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4.1 RATIONAL METHOD
Small Catchments A small catchment is described by the following features:
A catchment possessing some or all of the above properties is small in a hydrologic sense. Its runoff response may be described using relatively simple parametric or empirical methods, which lump all the relevant hydrologic processes into a few key descriptors such as rainfall intensity and catchment area. When increased detail is required, small catchments may be analyzed using the more complex overland flow techniques, which can be either spatially lumped (the conceptual storage model, or storage concept) or distributed (a deterministic model of the kinematic or diffusion wave types). For routine applications, all that is usually required is the simple parametric approach. Exceptions may be warranted in certain specialized applications, e.g., when coupling water quantity and water quality models. It is difficult to set the upper limit of a small catchment without being arbitrary to some degree. Given the natural variability in catchment slopes, vegetation cover, and so on, no single value is universally applicable. In practice, both time of concentration and catchment area have been used to define the upper limit of a small catchment. Some authorities regard a catchment with a time of concentration of 1 h or less as a small catchment. For others, a catchment of 2.5 km^{2} or less is considered small. Any such limit is bound to be arbitrary, reflecting the accumulated body of experience in runoff response. Rational Method The rational method is the most widely used method for the analysis of runoff response from small catchments. It has particular application in urban storm drainage, where it is used to calculate peak runoff rates for the design of storm sewers and small drainage facilities. The popularity of the rational method is attributed to its simplicity, although reasonable care is necessary in order to use the method effectively. The rational method takes into account the following hydrologic characteristics or processes:
In general, the rational method provides only a peak discharge, although in the absence of runoff diffusion it is possible to obtain an isoscelestriangleshaped runoff hydrograph. The peak discharge is the product of:
All processes are lumped into these three parameters. Rainfall intensity contains information on rainfall duration and frequency. In turn, rainfall duration is related to time of concentration, i.e., to the runoff concentration properties of the catchment. The runoff coefficient accounts for hydrologic abstractions and runoff diffusion, and may also be used to account for frequency. In this way, all the major hydrologic processes responsible for runoff response are embodied in the rational formula. The rational method does not take into account the following characteristics or processes:
In addition, the rational method does not explicitly account for the catchment's antecedent moisture condition; however, the latter may be implicitly accounted for by varying the runoff coefficient. The above conditions dictate that the rational method be restricted to small catchments. To start, the assumption of constant rainfall in space and time can only be justified for small catchments. Furthermore, in a small catchment, storm duration typically exceeds the time of concentration. Finally, in a small catchment, surface runoff processes are usually dominated by overland flow. There is no consensus regarding the upper limit of a small catchment. Values ranging from 0.625 to 12.5 km^{2} have been quoted in the literature [2, 26]. The current trend is to use 1.25 to 2.5 km^{2} as the upper limit for the applicability of the rational method. There is no theoretical lower limit, however, and catchments as small as 1 ha or less may be analyzed by the rational method. The rational method is based on the following formula:
in which Q_{p} = peak discharge corresponding to a given rainfall intensity, duration, and frequency; C = runoff coefficient, a dimensionless empirical coefficient related to the abstractive and diffusive properties of the catchment; I = rainfall intensity, averaged in time and space; and A = catchment area. In SI Units, for rainfall intensity in millimeters per hour, catchment area in square kilometers, and peak discharge in cubic meters per second, the formula for the rational method is the following:
For rainfall intensity in millimeters per hour, catchment area in hectares, and peak discharge in liters per second, the formula is:
In U.S. Customary units, for rainfall intensity in inches per hour, catchment area in acres, and peak discharge in cubic feet per second, the formula is:
The unit conversion coefficient 1.008 is usually neglected on practical grounds. Methodology The first requirement of the rational method is that the catchment be small. Once the size requirement has been met, the three components of the formula are evaluated. The catchment area is determined by planimetering or other suitable means. Boundaries may be established from topographic maps or aerial photographs. The drainage area survey should also include:
The evaluation of rainfall intensity is a function of several factors. First, it is necessary to determine the time of concentration. Normally, this is accomplished either:
Procedures to calculate time of concentration are not very well defined, often involving crucial assumptions such as flow level, channel shape, friction coefficients, and so on. Nevertheless, a value of time of concentration can usually be developed for practical use. For urban stormsewer design, time of concentration at a point is the sum of two parts:
Inlet time is the longest time required for runoff to flow over the catchment surface to the nearest sewer inlet (Fig. 41). Time of flow in the sewer, from inlet to point of interest, is calculated using hydraulic flow formulas.
Once time of concentration has been determined, the design storm duration is made equal to the time of concentration. This amounts to an assumption of concentrated catchment flow (Section 2.4). Subsequently, a rainfall frequency applicable to the given design condition is chosen. Frequencies (and return periods) vary with the type of project and degree of protection desired. Commonly used return periods are:
The size and importance of the project, as well as design criteria established by federal, state and local agencies, have a bearing in the selection of design frequency. The longer the return period (i.e., the smaller the frequency), the greater the peak discharge calculated by the rational formula.
Once rainfall duration and frequency have been determined, the corresponding rainfall intensity is obtained from the appropriate intensitydurationfrequency (IDF) curve. An example of IDF curve is shown in Fig. 42. The applicable curve can usually be obtained from cognizant government agencies. Where IDF curves are nonexistent, they can be developed from regional isopluvial maps containing depthdurationfrequency data. These maps are published by the National Weather Service [20, 21, 28].
Due to the hyperbolic nature of the intensityduration curve, an error in rainfall duration causes an error of opposite sign in rainfall intensity. For instance, if the rainfall duration is too long (i.e., time of concentration too long), the calculated rainfall intensity will be too low, and vice versa.
Once rainfall intensity and catchment area have been obtained, a runoff coefficient applicable to the given design condition is selected.
Runoff coefficients are theoretically restricted in the range
In urban drainage design, hydrologic abstractions include interception, infiltration, and surface storage (Section 2.2). Runoff diffusion is a measure of the catchment's ability to attenuate the flood peaks (Section 2.4).
In essence, the runoff coefficient is the ratio of the actual
(calculated) peak runoff rate to the maximum possible runoff rate.
For C = 1, the calculated peak discharge is equal to the maximum possible discharge.
Typical values of runoff coefficients for a wide variety of conditions are given in design manuals and other reference books; see for instance Tables 41 (a) and (b).
These values reflect the reduction in peak runoff that is likely to be produced by a given combination of rainfall abstraction and runoff diffusion.
For instance, in Table 41 (a), a lawn with a steep gradient (greater than 7 percent) in a heavy (clayey) soil might have C = 0.3,
but a lawn with a mild gradient (less than 2 percent) in a sandy soil might have C = 0.1, reflecting the prevailing abstractive and diffusive rates.
Furthermore, an asphaltic street (of negligible abstractive capability) might have
The runoff coefficients shown in Table 41 (a) are applicable to storms of 5 to 10y return period.
Less frequent storms (e.g., 50y return period) require the use of higher coefficients because infiltration and other abstractions have a reduced role for the larger storms.
The coefficients shown in Table 41 (a) represent average antecedent moisture conditions and are not designed to account for multiple storms, or storms of very long duration.
Special design cases usually warrant the use of higher runoff coefficients to simulate the existence of wet antecedent moisture conditions in the catchment.
Experimental evidence has shown that runoff coefficients tend to increase from one storm to another occurring shortly
thereafter, with runoff coefficients tending to increase with storm duration.
Design values of runoff coefficients are usually a function of rainfall intensity and, therefore, of rainfall frequency. Higher values of runoff coefficient are applicable for higher values of rainfall intensity and return period. A typical C versus I curve is shown in Fig. 43 [5]. Alternate ways of expressing the variation of runoff coefficient with rainfall frequency are shown in Figs. 44 and 45 [6, 25].
With runoff coefficient, rainfall intensity, and catchment area determined, the peak discharge is calculated by Eq. 41. The apparent simplicity of the procedure, however, is misleading. For one thing, there is a range of possible runoff coefficients for each surface condition. Therefore, the chosen C value is usually based on additional field information or designer's experience. The effect of frequency and/or antecedent moisture condition needs to be evaluated carefully. Furthermore, there is no absolute certainty that the calculated time of concentration (and, therefore, the rainfall duration) is correct, or even that it remains constant throughout the range of possible frequencies. In fact, since larger flows generally travel with greater velocities (perhaps excepting the case of mild overbank flows, see Fig. 93), time of concentration tends to decrease with an increase in return period. Notwithstanding these complexities, the rational method remains a practical way to calculate peak discharge for small catchments based on a few relevant parameters.
Theory of the Rational Method The rational method is based on the principles of runoff concentration and diffusion. For simplicity, the process can be explained in two parts:
Runoff Concentration Without Diffusion. In the absence of diffusion, a catchment concentrates the flow at the outlet, attaining the maximum possible flow (i.e., the equilibrium flow rate) at the time of concentration. By setting the design rainfall duration equal to the time of concentration, concentrated catchment flow is obtained at the outlet. Since there is no diffusion, the method gives not only a peak flow but also a hydrograph corresponding to that of concentrated catchment flow (Fig. 46), with recession time equal to rising time. The runoff coefficient is then simply the ratio of effective rainfall to total rainfall. A mass balance of effective rainfall and runoff leads to:
in which V_{r} = runoff volume; I_{e} = effective rainfall; and t_{r} = rainfall duration (either effective or total). Equation 45 leads to:
I_{e}
in which C_{a} = runoff coefficient due only to abstraction (C_{a} ≤ 1).
Runoff concentration without diffusion is typical of steep catchments, with slopes greater than 0.01, where the momentum balance is dominated by the gravitational and frictional forces. For catchments of milder slope (say, less than 0.001), the role of the flow depth gradient increases, and runoff diffusion becomes increasingly important. In the extreme case, for a hypothetical catchment of zero ground slope, the diffusion effect is theoretically the only one present.
Runoff Concentration With Diffusion.
When diffusion is present, the rational method accounts for it in the runoff coefficient.
Thus, the runoff coefficient is used to model not only abstraction but also diffusion.
Diffusion modifies the catchment response in such a way as to increase the recession time and decrease the peak flow.
Therefore, a hydrograph shape can no longer be obtained directly from a mass balance as in the
case of runoff concentration without diffusion.
The lack of a hydrograph shape does not impede the use of the rational method,
because the diffusion can be represented directly in the peak flow formula, by lowering the runoff
coefficient below that due only to abstraction The reduction in the runoff coefficient amounts to:
in which C = runoff coefficient and C_{d} = component of runoff coefficient accounting only for diffusion (C_{d} ≤ 1).
The question of whether the peak is reached before, at, or after the time of concentration, as
In the absence of diffusion, C_{d} = 1 and C = C_{a}.
Likewise, in the absence of abstraction, C_{a} = 1 and
In practice, no quantitative distinction is made between the abstractive and diffusive components of the runoff coefficient. Usage, however, reflects the fact that runoff diffusion is implicitly being considered; see, for instance, the marked change in runoff coefficient with surface slope shown in Table 41. Further Developments Attempts to analyze the behavior of the rational method have led to the concept of peak flow per unit area [27]:
Q_{p}
in which q_{p} = peak flow per unit area. Rainfall intensity varies with rainfall duration and frequency. Likewise, runoff coefficient also varies with rainfall duration and frequency. Therefore, a relation linking peak flow per unit area to rainfall duration and frequency can be obtained:
in which T = return period. Another approach is based on expressing the rational formula in the following form:
Q_{p}
where now the runoff coefficient may be interpreted as dimensionless peak flow, or peak flow per unit area per unit rainfall intensity. It follows that dimensionless peak flow is related to the abstractive and diffusive properties of the catchment. A similar concept is used in the Natural Resources Conservation Service TR55 method (Section 5.3). In this method, a unit peak flow is defined as the peak flow per unit area per unit rainfall depth. In the TR55 graphical method included, the unit peak flow is a function of time of concentration, abstraction parameter, and temporal storm pattern. The fact that unit peak flow is a function of temporal storm pattern qualifies the TR55 graphical method as an extension of the rational method to midsize catchments. While no upper limit to catchment size is indicated, the method is restricted to a time of concentration less than or equal to 10 h. Applications of the Rational Method Relation Between Runoff Coefficient and φindex. The runoff coefficient can be related to total rainfall intensity and φindex, provided the following assumptions are satisfied:
The first assumption is valid for steep catchments, whereas the second assumption is implicit in the application of the rational method. For catchment response without diffusion:
I_{e}
For constant rainfall intensities:
Combining Eqs. 411 and 412:
I  φ
Areal Weighing of Runoff Coefficients. Values of runoff coefficients may vary within a given catchment. When a clear pattern of variation is apparent, a weighted value of runoff coefficient should be used. For this purpose, the individual subareas are delineated and their respective runoff coefficients are identified. The weighted value is obtained by weighing the runoff coefficients in proportion to their respective subareas. This leads to:
in which C_{i} = runoff coefficient of i th subarea and A_{i} = drainage area of i th subarea. Applicable units are those of Eq. 42. Composite Catchments. A composite catchment is one that drains two or more adjacent subareas of widely differing characteristics. For instance, assume that a catchment has two subareas A and B with times of concentration t_{A} and t_{B}, respectively, with t_{A} being much less than t_{B} (Fig. 48).
To apply the rational method to this composite catchment, several rainfall durations are chosen, ranging from t_{A} to t_{B}. in suitable increments. The calculation proceeds by trial and error, with each trial associated with each rainfall duration. To calculate the partial contribution from subarea B, an assumption must be made regarding the rate at which the flow is concentrated at the catchment outlet. The rainfall duration that gives the highest combined peak flow (A plus B ) is taken as the design rainfall duration. The procedure is illustrated by the following example.
Effect of Catchment Shape. The rational method is suited to catchments where drainage area increases more or less linearly with catchment length. If this is not the case, the peak flow may not increase with an increase in catchment area. To illustrate, take the catchment shown in Fig. 49. The time of concentration to point A is t_{A}; the time of concentration to point B is t_{B}; and t_{B} is greater than t_{A}. Therefore, I_{A} is greater than I_{B}.
The drainage area to point A is A_{A}, and the drainage area to point B is A_{B}, and A_{B} is greater than A_{A}. Assuming the same runoff coefficient for the partial area (to point A) and the total area (to point B), the peak flow at A is: Q_{pA} = CI_{A}A_{A}. Likewise, the peak flow at B is: Q_{pB} = CI_{B}A_{B}. For Q_{pB} to be greater than Q_{pA}, it is necessary that (A_{B} / A_{A}) be greater than (I_{A} / I_{B}). In other words, the drainage area must grow in the downstream direction at least as fast as the decrease in corresponding rainfall intensity. Otherwise, the peak discharge at A would be greater than that at B. The situation is illustrated by the following example.
Modified Rational Method. The application of the rational method to large urban catchments, i.e., those featuring welldefined conveyance channels and drainage areas greater than 1.3 km^{2} but less than 2.5 km^{2}, requires special techniques. For one thing, the flow is likely to vary widely along the main channel, ranging from small at the upstream reaches to larger at the downstream reaches. In this case, it may be difficult to determine an average value of time of concentration. An alternative is to apply the rational method incrementally, using a technique known as the modified rational method. The method requires the subdivision of the catchment into several subcatchments, as shown in Fig. 410. First, the time of concentration t_{A} is estimated and used to calculate the peak flow Q_{pA} at A, using Eq. 42. With the aid of open channel flow formulas, Q_{pA} is conveyed through the main channel from A to B, and the travel time t_{AB} calculated. The time of concentration, t_{B} = t_{A} + t_{AB}, is used to calculate the peak flow Q_{pB} at B, again using Eq. 42. The procedure continues in the downstream direction until the peak flow Q_{pE} is calculated. If the runoff coefficients are different for each subcatchment, Eq. 414 can be used in lieu of Eq. 42. While the procedure is relatively straightforward, it may result in peak flows decreasing in the downstream direction (due to the effect of catchment shape).
Application to Stormsewer Design. A typical plan for design of a small stormsewer project is shown in Fig. 411. Table 42 shows a summary of the computations illustrating the application of the rational method to determine design flows. The example is based on the following conditions:
This example is extracted from Design and Construction of Sanitary and Storm Sewers, ASCE Manual of Engineering Practice No. 37, 1960 [2]. 4.2 OVERLAND FLOW
Overland flow is surface runoff that occurs in the form of sheet flow on the land surface without concentrating in clearly defined channels. This type of flow is the first manifestation of surface runoff, since the latter occurs first as overland flow before it has a chance to flow into channels and become streamflow. Overland flow theory uses deterministic methods to describe surface runoff in overland flow planes. The theory is based on established principles of fluid mechanics such as laminar and turbulent flow, mass and momentum conservation, and unsteady free surface flow. The spatial and temporal description leads to differential equations and to their solution by either analytical or numerical means. For certain applications, simplified conceptual models can be developed for practical use. Overland flow theory seeks to find an answer to the problem of catchment response: What is the hydrograph that will be produced at a catchment's outlet, subject to a given effective rainfall? In overland flow applications, effective rainfall is also referred to as rainfall excess. Unlike the rational method, which generally does not produce a hydrograph, overland flow models have the capability to account not only for runoff concentration but also for runoff diffusion. Another advantage of overland flow models is their distributed nature, i.e., the fact that rainfall excess can be allowed to vary in space and time if necessary. Overland flow models, then, are a more powerful tool than parametric models such as the rational method. However, the complexity increases in direct relation to their greater level of detail. As with the rational method, a question that must be addressed at the outset is the following: What size catchment can be analyzed with overland flow techniques? Here again, the answer is not very well defined. Intuitively, overland flow computations should be applicable to small catchments, primarily because overland flow is the main surfaceflow feature of small catchments. The method, however, is not necessarily restricted to small catchments. Midsize catchments may also benefit from the increased detail of overland flow models. The actual limit is a practical one. Computations need to be performed in modules of relatively small size; otherwise, it is likely that the terrain's topographic, frictional, and vegetative features will not be properly represented in the overland flow model. In practice, overland flow techniques are restricted to catchments for which the surface features can be adequately represented within the model's topological structure. Otherwise, the amount of lumping introduced (i.e., temporal and spatial averaging) would interfere with the method's ability to predict the occurrence of flows in a distributed context. Overland flow techniques often form part of computer models that simulate all relevant phases of the hydrologic cycle. These models use overland flow techniques in their catchment routing component. The fundamentals of overland flow theory are presented here. Catchment routing methods are described in Chapter 10. Overland Flow Theory
The mathematical description of overland flow begins with the equation of mass conservation of fluid mechanics, also referred to as the continuity equation.
In onedimensional flow, this equation states that the change in flow per unit length
in a control volume is balanced by the change in fiow area per unit time:
∂Q ∂A
This equation does not include sources or sinks. Inclusion of the latter leads to:
∂Q ∂A
in which q_{L} = lateral inflow or outflow (inflow positive, outflow negative), or net lateral flow per unit length, in L^{2} T^{ 1} units. In small catchment hydrology, overland flow is assumed to take place on the overland flow plane. This is a plane of length L (in the flow direction), slope S_{o}, and of sufficiently large width W (Fig. 412). Therefore, a unitwidth analysis is appropriate. For a unit width, Eq. 417 is converted to:
∂q ∂h
in which q = flow rate per unit width; h = flow depth; and i = lateral inflow (rainfall excess), or inflow per unit area, in L T ^{1} units. While lateral inflow can vary in time and space, a first approximation is to consider it constant.
The process is depicted in Fig. 413. The flow from start to equilibrium is called the rising limb of the overland flow hydrograph. The flow from equilibrium back to zero is called the receding limb of the hydrograph. The equilibrium outflow can be calculated by recognizing that at equilibrium state, the outflow must equal the inflow (i.e., rainfall excess). Therefore,
in which q_{e} = equilibrium outflow, in liters per second per meter; i = rainfall excess, in millimeters per hour; and L = plane length, in meters.
Equation 419 is essentially a statement of runoff concentration, similar to Eq. 258 or to Eq. 41 with C = 1. Whether the flow actually does concentrate and reach its equilibrium value will depend upon the duration of the rainfall excess t_{r} relative to the time t_{e} required to reach equilibrium. If t_{r} > t_{e}, equilibrium is reached. The volume of storage is the area below the line q = q_{e}, and above the rising limb of the overland flow hydrograph, as shown in Fig. 413. As a first approximation, the shaded area above the rising limb may be assumed to be equal to the area below the rising limb. In this case, the equilibrium storage volume is:
in which S_{e} = equilibrium storage volume, in liters per meter; q_{e} = equilibrium outflow, in liters per second per meter; and t_{e} = time to equilibrium, in seconds. In practice, surface and other irregularities cause the equilibrium state to be approached asymptotically and, therefore, the actual time to equilibrium is not clearly defined. A value of time t corresponding to q = 0.98q_{e} may be taken as a practical measure of t_{e}. Then, Eq. 420 is only an approximation of the actual storage volume. The equation of continuity, Eq. 418, can also be expressed in the following form:
in which u = q/h = mean velocity. The value of equilibrium outflow was obtained from Eq. 419 based on continuity considerations. However, the shape of the rising and receding limbs and the time to equilibrium remain to be elucidated. This can be obtained through the equation of momentum conservation (or equation of motion), following established principles of unsteady open channel flow [3, 9, 18]. The equation of motion, however, is a nonlinear partial differential equation. A form of this equation with u and h as dependent variables is [18]:
in which S_{f} = friction slope, S_{o} = plane slope, g = gravitational acceleration, and all other terms have been previously defined. All terms in Eqs. 421 and 422 are dimensionless. The solution of Eqs. 421 and 422 can be attempted in a variety of ways. Analytical solutions are usually based on an assumption of linearity [1, 22]. Numerical solutions have been extensively applied to stream and river flow problems [16, 18]. To date, overland flow problems have been solved with one of the following approaches:
The storage concept is similar to that used in reservoir routing (Chapter 8). The kinematic wave technique simulates runoff concentration in the absence of diffusion. The diffusion wave technique simulates runoff concentration in the presence of small amounts of diffusion. The dynamic wave technique solves the complete set of governing equations, Eqs. 421 and 422, including runoff concentration, diffusion, and dispersion (third order) processes [23]. For practical applications, the storage concept and kinematic and diffusion wave techniques may be shown to be useful approximations to the complete equations.
In principle, the kinematic wave is an improvement over the storage concept;
in turn, the diffusion wave is an improvement over the kinematic wave, whereas the dynamic wave is an improvement over the diffusion wave.
Invariably, the effort involved in obtaining a solution increases in direct relation
to the complexity of the equations being solved, including initial and boundary conditions.
Overland Flow Solution Based on Storage Concept Early approaches to solve the overland flow problem are attributed to Horton [10] and Izzard [13, 14]. In particular, Horton noticed that experimental data justified a relationship between equilibrium outflow and equilibrium storage volume of the following form:
in which a and m are empirical constants. A mean flow depth h_{e} is defined in the following way:
S_{e}
Combining Eqs. 423 and 424:
in which b = aL^{m}, another constant. The value of the exponent m is a function of flow regime, depending on whether the latter is laminar, turbulent (either Manning or Chezy), or mixed laminarturbulent. Typical values of m are shown in Table 43.
A conceptual estimate of time to equilibrium can be obtained by combining Eqs. 420 and 423 and solving for t_{e}:
For laminar flow conditions, b = aL^{m} = C_{L}, where C_{L} is defined as follows [3]:
and ν = kinematic viscosity, a function of water temperature (see Tables A1 and A2, Appendix A).
The units of C_{L} are L^{1}T^{ 1}.
Furthermore, with q_{e} = iL, Eq. 426 reduces to the following for the case of
in which t_{e} = time to equilibrium, in seconds; L = length of overland flow plane, in meters; and For turbulent Manning flow conditions, b = aL^{m} = (1/n) S_{o}^{1/2}, in which n is the Manning friction coefficient. With q_{e} = iL, Eq. 426 reduces to the following for the case of mixed laminarturbulent flow (5/3 < m < 3):
with the same units as Eq. 428 (t_{e} in seconds, L in meters, i in meters per second). As expected, time to equilibrium increases with bottom friction and plane length, and decreases with effective rainfall intensity and plane slope. Equation 429 was developed by assuming the Manning formula in the rating. Therefore, it is strictly applicable only to m = 5/3. In practice, however, this equation is also used for mixed laminarturbulent flow (5/3 < m < 3). In addition, the very shallow depths that usually prevail in overland flow calculations result in a substantial increase in friction. These differences are accounted for by using an effective roughness parameter N in lieu of the Manning friction coefficient [11]. Typical values of N are given in Table 44.
Rising limb of the overland flow hydrograph. The HortonIzzard solution to the overland flow problem is based on the assumption that Eq. 423 is valid not only at equilibrium but also at any other time:
in which q = outflow at time t, and S = storage volume at time t. This assumption is convenient because it allows an analytical solution for the shape of the overland flow hydrograph. Equation 430 is the formula for a nonlinear reservoir, i.e., a function relating outflow and storage volume in a nonlinear way (m ≠ 1). Therefore, a nonlinear reservoir is being used to model the equation of motion, Eq. 422. In essence, a deterministic model (Eq. 422) has been replaced by a conceptual model (Eq. 430). Equation 416 can be expressed in one space increment Δx to yield:
in which I = inflow to the control volume, O = outflow from the control volume, and dS/dt = rate of change of storage in control volume (Fig. 414).
For the overland flow case, I = iL and
which through Eqs. 419, 423, and 430 leads to:
Integrating Eq. 433:
and, through additional algebraic manipulation [1]:
Using Eq. 426, Eq. 435 reduces to:
For m = 2, which describes a flow regime that is 75% turbulent (between laminar, for which m = 3, and 100% turbulent Manning, for which m = 5/3), the solution of Eq. 436 is:
which was expressed by Horton with time as independent variable as follows [1]:
For m = 3 (laminar flow), the solution of Eq. 436 is [24]:
With the aid of Eq. 438, and Eq. 419 for q_{e} and Eq. 429 for t_{e},
the rising limb of the overland flow hydrograph for m = 2 can be calculated.
Likewise, with the aid of Receding limb. For m > 1, the receding limb of the overland flow hydrograph can be calculated by the following formula [24]:
where q /q_{e} = 1 for t /t_{e} = 0, i.e., the outflow is at equilibrium at the start of the recession. Likewise, for m = 1, the hydrograph recession is as follows [24]:
where q /q_{e} = 1 for t /t_{e} = 0. Limitations. Several assumptions limit the applicability of the HortonIzzard solution. The most important one is the nonlinear form of the storage equation, Eqs. 423 and 430. Izzard has suggested that the method should be restricted to cases where the product of rainfall intensity (in millimeters per hour) and plane length (in meters) (iL) does not exceed 3000. Notwithstanding the apparent limitations, the HortonIzzard solution of overland flow has been extensively used in the past, particularly in the design of airport drainage [3, 7]. Overland Flow Solution Based on Kinematic Wave Theory According to this theory, Eq. 422 can be approximated by a singlevalued flowdepth rating at any point, leading to:
in which b and m are constants analogous to those of Eq. 425. Unlike Horton's approach, which bases the rating on the storage volume on the entire plane (Eqs. 423 and 430), the kinematic wave approach bases the rating on flow depths at individual cross sections. This difference has substantial implications for computer modeling because whereas the Horton approach is lumped in space, the kinematic approach is not, and therefore it is better suited to distributed computation.
In the Horton approach, outflow is related to storage volume and, by extension, to the mean flow depth on the overland plane. In contrast, in the kinematic wave approach, outflow is related to the outflow depth. Since the typical overland flow problem has a nonzero water surface slope, it is more likely to behave as a channel rather than a reservoir. Therefore, it would appear that the kinematic wave approach is a better model of the physical process than the storage concept. Further analysis has shown, however, that while the kinematic wave lacks diffusion, the storage concept does not. In this sense, the storage concept may well be a better model than the kinematic wave approach for cases featuring significant amounts of runoff diffusion. The kinematic wave assumption, Eq. 442, amounts to substituting a uniform flow formula (such as Manning's) for the equation of motion, Eq. 422. In essence, it says that as far as momentum is concerned, the flow is steady. The unsteadiness of the phenomena, however, is preserved through the continuity equation, Eq. 418, or Eq. 421. The implication of the kinematic wave assumption is that unsteady flow may be visualized as a succession of steady uniform flows, with the water surface slope remaining constant at all times. This, of course, can be reconciled with reality only if the flow unsteadiness is very mild; that is, if the changes in stage occur very gradually. In practice, a necessary condition for the applicability of Eq. 442 to unsteady flows is that the changes in momentum be negligible compared to the force driving the steady flow, i.e., gravity (the plane or channel slope). The kinematic flow number used in overland flow applications (Eq. 456) serves as a quantitative measure of how kinematic a given unsteady flow condition is; that is, of the extent to which Eq. 442 a good surrogate of Eq. 422 and, therefore, a valid description of the unsteady flow phenomena. Kinematic wave solution. The application of kinematic wave theory to the overland flow problem begins with Eq. 418, repeated here for convenience:
∂q ∂h
Differentiating Eq. 442 with respect to flow depth, assuming that b and m are constants (a wide channel of constant friction) gives:
in which c = celerity of a kinematic wave. Since in overland flow m > 1 (Table 43), the celerity of a kinematic wave is greater than the mean flow velocity. Multiplying Eqs. 418 and 443 and using the chain rule,
which is a form of the kinematic wave equation with q as the dependent variable. Using the same approach, an expression for kinematic flow in terms of flow depth can be derived:
With c = dx/dt, i.e., the slope of the characteristic lines on an xt plane (Fig. 416), the left side of
In particular, Eq. 448 can be integrated to yield:
which implies that the flow depth at any point along the plane increases linearly with time, provided that rainfall excess i remains constant. The overland flow solution under the kinematic wave assumption resembles that of the storage concept, with a rising limb, an equilibrium state, and a receding limb. Although the equilibrium flow is the same (q_{e} = i L), the time to equilibrium is notably different, as is the shape of rising and receding limbs. To derive the kinematic wave solution, Eq. 442 is expressed in terms of equilibrium outflow:
in which, unlike in Eqs. 424 and 425, h_{e} is now interpreted as the equilibrium flow depth at the catchment outlet. Dividing Eq. 442 by Eq. 450 leads to:
Since h = it (Eq. 450), this leads to:
in which t_{k} = kinematic time parameter, defined as:
Equation 453 is applicable for t ≤ t_{k}. Otherwise, the flow would exceed the equilibrium value, which is clearly a physical impossibility. Thus, the kinematic time parameter may be interpreted as a kinematic timetoequilibrium. With Eq. 451, and since q_{e} = iL and b = (1/n) S_{o}^{1/2} (for turbulent Manning friction in wide channels), the kinematic time parameter may be expressed as follows:
which is the same as Eq. 429, albeit without the factor 2. In other words, the analytical solution to the kinematic wave equation is a parabola (Eq. 453), whereas the analytical solution of the storage concept is a hyperbolic trigonometric function (see, for instance, Eq. 438, applicable for m = 2). Wooding's open book. The kinematic approach to the solution of the overland flow problem was studied in detail by Iwagaki [12], Henderson and Wooding [8], Wooding [29, 30, 31], and Woolhiser and Liggett [32], among others. Wooding developed the concept of the open book shown in Fig. 417, which has been extensively used in catchment modeling (Chapter 10). The open book is formed by two overland flow planes; the outflow from the planes is lateral inflow to the channel, which conveys the flow to the catchment outlet.
Applicability of kinematic waves. Woolhiser and Liggett calculated the shape of the rising hydrograph under kinematic flow. Furthermore, they established the limit for the applicability of the kinematic wave in terms of the kinematic flow number, defined as follows:
in which K = kinematic flow number, a dimensionless number; F = Froude number corresponding to the equilibrium flow at the outlet; and h_{o} = equilibrium flow depth at the outlet (i.e., h_{e}). Values of K greater than 20 describe kinematic flow, while lower values do not [18]. In other words, for low K values, Eq. 441 is no longer a good approximation of Eq. 422. In particular, since S_{o} is likely to vary within a wider range than either L, F, or h_{o}, Eq. 456 could be interpreted to mean that the property of a wave being kinematic is directly related to plane slope: the steeper the slope, the greater the K value and the more kinematic the resulting flow should be. Conversely, the milder the slope, the lower the K value and the less kinematic the flow. The reason for this inability of the kinematic wave to account for a wide range of slopes is apparent from the nature of Eq. 422. The kinematic wave accounts only for friction and plane slopes. All other terms are excluded from the formulation and are, therefore, absent from the solution. For very mild plane slopes, the importance of these terms may be promoted to the point where neglecting them is no longer justified. While this appears to impose stringent limitations, the situation in practice is quite different. Most overland flow problems have steep slopes, on the order of S_{o} = 0.01 or more, resulting in the flow being essentially kinematic, as confirmed by their kinematic flow number (Eq. 456). However, much milder slopes, less than S_{o} = 0.001, result in very low kinematic flow numbers; for these cases, the kinematic wave solution may not be justified.
Kinematic shock.
A source of complexity in kinematic wave solutions arises from the fact that the wave celerity in
In nature, small amounts of diffusion and other irregularities usually act in such a way as to control and arrest shock development. An analytical solution, however, has no such imperfections. In essence, the total absence of diffusion in the analytical solution permits the uncontrolled development of the kinematic shock. Numerical solutions, however, usually have small amounts of diffusion and are, therefore, not perfect, with the shock being a rare occurrence in this case. This fact gives rise to practical implications for stream channel and catchment routing (Chapters 9 and 10). Comparison of storage and kinematic wave solutions. Figure 418 shows dimensionless rising hydrographs of storageconcept overland flow solutions for values of 1 ≤ m ≤ 3. Note that dimensionless discharge is defined as: q_{*} = q /q_{e}, and dimensionless time as: t_{*} = t /t_{e}. For comparison, the rising hydrograph of the kinematic wave (KW) is also shown in Fig. 418. This figure clearly shows that the response of the kinematic wave solution is about twice as fast as that of the storage concept (compare Eq. 429 for the storage concept with Eq. 454 for the kinematic wave).
Figure 419 shows dimensionless receding hydrographs of storageconcept overland flow solutions for values of 1 ≤ m ≤ 3.
Overland Flow Solution Based on Diffusion Wave Theory According to diffusion wave theory, the flow depth gradient ∂h/∂x in Eq. 422 is largely responsible for the diffusion mechanism, which is naturally present in unsteady free surface flows. Therefore, its inclusion in the analysis should provide runoff concentration with diffusion. The inclusion of the depth gradient term substantially increases the difficulty of obtaining a solution. An established procedure is to linearize the governing equations, Eqs. 421 and 422, around reference flow values. This entails using smallperturbation theory to develop linear analogs of these equations. This procedure, while heuristic, has worked well in a number of applications. Following Lighthill and Whitham [17], the linear analogs of Eqs. 421 and 422, neglecting the inertia terms (the first two terms of Eq. 422) and the momentum source term for simplicity, are, respectively,
The coefficients of Eq 457 are constant, being the reference flow velocity u_{o} and depth h_{o}, respectively. The second term of Eq. 458 represents a combined linear version of the friction slope S_{f} and channel slope S_{o}. Furthermore, in Eq. 458 friction is described by the Manning formula. If the Chezy formula is used, the factor 4/3 is replaced by 1. Differentiating Eq. 458 with respect to space and eliminating the term ∂u/∂x from the resulting equation (with the aid of Eq. 457), the following is obtained:
A similar expression with discharge as dependent variable may also be derived, albeit at the cost of increased algebraic manipulation (Chapter 9). Unlike the kinematic wave equation (Eq. 444), which has no secondorder term, the diffusion wave equation (Eq. 459) has a secondorder term. Therefore, the latter can describe not only runoff concentration but also diffusion. Equation 459 can be expressed in the following form:
in which
is the celerity of a diffusion wave, for turbulent Manning friction in hydraulically wide channels (overland flow), and
is the hydraulic diffusivity, or channel diffusivity. Equation 459 implies that the diffusion component (secondorder term) is small compared with the concentration component (firstorder terms) and that the channel diffusivity controls the diffusion contribution to the flow. In effect, for very low values of S_{o}, channel diffusivity is very large. In the limit, as channel slope approaches zero, channel diffusivity grows unbounded and Eq. 459 is no longer applicable. However, for realistic mild slopes (approximately in the range 0.0010.0001), the contribution of the diffusion term can be quite significant. Diffusion wave theory, then, applies for the milder slopes for which kinematic wave theory is not sufficient. Its application to catchment routing and hydrologic models of overland flow is discussed in Chapter 10. In summary, overland flow techniques can provide more detail than the rational method in describing the peak and timing of outflow hydrographs from small catchments. The increase in detail, however, is invariably associated with greater complexity. In practice, it is reckoned that rainfall and abstraction modules must be coupled with the overland flow module in order to arrive at a meaningful model. Ideally, these modules should be as detailed as the overland flow module. However, due to the complex spatial and temporal variability of rainfall and hydrologic abstractions, this consistency between model components is not always possible (Fig. 420).
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