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This paper presents a straightforward design discharge estimation procedure for small urban districts. In this approach, the configuration of the small urban district was simplified into either a rectangular porous overland plane or a V-shaped porous overland model. The Horton equation was used to estimate the infiltration on the overland planes, and the kinematic-wave approximation was applied for runoff simulations. In deriving the runoff equation, the Darcy–Weisbach coefficient was adopted to account for the influences of spatially various flow depths on overland flow resistance. The theoretical derivations were applied to a series of cases and verified by numerical models to demonstrate the applicability of the proposed method. The results showed that the design discharges obtained from the analytical equations are close to those simulated by the numerical model. The analysis procedure developed herein can provide a convenient way for engineers to conduct urban sewer designs.

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  • A concise formula for design discharge is derived using the time of concentration equation and the rainfall intensity–duration relationship, providing a practical tool for engineers.

  • The paper offers a straightforward and effective approach to stormwater drainage system design tailored specifically for small urban districts, providing valuable insights and tools for practical implementation.

analytical solution, design discharge, kinematic-wave approximation, time of concentration, urban watershed
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The runoff at a watershed outlet can reach its maximum rate if the rainfall duration is greater than or equal to the time of concentration of the watershed. The design discharge from the rational method usually relies on the time of concentration, a loosely defined parameter calculated rather subjectively in practice. Numerous empirical time of concentration equations for design discharge estimations can be found in previous literature, such as Kirpich (1940), Kerby (1959), as well as Huggins & Burney (1982). The above empirical equations vary from one watershed to another. McCuen et al. (1984) reported that the values of time of concentration estimated by different empirical equations show wide variations and generate the design discharges in significant differences. Therefore, empirical equations should not be applicable beyond the watershed conditions in which they were developed.

Based on the kinematic-wave approximation, Henderson & Wooding (1964) derived the time of concentration equations for a rectangular overland plane, and Wooding (1965) subsequently derived the travel time equation on a compound V-shaped overland plane. In their theoretical derivations, rainfall intensity is first included, and later, the rainfall intensity is found to be the most important input parameter for the time of concentration estimation (McCuen et al. 1984). The application of the kinematic-wave approximation for watershed routing has been considered acceptable for small watersheds, provided the watershed slope is not too flat (Ponce 1989; Fread 1993; McCuen & Spiess 1995). By applying the kinematic-wave method for flows on pervious planes, Chen & Evans (1977) used a constant infiltration rate to abstract rainfall intensities for peak discharge estimation. Akan (1985, 1986) developed a finite-difference kinematic-wave overland flow model and applied Green-Ampt infiltration to estimate the time of concentration. Baiamonte & Singh (2016) derived an analytical solution for the time of concentration on a rectangular overland plane using the kinematic-wave equation under the Green-Ampt infiltration.

Most previous time of concentration derivations on the overland plane used Manning's formula to describe the flow resistance, although some studies reported that the Manning coefficient varies with the Reynolds number (Emmett 1970; Taken & Govers 2000; Hessel et al. 2003; Zhang et al. 2010; Shen et al. 2023), Manning's coefficient was usually treated mainly depending on surface roughness and the flow was in turbulent regime. Nevertheless, as reported by Yu & McNown (1964), for Reynolds numbers up to 104, the flow on concrete surfaces is still under laminar or transitional flow regimes. Chen (1976) showed that the flow could be entirely laminar on the grass surface for the same range of the Reynolds number. Butler (1988) pointed out that the estimating error for the time of concentration may be significant if the flow regime is inappropriately classified into turbulence. Subsequently, Chen & Wong (1993) used a typical relationship between the Darcy–Weisbach resistance coefficient and the Reynolds number for overland flow analysis to cover various flow regimes.

This study adopts the Darcy–Weisbach resistance coefficient for kinematic-wave overland flow analysis. Flow conditions from laminar through transitional even to turbulence can be considered. As shown in Figure 1, a small urban district is conceptually treated as a V-shaped overland model, in which runoff is collected by the two overland planes to the central channel. The varying flow depths along the overland plane and channel are used to calculate the Reynolds number. Thus, the flow resistance has incorporated the effects of viscosity, slope, and varying flow depths along the flow path. In considering the infiltrability of the porous overland planes, we applied the Horton infiltration equation to estimate the time-varying rainfall excess. Moreover, the analytical solutions for peak discharge estimation are further verified using numerical model simulations for different cases. The theoretical derivations are promising to provide a convenient method of sewer size design in small urban districts.
Figure 1
Schematic configuration of a small urban district.
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Schematic configuration of a small urban district.

Figure 1
Schematic configuration of a small urban district.
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Schematic configuration of a small urban district.

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The time of concentration indicates the time it takes for flow to travel from the hydraulic remotest point in a watershed to the outlet. Hence, the discharge can reach a peak at the time of concentration while the entire watershed contributes flow to the watershed outlet. The usefulness of the kinematic-wave method in rainfall–runoff modeling lies in the feasibility of obtaining analytical solutions. As shown in Figure 2(a), for a simple rectangular overland plane, the continuity equation and simplified momentum equation are
(1)
(2)
where x is the flow direction, yo is the overland flow depth, qo is the overland flow discharge per unit width, ie is the rainfall excess, m is a constant, and is a parameter reflecting the hydraulic characteristics of the overland flow. The constant m can be recognized as 5/3 from Manning's equation. The parameter can be estimated as from Manning's equation, in which So is the plane slope and no is the effective roughness coefficient for the overland plane. If the Darcy–Weisbach friction formula is used to describe the overland flow, the constant m is equal to 3/2 and can be expressed as
(3)
where g is the gravitational acceleration, and fo is the Darcy–Weisbach resistance coefficient for overland flow. The Darcy–Weisbach resistance coefficient can be related to the Reynolds number Ro as follows:
(4)
where c and d are constants according to boundary roughness and flow condition, respectively. The Reynolds number in Equation (4) is defined as
(5)
where Vo is the overland flow velocity, is the kinematic viscosity of flow. Combining Equations (3), (4) and (5), the parameter can be further expressed as
(6)
where Co and A1 are parameters that can be expressed as
(7)
(8)
Figure 2
Watershed conceptual models.
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Watershed conceptual models.

Figure 2
Watershed conceptual models.
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Watershed conceptual models.

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As given in Equation (6), the parameter is related to overland flow depth on the plane; nevertheless, it would only be related to the roughness and slope of the overland plane if one adopts Manning's formula. Combining Equations (1) and (2), the continuity equation can be rewritten as
(9)
During the rising stage, the water depth yo can be obtained by integrating Equation (9) as
(10)
By applying the method of characteristics, the kinematic-wave celerity for overland flow can be derived as:
(11)
where A2 and A3 are parameters that can be expressed as
(12)
(13)
Equation (11) can be integrated separately with respect to x and t as
(14)
where xio and tio are the position and time for the origin of the characteristic curve, respectively. The result of Equation (14) gives the characteristic curves as
(15)
If the initial time is equal to zero and the term is equal to the length of the overland plane ⁠, the time for a wave traveling from the upper edge to the end of the overland plane is
(16)

The toc is recognized as the time of concentration of the rectangular overland plane.

For a watershed configuration composed of two identical rectangular planes as a V-shaped type (as shown in Figure 2(b)), the lateral inflow contributes to the central channel from both sides. If the rainfall falling directly onto the channel is assumed to be negligible, for a broad channel with a width B, the lateral inflow in the equilibrium state contributing to the channel is ⁠. Hence, the continuity equation and simplified momentum equation for the channel flow are
(17)
(18)
where yc is the channel-flow depth; qc is the channel-flow discharge per unit width of the channel, and is a parameter reflecting the hydraulic characteristics of the channel flow, which can be expressed as
(19)
where Sc is the channel slope and fc is the Darcy–Weisbach resistance coefficient for channel flow. The Darcy–Weisbach resistance coefficient can also be related to the Reynolds number for channel flow as
(20)
where c and d are constants as shown in Equation (4) according to boundary roughness and flow condition. The Reynolds number for channel flow in Equation (20) is defined as
(21)
where Vc is the channel flow velocity. Combining Equations (19), (20), and (21), the parameter for the channel flow is
(22)
where Cc is parameters that can be expressed as
(23)
Similarly, the characteristics of the channel flow can be derived as
(24)
where xic and tic are the position and time for the origin of the characteristic curve for channel flow. If the initial time tic is equal to zero and the term is equal to the length of the channel Lc, the equilibrium time for the channel flow in a V-shaped overland model is
(25)
The tcc is usually recognized as the time of concentration for the channel. Therefore, combining Equations (16) and (25), the time of concentration for the V-shaped overland model is
(26)

Two differences exist between the time of concentration equation derived herein and the conventional derivations, which used Manning's roughness coefficient. One is that the equation derived in this study can be applied from laminar through transitional, even extended to turbulence, by setting the d value between 0 and 1. On the contrary, those equations derived by using Manning's coefficient can only be applied to turbulence. The other is that the influence of flow depth on the resistance coefficient has been implicitly included in the above derivation.

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Intensity–duration curves are usually expressed as regression equations to avoid having to read the design rainfall intensity from graphs. For a porous overland plane, the intensity–duration relationship of the rainfall excess will differ from that of the original storm record. All the rainfall may infiltrate in the early stages of the storm because of the high infiltrability of initially unsaturated soil. The rainfall excess and, consequently, the overland flow will commence after the soil infiltrability drops below the rainfall intensity. Ponding time refers to the elapsed time when the rainstorm starts and the rainwater begins to accumulate on the soil surface. As shown in Figure 3, if the initial infiltrability, fi, is larger than the rainfall intensity, and the ponding time without surface runoff is ⁠, the rainfall excess duration can be defined as
(27)
where is the rainfall excess duration, is the recorded rainstorm duration, and is the ponding time. Hence, the average intensity for the rainfall excess is
(28)
where f(t) is the infiltration rate at time t; ir is the recorded storm intensity.
Figure 3
Rainfall intensity and infiltration.
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Rainfall intensity and infiltration.

Figure 3
Rainfall intensity and infiltration.
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Rainfall intensity and infiltration.

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In this study, the Horton infiltration equation was adopted to evaluate rainwater losses due to infiltration. It has been shown elsewhere that the Horton equation has a well-accepted physical basis, and its results agree well with various infiltration situations (Eagleson 1970; Raudkivi 1979). The infiltrability of the Horton equation can be expressed as
(29)
where f(t) is the infiltration rate, fi is the initial infiltrability, fe is the equilibrium infiltrability, and k is a constant. The ponding time in Equation (27) is equal to zero for ⁠, and for ⁠, it can be expressed as (Chow et al. 1988)
(30)
Once the parameters of the infiltration equation are obtained from soil characteristics analysis, the values of te and ie can be calculated by Equations (27) and (28). After deducing the recorded storm duration tr to the rainfall excess duration te, and calculating the recorded storm intensity ir to the rainfall excess intensity ie, the intensity–duration relationship for a specified return period can be expressed as
(31)
where a and b are regression coefficients. The rainfall excess intensity in Equation (31) is usually in mm/h, and the time is in min. Instead of adopting an extra constant in the denominator, the simplified form is due to the convenience of later derivation. Hence, the intensity–duration curve is suggested to separate into several intervals for sets of a and b values to account for the drawback.
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The design discharge of a watershed refers to the maximum flow rate expected to occur under a specified return period condition. It is typically determined based on the frequency analysis of rainfall records to account for the potential damage to infrastructure and property so authorities can ensure public safety and minimize environmental impacts. As indicated in Equation (31), the intensity of the rainfall excess is governed by the rainfall duration. For a given rectangular porous overland plane with length Lo, by setting te=toc and through the simultaneous solution of Equations (16) and (31), the design rainfall excess for the rectangular overland plane can be expressed in a meter–kilogram–sec system as follows
(32)
where Co is as indicated in Equation (7), ie is the rainfall excess intensity in m/s, Lo is the overland plane length in m. A4 is a parameter as
(33)
The constant K is used to transfer the unit from mm/h for ie and min for te in Equation (31) to the meter-kilogram-second system, which can be expressed as
(34)
Substituting Equation (32) into Equation (16), the design time of concentration for the porous overland plane can be expressed analytically as
(35)
where toc is the time of concentration in seconds, and Lo is in m.
For the case of the rainfall excess duration, te, greater than or equal to the time of concentration of the rectangular plane, the discharge can reach its maximum value as
(36)
where qop is the design discharge per unit width of the rectangular porous overland plane in m2/s.
For a given V-shaped porous overland model, by setting te=tc and then through Equations (26) and (31), the design rainfall excess for the V-shaped model can be expressed as
(37)
where ie is in m/s; Lo and Lc are in m. Cc and K are shown in Equations (23) and (34), respectively. Substituting Equation (37) into Equation (26), the design time of concentration for the V-shaped porous overland model is
(38)
where tc is the time of concentration of the V-shape model in sec, Lo and Lc are in m.
Likewise, for the case of the rainfall excess duration te greater than or equal to the time of concentration of the V-shaped porous overland model, the maximum discharge can be estimated as
(39)
where qcp is the design discharge per unit width of the channel flow at the outlet of the V-shaped model in m2/s; Lo, Lc, and B are in m. The above discharge derivations (Equations (36) and (39)) are restricted to apply to small watersheds. A small watershed indicates that (a) the rainfall is uniformly distributed in the watershed, (b) the rainfall duration is longer than the time of concentration of the watershed, and (c) the runoff is mainly in the form of overland flow. It is not recommended to use the derived equations for those watersheds whose hydrological characteristics are inconsistent with the above conditions.
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We applied the derived equations to rectangular porous overland planes and V-shaped porous overland models to demonstrate the potential use of the proposed method for urban sewer design. For both cases, the overland and channel slopes were set as 0.005, and the lengths of the overland plane and channel were tested from 100 to 500 m. The 5-year rainfall intensity–duration formula for Taipei City of Taiwan was used in the case study. To account for the infiltration losses for rainfall excess estimations, the corresponding infiltration parameters in Equation (29) are fi = 60.7 mm/h for the initial infiltrability, fe = 5.08 mm/h for the equilibrium infiltrability, and k = 3.085 h−1 for the infiltration constant (Yen 1989). Figure 4 shows the storm record and rainfall excess corresponding to the 5-year return period condition, and the regression coefficients of Equation (31) for the two datasets are listed in Table 1.
Table 1

Regression coefficients of the intensity–duration equation under the 5-year return period condition

Duration (min)Storm record
Rainfall excess
aBab
 305.66 0.346 200.06 0.352 
 521.93 0.502 269.94 0.439 
 639.33 0.550 451.24 0.561 
Duration (min)Storm record
Rainfall excess
aBab
 305.66 0.346 200.06 0.352 
 521.93 0.502 269.94 0.439 
 639.33 0.550 451.24 0.561 
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Figure 4
Five-year intensity–duration curves for Taipei City.
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Five-year intensity–duration curves for Taipei City.

Figure 4
Five-year intensity–duration curves for Taipei City.
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Five-year intensity–duration curves for Taipei City.

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Regarding the Darcy–Weisbach resistance coefficient shown in Equation (4), Chow (1959) reported that d equals unity for laminar flow. By using experimental data from Izzard (1942–43) and the U.S. Army Corps of Engineers (1954), Chen & Wong (1990, 1993) stated that for the Reynolds number up to 104, the value of the constant d is between 0.5 and 1.0 for various surface conditions. For illustration purposes, values of c = 4 and d = 0.5 (in Equation (4)) are used in the case study according to the local condition. Hence, for a rectangular overland plane with a length of 100 m, So = 0.005. The parameters are set as a = 200.06, b = 0.352 (Equation (31)), and m = 3/2 (Equation (2)); consequently, the design rainfall intensity is 101.91 mm/h (Equation (32)), the time of concentration is 6.80 min (Equation (35)), and the design discharge is qop = 0.00283 m2/s according to Equation (36). Figure 5 shows the design discharges for different lengths of overland planes from 100 to 500 m. The same analytical procedure was applied to the V-shaped porous overland models, in which the channel length was set equal to the overland plane length. The peak discharges estimated by Equation (39) for the V-shaped overland models are shown in Figure 6.
Figure 5
Design discharges for the rectangular overland planes with various lengths.
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Design discharges for the rectangular overland planes with various lengths.

Figure 5
Design discharges for the rectangular overland planes with various lengths.
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Design discharges for the rectangular overland planes with various lengths.

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Figure 6
Design discharges for the V-shape overland models with various lengths.
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Design discharges for the V-shape overland models with various lengths.

Figure 6
Design discharges for the V-shape overland models with various lengths.
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Design discharges for the V-shape overland models with various lengths.

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To verify the applicability of the derived analytical equations for design discharge estimation in small urban districts, we also conducted numerical models for comparison. Backward finite-difference schemes (Chow et al. 1988) were used to establish the kinematic-wave numerical model. Details of the numerical schemes can be found in the online Appendix. Table 2 shows the design rainfalls, time of concentration, and peak discharges calculated by the derived analytical equations and the numerical models for the rectangular overland planes and V-shaped models using the same model parameter dataset. As shown in Figures 5 and 6, only slight differences between the analytical solutions and the numerical results can be found. The first reason for the differences is the truncation error when using the finite-difference scheme to solve the governing equations. The second reason for the differences mainly results from the use of average rainfall excess intensity (Equation (28)) in deriving the peak discharge (Equations (36) and (39)); however, the time-varying rainfall excess intensities (as shown in Figure 3 and Equation (A.4) in the online Appendix) are input into the numerical model for simulations. The close fit of the results estimated by the derived analytical equations and the numerical models preserves the reliability of the proposed method for sewer design in small urban districts.

Table 2

Simulation results of the analytical solution and numerical model for different sizes of rectangular overland planes and V-shaped overland models

LengthRectangular overland plane
V-shaped overland model
Lo or Lc (m)ie (mm/h)toc (min)(qop)a (m2/s)(qop)b (m2/s)ie (mm/h)tc (min)(qcp)a (m2/s)(qcp)b (m2/s)
100 101.91 6.80 0.00283 0.00284 97.84 7.63 0.27179 0.27188 
200 87.89 10.35 0.00488 0.00490 85.36 11.24 0.94843 0.94844 
300 80.60 13.24 0.00672 0.00674 78.69 14.17 1.96715 1.96725 
400 75.79 15.76 0.00842 0.00844 74.23 16.72 3.29906 3.29911 
500 72.26 18.04 0.01004 0.01009 70.93 19.03 4.92545 4.92569 
LengthRectangular overland plane
V-shaped overland model
Lo or Lc (m)ie (mm/h)toc (min)(qop)a (m2/s)(qop)b (m2/s)ie (mm/h)tc (min)(qcp)a (m2/s)(qcp)b (m2/s)
100 101.91 6.80 0.00283 0.00284 97.84 7.63 0.27179 0.27188 
200 87.89 10.35 0.00488 0.00490 85.36 11.24 0.94843 0.94844 
300 80.60 13.24 0.00672 0.00674 78.69 14.17 1.96715 1.96725 
400 75.79 15.76 0.00842 0.00844 74.23 16.72 3.29906 3.29911 
500 72.26 18.04 0.01004 0.01009 70.93 19.03 4.92545 4.92569 

Notes:

1 (qop)a and (qop)b indicate the peak discharges in the rectangular overland plane estimated using the derived analytical equation and numerical model, respectively.

2 (qcp)a and (qcp)b indicate the peak discharges in the V-shaped overland model estimated using the derived analytical equation and numerical model, respectively.

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Drainage design in mid-size or large urban districts usually starts by using the rational method to estimate the outflow from each block and applying the contributing-area method (Wanielista et al. 1997) or a SWMM model to simulate all the block outflows into the sewer system. However, the design work can be relatively straightforward in small urban districts due to the comparatively small ditch and the simplicity of the drainage system.

In this study, we provide an analytical approach to determine the time of concentration and design discharge for small urban districts only based on rainfall–duration relationships and watershed geometry factors. The method solves the kinematic-wave approximation using the Darcy–Weisbach friction coefficient, which is more theoretically justified and can be explicitly expressed when determining the time of concentration. Consequently, the peak discharges under a specified return period can be analytically derived for a rectangular porous overland plane and a V-shaped porous overland model. To verify the reliability of the proposed analytical equations, we further applied numerical models for comparison. The results showed that the analytical solutions and numerical simulations were in good agreement for the five test sizes. Hence, the proposed method for design discharge estimation is more reliable and convenient than the conventional methods, which are mainly based on empirical time of concentration equations.

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All relevant data are included in the paper or its Supplementary Information.

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The authors declare there is no conflict.

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