Evaluating Curb Inlet E ffi ciency for Urban Drainage and Road Bioretention Facilities

An updated two-dimensional flow simulation program, FullSWOF-ZG, which fully (Full) solves shallow water (SW) equations for overland flow (OF) and includes submodules modeling infiltration by zones (Z) and flow interception by grate-inlet (G), was tested with 20 locally depressed curb inlets to validate the inlet efficiency (Eci), and with 80 undepressed curb inlets to validate the inlet lengths (LT) for 100% interception. Previous curb inlet equations were based on certain theoretical approximations and limited experimental data. In this study, 1000 road-curb inlet modeling cases from the combinations of 10 longitudinal slopes (S0, 0.1–1%), 10 cross slopes (Sx, 1.5–6%), and 10 upstream inflows (Qin, 6–24 L/s) were established and modeled to determine LT. The second 1000 modeling cases with the same 10 S0 and 10 Sx and 10 curb inlet lengths (Lci, 0.15–1.5 m) were established to determine Eci. The LT and Eci regression equations were developed as a function of input parameters (S0, Sx, and Qin) and Lci/LT with the multiple linear regression method, respectively. Newly developed regression equations were applied to 10,000 inlet design cases (10 S0, 10 Sx, 10 Qin, and 10 Lci combinations) and comprehensively compared with three equations in previous studies. The 100% intercepted gutter flow (Qg100) equations were derived, and over-prediction of Qg100 from previous methods was strongly correlated to smaller S0. Newly developed equations gave more accurate estimations of LT and Eci over a wide range of input parameters. These equations can be applied to designing urban drainage and road bioretention facilities, since they were developed using a large number of simulation runs with diverse input parameters, but previous methods often overpredict the gutter flow of total interception when the longitudinal slope S0 is small.


Introduction
The urban drainage system is designed and built to effectively convey the rainfall runoff out of the urban area to prevent inundation and local flooding [1], which can cause property damage and affect traffic and human safety. Curb inlets effectively intercept surface runoff into underground drainage pipes or bioretention facilities. As an important and typical practice, road bioretention facilities, which combine green/gray infrastructures to facilitate road runoff control through infiltration and storage, remove certain contaminants and sediments, and decrease roads' local flood inundation risk, are widely used in the pilot Sponge City construction in China [2] and all over the world. Li et al. [3] found that the curb inlet could be the bottleneck of road bioretention facilities that impedes the runoff The hydraulic performance of curb inlets for roadway drainage has been studied for more than 60 years, which was reviewed and summarized by Izzard [8], Li [9], and presently systematically documented in the Hydraulic Engineering Circular No. 22 (HEC-22) by Brown et al. [10]. In 1979, the Federal Highway Administration [11] first published a technical guide for the design of urban highway drainage, and then updated it in 1984 into a HEC-12 entitled "Drainage of Highway Pavement'' [12]. HEC-12 summarizes a semi-theoretical method developed for estimating street hydraulic capacities and procedures for sizing street inlets. The most recent HEC-22 [10] was published and widely used in the USA, and refined the design procedures stated in HEC-12.
Izzard [8] developed equations to calculate the normal depth of gutter flow and the curb opening length (LT) (Equation (1)) required to intercept 100% of the gutter flow. Izzard [8] assumed that the The hydraulic performance of curb inlets for roadway drainage has been studied for more than 60 years, which was reviewed and summarized by Izzard [8], Li [9], and presently systematically documented in the Hydraulic Engineering Circular No. 22 (HEC-22) by Brown et al. [10]. In 1979, the Federal Highway Administration [11] first published a technical guide for the design of urban highway drainage, and then updated it in 1984 into a HEC-12 entitled "Drainage of Highway Pavement" [12]. HEC-12 summarizes a semi-theoretical method developed for estimating street hydraulic capacities and procedures for sizing street inlets. The most recent HEC-22 [10] was published and widely used in the USA, and refined the design procedures stated in HEC-12.
Izzard [8] developed equations to calculate the normal depth of gutter flow and the curb opening length (L T ) (Equation (1)) required to intercept 100% of the gutter flow. Izzard [8] assumed that the transverse velocity of the approach flow to an inlet was zero and that the hydraulic head decreased linearly along the curb inlet. Izzard (1950) applied the usual form of Manning's equation in a local sense, and calculated the total gutter flow via integrating dQ at each section and across the flow area with a uniform cross slope. Izzard (1950) compared his theoretical equation to unpublished data from experiments with less than 100% efficiency on undepressed curb inlets conducted at the University of Illinois, and determined the coefficient value in Equation (1): 7/16 in S 9/32 0 /(nS x ) 9/16 (1) where L T is in m, Q in is the upstream inflow (m 3 /s), S 0 is the longitudinal slope of the road or gutter, and S x is the cross slope (Figure 1), and n is Manning's roughness coefficient. Izzard [8] developed Equation (2) to calculate the efficiency (E ci ) of a curb inlet, which is the ratio of flow intercepted by the curb inlet (Q ci ) and the gutter flow Q in of undepressed curb inlet with the opening length of L ci (m).
For the undepressed curb inlet, Equations (3) and (4) were adapted by HEC-22 [10] to determine L T and E ci .
where α is not a constant, but calculated with an equation α = 0.42 S 0.42 x . When S x ranges from 1.5% to 6%, α changes from 2.45 to 1.37.
Guo and MacKenzie [14] stated that the HEC-22 procedure overestimates the capacity of a curb-opening inlet when water depth is shallow, and then becomes underestimating when water depth exceeds 7 inches for 3 ft curb inlet in the sump. To determine inlet efficiency and performance, the reduced-scale physical models were used and based on Froude number scaling-that is, a 3:4 scale model for Hammonds and Holley's study [7], 1:4 for Uyumaz's research [15], and 1:3 for Guo and Mackenzie's tests [14]; only Schalla et al. [16] conducted full-scale experiments for the curb inlet. The scaling effect has recently been discussed as a possible reason for significant discrepancies in interception efficiency between field measurements and predictions of the regression equations based on scaled laboratory experiments [17,18].
Much more attention and focus should be paid on studying the interception efficiency of a curb inlet and its design because it is an important component for urban drainage and road bioretention facilities. Most of the currently used curb inlet E ci equations were based on simple theoretical derivation with assumptions and revised coefficients or exponents determined using experimental data. In this study, the two-dimensional numerical models were first developed to represent 20 full-scale laboratory experiments of locally depressed curb inlets conducted by Hammonds and Holley [7] and 80 experiments for undepressed curb inlets by Wesley [19] using high-resolution digital elevation models (DEMs) for the model validation study. The overland flow on the road with curb inlets and upstream inflow was simulated using the FullSWOF-ZG program [3], which was updated/reconfigured from an open-source two-dimensional overland flow program, FullSWOF_2D [20]. One thousand modeling cases of a road with an undepressed curb inlet with 10 S 0 , 10 S x , and 10 Q in were then established and modeled to develop L T for the 100% interception of gutter flow, and then a regression equation of L T as a function of input parameters was developed by the multiple linear regression method. The second 1000 modeling cases of the road with 10 S 0 , 10 S x , and 10 curb inlet lengths L ci were established and simulated to determine E ci of different L ci , and a regression equation of E ci as a function of L ci /L T was also developed. The simulation results of L T and E ci were discussed and comprehensively compared with calculated/predicted results from HEC-22 [10], Izzard [8], and Muhamad [13]. In this study, the height of the opening of the curb inlets was not directly considered when a two-dimensional model was used. Only under severe flood situations was the height of the opening found to play a role on flow interception and inlet efficiency.

Materials and Methods
Fang et al. [21] used a three-dimensional computational fluid dynamics (CFD) software, Flow-3D, to develop the numerical models simulating unsteady, free-surface, shallow flow through Type C and Type D [7] curb-opening inlets. They demonstrated that an advanced CFD model could be used as a virtual laboratory to evaluate the performance of curb inlets with different geometry and inflow conditions. In this study, the two-dimensional open source FullSWOF_2D (version 1.07, Dieudonné Laboratory J.A., Polytech Nice Sophia, Nice, France) [20] program was updated to simulate the complex flow through an inlet to determine the inlet-opening length L T of 100% (total) interception, and the efficiency E ci of an undepressed curb inlet.
The FullSWOF_2D program fully solves shallow-water equations (SWEs) [20], depth-integrating the Navier-Stokes equations [22] on a structured mesh (square cells) in two-dimensional domains using the finite volume method [23], and is programmed using C++ to fully describe the rainfall-runoff and flow distribution progress on the surface [24]. As a Saint-Venant system [22], the SWEs model is widely used to simulate the incompressible Navier-Stokes flow occurring in rivers, channels, ocean, and land surfaces [25]. It is derived with two assumptions: the water depth is small with respect to the horizontal (x, y) dimensions, and the pressure of the fluid is hydrostatic (∂p/∂z = −g), which means the pressure field could be calculated with simple integration along the vertical (z) direction [26]. A well-balanced numerical scheme was adapted to guarantee the positivity of water height and the preservation of steady states for specific hydrological features, such as during wet-dry transitions and tiny water depth [27]. Different boundary conditions, friction laws, and numerical schemes were developed, which make the program a very powerful overland flow simulation software [20].
The FullSWOF_2D program, which applies the uniform rainfall and infiltration parameters to the whole simulation domain, was revised by Li et al. [3] to include 2D plane zones (Z) with different rainfall and infiltration parameters and a 2D-1D grate-inlet (G) drainage module. Therefore, the updated FullSWOF-ZG program can simulate impervious and pervious surfaces (different infiltration parameters/capabilities in different zones) in the road bioretention domain simultaneously under rainfall events. The 2D-1D grate-inlet drainage submodule enables the program to simulate the 2D overland runoff flowing into a grate inlet, and then to a 1D underground drainage pipe using the weir equation [28]. In Li's study [3], the FullSWOF-ZG was used to evaluate the performance of a road bioretention facility and explore/understand key parameters of continuous road bioretention design. It was found that the curb inlet becomes the bottleneck of the road bioretention strip system that could impede the runoff flowing into the bioretention strip for detention and infiltration to improve the stormwater quality [3].

FullSWOF-ZG Validation Cases
For the reduced-scale and full-scale laboratory experiments, the curb inlet efficiency was calculated with flow intercepted by the curb inlet divided by the total upstream inflow, which did not consider the rainfall-runoff generation and concentration process. It was not meant to understand the performance of the curb inlet under a rainfall event, but to provide the information for engineering design of the curb inlet as a function of upstream inflow. Inflows with different magnitudes and spreads for curb inlets include runoff from upstream and surrounding lands and runoff produced from the roadway. Therefore, these experimental studies are valuable, and numerical model studies under the same experimental conditions were used to validate the FullSWOF-ZG model to see how well the model can predict the curb inlet efficiency. Twenty locally depressed curb inlet cases, which were tested in a laboratory by Hammonds and Holley [7], were used to validate FullSWOF-ZG for curb inlet efficiency simulation. Eighty undepressed curb inlet cases, which were tested in a laboratory by Wesley [19], were used to validate FullSWOF-ZG for curb inlet lengths of 100% interception.

Modeling Cases to Validate Curb Inlet Efficiency
The FullSWOF-ZG program was previously tested and verified for overland flow on pervious surfaces [3,29]. In a previous study [3], the FullSWOF-ZG program was tested with 20 type C curb inlet cases. The coefficient of determination (R 2 ) of the linear relationship between the simulated and observed curb inlet interception efficiencies was 0.94 for type C curb inlet test cases. The differences between the simulated and observed interception efficiencies (∆E) ranged from −3.2% to 13.2%, with an average ± standard deviation of 3.5 ± 3.5%. In this study, FullSWOF-ZG was first tested using 20 locally depressed curb inlets (type D, Figure 1), which was tested in laboratory experiments by Hammonds and Holley [7]. The simulation domain was represented by a detailed and high-resolution DEM ( Figure 1b) with a cell size equal to 0.076 m (0.25 ft). The elevation of every computation cell was calculated using a user-developed MATLAB r2017a (MathWorks, Natick, MA, United States) [30] code with consideration of the road's longitudinal slope, cross slope, locally depressed cross slope of the curb inlet, and the slopes of the inlet's upstream and downstream transition parts. The longitudinal (x-direction) and cross (y-direction) slopes for the simulation domain are from left to right and bottom to top, respectively ( Figure 1). Manning's law in FullSWOF-ZG was used in the simulation, and the roughness coefficient determined for the laboratory roadway was 0.018 [7].
The imposed discharge condition in FullSWOF-ZG was chosen as the left or upstream boundary condition of the domain. The imposed discharge for the boundary cells within the spread (T) was approximately assumed as the total inflow rate (Q in ) divided by the number of the cells within the spread and set to be equal to 0 for other boundary cells outside of the spread. The top and right (downstream) boundary of the simulation domain was set as a Neumann condition that allows the flow to get out of the simulation domain. At the top of the simulation domain, those cells on the curb had higher elevations to prevent the outflow. The bottom boundary of the simulation domain ( Figure 1b) had the highest elevation along the y-direction, and was set as a wall boundary condition to guarantee that the flow would not pass through the bottom boundary.

Modeling Cases to Simulate/Validate Curb Inlet Length of 100% Interception
Wesley [19] conducted a series of full-scale experiments to determine the 100% intercepted curb inlet lengths with different longitudinal and cross slopes for undepressed curb inlets. The experiment facility had a triangular cross-section with the curb side being nearly vertical, and placed on a continuous grade with no local depression on the channel bottom. The length of the curb opening was sufficient to allow for interception of all the flow from the upstream road. The experiment facility was 50 ft (15.24 m) in overall length and 6 ft (1.83 m) in width. At 32 ft (9.75 m) from the upstream end, the curb inlet opening began. This upstream length (32 ft) is sufficient for the development of a uniform flow condition. The 100% intercepted curb inlet length was then experimentally determined using the observed distribution of water depth along the curb [19].
The simulation domain was represented by detailed and high-resolution DEM with a smaller cell size equal to 0.05 ft (0.015 m), similar to Figure 1 without local depressions. The Manning's value was 0.01, which is the same as the experiment facility. For eighty experimental cases, S 0 ranged from 0.005 to 0.05, S x ranged from 0.01 to 0.08, and upstream inflow Q in ranged from 0.18 L/s to 84.38 L/s, which were simulated to determine the 100% interception curb inlet lengths.

Modeling Cases to Evaluate 100% Interception Length and Curb Inlet Efficiency
After the FullSWOF-ZG model was validated to be able to accurately simulate flow over the curb inlet, 1000 modeling cases were selected and modeled to determine the curb inlet length L T of 100% interception under different S 0 , S x , and Q in . The length and width of the simulation domain for these 1000 modeling cases were 12 m (x-direction) including the 10 m road surface before the inlet and 6.7 m (y-direction, Figure 1 Commonly used S 0 and S x values in stormwater drainage design were chosen from the HEC-22 [10] for 1000 modeling cases, which are the combinations of 10 longitudinal slopes S 0 , 10 cross slopes S x , and 10 upstream inflows Q in (Table 1). Ten S 0 slopes ranged from 0.1 to 1% with an increase interval of 0.1% and ten S x from 1.5 to 6% with an increase interval of 0.5%, respectively. Ten upstream inflows which ranged from 6 to 24 L/s with an interval increase of 2 L/s were adapted for the simulation. The case number was named using the sequence number (1 to 10) of the parameter's choice of S 0 , S x , and Q in ; for example, the modeling case O1X1Q1 (Table 1) meant the road had S 0 = 0.001 (0.1%) and S x = 0.015 (1.5%) with Q in = 6 L/s for upstream inflow. The curb length was set to be large enough to intercept 100% inflow for all 1000 modeling cases. Table 1. Sequence numbers and corresponding geometry and inflow parameters of modeling cases used for FullSWOF-ZG simulations, and the results for the 20 sample cases.

Sequence
No. Note: 1 -modeling case for determining curb inlet length L T of 100% interception, 2 -modeling cases for determining inlet efficiency E ci of different length L ci when Q in = 10 L/s, and 3 -L ci in m is given outside of brackets and the ratio L ci /L T is given inside of brackets (dimensionless).
To evaluate the curb inlet efficiency E ci at different inlet lengths, the second 1000 modeling cases were selected using 10 choices of L ci and the same 10 choices for S 0 and S x , which were used for the 100 modeling cases to determine L T . Ten curb inlet lengths L ci ranged from 0.15-1.5 m (6-60 inches) with an increase of 0.15 m (6 inches), which was adapted based on the curb inlet survey conducted by Stoolmiller et al. [5]. The imposed upstream inflow Q in was chosen as 10 L/s for the left boundary condition of the domain for the second 1000 modeling cases, and a part of the inflow was intercepted by the curb inlet-that is, Q ci in Figure 1-and the remainder of the inflow was discharged downstream along the road (Q bp ), where the inlet length L ci was less than L T for 100% interception.
All cells' elevations were calculated using a MATLAB program when the bottom-left corner reference cell's elevation (the highest in the domain) was assumed to be 10 m (Figure 1b). The road surface ground elevations, therefore, varied with longitudinal and cross slopes set for each modeling case ( Figure 1b). All cells for the 0.1 m curb were set 0.2 m higher than the road surface cells. The cell's elevations inside the curb inlet cells were calculated using the same cross slope of the road surface, which helps and allows the runoff to flow out the road surface. The total simulation duration was 120 s (1.5 min) for reaching an equilibrium condition to determine E ci .

FullSWOF-ZG Validation Results of Curb Inlet Efficiency
For the 20 modeling cases conducted for the type D inlet, geometry (S 0 , S x , and L ci ) and flow (Q in and T) parameters have been listed in Table 2, and the same for the experimental conditions [7,21], which cover five longitudinal slopes (0.004-0.06), two cross slopes (0.0208 and 0.0407), 17 spreads (1.05-4.27 m), and 19 upstream inflows (0.0285-0.2597 m 3 /s). Table 2 shows that simulated intercepted flows (Q cis ) and inlet efficiencies (E cis ) matched well with the observed results (Q cio and E cio ) from the laboratory experiments conducted by Hammonds and Holley [7]. The coefficient of determination (R 2 ) of the linear relationship between simulated and observed E ci is 0.99. The differences (∆E in Table 2) of simulated and observed E ci ranged from −2.28% to 4.21% with average ± standard deviation as 1.10% ± 1.67%. The percent differences (PD E ) of simulated and observed E ci ranged from −3.15% to 5.17% with average ± standard deviation as 1.57% ± 2.29%. Therefore, the FullSWOF-ZG program can accurately simulate the overland flow through the road surface, gutter, local depressions (transition), and the flow interception over Type D curb inlets, and predict the curb inlet interception efficiency well. Table 2. Geometry and inflow parameters and simulation results of 20 Type D locally depressed curb inlet modeling cases.
The simulation results in Table 2 were first developed using the cell size of 0.076 m (0.25 ft) for square computational grids. Three other cell sizes (0.05 m, 0.025 m, and 0.01 m) were then used for 10 modeling cases of undepressed inlets (O10X10L1-O10X10L10 in Table 1)

Validation Results of 100% Intercepted Curb Inlet Length
The simulated water surface profile along the curb inlet was outputted and used by a MATLAB code to determine the 100% interception curb inlet length L T . Figure 2a shows six examples of the simulated water surface profile along the curb for six selected L T validation cases (WS11, 23, 34, 47, 56, and 73) out of 80 experiment tests (WS1-WS80) conducted by Wesley [19]. Because the curb inlet opening starts at 9.75 m, the water depth drops sharply at the first 0.2 m of the curb inlet opening and then decreases slowly and linearly along the curb inlet. Finally, the water depth becomes very small (a thin layer of water) across the remaining inlet length. These water surface profiles along the curb inlet show similar variations with distance as reported by other research studies, such as Schalla [16], Hodges et al. [31], and Muhammad [13], but are significantly different from the linear decrease assumption made and used by Izzard [8] to develop the L T equation.

Validation Results of 100% Intercepted Curb Inlet Length
The simulated water surface profile along the curb inlet was outputted and used by a MATLAB code to determine the 100% interception curb inlet length LT. Figure 2a shows six examples of the simulated water surface profile along the curb for six selected LT validation cases (WS11, 23, 34, 47, 56, and 73) out of 80 experiment tests (WS1-WS80) conducted by Wesley [19]. Because the curb inlet opening starts at 9.75 m, the water depth drops sharply at the first 0.2 m of the curb inlet opening and then decreases slowly and linearly along the curb inlet. Finally, the water depth becomes very small (a thin layer of water) across the remaining inlet length. These water surface profiles along the curb inlet show similar variations with distance as reported by other research studies, such as Schalla [16], Hodges et al. [31], and Muhammad [13], but are significantly different from the linear decrease assumption made and used by Izzard [8] to develop the LT equation.  Five depth limits (0.05 mm, 0.1 mm, 0.2 mm, 0.3 mm, and 0.5 mm) were tested to determine the end point of the 100% interception curb inlet length. The location of water depth, equal to the depth limit minus the curb inlet start location, was regarded as the 100% interception curb inlet length. The green solid vertical lines in Figure 2a show the location of the water depth equal to the depth limit of 0.2 mm for these six example cases. The simulated curb inlet lengths of total interception for all 80 modeling cases of Wesley's tests were determined using the five water depth limits and compared to L T observed and determined in laboratory tests (Figure 2b). The final water depth limit was chosen when the root-mean-square error (RMSE) and mean absolute percentage error (MAPE) of simulated and observed curb inlet lengths of total interception were the smallest. Simulated L T values for the 80 cases/tests ranged from 1.30 m (4.3 ft) to 5.33 m (17.5 ft). Figure 2b shows that the simulated L T best matched with observed data when the depth limit for determining L T was 0.2 mm. The RMSEs for the simulated L T results with depth limits equal to 0.05, 0.1, 0.2, 0.3, 0.5 mm were 0.30 m, 0.28 m, 0.27 m, 0.29 m, 0.31 m, and 0.34 m, respectively. The corresponding MAPEs for the five depth limits are 7.32%, 6.46%, 6.04%, 6.08%, 6.34%, and 6.80%, respectively. Since the smallest RMSE and MAPE were for the depth limit of 0.2 mm, therefore, the water depth limit 0.2 mm was used for the one thousand modeling cases (O1X1Q1-O10X10Q10) to determine the 100% interception curb inlet lengths summarized in Section 3.3. Figure 3 shows Five depth limits (0.05 mm, 0.1 mm, 0.2 mm, 0.3 mm, and 0.5 mm) were tested to determine the end point of the 100% interception curb inlet length. The location of water depth, equal to the depth limit minus the curb inlet start location, was regarded as the 100% interception curb inlet length. The green solid vertical lines in Figure 2a show the location of the water depth equal to the depth limit of 0.2 mm for these six example cases. The simulated curb inlet lengths of total interception for all 80 modeling cases of Wesley's tests were determined using the five water depth limits and compared to LT observed and determined in laboratory tests (Figure 2b). The final water depth limit was chosen when the root-mean-square error (RMSE) and mean absolute percentage error (MAPE) of simulated and observed curb inlet lengths of total interception were the smallest. Simulated LT values for the 80 cases/tests ranged from 1.30 m (4.3 ft) to 5.33 m (17.5 ft). Figure 2b shows that the simulated LT best matched with observed data when the depth limit for determining LT was 0.2 mm. The RMSEs for the simulated LT results with depth limits equal to 0.05, 0.1, 0.2, 0.3, 0.5 mm were 0.30 m, 0.28 m, 0.27 m, 0.29 m, 0.31 m, and 0.34 m, respectively. The corresponding MAPEs for the five depth limits are 7.32%, 6.46%, 6.04%, 6.08%, 6.34%, and 6.80%, respectively. Since the smallest RMSE and MAPE were for the depth limit of 0.2 mm, therefore, the water depth limit 0.2 mm was used for the one thousand modeling cases (O1X1Q1-O10X10Q10) to determine the 100% interception curb inlet lengths summarized in Section 3.3. Figure 3 shows

Simulated Curb Inlet Lengths of 100% Interception
After LT was determined for the 1000 modeling cases using FullSWOF-ZG, a generalized power relation, Equation (7) was chosen to develop the regression equation for LT as a function of four input parameters (Qin, S0, n, and Sx). Table 1 also shows LT determined for 10 example cases (OmXmQm, m = 1, 2, …, 10) using FullSWOF-ZG. In Equation (7) the Manning's value n and cross slope Sx were grouped as one combined variable, the same as the HEC-22 Equation (3).
where LT is the curb inlet length in m for 100% interception; Sx and S0 are the cross slope and longitudinal slopes of the road/street (Table 1), Qin is the upstream inflow rate from the road/street

Simulated Curb Inlet Lengths of 100% Interception
After L T was determined for the 1000 modeling cases using FullSWOF-ZG, a generalized power relation, Equation (7) was chosen to develop the regression equation for L T as a function of four input parameters (Q in , S 0 , n, and S x ). Table 1 also shows L T determined for 10 example cases (OmXmQm, m = 1, 2, . . . , 10) using FullSWOF-ZG. In Equation (7) the Manning's value n and cross slope S x were grouped as one combined variable, the same as the HEC-22 Equation (3).
where L T is the curb inlet length in m for 100% interception; S x and S 0 are the cross slope and longitudinal slopes of the road/street (Table 1), Q in is the upstream inflow rate from the road/street surface to the curb inlet in m 3 /s (0.006-0.024, Table 1), and n (-) is Manning's roughness of the road surface. The variation inflation factors (VIF) among three input variables (Q in , S 0 , and nS x ) were calculated with MATLAB before developing the equation. The VIFs among the three variables are all equal to 1. This means the predictors are more related to the target variable L T than they are to each other [32], and the multicollinearity of three variables are not significant. The coefficient k and exponents (a, b, and c) were estimated using the multiple linear regression (MLR) method after the log transformation of Equation (7), and the resulting regression equation of L T was: The  Figure 4a shows the comparison of fitted L T calculated using Equation (8), and simulated L T . The R 2 and RMSE between fitted and simulated L T are 0.99 (Figure 4a) and 0.13 m, respectively. The MAPE between fitted and simulated L T is 2.34% for all 1000 cases. It shows the fitted L T or predicted by the regression equation matched well with the simulated L T when L T < 5 m, while the difference between simulated and predicted L T becomes larger when L T > 5 m. The ratio (R lt ) of fitted L T calculated using Equation (8) and simulated L T was computed for all 1000 modeling cases; then, the mean R lt and standard deviation for each 100 cases with the same S 0 were also calculated and plotted in Figure 5. The ratio R lt ranged from 0.89 to 1.06 (maximum of 11% underestimate and 6% overestimate), and the 915 fitted L T values were within 5% from simulated L T (0.95 < R lt ≤ 1.05) for Equation (8).  Table 1), and n (-) is Manning's roughness of the road surface.
The variation inflation factors (VIF) among three input variables (Qin, S0, and nSx) were calculated with MATLAB before developing the equation. The VIFs among the three variables are all equal to 1. This means the predictors are more related to the target variable LT than they are to each other [32], and the multicollinearity of three variables are not significant. The coefficient k and exponents (a, b, and c) were estimated using the multiple linear regression (MLR) method after the log transformation of Equation (7), and the resulting regression equation of LT was: The 95% confidence intervals for the coefficient k and exponents a, b, and c (3), the exponent of S0 in Equation (8) Figure 4a shows the comparison of fitted LT calculated using Equation (8), and simulated LT. The R 2 and RMSE between fitted and simulated LT are 0.99 (Figure 4a) and 0.13 m, respectively. The MAPE between fitted and simulated LT is 2.34% for all 1000 cases. It shows the fitted LT or predicted by the regression equation matched well with the simulated LT when LT < 5 m, while the difference between simulated and predicted LT becomes larger when LT > 5 m. The ratio (Rlt) of fitted LT calculated using Equation (8) and simulated LT was computed for all 1000 modeling cases; then, the mean Rlt and standard deviation for each 100 cases with the same S0 were also calculated and plotted in Figure 5. The ratio Rlt ranged from 0.89 to 1.06 (maximum of 11% underestimate and 6% overestimate), and the 915 fitted LT values were within 5% from simulated LT (0.95 Rlt 1.05) for Equation (8).

100% Intercepted Gutter Flow for Drainage and Road Bioretention Design
The simulated LT ranged from 1.61 to 7.56 m for the 1000 modeling cases (Figure 4a). In urban drainage design, the inlet opening lengths for various types of curb inlets standardized by municipalities and transportation agencies have only a few preset/pre-cast/manufactured lengths; for example, Texas Type C and Type D curb inlets [7] and TxDOT precast curb inlet outside roadway (PCO) on-grade curb inlets [31] have three opening lengths of 1.52 m (5 ft), 3.05 m (10 ft), and 4.57 m (15 ft). When the calculated curb inlet length for 100% interception is large under design gutter flow, none of the very large opening curb inlets are actually used, but continuously depressed gutter or locally depressed inlets with necessary transient lengths are typically designed and built. Therefore, Equation    (8), which indicate that the Equation (8) matches very well with simulated L T for all S 0 conditions. For HEC-22 and Muhammad methods, R lt < 0.95 for all S 0 situations and the Muhammad method has a larger standard deviation (0.05-0.07), which means the HEC-22 and Muhammad methods underestimate L T in a greater extent under smaller S 0 situations. For the Izzard method, most of R lt > 1 (overestimates L T ) when S 0 ≥ 0.3%, and it underestimates L T when S 0 < 0.3%. For three previous methods, the mean R lt strongly correlates with S 0 as a power function with R 2 > 0.97 ( Figure 5): R lt increases with the increase of S 0 .

100% Intercepted Gutter Flow for Drainage and Road Bioretention Design
The simulated L T ranged from 1.61 to 7.56 m for the 1000 modeling cases (Figure 4a). In urban drainage design, the inlet opening lengths for various types of curb inlets standardized by municipalities and transportation agencies have only a few preset/pre-cast/manufactured lengths; for example, Texas Type C and Type D curb inlets [7] and TxDOT precast curb inlet outside roadway (PCO) on-grade curb inlets [31] have three opening lengths of 1.52 m (5 ft), 3.05 m (10 ft), and 4.57 m (15 ft). When the calculated curb inlet length for 100% interception is large under design gutter flow, none of the very large opening curb inlets are actually used, but continuously depressed gutter or locally depressed inlets with necessary transient lengths are typically designed and built. Therefore, Equation (8) (5) from  were rearranged to determine the gutter flow for 100% interception (Q g100 ) when the curb inlet length is given. All the above equations are for the International System of Units (SI) where L ci is in m and Q g100 is m 3 /s. Determining Q g100 helps us to reevaluate these L T equations. For urban drainage design, design discharge for the gutter was calculated first based on the catchment area, runoff coefficients of land use, and design rainfall intensity; then, the distance between two curb inlets and the curb inlet opening were calculated and selected/specified for the design. Figure 6 shows the comparison of predicted Q g100 from Equations (10)- (12) and from Equation (9) for undepressed curb inlets with L ci = 5 ft (1.524 m). Fourteen S 0 values were used for Figure 6 and from 0.003 (0.3%) to 0.04 (4%), and two small S 0 (0.001 and 0.002) in Table 1 were not used (HEC-22 recommends S 0 > 0.3%), and six larger S 0 (0.015, 0.03, 0.025, 0.03, 0.035, and 0.04) were added to represent the steep-slope roads for urban drainage and road bioretention curb inlet design. The 14 S 0 and 10 S x (Table 1, 1.5-6%) formed 140 slope combinations that were used to compare/evaluate the above Q g100 equations in Figure 6. Even the overall R 2 for linear correlations of Q g100 predicted from three previous studies and the newly developed Equation (9) are greater than 0.80, where Q g100 predicted by HEC-22 (2009) and Muhammad (2018) is much larger than Q g100 predicted by Equation (9). Muhammad's equation has the largest mean absolute percent deviation (MAPD), equal to 180.5%, which always overestimates Q g100 (Figure 6), in comparison to Q g100 from the current study. The MAPD for Izzard (1950) All the above equations are for the International System of Units (SI) where Lci is in m and Qg100 is m 3 /s. Determining Qg100 helps us to reevaluate these LT equations. For urban drainage design, design discharge for the gutter was calculated first based on the catchment area, runoff coefficients of land use, and design rainfall intensity; then, the distance between two curb inlets and the curb inlet opening were calculated and selected/specified for the design. Figure 6 shows the comparison of predicted Qg100 from Equations (10)- (12) and from Equation (9) for undepressed curb inlets with Lci = 5 ft (1.524 m). Fourteen S0 values were used for Figure 6 and from 0.003 (0.3%) to 0.04 (4%), and two small S0 (0.001 and 0.002) in Table 1 were not used (HEC-22 recommends S0 > 0.3%), and six larger S0 (0.015, 0.03, 0.025, 0.03, 0.035, and 0.04) were added to represent the steep-slope roads for urban drainage and road bioretention curb inlet design. The 14 S0 and 10 Sx (Table 1, 1.5-6%) formed 140 slope combinations that were used to compare/evaluate the above Qg100 equations in Figure 6. Even the overall R 2 for linear correlations of Qg100 predicted from three previous studies and the newly developed Equation (9)   Hodges et al. [31] compared Q g100 predicted using HEC-22 equation and their experimental measurements. When S x was fixed at 6% and S 0 changed from 4% to 0.1%, the Q g100 predicted using the HEC-22 equation increased about 6 times, but experimental measurements only increased less than 1.6 times. The purple line on Figure 6b shows the comparison of HEC-22 and Equation (9) predicted Q g100 results when S x = 0.06 (6%). When S 0 decreased from 4% to 0.3% at S x = 6%, Q g100 calculated using HEC-22 increased 6.4 times but Q g100 from Equation (9) only increased two times, which is similar to results from the physical model by Hodges et al. [31]. The smaller exponent 0.269 for S 0 in Equation (9) for the current study seems to give a better prediction on Q g100 compared to HEC-22 Equation (11).
In Figure 6b, 10 red filled squares gave Q g100 predicted from HEC-22 for S 0 = 0.3%, and S x increased from 1.5% to 6%, and Q g100 increased from 1.8 L/s to 13.6 L/s that was, on average, 2.83 times larger than Q g100 (0.62-5.1 L/s) from Equation (9). At S 0 = 0.3%, the ratio of Q g100 predicted from HEC-22 to Equation (9) ranged from 2.69 to 3.03 with a standard deviation from the mean of 0.11, which graphically shows as a perfect linear relation of these red filled squares on Figure 6b. This strong linear correlation of Q g100 predicted from three previous studies versus Equation (9) exists for all other S 0 when S x is changed. The ratio of Q g100 predicted from HEC-22 (2009) versus Equation (9) ranges from 0.85 to 3.03, and has a strong correlation with the longitudinal slope S 0 : a power function Q g100-HEC-22 /Q g100-Equation-9 = 0.213 S 0 −0.445 (R 2 = 0.99). Actually, the power function can be approximately derived by dividing Equation (11) to Equation (9). The power function clearly indicates that over-prediction occurs in smaller S 0 because of the negative exponent −0.445; for example, S 0 = 0.3%, as shown by the red filled squares. The blue filled squares on Figure 6 show results for 60 cases with S 0 > 1% and S x from 1.5% to 4%. From the power function, when S 0 is larger, the ratio of Q g100 is smaller, as is also clearly shown in Figure 6 by the blue filled squares for all three methods. Figure 6b shows that the predicted Q g100 from HEC-22 matched very well with the ones predicted from Equation (9) (9) also seems to correlate with the ratio S x /S 0 and becomes larger (over-prediction) when the ratio S x /S 0 increases. The power function of the Q g100 ratio versus the ratio S x /S 0 has a determination coefficient of 0.69, which is much weaker than the correlation with S 0 only (R 2 = 0.99). Hodges et al. [31] also show that, when the curb length L ci = 10 ft, S x was fixed at 6%, and S 0 changed from 4% to 0.1%, Q g100 calculated using HEC-22 equation over-predicts by an average factor of 1.51 when compared to measured Q g100 from their physical model. The ratio of Q g100 predicted from HEC-22 (2009) versus measurements actually ranges from 0.81 to 3.0, and the ratio of Q g100 predicted from HEC-22 versus Equation (9) also ranges from 0.85 to 2.37 ( Figure 6), and these two results are very similar. The comparison of predicted 100% intercepted gutter flow with Equations (10)- (12) to Equation (9) for curb length L ci = 10 ft (3.048 m) and L ci = 15 ft (4.572 m) also gives similar results discussed above for L ci = 5 ft (1.524 m) cases, and are therefore not repeated here. Table 1 shows E ci determined for 10 example cases (OmXmLm, m = 1, 2, . . . , 10), which range from 15.3-75.7%. E ci was calculated as the curb inlet outflow divided by the upstream inflow (Figure 1) when the flow through the curb inlet reaches the equilibrium-in other words, the E ci change is less than 0.0005. For the second set of 1000 modeling cases (O1X1L1 to O10X10L1) of 10 different L ci values (Table 1), the mean and standard deviations of simulated E ci at the same L ci were calculated. The mean E ci increased from 12.8% to 84.2%, and the corresponding standard deviation increased from 5.2% to 11.8% when L ci increased from 0.15 m to 1.50 m. This means inlets in the Philadelphia area (the survey conducted by Stoolmiller et al. [5]) could intercept different amounts of stormwater runoff to road bioretention facilities, and some of them were under-designed and had flooding risks on the road.

Simulated Curb Inlet Efficiency and Evaluation Equation
The format of Equation (6) was used to develop a new relationship between E ci and L ci /L T using 1000 simulated E ci for 10 different L ci when Q in was a constant of 10 L/s. Even E ci has a strong correlation with L ci (R 2 = 0.86 for a power function), but L ci /L T is used in Equation (13) so that it can be applied to other flow rates Q in from upstream road or watershed, since L T (Equation (8)) links with and has the impact of the input parameters S 0 , S x , Q in , and n.
The exponent α was determined based on the simulated E ci results of 1000 cases with the MLR method. The 95% confidence intervals for the exponents α is [2.408, 2.4358] with p-value < 0.0001. Figure 7a shows how the comparison of fitted and simulated E ci ; and fitted E ci matches well with the simulated E ci with R 2 = 0.98. Figure 7b shows E ci versus L ci /L T from the current study (fitted Equation Eci increased from 12.8% to 84.2%, and the corresponding standard deviation increased from 5.2% to 11.8% when Lci increased from 0.15 m to 1.50 m. This means inlets in the Philadelphia area (the survey conducted by Stoolmiller et al. [5]) could intercept different amounts of stormwater runoff to road bioretention facilities, and some of them were under-designed and had flooding risks on the road. The format of Equation (6) was used to develop a new relationship between Eci and Lci/ LT using 1000 simulated Eci for 10 different Lci when Qin was a constant of 10 L/s. Even Eci has a strong correlation with Lci (R 2 = 0.86 for a power function), but Lci/LT is used in Equation (13) so that it can be applied to other flow rates Qin from upstream road or watershed, since LT (Equation (8)) links with and has the impact of the input parameters S0, Sx, Qin, and n.
The exponent was determined based on the simulated Eci results of 1000 cases with the MLR method. The 95% confidence intervals for the exponents is [2.408, 2.4358] with p-value < 0.0001. Figure 7a shows how the comparison of fitted and simulated Eci; and fitted Eci matches well with the simulated Eci with R 2 = 0.98. Figure 7b shows Eci versus Lci/LT from the current study (fitted Equation      (8) and (13) for 10,000 design cases that are all combinations for 10 S 0 , 10 S x , 10 Q in , and 10 L ci listed in Table 1. The corresponding L T equation for each method was applied first before L ci /L T , and then E ci were calculated. When applying E ci equations, E ci was assumed to be 100% when L ci is greater than L T or L ci /L T > 1. Among all 10,000 cases, there are 21, 228, and 523 cases with L ci /L T > 1 for Izzard, HEC-22, and the Muhammad method when the predicted L T is smaller than L ci .

Summary and Conclusions
In this study, the updated FullSWOF-ZG program based on the open-source overland flow simulation program FullSWOF_2D was tested with 20 different locally depressed Texas type D curb inlet cases to simulate inlet efficiency. The differences between simulated and observed E ci ranged from −2.28% to 4.21% with the average ± standard deviation being 1.10% ± 1.67%. The FullSWOF-ZG program was also validated using 80 laboratory tests to simulate the curb inlet length of 100% interception with an RMSE equal to 0.27 m and MAPE equal to 6.04%. These validation runs indicated that the FullSWOF-ZG program can accurately simulate the overland flow through the curb inlets with the high agreement and small error with observed ones so that it can be used to determine L T and E ci . One thousand undepressed curb inlet modeling cases of the road with 10 S 0 , 10 S x , and 10 Q in were established and modeled to determine L T , and then a new estimation equation of L T was developed by the regression with the input parameters. The second set of 1000 road modeling cases of undepressed curb inlets with 10 S 0 , 10 S x , and 10 L ci and a constant Q in (10 L/s) were established and modeled to determine E ci , and then a new estimation equation of E ci was developed as a function of L ci /L T . The newly developed L T equation was compared with three previous methods, including Izzard (1950), HEC-22 (2009), and Muhammad (2018) for predicting L T for 1000 undepressed curb inlet cases. Finally, L T and E ci equations for four methods were applied together to predict E ci for 10,000 curb-inlet cases of all ten S 0 , S x , Q in , and L ci combinations listed in Table 1. Predicted E ci values for all 10,000 cases from Izzard (1950), HEC-22 (2009), and Muhammad (2018) method had RMSE < 8.5% and MAPD < 13% in comparison with ones from the newly developed L T and E ci equations in this study. The newly developed equations gave more accurate estimations of L T and E ci over a wide range of input parameters. These equations can be applied to design urban drainage and road bioretention facilities since they were developed using a large number of simulation runs with diverse input parameters, but previous methods often overpredict the gutter flow Q g100 of total interception when longitudinal slope S 0 is small. Also, simulation runs were done after the FullSWOF-ZG program for overland flow simulation was comprehensively validated with 100 laboratory tests. In future studies, the equations used to evaluate efficiency of locally depressed and continuously depressed curb inlets can be developed using simulation results from the FullSWOF-ZG program. The hydraulic-performance-based equations for different types of curb inlets should be promoted for the design of road bioretention facilities, instead of only considering landscape and safety perspective.
Author Contributions: X.L. updated the FullSWOF_2D program, conducted the simulations and analysis of the results, prepared the manuscript draft. X.F. supervised the model development, simulation runs, data analysis; and revised the manuscript. G.C., Y.G., J.W. and J.L. provided inputs on the writing, data analysis, and revised the manuscript. All authors made contributions to the study and writing the manuscript.