5.1 MIDSIZE CATCHMENTS
A midsize catchment is described by the following features:
A catchment possessing some or all of the above properties is midsize in a hydrologic sense. Since rainfall intensity varies within the storm duration, catchment response is described by methods that take explicit account of the temporal variation of rainfall intensity. The most widely used method to accomplish this is the unit hydro graph technique. In an nutshell, it consists of deriving a hydrograph for a unit storm (the unit hydrograph) and using it as a building block to develop the hydro graph corresponding to the actual effective stonn hyetograph. In unit hydrograph analysis, the duration of the unit hydrograph is usually a fraction of the time of concentration. The increase in time of concentration is due to the larger drainage area and the associated reduction in overall catchment gradient.The latter has the effect of increasing runoff diffusion. The assumption of uniform spatial distribution of rainfall is a characteristic of midsize catchment analysis. This assumption allows the use of a lumped method such as the unit hydrograph. Unlike midsize catchments, for large catchments rainfall is likely to vary spatially, either as a general storm of concentric isohyetal distribution covering the entire catchment with moderate rainfall or as a highly intensive local storm (thunderstorm) covering only a portion of the catchment. An important feature of large catchments that sets them apart from midsize catchments is their substantial capability for channel storage. Channel storage processes act to attenuate the flows while in transit in the river channels. Attenuation can be due either to longitudinal storage (for inbank flows) or to lateral storage (for overbank flows). In the first case, the storage amount is largely controlled by the slope of the main channel. For catchments with mild channel slopes, channel storage is substantial; conversely, for catchments with steep channel slopes, channel storage is negligible. Since large catchments are likely to have mild channel slopes, it follows that they have a substantial capability for channel storage. In practice, this means that large catchments cannot be analyzed with spatially lumped methods such as the unit hydrograph , since these methods do not take explicit account of channel storage processes. Therefore, unlike for midsize catchments, for large catchments it is necessary to use channel routing (Chapter 9) to account for the expanded role of river flow in the overall runoff response. As with the limit between small and midsize catchments, the limit between midsize and large catchments is not immediately apparent. For midsize catchments, runoff response is primarily a function of the characteristics of the storm hyetograph, with concentration time playing a secondary role. Therefore, the latter is not well suited as a descriptor of catchment scale. Values ranging from 100 to 5000 km2 have been variously used to define the limit between midsize and large catchments. While there is no consensus to date, the current trend is toward the lower limit. In practice, it is likely that there would be a range of sizes within which both midsize and large catchment techniques are applicable. However, the larger the catchment area, the less likely it is that the lumped approach is able to provide the necessary spatial details. It should be noted that the techniques for midsize and large catchments are indeed complementary. A large catchment can be viewed as a collection of midsize subcatchments. Unit hydrograph techniques can be used for subcatchment runoff generation, with channel routing used to connect streamflows in a typical dendritic network fashion (Fig. 51). Such a computationally intensive procedure is ideally suited to solution with the aid of a computer. Examples of hydrologic computer models using the network concept are HEC1 of the U.S. Army Corps of Engineers and TR20 of the USDA Natural Resources Conservation Service. These and other computer models are described in Chapter 13. In practice, channelrouting techniques are not necessarily restricted to large catchments. They can also be used for midsize catchments and even for small catchments. However, the routing approach is considerably more complicated than the unit hydrograph technique. The routing approach is applicable to cases where an increased level of detail is sought, above that which the unit hydrograph technique is able to providefor instance, when the objective is to describe the temporal variation of streamflow at several points inside the catchment. In this case, the routing approach may well be the only way to accomplish the modeling objective. The hydrologic description of midsize catchments consists of two processes:
This chapter focuses on a method of rainfall abstraction that is widely used for hydrologic design in the United States: Natural Resources Conservation Service (NRCS) runoff curve number method. Other rainfall abstraction procedures used by e Isting computer models are discussed in Chapter 13. With regard to hydrograph generation, this chapter centers on the unit hydrograph technique, which is a defacto standard for midsize catchments, having been
used extensively throughout the world. The NRCS TR55 method, also included in this chapter, has peak flow and hydrograph generation capabilities and is applicable to small and midsize urban catchments with concentration time in the range 0.110.0 h. The TR55 method is. based on the runoff curve number method, unit hydrograph techniques, and simplified stream channel routing procedures. 5.2 RUNOFF CURVE NUMBER METHOD
5.3 UNIT HYDROGRAPH TECHNIQUES
The concept of unit hydrograph, originated by Sherman [14], is used in midsize catchment analysis as a means to develop a hydrograph for any given storm. The word unit is normally taken to refer to a unit depth of effective rainfall or runoff. However, it should be noted that Sherman first used the word to describe a unit depth of runoff (1 cm or 1 in.) lasting a unit increment of time (i.e., an indivisible increment). The unit increment of time can be either 1h, 3h, 6h, 12h, 24h, or any other suitable duration. For midsize catchments, unit hydrograph durations from 1 to 6 h are common. The unit hydrograph is defined as the hydrograph produced by a unit depth of runoff uniformly distributed over the entire catchment and lasting a specified duration. To illustrate the concept of unit hydrograph, assume that a certain storm produces 1 cm of runoff and covers a 50km^{2} catchment over a period of 2 h. The hydrograph measured at the catchment outlet would be the 2h unit hydrograph for this 50km^{2} catchment (Fig. 54).
A unit hydrograph for a given catchment can be calculated either
While both methods can be used for gaged catchments, only the latter method is appropriate for ungaged catchments. Since a unit hydrograph has meaning only in connection with a given storm duration, it follows that a catchment can have several unit hydrographs, each for a different rainfall duration. Once a unit hydrograph for a given duration has been determined, other unit hydrographs can be derived from it by using one of the following methods:
Two assumptions are crucial to the development of the unit hydrograph technique. These are the principles of linearity and superposition. Given a unit hydrograph, a hydrograph for a runoff depth other than unity can be obtained by simply multiplying the unit hydrograph ordinates by the indicated runoff depth (linearity). This, of course, is possible only under the assumption that the time base remains constant regardless of runoff depth (Fig. 55(a)). The time base of all hydrographs obtained in this way is equal to that of the unit hydrograph. Therefore, the procedure can be used to calculate hydrographs produced by a storm consisting of a series of runoff depths, each lagged in time one increment of unit hydrograph duration (Fig. 55(b)). The summation of the corresponding ordinates of these hydrographs (superposition) allows the calculation of the composite hydrograph (Fig. 55(c)). The procedure depicted in Fig. 55 is referred to as the convolution of a unit hydrograph with an effective storm hyetograph. In essence, the procedure amounts to stating that the composite hydrograph ordinates are a linear combination of the unit hydrograph ordinates, while the composite hydrograph time base is the sum of the unit hydrograph time base minus the unit hydrograph duration plus the storm duration. The assumption of linearity has long been considered one of the limitations of unit hydrograph theory. In nature, it is unlikely that catchment response will always follow a linear function. For one thing, discharge and mean velocity are nonlinear
functions of flow depth and stage. In practice, the linear assumption provides a convenient means of calculating runoff response without the complexities associated with nonlinear analysis. More recent developments of unit hydrograph theory [1, 4, 15] have relaxed the linear assumption. Methods accounting for the nonlinearity of runoff response constitute what is known as nonlinear unit hydrograph theory. The upper limit of applicability of the unit hydrograph is not very well defined. Sherman [14] used it in connection with basins varying from 1300 to 8000 km^{2}. Linsley et al. [10] mention an upper limit of 5000 km^{2} in order to preserve accuracy. More recently, the unit hydrograph has been linked to the concept of midsize catchment, i.e. , greater than 2.5 km^{2} and less than 250 km^{2} This cenainly does not preclude the unit hydrograph technique from being applied to catchments larger than 250 km^{2}, although overall accuracy is likely to decrease with an increase in catchment area. Development of Unit Hydrographs: Direct Method To develop a unit hydrograph by the direct method it is necessary to have a gaged catchment, i.e., a catchment equipped with rain gages and a stream gage at the outlet, and adequate sets of corresponding rainfallrunoff data. The rainfallrunoff records should be screened to identify storms suitable for unit hydrograph analysis. Ideally, a storm should have a clearly defined duration, with no rainfall preceding it or following it. The selected storms should be of uniform rainfall intensity both temporally and spatially. In practice, the difficulty in meeting this latter requirement increases with catchment size. As catchment scale grows from midsize to large, the requirement of spatial rainfall uniformity in particular is seldom met. This limits unit hydrograph development by the direct method to midsize catchments. Catchment Lag. The concept of catchment lag, basin lag, or lag time is central to the development of unit hydrograph theory. It is a measure of the time elapsed between the occurrence of unit rainfall and the occurrence of unit runoff. Catchment lag is a global measure of response time, encompassing hydraulic length, catchment gradient, drainage density, drainage patterns, and other related factors. There are several definitions of catchment lag, depending on what particular instant is taken to describe the occurrence of either unit rainfall or runoff. Hall [7] has identified seven definitions, shown in Fig. 56. Among them the T_{2} lag, defined as the time elapsed from the centroid of effective rainfall to the peak of runoff, is the most commonly used definition of catchment lag. In unit hydrograph analysis, the concept of catchment lag is used to characterize the catchment respone time. Because runoff volume must be conserved (i.e., runoff volume should equal one unit of effective rainfall depth), short lags result in unit responses featuring hig peaks and relatively short time bases; conversely, long lags result in unit responses showing low peaks and long time bases. In practice, catchment lag is empirically related to catchment characteristics. A general expression for catchment lag is the following:
in which t_{t} = catchment lag; L = catchment length (length measured along the main
stream from outlet to divide); L_{c} = length to catchment centroid (length measured along the main stream from outlet to a point located closest to the catchment centroid); S = a weighted measure of catchment slope, usually taken as the S_{2} channel slope (Chapter 2); and C and N are empirical parameters. The parameter L describes length, L_{c} is a measure of shape, and S relates to relief. Methodology. In addition to the requirements of uniform rainfall intensity in time and space, storms suitable for unit hydrograph analysis should be of about the same duration. The duration should lie between 10 percent to 30 percent of catchment lag. The latter requirement implies that runoff response is of the sub concentrated type, with rainfall duration less than time of concentration. Indeed, sub concentrated flow is a characteristic of midsize catchments. For increased accuracy, direct runoff should be in the range O.5 to 2.0 units (usually centimeters or inches) Several individual storms (at least five events) should be analyzed to assure consistency. The following steps are applied to each individual storm:
The catchment unit hydrograph is obtained by averaging the unit hydrograph ordinates obtained from each of the individual storms and averaging the respective unit hydrograph durations. Minor adjustments in hydrograph ordinates may be necessary to ensure that the volume under the unit hydrograph is equal to one unit of runoff depth. Hydrograph Separation. Only the direct runoff component of the measured hydrograph is used in the computation of the unit hydrograph. Therefore, it is necessary to separate the measured hydrograph into its direct runoff and baseflow components. Interflow is usually included as part of baseflow. Procedures for baseflow separation are usually arbitrary in nature. First, it is necessary to identify the point in the receding limb of the measured hydrograph where direct runoff ends. Generally, this ending point is located in such a way that the receding time up to that point is about 2 to 4 times the timetopeak (Fig. 57). For large basins, this multiplier may be greater than 4. As far as possible, the location of the ending point should be such that the time base is an even multiple of the unit hydrograph duration. A common assumption is that baseflow recedes at the same rate as prior to the storm until the peak discharge has passed and then gradually increases to the ending point P in the receding limb, as illustrated by line a in Fig. 57. If a stream and groundwater table are hydraulically connected (Fig. 58), water infiltrates during the rising limb, reducing baseflow, and exfiltrates during the receding limb, increasing
baseflow, as shown by line b in Fig. 57 [5]. The most expedient assumption for baseflow separation is a straight line from the start of the rising limb to the ending point, as shown by line c. Differences in baseflow due to the various separation techniques are likely to be small when compared to the direct runoff hydrograph volume. Additional methods for hydrograph separation and baseflow recession are described in Chapter 11.
Development of Unit Hydrographs: Indirect Method In the absence of rainfallrunoff data, unit hydrographs can be derived by synthetic means. A synthetic unit hydrograph is a unit hydro graph derived following an established formula, without the need for rainfallrunoff data analysis. The development of synthetic unit hydrographs is based on the following principle: Since the volume under the hydrograph is known (volume is equal to catchment area multiplied by 1 unit of runoff depth), the peak discharge can be calculated by assuming a certain unit hydrograph shape. For instance, if a triangular shape is assumed (Fig. 59), the volume is equal to:
in which V = volume under the triangular unit hydrograph; Q_{p} = peak flow; T_{bt} = time base of the triangular unit hydrograph; A = catchment area; and (1) = one unit of runoff depth. From Eq. 517:
Synthetic unit hydrograph methods usually relate time base to catchment lag. In turn, catchment lag is related to the timing response characteristics of the catchment, including catchment shape, length, and slope. Therefore, catchment lag is a fundamental parameter in synthetic unit hydrograph determinations. Several methods are available for the calculation of synthetic unit hydrographs. Two widely used methods, the Snyder and the Soil Conservation Service methods, are described here. The Clark method, also widely used, is based on catchment routing techniques; therefore, it is described in Chapter 10. Snyder's Synthetic Unit Hydrograph In 1938, Snyder [17] introduced the concept of synthetic unit hydrograph. The analysis of a large number of hydrographs from catchments in the Appalachian region led to the following formula for lag:
in which t_{l} = catchment or basin lag in hours, L = length along the mainstream from outlet to divide, L_{c} = length along the mainstream from outlet to point closest to catchment centroid, and C_{t} = a coefficient accounting for catchment gradient and associated catchment storage. With distances L and L_{c} in kilometers, Snyder gave values of C_{t} varying in the range 1.35 to 1.65, with a mean of 1.5. With distances L and L_{c} in miles, the corresponding range of C_{t} is 1.8 to 2.2, with a mean of 2. Snyder's formula for peak flow is:
which when compared with Eq. 518 reveals that
is an empirical coefficient relating triangular time base to lag. Snyder gave values of C_{p} in the range 0.56 to 0.69, which are associated with T_{bt} / t_{l} ratios in the range 3.57 to 2.90. The lower the value of C_{p} (i.e., the lower the peak flow), the greater the value of T_{bt} / t_{l} and the greater the capability for catchment storage. In SI units, Snyder's peak flow formula is:
in which Q_{p} = unit hydrograph peak flow corresponding to 1 cm of effective rainfall, in cubic meters per second; A = catchment area, in square kilometers; and t_{l} = lag, in hours. In U.S. customary units, Snyder's peak flow formula is
in which Q_{p} = unit hydrograph peak flow corresponding to 1 in. of effective rainfall in cubic feet per second; In Snyder's method, the unit hydrograph duration is a linear function of lag:
in which t_{r} = unit hydrograph duration. In applying the procedure to flood forecasting, Snyder recognized that the actual duration of the storm is usually greater than the duration calculated by Eq. 524. Therefore, he devised a formula to increase the lag in order to account for the increased storm duration. This led to:
in which t_{lR} is the adjusted lag corresponding to a duration t_{R}. Assuming uniform effective rainfall for simplicity, the unit hydrograph timetopeak is equal to onehalf of the storm duration plus the lag (Fig. 57). Therefore, the timetopeak in terms of the lag is:
When calculating the actual time base of the unit hydrograph, Snyder included interflow as part of direct runoff. This results in a longer time base than that corresponding only to direct runoff. Snyder's formula for actual time base is the following:
in which T_{b} = actual unit hydrograph time base (including interflow) in hours and t_{l} = lag in hours.
For a 24h lag, this formula gives T_{b} / t_{l} = 6, which is a reasonable value considering that interflow is being included in the calculation.
For smaller lags, however, Eq. 527 gives unrealistically high values of T_{b} / t_{l}. For instance, for a 6h lag, T_{b} / t_{l} = 15.
For midsize catchments, and excluding interflow, experience has shown that values of The Snyder method gives peak flow (Eq. 522), timetopeak (Eq. 526), and time base (Eq. 527) of the unit hydrograph. These values can be used to sketch the unit hydrograph, adhering to the requirement that unit hydrograph volume should equal 1 unit of runoff depth. Snyder gave a distribution chart (Fig. 510) to aid in plotting the unit hydrograph ordinates, but cautioned against the exclusive reliance on this graph to develop the shape of the unit hydrograph. The Snyder method has been extensively used by the U.S. Army Corps of Engineers. Their experience has led to two empirical formulas that aid in determining the shape of the Snyder unit hydrograph [20]:
in which W_{50} = width of unit hydrograph at 50 percent of peak discharge in hours; W_{75} = width of unit hydrograph at 75 percent of peak discharge in hours; Q_{p} = peak discharge in cubic meters per second; and A = catchment area in square kilometers (Fig. 511). These time widths should be proportioned in such a way that onethird is located before the peak and twothirds after the peak. Snyder cautioned that lag may tend to vary slightly with flood magnitude and that synthetic unit hydrograph calculations are likely to be more accurate for fanshaped catchments than for those of highly irregular shape. He recommended that the coefficients C_{t}, and C_{p} be determined on a regional basis. Examination of Eq. 519 reveals that C, is largely a function of catchment slope,
since both length and shape have already been accounted for in L and L_{c} , respectively. Since Eq. 519 was derived empirically, the actual value of Cr depends on the units of L and L_{c}. Furthermore, Eq. 519 implies that when the product of (LL_{c} is equal to 1, the lag is equal to C_{t}. Since for two catchments of the same size, lag is a function of slope, it is unlikely that C_{t} is a constant. To give an example, an analysis of 20 catchments in thep.orthand middle Atlantic United States [19] led to: C_{t} = 0.6/S^{1/2}. A similar conclusion is drawn from Eq. 516. Therefore, values of C_{t} have regional meaning, in general being a function of catchment slope. Values of C_{t} quoted in the literature reflect the natural variability of catchment slopes. The parameter C_{p} is dimensionless and varies within a narrow range. In fact, it is readily shown that the maximum possible value of C_{p} is 11/12. Since triangular time base cannot be less than twice the timetopeak (otherwise, runoff diffusion would be negative, clearly a physical impossipility), it follows that in the limit (Le., absence of runoff diffusion), T_{bt} = 2t_{p}; and, therefore, C_{p} = t_{l} / t_{p} = 11/12. In practice, triangular time base is usually about 3 times the timetopeak. For T_{bt} = 3t_{p} , a similar calculation leads to: C_{p} = 0.61, which lies approximately' in the middle of Snyder's data (0.560.69). Since C_{t} increases with catchment storage and C_{p} decreases with catchment storage, the ratio C_{t} / C_{p} can be directly related to catchment storage. Furthermore, the reciprocal ratio (C_{p} / C_{t}) can be directly related to extent of urban development, since the latter usually results in a substantial reduction in the catchment's storage capability [26].
NRCS Synthetic Unit Hydrograph The NRCS Synthetic unit hydrograph is the dimensionless unit hydrograph developed by Victor Mockus in the 1950s and described in NEH4 [21]. This hydrograph was developed based on the analysis of a large number of natural unit hydrographs from a wide range of catchment sizes and geographic locations. The method has come to be recognized as the NRCS synthetic unit hydrograph and has been applied to midsize catchments throughout the world.
The method differs from Snyder's in that it uses a constant ratio of triangular time base to timetopeak, To calculate catchment lag (the T2 lag), the NRCS Method uses the following two methods:
The curve number method is limited to catchments of areas less than 8 km^{2} (2000 ac), although recent evidence suggests that it may be extended to catchments up to 16 km^{2} (4000 ac) [11]. In the curve number method, the lag is expressed by the following formula:
in which t_{l} = catchment lag in hours; L = hydraulic length (length measured along principal watercourse) in meters; CN = runoff curve number; and Y = average catchment land slope in meters per meter. In U.S. customary units, the formula is:
The velocity method is used for catchments larger than 8 km^{2}, or for curve numbers outside of the range 50 to 95. The main stream is divided into reaches, and the 2y flood (or alternatively the bankfull discharge) is estimated. In certain cases it may be desirable to use discharges corresponding to 10y frequencies or more. The mean velocity is computed, and the reach concentration time is calculated by using the reach valley length (straight distance). The sum of the concentration time for all reaches is the concentration time for the catchment. The lag is estimated as follows:
in which t_{l} = lag and t_{c} = concentration time NRCS experience has shown that this ratio is typical of midsize catchments [21].
which is close to Snyder's ratio of 6. Assuming uniform effective rainfall for simplicity, the timetopeak is by definition equal to
Eliminating t_{r} from Eqs. 533 and 534, leads to
Therefore:
and
To derive the NRCS unit hydrograph peak flow formula, the ratio Tb/tp = 8/3 is used in Eq. 518, leading to
In SI units, the peak flow formula is:
in which Q_{p} = unit hydrograph peak flow for 1 cm of effective rainfall in cubic meters per second; A = catchment area in s uare kilometers; and t_{p} = timetopeak in hours. In U.S. customary units, the NRCS peak flow formula is:
in which Q_{p} = unit hydrograph peak flow for 1 in. of effective rainfall; A = catchment area in square miles; and t_{p} = timetopeak in hours. Given Eqs. 532 and 534, the timetopeak can be readily calculated as follows: t_{p} = 0.5t + 0.6t_{c} Once t_{p} and Q_{p} have been determined, th NRCS imensionless umt hydrograph (Fig. 512) is used to calculate the unit hydrograph ordinates. The shape of the dimensionless unit hydrograph is more in agreement with unit hydrographs that are likely to occur in nature than the triangular shape ( T_{bt} / t_{p} = 8/3 ) used to develop the
peak flow value. The dimensionless unit hydrograph has a value of T_{b} / t_{p} = 5. Values of NRCS dimensionless unit hydrograph ordinates at intervals of 0.2 (t / t_{p}) are given in Table 56. The NRCS method provides a unit hydrograph shape and therefore leads to more reproducible results than the Snyder method. However, the ratio T_{bt} / t_{p} is kept constant and equal to 8/3. Also, when lag is calculated by the velocity method, the ratio t_{l} / t_{c} is kept constant and equal to 6/10. Although these assumptions are based on a wide range of data, they render the method inflexible in certain cases. In particular, values of T_{bt} / t_{p} other than 8/3 may lead to other shapes of unit hydrographs. Larger values of T_{bt} / t_{p} (equivalent to lower values of C_{p} in the Snyder method) imply greater catchment storage. Therefore, since the NRCS method fixes the value of T_{bt} / t_{p} it should be limited to midsize catchments in the lower end of the spectrum (2.5250 km^{2}). The Snyder method, however, by providing a variable Tb,ltp, may be used for larger catchments (from 250 up to 5000 km^{2}) [10]. Efforts to extend the range of applicability of the NRCS method have led to the relaxation of the T_{bt} / t_{p} ratio. It can be shown that the ratio p of volumetopeak (volume under the rising limb of the triangular unit hydrograph) to the triangular unit hydrograph volume is the reciprocal of the ratio T_{bt} / t_{p}. For instance, in the case of the standard NRCS Synthetic unit hydrograph, T_{bt} / t_{p} = 8/3, and p = 3/8. In terms of p , Eq. 538 can be expressed as follows:
which converts the NRCS method into a twoparameter model like the Snyder method, thereby increasing its flexibility. Other Synthetic Unit Hydrographs The Snyder and NRCS methods base their calculations on the following properties:
In addition, the NRCSmethod specifies a gamma function for the shape of the unit hydrograph. Many other synthetic unit hydrographs have been reported in the literature [16]. In general , any procedure defining geometric properties and hydrograph shape can be used to develop a synthetic unit hydrograph.
Change in Unit Hydrograph Duration A unit hydrograph, whether derived by direct or indirect means, is valid only for a given (effective) storm duration. In certain cases, it may be necessary to change the duration of a unit hydrograph. For instance, if an Xhour unit hydrograph is going to be used with a storm hyetograph defined at Yhour intervals, it is necessary to convert the Xhour unit hydrograph into a Xhour unit hydrograph. In general, once a unit hydrograph of a given duration has been derived for a catchment, a unit hydrograph of another duration can be calculated. There are two methods to change the duration of unit hydrographs:
The superposition method converts an Xhour unit hydrograph into a nXhour unit hydrograph, in which n is an integer. The Shydrograph method converts an Xhour unit hydrograph into a Yhour unit hydrograph, regardless of the ratio between X and Y. Superposition Method. This method allows the conversion of an Xhour unit hydrograph into a nXhour unit hydrograph, in which n is an integer. The procedure
consists of lagging nXhour unit hydrographs in time, each for an interval equal to X hours, summing the ordinates of all n hydrographs, and dividing the summed ordinates by n to obtain the nXhour unit hydrograph. The volume under Xhour and nXhour unit hydrographs is the same. If T_{b} is the time base of the Xhour hydrograph, the time base of the nXhour hydrograph is equal to T_{b} + (n  l)X. The procedure is illustrated by the following example.
SHydrograph Method. The Shydrograph method allows the conversion of an Xhour unit hydrograph into a Yhour unit hydrograph, regardless of the ratio between X and Y. The procedure consists of the following steps:
The volume under Xhour and Yhour unit hydrographs is the same. If T_{b} is the time base of the Xhour unit hydrograph, the time base of the Yhour unit hydrograph is T_{b}  X + Y.
Minor errors in unit hydrograph ordinates may often lead to errors (i.e., undesirable oscillations) in the resulting Shydrograph. In this case, a certain amount of smoothing may be required to achieve the typical Sshape (Fig. 513). Convolution and Composite Hydrographs The procedure to derive a composite or flood hydrograph based on a unit hydro graph and an effective storm hyetograph is referred to as hydrograph convolution. This technique is based on the principles of linearity and superposition. The volume under the composite hydrograph is equal to the total volume of the effective rainfall. If T_{b} is the time base of the Xhour unit hydrograph and the storm consists of n Xhour intervals, the time base of the composite hydrograph is equal to T_{b}  X + nX = T_{b} + (n  1)X. The convolution procedure is illustrated by the following example.
Unit Hydrographs from Complex Storms The convolution procedure enables the calculation of a storm hydrograph based on a unit hydrograph and a storm hyetograph. In theory, the procedure can be reversed to allow the calculation of a unit hydrograph for a given storm hydrograph and storm hyetograph. Method of Forward Substitution. The unit hydrograph can be calculated directly due to the banded property of the convolution matrix (see Table 510).
With m = number of nonzero unit hydrograph ordinates, n = number of intervals of effective rainfall, and
Therefore:
By elimination and back substitution, the following formula can be developed for the unit hydrograph ordinates u_{i} as a function of storm hydrograph ordinates q_{i} and effectiverainfall depths r_{k} :
for i varying from 1 to m. In the summation term, j decreases from i1 to 1, and k increases from 2 up to a maximum of n. This recursive equation allows the direct calculation of a unit hydrograph based on hydrographs from complex storms. In practice, however, it is not always feasible to arrive at a solution because it may be difficult to get a perfect match of storm hydrograph and effective rainfall hyetograph (due to errors in the data). For one thing, the measured storm hydrograph would have to be separated into direct runoff and baseflow before attempting to use Eq. 544. The uncertainties involved have led to the use of the least square technique. In this technique, rainfallrunoff data (r, h) for a number of events are used to develop a set of average values of u using statistical tools [12]. Other methods to derive unit hydrographs for complex storms are discussed by Singh [16].
5.4 NRCS TR55 METHOD
The TR55 method is a collection of simplified procedures developed by the USDA Natural Resources Conservation Service (formerly Soil Conservation Service) to calculate peak discharges, storm hydrographs, and stormwater storage volumes in small/midsize urban catchments [22]. It consists of three methodologies:
The graphical method calculates a flood peak discharge for a hydrologically homogeneous catchment, i.e., that which can be represented by a single area, of given slope and curve number. The tabular method calculates a flood hydrograph for a hydrologically heterogeneous catchment, which is better analyzed by dividing it into several homogeneous subareas, each of given slope and curve number. These methods were developed based on information obtained with the NRCS TR20 hydrologic computer model (Section 13.4). They are designed to be used in cases where their applicability can be clearly demonstrated, in lieu of more elaborate techniques. Whereas TR55 does not specify catchment size, the graphical method is limited to catchments with time of concentration in the range 0.110 h. This encompasses most small and midsize catchments in the terminology used in this book. Likewise, the tabular method is limited to catchments with time of concentration in the range 0.12 h. The graphical method is described in this section. The tabular method is described in the original reference [22]. The detentionbasin method is described in Section 8.5. TR55 Storm, Catchment and Runoff Parameters Rainfall in TR 55 is described in terms of total rainfall depth and one of four standard 24h temporal rainfall distributions: type I, type lA, type II, and type III (Fig. 514). Type I appltes to California (south of the San Francisco Bay area) and Alaska; type IA applies to the Pacific Northwest and Northern California; type III applies to the Gulf Coast states; and type II applies everywhere else within the contiguous United States (FIg. 515 and Chapter 13). The duration of these rainfall distributions is 24 h. This constant duration was selected because most rainfall data is reported on a 24h basis. Rainfall intensities corresponding to durations shorter than 2 h are contained within the NRCS distributions . For instance, if a lOy 24h rainfall distribution is used, the 1h period with the most intense rainfall corresponds to the 10y 1h rainfall depth. TR55 uses the runoff curve number method (Section 5.1) to abstract total rainfall depth and calculate runoff depth. The abstraction procedure follows the
guidelines established in NEH4 [21], with extensions to account for curve numbers applicable to urban areas. In addition, TR55 includes procedures to determine time of concentration for the following types of surface flow:
Shallow concentrated flow is a type of flow of characteristics in between those of overland flow and streamflow. Applicability of TR55 When using TR55, there is a choice between graphical or tabular method. The graphical method gives only a peak discharge, whereas the tabular method provides a flood hydro graph. The graphical method should be used for hydrologically homogeneous catchments; the tabular method should be used for hydrologically heterogeneous catchments, for which catchment subdivision is necessary. The primary objective of TR55 is to provide simplified techniques, thereby reducing the effort involved in routine hydrologic calculations. The potential accuracy of the method is less than that which could be obtained with more elaborate techniques. The method is strictly applicable to surface flow and should not be used to describe flow properties in underground conduits.
Selection of Runoff Curve Number To estimate curve numbers for urban catchments, TR55 defines two types of areas:
Once pervious and impervious areas are delineated, the percent imperviousness can be determined. Impervious areas are of two kinds:
The question is: Do the impervious areas connect directly to the drainage system, or do they discharge onto lawns or other pervious areas where infiltration can occur? An impervious area is considered connected
An impervious area is considered unconnected if runoff from it spreads over a pervious area as overland (sheet) flow. Table 52(a) shows urban runoff curve numbers for connected impervious areas. The curve numbers shown are for typical values of average percent impervious area (second column). These composite curve numbers were developed based on the following assumptions:
Tables 52(b), (c), and (d) show runoff curve numbers for cultivated agricultural lands, other agricultural lands, and arid and semiarid rangelands, respectively. Figure 516 is used in lieu of Table 52(a) when the average percent (connected) impervious area and/or pervious area land use assumptions are other than those shown in the table. For example, Table 52(a) gives a CN = 70 for a 1/2 acre lot in hydrologic soil group B, assuming a 25 percent impervious area. If the lot has a different percent impervious area, say 20 percent, but the pervious area land use is the same as that assumed in Table 52(a) (open space in good hydrologic condition), then the pervious area CN is 61 (for hydrologic soil group B) and the composite curve number obtained from Figure 516 with 20 percent impervious area and pervious area CN = 61 is: CN = 69. The difference between 70 and 69 reflects the difference in percent impervious area only (25 vs 20 percent). Figure 517 is used to determine a composite eN when all or part of the impervious area is unconnected and the percent imperviousness is 30 percent or less. However, when the percent imperviousness is more than 30 percent, Fig. 516 is used instead to determine the composite CN, since the absorptive capacity of the remaining pervious areas (less than 70 percent) will not significantly affect runoff. In Figure 517, enter the rightside figure with percent imperviousness to the line matching the ratio of unconnected impervious to total impervious area. Then, move horizontally to the leftside figure to match the pervious area CN, and vertically down to find the composite CN. For example, for a 1/2acre lot with 20 percent imperviousness, 75 percent of which is unconnected, and pervious CN = 61, the composite CN (from Fig. 517) is: CN = 66. If all of the impervious area is connected (i.e., zero percent unconnected), the resulting CN (from Fig. 517) is: CN = 69. This value matches the example of the previous paragraph. Travel Time and Time of Concentration For any reach or subreach, travel time is the ratio of flow length to flow velocity. The time of concentration is the sum of travel times through the individual subreaches. For overland (sheet) flow with length less than 300 ft, TR55 uses the following formula for travel time:
in which t_{t}, = travel time in hours; n = Manning n; L = flow length, in feet;
p_{2} = 2y 24h rainfall depth in inches; and S = average land slope, in feet per foot. In SI units, this equation is:
in which L is given in meters; p_{2}, in centimeters; S, in meters per meter; and the remaining terms are the same as in Eq. 545. TR55 values of Manning n applicable to overland flow are given in Table 511. Overland flow lengths over 300 ft (90 m) lead to a form of surface flow referred to as shallow concentrated flow. In this case, the average flow velocity is determined from Fig. 518. For streamflow, the Manning equation (Eq. 289) can be used to calculate average flow velocities. Values of Manning n applicable to open channel flow are obtained from standard references [2, 3, 6].
TR55 Graphical Method The TR55 graphical method calculates peak discharge based on the concept of unit peak flow. The unit peak flow is the peak flower unit area, per unit runoff depth. In TR55, unit peak flow is a function of
Peak discharge is calculated by the following formula:
in which Q_{p} = peak discharge in L^{3}T^{1} units; q_{u} = unit peak flow in T^{1} units; A = catchment area in L^{2} units; Q = runoff depth in L units; and F = surface storage correction factor (dimensionless). To use the graphical method, it is first necessary to evaluate the catchment flow type and to calculate the time of concentration assuming either overland flow , shallow concentrated flow , or streamflow. The runoff curve number is determined from either Table 52. Fig. 516, or Fig. 517. A flood frequency is selected, and an appropriate rainfall map (depthdurationfrequency) is used to determine the rainfall depth for the 24h duration and the chosen frequency. With the rainfall depth P and the CN. The runoff depth Q is determined using either Fig. 52, Eqs. 58, or 59. The initial abstraction is calculated by combining Eqs. 54 and 57 to yield:
in which I_{a} = initial abstraction, in inches. The equivalent SI formula is:
in which I_{a} is given in centimeters. The surface storage correction factor F is obtained from Table 512 as a function of the percentage of pond and swamp areas. With time of concentration t_{c}, ratio I_{a} / P, and storm type (either I, lA, II, or III). Fig. 519 is used to determine the unit peak flow in cubic feet per second per square mile per inch. Interpolation can be used for values of I_{a} / P different than those shown in Fig. 519. For values of I_{a} / P outside of the range shown in Fig. 519, the maximum (or minimum) value should be used. To obtain unit peak flow in cubic meters per second per square kilometer per centimeter, the unit peak flow values obtained from Fig. 519 are multiplied by 0.0043. Peak discharge is calculated by Eq. 547 as a function of unit peak flow, catchment area, runoff depth, and surface storage correction factor. The TR55 graphical method is limited to runoff curve numbers greater than 40, with time of concentration in the range 0.1 to 10.0 h, and surface storage areas spread throughout the catchment and covering less than 5 percent of it.
Assessment of TR55 Graphical Method The TR55 graphical method provides peak discharge as a function of unit peak flow, catchment area, runoff depth, and surface storage correction factor. The unit peak . flow is a function of tim of concentration ,abstraction parameter I_{a}/P, and storm type. The runoff depth is a function of total rainfall depth and runoff curve number. In the TR55 graphical method , time of concentration accounts for both runoff concentration and runoff diffusion. From Fig. 519, it is seen that unit peak flow decreases with time of concentration, implying that the longer the time of concentration ,the greater the catchment storage and peak flow attenuation. The parameter I_{a}/P is related to the catchment's abstractive properties. The greater the curve number, the lesser the value of I_{a}/P and the greater the unit peak flow . The surface storage correction factor F reduces the peak discharge to account for additional runoff diffusion caused by surface storage features typical of low relief catchments (i.e., ponds and swamps). The geographical locaation and associated storm type is accounted for by the four standard NRCS temporal storm distributions. Therefore, the TR55 graphical method accounts for hydrologic abstraction, runoff concentration and diffusion, geographical location and type of storm, and the additional surface storage of lowrelief catchments. The TR55 graphical method can be considered an extension of the rational method to midsize catchments. The unit peak flow used in the graphical method is similar in concept to the runoff coefficient of the rational method. However, unlike the latter, the TR55 graphical method includes runoff curve number and storm type and is applicable to midsize catchments with time of concentration to 10 h. The unit values of catchment area, runoff depth, and time of concentration can be used to provide a comparison between the TR55 graphical method and the rationa method. To illustrate, assume a catchment area of 1 mi^{2} (640 ac), time of concentration 1 h, and corresponding rainfall intensity 1 in./h, and runoff coefficient C = 0.95 (the maximum practicable value). A calculation by Eq. 44 gives a peak discharge of Q_{p}= 613ft^{3}/ s. A calculation with the TR55 graphical method, using the lowest possible value of abstraction for comparison purposes (I_{a}/P = 0.10), gives the following: For storm type 1, 200 ft^{3}/s; type lA, 108 ft^{3}/s; type 11,360 ft^{3}/s; and type III, 295 ft^{3}/s. This example shows the effect of regional storm hyetograph on the calculated peak discharge. It also shows that the TR5S graphical method generally gives lower peak flows than the rational method. This may be attributed to the fact that the TRS5 method accounts for runoff diffusion in a somewhat better way than the rational method. However, it should be noted that the peak discharges calculated by the two methods are not strictly comprable, since the value of I_{a}/P = 0.1 does not correspond exactly to C = 0.95. QUESTIONS
PROBLEMS
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