Simulation of Water Flow Path Length (WFPL) and Water Film Depth (WFD) for Wide Expressway Asphalt Pavement

Abstract

This paper simulates actual rainfall conditions and raindrops flowing to form a water flow path (WFP) on the pavement surface of the wide expressway. Then, the different linear combination conditions, including longitudinal slope (LS), transverse slope (superelevation, TS), gradual change rate of TS, and pavement width (PW), were simulated and analyzed. The results show that (1) the influence of each linear index on the maximum water film path length (WFPL_max) and maximum water flow depth (WFD_max) differs (according to the absolute values of Beta, LS has the greatest influence on WFPL_max, and PW has the greatest influence on the WFD_max for both straight-line and circular-curve sections); (2) when the design value of LS is between 1.1% and 4%, the WFD_max can be effectively reduced by lowering the value of LS; (3) in the case of a high design value of LS, it can be considered to increase the TS of the pavement arch from 2% to 2.5% to effectively reduce the WFPL_max, and the wider PW, the better the reducing effect; (4) while widening the expressway, adjusting the TS from 2% to 2.5% can effectively offset the increasing effect of PW on the WFD_max. This research aims to fill the research gap in the simulation of runoff characteristics of wide expressway asphalt pavements and to improve the alignment design of expressways from the drainage perspective for the improvement of driving safety.

1. Introduction

With the expansion of the expressway, the early two-way four lanes have gradually changed to two-way eight lanes or more. The increase in pavement width increases the water flow path length (WFPL) in rainy conditions, resulting in an increase in the water film depth (WFD) and the occurrence of hydroplaning, threatening driving safety [1,2,3]. The formation of water on expressway pavements depends entirely on the WFD, and the definition of the WFD of pavement needs to be clarified due to the differences in the texture depth. The WFD is defined as the sum of the WFD of the surface and the average texture depth [4,5]. Factors affecting the WFD include rainfall intensity, the WFPL, and the geometric linear index of the road [6,7]. Among them, the geometric characteristics of the pavement are determined by the plane line, longitudinal line, and cross-section, and the combination of plane and longitudinal line and cross-section are not completely independent, and their interaction also determines whether the geometric indexes meet the linear requirements. The WFP refers to the rainwater that falls on the pavement surface and begins to flow after a certain period. The water passes through the pavement surface along the synthetic slope direction under the action of gravity, generating a certain water film thickness [8,9]. The geometric characteristics of the pavement are the key factors determining the runoff behavior, which determines the WFP, including the runoff direction, the runoff velocity, and the length of the drainage path [10]. The geometry indexes related to the synthetic slope mainly include the longitudinal slope (LS), transverse slope (superelevation, TS), gradual change rate of TS, and geometry indicators related to the pavement surface, which is mainly pavement width (PW). The analysis of runoff characteristics should consider the influence of properties of the road itself, and the two-dimensional model of the pavement surface can describe the time-varying process of the runoff more completely compared with the one-dimensional model [11,12,13,14,15].

2. Literary Review

A series of results and findings have been achieved in the research on the simulation and calculation of asphalt pavement runoff path and the WFD. Luo et al. stated that the simulation and analysis of pavement runoff should be combined with the attribute characteristics of the pavement itself [16]. Zhang et al. analyzed the effect of road slope on the WFD and found that the effect of TS on WFD was greater than that of LS [17]. Richard et al. proposed formulae for calculating the WFD_max for straight lines, circular curves, and vertical curves [18]. Guo et al. obtained a prediction model of the runoff of road surface through simulation and calculation and analyzed the effects of rainfall intensity, LS, TS, and roadside drainage mode on the runoff [19]. Qi et al. used a combination of theoretical analysis, outdoor tests, and numerical simulation means to study runoff behavior under different alignment combinations at a certain rainfall condition. The linear combination threshold of a two-way six-lane road with a design speed of 120 km/h under rainy conditions was obtained, which does not affect driving safety [10].

Regarding the prediction model of the WFD on pavement surface under rainfall conditions, a series of studies have been carried out, and a variety of WFD prediction models have been established, mainly including regression models [16], mathematical–physical models [18], and neural network models [20]. Rainfall is the main water source of pavement runoff, and the existing models of the relationship between rainfall and runoff of the road surface can be divided into empirical models and hydrodynamic models [21,22]. Empirical models are easily restricted by testing conditions and cannot express the continuous time-varying characteristics of the flow field, while hydrodynamic models, including kinematic waves, power waves, and diffusion wave models, are widely applicable [23]. Cea et al. analyzed the WFD of the pavement surface in an urban area using a two-dimensional average depth prediction model and found that the two-dimensional model can better express the changes in the flow field [24]. Staufer et al. analyzed the changing characteristics of runoff under the influence of different factors and simulated the WFD of runoff by using a complete power wave [25]. Ma et al. predicted the WFD using an artificial neural network model, but their results were desirable when the input and output variables were within the range of the training samples [6]. Li et al. found that pavements go through the stages of wetting, flow production, steady state confluence, and receding in sequence for natural rainfall events, which is a nonlinear time-varying process on a two-dimensional space [26,27,28].

Currently, simulation and calculation studies of the WFPL and WFD are mostly based on two-way four/six-lane roads, and the applicability to wide expressways needs further investigation. In addition, measurements of WFD are based on indoor scale model experiments that are difficult to perform in the actual field. Therefore, in this paper, the DPM model in Fluent fluid simulation software is used to simulate the actual rainfall conditions, and the EWF model is applied to simulate the raindrops flowing on the road surface. Different linear combination conditions of wide expressway, including the LS, TS, gradual change rate of TS, and PW, were considered and simulated. The simulation results of the WFPL and WFD distribution, the WFPL_max, and the WFD_max of the straight-line section or circular-curve section are analyzed, and on this basis, the optimization suggestions of linear indexes for drainage requirements of wide expressways are proposed. This study aims to fill the research gap in the simulation of runoff characteristics of wide expressway pavements and to improve the alignment design of expressways from the drainage perspective. The novelty of this research lies in the extension of the simulation and calculation studies of WFPL and WFD to the wide expressway asphalt pavement, which has eight/ten/twelve-lane roads. In addition, the DPM model and the EWF model are applied to simulate actual rainfall conditions and the raindrops flowing on the road surface.

Next, Section 3 introduces the methodology with which to simulate the pavement runoff characteristics in rainy conditions and predict the WFD values of straight-line sections and circular-curve sections. Section 4 shows and discusses the simulation results of the distribution of WFPL and WFD and the WFPL_max of the straight-line section and the circular-curve sections under different linear combination conditions. Section 5 gives the conclusions of this paper based on the simulation results and analysis obtained from Section 4.

3. Methodology

To simulate the pavement runoff characteristics in rainy conditions, the Fluent software was selected, and the DPM model was applied. To predict the WFD values of straight-line sections and circular-curve sections of wide expressway pavement, a WFD prediction model was proposed, and the different linear combination conditions were selected.

3.1. Modelling of the Pavement

In this study, ANSYS Fluent 2022 was selected for the simulation of pavement runoff characteristics in rainy conditions. The used model in this study consists of a pavement triangular mesh surface model and a rainfall model. The pavement surface model consists of several triangular mesh planes with 1 m side length (see Figure 1), and the triangular mesh is generated by cutting the model generated from the HintCAD 8.0.5 version.

The formation of runoff on the pavement surface after rainfall is a complex two-phase interaction process which can be decomposed into two processes: the formation of raindrops and the falling; and the formation of runoff after contact with the pavement surface. The rainfall area was constructed in the form of a rectangle, with the top surface of the rectangle defined as the rainfall inlet plane (velocity inlet), and the rest of the surface defined as the water outlet (pressure outlet), as shown in Figure 2. The pavement model is located in the rainfall region (see the light blue area in Figure 3), which forms a whole with the rectangular body through shared topology. Raindrop particles are shot in from the top surface of the model and fall on the pavement surface to form a water film, which flows under gravity and surface tension and is removed from the pressure outlet.

3.2. Selection of Simulation Models

3.2.1. Rainfall Simulation

Rainfall simulation was performed through the simulation of raindrop particle injection with a certain rainfall intensity on the surface of a pavement geometry model. Conditions to be met include the following:

• The flow input is stable and consistent with the set rainfall intensity;
• The range of simulated rainfall needs to completely cover the surface of the pavement geometry model, and the spatial distribution is uniform and reasonable;
• The flow inputs in the form of raindrops during rainfall should be reproduced in a granular manner as far as possible;
• The duration of rainfall can be controlled to simulate rainfall within a certain time interval.

3.2.2. Runoff Simulation

Runoff simulation was conducted via simulation of the water flow on the pavement surface and the distribution of WFD in each area of the pavement, etc. Conditions to be met include the following:

• Can be computed with the DPM model to collect DPM fluid particles and form a water film on the surface of the pavement geometry model;
• Able to simulate the flow of runoff on the pavement surface, tracking the water film in the selected area under the action of gravity, surface tension, etc., to form runoff and flow out of the pavement width dissipation;
• Can visualize the depth distribution of water on the pavement surface at different locations in the form of WFD.

Combining the above simulation needs, the DPM model was chosen to simulate rainfall, reflecting the raindrop size by the diameter of DPM particles and describing the intensity of rainfall by setting the second-phase flow rate. The EWF model was used to collect the DPM raindrop particles and realize the fluid flow on the surface of the pavement model, tracking the flow of the water film in the selected area after it had been aggregated into the runoff.

3.3. Rainfall Condition Simulation

The DPM model is a simulation of the rainfall process. Based on selecting flow as turbulence and discrete-phase particles as liquid water, raindrop particle diameter, rainfall start time and stop time, rainfall speed, and rainfall intensity need to be determined [29]. The diameter of raindrops is not constant during rainfall, and the average diameter of raindrops will change with the real-time change in rainfall intensity. In relevant studies, the average diameter of raindrops under different rainfall intensities is shown in

Table 1. Hourly rainfall intensity and mean diameter of raindrop.

Rainfall Intensity (mm/h)	Mean Diameter of a Raindrop (mm)	Hourly Rainfall Intensity (mm/h)	Mean Diameter of a Raindrop (mm)
0.25	0.75–1.00	25.4	2.00–2.25
1.27	1.00–1.25	50.8	2.25–2.75
2.54	1.25–1.50	101.6	2.75–3.00
12.70	1.75–2.00	152.4	3.00–3.25

.

In this study, the average diameter of raindrops was 2.86 mm via interpolation method according to the rainfall intensity of 60 min rainfall duration once in five years. Concerning the previous setting of the boundary, the rainfall duration of 60 min was selected to take the rainfall intensity as 0.0242 mm/s within 120 s after the start of rainfall (start time was 0 s and stop time was 120 s).

3.4. WFD Prediction Model

This paper analyzed and compared the regression formulae of the existing more-classical WFD prediction models, mainly including the model proposed by Luo, the model proposed by Ji, the Gallway model, the Wambold model, etc., to determine the parameters of the WFD prediction model of this paper and its experimental conditions, as follows:

(1)

The model proposed by Luo et al. [16]:

$$W_D=0.068\times \frac{L^{0.32}·I^{0.41}·T_{XD}^{1.17}}{S^{0.31}}.$$

(1)

(2)

The model proposed by Ji et al. [30]:

$$h=0.1258·l^{0.6715}·i^{-0.3147}·r^{0.7786}·{TD}^{0.7261}.$$

(2)

(3)

The Gallaway model [31]:

$$d=0.1664·T^{0.11}·L^{0.43}·I^{0.59}·S^{-0.42}-T.$$

(3)

(4)

The Wambold model [32]:

$$d=5.979\times {10}^{-3}·T^{0.11}·I^{0.59}·S^{-0.42}-T.$$

(4)

From these four models, it can be seen that the factors affecting the WFD in the regression models include the four parameters: rainfall intensity, road slope, WFPL, and textile depth. It can be seen from the WFD prediction model that rainfall intensity is the only environmental factor that affects the WFD considered in the above four models, which is one of the assumptions of the above model; i.e., it is assumed that rainfall intensity is a stable variable that does not change over time. The model structure of the WFD model in this study was similar to the above model, but the parameters need to be comprehensively determined based on the indoor and outdoor simulated rainfall tests and are assumed to be modeled as follows:

$$d=K·I^{k_1}·L^{k_2}·S^{k_3} ·T^{k_4},$$

(5)

where d—WFD (mm); L—WFPL (m); S—road slope (%); I—rainfall intensity (mm/min); T—texture depth (mm); and K, k₁, k₂, k₃, k₄—regression coefficient.

A multivariate linear regression equation is obtained by transforming it logarithmically:

$$\ln d=\ln K+k_1\ln I+k_2\ln L+k_3\ln S+k_4\ln T.$$

(6)

After the multiple linear regression equations were analyzed, the regression equations and regression coefficients were tested for significance, where the F test was used for the regression equation and t testing was used for the significance of the regression coefficients.

3.5. Linear Combination Conditions

3.5.1. Straight-Line Section

In the simulation of the straight-line section, the right side of the straight-line section with 2% TS arch and without LS was first selected to verify the validity of the simulation. Then, the following linear combination conditions (

Table 2. Linear combination conditions of straight-line sections.

Number of Lanes per Side	Section Length (m)	LS (%)	TS (%)
4	300	−0.3	2
		−0.5
5		−1.0
		−2.0	2.5
6		3.0
		−4.0

), including the number of lanes per side, LS, and TS, were selected for simulation and analysis according to the current specification requirement [33].

3.5.2. Circular-Curve Section

In the circular-curve section, the inner and outer sides of the roadway reach the same TS. For the inner section of the left-deflected circular curve, the synthetic LS direction is pointing from the middle line to the shoulder, which is consistent with the synthetic slope situation of the arch in the straight section, and the WFD_max occurs on the shoulder side of the expressway. The water film of the inner lane is relatively small, and it will not affect driving safety. As for the outer section of the circular curve, the synthetic LS direction points from the shoulder to the median strip, and the endpoint of WFP is located in the left curb strip, which may lead to the high WFD value of the inner lane and affect driving safety. In addition, under the condition of a design speed of 120 km/h, whether the minimum TS of 2% for circular curves suggested by the specification applies to wide circular-curve sections needs to be examined and discussed. Based on the above reasons, the following linear combination conditions were selected for simulation analysis (

Table 3. Linear combination condition of circular-curve sections.

Number of Lanes per Side	Section Length (m)	Radius of the Circular Curve (m)	LS (%)	TS (%)
4	300	4500	−0.3%	2.0%
			−0.5%
5			−1.0%
			−2.0%	2.5%
6			−3.0%
			−4.0%

).

4. Results and Discussion

According to the simulation, the results for the distribution of WFPL and WFD, as well as the WFPL_max of the straight-line section and the circular-curve sections under different linear combination conditions, were presented and analyzed, respectively. The results show that LS has the greatest influence on the WFPL_max, while PW has the greatest influence on the WFD_max.

4.1. Simulation Results and Discussion of Straight-Line Sections

4.1.1. Distribution of the WFPL and WFD

For the straight-line section without the LS, the right side of the pavement with eight lanes in both directions and a 2% TS was selected to verify the validity of the simulation. The WFPL and WFD distribution simulation results are shown in Figure 4. It can be seen that the direction of runoff on the pavement surface is the same as that in the direction of TS (Figure 4a). With a WFPL of 18.75 m, the WFD shows an increasing trend from the inner side of the travel lanes to the outer side (Figure 4b). The simulation results are consistent with the actual situation, and the validity of the simulation was verified. Thus, the simulation can be considered effective.

For the straight-line section with LS, WFPL and WFD distribution maps were obtained from the simulation of a two-way eight-lane section with a section length of 300 m, LS of 0.5%, and TS of 2%. The WFPL and WFD distribution simulation results are shown in Figure 5, which shows that the WFP flow direction is the same as that of the synthetic LS direction, and the WFD shows an incremental tendency from the inner side of the traveled roadway to the outer side. The results of the above simulation lead to the preliminary conclusion that the WFP flows from the inside of the roadway to the outside of the straight roadway, and the WFD gradually increases as the WFP moves away from the center median.

4.1.2. Maximum Water Film Path Length (WFPL_max)

Under different LS and TS combinations, the WFPL_max of the straight-line sections of the two-way eight-lane, ten-lane, and twelve-lane road sections were simulated, and the results of all the linear combinations of conditions are shown in

Table 4. WFPL_max (m) under different LS, TS, and PW conditions at 120 km/h.

Number of Lanes in Both Sides		LS (%)	0.3	0.5	1.0	2.0	3.0	4.0
	TS (%)
8	2.0		15.93	16.23	17.61	22.27	28.39	35.22
	2.5		15.86	16.06	16.96	20.17	24.60	29.72
10	2.0		19.72	20.10	21.80	27.58	35.15	43.60
	2.5		19.64	19.89	21.00	24.97	30.46	36.79
12	2.0		23.51	23.97	25.99	32.88	41.91	51.99
	2.5		23.42	23.71	25.04	29.77	36.32	43.87

. When the PW and TS are the same, the WFPL_max values are positively correlated with LS. When the arch TS is 2%, the WFPL of 4% LS under two-way eight lanes is 2.23 times that of 2% LS. The WFPL of 4% LS under two-way ten lanes is 2.19 times that of 2% LS. The WFPL of 4% LS under two-way twelve lanes is 2.20 times that of 2% LS. Although the increase in LS allows for an increase in the synthetic slope, its direction is gradually biased in the direction of LS and leads to an increase in the WFPL_max. In addition, increasing the TS from 2% to 2.5% under the same LS condition effectively reduces the WFPL_max. This can be explained by the fact that the increase in TS increases the synthetic slope and, at the same time, makes the direction of the WFP gradually biased in the direction of TS, and since the transverse width of the pavement is much smaller than the longitudinal length, this makes the WFPL decrease, effectively reducing the WFPL_max. Furthermore, under a certain TS and LS, the greater the number of lanes, the longer the WFPL_max. This is because under a certain TS and LS, the direction of the WFP remains unchanged, while the increase in the number of lanes makes the PW increase. The WFPL becomes longer, so the WFPL_max also becomes longer accordingly.

The results of the WFPL_max reduction since the change of TS from 2% to 2.5% for different numbers of lanes at 120 km/h are shown in

Table 5. WFPLmax reduction (m) of TS from 2% to 2.5% for different number of lanes at 120 km/h.

	Number of Lanes on Both Sides	8	10	12
LS (%)
0.3		0.07	0.08	0.09
4		5.50	6.81	8.12
Multiple		78	85	90

. For a certain number of lanes, the larger the LS, the more significant the reduction of the WFPL_max when the TS is changed from 2% to 2.5%. For a two-way eight-lane pavement, changing TS from 2% to 2.5%, the reduction of WFPL_max at 4% LS is 5.50 m, which is 78 times the decrease corresponding to 0.3% LS (0.07 m). For the two-way ten-lane pavement, changing TS from 2% to 2.5%, the reduction of WFPL_max at 4% LS is 6.81 m, which is 85 times the decrease in the corresponding 0.3% LS. For the two-way twelve-lane pavement, the reduction of WFPL_max at 4% LS is 90 times the decrease corresponding to 0.3% LS. Moreover, the reduction of WFPL_max by changing TS from 2% to 2.5% is not only obvious with the increase in LS but also significant with the increase in PW. It can be seen that in the case of limited-alignment and large-design LS, it can be considered to increase the TS of the pavement arch from 2% to 2.5% to effectively reduce the WFPL_max, and the wider the PW, the better the improvement effect.

Considering that in the above analysis, there is mutual influence among the TS, LS, and PW indicators, it is not possible to accurately rank the influence degree of each indicator on the WFPL_max; therefore, a linear regression analysis was carried out by using SPSS to compare using the standardized coefficient absolute value, and the results of the analysis are shown in

Table 6. Multiple regression model coefficient of WFPL_max.

Model	Unstandardized Coefficient		Standard Coefficient	t	Significance	Covariance Statistic
	B	Standard Error	Beta			Tolerance	VIF
Constant	2.26	1.28	/	1.77	0.078	/	/
LS	−5.80	0.11	−0.78	−54.48	0.00	1.00	1.00
TS	−6.23	0.46	−0.20	−13.56	0.00	1.00	1.00
PW	1.46	0.037	0.56	38.94	0.00	1.00	1.00

. As can be seen from Table 6, when using the absolute value of the standard coefficient for comparison, LS has the greatest influence on WFPL_max, followed by PW, and TS has the least influence.

4.1.3. Maximum Water Flow Depth (WFD_max)

The results of the WFD_max, WFD on the first lane middle line, and WFD on the second lane middle line of the straight-line sections under different LS, TS, and PW conditions at 120 km/h are shown in

Table 7. WFD_max (mm) under different LS, TS, and PW conditions at 120 km/h.

Number of Lanes on Both Sides		LS (%)	0.3	0.5	1.0	2.0	3.0	4.0
	TS (%)
8	2.0		3.20	3.21	3.23	3.29	3.36	3.42
	2.5		3.01	3.01	3.03	3.07	3.12	3.17
10	2.0		3.46	3.46	3.49	3.56	3.63	3.70
	2.5		3.25	3.25	3.27	3.32	3.37	3.43
12	2.0		3.69	3.69	3.72	3.79	3.87	3.94
	2.5		3.46	3.47	3.48	3.53	3.59	3.65

and

Table 8. WFD (mm) on the middle line of the first lane and the second lane under different LS, TS, and PW conditions at 120 km/h.

Number of Lanes on Both Sides		LS (%)	The First Lane						The Second Lane
	TS (%)		0.3	0.5	1.0	2.0	3.0	4.0	0.3	0.5	1.0	2.0	3.0	4.0
8	2.0		1.67	1.68	1.69	1.73	1.76	1.80	2.31	2.31	2.33	2.38	2.42	2.47
	2.5		1.57	1.58	1.59	1.60	1.64	1.66	2.16	2.17	2.19	2.22	2.25	2.28
10	2.0		1.68	1.68	1.68	1.73	1.75	1.80	2.32	2.32	2.33	2.38	2.43	2.47
	2.5		1.58	1.59	1.59	1.61	1.65	1.67	2.17	2.18	2.19	2.22	2.25	2.29
12	2.0		1.67	1.68	1.69	1.73	1.76	1.81	2.31	2.32	2.33	2.39	2.44	2.47
	2.5		1.58	1.59	1.59	1.61	1.65	1.68	2.17	2.19	2.19	2.22	2.26	2.29

. For the 120 km/h design speed, the WFD_max and WFD values on the first/second lane middle line are not greater than 5.03 mm, being lower than the safe WFD thresholds and meeting the needs of traffic safety. When PW is the same and the TS is fixed at 2% or 2.5%, the WFD_max and WFD values are positively correlated with LS; i.e., the increase in LS makes the WFD generally increase. This is because when the TS is the same, the increase in LS makes the synthetic slope direction biased towards the longitudinal direction of the pavement, the WFPL_max becomes longer, and the retention time of the surface water in the pavement becomes longer, which increases the WFD.

Moreover, when the LS is the same, increasing the TS from 2% to 2.5% can effectively reduce the WFD. As the LS and PW increase, increasing the TS is more effective in the WFD. This not only increases the synthetic slope of the pavement and accelerates the drainage process but also gradually biases the direction of the WFP in the direction of the TS, which shortens the WFPL because the transversal width of the pavement is much smaller than the longitudinal length, which reduces the WFPL, and the retention time in the pavement surface is shortened, so the WFD is greatly reduced.

The relationship between the change rate of the WFD_max and the LS is shown in Figure 6, and it can be seen that the WFD_max keeps increasing with the increase in LS. The trend of the WFD_max change with LS under different PW conditions is relatively close; i.e., the change rate shows a process of slow to fast and then to slow. In the case of LS less than 2.2%, the increase rate of the WFD_max becomes faster as LS increases. For LS greater than or equal to 2.2%, the WFD_max change rate becomes slower with the increase in LS. Therefore, when LS is between 1.1% and 4%, the change in LS has a greater effect on the change rate of the WFD_max than when LS is between 0.3% and 1.1%. It can be concluded that the WFD_max can be effectively reduced by decreasing the LS when LS is between 1.1–4%, and it is not recommended to prioritize the adjustment of LS to reduce the WFD_max when the LS is less than 1.1%.

The following linear regression analysis was carried out using SPSS to rank the effect degree of the TS, LS, and PW on the WFD_max regarding the absolute value of the standardized coefficient, and the results of the analysis are shown in

Table 9. Multiple regression model coefficient of the WFD_max.

Model	Unstandardized Coefficient		Standard Coefficient	t	Significance	Covariance Statistic
	B	Standard Error	Beta			Tolerance	VIF
Constant	3.19	0.011		284.30	0.00
LS	−0.06	0.001	−0.27	−63.60	0.00	1.00	1.00
TS	−0.48	0.004	−0.50	−119.31	0.00	1.00	1.00
PW	0.06	0.000	0.82	194.94	0.00	1.00	1.00

. As can be seen from Table 9, when using the absolute value of the standard coefficient for comparison, the PW has the greatest impact on the WFD_max, followed by the TS, and the LS has the least impact.

4.2. Simulation Results and Discussion of Circular-Curve Sections

4.2.1. Distribution of the WFPL and WFD

Under the conditions of −1% LS and 2% TS, the WFPL and WFD distribution results of the two-way eight lanes of the circular-curve sections are shown in Figure 7. It can be seen that the direction of the WFP and the synthetic slope direction are similar. The WFD increases from the outside to the inside of the lane, with the maximum value distributed on the left curb. Preliminary conclusions can be drawn from the results of the above simulation: due to the presence of the TS in the outside of the circular-curve section, the water flows from the outer to the inner side of the pavement, and the WFD gradually decreases as the WFP moves away from the center.

4.2.2. Maximum Water Film Path Length (WFPL_max)

The WFPL_max of two-way eight-lane, ten-lane, and twelve-lane circular-curve sections are simulated under different linear combinations of TS and LS. The results are shown in

Table 10. WFPL_max under different LS, TS, and PW conditions at 120 km/h.

Number of Lanes on Both Sides		LS (%)	0.3	0.5	1.0	2.0	3.0	4.0
	TS (%)
8	2.0		18.96	19.33	20.96	26.52	33.80	41.93
	2.5		18.88	19.12	20.19	24.01	29.29	35.38
10	2.0		22.75	23.19	25.16	31.82	40.56	50.31
	2.5		22.66	22.95	24.23	28.81	35.15	42.45
12	2.0		26.54	27.06	29.35	37.12	47.32	58.70
	2.5		26.44	26.77	28.27	33.62	41.00	49.53

. At a certain value of PW and TS, WFPL_max values are positively correlated with LS. With the arch TS is 2%, WFPL of 4% LS under two-way eight lanes is 2.21 times that of 0.3% LS. WFPL of 4% LS under two-way ten lanes is 2.21 times that of 2% LS. WFPL of 4% LS under two-way twelve lanes is 2.22 times that of 2% LS. The increase in LS, although allowing for an increase in the synthetic slope of the pavement, is biased in the direction of LS, while the longitudinal length of the pavement is quite long compared to the transverse width, which increases WFPL.

Increasing TS from 2% to 2.5% for circular-curved sections with the same LS can effectively reduce the WFPL_max. This not only increases the synthetic slope but also causes the synthetic LS direction to gradually deviate from the TS direction, which makes the WFPL decrease greatly because the transverse width of the pavement is quite short relative to the longitudinal length. When LS is the same, WFPL_max continues to become longer as PW becomes larger. This is due to certain conditions of the same TS and LS. The direction of water flowing is unchanged, and the number of lanes increases so that the PW increases. The water flow is correspondingly longer, so the WFPL increases greatly.

As shown in

Table 11. WFPL_max reduction (m) of TS from 2% to 2.5% for different number of lanes at 120 km/h.

	Number of Lanes on Both Sides	8	10	12
LS (%)
0.3		0.08	0.09	0.10
4		6.55	7.86	9.17
Multiple		82	87	92

, it can be seen that when the number of lanes is certain, the larger the LS, the better the reduction effect of WFPL_max through changing TS from 2% to 2.5%. For a two-way eight-lane pavement, changing TS from 2% to 2.5%, the reduction of WFPL_max at 4% LS is 6.55 m, which is 82 times the decrease corresponding to 0.3% LS (0.08 m). For the two-way ten-lane pavement, changing TS from 2% to 2.5%, the reduction of WFPL_max at 4% LS is 7.86 m, which is 87 times the decrease in the corresponding 0.3% LS (0.09 m). For the two-way twelve-lane pavement, the reduction of WFPL_max at 4% LS is 9.17 m, which is 92 times the decrease corresponding to 0.3% LS. From Table 11, it can be found that the reduction of WFPL_max, when changing TS from 2% to 2.5%, is not only obvious with the increase in LS, but is significant with the increase in PW. It can be concluded that in the case of a wide circular-curve section with limited alignment and large design LS, it can be considered to increase TS from 2% to 2.5% to reduce WFPL_max, and the wider the pavement width, the better the improvement effect.

The following linear regression analysis was carried out using SPSS to rank the effect degree of the three indicators of TS, LS, and PW on WFPL_max regarding the absolute value of the standard coefficient, and the results are shown in

Table 12. Multiple regression model coefficient of WFPL_max.

Model	Unstandardized Coefficient		Standard Coefficient	t	Significance	Covariance Statistic
	B	Standard Error	Beta			Tolerance	VIF
Constant	7.37	1.47	/	5.01	0.00	/	/
LS	6.46	0.12	0.82	55.3	0.00	1.00	1.00
TS	−6.90	0.53	−0.19	−13.02	0.00	1.00	1.00
PW	1.43	0.04	0.49	33.03	0.00	1.00	1.00

. From Table 12, it can be seen that when using the absolute value of the standard coefficient for comparison, the LS has the greatest influence on the WFPL_max, followed by the PW, and the TS has the smallest influence.

4.2.3. Maximum Water Flow Depth (WFD_max)

Results of the WFD_max, the WFD on the first lane middle line, and the WFD on the second lane middle line of circular-curve sections under different LS, TS, and PW conditions at 120 km/h are shown in

Table 13. WFD_max (mm) under different LS, TS, and PW conditions at 120 km/h.

Number of Lanes on Both Sides		LS (%)	0.3	0.5	1.0	2.0	3.0	4.0
	TS (%)
8	2.0		3.41	3.42	3.44	3.51	3.58	3.65
	2.5		3.21	3.21	3.22	3.27	3.33	3.38
10	2.0		3.64	3.65	3.67	3.75	3.82	3.89
	2.5		3.42	3.43	3.44	3.49	3.55	3.61
12	2.0		3.85	3.86	3.88	3.96	4.04	4.12
	2.5		3.62	3.62	3.64	3.69	3.75	3.81

and

Table 14. WFT (mm) on the middle line of the first lane and the second lane under different LS, TS, and PW conditions at 120 km/h.

Number of Lanes on Both Sides		LS (%)	The First Lane						The Second Lane
	TS (%)		0.3	0.5	1.0	2.0	3.0	4.0	0.3	0.5	1.0	2.0	3.0	4.0
8	2.0		3.23	3.24	3.26	3.32	3.39	3.45	3.23	3.24	3.26	3.32	3.39	3.45
	2.5		3.04	3.04	3.05	3.10	3.15	3.20	3.04	3.04	3.05	3.10	3.15	3.20
10	2.0		3.48	3.49	3.51	3.58	3.66	3.72	3.48	3.49	3.51	3.58	3.66	3.72
	2.5		3.27	3.28	3.29	3.34	3.40	3.45	3.27	3.28	3.29	3.34	3.40	3.45
12	2.0		3.71	3.71	3.74	3.81	3.89	3.96	3.71	3.71	3.74	3.81	3.89	3.96
	2.5		3.48	3.49	3.50	3.56	3.61	3.67	3.48	3.49	3.50	3.56	3.61	3.67

. For circular-curve sections under the design speed of 120 km/h, the WFD_max and the WFD on the first/second lane middle line are not greater than 5.03 mm, are lower than the safe WFD thresholds, thus meeting the needs of traffic safety. When PW and TS are certain, the WFD_max and WFD on the first/second lane middle line increase with the increase in LS. For two-way eight lanes under 2% TS and 4% LS, the WFD_max is 3.65 mm relative to a 2% LS increase of 0.14 mm. For two-way ten lanes under 2% TS and 4% LS, the WFD_max is 3.89 mm relative to the 2% LS increase of 0.14 mm. For the two-way ten lanes under 2% TS and 4% LS, the WFD_max is 4.12 mm relative to the 2% LS increase of 0.16 mm. This is because when the TS is fixed, the increase in LS makes the synthetic slope direction favor the longitudinal direction of the pavement, the WFPL_max becomes bigger, and the retention time of the water flow on the surface becomes longer, which leads to the increase in the WFD.

When the LS is the same, increasing the TS from 2% to 2.5% can effectively reduce the WFD. As the LS and PW increase, increasing the TS has a better effect on reducing the WFD. The increase in TS can not only increase the synthetic slope of the pavement and accelerate the water drainage but also make the direction of the WFP gradually biased in the direction of TS because the length of the transverse width of the pavement is much smaller than the length of the longitudinal direction, which makes the pavement surface water in a certain width of the pavement flow through a reduced distance, and the WFPL is shortened so that the WFD is greatly reduced. Moreover, the WFD_max and the WFD on the first and second lanes become larger as the PW increases at a certain TS and LS. This is due to the fact that as the number of lanes increases, the pavement becomes wider, the WFPL_max becomes longer, the corresponding WFPL also becomes longer, and the retention time of the surface water on the pavement becomes longer, resulting in the accumulation of water film, and the WFD increases accordingly.

The relationship between the change rate of the WFD_max and LS is shown in Figure 8. From Figure 8, it can be seen that when LS is certain, the change rate of the WFD_max (Figure 8a), the change rate of the WFD in the first travel lane middle (Figure 8b), and the change rate of the WFD in the second lane middle (Figure 8c) decrease successively, indicating that the further the distance from the central median, the smaller the change rate of the WFD affected by LS. Moreover, the change trends of the WFD_max and the WFDs in the first/second lanes with the growth of the LS are consistent, and all of them have the maximum value at LS of 2.1%. In cases where the LS is less than 2.1%, the WFD change rate becomes faster as LS increases, while in cases where the LS is greater than or equal to 2.1%, the WFD change rate becomes slower with the LS growth. In the case of LS between 1.1% and 4%, the change in LS has a greater effect on the WFD than in the case of LS between 0.3% and 1.1%. It can be concluded that it is more effective to reduce the WFD by adjusting LS when the LS is between 1.1–4%, and it is not recommended to prioritize adjusting the LS metrics to reduce the WFD_max when the LS is less than 1.1%.

Moreover, widening of two-way ten lanes to twelve lanes while adjusting TS from 2% to 2.5%, or widening of two-way eight lanes to ten lanes while adjusting TS from 2% to 2.5%, can both reduce the WFD_max, and the wider the pavement, the greater the magnitude of the reduction of the WFD_max. The maximum reduction can reach 0.08 mm. Therefore, in the design of the widening of the old road, 2.5% can be considered the minimum TS to effectively alleviate the problem of the WFD on the inside of the road becoming larger as a result of road widening.

The linear regression analysis was carried out using SPSS to rank the influence degree of the three indicators of the TS, LS, and PW on the WFD_max regarding the absolute value of the standard coefficient, and the results are shown in the table below. As can be seen from

Table 15. Multiple regression model coefficient of WFD_max.

Model	Unstandardized Coefficient		Standard Coefficient	t	Significance	Covariance Statistic
	B	Standard Error	Beta			Tolerance	VIF
Constant	3.54	0.01	/	312.76	0.00	/	/
LS	0.06	0.00	0.30	68.03	0.00	1.00	1.00
TS	−0.51	0.00	−0.55	−123.73	0.00	1.00	1.00

, when using the absolute value of the standard coefficients for comparison, PW has the greatest impact on WFD_max, followed by TS and LS.

5. Conclusions

Based on the simulation results and analysis of straight-line and circular-curve sections under different linear combination conditions, the following conclusions can be drawn:

• According to the absolute values of Beta, the influence of each linear index on the WFPL_max and WFD_max for both straight-line and circular-curve sections is different: LS has the greatest influence on the WFPL_max, followed by the PW; while the PW has the greatest influence on the WFD_max, and LS has the least effect.
• WFD_max is positively correlated with the WFPL_max, and the increase in the WFPL makes the retained time of water flow in the pavement longer, increasing the WFD. The WFD increases gradually along the direction of the WFPL_max and reaches the maximum value on the inside of the road curb.
• When the design value of LS is between 1.1% and 4%, the WFD_max can be effectively reduced by lowering the LS; and when the LS is less than 1.1%, it is not recommended to prioritize the adjustment of the LS to reduce the WFD_max.
• In the case of large design values of the LS, it can be considered to effectively reduce the WFPL_max by increasing the arch TS from 2% to 2.5% for straight-line and circular-curve sections, and the wider PW, the better the improvement effect.
• While the pavement width is widened, adjusting TS from 2% to 2.5% can effectively offset the increasing effect of the PW on the WFD_max.

Finally, this research aims to provide a better understanding of the runoff characteristics of wide expressway asphalt pavements. Future studies will be focused on the simulation of S-curve sections under different linear combination conditions, establishing a three-dimensional numerical model of a wide expressway asphalt pavement to simulate the rainfall catchment process and field validation.