Estimating Time of Concentration for Overland Flow on Pervious Surfaces by Particle Tracking Method

Abstract

The particle tracking method (PTM) module was added into the open source Full Shallow-Water equations for Overland Flow in a two-dimensional (FullSWOF_2D) program, which has coupled rainfall–runoff and infiltration modules to determine the time of concentration (T_c) for impervious (T_ci) and pervious (T_cp) surfaces. The updated program FullSWOF-PTM was tested using observed rainfall events with Nash–Sutcliffe efficiencies ranging from 0.60 to 0.95 (average of 0.75) for simulated runoff hydrographs. More than 400 impervious modeling cases with different surface slope (S₀), roughness coefficient (n), length (L), and rainfall intensity (i) combinations were developed and simulated to obtain the T_ci for developing the regression equation of T_ci as a function of the four input parameters. More than 700 pervious modeling cases with different combinations of S₀, n, L, i, and infiltration parameters including the saturated hydraulic conductivity, suction head, and moisture deficit were simulated to estimate the T_cp based on the travel time of 85% of particles arriving at the outlet and the ponding time. The regression equation of T_cp was developed as the sum of T_ci and additional travel time as a function of infiltration parameters and i. The T_cp equation can be applied to wide ranges of input parameters in comparison to Akan’s equation.

1. Introduction

Mulvany [1] first put forward the concept of time of concentration (T_c) and Kuichling [2] defined T_c as the time needed for the runoff from the most remote part of a catchment to travel to the outlet during the rainfall–runoff process. It is widely used to design the highway and urban stormwater drainage facilities [3] using T_c as design rainfall duration [4]. There are dozens of studies where researchers developed and tested/compared T_c equations [5,6]. They obtained the T_c estimation using hydrograph analysis for laboratory plots/watersheds [7,8,9], theoretical derivation based on kinematic wave theory [4,10,11,12], and distributed physically-based numerical simulation programs utilizing topographic elevation and geometric data [13,14,15,16]. Izzard [7] developed a method to calculate the runoff hydrography and the time necessary to substantially reach an equilibrium of flow resulting from given rainfall intensity, roughness, slope, and the length of the overland flow plan based on the laboratory experiments. Compared to laboratory analysis and theory deduction, the distributed and physically-based numerical models solving the shallow-water equations (SWEs) are more and more widely used in overland flow simulation for its better performance dealing with mixed subcritical and supercritical flow compared to kinematic wave and diffusive wave approach or other approximation [17,18]. Su and Fang [15] established a two-dimensional numerical model based on shallow-water equations to estimate traveling time for different rainfall intensity, roughness, length, and slope modeling cases and developed the travel time estimation equation for relatively steep and very flat watersheds. Recently, more and more researchers moved their focus to the overland flow of pervious surfaces [19,20,21,22]. Hjelmfelt [23] analyzed the infiltration influence on the overland flow by combining the storage–depletion model of the U.S. Soil Conservation Service with the kinematic wave equations and found the variation of infiltration rate during a storm had a significant effect on the time of concentration and on the shape of the runoff hydrograph. In Guo’s study [4], the Wooding’s solution was expanded to overland flow on pervious surfaces by coupling the kinematic wave equations with the Horton infiltration model. Only Akan developed a time of concentration calculation chart [10] and a formula [24] using Manning’s friction law on rectangular pervious plots under constant-intensity rainfall based on the kinematic wave equations and Green–Ampt (GA) infiltration model. The time of concentration in Akan’s study was measured from the beginning of the rainfall event, which meant the ponding time (t_p) and runoff travel time were lumped together in the formula. The chart and formula are mainly appropriate for the cases when Manning’s friction law is acceptable and there are limited ranges of rainfall and soil infiltration parameters. Further work is still needed, e.g., to expand the formula to other flow resistance laws and wide ranges of rainfall and soil infiltration parameters.

Conceptually, the time of concentration is when the entire catchment becomes contributory to the runoff at the outlet, but there are various methods that have been developed/used to estimate T_c. T_c for impervious areas (T_ci) was typically estimated from hydrograph analysis, e.g., T_ci as lapsed time from the beginning of a rainfall event to the outlet flow reaching 98% of the peak discharge, which is called T_{c_q98} hereafter, since the runoff starts immediately after the rainfall, and T_c and the runoff equilibrium time are basically the same for impervious surfaces. The above method does not work for determining the time of concentration on pervious surfaces (T_cp) since the runoff does not start before the ponding time (t_p) and then discharge increases asymptotically to peak or equilibrium discharge under constant rainfall intensity even after a long period of simulation. Guo [4] suggested evaluating T_cp of a small catchment by velocity-based methods rather than those empirical formulas developed for and calibrated by the observed hydrographs. The particle tracking method (PTM) is popular for generating path lines and travel time information since it directly utilizes the simulated velocity field results [25,26,27,28]. KC and Fang [29] developed a quasi-two-dimensional (2D) diffusion wave model (DWM) coupled with particle tracking to determine the time parameters including the travel time for 85%, 95%, and 100% of particles to arrive at the outlet (T_{r_p85}, T_{r_p95}, T_{r_p100}) of overland flow on impervious surfaces. These travel times have significant linear correlations with each other and a significant agreement between the T_{r_p85} and T_{c_q98} was found.

To understand and estimate T_cp for pervious surfaces is particularly important and useful to smart stormwater management using the lower impact development (LID) and green infrastructures (GI) that promote the infiltration [22]. Therefore, the goal of this study is to develop a T_cp equation for pervious surfaces that can be applied to wide ranges of rainfall, watershed, and soil parameters. First, the PTM module was added into the open source Full Shallow-Water equations for Overland Flow in a two-dimensional (FullSWOF_2D, version 1.07, Lab. J. A. Dieudonné & EPU Nice Sophia, Nice, France) [30] program for determining T_cp for pervious surfaces. The FullSWOF_2D program has already coupled the rainfall–runoff modules with the infiltration module [29] for possibly exploring T_cp after adding the PTM module (called FullSWOF-PTM). A total of 750 pervious modeling cases that are combinations of diverse values of rainfall intensity i (m/s or mm/h), watershed slope S₀, Manning’s roughness coefficient n, length L (m), hydraulic conductivity K (m/s), suction head φ (m), and moisture deficit Δθ of pervious surfaces were generated considering different types of soil groups. Travel time for 85% of particles arriving at the outlet from the beginning of rainfall, which is determined using FullSWOF-PTM and called T_{r_p85} hereafter, was used directly to evaluate T_ci for impervious surfaces, and the T_{r_p85} subtracting the ponding time t_p was used to evaluate T_cp for pervious surfaces, which is consistent with Guo’s study [4] but different from Akan’s equation [24]. The multiple linear regression (MLR) method was used to derive the T_ci and T_cp equations as a function of input parameters.

2. Materials and Methods

2.1. Shallow Water Equations (SWEs) and FullSWOF_2D

As a Saint-Venant system [31], the simplified SWEs model is widely used to simulate the incompressible Navier–Stokes flow occurring in rivers, channels, ocean, and land surfaces [32]. It is derived with two assumptions: one is that the fluid velocity is constant along the vertical (z) direction and that the water depth is small with respect to the horizontal (x, y) dimensions; another is that the pressure of the fluid is hydrostatic ($\partial p/\partial z=-g$), which means the pressure field could be calculated with simple integration along the vertical (z) direction [30,33]. The conservative form of the 2D SWEs including the continuity equation and two momentum equations for x and y directions are stated as the following equations:

$$\frac{\partial h}{\partial t}+\frac{\partial hu}{\partial x}+\frac{\partial hv}{\partial y}=R-I$$

(1)

$$\frac{\partial hu}{\partial t}+\frac{\partial }{\partial x}\left(h{u}^2+\frac{g{h}^2}{2}\right)+\frac{\partial }{\partial y}\left(huv\right)=-gh\left(\frac{\partial z}{\partial x}+S_{fx}\right)$$

(2)

$$\frac{\partial hv}{\partial t}+\frac{\partial }{\partial x}\left(huv\right)+\frac{\partial }{\partial y}\left(h{v}^2+\frac{g{h}^2}{2}\right)=-gh\left(\frac{\partial z}{\partial y}+S_{fy}\right)$$

(3)

where R (m/s) is the rainfall intensity; I (m/s) is the infiltration rate; h (m) is the cell water height (depth); z (m) is the cell topography elevation; u (m/s) and v (m/s) are the cell depth-averaged velocities in x and y directions, respectively; S_fx and S_fy are the cell friction slopes in x and y directions, respectively; g (m/s²) is gravity acceleration; t (s) is time.

The FullSWOF_2D program fully solves SWEs on a structured mesh in two space dimensions using the finite volume method (FVM) which ensures mass conservation compared to the finite difference method (FDM) [34]. A well-balanced scheme was adapted to guarantee the positivity of water depth and the preservation of steady states for specific hydrological features such as during wet-dry transitions and tiny water depth [30,35]. Different boundary conditions, friction laws, and numerical schemes were developed which make the program a very powerful overland flow simulation software. The parallelization strategies of FullSWOF_2D were also examined to improve its simulation efficiency dealing with large-scale cases [35]. A modified bi-layer (crust- and soil-layer) Green–Ampt (GA) infiltration model [36] to calculate I for Equation (1) was coupled in the FullSWOF_2D [34] which enables the program to simulate overland flow on impervious and pervious surfaces.

FullSWOF_2D has five boundary condition choices including the imposed discharge and water height case, wall condition, Neumann boundary (open boundary) condition, periodic variations of discharge and water height, and imposed discharge condition. It has three options of friction formulas including the Manning’s equation, the Darcy–Weisbach equation, and the laminar law, as well as the no-friction setting. The simulation domain could be set nonuniformly by defining the friction value of every computational cell with an input file. The Rusanov flux, Harten–Lax–Van Leer (HLL) flux, Harten–Lax–Van Leer with Contact surface (HLLC) flux, HLL2, and HLLC2 [30] methods are provided to calculate every time level flux between computational cells. Three linear reconstruction methods include the MUSCL, ENO, and modified ENO, as well as three slope limiters, including the classical Minmod slope limiter, Van Albada limiter, and Van Leer’s limiter, which are used in the reconstruction part of the 2nd order numerical scheme. The details about the numerical flux, reconstruction methods, and the limiters could be found in Bouchut’s book [37].

The FullSWOF_2D has been validated using several analytical solutions and benchmarks of the steady-state solutions and the transitory solutions. The steady-state solutions validated by FullSWOF_2D include the emerged bump at rest and Mac Donald test cases with different settings. The transitory solutions include the dam break on a dry domain and Thacker test case with a planar surface in paraboloid [38]. It is also widely used in the river flood simulation in a complex environment based on high-resolution topographic data [39] and spatial global sensitivity analysis of high-resolution classified topographic data in 2D urban flood modeling [40].

2.2. Particle Tracking Method (PTM) and FullSWOF-PTM

The PTM is a powerful method to study the characteristics of complex flow velocity fields during steady and transient state using simulated velocities from flow-governing equations, e.g., Equations (1)–(3) for shallow overland flow. It is widely used in different research areas, especially the groundwater flow and pollutant transport study [41]. Most of the commonly used PTMs provide satisfactory results for steady-state analysis [42]. PTMs could also be used for transient analysis under the assumption that the velocity field does not significantly change during the simulation duration. In Cheng’s study [28], a PTM was developed based on the Lagrangian–Eulerian finite element method (FEM) which could reduce the numerical errors considerably and enable the PTM to trace fictitious particles in a complex flow field. It is suggested that the PTM could be extended to transient simulations by tracking velocity calculated with the velocity field of previous time levels and current time levels (stepwise approximation) when the FEM is used to solve the transport equations. Bensabat et al. [27] and Lu [43] developed a linear temporal interpolation scheme instead of a stepwise temporal approximation in order to count for the changes in velocity during a time step in complex unsteady flow while it is only suited to the FDM rather than FEM or FVM. The travel time and the pathlines could be generated using PTM directly incorporated with the overland flow simulation velocity field results.

In the previous study [29], the PTM using simulated velocities from DWM was developed to determine the travel time of different percentage particles’ arrival at the outlet. The travel time of each particle in the simulation domain is computed using the PTM module that uses flow velocity fields simulated by the quasi-two-dimensional DWM at every time level. Over each time step, the particle travel distance is determined by the product of the appropriate tracking velocity (interpolated by the spatial linear method) and time step interval. In this study, a PTM module was incorporated with FullSWOF_2D using a simulated velocity field at each time step, and the updated program became FullSWOF-PTM. A fourth-order Runge–Kutta (RK4) spatial interpolation scheme rather than linear spatial interpolation scheme [42] was adapted to get the particle velocity at each time step at different locations of the simulation domain. The temporal change of all particles was calculated and updated using the simulated particle velocity and time interval at every time step.

The following algorithm is implemented in the PTM code of FullSWOF-PTM at each time step: (1) The particle location is checked to determine whether it is within the simulation domain. If the particle arrives at the outlet cell, it is ignored and the tracking process moves on to the next particle. (2) The computational cell that the particle locates in and the adjacent cell for each particle are determined based on the particle location. (3) The particle velocities in x- and y-directions are spatially interpolated using a RK4 scheme based on the simulated x and y velocities of the cell that the particle locates in and adjacent computational cells at the time step. (4) The new locations of all particles are calculated and updated using previous locations, current particle velocities, and time step intervals. (5) The particle with the new location is checked again to determine whether it stays in the simulation domain or arrives at the outlet cell and gets out of the domain. (6) The percentage of particles remaining in the simulation domain is counted. The particle evolution information of each time step including the total number and percentage of particles remaining in the simulation domain is outputted during the whole simulation period with FullSWOF-PTM. A user interface for FullSWOF-PTM was developed using MATLAB r2017a (MathWorks, Natick, MA, United States) [44] to run all impervious and pervious cases in batches.

2.3. Modeling Cases

Three kinds of modeling cases were developed and simulated in this study: 11 testing cases, 446 impervious cases, and 750 pervious cases, and details are described below.

2.3.1. FullSWOF-PTM Testing Cases

FullSWOF-PTM was validated using 11 rainfall events as testing cases to demonstrate that it can be used for accurate overland flow simulations. In Esteves’s study [36], a 2D overland flow model solving SWEs based on an explicit FDM and coupled with the GA infiltration module was developed and calibrated/validated with the observed data on a natural hillslope plot. These observed rainfall and runoff data from 11 events plus the plot topography and soil infiltration data were obtained from Dr. Esteves and first used to test the FullSWOF-PTM model. The plot was 14.25 m long and 5 m wide, bordered by 150 mm wide cement blocks driven about 50 mm into the ground [45]. The cell size used in the simulation was 0.25 m based on a detailed topographic survey. The Darcy–Weisbach friction law (friction coefficient = 0.25) was used in the simulation. The plot was crusted, almost without vegetation. The infiltration model parameters including the saturated hydraulic conductivity (K, 0.0162 mm/h for crust and 77.4 mm/h for soil below), saturated water content (θs, 0.245 for crust and 0.296 for soil), and suction head (φ) of crust layer and soil, as well as the crust layer thickness (0.005 m) were all the same as the calibrated parameters used in Esteves’s study [36].

The plot slope in x- and y-directions were 0.0640 $\pm$ 0.0292 (from the right to left boundary) and 0.0196 $\pm$ 0.0155 (from the top to bottom boundary), respectively. The right, top, and bottom boundaries of the study plot were all set as wall condition and the left (downstream) boundary was as Neumann (open) condition based on the field situation. The HLLC flux choice in FullSWOF-PTM was selected from the 1st order numerical scheme to calculate the new time level flux of each computational cell in the simulation. The Courant–Friedrichs–Lewy (CFL) condition (CFL = 0.4) was used to guarantee the numerical stability and calculate the time step interval for the simulation.

The rainfall data was measured with an electronic tipping-bucket recording rain gauge (each tip corresponding to 0.5 mm of rainfall). The discharge at the plot outlet was measured through a triangular 20° V-notch weir at every 5-second interval. The initial water content of the soil at the beginning of the rainfall was measured using a neutron-probe access tube located in the center of the plot [36]. The moisture deficit was calculated as the difference of saturation water content (${\theta }_s=\mathrm{prosity}$) and initial moisture content (θ_i).

The goodness of fit for the simulated hydrograph is evaluated using the Nash–Sutcliffe Efficiency (NSE) coefficient [46].

$$NSE=1-\frac{{{\displaystyle \sum }}_{j=1}^m{\left(Q_{oj}-Q_{si}\right)}^2}{{{\displaystyle \sum }}_{j=1}^m{\left(Q_{oj}-\overline{Q}\right)}^2}$$

(4)

where $Q_{oj}$ (m³/s) is the jth observed runoff rate, $Q_{sj}$ (m³/s) is the corresponding simulated runoff rate, $\overline{Q}$ (m³/s) is the mean observed runoff rate, and m [-] is the total number of observed runoff rates. The NSE values for 11 rainfall events were calculated and compared with those reported by Esteves’s program to evaluate the FullSWOF-PTM. The FullSWOF-PTM’s performance was further evaluated by comparing simulated and observed runoff depth and peak discharge at the outlet.

2.3.2. Impervious Modeling Cases

Using FullSWOF-PTM to model the overland flow on impervious surfaces has two purposes: (1) validating FullSWOF-PTM since calculated T_ci can be compared with many previous studies or established equations, and (2) T_ci will be used to develop T_cp equation. A total of 446 impervious modeling cases with different combinations of S₀, n, L, and i were simulated. In this study, the cell sizes in x- and y-directions and simulation domain width used for impervious and pervious modeling cases were 0.25, 0.25, and 1 m (4 cells in the y-direction, no cross slope), respectively. The Manning’s friction formula was selected among three friction formulas of FullSWOF-PTM. Figure 1 shows the model-parameter value distributions of the 446 cases, which also prove the parameter values are representative and set in the commonly used ranges [3].

For 446 impervious modeling cases, the plot longitudinal slope S₀ ranges from 0.0005 to 0.1 with the average slope of 0.0133 and standard deviation of 0.0175, and 86% of the cases have S₀ ≤ 2%. The Manning’s roughness n ranges from 0.01 to 0.8 with 86% of n values less than 0.1. The plot length L ranges from 20 to 250 m; the rainfall intensity i ranges from 10.2 to 256.5 mm/h with the average value and standard deviation of 80.8 and 64.0 mm/h, respectively. There are 384 modeling cases with S₀ ≤ 0.02, 395 cases with n ≤ 0.15, 435 cases with L ≤ 150 m, and 381 cases with i ≤ 150 mm/h.

The topography file, rainfall file, particle initialization file, and model parameter file are required for FullSWOF-PTM in the impervious modeling cases, which were created using user-developed MATLAB code. The 2nd order numerical scheme in FullSWOF-PTM including the numerical flux methods (5 choices), linear reconstruction settings (3 choices), and slope limiters (3 choices) were tested using one impervious modeling case (S₀ = 0.05, L = 35 m, n = 0.01, and i = 12.7 mm/h), which aims to identify the best numerical scheme among the 45 combinations. After performing the tests, all other impervious modeling cases were run in batches with MATLAB code using the tested best numerical scheme combination.

2.3.3. Pervious Modeling Cases

The GA infiltration parameters for different soil types of pervious surfaces were adapted from the research conducted by Rawls et al. [47]. The soil was categorized in this study into three groups: sand (K ranging from 25.4 to 127.0 mm/h), loam (K ranging from 2.54 to 12.7 mm/h), and clay (K ranging from 0.23 to 1.52 mm/h) groups depending upon the saturated hydraulic conductivity. There were 204, 350, and 196 modeling cases for sand, loam, and clay soil, respectively. A total of 750 modeling cases were developed and simulated while one T_cp equation was derived from the results of all pervious surfaces rather than developing three equations for different soil groups. The crust thickness of the pervious plots is set equal to zero because this study focused on the influence of soil property on the T_cp for pervious surfaces. The soil infiltration parameters (saturated hydraulic conductivity K, suction head φ, and moisture deficit Δθ) used for the 750 modeling cases are shown in Figure 2. A dimensionless saturated hydraulic conductivity K′ (=K/i), which was used by Akan in the derivation of the T_cp equation for pervious surfaces [24], was calculated and ranged from 0.001 to 0.97 for the 750 modeling cases (Figure 2). Since K′ is less than 1, it means K < i and all pervious modeling cases should produce runoff eventually when the rainfall duration is long enough. Equation (5) developed by Akan [24] is limited to K′ ≤ 0.4, i.e., i ≥ 2.5 K, which means Akan’s equation does not apply to relatively small rainfall intensity in comparison to K, but in reality i can be larger or even smaller than K. When i ≤ K, it is not applicable to determine the T_cp for pervious surfaces since there is no surface runoff as all rainfall is infiltrated.

$$T_{cp\_A}={(\frac{Ln}{\sqrt{S_0}})}^{0.60}\frac{1}{{\left(i-K\right)}^{0.40}}+3.1\frac{K^{1.33}\phi \Delta \theta }{i^{2.33}}(\mathrm{For}{K}^{\prime }\le 0.4)$$

(5)

The representative values of infiltration parameters of three soil groups were selected from literature [47] for 750 modeling cases. The saturation hydraulic conductivity ranges from 0.23 to 127 mm/h. The soil dry suction head ranges from 0.0457 to 0.3238 m. The moisture deficit ranges from 0.01 to 0.45.

3. Results and Discussion

3.1. FullSWOF-PTM Testing Results

The simulated discharges from FullSWOF-PTM were divided by the drainage area and compared to the observed runoff data (mm/h) for all 11 testing events. The comparison of observed and simulated discharge hydrographs on 24 August and 4 September, 1994, are shown in Figure 3 as sample results, and the hydrographs closely follow rainfall variations. It shows that the FullSWOF-PTM simulation results have strong consistency with the observed data during the whole rainfall period.

For 11 rainfall events, the initial moisture content θ_i ranged from 0.048 to 0.106 (

Table 1. Comparison of simulated and observed discharge parameters for 11 rainfall events.

Events [θi, RD]	NSE 1	NSE 2	Dob 2 (mm)	Dsi 1 (mm)	∆D (%)	Qpo 2 (mm/h)	Qps 1 (mm/h)	∆Qp (%)
Jul 20 [0.048, 51.5]	0.81	0.85	29.0	32.4	11.7	144.5	165.5	14.5
Jul 21 [0.103, 59.5]	0.68	0.66	26.7	30.1	12.7	77.0	85.2	10.6
Aug 7 [0.100, 30.0]	0.71	0.90	22.0	20.8	−5.5	179.4	138.3	−22.9
Aug 10 [0.106, 11.0]	0.75	0.90	5.6	5.6	0.0	59.1	52.4	−11.3
Aug 20 (1) [0.067, 22.0]	0.69	0.63	9.7	7.2	−25.8	42.0	33.1	−21.2
Aug 20 (2) [0.095, 4.0]	0.70	0.90	2.2	1.7	−22.7	50.5	37.0	−26.7
Aug 24 [0.088, 31.0]	0.90	0.88	18.8	21.0	11.7	142.2	159.0	11.8
Aug 25 [0.085, 24.0]	0.82	0.89	14.3	14.3	0.0	91.6	86.7	−5.3
Sep 4 [0.082, 62.0]	0.95	0.93	37.8	35.1	−7.1	123.5	113.7	−7.9
Sep 8 [0.082, 31.0]	0.62	0.46	15.3	20.4	33.3	86.7	107.3	23.8
Sep 16 [0.061, 9.5]	0.60	0.66	2.4	2.7	12.5	48.4	41.8	−13.6

Note: θ_i is initial soil moisture content, R_D is rainfall depth in mm, ¹ is calculated using FullSWOF-PTM simulated and observed data, ² is adapted from Esteves’s paper after comparing with observed data [36], D_ob (mm) is the observed total runoff depth, Dsi (mm) is the simulated total runoff depth, ∆D (%) is the percent error of simulated runoff depth = (D_si − D_ob)/D_ob $\times$ 100%, Q_po (mm/h) is the observed peak runoff rate, Q_ps (mm/h) is the simulated peak runoff rate, ∆Q_p (%) is the percent error of simulated peak discharge = (Q_ps − Q_po)/Q_po $\times$ 100%.

). Since the soil porosity of the field was 0.296, the moisture deficit Δθ ranged from 0.190 to 0.248. The discharge NSE values of FullSWOF-PTM and Esteves’s programs ranged from 0.64 to 0.95 (average $\pm$ standard deviation as 0.75 $\pm$ 0.11) and from 0.46 to 0.93 (0.79 $\pm$ 0.15), respectively; and this indicates FullSWOF-PTM performed as well as Esteves’s program in simulating overland flows. Table 1 also gives the comparison of simulated and observed runoff depths (mm) and peak discharges (mm/h). The percent errors of simulated runoff depths and peak discharges are 1.9% $\pm$ 15.9% (average $\pm$ standard deviation) and −4.4% $\pm$ 16.3%, respectively. The results for all 11 rainfall events (Table 1 and Figure 3) show that the FullSWOF-PTM program predicts the rainfall–runoff process of overland flows on a pervious surface with reasonable accuracy.

3.2. Time of Concentration (T_ci) of Impervious Surfaces

In previous studies [2,7,9,15,48], T_c was evaluated as the time when discharge at the outlet reaches a specific percentage of the equilibrium discharge, e.g., 90%, 95%, or 98% of peak discharge (Q_p). However, it is difficult to evaluate T_c for pervious surfaces using the fixed percent Q_p because there is almost no equilibrium discharge for pervious surfaces. In this study, the travel time for 85% of particles arriving at the drainage outlet (T_r__p85) was used to evaluate the T_c for both impervious and pervious surfaces based on the previous PTM studies [29].

Figure 4 shows simulated outlet discharge and the in-domain particle percentage versus time under eight Manning’s roughness coefficients but the same plot slope, length, and rainfall intensity. The 2nd order numerical scheme combinations HLL2 for the numerical flux, ENO for the linear reconstruction, and Vanleer for the slope limiter [37] were used for the simulation because the simulated Q_p was exactly the same as Q_p calculated by the rational method and the run time for the program is the shortest. Simulated discharges verses time by FullSWOF-PTM give S-hydrographs [49] in Figure 4 under the constant rainfall intensities over a long period of time. The outlet discharge increases and the in-domain particle percentage decreases as the constant rainfall continues, and finally both reach the equilibriums. Based on the rational equation, the peak discharge (Q_pr) for all eight runs should be the same and equal to 0.864 L/s, which is the same as FullSWOF-PTM simulated Q_p (Figure 4).

On Figure 4, T_{c_q98} is the T_c defined or calculated as the travel time when the runoff reaches the 98% percent of the equilibrium discharge Q_pr, and T_{r_p85} is the travel time when the in-domain particle percentage is 15%. Both T_{c_q98} and T_{r_p85} increase with the increase of the roughness (Figure 4) since the flow velocity is smaller with higher roughness. Figure 4 indicates that T_{c_q98} is somewhat smaller than T_{r_p85} for each case, which is the same as the conclusion of the previous research [29].

A generalized power relation, Equation (6) was chosen to develop the regression equation for T_ci as a function of four input/influencing parameters (L, n, S_o, and i). The three influencing parameters L, n, and S_o that describe/characterize the overland flow surface were grouped as a combined parameter (Ln/$\sqrt{S_0}$) because the Manning’s equation was used in FullSWOF-PTM as friction formula for the overland flow resistance [24]. The T_{r_p85} obtained from FullSWOF-PTM was considered as T_ci or T_{ci_p85} (Figure 5), and T_{r_p85} values for all 446 impervious modeling cases were used to develop the T_ci regression equation.

$$T_{ci}={(\frac{Ln}{\sqrt{S_0}})}^{k_1}\frac{1}{i^{k_2}}$$

(6)

The exponents (k₁ and k₂) were estimated using the MLR method after the log transformation of Equation (6), and the resulting regression equations of T_{ci_p85} are:

$$T_{ci}={(\frac{nL}{\sqrt{S_0}})}^{0.608}\frac{1}{i^{0.422}}(R^2=0.996)$$

(7)

$$T_{ci}=9.741\times {(\frac{nL}{\sqrt{S_0}})}^{0.608}\frac{1}{i^{0.422}}$$

(8)

where L is in (m), n is in [-], S₀ is in [-], T_ci of Equation (7) is in seconds and Equation (8) in minutes, respectively, when i for Equation (7) is in m/s and for Equation (8) in mm/h. FullSWOF-PTM uses i in m/s for all computations (1 m/s = 3,600,000 mm/h). The 95% confidence intervals for k₁ and k₂ are [0.606, 0.610] and [0.421, 0.423] with p-value < 0.0001, respectively. The average difference between T_ci calculated using Equation (7) and simulated T_{ci_p85} is 0.11 min with a standard deviation of 1.45 min (Figure 5).

Figure 5 shows calculated T_ci from Equation (8) and five other equations [50,51,52,53,54] versus simulated T_{ci_p85}. The calculated T_ci from all six equations (

Table 2. Six T_ci equations and statistical results when compared with simulated T_{ci_p85} (Figure 5).

Source	Tci Formula (min)	R2	RMSE (min)
Tci (Equation (8))	$T_{ci}=9.741\times {\left(nL/\sqrt{S_0}\right)}^{0.608}/i^{0.422}$	0.996 (1.00 × Tci_p85) 1	1.45
Linsley (1958)	$T_{ci}=6.82{\left(nL/\sqrt{S_0}\right)}^{0.633}/i^{0.398}$	0.998 (0.87 × Tci_p85)	4.15
Henderson & Woods (1964)	$T_{ci}=6.98{\left(nL/\sqrt{S_0}\right)}^{0.60}/i^{0.40}$	0.997 (0.74 × Tci_p85)	8.04
Morgali (1965)	$T_{ci}=7.05L^{0.593}{n}^{0.605}/\left(i^{0.388}{S}_0^{0.38}\right)$	0.995 (1.19 × Tci_p85)	6.08
KC et al., (2014)	$T_{ci}=8.67\times L^{0.541}{n}^{0.649}/\left(i^{0.391}{S}_0^{0.359}\right)$	0.994 (0.96 × Tci_p85)	2.33
KC & Fang (2015)	$T_{ci}=9.25\times L^{0.599}{n}^{0.609}/\left(i^{0.399}{S}_0^{0.303}\right)$	0.997 (0.98 × Tci_p85)	1.38

Note: ¹—the fitted equation between each equation-calculated T_ci and T_{ci_p85} is given inside brackets after R².

) linearly correlates well with the simulated T_{ci_p85} as indicated by R² > 0.994 and regression equations in Table 2. This proves Equation (8) developed from T_{ci_p85} predicts well T_c for impervious surfaces and the travel time for 85% of particles to arrive at the outlet can be considered as T_ci with reasonable accuracy. The root-mean-square error (RMSE) value of the KC and Fang’s equation (2015), also based on the travel time for 85% of particles to arrive at the outlet [29], is the smallest (1.38 min, Table 2) and the RMSE of Equation (8) is just slightly larger. In overall, the Henderson & Woods and Linsley equations underestimate T_{ci_p85} while the Morgali equation overestimates, and the remaining three equations predict T_ci well with lower RMSE for impervious surfaces.

3.3. Time of Concentration (T_cp) of Pervious Surfaces

The responding runoff hydrographs and in-domain particle percentages of eight saturation hydraulic conductivity values for pervious surfaces (S₀ = 0.01, L = 50 m, φ = 0.06 m, Δθ = 0.18) are summarized and compared in Figure 6 under constant rainfall intensity (i = 105.2 mm/h). Figure 6 clearly indicates that the PTM is the only choice to evaluate T_cp for pervious surfaces because it is not practical to run the simulation for very long periods for the runoff to reach acceptable equilibrium. For a pervious surface, the rainfall in the early period will completely infiltrate when the rainfall intensity is smaller than the infiltration capacity [55] (Figure 6). The time prior to ponding or the start of surface runoff is defined as the ponding time, which depends on both the rainfall intensity and infiltration rate. When the Green–Ampt infiltration model is used, the ponding time (t_p) under constant rainfall intensity is calculated using Equation (9) [56].

$$t_p=\frac{\phi \Delta \theta K}{i\left(i-K\right)}=\frac{\phi \Delta \theta K^{\prime }}{i\left(1-K^{\prime }\right)}$$

(9)

For eight modeling cases on Figure 6, K ranges from 7.06 × 10⁻⁶ to 19.4 × 10⁻⁶ m/s (25.4–69.8 mm/h), which belongs to sandy loam or sand soil groups, and the ponding time from Equation (9) ranges from 1.96 to 12.2 min, which is the same as predicted by FullSWOF-PTM. Equation (9) is valid when i > K, but if t ≤ t_p even when i > K, there is still no surface runoff and then it is not necessary and not meaningful to determine T_cp for pervious surfaces. For impervious surfaces, T_ci always exists for any non-zero rainfall intensity (Table 2).

For pervious surfaces, the outlet discharge decreases with the increase of K (Figure 6) because of more infiltration. The outlet discharge and in-domain particle percentage are postponed more and more when K and the soil infiltration capacity increases. The in-domain particle percentage is 100% when t ≤ t_p. The travel time for 85% of particles to arrive at the outlet T_{r_p85} was calculated by FullSWOF-PTM from the beginning of the rainfall event and ranges from 10.7 to 27.8 min (open squares on Figure 6) for these eight cases.

In this paper, the time of concentration for a pervious surface T_cp is considered as T_{r_p85} of pervious surfaces determined from FullSWOF-PTM minus the ponding time t_p determined using Equation (9), and presented in Equation (10). Figure 7 plots cumulative distributions of t_p calculated from Equation (9), simulated T_{r_p85}, and T_cp for all 750 modeling cases, and their statistical summary is given in

Table 3. Statistical summary of t_p, T_{r_p85}, and T_cp of pervious surfaces for 750 modeling cases.

Parameter	tp (min)	Tr_p85 (min)	Tcp (min)
Minimum	0.02	3.2	3.2
50 percentiles	3.3	23.2	17.3
75 percentiles	18.0	56.6	35.8
90 percentiles	56.0	104.7	60.3
95 percentiles	69.1	136.3	82.6
Maximum	180.1	351.6	227.8

Note: t_p is the ponding time, T_{r_p85} is the travel time for 85% of particles to arrive at the outlet from the beginning of rainfall, and T_cp is the time of concentration for pervious surfaces, which is equal to T_{r_p85} − t_p.

. The modeling cases with large t_p and T_{r_p85} have small differences between i and K. For 750 cases, t_p is 0.2% to 82.6% of T_{r_p85}, and 50% of t_p is less than 18.8% of T_{r_p85}. Excluding t_p from T_{r_p85}, it means T_cp is not counted from the beginning of the rainfall but the commencing of the runoff, and this is consistent with Guo’s study [4]. The purpose of subtracting t_p from T_{r_p85} is also to advise engineers and designers that it is not meaningful to determine T_cp for pervious surfaces when the rainfall duration is less than t_p. It is recommended that t_p should be calculated, e.g., using Equation (9), before T_cp for a pervious surface is calculated.

Following Akan’s study [24], T_cp is considered as the sum of the time of concentration of an equivalent impervious surface (T_ci) using (i − K) as effective rainfall and additional travel time due to infiltration (T_rs) related to the soil infiltration properties. A generalized power relation in Equation (10) was used to develop a regression equation for T_rs and then for T_cp equation of pervious surfaces based on the simulated T_{r_p85} of 750 pervious modeling cases.

$$T_{cp}=T_{r\_p85}-t_p=T_{ci}+T_{rs}={(\frac{Ln}{\sqrt{S_0}})}^{0.608}\frac{1}{{\left(i-K\right)}^{0.422}}+c_1\frac{K^{c_2}{\phi }^{c_3}\Delta {\theta }^{c_4}}{i^{c_5}}$$

(10)

The exponents (C₁, C₂, C₃, C₄, and C₅) in Equation (10) were determined using the MLR regression between y = T_{r_p85} − t_p − T_ci and x = (K, φ, Δθ, and i) after log-transformed. The fit results of five exponents were summarized in

Table 4. MLR fitted exponents C₁ to C₅ of Equation (10) and corresponding 95% confidence intervals.

Parameter	Value	95% Confidence Interval
C1	2.162	[0.875, 5.340]
C2	0.535	[0.488, 0.582]
C3	0.161	[0.030, 0.292]
C4	0.645	[0.591, 0.699]
C5	1.213	[1.172, 1.255]

and Equation (11).

$$T_{cp}={(\frac{Ln}{\sqrt{S_0}})}^{0.608}\frac{1}{{\left(i-K\right)}^{0.422}}+2.162\frac{K^{0.535}{\phi }^{0.161}\Delta {\theta }^{0.645}}{i^{1.213}}$$

(11)

where T_cp is in seconds when L is in m; n, S₀, and Δθ are dimensionless, i and K are in m/s, and φ in m. The regression equation for T_cp has a p-value < 0.0001, and the residuals between calculated T_cp from Equation (11) and T_{r_p85} − t_p range from −15.4 to 53.3 min with average residual of 1.12 min (standard deviation 6.13 min). One can see that Equation (11) developed from this study can be used to determine T_c of overland flows for both impervious (setting K = 0) and pervious surfaces.

The comparison of Equation (11) and Akan’s Equation (5) was shown in Figure 8: (a) for K′ ≤ 0.4 (427 modeling cases) and (b) for all 750 modeling cases since Akan [24] developed Equation (5) for K′ ≤ 0.4. Akan’s time of concentration for pervious surfaces (T_{cp_A}) is counted from the beginning of the rainfall event and includes the period before the ponding time; therefore, T_cp from Equation (11) and T_{cp_A} − t_p are plotted against T_{r_p85} − t_p on Figure 8. Akan’s T_{cp_A} − t_p is linearly correlated well with T_{r_p85} − t_p when the overall R² is 0.992, but underestimated since T_{cp_A} − t_p = 0.664 (T_{r_p85} − t_p) (Figure 8) and RMSE is 13.7 min. The RMSEs of Equation (11) is 4.8 min and the overall R² are 0.993.

Figure 8b shows the comparison of calculated T_cp from Equation (11) for 750 modeling cases and T_{cp_A} − t_p for 698 cases with T_{cp_A} − t_p > 0 against T_{r_p85} − t_p. When Akan’s Equation (5) is applied to 323 modeling cases with K′ > 0.4, there are 52 cases having T_{cp_A} − t_p < 0 or T_{cp_A} < t_p, which were not shown on the log scale plot (Figure 8). The R² of Equation (11) and Akan’s T_{cp_A} − t_p are 0.979 for 750 cases and 0.792 for 698 cases with T_{cp_A} − t_p > 0, respectively. The RMSE of Equation (11) and Akan’s T_{cp_A} − t_p are 6.2 min and 18.0 min, respectively. Figure 8 and the above analysis show Akan’s Equation (5) should not be applied to beyond its limits: K′ ≤ 0.4 and P′ < 9, where P′ = φΔθ/iT_ci and T_ci used by Akan in Equation (5) is the same as Henderson and Woods’ T_ci equation [24]. Akan introduced P′ for solving non-dimensional flow and infiltration equations but did not explain any physical meaning of P′ [10,19,24]. One can see that iT_ci is rainfall depth over T_ci of impervious surface of the same geometry and the ponding time increases with φΔθ indicated by Equation (9). Akan [24] indicated K′ ≤ 0.4 and P′ < 9 are for most practical applications without providing any reason. For 750 modeling cases studied here, the limits of K′ values are [0.001, 0.97] and of P′ values are [0.18, 55.27]. Therefore, Equation (11) developed from this study can be used to estimate T_cp for pervious surfaces with reasonable accuracy and over wide ranges of input parameters, especially for small rainfall intensities in comparison to K.

4. Summary and Conclusions

The particle tracking method (PTM) module was added into the 2D overland flow simulation program based on the open-source program FullSWOF_2D that can be used to estimate the time of concentration for impervious and pervious surfaces. The FullSWOF-PTM program was tested using published rainfall and runoff data and simulated hydrographs match well with observed data (Table 1), which proves it can predict the overland flow accurately. Four hundred and forty-six impervious modeling cases were developed and simulated to explore T_ci of overland flow on impervious surfaces. The travel time of 85% of particles to arrive at the drainage outlet (T_{r_p85}) was calculated by FullSWOF-PTM for determining the time of concentration of impervious and pervious surfaces in this study. A regression equation of T_ci, Equation (7) was derived using the MLR regression method and as a power function of $Ln/\sqrt{S_0}$ and i. The derived impervious surface T_ci equation matches well with T_{r_p85} and correlates well with T_ci from other five published equations, which further proves FullSWOF-PTM can be used to estimate T_cp of overland flow on pervious surfaces.

Seven hundred and fifty pervious modeling cases were developed and simulated to explore the T_cp equation. In this study, T_cp is considered as T_{r_p85} of pervious surfaces determined from FullSWOF-PTM minus the ponding time t_p determined using Equation (9). It means T_cp is not counted from the beginning of the rainfall but the commencing of the runoff. Engineers and designers should calculate t_p first, e.g., using Equation (9), before T_cp for pervious surface is calculated because it is not meaningful to determine T_cp for pervious surfaces when the rainfall duration is less than t_p. A regression equation for T_cp, Equation (11) was developed using simulated T_{rp_p85} and calculated t_p from 750 pervious modeling cases. Equation (11) includes T_ci for an equivalent impervious surface using (i − K) as effective rainfall and additional travel time due to infiltration (T_rs) as a function of rainfall intensity and the soil infiltration parameters (K, φ, and Δθ). Therefore, Equation (11) can be used for both impervious and pervious surfaces. The derived T_cp equation has higher R² and smaller RMSE compared to Akan’s equation as well as wide ranges of input parameters.