Runoff Models

Runoff models predict the temporal distribution of runoff at a catchment outlet based on the tem- poral distribution of effective rainfall and the catchment characteristics. The effective rainfall is defined as the incident rainfall minus the abstractions and is sometimes referred to as the rain- fall excess. The most important abstractions are usually infiltration and depression storage, and the catchment characteristics that are usually most important in translating the effective rainfall distribution to a runoff distribution at the catchment outlet are those related to the topography and surface cover of the catchment. Runoff models are classified as either distributed-parameter models or lumped-parameter models. Distributed-parameter models account for runoff processes

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on scales smaller than the size of the catchment, such as accounting for the runoff from every roof, over every lawn, and in every street gutter, while lumped-parameter models consider the entire catchment as a single hydrologic element, with the runoff characteristics described by one or more (lumped) parameters.

In cases where the surface runoff flows into an unlined drainage channel that penetrates an aquifer, the flow in the drainage channel originates from both surface-water runoff and ground- water inflow to the channel, and these flow components must generally be modeled separately. The flow resulting from surface runoff is called direct runoff, and the flow resulting from ground-water inflow to the channel is subdivided into base flow and interflow, which is sometimes referred to as throughflow. Base flow is typically (quasi-) independent of the rainfall event, is equal to the flow of ground-water into the drainage channel, and depends on the difference between the ground-water elevation and the water-surface elevation in the drainage channel (Chin, 1991). Interflow is the inflow to the drainage channel that occurs between the ground surface and the water table and is typically caused by a low-permeability subsurface layer that impedes the vertical infiltration of rainwater along with large pores left by rotting tree roots and burrowing animals. Interflow contributions to river flows can be significant in forested areas. Various methods have been adopted for computationally separating the direct runoff and baseflow components of observed hydrographs (Singh, 1992; Tallaksen, 1995; McCuen, 1998). The simplest approach is to assume a constant baseflow equal to the discharge just before the rain begins (Wurbs and James, 2002). The direct runoff resulting from a storm event is added to the base flow and interflow to yield the flow hydrograph in the drainage channel. Baseflow contributed by infiltrating ground water is a relatively slow process compared with overland flow and interflow to a channel and, as a consequence, overland flow and interflow are sometimes collectively referred to as quickflow (Fitts, 2002).

A fundamental hypothesis that was originally made by Horton (1933b; 1945) is that overland flow occurs when the rainfall rate exceeds the infiltration capacity of the soil. This type of overland flow is commonly referred to as Hortonian overland flow. In modern hydrologic practice, it is gen- erally recognized that rainfall rates rarely exceed the infiltration capacities of soils, and runoff does occur when rainfall rates are less than the soil infiltration capacities. Betson (1964) proposed that, whereas surface runoff may be generated by a Hortonian mechanism, within a catchment there are only limited areas that contribute overland flow to a runoff hydrograph, and this is referred to as the partial-areas concept. Hewlett and Hibbert (1967) were the first to suggest that there may be a process other than the Hortonian process that is responsible for the generation of overland flow. In modern engineering practice, it is generally recognized that the two primary hydrological mech- anisms that generate overland flow are infiltration excess and saturation excess. Saturation excess is fundamentally different from infiltration excess in that overland flow is generated at locations where the soil is saturated at the surface. Unlike Hortonian runoff, where soil type and land use play a controlling role in runoff generation, landscape position, local topography, and soil depth are some of the primary controls in saturation-excess runoff. Saturation excess is the basis of the concept of variable source-area hydrology that acknowledges that the spatial extent of saturation will vary seasonally, depending on the relative rates of rainfall and evapotranspiration. In the saturation-excess process, rainfall causes a thin layer of soil on some parts of the basin to saturate upward from some restricting boundary to the ground surface, especially in zones of shallow, wet, or less-permeable soil. This process occurs frequently on the footslopes of hills, bottoms of valleys, swamps, and shallow soils. Determining which process dominates is fundamental identifying ap- propriate methods for describing the rainfall-runoff relation. Unfortunately, it is all too common

in hydrologic practice to assume, without evidence, that the runoff mechanism is Hortonian and apply models that may not reflect reality. In cases where significant surface runoff is observed while estimated infiltration capacities and rainfall intensities indicate that significant Hortonian runoff should not occur, saturation-excess runoff is likely to be the primary runoff mechanism (Walter et al., 2003).

A wide variety of models are available for calculating runoff from rainfall, and the applicability of these models must be assessed in light of the fundamental rainfall-runoff process. The applicability of various runoff models can be broadly associated with the scale of the catchment, which can be classified as small, midsize, or large (Ponce, 1989). In small catchments, the response to rainfall events is sufficiently rapid and the catchment is sufficiently small that runoff during a relatively short time interval can be adequately modeled by assuming a constant rainfall in space and time. The rational method is the most widely used runoff model in small catchments. In midsize catchments, the slower response requires that the temporal distribution of rainfall be accounted for; however, the catchment is still smaller than the characteristic storm scale, and the rainfall can be assumed to be uniform over the catchment. Unit hydrograph models are the most widely used runoff models in midsize catchments. In large catchments, both the spatial and temporal variations in precipitation events must be incorporated in the runoff model, and models that explicitly incorporate routing methodologies are the most appropriate. Runoff regimes within a catchment vary from overland flow at the smallest scales to river flow at the largest scales, and runoff models must necessarily accommodate this scale effect. Small catchments have predominantly overland flow runoff, while large catchments typically have a significant amount of runoff in identifiable river or drainage channels. As a consequence, the channel storage characteristics increase significantly from small catchments to large catchments.

5.4.1 Time of Concentration

The parameter that is most often used to characterize the response of a catchment to a rainfall event is the time of concentration. The time of concentration is defined as the time to equilibrium of a catchment under a steady rainfall excess. Alternatively, the time of concentration is sometimes defined as the longest travel time that it takes surface runoff to reach the discharge point of a catchment (Wanielista et al., 1997). Most equations for estimating the time of concentration, tc,

express tcas function of the rainfall intensity, i, catchment length scale, L, average catchment slope,

So, and a parameter that describes the catchment surface, C, hence the equations for tc typically

have the functional form

tc = f (i, L, So, C) (5.82)

The time of concentration of a catchment includes the time of overland flow and the travel time in drainage channels leading to the catchment outlet.

5.4.1.1 Overland Flow

There are several equations that are commonly used to estimate the time of concentration for overland flow. The most popular equations are described here.

Kinematic-Wave Equation. A fundamental expression for the time of concentration in overland flow can be derived by considering the one-dimensional approximation of the surface-runoff process

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illustrated in Figure 5.31. The boundary of the catchment area is at x = 0, ie is the rainfall-excess

y Catchment boundar y Runof f i_e Dx _u x 5 0 Rainfall excess Control volume q x

Figure 5.31: One-Dimensional Approximation of Surface-Runoff Process

rate, y is the runoff flow depth, and q is the volumetric flow rate per unit width of the catchment area. Within the control volume of length ∆x, the law of conservation of mass requires that the net mass inflow is equal to the rate of change of mass within the control volume. This law can be stated mathematically by the relation

 (ρq) − ∂(ρq)_∂x ∆x₂  + [ieρ∆x] −  (ρq) + ∂(ρq) ∂x ∆x 2  = ∂y ∂tρ∆x (5.83) where the first term in square brackets is the inflow into the control volume, the second term is the rainfall excess entering the control volume, the third term is the outflow, and the righthand side of Equation 5.83 is equal to the rate of change of fluid mass within the control volume. Taking the density as being constant and simplifying yields

∂y ∂t +

∂q

∂x = ie (5.84) This equation contains two unknowns, q and y, and a second relationship between these variables is needed to solve this equation. Normally, the second equation is the momentum equation; however, a unique relationship between the flow rate, q, and the flow depth, y, can be assumed to have the form

q = αym (5.85) where α is a proportionality constant. The assumption of a relationship such as Equation 5.85 is justified in that equations describing steady-state flow in open channels, such as the Manning and Darcy-Weisbach equations, can be put in the form of Equation 5.85. Combining Equations 5.84 and 5.85 leads to the following differential equation for y

∂y

∂t + αmym−1 ∂y

∂x = ie (5.86) Solution of this equation can be obtained by comparing it with

dy dt = ∂y ∂t + dx dt ∂y ∂x = ie (5.87)

which gives the rate of change of y with respect to t observed by moving at a velocity dx/dt. Consequently, Equation 5.86 is equivalent to the following pair of equations

dx

dt = αmy

m−1 _(5.88)

dy

dt = ie (5.89) where dx/dt is called the wave speed, and Equation 5.87 is called the kinematic-wave equation. Solution of Equation 5.89 subject to the boundary condition that y = 0 at t = 0 yields

y = iet (5.90)

Substituting this result into Equation 5.88 and integrating subject to the boundary condition that x = 0 at t = 0 yields

x = αim−1_e tm (5.91) Equations 5.90 and 5.91 are parametric equations describing the water surface illustrated in Figure 5.31, and the discharge at any location along the catchment area can be obtained by combining Equations 5.85 and 5.90 to yield

q = α(iet)m (5.92)

Defining the time of concentration, tc, of a catchment as the time required for a kinematic wave to

travel the distance L from the catchment boundary to the catchment outlet, then Equation 5.91 gives the time of concentration as

tc =  _L αim−1e 1 m (5.93) If the Manning equation is used to relate the runoff rate to the depth, then Equation 5.93 can be written in the form (ASCE, 1992)

tc = 6.99

(nL)0.6 i0.4

e So0.3

(5.94) where tc is in minutes, iein mm/h, and L in m, n is the Manning roughness coefficient for overland

flow, and So is the ground slope. Estimates of the Manning roughness coefficient for overland flow

are given in Table 5.17, where it should be noted that the surface types are ordered with increasing roughness. On the basis of Equation 5.94, the time of concentration for overland flow should be regarded as a function of the rainfall-excess rate (ie), the catchment-surface roughness (n), the flow

length from the catchment boundary to the outlet (L), and the slope of the flow path (So).

Equation 5.94 assumes that the surface runoff is described by the Manning equation, which is only valid for turbulent flows; however, at least a portion of the surface runoff will be in the laminar and transition regimes (Wong and Chen, 1997). This limitation associated with using the Manning equation can be addressed by using the Darcy-Weisbach equation, which yields

α = _8gS o Cνk  1 (2−k) , and m = 3 2 − k (5.95)

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Table 5.17: Manning’s n for Overland Flow

Surface Type Manning n Range Smooth concrete 0.011 0.01 –0.014 Bare sand 0.01 0.01 –0.016 Graveled surface 0.012 0.010–0.018 Asphalt 0.012 0.010–0.018 Bare clay 0.012 0.010–0.016 Smooth earth 0.018 0.015–0.021 Bare clay-loam (eroded) 0.02 0.012–0.033 Bare smooth soil 0.10 — Range (natural) 0.13 0.01 –0.32 Sparse vegetation 0.15 — Short grass 0.15 0.10 –0.25 Light turf 0.20 — Woods, no underbrush 0.20 0.1 –0.3 Dense grass 0.24 0.15 –0.35 Lawns 0.25 0.20 –0.30 Dense turf 0.35 0.30 –0.35 Pasture 0.35 0.30 –0.40 Dense shrubbery and forest litter 0.40 — Woods, light underbrush 0.40 0.3 –0.5 Bermuda grass 0.41 0.30 –0.50 Bluegrass sod 0.45 0.39 –0.63 Woods, dense underbrush 0.80 0.6 –0.95

Sources: ASCE (1992); Wurbs and James (2002); Crawford and Linsley (1966); Engman (1986); McCuen et al. (1996); McCuen (2005).

where ν is the kinematic viscosity of water, C and k are parameters relating the Darcy-Weisbach friction factor, f , to the Reynolds number, Re:

f = C

Rek (5.96)

where k = 0 for turbulent flow, k = 1 for laminar flow, 0 < k < 1 for transitional flow, and the Reynolds number is defined by

Re = q

ν (5.97)

Laminar flow typically occurs where Re < 200, turbulent flow where Re > 2,000, and transition flow where 200 < Re < 2,000. Values of C for overland flow have not been widely published, but Radojkovic and Maksimovic (1987) and Wenzel (1970) indicate that for concrete surfaces C values of 41.8, 2, and 0.04 are appropriate for laminar, transition, and turbulent flow regimes, respectively. Combining the kinematic-wave expression for tc, Equation 5.93, with the Darcy-Weisbach equation,

Equations 5.95 to 5.97, yields tc = " 0.21(3.6 × 106ν)kCL2−k Soi1+ke #1 3 (5.98)

where tc is in minutes, ν is in m2/s, L is in meters, and ie in mm/h. Equation 5.98, called the

Chen and Wong formula (Wong, 2005), can be used to account for various flow regimes in overland flow, and it has been shown that assuming a single flow regime (laminar, transition, or turbulent) will tend to underestimate the time of concentration (Wong and Chen, 1997). Indications are that overland flow is predominantly in the transition regime and that Equation 5.98 may be most applicable using k ≈ 0.5.

NRCS Method. NRCS (SCS, 1986) proposed that overland flow consists of two sequential flow regimes: sheet flow and shallow concentrated flow. Sheet flow is characterized by runoff that occurs as a continuous sheet of water flowing over the land surface, while shallow concentrated flow is characterized by flow in isolated rills and then gullies of increasing proportions. Ultimately, most surface runoff enters open channels and pipes, which is the third regime of surface runoff included in the time of concentration. In many cases, the time of concentration of a catchment is expressed as the sum of the travel time as sheet flow plus the travel time as shallow concentrated flow plus the travel time as open-channel flow. The flow characteristics of sheet flow are sufficiently different from shallow concentrated flow that separate equations are recommended. The flow length of the sheet flow regime should generally be less than 100 m, and the travel time, tf (in hours), over a

flow length, L (in m), is estimated by (SCS, 1986) tf = 0.0288

(nL)0.8 P0.5

2 So0.4

(5.99) where n is the Manning roughness coefficient for overland flow (Table 5.17), So is the land slope,

and P2 is the two-year 24-hour rainfall (in cm). Equation 5.99 was developed from the kinematic-

wave equation (Equation 5.94) by Overton and Meadows (1976) using the following assumptions: (1) The flow is steady and uniform with a depth of about 3 cm; (2) the rainfall intensity is uniform over the catchment; (3) the rainfall duration is 24 hours; (4) infiltration is neglected; and (5) the maximum flow length is 100 m. In considering the validity of these assumptions, it should be noted that overland flow may be significantly different than 3 cm in many areas, the rainfall duration may differ from 24 hours, and the actual travel time can increase if there is a significant amount of infiltration in the catchment. By limiting the maximum flow length to 100 m, the catchment is necessarily small, and the assumption of a spatially uniform rainfall distribution is reasonable. After a maximum distance of 100 m, sheet flow usually becomes shallow concentrated flow, and the average velocity, Vsc, is taken to be a function of the slope of the flow path and the type of land

surface in accordance with the Manning equation Vsc = 1 nR 2 3S 1 2 o (5.100)

where n is the roughness coefficient, R is the hydraulic radius, and So is the slope of the flow path.

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n = 0.025 and R = 6 cm. Equation 5.100 can also be expressed in the form Vsc = kS

1 2

o (5.101)

where k (= R2/3/n) is called the intercept coefficient, and several suggested values of k are given in Table 5.18. In addition to intercept coefficients for shallow concentrated flow, Table 5.18 also gives

Table 5.18: Intercept Coefficient for Overland-Flow Velocity versus Slope

Land Cover/Flow Regime k

(m/s) Forest with heavy ground litter; hay meadow (overland flow). 0.76 Trash fallow or minimum tillage cultivation; contour or strip 1.52 cropped; woodland (overland flow).

Short grass pasture (overland flow). 2.13 Cultivated straight row (overland flow). 2.74 Nearly bare and untilled (overland flow); alluvial fans in 3.05 western mountain regions.

Grassed waterway (shallow concentrated flow). 4.57 Unpaved (shallow concentrated flow). 4.91 Paved area (shallow concentrated flow); small upland gullies. 6.19

Source: U.S. Federal Highway Administration (1996).

bulk intercept coefficients for the entire overland flow, including sheet flow and shallow concentrated flow regimes. The average velocity, Vsc, derived from Equation 5.101 is then combined with the

flow length, Lsc, of shallow concentrated flow to yield the flow time, tsc, as

tsc=

Lsc

Vsc

(5.102) The total time of concentration, tc, of overland flow is taken as the sum of the sheet flow time,

tf, given by Equation 5.99, and the shallow concentrated flow time, tsc, given by Equation 5.102.

The overland flow time of concentration is added to the channel flow time to obtain the time of concentration of the entire catchment.

Kirpich Equation. An empirical time of concentration formula that is especially popular is the Kirpich formula (Kirpich, 1940) given by

tc = 0.019

L0.77

S0.385 o

where tcis the time of concentration in minutes, L is the flow length in meters, and Sois the average

slope along the flow path. Equation 5.103 was originally developed and calibrated from NRCS data reported by Ramser (1927) on seven partially-wooded agricultural catchments in Tennessee, ranging in size from 0.4 to 45 ha, with slopes varying from 3% to 10%; it has found widespread use in urban applications to estimate both overland flow and channel flow times. Equation 5.103 is most applicable for natural basins with well-defined channels, bare-earth overland flow, and flow in mowed channels including roadside ditches (Debo and Reese, 1995). Rossmiller (1980) has reviewed field applications of the Kirpich equation and suggested that for overland flow on concrete or asphalt surfaces, tc should be multiplied by 0.4; for concrete channels, multiply tc by 0.2; and for general

overland flow and flow in natural grass channels, multiply tcby 2. According to Prakash (1987), the

Kirpich equation yields relatively low estimates of the time of concentration. The Kirpich formula is usually considered applicable to small agricultural watersheds with drainage areas less than 80 ha.

Izzard Equation. The Izzard equation (Izzard, 1944; 1946) was derived from laboratory exper- iments on pavements and turf where overland flow was dominant. The Izzard equation is given by

tc =

530KL1/3 i2/3e

, where ieL < 3.9 m2/h (5.104)

where tc is the time of concentration in minutes, L is the overland flow distance in meters, ieis the

effective rainfall intensity in mm/h, and K is a constant given by K = 2.8 × 10−6ie+ cr

So1/3

(5.105) where cr is a retardance coefficient that is determined by the catchment surface as given in Table

5.19 and So is the catchment slope.

Table 5.19: Values of cr in the Izzard Equation

Surface cr

Very smooth asphalt 0.0070 Tar and sand pavement 0.0075 Crushed-slate roof 0.0082 Concrete 0.012 Tar and gravel pavement 0.017 Closely clipped sod 0.016 Dense bluegrass 0.060

Source: Izzard (1944; 1946).

Kerby Equation. The Kerby equation (Kerby, 1959) is given by

tc = 1.44  _Lr √ So 0.467 , where L < 365 m (5.106)

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where tc is the time of concentration in minutes, L is the length of flow in meters, r is a retar-

dance roughness coefficient given in Table 5.20, and So is the slope of the catchment. The Kerby

Table 5.20: Values of r in the Kerby Equation

Surface r

Smooth pavements 0.02 Asphalt/concrete 0.05–0.15 Smooth bare packed soil, free of stones 0.10 Light turf 0.20 Poor grass on moderately rough ground 0.20 Average grass 0.40 Dense turf 0.17–0.80 Dense grass 0.17–0.30 Bermuda grass 0.30–0.48 Deciduous timberland 0.60 Conifer timberland, dense grass 0.60

Sources: Kerby (1959); Westphal (2001).

equation is an empirical relation developed by Kerby (1959) using published research on airport drainage done by Hathaway (1945), consequently, the Kerby equation is sometimes referred to as the Kerby/Hathaway equation. Catchments with areas less than 4 ha, slopes less than 1%, and retardance coefficients less than 0.8 were used in calibrating the Kerby equation, and application of this equation should also be limited to this range. In addition, since the Kerby equation applies

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