduction w13060843 In recent years, permeable pavements have been used widely, and they have played an essential role in controlling rainwater quantity and quality in urban areas [1–3]. There are Academic Editors: Enedir Ghisi and different types of permeable pavements, such as porous asphalt, pervious concrete, porous Liseane Padilha Thives turf, plastic geo-cell, open-jointed block, open-celled grid, and permeable friction course (PFC) pavement [4]. The PFC pavement or open-graded friction course pavement refers Received: 22 January 2021 to a type of porous asphalt laid on the surface layer of conventional impermeable asphalt Accepted: 15 March 2021 Published: 19 March 2021 pavement, and it is increasingly becoming popular in many countries [5]. As the porous asphalt is designed to have a high , the PFC pavement allows rainwater to infiltrate

Publisher’s Note: MDPI stays neutral its pores and drain out laterally. As a result, PFC pavement tends to reduce the peak flow with regard to jurisdictional claims in of and pollutants in rainwater compared to impermeable asphalt pavement [6–8]. published maps and institutional affil- The hydrologic characteristic of a PFC pavement depends not only on the properties iations. of the porous asphalt material (such as porosity and permeability) but also on the geometric design of the pavement, such as the longitudinal slope (Sy), cross slope (Sx), 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 [9–11]. Ranieri [10] used a rainfall Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. simulator to evaluate the hydrologic characteristic of a PFC pavement having dimensions × × This article is an open access article of 1.45 0.75 0.07 m. It was tested for various longitudinal slopes (Sy) and rainfall distributed under the terms and intensities (I). were installed within the specimen for recording the head of conditions of the Creative Commons water flow within the body of the PFC pavement. The results of the rainfall simulation Attribution (CC BY) license (https:// experiments were in a good agreement with those from theoretical calculations. Based on creativecommons.org/licenses/by/ the results of the experimental tests, they provided a chart that described the relationship 4.0/). between the ratio of the maximum head of water flow to the pavement length (H/L) and the

Water 2021, 13, 843. https://doi.org/10.3390/w13060843 https://www.mdpi.com/journal/water Water 2021, 13, 843 2 of 14

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 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 (Sx, Sy, 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 (Ia) 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 (Ia) was established. The results also showed that as the cross slope (Sx) and the ratio of thickness to width (T/W) increased, the effect of the longitudinal slope (Sy) 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 (tp), 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 ) 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 (tp) 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. The literature showed that the hydrologic characteristic of PFC pavement is influenced by rainfall intensity, geometric design, and porous asphalt properties. However, there is a dearth of literature on how long it takes until surface ponding occurs or the time taken for water flow over the PFC pavement surface and how the PFC fails to accommodate the excess rainfall. The aim of this study is to determine the time taken for water flow over the PFC pavement surface and the effects of the geometric design and rainfall intensity on it using numerical analysis. PFC pavements with different geometric designs were simulated for a range of rainfall intensities to obtain the time taken for water to flow over the PFC pavement surface. Based on the results of this study, the hydrologic characteristic of the PFC pavement was evaluated and compared. Water 2021, 13, x FOR PEER REVIEW 3 of 14

the PFC pavement surface. Based on the results of this study, the hydrologic characteristic of the PFC pavement was evaluated and compared.

2. Water Flow in a PFC Pavement Water 2021, 13, 843 3 of 14 Water flow from rainfall in a PFC pavement consists of two components: flow within the body and flow over the surface of the pavement. Generally, under the dry condition, with constant rainfall intensity, the water flow can be described as follows: water initially 2.infiltrates Water Flow the pores in a PFC of the Pavement PFC pavement and then begins to flow out of the body laterally withoutWater surface flow fromponding. rainfall The in water a PFC flow pavement within consists a PFC body of two is components:considered to flow be laminar within theor between body and laminar flow over and the turbulent surface [10,18]. of the pavement. The head of Generally, the water under flow the(H) dryin a condition,PFC body withis illustrated constant in rainfall Figure intensity, 1. This value the water is a function flow can of be the described rainfall intensity as follows: and water time, initially as well infiltratesas the permeability the pores ofand the geometric PFC pavement design and of the then PFC. begins During to flow the outrainfall, of the the body pores laterally in the withoutPFC pavement surface ponding.are filled with The water water, flow and within the head a PFC of water body isflow considered increases to gradually. be laminar or betweenWhen laminar the head and of turbulent water flow [10 reaches,18]. The the head surface of the of waterthe PFC flow pavement, (H) in a i.e., PFC when body the is illustratedhead of water in Figure flow1 .exceeds This value the isthickness a function of ofthe the PFC rainfall pavement intensity (H andequals time, to asT), well surface as theponding permeability occurs. andAt this geometric moment, design water of can the PFC.flow over During the the PFC rainfall, pavement the pores surface, in the and PFC the pavementpavement aretends filled to withbe almost water, saturated. and the head of water flow increases gradually.

FigureFigure 1.1.Water Water flowflow in in a a PFC PFC body body [ 10[10]]. .

3. ModelingWhen the of head Water of waterFlow in flow a PFC reaches Pavement the surface and Analysis of the PFC Parameters pavement, i.e., when the head of water flow exceeds the thickness of the PFC pavement (H equals to T), surface 3.1. Transient Unsaturated/Saturated Seepage ponding occurs. At this moment, water can flow over the PFC pavement surface, and the pavementIn this tends study, to bea finite almost element saturated. program, SV Flux 2D [19], was used to model the tran- sient unsaturated/saturated seepage in the PFC pavement. Assuming that water has a con- 3.stant Modeling volume of and Water is incompressible, Flow in a PFC Pavement the governing and Analysis equation Parameters of two-dimensional water 3.1.flow Transient is expressed Unsaturated/Saturated in Equation (1): Seepage ∂ ∂ ∂u ∂  ∂ ∂u  ∂ In this study,h + a finitew element + program,h + SV Fluxw = 2D−γ [19],w wash used to model the k x k vd  k y k vd  w m 2 (1) transient∂x unsaturated/saturated ∂x ∂x  ∂ seepagey  ∂ iny the PFC∂y pavement. Assuming∂t that water has a constant volume and is incompressible, the governing equation of two-dimensional water where kx and ky are the hydraulic conductivities in the horizontal and vertical directions, flow is expressed in Equation (1): respectively, h is the total water head, kvd is the vapor conductivity, uw is the pore water w pressure, γw is the∂  unit∂h weight∂ uof water, ∂ m ∂ish the coefficient∂u  of water∂h storage obtained k + k w + k 2 + k w = −γ mw (1) from the derivative∂x ofx ∂ thex –wvd ∂xater characteristic∂y y ∂y curve,vd ∂y and t is wthe2 time.∂t The soil–water characteristic curve (SWCC) represents the nonlinear relationship be- wheretween kthex and volumetricky are the water hydraulic content conductivities and the suction in the in the horizontal soil. Van and Genuchten’s vertical directions, equation respectively, h is the total water head, kvd is the vapor conductivity, uw is the pore water [20,21] was used in this study to model the wSWCC, as shown in Equation (2): pressure, γw is the unit weight of water, m is the coefficient of water storage obtained − 2 from the derivative of= the soil–water+ θ s characteristicθ r curve, and t is the time. θ θ r m (2) The soil–water characteristic[]1 + () curveaψ n (SWCC) represents the nonlinear relationship between the volumetric and the suction in the soil. Van Genuchten’s equa-

tionwhere [20 ,θ21 is] wasthe volumetric used in this water study content, to model θs the is the SWCC, saturated as shown volu inmetric Equation water (2): content, θr is the residual volumetric water content, Ψ is the soil suction, and a, n, and m are the ma- θs − θr terial (fitting) parameters. θ = θr + (2) [1 + (aψn)]m

where θ is the volumetric water content, θs is the saturated volumetric water content, θr is the residual volumetric water content, Ψ is the soil suction, and a, n, and m are the material (fitting) parameters. Water 2021, 13, 843 4 of 14 Water 2021, 13, x FOR PEER REVIEW 4 of 14

3.2. Geometric Dimensions 3.2. GeometricIn a PFC Dimensions pavement, a majority of the water flows along the steepest slope, which is the resultantIn a PFC Spavement,R of the cross a majority slope Sx ofand the the water longitudinal flows along slope theSy ,steepest as presented slope, in which Figure is2. theThe resultant flow of waterSR of the for across PFC slope pavement Sx and in the a three-dimensional longitudinal slope direction Sy, as presented can be represented in Figure 2.by The the flow flow of of water water for onea PFC with pavement a two-dimensional in a three-dimensional direction having direction an equivalent can be repre- water sentedflow path by the consisting flow of water of the for length one with (LR) a and two- slopedimensional (SR). These direction variables having can an be equivalent calculated waterfrom theflow width path Wconsisting, length L of, cross the length slope S (xL,R and) and longitudinal slope (SR). These slope Svariablesy of the PFC can pavementbe calcu- latedbased from on Equation the width (3) W and, length Equation L, cross (4) [ 10slope]: Sx, and longitudinal slope Sy of the PFC pavement based on Equation (3) and Equation (4) [10]: q 2 2 2 2 S S=R =S S+x S+ Sy (3)(3) R x y

s 2  S  S 2 L = W 1 + y y (4) R LR = W 1 +S  (4)  x Sx

FigureFigure 2. 2. EquivalentEquivalent water water flow flow path: path: length length (L (LR)R and) and slope slope (S (RS).R ).

3.3.3.3. Boundary Boundary Conditions Conditions ThreeThree main main types of boundaryboundary conditionsconditions were were applied applied in in the the analysis: analysis: climate, climate, review, re- view,and zero-fluxand zero-flux conditions. conditions. Figure Figure1 illustrates 1 illustrates the application the application of the of boundarythe boundary conditions condi- tionsin modeling in modeling the PFC the pavement.PFC pavement. The surfaceThe surface of the of PFC the pavement,PFC pavement, i.e., the i.e., AB the line, AB alongline, alongwhich which rainfall rainfall occurred, occurred, was assigned was assigned the climate the climate boundary boundary condition. condition. The outlet The outlet of the ofPFC the pavement,PFC pavement, i.e., the i.e., BC the line, BC line, from from where where the water the water drained drained out ofout the of pavement,the pavement, was assigned the review boundary condition. Finally, the bottom CD line and the upper AD was assigned the review boundary condition. Finally, the bottom CD line and the upper line, where the water could not drain out, were assigned zero-flux boundary conditions to AD line, where the water could not drain out, were assigned zero-flux boundary condi- model the impermeable characteristics. tions to model the impermeable characteristics. 3.4. Analysis Cases 3.4. Analysis Cases Based on the conventional dimension of pavement in the AASHTO manual, “Policy on GeometricBased on Design the conventional of Highways dimension and Streets” of [ 22pavement], the equivalent in the AASHTO water flow manual, path including “Policy on Geometric Design of Highways and Streets” [22], the equivalent water flow path in- the length LR and slope SR of the PFC pavement was determined according to Equation (3) cluding the length LR and slope SR of the PFC pavement was determined according to and Equation (4). The equivalent length LR, ranging from 10 to 30 m, and the equivalent Equation (3) and Equation (4). The equivalent length LR, ranging from 10 to 30 m, and the slope SR, ranging from 0.5% to 8%, as listed in Table1, were selected for observation since equivalentthe other values slope S areR, ranging exaggerated fromfor 0.5% a PFC to 8%, pavement. as listed Forin Table each 1, geometric were selected model, for a rainfallobser- vationintensity since of 10the mm/h other wasvalues simulated are exaggerated first, and this for valuea PFC was pavement. increased For in steps each ofgeometric 10 mm/h model,up to aa maximumrainfall intensity value of 10 120 mm/h mm/h. was For simulated each iteration, first, and the this time value taken was for increased water flow in stepsover of the 10 PFC mm/h pavement up to a surfacemaximum (tp) value was evaluated. of 120 mm/h. For each iteration, the time taken for water flow over the PFC pavement surface (tp) was evaluated. Table 1. Analysis cases.

Equivalent Length, LR (m) Equivalent Slope, SR (%) Rainfall Intensity, I (mm/h) 10, 20, 30, 40, 50, 60, 70, 80, 90, 10, 15, 20, 30 0.5, 2, 4, 6, 8 100, 110, 120 Water 2021, 13, x FOR PEER REVIEW 5 of 14

Table 1. Analysis cases.

Equivalent Length, LR (m) Equivalent Slope, SR (%) Rainfall Intensity, I (mm/h) Water 2021, 13, 843 5 of 14 10, 20, 30, 40, 50, 60, 70, 80, 10, 15, 20, 30 0.5, 2, 4, 6, 8 90, 100, 110, 120

3.5.3.5. Material Material Parameters Parameters TheThe horizontal horizontal and and vertical vertical permeabilities permeabilities of of the the porous porous asphalt asphalt used used in thein the numerical numeri- analysiscal analysis were were determined determined from from the the laboratory laboratory experiment. experiment. Specimens Specimens with with dimensions dimensions ofof 0.3 0.3× × 0.3 × 0.050.05 m m and and porosity porosity of of 19.6% 19.6% were were tested. tested. The The horizontal horizontal permeability permeability was wasevaluated evaluated according according to a toprocedure a procedure developed developed by Ahn by et Ahn al. et[23], al. and [23], the and vertical the vertical perme- permeabilityability was examined was examined using using a constant a constant head head procedure procedure presented presented in inAhn Ahn et etal. al. [24]. [24 ].A Asummary summary of of the the two two test test procedures procedures is is presented presented in in Figures Figures 3 and 44..

(a) Test specimen (b) Stepper for adjusting the water level

(c) Specimen setup (d) Constant head test

FigureFigure 3. 3.Horizontal Horizontal permeability permeability test test for for porous porous asphalt asphalt specimens specimens (after (after Ahn Ahn et et al. al. [23 [23]]). ). From the experimental results, the horizontal and vertical permeabilities were 11.1 and 9.2 mm/s, respectively. The horizontal permeability of the porous asphalt was found to be slightly higher than the vertical permeability. In this study, the permeability of the PFC pavement was assumed to be isotropic and equal to 10 mm/s. The SWCC parameters for a pervious concrete given in Lim and Kim [25] were used for those of the PFC pavement in this study. They obtained the SWCC parameters through the equation of Fredlund and Xing [26] using Fredlund’s device [27]. These parameters were substituted in Equation (2) in this study. The SWCC parameters and curve are presented in Table2 and Figure5, respectively. Water 2021, 13, 843 6 of 14 Water 2021, 13, x FOR PEER REVIEW 6 of 14

(a) Specimen fabrication (b) Sealing gap using silicone glue

(c) Specimen mold assemblage (d) Constant head test

FigureFigure 4. VerticalVertical permeability test for porous asphalt specimen (after Ahn et al. [[24]24]).).

TableFrom 2. SWCC the parametersexperimental for theresults, PFC pavement the horizontal (after Lim and and vertical Kim [25 permeabilities]). were 11.1 and 9.2 mm/s, respectively. The horizontal permeability of the porous asphalt was found to beVolumetric slightly higher Water than the Residualvertical permea Volumetricbility. InMaterial this study, Parameters, the permeabilitySoil Suction, of the PFC pavementContent, θ wass (%) assumed Waterto be isotropic Content, θ andr (%) equal to a10 mm/s. n Ψ (kPa) The SWCC20 parameters for a pervious 0.001 concrete given 2.23 in Lim and 1.63 Kim [25] were 0.01 used for those of the PFC pavement in this study. They obtained the SWCC parameters through the equationAccording of toFredlund previous and studies, Xing PFC[26] pavementsusing Fredlund’s absorb device the surrounding [27]. These air parameters to reduce werethe noise substituted from moving in Equation vehicle (2) tires in this [4]. study. In the The case SWCC of no parameters rainfall, PFC and pavement curve are has pre- a sentedsuction in force Table and 2 and a negative Figure 5, value respectively. for the [28]. Therefore, in this study, the initial pore water pressure was set to a negative value within the range of the Tableresidual 2. SWCC zone. parameters for the PFC pavement (after Lim and Kim [25]). In the numericalResidual analysis, Volumetric it was noted Water that theMaterial pore water Parameters, pressure values within the Volumetric Water Soil Suction, Ψ range of the residual zoneContent, had no θ significantr (%) effect on the allowable rainfall intensity and Content, θs (%) a n (kPa) time taken for water flow over the PFC pavement surface. Based on Figure5, −200 kPa was selected20 as the initial value of the0.001 pore water pressure2.23 in the PFC pavement1.63 to indicate0.01 that the pavement was completely dry before rainfall. Data for inflow and outflow volumes and degree of saturation with respect to time were continually recorded during the analysis. Water 2021, 13, 843 7 of 14 Water 2021, 13, x FOR PEER REVIEW 7 of 14

25

20

15

10

5 Volumetric water water Volumetric content (%)

0 0.01 0.1 1 10 100 1000 10000 100000 Pavement suction (kPa)

FigureFigure 5. 5.SWCC SWCC curve curve of of the the PFC PFC pavement. pavement.

4. ResultsAccording and Discussion to previous studies, PFC pavements absorb the surrounding air to reduce 4.1.the Timenoise Taken from for moving Water Flowvehicle over tires the PFC [4]. PavementIn the case Surface of no rainfall, PFC pavement has a suctionVarious force values and aof negative rainfall value intensity, for whichthe pore ranged water from pressure 10 to [28]. 120 mm/h,Therefore, were in con- this sideredstudy, the in the initial numerical pore water analysis pressure to evaluate was set the to timea negative taken value for water within flow the over range the of PFC the pavementresidual zone. surface. The results are presented in Table3. The cells with no numerical values indicateIn the that numerical at that given analysis, rainfall it was intensity, noted surface that the ponding pore water did pressure not occur, values i.e., the within rainfall the intensityrange of wasthe residual not large zone enough had tono trigger significant surface effect ponding on the inallowable these cases. rainfall intensity and time taken for water flow over the PFC pavement surface. Based on Figure 5, −200 kPa

Tablewas selected 3. Time taken as the for initial water value flow overof the the pore PFC water pavement pressure surface in ( tthep). PFC pavement to indicate that the pavement was completely dry before rainfall. Data for inflow and outflow vol- umes and degree of saturationTime Taken with for Waterrespect Flow to time over were PFC Pavement continually Surface, recordedtp (min) during the LR SR I = 10 analysis.(m) (%) 20 30 40 50 60 70 80 90 100 110 120 (mm/h) 4. Results0.5 and Discussion 60 30 20 14 11 9 8 7 6 5 5 4 4.1. Time Taken2 for Water 85 Flow 35over the 22 PFC15.5 Pavement 12.5 10 Surface 8.5 7.5 6.5 5.5 5.5 5 10 Various4 values of - rainfall 49 intensity, 26 18which 14 ranged 11 from 9 10 8to 120 7 mm/h, 6 were 5.5 consid- 5 ered in the6 numerical - analysis - to 35evaluate 21 the 15 time 12 taken 10 for 8water 7 flow 6 over 6 the 5PFC pavement surface. The results are presented in Table 3. The cells with no numerical values 8 - - - 26 18 13.5 11 9 8 7 6 5.5 indicate that at that given rainfall intensity, surface ponding did not occur, i.e., the rainfall intensity 0.5was not large 60 enough 30 to trigger 19 14 surface 11 ponding 9 8 in these 7 cases. 6 5 5 4 PFC pavements2 75 with different33.5 21 equivalent15.5 12 water 10 flow 8.5 paths 7.5 (lengths 6 L 5.5R and 5slopes 4.5 SR) resulted15 in4 various times - taken. 41 Generally, 24 17 as 13 the10.5 rainfall9 intensity 7.5 6.5 increased, 6 5.5 the 4.5time taken for water flow over the PFC pavement surface decreased. This finding confirms that 6 - 55.5 28 19 14 11.5 9.5 8 7 6 5.5 5 the hydrologic characteristic is dependent on rainfall intensity and geometric design. 8 - - 34 21 15.5 12 10 8.5 7.5 6.5 6 5

Table 3. Time0.5 taken for 60 water flow 30 over 19 the PFC 14 pavement 11 9 surface 8 (tp). 7 6 5 5 4

LR 2 71Time Taken 33 for 21 Water 15 Flow 12 over 10PFC Pavement 8 7 Surface, 6 t 5p (min) 5 4 SR (%) (m)20 4I = 10 (mm/h) 104 20 38 3023 40 16 1250 10 60 8 70 780 690 5100 5110 4120 0.5 6 60 - 30 45 20 25 14 17 1311 119 9 8 87 76 55 55 44 2 85 35 22 15.5 12.5 10 8.5 7.5 6.5 5.5 5.5 5 8 - 59 29 19 14 11 9 8 7 6 5 5 10 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 Water 2021, 13, 843 8 of 14

Water 2021, 13, x FOR PEER REVIEW 8 of 14 Table 3. Cont.

Time Taken for Water Flow over PFC Pavement Surface, tp (min) LR SR 0.5 I = 10 60 30 19 14 11 9 8 7 6 5 5 4 (m) (%) 20 30 40 50 60 70 80 90 100 110 120 2 (mm/h) 75 33.5 21 15.5 12 10 8.5 7.5 6 5.5 5 4.5 150.5 4 59 - 30 41 19 24 1417 1113 910.5 89 77.5 6.5 6 6 5 5.5 5 4.5 4 6 - 55.5 28 19 14 11.5 9.5 8 7 6 5.5 5 28 67 - 32 - 20 34 15 21 1215.5 9 12 8 10 78.5 7.5 6 6.5 5 6 5 5 4 30 40.5 84 60 35 30 2119 1514 1211 109 8 8 77 66 5 5 5 4 4 2 71 33 21 15 12 10 8 7 6 5 5 4 6 115 39 23 16 12 10 8 7 6 5 5 4 20 4 104 38 23 16 12 10 8 7 6 5 5 4 86 - - 45 45 2525 1717 1313 1011 99 78 67 5 5 5 4 4 8 - 59 29 19 14 11 9 8 7 6 5 5 0.5 59 30 19 14 11 9 8 7 6 5 5 4 PFC pavements with different equivalent water flow paths (lengths L and slopes S ) 2 67 32 20 15 12 9 8 7 6 R 5 5 4 R resulted in30 various4 times 84 taken. Generally, 35 21 as15 the rainfall12 10 intensity8 7 increased, 6 5 the time5 taken4 for water flow6 over the 115 PFC pavement 39 23 surface 16 decreased.12 10 This8 finding7 6 confirms5 5 that4 the hydrologic characteristic8 - is dependent 45 25 on rainfall17 13 intensity 10 and9 geometric7 6 design.5 5 4

4.2. Effect4.2. of Effect Geometric of Geometric Design Design The relationshipsThe relationships between between the the time time taken taken forfor water water flow flow over over the thePFCPFC pavement pavement sur- surfaceface and and the the rainfall rainfall intensity intensity according according toto the different different geometric geometric designs designs are aredepicted depicted in in FigureFigure6. Based 6. Based on on the the results, results, it it isis observed that that tp decreasestp decreases with withthe increase the increase in the length in the of LR. For instance, for I = 40 mm/h and SR = 8%, as LR increased by 20 m (from 10 to 30 m), length of LR. For instance, for I = 40 mm/h and SR = 8%, as LR increased by 20 m (from tp decreased by 9 min (from 26 to 17 min). These results indicate that surface ponding 10 to 30 m), tp decreased by 9 min (from 26 to 17 min). These results indicate that surface occurred earlier for the PFC pavement that had a longer length LR. ponding occurred earlier for the PFC pavement that had a longer length LR.

(a) LR = 10 m (b) LR = 15 m

(c) LR = 20 m (d) LR = 30 m

Figure 6. Relationship between the time taken for the water flow over the PFC pavement surface (tp) and rainfall intensity (I) with different slope (SR) values. Water 2021, 13, 843 9 of 14

In this study, it was also found that the slope had a significant effect on the time taken for water flow over the PFC pavement surface. The increase in SR results in a corresponding increase in tp, as reported by Tan et al. [9]. For example, for LR = 10 m and I = 40 mm/h, tp increases by 12 min (from 14 to 26 min) when SR increases by 7.5% (from 0.5% to 8%). In Figure6d, it is noticed that as LR = 30 m, different values of SR result in correspond- ingly similar values of tp. The finding highlights that the hydrologic characteristic of the PFC pavement is less sensitive to the slope SR when it has a longer length LR. 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

In this study, the allowable rainfall intensity Ia 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 Table4.

Table 4. Allowable rainfall intensity, Ia.

Length, LR Slope, SR Allowable Rainfall Intensity, Ia (m) (%) (mm/h) 0.5 2.5 2 7.5 10 4 15 6 22.5 8 30 0.5 1.5 2 5 15 4 10 6 15 8 20 0.5 1 2 2.5 20 4 5 6 10 8 15

In Figure7, the allowable rainfall intensities Ia according to the slope SR and length LR are presented. It can be seen that when SR increases, there is an upward trend in Ia. This observation is consistent with that of Tan et al. [9]. In addition, Ia decreases when LR increases, which supports the results reported by Ranieri [29]. This observation implies that a PFC pavement with a shorter LR and longer SR exhibits higher drainage capacity. A PFC pavement with a shorter length LR is more sensitive to variations in the slope SR. For example, when the value of SR increases from 0.5% to 8%, for LR = 10 m, Ia significantly increases by 27.5 mm/h (an increase from 2.5 to 30 mm/h), but for LR = 15 m, I¬¬a increases by only 19 mm/h. However, for LR = 20 m, Ia slightly increases by 5 mm/h.

4.4. Effects of Thickness and Porosity The effects of the thickness T and porosity n of the PFC pavement on its hydro- logic characteristic were investigated. Observations were made for PFC pavements with T = 0.05 m and 0.1 m, and n = 10% and 20%. All the PFC pavements had the same water flow path of LR = 10 m and SR = 2%. The results are displayed in Tables5 and6. WaterWater2021 2021, 13, ,13 843, x FOR PEER REVIEW 1010 of of 14 14

FigureFigure 7. 7.Allowable Allowable rainfall rainfall intensity intensityIa Iaccordinga according to to the the slope slopeS RSRand and lengths lengthsL RL.R.

Table 5. Time taken for water flow over the PFC pavement surface (tp) for I = 50 mm/h, LR = 10 m, A PFC pavement with a shorter length LR is more sensitive to variations in the slope and SR = 2%. SR. For example, when the value of SR increases from 0.5% to 8%, for LR = 10 m, Ia signifi- cantlyThickness, increasesT by 27.5 mm/hPorosity, (ann increase fromTime 2.5 Takento 30 formm/h), Water but Flow for over LRPFC = 15 m, Ia increases(m) by only 19 mm/h. However,(%) for LR = 20 m, IPavementa slightly increases Surface, tp by(min) 5 mm/h. 10 6 0.05 4.4. Effects of Thickness and Porosity20 12.5 The effects of the thickness10 T and porosity n of the PFC pavement 13.5 on its hydrologic 0.1 characteristic were investigated.20 Observations were made for 27.5PFC pavements with T = 0.05 m and 0.1 m, and n = 10% and 20%. All the PFC pavements had the same water flow path of LR = 10 m and SR = 2%. The results are displayed in Tables 5 and 6. Table 6. Time taken for water flow over the PFC pavement surface (tp) for I = 100 mm/h, LR = 10 m,

andTableSR = 5. 2%. Time taken for water flow over the PFC pavement surface (tp) for I = 50 mm/h, LR = 10 m, and SR = 2%. Thickness, T Porosity, n Time Taken for Water Flow over PFC Thickness,(m) T Porosity,(%) n Time TakenPavement for Water Surface, Flowtp (min)over PFC Pave- (m) 10(%) ment Surface, 2.5 tp (min) 0.05 2010 5.56 0.05 1020 12.5 6 0.1 2010 1213.5 0.1 20 27.5 From the Tables, it can be seen that the thickness T and porosity n have a significant effectTable on 6. theTime hydrologic taken for water characteristic flow over ofthe a PFC PFC pavement pavement. surface As the (tp) thickness for I = 100 Tmm/h,and porosityLR = 10 m, n andincrease SR = 2%. by a factor of two, the time taken for the water flow over the PFC pavement surface t doubles. According to Hou et al. [11], this could be attributed to the storage Thickness,p T Porosity, n Time Taken for Water Flow over PFC Pave- capacity of the PFC pavement, a PFC pavement with a higher porosity n has a higher water (m) (%) ment Surface, tp (min) storage capacity. 10 2.5 0.05 4.5. Design Charts for PFC Pavement20 5.5 10 I L 6 S Five variables,0.1 namely, rainfall intensity , length R, and slope R for the water flow path, the time taken for the20 water flow over the PFC pavement12 surface tp, and the permeability of porous asphalt k, were used to develop design charts in terms of the LR I hydrologicFrom characteristic the Tables, it forcan the be PFCseen pavement.that the thickness For each T and value porosity of T , the n have values a significant of k are t ×k presentedeffect on asthe a hydrologic function of characteristicp . Table7 below of a PFC gives pavement. the values As ofthe the thickness five variables. T and porosity n increase by a factor of two,LR the time taken for the water flow over the PFC pavement surface tp doubles. According to Hou et al. [11], this could be attributed to the storage Water 2021, 13, 843 11 of 14

Table 7. Normalized variables for PFC pavement design.

tp ×k LR LR T SR (%) 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 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 0.3 200 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 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 0.135 300 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 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 0.12 400 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 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 0.08 600 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

It is possible to estimate the time taken for water to flow over the PFC pavement tp at the design rainfall intensity I using the design charts in Figure8. The use of these charts is described in the following example. A PFC pavement with an equivalent water flow path has a length LR = 10 m, a thickness T = 0.05 m, and LR/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 tp for different slopes SR can be determined as follows (Equation 5):

I 50 = = 0.00208 (5) k 24000

tp×k The values of are defined in Figure8. Consequently, the values of tp were calcu- LR lated and are listed in Table8. The results from the design charts demonstrate that the time taken for the water flow over the PFC pavement surface (tp) can be determined efficiently.

Table 8. Results of the time taken for water flow over the PFC pavement surface, tp.

tp×k Slope, SR (%) Time, tp (h) Time, tp (min) LR 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 Water 2021, 13, 843 12 of 14 Water 2021, 13, x FOR PEER REVIEW 12 of 14

(a) LR/T = 200 (b) LR/T = 300

(c) LR/T = 400 (d) LR/T = 600

Figure 8.FigureDesign 8. charts.Design charts.

It is possible to estimate the time taken for water to flow over the PFC pavement tp 5. Conclusionsat the design rainfall intensity I using the design charts in Figure 8. The use of these charts Inis this described study, the in hydrologicthe following characteristic example. A of PFC PFC pavement pavements with of various an equivalent lengths water and flow slopes waspath determinedhas a length via LR numerical = 10 m, a analysis.thickness A T series = 0.05 of m, analyses and LR/T was = conducted200. Assuming using that the lengthpermeability values of 10, k 15, = 6.6 20, mm/s and 30 = m24,000 and slopemm/h, values at a design of 0.5%, rainfall 2%, 4%,intensity 6%, and I = 8%50 mm/h, for the the equivalentresults of water the time flow taken path. for The the PFC water pavements flow over were the PFC simulated pavement for varioussurface t valuesp for different of rainfallslopes intensity, SR can whichbe determined ranged from as follows 10 to 120(Equation mm/h, 5): to determine the time taken for water to flow over the PFC pavement surface. The following conclusions are drawn: The hydrologic characteristic of a PFCI pavement50 significantly depends on the rainfall intensity and geometric design of the pavement.= PFC pavements= 0.00208 with different equivalent (5) k 24000 water flow paths resulted in different times for water overflow.Generally, as the rainfall intensity increases, the time for× water to flow over the PFC pavement surface decreases. t p k The observationThe values of of the effect of are the defined geometric in Figure design 8. on Co thensequently, hydrologic the characteristic values of tp were demonstrates following. As theLR length of the water flow path increases, the time for water overflowcalculated decreases. and However, are listed whenin Table the 8. slope The resu increases,lts from it increasesthe design significantly. charts demonstrate Based that on thesethe results, time taken it appears for the that water the flow hydrologic over the characteristic PFC pavement of surface the PFC (tp pavement) can be determined that has a longerefficiently. water flow path is less sensitive to variations in the slope. In addition, a PFC pavement having a shorter water flow path or steeper slope exhibits higher drainage capacity. Thickness and porosity have significant effects on the hydrologic characteristic of Water 2021, 13, 843 13 of 14

a PFC pavement. As the thickness and porosity increase by a factor of two, the time for water overflow is doubled. A series of design charts was developed based on the results of the numerical analysis of the PFC pavement. These charts can be used to determine the time for water overflow over the PFC pavement surface at specified rainfall intensities. Since this study was conducted based on numerical analysis, sequential experimental studies are necessary to verify the results presented.

Author Contributions: Conceptualization, T.H.N. and J.A.; methodology, J.A.; , J.A.; valida- tion, J.A. and T.H.N.; formal analysis, T.H.N.; investigation, T.H.N.; resources, J.A.; data curation, J.A.; writing—original draft preparation, T.H.N.; writing—review and editing, J.A. and T.H.N.; visu- alization, J.A.; supervision, J.A.; project administration, J.A.; funding acquisition, J.A. All authors have read and agreed to the published version of the manuscript. Funding: This work is supported by the Korea Agency for Infrastructure Technology Advancement (KAIA) grant funded by the Ministry of Land, Infrastructure and Transport (Grant 21CTAP-C152124-03). Data Availability Statement: Not applicable. Acknowledgments: The authors would like to thank the Ministry of Land, Infrastructure, and Transport of the Korean government for the grant from the Technology Advancement Research Program (Grant Number 21CTAP-C152124-03). Conflicts of Interest: The authors declare no conflict of interest.

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