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CHAPTER 5: 
HYDROLOGY OF 
MIDSIZE CATCHMENTS 

"The concept of 100-yr flood was taken from TVA's "intermediate regional flood," which seemed a moderately
reasonable figure. The term "catastrophic flood" is used for events of much lesser frequency."
Gilbert F. White (1993)


This chapter is divided into four sections. Section 5.1 describes midsize catchments and its properties. Section 5.2 describes the runoff curve number method. Section 5.3 discusses unit hydrograph techniques, including unit hydrographs derived from measured data and synthetic unit hydrographs. Section 5.4 deals with the TR-55 graphical method for peak discharge determinations.


5.1  MIDSIZE CATCHMENTS

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A midsize catchment is described by the following features:

  1. Rainfall intensity varies within the storm duration,

  2. Storm rainfall can be assumed to be uniformly distributed in space,

  3. Runoff is by overland flow and stream channel flow, and

  4. Channel slopes are steep enough so that channel storage processes are small.

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 hydrograph 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 hydrograph corresponding to the actual effective storm 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 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 may be 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 time of concentration 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 detail.

It should be noted that the techniques for midsize and large catchments are indeed complementary. A large catchment may 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. 5-1). An example of a hydrologic computer model using the network concept is the HEC-HMS model (Hydrologic Modeling System) of the U.S. Army Corps of Engineers. This and other computer models are described in Chapter 13.

An intensity-duration-frequency curve.

Figure 5-1  Subdivision of a large catchment into several midsize catchments.

In practice, channel-routing 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 provide; for 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:

  1. Rainfall abstraction, and

  2. Hydrograph generation.

This chapter focuses on a method of rainfall abstraction that is widely used for hydrologic design in the United States: the Natural Resources Conservation Service (NRCS) runoff curve number method. Other rainfall abstraction procedures used by existing 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 TR-55 method, also included in this chapter, has peak flow and hydrograph generation capabilities and is applicable to small and midsize urban catchments with time of concentration in the range 0.1-10 h. The TR-55 method is based on the runoff curve number method, unit hydrograph techniques, and simplified stream channel routing procedures.


5.2  RUNOFF CURVE NUMBER

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The runoff curve number method is a procedure for hydrologic abstraction of storm rainfall developed by the U.S. Natural Resources Conservation Service (formerly Soil Conservation Service) [21]. In this method, total storm runoff depth is a function of total storm rainfall depth and an abstraction parameter referred to as runoff curve number, curve number, or CN. The curve number varies in the range 1 ≤ CN ≤ 100, being a function of the following runoff-producing catchment properties:

  1. Hydrologic soil type,

  2. Land use and treatment types,

  3. Hydrologic surface condition, and

  4. Antecedent moisture condition.

The runoff curve number method was developed based on daily rainfall P (in.) and its corresponding runoff Q (in.) for the annual floods at a given site. It limits itself to the calculation of runoff depth and does not explicitly account for temporal variations of rainfall intensity. In midsize catchment analysis, the temporal rainfall distribution is introduced at a later stage, during the generation of the flood hydrograph, by means of the convolution of the unit hydrograph with the effective storm hyetograph (Section 5.3).

Runoff Curve Number Equation

In the runoff curve number method:

  • Potential runoff (i.e., total rainfall) is referred to as P,

  • Actual runoff is referred to as Q,

  • Potential retention, or, in NRCS use, potential maximum retention, is referred to as S, and

  • Actual retention is defined as (P - Q ), with (P - Q ) ≤ S.

The method assumes a proportionality between retention and runoff, such that the ratio of actual retention to potential retention is equal to the ratio of actual runoff to potential runoff:

 P - Q          Q
_______  =  ____
    S             P
(5-1)

This assumption underscores the conceptual basis of the runoff curve number method, namely the asymptotic behavior of actual retention toward potential retention for sufficiently large values of potential runoff.

For practical applications, Eq. 5-1 is modified by reducing the potential runoff by an amount equal to the initial abstraction Ia. The latter consists mainly of interception, surface storage, and some infiltration, which take place before runoff begins.

 P - Ia - Q              Q
___________  =  _________
       S                  P - Ia
(5-2)

Solving for Q  from Eq. 5-2:

           ( P - Ia )2
Q = ______________
           P - Ia + S
(5-3)

which is physically subject to the restriction that P > Ia, i.e., the potential runoff minus the initial abstraction cannot be negative.

To simplify Eq. 5-3, the initial abstraction is linearly related to the potential maximum retention as follows:

Ia  =  0.2S (5-4)

This relation was obtained based on rainfall-runoff data from small experimental watersheds. The coefficient 0.2 has been subjected to wide scrutiny. For instance, Springer et al. [18] evaluated small humid and semiarid catchments and found that the coefficient in Eq. 5-4 varied in the range 0.0 to 0.26. Nevertheless, 0.2 is the standard initial abstraction coefficient recommended by NRCS [21]. For research applications and particularly when warranted by field data, it is possible to consider the initial abstraction coefficient as an additional parameter in the runoff curve number method. In general:

Ia = λ S (5-5)

in which λ = initial abstraction coefficient.

With Eq. 5-4, Eq. 5-3 reduces to:

          ( P - 0.2S )2
Q = _______________
            P + 0.8S
(5-6)

which is subject to the restriction that P ≥ 0.2S.

Since potential retention varies within a wide range (0 ≤ S < ∞), it has been conveniently mapped in terms of a runoff curve number, an integer varying in the range 0-100. The chosen mapping equation is:

          1000
S = _________  -  10
           CN
(5-7)

in which CN is the runoff curve number (dimensionless) and S, 1000 and 10 are given in inches. To illustrate, for CN = 100, S = 0 in.; and for CN = 1, S = 990 in. Therefore, the catchment's capability for rainfall abstraction is inversely proportional to the runoff curve number. For CN = 100, no abstraction is possible, with runoff being equal to total rainfall. On the other hand, for CN = 1 practically all rainfall would be abstracted, with runoff being essentially close to zero.

With Eq. 5-7, Eq. 5-6 can be expressed in terms of CN:

              [ CN ( P + 2 ) - 200 ] 2
Q =   ___________________________
            CN [ CN ( P - 8 ) + 800 ]
(5-8)

which is subject to the restriction that P ≥ ( 200/ CN ) - 2. In Eq. 5-8, P and Q are given in inches. In SI units, the equation is:

             R [ CN ( P/R + 2 ) - 200 ] 2
Q =   _______________________________
            CN [ CN ( P/R - 8 ) + 800 ]
(5-9)

which is subject to the restriction that PR [ ( 200/CN ) - 2 ]. With R = 2.54 in Eq. 5-9, P and Q are given in centimeters.

For a variable initial abstraction, Eq. 5-8 is expressed as follows:

                         [ CN ( P + 10 λ ) - 1000 λ ] 2
Q =   _______________________________________________
            CN { CN [ P - 10 ( 1- λ ) ] + 1000 ( 1 - λ ) }
(5-10)

which is subject to the restriction that P ≥ ( 1000 λ / CN ) - 10 λ. An equivalent equation in SI units is:

                      R [ CN ( P/R + 10 λ ) - 1000 λ ] 2
Q =   ________________________________________________
            CN { CN [ P/R - 10 ( 1- λ ) ] + 1000 ( 1 - λ ) }
(5-11)

A plot of Eqs. 5-8 and 5-9 is shown in Fig. 5-2. This figure is applicable only for the standard initial abstraction value, Ia = 0.2 S. If this condition is relaxed, as in Eqs. 5-10 and 5-11, Fig. 5-2 has to be modified appropriately.

Direct runoff as a function of rainfall and runoff curve number.

Figure 5-2  Direct runoff as a function of rainfall and runoff curve number [21].

Estimation of Runoff Curve Number From Tables

With rainfall P and runoff curve number CN, the runoff Q can be determined by either Eq. 5-8 or Eq. 5-9, or from Fig. 5-2.

For ungaged watersheds, estimates of runoff curve numbers are given in tables supplied by federal agencies (NRCS, USDA Forest Service) and local city and county departments. Tables of runoff curve numbers for various hydrologic soil-cover complexes are widely available. The hydrologic soil-cover complex describes a specific combination of hydrologic soil group, land use and treatment class, hydrologic surface condition, and antecedent moisture condition. All these have a direct bearing on the amount of runoff produced by a watershed.

The hydrologic soil group describes the type of soil. The land use and treatment class describes the type and condition of vegetative cover. The hydrologic surface condition refers to the ability of the watershed surface to enhance or impede direct runoff. The antecedent moisture condition accounts for the recent history of rainfall and, consequently, it is a measure of the amount of moisture stored by the catchment.

 Example 5-1.

A certain catchment experiences P = 4 in of total rainfall. The runoff curve number is CN = 80. Determine the direct runoff Q.


From Fig. 5-2, for the given P and CN : Q = 2.05 in. Using Eq. 5-8: Q = 2.04 in.

calculator image

ONLINE CALCULATION. Using ONLINE CURVE NUMBER, the direct runoff is Q = 2.04 in.


Hydrologic Soil Groups. All soils are classified into four hydrologic soil groups of distinct runoff-producing properties. These groups are labeled A, B, C, and D (Table 5-1).

Group A consists of soils of low runoff potential, having high infiltration rates even when wetted thoroughly. They are primarily deep, very well drained sands and gravels, with a characteristically high rate of water transmission.

Group B consists of soils with moderate infiltration rates when wetted thoroughly, primarily moderately deep to deep, moderately drained to well drained, with moderately fine to moderately coarse textures. These soils have a moderate rate of water transmission.

Group C consists of soils with slow infiltration rate when wetted thoroughly, primarily soils having a layer that impedes downward movement of water or soils of moderately fine to fine texture. These soils have a slow rate of water transmission.

Group D consists of soils of high runoff potential, having very slow infiltration rates when wetted thoroughly. They are primarily clay soils with a high swelling potential, soils with a permanent high water table, soils with a clay layer near the surface, and shallow soils overlying impervious material. These soils have a very slow rate of water transmission.


Table 5-1  NRCS Hydrologic Soil Groups.
Hydrologic
soil group
Rate of
water transmission
Texture
A High Gravel, sand, sandy loam
B Moderate Silty loam, loam
C Slow Sandy clay loam
D Very slow Clay soils

Maps showing the geographical distribution of hydrologic soil types for most areas in the United States are available either directly from NRCS or from pertinent local agencies. Additional detail on U.S. soils and their hydrologic soil groups can be found in NRCS publications [21].

Land Use and Treatment. The effect of the surface condition of a watershed is evaluated by means of land use and treatment classes. Land use pertains to the watershed cover, including every kind of vegetation, litter and mulch, fallow (bare soil), as well as nonagricultural uses such as water surfaces (lakes, swamps, and so on), impervious surfaces (roads, roofs, and the like), and urban areas. Land treatment applies mainly to agricultural land uses, and it includes mechanical practices such as contouring or terracing and management practices such as grazing control and crop rotation. A class of land use/treatment is a combination often found in a catchment.

The runoff curve number method distinguishes between cultivated land, grasslands, and woods and forests. For cultivated lands, it recognizes the following land uses and treatments: fallow, row crop, small grain, close-seed legumes, rotations (from poor to good), straight-row fields, contoured fields, and terraced fields. Additional detail on these land use and treatment classes can be found in reference [21].

Hydrologic Condition. Grasslands are evaluated by the hydrologic condition of native pasture. The percent of areal coverage by native pasture and the intensity of grazing are visually estimated. A poor hydrologic condition describes less than 50 percent areal coverage and heavy grazing. A fair hydrologic condition describes 50 to 75 percent areal coverage and medium grazing. A good hydrologic condition describes more than 75 percent areal coverage and light grazing.

Woods are small isolated groves or trees being raised for farm or ranch use. The hydrologic condition of woods is visually estimated as follows:

  • Poor:  Heavily grazed or regularly burned woods, with very little litter and few shrubs,

  • Fair:  Grazed but not burned, with moderate litter and some shrubs, and

  • Good:  Protected from grazing, with heavy litter and many shrubs covering the surface.

Runoff curve numbers for forest conditions are based on guidelines developed by the U. S. Forest Service. The publication Forest and Range Hydrology Handbook [24] describes the determination of runoff curve numbers for national and commercial forests in the eastern United States. The publication Handbook of Methods for Hydrologic Analysis [25] is used for curve number determinations in the forest-range regions in the western United States.

Antecedent Moisture Condition. The runoff curve number method has three levels of antecedent moisture: AMC I, AMC II, and AMC III. The dry antecedent moisture condition (AMC I) has the lowest runoff potential, with the soils being dry enough for satisfactory plowing or cultivation to take place. The average antecedent moisture condition (AMC II) has an average runoff potential. The wet antecedent moisture condition (AMC III) has the highest runoff potential, with the catchment being practically saturated by antecedent rainfalls. Prior to 1993, the appropriate AMC level was based on the total 5-d antecedent rainfall, for dormant or growing season, as shown in Table 5-1. The current version of Chapter 4, NEH-4, released in 1993 [23], no longer supports Table 5.2, which is included here only for the sake of completeness. Applicable levels of AMC, including fractional values (see Table 5-5), have been developed on a regional basis.

Table 5-2  Seasonal rainfall limits for three levels of antecedent moisture condition (AMC) [21].
AMC Total 5-d Antecedent rainfall (cm)
Dormant Season Growing Season
I Less than 1.3 Less than 3.6
II 1.3 to 2.8 3.6 to 5.3
III More than 2.8 More than 5.3
Note: This table was developed using data from the midwestern United States. Therefore, caution is recommended when using the values supplied in this table for AMC determinations in other geographic or climatic regions.

Tables of runoff curve numbers for various hydrologic soil-cover complexes are in current use.

  • Table 5-3 (a) shows runoff curve numbers for urban areas.

  • Table 5-3 (b) shows runoff for cultivated agricultural areas.

  • Table 5-3 (c) shows runoff for other agricultural lands.

  • Table 5-3 (d) shows runoff for arid and semiarid rangelands.

Runoff curve numbers shown in these tables are for the average AMC II condition. Corresponding runoff curve numbers for AMC I and AMC III conditions are shown in Table 5-4.

Table 5-3 (a)  Runoff curve numbers for urban areas1 [22].
Cover Description Hydrologic Soil Group
Cover Type and
Hydrologic Condition
Average Percent
Impervious Area2
A B C D
Fully developed urban areas
( vegetation established )
         
Open space (lawns, parks, golf courses, cemeteries, etc.)3          
   Poor condition
   (grass cover less than 50%)
  68 79 86 89
   Fair condition
   (grass cover 50 to 75%)
  49 69 79 84
   Good condition
   (grass cover greater than 75%)
  39 61 74 80
Impervious areas          
   Paved parking lots, roofs, driveways,
   etc. (excluding right of way):
  98 98 98 98
   Streets and roads:
   Paved: curves and storm sewers
   (excluding right of way)
  98 98 98 98
   Paved: open ditches
   (including right of way)
  74 89 92 93
   Gravel
   (including right of way)
  76 85 89 91
   Dirt
   (including right of way)
  72 82 87 89
Western desert urban areas
   Natural desert landscaping
   (pervious areas only)4
  63 77 85 88
   Artificial desert landscaping
   (impervious weed barrier, desert
   shrub with 1 to 2 in. sand or gravel
   mulch and basin borders)
  96 96 96 96
Urban districts:
   Commercial and business 85 89 92 94 95
   Industrial 72 81 88 91 93
Residential districts by average lot size:
   1/8 ac. or less (town houses) 65 77 85 90 92
   1/4 ac. 38 61 75 83 87
   1/3 ac. 30 57 72 81 86
   1/2 ac. 25 54 70 80 85
   1 ac. 20 51 68 79 84
   2 ac. 12 46 65 77 82
Developing urban areas
Newly graded areas
(pervious areas only, no vegetation)5
77 86 91 94
Idle lands (curve numbers (CNs) are determined using cover types similar to thos in Table 5-3(c)).
Notes:

1 Average antecedent moisture condition and Ia = 0.2S.

2 The average percent impervious area shown was used to develop the composite CNs. Other assumptions are as follows: Impervious areas are directly connected to the drainage system; impervious areas have a CN = 98; and pervious areas are considered equivalent to open space in good hydrologic condition. CNs for other combinations of conditions may be computed using Fig. 5-17 or 5-18.

3 CNs shown are equivalent to those of pasture. Composite CNs may be computed for other combinations of open space cover type.

4 Composite CNs for natural desert landscaping should be computed using Figs. 5-17 or 5-18 based on the impervious area percentage (CN = 98) and the pervious area CN. The pervious area CNs are assumed equivalent to desert shrub in poor hydrologic condition.

5 Composite CNs to use for the design of temporary measures during grading and construction should be computed using Figs. 5-17 or 5-18, based on the degree of development (impervious area percentage) and the CNs for the newly graded pervious areas.

Table 5-3 (b)  Runoff curve numbers for cultivated agricultural lands1 [22].
Cover Description Hydrologic Soil Group
Cover Type Treatment2 Hydrologic
Condition3
A B C D
Fallow Bare Soil ___ 77 86 91 94
Crop residue cover (CR) Poor 76 85 90 93
Good 74 83 88 90
Row crops Straight row (SR) Poor 72 81 88 91
Good 67 78 85 89
SR + CR Poor 71 80 87 90
Good 64 75 82 85
Contoured (C) Poor 70 79 84 88
Good 65 75 82 86
C + CR Poor 69 78 83 87
Good 64 74 81 85
Contoured and terraced (C&T) Poor 66 74 80 82
Good 62 71 78 81
C&T + CR Poor 65 73 79 81
Good 61 70 77 80
Small grain SR Poor 65 76 84 88
Good 63 75 83 87
SR + CR Poor 64 75 83 86
Good 60 72 80 84
C Poor 63 74 82 85
Good 61 73 81 84
C + CR Poor 62 73 81 84
Good 60 72 80 83
C&T Poor 61 72 79 82
Good 59 70 78 81
C&T + CR Poor 60 71 78 81
Good 58 69 77 80
Close-seeded or broadcast legumes or rotation meadow SR Poor 66 77 85 89
Good 58 72 81 85
C Poor 64 75 83 85
Good 55 69 78 83
C&T Poor 63 73 80 83
Good 51 67 76 80
Notes:

1 Average antecedent moisture condition and Ia = 0.2S.

2 Crop residue cover applies only if residue is on at least 5% of the surface throughout the year.

3 Hydrologic condition is based on combination of factors that affect infiltration and runoff, including:

  1. Density and canopy of vegetated areas;

  2. Amount of year-round cover;

  3. Amount of grass or close-seeded legumes in rotation;

  4. Percent of residue cover on the land surface (good hydrologic condition is greater than or equal to 20%); and

  5. Degree of surface roughness.

  Poor:  Factors impair infiltration and tend to increase runoff.

  Good:  Factors encourage average and better than average infiltration and tend to decrease runoff.

Table 5-3 (c)  Runoff curve numbers for other agricultural lands1 [22].
Cover Description Hydrologic Soil Group
Cover Type Hydrologic
Condition
A B C D
Pasture, grassland, or range: continuous forage for grazing2 Poor 68 79 86 89
Fair 49 69 79 84
Good 39 61 74 80
Meadow: continuous grass, protected from grazing and generally mowed for hay ___ 30 58 71 78
Brush__brush-weed grass mixture with brush being the major element3 Poor 48 67 77 83
Fair 35 56 70 77
Good 304 48 65 73
Woods__grass combination (orchard or tree farm)5 Poor 57 73 82 86
Fair 43 65 76 82
Good 32 58 72 79
Woods6 Poor 45 66 77 83
Fair 36 60 73 79
Good 304 55 70 77
Farmsteads__buildings, lanes, driveways, and surrounding lots __ 59 74 82 86
Notes:

1 Average antecedent moisture condition and Ia = 0.2S.

2 Poor : less than 50% ground cover on heavily grazed with no mulch.
  Fair : 50 to 75% ground cover and not heavily grazed.
  Good : more than 75% ground cover and lightly or only occasionally grazed.

3 Poor : less than 50% ground cover.
  Fair : 50 to 75% ground cover.
  Good : more than 75% ground cover.

4 Actual curve number is less than 30; use CN = 30 for runoff computations.

5 CNs shown were computed for areas with 50% woods and 50% grass (pasture) cover.
  Other combinations of conditions may be computed from the CNs for woods and pasture.

6 Poor : Forest litter, small trees, and brush are destroyed by heavy grazing or regular burning.
  Fair : Woods are grazed but not burned, and some forest litter covers the soil.
  Good : Woods are protected from grazing, and litter and brush adequately cover the soil.

Table 5-3 (d)  Runoff curve numbers for arid and semiarid rangelands1 [22].
Cover Description Hydrologic Soil Group
Cover Type Hydrologic
Condition
2
A3 B C D
Herbaceous__mixture of grass, weeds, and low-growing brush, with brush the minor element Poor 80 87 93
Fair 71 81 89
Good 62 74 85
Oak-aspen__mountain brush mixture of oak brush, aspen, mountain mahogany, bitter brush, maple, and other brush Poor 66 74 79
Fair 48 57 63
Good 30 41 48
Pinyon juniper __pinyon, juniper, or both; grass understory Poor 75 85 89
Fair 58 73 80
Good 41 61 71
Sagebrush with grass understory Poor 67 80 85
Fair 51 63 70
Good 35 47 55
Desert shrub__major plants include saltbrush, greasewood, creosotebush, blackbrush, bursage, palo verde, mesquite, and cactus Poor 63 77 85 88
Fair 55 72 81 86
Good 49 68 79 84
Notes:

1 Average antecedent moisture condition and Ia = 0.2S.
   For range in humid regions, use Table 5-3 (c).

2 Poor : less than 30% ground cover .
  Fair : 30 to 70% ground cover.
  Good : more than 70% ground cover.

3 Curve numbers for group A have been developed only for desert shrub.

Table 5-4  Corresponding runoff curve numbers
for three AMC conditions [21].

AMC II AMC I AMC III   AMC II AMC I AMC III
100 100 100  60 40 78
99 97 100  59 39 77
98 94 99  58 38 76
97 91 99  57 37 75
96 89 99  56 36 75
95 87 98  55 35 74
94 85 98  54 34 73
93 83 98  53 33 72
92 81 97  52 32 71
91 80 97  51 31 70
90 78 96  50 31 70
89 76 96  49 30 69
88 75 95  48 29 68
87 73 95  47 28 67
86 72 94  46 27 66
85 70 94  45 26 65
84 68 93  44 25 64
83 67 93  43 25 63
82 66 92  42 24 62
81 64 92  41 23 61
80 63 91  40 22 60
79 62 91  39 21 59
78 60 90  38 21 58
77 59 89  37 20 57
76 58 89  36 19 56
75 57 88  35 18 55
74 55 88  34 18 54
73 54 87  33 17 53
72 53 86  32 16 52
71 52 86  31 16 51
70 51 85  30 15 50
69 50 84 
68 48 84  25 12 43
67 47 83  20 9 37
66 46 82  15 6 30
65 45 82  10 4 22
64 44 81  5 2 13
63 43 80  0 0 0
62 42 79 
61 41 78 

AMC correlations. Using Eq. 5-7, Hawkins et al [8] have expressed the values in Table 5-4 in terms of potential maximum retention. They correlated the values of potential maximum retention for AMC I and III with those of AMC II and found the following ratios to be a good approximation:

SI          SII
____  ≅  ____  ≅ 2.3
SII          SIII
(5-12)

This led to the following relationships:

                        CNII
CNI  =  ____________________
                2.3 - 0.013 CNII
(5-13)

                           CNII
CNIII  =  _______________________
                 0.43 + 0.0057 CNII
(5-14)

which can be used in lieu of Table 5-4 to calculate runoff curve numbers for AMC I and AMC III in terms of the AMC II value.

Estimation of Runoff Curve Numbers from Data

The runoff curve number method was developed primarily for design applications in ungaged catchments and was not intended for simulation of actual recorded hydrographs. In the absence of data, the nationwide tables (Table 5-3) are generally applicable. Where rainfall-runoff records are available, estimations of runoff curve numbers can be obtained directly from data. These values complement and in certain cases may even replace the information obtained from tables.

To estimate runoff curve numbers from data, it is necessary to assemble corresponding rainfall-runoff data sets for several events occurring individually. As far as possible, the events should be of constant intensity and should uniformly cover the entire catchment. The selected set should encompass a wide range of antecedent moisture conditions, from dry to wet. In principle, daily rainfall-runoff data corresponding to the annual floods at a site would result in runoff curve numbers emulating those obtained in the method's original development. Thus, a recommended procedure is to select events that correspond to annual floods. In the absence of a long annual flood series, less selective criteria have been used for candidate storm events, including those of return period less than 1 yr. This choice results in considerable more data for analysis, as well as in curve numbers which are slightly higher than those obtained using an annual flood series. The choice of frequency for candidate storm events is the subject of continuing research.

For each event, a value of P, total rainfall depth, is identified. The associated direct runoff hydrograph is integrated to obtain the direct runoff volume. This runoff volume is divided by the catchment area to obtain Q, the direct runoff depth (in centimeters or inches). The values of P and Q are plotted on Fig. 5-2 and a corresponding value of CN is identified. The procedure is repeated for all events, and a CN value is obtained for each event, as shown in Fig. 5-3. In theory, the AMC II runoff curve number is that which separates the data into two equal groups, with half of the data plotting above the line and half below it. The AMC I runoff curve number is the curve number that envelopes the data from below. The AMC III runoff curve number is the curve number that envelopes the data from above (see Fig. 5-3).

Estimation of runoff curve numbers from measured data.

Figure 5-3  Estimation of runoff curve numbers from measured data.

Assessment of Runoff Curve Number Method

The positive features of the runoff curve number method are its simplicity and the fact that runoff curve numbers are related to the major runoff producing properties of the watershed, such as soil type, vegetation type and treatment, surface condition, and antecedent moisture. The method is used in practice to determine runoff depths based on rainfall depths and curve numbers, with no explicit account of rainfall intensity and duration.

A considerable body of experience has been accumulated on the runoff curve number method. Publications continue to appear in the literature either to augment the already extensive experience or to examine critically the applicability of the method to individual situations. For best results, however. the method should be used judiciously, with particular attention paid to its capabilities and limitations.

Experience with the method has shown that results are sensitive to curve number. This stresses the importance of an accurate estimation of curve number to minimize the variance in runoff determinations. The standard tables provide helpful guidelines, but local experience is recommended for increased accuracy. Typical runoff curve numbers used in design are in the range 50 ≤ CN ≤ 98.

Closely associated with the method's sensitivity to runoff curve number is its sensitivity to antecedent moisture. Since runoff curve number varies with antecedent moisture, markedly different results can be obtained for each of the three levels of antecedent moisture. At first, this appears to be a limitation; however, closer examination reveals that runoff is indeed a function of antecedent moisture, with the method's sensitivity to AMC reflecting the conditions likely to prevail in nature. Hjelmfelt et al. [9] attached a probability meaning to AMC, with AMC I corresponding to 10 percent probability of exceedence, AMC II to 50 percent, and AMC III to 90 percent. This may help explain why practical enveloping curves to determine AMC I and AMC III usually do not encompass all the data.

The popularity of the runoff curve number method is largely due to its simplicity, although proper care is necessary to use the method correctly. The method is essentially a conceptual model to estimate storm runoff volume based on established hydrologic abstraction mechanisms, with the effect of antecedent moisture taken in a probability context. In practice, (average) AMC II describes a typical design condition. When warranted, other antecedent moisture conditions, including those intermediate between I, II, and III, may be considered. An example of regional practice is given in Table 5-5.


Table 5-5  Antecedent moisture condition versus design storm frequency.
Design Frequency Location
Coast Foothills Mountains Desert
5 - 35 y 1.5 2.5 2.0 1.5
35 - 150 y 2.0 3.0 3.0 2.0
Source : San Diego County Hydrology Manual.

Experience with the runoff curve number method has shown that the curve numbers obtained from Table 5-3 tend to be conservative (i.e., too high) for large catchments, especially those located in semiarid and arid regions. Often this is due to the fact that these large catchments have additional sources of hydrologic abstraction, in particular, channel transmission losses, not accounted for by the tables. In this case it is necessary to perform a separate evaluation of the effect of channel abstractions on the quantity of surface runoff.

While the applicability of the runoff curve number procedure appears to be independent of catchment scale, its indiscriminate use for catchments in excess of 250 km2 (100 mi2) without catchment subdivision is generally not recommended. The runoff curve number was originally developed by SCS for use in midsize rural watersheds. Subsequently, the method was applied to small and midsize urban catchments (the TR-55 method). Therefore, its extension to large basins requires considerable judgment.

 Example 5-2.

A certain catchment experiences 12.7 cm of total rainfall. The catchment is covered by pasture with medium grazing, and 32 percent of B soils and 68 percent of C soils. This event has been preceded by 6.35 cm of rainfall in the last 5 d. Following the SCS methodology, determine the direct runoff for the 12.7 cm rainfall event.


A fair hydrologic condition is chosen for pasture with medium grazing. From Table 5- 2 (c), the runoff curve numbers for pasture with fair hydrologic condition are CN = 69 for B soils, and CN = 79 for C soils. The applicable CN is a weighted value:

CN = (69 x 0.32) + 79 x 0.68) = 76 (5-15)

Since this event has been preceded by a substantial amount of moisture in the last few days, AMC III is chosen. From Table 5-4, for AMC II CN = 76, AMC III CN = 89. From Eq. 5-9 or Fig. 5-2, with CN = 89 and P = 12.7 mm (5 in.), a value of Q = 9.58 cm (3.77 in.) is obtained as the direct runoff for this event.


5.3  UNIT HYDROGRAPH

[TR-55 Method]   [Questions]   [Problems]   [References]      [Top]   [Midsize Catchments]   [Runoff Curve Number]  

The concept of unit hydrograph, attributed to Sherman [14], is used in midsize catchment analysis as a means to develop a hydrograph for a 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 1-h, 3-h, 6-h, 12-h, 24-h, or any other suitable duration. For midsize catchment analysis, 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 50-km2 catchment over a period of 2 h. The hydrograph measured at the catchment outlet would be the 2-h unit hydrograph for this 50-km2 catchment (Fig. 5-4).

Concept of a unit hydrograph

Figure 5-4  Concept of a unit hydrograph.

A unit hydrograph for a given catchment can be calculated either:

  1. Directly, by using rainfall-runoff data for selected events, or

  2. Indirectly, by using a synthetic unit hydrograph formula.

While both methods may be used for gaged catchments, the latter method is appropriate only 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:

  1. Superposition method, and

  2. S-hydrograph method.

Two assumptions are crucial to the development of the unit hydrograph. 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), as shown in Fig. 5-5 (a). This, of course, is possible only under the assumption that the time base remains constant regardless of runoff depth.

Concept of linearity

Figure 5-5 (a)  Unit hydrograph properties: Linearity.

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, as shown in Fig. 5-5 (b).

Concept of linearity

Figure 5-5 (b)  Unit hydrograph properties: Lagging.

The summation of the corresponding ordinates of these hydrographs (superposition) allows the calculation of the composite hydrograph, as shown in Fig. 5-5 (c). The procedure depicted in Fig. 5-5 (a), (b), and (c) 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. The composite hydrograph time base is the sum of the unit hydrograph time base minus the unit hydrograph duration plus the storm duration.

Concept of linearity

Figure 5-5 (c)  Unit hydrograph properties: Superposition.

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, however, the linear assumption provides a convenient means of calculating runoff response without the complexities associated with nonlinear analysis [1, 4, 15].

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 km2. Linsley et al. [10] mention an upper limit of 5000 km2 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 km2 and less than 250 km2. This certainly does not preclude the unit hydrograph technique from being applied to catchments larger than 250 km2, although overall accuracy is likely to decrease with an increase in catchment size.

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 raingages and a stream gage at the outlet, and adequate sets of corresponding rainfall-runoff data; see, for instance, Fig. 5-6.

The rainfall-runoff 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 effectively limits unit hydrograph development by the direct method to midsize catchments.

Concept of linearity

Figure 5-6  The ARS Walnut Gulch Experimental Watershed, near Tombstone, Arizona.

Catchment Lag. The concept of catchment lag, basin lag, or lag time is central to unit hydrograph analysis. Catchment lag is a measure of the time elapsed between the occurrence of unit rainfall and the occurrence of unit runoff. It 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. 5-7. The T2 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.

Alternate definitions of catchment lag

Figure 5-7  Alternate definitions of catchment lag.

In unit hydrograph analysis, the concept of catchment lag is used to characterize the catchment response time. Runoff volume must be conserved (i.e., runoff volume should equal one unit of effective rainfall depth). Therefore, short lags result in unit responses featuring high 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:

                L Lc     N
tl  =  C (_______)
                 S 1/2
(5-16)

in which tl = catchment lag; L = catchment length (length measured along the main stream from outlet to divide); Lc = 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 S2 channel slope (Section 2.3); and C and N are empirical parameters. The parameter L describes length, Lc 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 must be of about the same duration. The duration should lie between 10 percent to 30 percent of the catchment lag. The latter requirement implies that runoff response is of the subconcentrated type, with rainfall duration less than time of concentration. Indeed, subconcentrated flow is a characteristic of midsize catchments.

For increased accuracy, direct runoff should be in the range 0.5 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:

  1. Separation of the measured hydrograph into direct runoff hydrograph (DRH) and baseflow (BF), following the procedures explained below.

  2. Calculation of direct runoff volume (DRV) by integrating the direct runoff hydrograph (DRH).

  3. Calculation of direct runoff depth (DRD) by dividing the direct runoff volume (DRV) by the catchment area.

  4. Calculation of unit hydrograph (UH) ordinates by dividing the ordinates of the direct runoff hydrograph (DRH) by the direct runoff depth (DRD).

  5. Estimation of the unit hydrograph duration.

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, if any, 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 time-to-peak (Fig. 5-8). 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.

Procedures for baseflow separation

Figure 5-8  Procedures for baseflow separation.

A common assumption is that baseflow recedes at the same rate as prior to the storm until the peak discharge has passed, and then it gradually increases to the ending point P in the receding limb, as illustrated by line a in Fig. 5-8. If a stream and groundwater table are hydraulically connected (Fig. 5-9), water infiltrates during the rising limb, reducing baseflow, and exfiltrates during the receding limb, increasing baseflow, as shown by line b in Fig. 5-8 [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.

Hydraulically connected stream and water table.

Figure 5-9  Hydraulically connected stream and water table.

Other techniques for hydrograph separation and baseflow recession are described in Chapter 11. The development of a unit hydrograph by the direct method is illustrated by Example 5-3.

 Example 5-3.

A unit hydrograph is to be developed for a 37.8-km2 catchment with a lag time of 12 h. A 2-h rainfall produced the following streamflow data:

Time (h) 0 2 4 6 8 10 12 14 16 18 20 22 24
Streamflow (m3/s) 2 1 3 5 9 8 7 6 5 4 3 1 1

Develop a unit hydrograph for this catchment.


A summary of the calculations is shown in Table 5-6. Columns 1 and 2 show time and measured streamflow, respectively. Baseflow is established by examining the measured streamflow. Since the hydrograph rise starts at 2 h and ends at 22 h, a value of baseflow equal to 1 m3/s appears reasonable. (In practice, a more detailed analysis as described in Section 11.5 may be necessary).

Column 3 shows the ordinates of the DRH obtained by substracting baseflow from the measured streamflow. To calculate direct runoff depth, the DRH is integrated numerically following Simpson's rule. Simpson's coefficients are shown in Col. 4. Column 5 shows the weighted ordinates obtained by multiplying Col. 3 by Col. 4. Summing up the weighted ordinates (Col. 5), a value of 126 m3/s is obtained. Since the integration interval is 2 h, the DRV (according to Simpson's rule) is DRV = (126 m3/s × 7200 seconds)/ 3 = 302,400 m3. The DRD is obtained by dividing DRV by the catchment area (37.8 km2) to yield: DRD = 0.8 cm. The unit hydrographs ordinates (Col. 6) are calculated by dividing the DRH ordinates (Col. 3) by DRD. To verify the calculations, the unit hydrograph shown in Col. 6 is integrated by multiplying Col. 4 times Col. 6 to obtain Col. 7. The sum of Col. 7 is 157.5 m3/s. It is verified that the ratio of DRV to unit hydrograph volume is indeed 0.8; i.e., (126 / 157.5) = 0.8. Finally, it is confirmed that the unit hydrograph duration (2 h) is an appropriate percentage (17 percent) of the lag time (12 h).

Table 5-6  Development of Unit Hydrograph:  Direct Method.
(1) (2) (3) (4) (5) (6) (7)
Time
(h)
Streamflow
(m3/s)
DRH
(m3/s)
Simpson's
coefficients
Volume UH
(m3/s)
Verification
0 2 __ __ __ __ __
2 1 0 1 0 0.00 0.00
4 3 2 4 8 2.50 10.00
6 5 4 2 8 5.00 10.00
8 9 8 4 32 10.00 40.00
10 8 7 2 14 8.75 17.50
12 7 6 4 24 7.50 30.00
14 6 5 2 10 6.25 12.50
16 5 4 4 16 5.00 20.00
18 4 3 2 6 3.75 7.50
20 3 2 4 8 2.50 10.00
22 1 0 1 0 0.00 0.00
24 1 __ __ __ __ __
Sum 126 157.50


Development of Unit Hydrographs:  Indirect Method

In the absence of rainfall-runoff data, unit hydrographs can be derived by synthetic means. A synthetic unit hydrograph is derived following an established formula, without the need for rainfall-runoff 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, the volume is equal to (Fig. 5-10):

         Qp Tbt
V = _________ = A × (1)
             2
(5-17)

in which V = volume under the triangular unit hydrograph; Qp = peak flow; Tbt = time base of the triangular unit hydrograph; A = catchment area; and (1) = one unit of runoff depth.

Triangular unit hydrograph.

Figure 5-10  Triangular unit hydrograph.

From Eq. 5-17:

           2A
Qp = ______
           Tbt
(5-18)

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 variable in synthetic unit hydrograph analysis.

Several methods are available for the calculation of synthetic unit hydrographs. Two widely used methods, the Snyder and the Natural Resources Conservation Service (NRCS) methods, are described here. The Clark unit hydrograph, 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:

tl = Ct  (L Lc) 0.3 (5-19)

in which tl = catchment or basin lag, in hours, L = length along the mainstream from outlet to divide, Lc = length along the mainstream from outlet to point closest to catchment centroid, and Ct = a coefficient accounting for catchment gradient and associated catchment storage. With distances L and Lc in kilometers, Snyder gave values of Ct varying in the range 1.35 - 1.65, with a mean of 1.5. With distances L and Lc in miles, the corresponding range of Ct is 1.8 - 2.2, with a mean of 2.

Snyder's formula for peak flow is:

          Cp A
Qp = _______
             tl
(5-20)

which when compared with Eq. 5-18 reveals that

             2
Cp = _______
           Tbt
          _____
             tl
(5-21)

is an empirical coefficient relating triangular time base to lag. Snyder gave values of Cp in the range 0.56 to 0.69, which are associated with Tbt /tl ratios in the range 3.57 to 2.90. The lower the value of Cp (i.e., the lower the peak flow), the greater the value of Tbt /tl and the greater the capability for catchment storage.

In SI units, Snyder's peak flow formula is:

           2.78 Cp A
Qp = _____________
                 tl
(5-22)

in which Qp = unit hydrograph peak flow corresponding to 1 cm of effective rainfall, in cubic meters per second; A = catchment area, in square kilometers; and tl = lag, in hours. In U.S. customary units, Snyder's peak flow formula is

           645 Cp A
Qp = ____________
                  tl
(5-23)

in which Qp = unit hydrograph peak flow corresponding to 1 in. of effective rainfall in cubic feet per second; A = catchment area in square miles; and tl = lag in hours.

In Snyder's method, the unit hydrograph duration is a linear function of the lag:

tr = (2/11) tl (5-24)

in which tr = 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. 5-24. Therefore, he devised a formula to increase the lag in order to account for the increased storm duration. This led to:

                  tR - tr
tlR = tl  +  ________
                     4
(5-25)

in which tlR is the adjusted lag corresponding to a duration tR.

Assuming uniform effective rainfall for simplicity, the unit hydrograph time-to-peak is equal to one-half of the storm duration plus the lag (Fig. 5-7). Therefore, the time-to-peak in terms of the lag is:

tp = (12/11) tl (5-26)

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:

Tb = 72  +  3tl (5-27)

in which Tb = actual unit hydrograph time base (including interflow), in hours and tl = lag, in hours. For a 24-h lag, this formula gives Tb /tl = 6, which is a reasonable value considering that interflow is being included in the calculation. For smaller lags, however, Eq. 5-27 gives unrealistically high values of Tb /tl. For instance, for a 6-h lag, Tb /tl = 15. For midsize catchments, and excluding interflow, experience has shown that values of Tb /tp  around 5 (corresponding to values of Tb /tl around 5.45) may be more realistic.

The Snyder method gives peak flow (Eq. 5-22), time-to-peak (Eq. 5-26), and time base (Eq. 5-27) 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 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 (Fig. 5-11).

Snyder's distribution chart for plotting unit hydrograph orinates

Figure 5-11  Snyder's distribution chart for plotting unit hydrograph ordinates [17].

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]:

                   6.33
W50 = _______________
              (Qp /A)1.08
(5-28)

                   3.58
W75 = ______________
             (Qp /A)1.08
(5-29)

in which W50 = width of unit hydrograph at 50 percent of peak discharge in hours; W75 = width of unit hydrograph at 75 percent of peak discharge in hours; Qp = peak discharge in cubic meters per second; and A = catchment area in square kilometers (Fig. 5-12). These time widths should be proportioned in such a way that one-third is located before the peak and two-thirds after the peak.

An intensity-duration-frequency curve.

Figure 5-12  Snyder's synthetic unit hydrograph widths:  W50 and W75.

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 fan-shaped catchments than for those of highly irregular shape. He recommended that the coefficients Ct and Cp be determined on a regional basis.

The examination of Eq. 5-19 reveals that Ct  is largely a function of catchment slope, since both length and shape have already been accounted for in L and Lc , respectively. Since Eq. 5-19 was derived empirically, the actual value of Ct  depends on the units of L and Lc. Furthermore, Eq. 5-19 implies that when the product LLc is equal to 1, the lag is equal to Ct. Since for two catchments of the same size, lag is a function of slope, it is unlikely that Ct is a constant. To give an example, an analysis of 20 catchments in the north and middle Atlantic United States led to [19]: Ct = 0.6/S1/2. A similar conclusion is drawn from Eq. 5-16. Therefore, values of Ct have regional meaning, in general being a function of catchment slope. Values of Ct quoted in the literature reflect the natural variability of catchment slopes.

The parameter Cp is dimensionless and varies within a narrow range. In fact, it is readily shown that the maximum possible value of Cp is 11/12. Since triangular time base cannot be less than twice the time-to-peak (otherwise, runoff diffusion would be negative, clearly a physical impossibility), it follows that in the limit (i.e., in the absence of runoff diffusion), Tbt = 2tp; and, therefore, Cp = tl / tp = 11/12. In practice, triangular time base is usually about 3 times the time-to-peak. For Tbt = 3tp, a similar calculation leads to: Cp = 0.61, which lies approximately in the middle of Snyder's data (0.56-0.69).

Since Ct increases with catchment storage and Cp decreases with catchment storage, the ratio Ct /Cp can be directly related to catchment storage. Furthermore, the reciprocal ratio (Cp /Ct) 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]. The calculation of Snyder's synthetic unit hydrograph is illustrated by the following example.

 Example 5-4.

Calculate the properties of a Snyder unit hydrograph using the following data: L = 25 km, Lc = 10 km, A = 400 km2, Ct = 1.5, and Cp = 0.61.


Using Eq. 5-19, tt = 7.86 h. From Eq. 5-21, solving for Tbt: Tbt = 25.77 h. Using Eq. 5-22, Qp = 86.3 m3/s. Using Eq. 5-24, tr = 1.43 h. Using Eq. 5-26, tp = 8.57 h. The time base calculated by Eq. 5-27 is Tb = 95.58 h. This is too high a value. Instead, assume time Tb = 5tp; then: Tb = 42.85 h. Using Eq. 5-28, W50 = 33.2 h; using Eq. 5-29, W75 = 18.8 h. The actual unit hydrograph is drawn primarily on the basis of Qp, tp and Tb, with the remaining values used as guidelines.


NRCS Synthetic Unit Hydrograph

The NRCS synthetic unit hydrograph is the dimensionless unit hydrograph developed by Victor Mockus in the 1950s [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 time-to-peak, Tbt /tp = 8/3, which implies that Cp = 0.6875. Unlike Snyder's method, the NRCS method uses a constant ratio of actual time base to time-to-peak, Tb/tp = 5. In addition, it uses a dimensionless hydrograph function to provide a standard unit hydrograph shape.

To calculate catchment lag (the T2 lag), the NRCS method uses the following two methods:

  1. The curve number method, and

  2. The velocity method.

The curve number method is limited to catchments of areas less than 8 km2 (2000 ac), although recent evidence suggests that it may be extended to catchments up to 16 km2 (4000 ac) [11]. In the curve number method, the lag is expressed by the following formula:

         L0.8 ( 2540 - 22.86CN )0.7
tl = _____________________________
              14104 CN 0.7Y 0.5
(5-30)

in which tl = 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:

          L0.8 ( 1000 - 9CN )0.7
tl = __________________________
            1900 CN 0.7Y 0.5
(5-31)

The velocity method is used for catchments larger than 8 km2, or for curve numbers outside of the range 50 - 95. The main stream is divided into reaches, and the 2-y flood (or, alternatively, the bankfull discharge) is estimated. In certain cases it may be desirable to use discharges corresponding to 10-y frequencies or more. The mean velocity is computed, and the reach time of concentration is calculated by using the reach valley length (straight distance). The sum of the time of concentration for all reaches is the time of concentration for the catchment. The lag is estimated as follows:

  tl          6
____ = _____
 tc         10
(5-32)

in which tl = lag, and tc = time of concentration. NRCS experience has shown that this ratio is typical of midsize catchments [21].

In the NRCS method, the ratio of time-to-peak to unit hydrograph duration is fixed at

 tp
___ = 5
 tr
(5-33)

which is close to Snyder's ratio of 6. Assuming uniform effective rainfall for simplicity, the time-to-peak is, by definition, equal to

         tr
tp = ____ + tl
         2
(5-34)

Eliminating tr from Eqs. 5-33 and 5-34, leads to

 tp       10
___ = _____
 tl         9
(5-35)

Therefore:

 tr         2
___ = ____
 tl         9
(5-36)

and

 tr         2
___ = _____
 tl        15
(5-37)

To derive the NRCS unit hydrograph peak flow formula, the ratio Tbt /tp = 8/3 is used in Eq. 5-18, leading to

          (3/4) A
Qp = _________
              tp
(5-38)

In SI units, the peak flow formula is:

          2.08 A
Qp = _________
              tp
(5-39)

in which Qp = unit hydrograph peak flow for 1 cm of effective rainfall in cubic meters per second; A = catchment area, in square kilometers; and tp = time-to-peak, in hours. In U.S. customary units, the NRCS peak flow formula is:

          484 A
tp = __________
             tp
(5-40)

in which Qp = unit hydrograph peak flow for 1 in. of effective rainfall; A = catchment area, in square miles; and tp = time-to-peak, in hours.

Given Eqs. 5-32 and 5-34, the time-to-peak can be readily calculated as follows: tp = 0.5tr + 0.6tc. Once tp and Qp have been determined, the NRCS dimensionless unit hydrograph (Fig. 5-13) 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 (Tbt / tp = 8/3) used to develop the peak flow value. The dimensionless unit hydrograph has a value of Tb / tp = 5. Values of NRCS dimensionless unit hydrograph ordinates at intervals of 0.2 (t /tp) are given in Table 5-7. The calculation of an NRCS synthetic unit hydrograph is illustrated by the following example.

NRCS dimensionless unit hydrograph

Figure 5-13  NRCS dimensionless unit hydrograph [21].

Table 5-7  NRCS dimensionless unit hydrograph ordinates.
t / tp Q / Qp t / tp Q / Qp t / tp Q / Qp t / tp Q / Qp t / tp Q / Qp
0.0 0.00
0.2 0.10 1.2 0.93 2.2 0.207 3.2 0.040 4.2 0.0100
0.4 0.31 1.4 0.78 2.4 0.147 3.4 0.029 4.4 0.0070
0.6 0.66 1.6 0.56 2.6 0.107 3.6 0.021 4.6 0.0030
0.8 0.93 1.8 0.39 2.8 0.077 3.8 0.015 4.8 0.0015
1.0 1.00 2.0 0.28 3.0 0.055 4.0 0.011 5.0 0.0000

 Example 5-5.

Calculate the NRCS synthetic unit hydrograph for a 6.42 km2 catchment with the following data: Hydraulic length L = 2204 m; runoff curve number CN = 62; average land slope Y = 0.02.


Using Eq. 5-30, tl = 1.8 h. Therefore: tr = 0.4 h; tp = 2 h; Tb = 10 h. Using Eq. 5-39, Qp = 6.68 m3/s. Using Table 5-7, the ordinates of the unit hydrograph are calculated as shown in Table 5-8.


Table 5-8  Unit hydrograph ordinates: Example 5-5
(Qp = 6.68 m3; tp = 2 h).

t /tp Q /Qp t
(h)
Q
(m3/s)
0.0 0.00 0.0 0.000
0.2 0.10 0.4 0.668
0.4 0.31 0.8 2.071
0.6 0.66 1.2 4.410
0.8 0.93 1.6 6.212
1.0 1.00 2.0 6.680
1.2 0.93 2.4 6.212
1.4 0.78 2.8 6.212
1.6 0.56 3.2 3.740
1.8 0.39 3.6 2.605
2.0 0.28 4.0 1.870
2.2 0.207 4.4 1.382
2.4 0.147 4.8 0.982
2.6 0.107 5.2 0.714
2.8 0.077 5.6 0.514
3.0 0.055 6.0 0.367
3.2 0.040 6.4 0.267
3.4 0.029 6.8 0.194
3.6 0.021 7.2 0.140
3.8 0.015 7.6 0.100
4.0 0.011 8.0 0.073
4.2 0.010 8.4 0.067
4.4 0.007 8.8 0.047
4.6 0.003 9.2 0.020
4.8 0.0015 9.6 0.010
5.0 0.0000 10.0 0.000


Two-parameter NRCS method. The NRCS method provides a unit hydrograph shape and, therefore, leads to more reproducible results than the Snyder method. However, the ratio Tbt /tp is kept constant and equal to 8/3. Also, when lag is calculated by the velocity method, the ratio tl /tc 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 Tbt /tp other than 8/3 may lead to other shapes of unit hydrographs. Larger values of Tbt /tp (equivalent to lower values of Cp in the Snyder method) imply greater catchment storage. Therefore, since the NRCS method fixes the value of Tbt /tp, it should be limited to midsize catchments in the lower end of the range (2.5 - 250 km2). The Snyder method, however, by providing a variable Tbt /tp may be used for larger catchments [10].

Efforts to extend the range of applicability of the NRCS method have led to the relaxation of the Tbt /tp ratio. It can be shown that the ratio p of volume-to-peak (volume under the rising limb of the triangular unit hydrograph) to the triangular unit hydrograph volume is the reciprocal of the ratio Tbt /tp. For instance, in the case of the standard NRCS synthetic unit hydrograph, Tbt /tp = 8/3, and p = 3/8. In terms of p, Eq. 5-38 can be expressed as follows:

          2 p A
tp = __________
             tp
(5-41)

which converts the NRCS method into a two-parameter 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:

  1. Catchment lag,

  2. Ratio of triangular time base to time-to-peak, and

  3. Ratio of actual time base to time-to-peak.

In addition, the NRCS method 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 X-hour unit hydrograph is going to be used with a storm hyetograph defined at Y-hour intervals, it is necessary to convert the X-hour unit hydrograph into a Y-hour 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:

  1. Superposition method, and

  2. S-hydrograph method.

The superposition method converts an X-hour unit hydrograph into a nX-hour unit hydrograph, in which n is an integer. The S-hydrograph method converts an X-hour unit hydrograph into a Y-hour unit hydrograph, regardless of the ratio between X and Y.

Superposition Method. This method allows the conversion of an X-hour unit hydrograph into a nX-hour unit hydrograph, in which n is an integer. The procedure consists of lagging nX-hour 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 nX-hour unit hydrograph. The volume under X-hour and nX-hour unit hydrographs is the same. If Tb is the time base of the X-hour hydrograph, the time base of the nX-hour hydrograph is equal to Tb + (n - 1)X. The procedure is illustrated by the following example.

 Example 5-6.

Use the superposition method to calculate the 2-h and 3-h unit hydrographs of a catchment, based on the following 1-h unit hydrograph:

Time (h) 0 1 2 3 4 5 6 7 8 9 10 11 12
Flow (m3/s) 0 100 200 400 800 700 600 500 400 300 200 100 0


The calculations are shown in Table 5-9. Column 1 shows the time in hours. Column 2 shows the ordinates of the 1-h unit hydrograph. Column 3 shows the ordinates of the 1-h unit hydrograph, lagged 1 h. Column 4 shows the ordinates of the 1-h unit hydrograph, lagged 2 h. Column 5 shows the ordinates of the 2-h unit hydrograph, obtained by summing the ordinates of Cols. 2 and 3 and dividing by 2. Column 6 shows the ordinates of the 3-h unit hydrograph, obtained by summing the ordinates of Cols. 2, 3, and 4, and dividing by 3. The sum of ordinates for 1-h, 2-h, and 3-h unit hydrographs is the same: 4300 m3/s. The time base of the 1-h unit hydrograph is 12 h, whereas the time base of the 2-h unit hydrograph is 13 h and the time base of the 3-h unit hydrograph is 14 h.

Table 5-9  Change in unit hydrograph duration, Superposition method:  Example 5-6.
(1) (2) (3) (4) (5) (6)
Time
(h)
1-h
UH
Lagged
1 h
Lagged
2 h
2-h
UH
3-h
UH
0 0 0 0 0 0
1 100 0 0 50 33
2 200 100 0 150 100
3 400 200 100 300 233
4 800 400 200 600 467
5 700 800 400 750 633
6 600 700 800 650 700
7 500 600 700 550 600
8 400 500 600 450 500
9 300 400 500 350 400
10 200 300 400 250 300
11 100 200 300 150 200
12 0 100 200 50 100
13 0 0 100 0 33
14 0 0 0 0 0
Sum 4300 4300 4299


S-Hydrograph Method. The S-hydrograph method allows the conversion of an X-hour unit hydrograph into a Y-hour unit hydrograph, regardless of the ratio between X and Y. The procedure consists of the following steps:

  1. Determine the X-hour S-hydrograph (Fig. 5-14). Note that the X-hour S-hydrograph is derived by accumulating the unit hydrograph ordinates at intervals equal to X.

  2. Lag the X-hour S-hydrograph by a time interval equal to Y  hours.

  3. Subtract ordinates of the two previous S-hydrographs.

  4. Multiply the resulting hydrograph ordinates by X/Y to obtain the Y-hour unit hydrograph.

Sketch of unit hydrograph and corresponding S-hydrograph.

Figure 5-14  Sketch of unit hydrograph and corresponding S-hydrograph.

The volume under X-hour and Y-hour unit hydrographs is the same. If Tb is the time base of the X-hour unit hydrograph, the time base of the Y-hour unit hydrograph is Tb - X + Y. The procedure is illustrated by the following example.

 Example 5-7.

For the 2-h unit hydrograph calculated in the previous example (Example 5-6), derive the 3-h unit hydrograph by the S-hydrograph method. Use this 3-h unit hydrograph to derive the 2-h unit hydrograph, confirming the applicability of the S-hydrograph method, regardless of the ratio between X and Y.


The calculations are shown in Table 5-10.

  • Column 1 shows the time in hours.

  • Column 2 shows the 2-h unit hydrograph ordinates calculated in the previous example.

  • Column 3 is the 2-h S-hydrograph, obtained by accumulating the ordinates of Col. 2 at intervals of X = 2 h.

  • Column 4 is the S-hydrograph of Col. 3 lagged Y = 3 h.

  • Column 5 is equal to Col. 3 minus Col. 4.

  • Column 6 is the product of Col. 5 times X/Y = 2/3. Column 6 is the 3-h unit hydrograph. Its sum is 4299 m3/s, the same as the sum of Col. 2, confirming that it contains a unit volume. The time base of the 2-h unit hydrograph is 13 h, and the time base of the 3-h unit hydrograph is 14 h.

  • Column 7 is the 3-h S-hydrograph, obtained by accumulating the ordinates of Col. 6 at intervals of X = 3 h.

  • Column 8 is the S-hydrograph of Col. 7 lagged Y = 2 h.

  • Column 9 is equal to Col. 7 minus Col. 8.

  • Column 10 is the product of Col. 9 times X/Y = 3/2. Column 10 is the 2-h unit hydrograph, and it is confirmed to be the same as that of Col. 2.

Table 5-10  Change in unit hydrograph duration, S-hydrograph method: Example 5-7.
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
Time
(h)
2-h
UH
2-h
SH
Lagged
3 h
Col.3
- Col.4
3-h
UH
3-h
SH
Lagged
2 h
Col.7
- Col.8
2-h
UH
0 0 0 0 0 0 0 0 0 0
1 50 50 0 50 33 33 0 33 50
2 150 150 0 150 100 100 0 100 150
3 300 350 0 350 233 233 33 200 300
4 600 750 50 700 467 500 100 400 600
5 750 1100 150 950 633 733 233 500 750
6 650 1400 350 1050 700 933 500 433 650
7 550 1650 750 900 600 1100 733 367 550
8 450 1850 1100 750 500 1233 933 300 450
9 350 2000 1400 600 400 1333 1100 233 350
10 250 2100 1650 450 300 1400 1233 167 250
11 150 2150 1850 300 200 1433 1333 100 150
12 50 2150 2000 150 100 1433 1400 33 50
13 0 2150 2000 50 33 1433 1433 0 0
14 0 2150 2150 0 0 1433 1433 0 0
Sum 4300 4299 4300

calculator image

ONLINE CALCULATION. Using ONLINE S-HYDROGRAPH, the calculated 3-h unit hydrograph is the same as the one calculated in Col. 6 of Table 5-10.


Minor errors in unit hydrograph ordinates may often lead to errors (i.e., undesirable oscillations) in the resulting S-hydrograph. In this case, a certain amount of smoothing may be required to achieve the typical S-shape (Fig. 5-14).

Convolution and Composite Hydrographs

The procedure to derive a composite or flood hydrograph based on a unit hydrograph 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 Tb is the time base of the X-hour unit hydrograph and the storm consists of n X-hour intervals, the time base of the composite hydrograph is equal to Tb - X + nX = Tb + (n - 1)X. The convolution procedure is illustrated by the following example.

 Example 5-8.

Assume that the following 1-h unit hydrograph has been derived for a certain watershed:

Time (h) 0 1 2 3 4 5 6 7 8 9
Flow (m3/s) 0 100 200 400 800 600 400 200 100 0

A 6-h storm with a total of 5 cm of effective rainfall covers the entire watershed and is distributed in time as follows:

Time (h) 0 1 2 3 4 5 6
Effective rainfall (cm) 0.1 0.8 1.6 1.2 0.9 0.4

Calculate the composite hydrograph using the convolution technique.


The calculations are shown in Table 5-11.

  • Column 1 shows the time in hours.

  • Col. 2 shows the unit hydrograph ordinates in cubic meters per second.

  • Column 3 shows the product of the first-hour rainfall depth times the unit hydrograph ordinates.

  • Column 4 shows the product of the second-hour rainfall depth times the unit hydrograph ordinates, lagged 1 h with respect to Col. 3.

  • The computational pattern established by Cols. 3 and 4 is the same for Cols. 5 to 8.

  • Column 9, the sum of Cols. 3 through 8, is the composite hydrograph for the given storm pattern.

Table 5-11  Composite hydrograph by convolution:  Example 5-8.
(1) (2) (3) (4) (5) (6) (7) (8) (9)
Time
(h)
UH
( m3/s)
0.1 ×
UH
0.8 ×
UH
1.6 ×
UH
1.2 ×
UH
0.9 ×
UH
0.4 ×
UH
Composite
hydrograph
( m3/s)
0 0 0 __ __ __ __ __ 0
1 100 10 0 __ __ __ __ 10
2 200 20 80 0 __ __ __ 100
3 400 40 160 160 0 __ __ 360
4 800 800 320 320 120 0 __ 840
5 600 60 640 640 240 90 0 1670
6 400 40 480 1280 480 180 40 2500
7 200 20 320 960 960 360 80 2700
8 100 10 160 640 720 720 160 2410
9 0 0 80 320 480 540 320 1740
10 __ __ 0 160 240 360 240 1000
11 __ __ __ 0 120 180 160 460
12 __ __ __ __ 0 90 80 170
13 __ __ __ __ __ 0 40 40
14 __ __ __ __ __ __ 0 0
Sum 2800 14,000

The sum of Col. 2 is 2800 m3/s and is equivalent to 1 cm of net rainfall. The sum of Col. 9 is verified to be 14,000 m3/s, and, therefore, the equivalent of 5 cm of effective rainfall. The time base of the composite hydrograph is Tb = 9 + (6 - 1) × 1 = 14 h.

calculator image

ONLINE CALCULATION. Using ONLINE CONVOLUTION, using effective rainfall (CN = 100), the calculated flood hydrograph is the same as that of Col. 9 of Table 5-11.


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 5-11). With m = number of nonzero unit hydrograph ordinates, n = number of intervals of effective rainfall, and N = number of nonzero storm hydrograph ordinates, the following relation holds:

N = m + n - 1 (5-42)

Therefore:

m = N - n + 1 (5-43)

By elimination and back substitution, the following formula can be developed for the unit hydrograph ordinates ui as a function of storm hydrograph ordinates qi  and effective rainfall depths rk :

                    k = 2, n
           qi  _   Σ     uj rk
                    j = i - 1, 1
ui  =  _______________________
                         r1
(5-44)

for i varying from 1 to m. In the summation term, j decreases from j -1 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. 5-44.

The uncertainties involved have led to the use of the least square technique. In this technique, rainfall-runoff 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].

 Example 5-9.

Use Eq. 5-44 and the storm hydrograph obtained in the previous example to calculate the unit hydrograph.


Since N = 13 and n = 6: m = 8.

  • The first ordinate is: u1 = q1/ r1 = 10 / 0.1 = 100

  • The second ordinate is: u2 = (q2 - u1r2) = (100 - 100 × 0.8) / 0.1 = 200

  • The third ordinate is:
    u3 = [q3 - (u2r2 + u1r3)] / r1 = [360 - (200 × 0.8 + 100 × 1.6)] / 0.1 = 400

  • The fourth ordinate is:
    u4 = [q4 - (u3r2 + u2r3 + u1r4) ] / r1 = [840 - (400 × 0.8 + 200 × 1.6 + 100 × 1.2)] / 0.1 = 800

  • The remaining ordinates are obtained in a similar way.


5.4  TR-55 METHOD

[Questions]   [Problems]   [References]      [Top]   [Midsize Catchments]   [Runoff Curve Number]   [Unit Hydrograph]  

The TR-55 method is a collection of simplified procedures developed by the USDA Natural Resources Conservation Service to calculate peak discharges, storm hydrographs, and stormwater storage volumes in small/midsize urban catchments [22]. It consists of three methodologies:

  1. A graphical method for flood peak discharge determination,

  2. A tabular method for hydrograph computation, and

  3. A detention-basin method to size stormwater storage facilities.

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 TR-20 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 TR-55 does not specify catchment size, the graphical method is limited to catchments with time of concentration in the range 0.1-10 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 0.1-2 h.

The graphical method is described in this section. The tabular method is described in the original reference [22]. The detention-basin method is described in Section 8.5.

TR-55 Storm, Catchment and Runoff Parameters

Rainfall in TR-55 is described in terms of total rainfall depth and one of four standard 24-h temporal rainfall distributions: Type I, Type IA, Type II, and Type III. These distributions are shown in Fig. 5-15. Type I applies 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, as shown in Fig. 5-16.

An intensity-duration-frequency curve.

Figure 5-15  NRCS 24-h rainfall distributions [22].

An intensity-duration-frequency curve.

Figure 5-16  Approximate geographical boundaries for NRCS rainfall distributions [22].

The duration of these rainfall distributions is 24 h. This constant duration was selected because most rainfall data is reported on a 24-h basis. Rainfall intensities corresponding to durations shorter than 24 h are contained within the NRCS distributions. For instance, if a 10-y 24-h rainfall distribution is used, the 1-h period with the most intense rainfall corresponds to the 10-y 1-h rainfall depth.

TR-55 uses the runoff curve number method (Section 5.1) to abstract total rainfall depth and calculate runoff depth. The abstraction procedure follows established guidelines [21], with extensions to account for curve numbers applicable to urban areas. In addition, TR-55 includes procedures to determine the time of concentration for the following types of surface flow:

  1. Overland flow,

  2. Shallow concentrated flow, and

  3. Streamflow.

Shallow concentrated flow is a type of flow of characteristics in between those of overland flow and streamflow.

Applicability of TR-55

When using TR-55, there is a choice between graphical or tabular method. The graphical method gives only a peak discharge, whereas the tabular method provides a flood hydrograph. 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 TR-55 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, TR-55 defines two types of areas:

  1. Pervious, and

  2. Impervious.

Once pervious and impervious areas are delineated, the percent imperviousness can be determined. Impervious areas are of two kinds:

  1. Connected, and

  2. Unconnected.

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:

  • If runoff from it flows directly into the drainage system, or

  • If runoff from it occurs as shallow concentrated flow which runs first over a pervious area and then into a drainage system.

An impervious area is considered unconnected if runoff from it spreads over a pervious area as overland (sheet) flow.

Table 5-3 (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:

  1. Impervious areas are directly connected to the drainage system and have a CN = 98; and

  2. Pervious areas are considered equivalent to pasture (open space in Table 5-3 (a)) in good hydrologic condition.

Tables 5-3 (b), (c), and (d) show runoff curve numbers for cultivated agricultural lands, other agricultural lands, and arid and semiarid rangelands, respectively.

Figure 5-17 is used in lieu of Table 5-3 (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 5-3 (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 5-3 (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 Fig. 5-17 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).

Composite <i>CN</i> as a function of impervious area percent and pervious area <i>CN</i>

Figure 5-17  Composite CN as a function of impervious area percent and pervious area CN [22].

Figure 5-18 is used to determine a composite CN 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. 5-17 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 Fig. 5-18, enter the right-side figure with percent imperviousness to the line matching the ratio of unconnected impervious to total impervious area. Then, move horizontally to the left-side figure to match the pervious area CN, and vertically down to find the composite CN. For example, for a 1/2-acre lot with 20 percent imperviousness, 75 percent of which is unconnected, and pervious CN = 61, the composite CN (from Fig. 5-18) is: CN = 66. If all of the impervious area is connected (i.e., zero percent unconnected), the resulting CN (from Fig. 5-17) is: CN = 69. This value matches the example of the previous paragraph.

Composite <i>CN</i> as a function of total impervious area percent, ratio of <br>unconnected impervious area to total impervious area, and pervious area <i>CN</i>

Figure 5-18  Composite CN as a function of total impervious area percent, ratio of
unconnected impervious area to total impervious area, and pervious area CN [22].

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, TR-55 uses the following formula for travel time:

          0.007 (nL)0.8
tt = _________________
            P2 0.5S 0.4
(5-45)

in which tt, = travel time, in hours; n = Manning n; L = flow length, in feet; P2 = 2-y 24-h rainfall depth in inches; and S = average land slope, in feet per foot. In SI units, this equation is:

         0.0288 (nL) 0.8
tt = __________________
             P2 0.5S 0.4
(5-46)

in which L is given in meters; P2 in centimeters; S in meters per meter; and the remaining terms are the same as in Eq. 5-45. TR-55 values of Manning n applicable to overland flow are given in Table 5-12.

Table 5-12  TR-55 Manning n values for overland flow [22].
Surface Description Manning n
Smooth surfaces (concrete, asphalt, gravel, or bare soil) 0.011
Fallow (no residue) 0.05
Cultivated ground
  Residue cover less than or equal to 20% 0.06
  Residue cover greater than 20% 0.17
Grass
  Short Prairie 0.15
  Dense 0.24
  Bermuda 0.41
  Range (natural) 0.13
Woods
  Light underbrush 0.40
  Dense underbrush 0.80
Note: Dense grass includes weeping lovegrass, bluegrass, buffalo grass, blue gamma grass, native grass mixture, alfalfa, and the like.

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. 5-19. For streamflow, the Manning equation (Eq. 2-65) 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].

Average velocities for estimating travel time for shallow concentrated flow.

Figure 5-19  Average velocitites for estimating travel time for shallow concentrated flow [22].

TR-55 Graphical Method

The TR-55 graphical method calculates peak discharge based on the concept of unit peak flow. The unit peak flow is the peak flow per unit area, per unit runoff depth. In TR-55, unit peak flow is a function of the following variables:

  1. Time of concentration,

  2. Ratio of initial abstraction to total rainfall, and

  3. Storm type.

Peak discharge is calculated by the following formula:

Qp = qu A Q F (5-47)

in which Qp = peak discharge in L3T-1 units; qu = unit peak flow in T-1 units; A = catchment area in L2 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: (1) overland flow, (2) shallow concentrated flow, or (3) streamflow. The runoff curve number is determined from either Table 5-3, Fig. 5-17, or Fig. 5-18. A flood frequency is selected, and an appropriate rainfall map (depth-duration-frequency) is used to determine the rainfall depth for the 24-h duration and the chosen frequency. With the rainfall depth P and the CN, the runoff depth Q is determined using either Fig. 5-2, Eq. 5-8, or Eq. 5-9.

The initial abstraction is calculated by combining Eqs. 5-4 and 5-7 to yield:

         200
Ia = ______  -  2
         CN
(5-48)

in which Ia = initial abstraction, in inches. The equivalent SI formula is:

         508
Ia = ______  -  5.08
         CN
(5-49)

in which Ia is given in centimeters.

The surface storage correction factor F is obtained from Table 5-13 as a function of the percentage of pond and swamp areas.


Table 5-13  TR-55 surface storage correction factor F [22].
Percentage of pond
and swamp areas
Surface storage correction factor F
0.0 1.00
0.2 0.97
1.0 0.87
3.0 0.75
5.0 0.72
Note: Pond and swamp areas should be spread throughout the catchment.

With time of concentration tc, ratio Ia/P, and storm type (either I, IA, II, or III), Fig. 5-20 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 Ia/P different than those shown in Fig. 5-20. For values of Ia/P outside of the range shown in Fig. 5-20, the maximum (or minimum) value should be used.

Conversion to SI Units. To obtain unit peak flow in cubic meters per second per square kilometer per centimeter, the unit peak flow values obtained from Fig. 5-20 are multiplied by the factor 0.0043.

Unit peak discharge in TR-55 graphical method.

Figure 5-20 (a)  Unit peak discharge in TR-55 graphical method:  NRCS Type I rainfall distribution [22].

Unit peak discharge in TR-55 graphical method.

Figure 5-20 (b)  Unit peak discharge in TR-55 graphical method:  NRCS Type IA rainfall distribution [22].

Unit peak discharge in TR-55 graphical method.

Figure 5-20 (c)  Unit peak discharge in TR-55 graphical method:  NRCS Type II rainfall distribution [22].

Unit peak discharge in TR-55 graphical method.

Figure 5-20 (d)   Unit peak discharge in TR-55 graphical method:  NRCS Type III rainfall distribution [22].

Peak discharge is calculated by Eq. 5-47 as a function of unit peak flow, catchment area, runoff depth, and surface storage correction factor. The TR-55 graphical method is limited to runoff curve numbers greater than 40, with time of concentration in the range 0.1 to 10 h, and surface storage areas spread throughout the catchment and covering less than 5 percent of it. The computational procedure is illustrated by the following examples.

 Example 5-10.

Calculate the 10-y peak flow by the TR-55 graphical method using the following data: catchment area 4 km2; total impervious area 0.8 km2; unconnected impervious area 0.6 km2; pervious area curve number CN = 70; storm type II ; time of concentration 1.5 h; 10-y rainfall P = 9 cm; percentage of pond and swamp areas, 1 percent.


Since there are unconnected impervious areas and the total impervious area amounts to less than 30 percent of the catchment, Fig. 5-17 is used to calculate the composite curve number. With total impervious area (20 percent), ratio of unconnected impervious to total impervious (0.75), and pervious CN (70) areas, the composite curve number from Fig. 5- 17 is CN = 74. The runoff depth (Eq. 5-9) is Q = 3.23 cm. The initial abstraction (Eq. 5-49) is Ia = 1.78 cm, and the ratio Ia/P = 0.2. From Fig. 5-20 (c) (storm type II), time of concentation 1.5 h, and Ia/P = 0.2 , the unit peak flow is 250 ft3/ (s-mi2-in.) or 250 × 0.0043 = 1.075 m3/ (s-km2-cm). From Table 5-13, F = 0.87. From Eq. 5-47, with qu = 1.075 m3/ (s-km2-cm); A = 4 km2; Q = 3.23 cm; and F = 0.87, the peak discharge is Qp = 12.08 m3/s.

calculator image

ONLINE CALCULATION. Using ONLINE TR-55, the 10-y peak flow discharge for the given data is: Qp = 12.09 m3/s. This result agrees closely with the hand calculation.

 Example 5-11.

Calculate the 25-y peak flow by the TR-55 graphical method using the following data:

  • Urban watershed of area A = 1.5 mi2 ;

  • Surface flow is shallow concentrated, paved, hydraulic length L = 4,320 ft;

  • Slope S = 0.014;

  • 26 percent of the watershed is 1/3-acre lots, 30 percent impervious, hydrologic soil group B;

  • 42 percent of the watershed is 1/2-acre lots, with lawns in fair hydrologic condition, 36 percent impervious, hydrologic soil group C;

  • 32 percent of the watershed is l/2-acre lots, with lawns in good hydrologic condition, 24 percent total impervious, 50 percent unconnected, hydrologic soil group C;

  • Storm Type I;

  • 25-y rainfall P = 5 in., and

  • 0.2 percent pond and swamps areas.

From Fig. 5-18, the average velocity along the hydraulic length is v = 2.4 ft/s; therefore, the time of concentration is tc = L / v = 0.5 h. For the 26 percent subarea, with 1/3-acre lots, 30 percent impervious, the curve number is obtained directly from Table 5-3(a): CN = 72. For the 42 percent subarea, with 1/2-acre lots, 36 percent impervious, first the pervious area CN is obtained from Table 5-3 (a) (open space in fair hydrologic condition, soil group C): CN = 79; then, the composite CN is obtained from Fig. 5-17: CN = 86. For the 32 percent subarea, with 1/2- acre lots, 24 percent total impervious, 50 percent unconnected, first the pervious area CN is obtained from Table 5-3 (a) (open space in good hydrologic condition, soil group C): CN = 79; then, the composite CN is obtained from Fig. 5-17: CN = 78. The composite CN for the entire watershed is: CN = (0.26 × 72) + (0.42 × 86) + (0.32 × 78) = 80. The runoff depth (Eq. 5.8) is: Q = 2.9 in. The initial abstraction (Eq. 5-48) is: Ia = 0.5 in.; then, the ratio Ia /P = 0.1. The unit peak: flow (Fig. 5-20 (a)) is: qu = 282 ft3/(s-mi2-in.). The surface storage correction factor (Table 5-13) is: F = 0.97. Finally, the peak: flow (Eq. 5-47) is: Qp = 282 × 1.5 × 2.9 × 0.97 = 1190 ft3/s.

calculator image

ONLINE CALCULATION. Using ONLINE TR-55, the 25-y peak flow discharge for the given data is: Qp = 1195.53 ft3/s. This result agrees closely with the hand calculation.


Assessment of TR-55 Graphical Method

The TR-55 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 time of concentration, abstraction parameter Ia/P, and NRCS storm type. The runoff depth is a function of total rainfall depth and runoff curve number.

In the TR-55 graphical method, time of concentration accounts for both runoff concentration and runoff diffusion. From Fig. 5-20, it is clearly 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 Ia/P is related to the catchment's abstractive properties. The greater the curve number, the lesser the value of Ia/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 location and associated storm type is accounted for by the four standard NRCS temporal storm distributions. Therefore, the TR-55 graphical method accounts for hydrologic abstraction, runoff concentration and diffusion, geographical location and type of storm, and the additional surface storage of low-relief catchments.

The TR-55 graphical method may 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 TR-55 graphical method includes runoff curve number and storm type and is applicable to midsize catchments with times of concentration up to 10 h.

The unit values of catchment area, runoff depth, and time of concentration may be used to provide a comparison between the TR-55 graphical method and the rational method. To illustrate, assume a catchment area of 1 mi2 (640 ac), time of concentration 1 h, corresponding rainfall intensity 1 in./h, and runoff coefficient C = 0.95 (the maximum practicable value). A calculation by Eq. 4-4 gives a peak discharge of Qp = 613 ft3/s.

A calculation with the TR-55 graphical method, using the lowest possible value of abstraction for comparison purposes (Ia/P = 0.10), gives the following results: For storm type I, 203 ft3/s; type IA, 108 ft3/s; type II, 360 ft3/s; and type III, 295 ft3/s. This example shows the effect of regional storm hyetograph on the calculated peak discharge. It also shows that the TR-55 graphical method generally gives lower peak flows than the rational method. This may be attributed to the fact that the TR-55 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 comparable, since the value of Ia/P = 0.1 does not correspond exactly to C = 0.95.

 Example 5-12.

Given for the following data: (a) catchment area A = 10 mi2; (b) 24-h rainfall depth P = 5.0 in.; (c) return period T = 10 y; (d) curve number CN = 80; (e) time of concentration tc = 1 h; (f) percentage of ponds and swamps 0%; and (g) Storm Type I. Calculate the peak flow discharge by the TR-55 graphical method, and compare with the online calculation using ONLINE TR-55.


Using Eq. 5-48, the initial abstraction is Ia = (200/80) - 2 = 0.5 in. The ratio Ia/P = 0.5/5.0 = 0.1. Using Fig. 5-20 (a), the unit peak flow is: qu = 203 ft3/(s-mi2-in.). Using Eq. 5-8, the runoff Q = 2.89 in. From Table 5-13, F = 1. Using Eq. 5-47, the 10-y peak flow discharge is: Qp = 203 × 10 × 2.893 × 1 = 5872.8 ft3/s.

calculator image

ONLINE CALCULATION. Using ONLINE TR-55, the 10-y peak flow discharge for the given data is: Qp = 5872.5 ft3/s. This result agrees closely with the hand calculation.


QUESTIONS

[Problems]   [References]      [Top]   [Midsize Catchments]   [Runoff Curve Number]   [Unit Hydrograph]   [TR-55 Method]  

  1. What catchment properties are used in estimating a runoff curve number? What significant rainfall characteristic is absent from the NRCS runoff curve number method?

  2. What is the antecedent moisture condition in the runoff curve number method? How is it estimated?

  3. What is hydrologic condition in the runoff curve number method? How is it estimated?

  4. Describe the procedure to estimate runoff curve numbers from measured data. What level of antecedent moisture condition will cause the greatest runoff? Why?

  5. What is a unit hydrograph? What does the word unit refer to?

  6. Discuss the concepts of linearity and superposition in connection with unit hydrograph theory.

  7. What is catchment lag? Why is it important in connection with the calculation of synthetic unit hydrographs?

  8. In the Snyder method of synthetic unit hydrographs, what do the parameters Ct and Cp describe?

  9. Compare lag, time-to-peak, time base, and unit hydrograph duration in the Snyder and NRCS synthetic unit hydrograph methods.

  10. What is the shape of the triangle used to develop the peak flow formula in the NRCS synthetic unit hydrograph method? What value of Snyder's Cp matches the NRCS unit hydrograph?

  11. What elements are needed to properly define a synthetic unit hydrograph?

  12. What is the difference between superposition and S-hydrograph methods to change unit hydrograph duration? In developing S-hydrographs, why are the ordinates summed up only at intervals equal to the unit hydrograph duration?

  13. What is hydrograph convolution? What assumptions are crucial to the convolution procedure?

  14. What is an unconnected impervious area in the TR-55 methodology? What is unit peak flow?

  15. Given the similarities between the TR-55 graphical method and the rational method, why is the former based on runoff depth while the latter is based on rainfall intensity?


PROBLEMS

[References]      [Top]   [Midsize Catchments]   [Runoff Curve Number]   [Unit Hydrograph]   [TR-55 Method]   [Questions]  

  1. An agricultural watershed has the following hydrologic characteristics: (1) a subarea in fallow, with bare soil, soil group B, covering 32 percent; and (2) a subarea planted with row crops, contoured and terraced, in good hydrologic condition, soil group C, covering 68 percent. Determine the runoff Q, in centimeters, for a 10.5-cm rainfall. Assume an AMC II antecedent moisture condition.

  2. A rural watershed has the following hydrologic characteristics:

    1. A pasture area, in fair hydrologic condition, soil group B, covering 22 percent,

    2. A meadow, soil group B, covering 55%, and

    3. Woods, poor hydrologic condition, soil group B-C, covering 23 percent.

    Determine the runoff Q, in centimeters, for a 12-cm rainfall. Assume an AMC III antecedent moisture condition.

  3. Rain falls on a 9.5-ha urban catchment with an average intensity of 2.1 cm/h and duration of 3 h. The catchment is divided into (1) business district (with 85 percent impervious area), soil group C, covering 20 percent; and (2) residential district, with 1/3-ac average lot size (with 30 percent impervious area), soil group C. Determine the tntal runoff volume, in cubic meters, assuming an AMC II antecedent moisture condition.

  4. Rain falls on a 950-ha catchment in a semiarid region. The vegetation is desert shrub in fair hydrologic condition. The soils are: 15 percent soil group A; 55 percent soil group B, and 30 percent soil group C. Calculate the runoff Q, in centimeters, caused by a 15-cm storm on a wet antecedent moisture condition. Assume that field data support the use of an initial abstraction parameter λ = 0.3.

  5. The hydrologic response of a certain 10-mi2 agricultural watershed can be modeled as a triangular-shaped hydrograph, with peak flow and time base defining the triangle. Five events encompassing a wide range of antecedent moisture conditions are selected for analysis. Rainfall-runoff data for these five events are as follows:

    Rainfall P
    (in.)
    Peak flow Qp
    (ft3/s)
    Time base
    (h)
    7.05 3100 12.
    6.41 3700 14.
    5.13 4100 13.
    5.82 4500 12.
    6.77 3500 14.

    Determine a value of AMC II runoff curve number based on the above data.

  6. The following rainfall-runoff data were measured in a certain watershed:

    Rainfall P
    (cm)
    Runoff Qp
    (cm)
    15.2 12.3
    10.5 10.1
    7.2 4.3
    8.4 5.2
    11.9 9.1

    Assuming that the data encompass a wide range of antecedent moisture conditions, estimate the AMC II runoff curve number.

  7. The following rainfall distribution was observed during a 6-h storm:

    Time (h) 0 2 4 6
    Intensity (mm/h) 10 15 12

    The runoff curve number is CN = 76. Calculate the φ-index.

  8. The following rainfall distribution was observed during a 12-h storm:

    Time (h) 0 2 4 6 8 10 12
    Intensity (mm/h) 5 10 13 18 3 10

    The runoff curve number is CN = 86. Calculate the φ-index.

  9. The following rainfall distribution was observed during a 6-h storm:

    Time (h) 0 2 4 6
    Intensity (mm/h) 18 24 12

    The φ-index is 10 mm/h. Calculate the runoff curve number.

  10. The following rainfall distribution was observed during a 24-h storm:

    Time (h) 0 3 6 9 12 15 18 21 24
    Intensity (mm/h) 5 8 10 12 15 5 3 6

    The φ-index is 4 mm/h. Calculate the runoff curve number.

  11. A unit hydrograph is to be developed for a 29.6-km2 catchment with a 4-h T2 lag. A 1-h rainfall has produced the following runoff data:

    Time (h) 0 1 2 3 4 5 6 7 8 9 10 11 12
    Flow (m3/s) 1 2 4 8 12 8 7 6 5 4 3 2 1

    Based on this data, develop a 1-h unit hydrograph for this catchment. Assume baseflow is 1 m3/ s.

  12. A unit hydrograph is to be developed for a 190.8-km2 catchment with a 12-h T2 lag. A 3-h rainfall has produced the following runoff data:

    Time (h) 0 3 6 9 12 15 18 21 24
    Flow (m3/s) 15 20 55 80 60 48 32 20 15

    Based on this data, develop a 3-h unit hydrograph for this catchment. Assume baseflow is 15 m3/s.

  13. Calculate a set of Snyder synthetic unit hydrograph parameters for the following data: catchment area A = 480 km2; L = 28 km; Lc = 16 km; Ct = 1.45; and Cp = 0.61.

  14. Calculate a set of Snyder synthetic unit hydrograph parameters for the following data: catchment area A = 950 km2; L = 48 km; Lc = 21 km; Ct = 1.65; and Cp = 0.57.

  15. Calculate an NRCS synthetic unit hydrograph for the following data: catchment area A = 7.2 km2; runoff curve number CN = 76; hydraulic length L = 3.8 km; and average land slope Y = 0.012.

  16. Calculate an NRCS synthetic unit hydrograph for the following data: catchment area (natural catchment) A = 48 km2; runoff curve number CN = 80; hydraulic length L = 9 km; and mean velocity along hydraulic length V = 0.25 m/s.

  17. Calculate the peak flow of a triangular SI unit hydrograph (1 cm of runoff) having a volume-to-peak to unit-volume ratio p = 3/10. Assume basin area A = 100 km2, and time to- peak tp = 6 h.

  18. Given the following 1-h unit hydrograph for a certain catchment. find the 2-h unit hydrograph using: (a) the superposition method, and (b) the S-hydrograph method.

    Time (h) 0 1 2 3 4 5 6
    Flow (ft3/s) 0 500 1000 750 500 250 0

  19. Given the following 3-h unit hydrograph for a certain catchment. find the 6-h unit hydrograph using: (a) the superposition method, and (b) the S-hydrograph method.

    Time (h) 0 3 6 9 12 15 18 21 24
    Flow (m3/s) 0 5 15 30 25 20 10 5 0

  20. Given the following 2-h unit hydrograph for a certain catchment, find the 3-h unit hydrograph. Using this 3-h unit hydrograph, calculate the 1-h unit hydrograph.

    Time (h) 0 1 2 3 4 5 6 7
    Flow (m3/s) 0 25 75 87.5 62.5 37.5 12.5 0

  21. Given the following 4-h unit hydrograph for a certain catchment, find the 6-h unit hydrograph. Using this 6-h unit hydrograph, calculate the 4-h unit hydrograph, verifying the computations.

    Time (h) 0 2 4 6 8 10 12 14 16 18 20 22 24
    Flow (m3/s) 0 10 30 60 100 90 80 70 50 40 20 10 0

  22. Given the following 4-h unit hydrograph for a certain catchment: (a) Find the 6-h unit hydrograph; (b) using the 6-h unit hydrograph, calculate the 8-h unit hydrograph; (c) using the 8-h unit hydrograph, calculate the 4-h unit hydrograph, verifying the computations.

    Time (h) 0 2 4 6 8 10 12 14 16 18 20
    Flow (m3/s) 0 10 25 40 50 40 30 20 10 5 0

  23. The following 2-h unit hydrograph has been developed for a certain catchment:

    Time (h) 0 2 4 6 8 10 12
    Flow (ft3/s) 0 100 200 150 100 50 0

    A 6-h storm covers the entire catchment and is distributed in time as follows:

    Time (h) 0 2 4 6
    Total rainfall (in./h) 1.0 1.5 0.5

    Calculate the composite hydrograph for the effective storm pattern, assuming a runoff curve number CN = 80.

  24. The following 3-h unit hydrograph has been developed for a certain catchment:

    Time (h) 0 3 6 9 12 15 18 21 24
    Flow (m3/s) 0 10 20 30 25 20 15 10 0

    A 12-h storm covers the entire catchment and is distributed in time as follows:

    Time (h) 0 3 6 9 12
    Total rainfall (mm/h) 6 10 18 2

    Calculate the composite hydrograph for the effective storm pattern, assuming a runoff curve number CN = 80.

  25. A certain basin has the following 2-h unit hydrograph:

    Time (h) 0 1 2 3 4 5 6 7 8 9 10 11 12 13
    Flow (m3/s) 0 5 15 30 60 75 65 55 45 35 25 15 5 0

    Calculate the flood hydrograph for the following effective rainfall hyetograph:

    Time (h) 0 3 6
    Effective rainfall (cm/h) 1.0 2.0

  26. Given the following flood hydrograph and effective storm pattern, calculate the unit hydrograph ordinates by the method of forward substitution.

    Time (h) 0 1 2 3 4 5 6 7 8 9 10 11 12
    Flow (m3/s) 0 5 18 46 74 93 91 73 47 23 9 2 0

    Time (h) 0 1 2 3 4 6 6
    Effective rainfall (cm/h) 0.5 0.8 1.0 0.7 0.5 0.2

  27. Using TR-55 procedures, calculate the time of concentration for a watershed having the following characteristics:

    • Overland flow, dense grass, length L = 100 ft, slope S = 0.01, 2-y 24-h rainfall P2 = 3.6 in.;

    • Shallow concentrated flow, unpaved, length L = 1400 ft, slope S = 0.01; and

    • Streamflow, Manning n = 0.05, flow area A = 27 ft2, wetted perimeter P = 28.2 ft, slope S = 0.005, length L = 7300 ft.

  28. Using TR-55 procedures, calculate the time of concentration for a watershed having the following characteristics:

    • Overland flow, bermuda grass, length L = 50 m, slope S = 0.02, 2-y 24-h rainfall P2 = 9 cm; and

    • Streamflow, Manning n = 0.05, flow area A = 4.05 m2, wetted perimeter P = 8.1 m, slope S = 0.01, length L = 465 m.

  29. A 250-ac watershed has the following hydrologic soil-cover complexes:

    1. Soil group B, 75 ac, urban, 1/2-ac lots with lawns in good hydrologic condition, 25 percent connected impervious;

    2. Soil group C, 100 ac, urban, 1/2-ac lots with lawns in good hydrologic condition, 25 percent connected impervious; and

    3. Soil group C, 75 ac, open space in good condition.

    Determine the composite runoff curve number.

  30. A 120-ha watershed has the following hydrologic soil-cover complexes:

    1. Soil group B, 40 ha, urban, 1/2-ac lots with lawns in good hydrologic condition, 35 percent connected impervious;

    2. Soil group C, 55 ha, urban, 1/2-ac lots with lawns in good hydrologic condition, 35 percent connected impervious; and

    3. Soil group C, 25 ha, open space in fair condition.

    Determine the composite runoff curve number.

  31. A 90-ha watershed has the following hydrologic soil-cover complexes:

    1. Soil group C, 18 ha, urban, 1/3-ac lots with lawns in good hydrologic condition, 30 percent connected impervious;

    2. Soil group D, 42 ha, urban, 1/3-ac lots with lawns in good hydrologic condition, 40"70 connected impervious; and

    3. Soil group D, 30 ha, urban, 1/3-ac lots with lawns in fair hydrologic condition, 30 poercent total impervious, 25% of it unconnected impervious area.

    Determine the composite runoff curve number.

  32. Use the TR-55 graphical method to compute the peak discharge for a 250-ac watershed, with 25-y 24-h rainfall P = 6 in., time of concentration tc = 1.53 h, runoff curve number CN = 75, and Type II rainfall.

  33. Use the TR-55 graphical method to calculate the peak discharge for a 960-ha catchment, with 50-y 24-h rainfall P = 10.5 cm, time of concentration tc = 3.5 h, runoff curve number CN = 79, type I rainfall, and 1 % pond and swamp areas.

  34. Calculate the 25-y peak flow by the TR-55 graphical method for the following watershed data:

    • Urban watershed, area A = 9.5 km2;

    • Surface flow is shallow concentrated, paved; hydraulic length L = 3850 m; slope S = 0.01;

    • 42 percent of watershed is 1/3-ac lots, lawns with 85% grass cover, 34% total impervious, soil group C;

    • 58 percent of the watershed is 1/3-ac lots, lawns with 95% grass cover, 24% total impervious, 25% of it unconnected, soil group C;

    • Pacific Northwest region, 25-y 24-h rainfall P = 10 cm; 1 percent ponding.


REFERENCES

   [Top]   [Midsize Catchments]   [Runoff Curve Number]   [Unit Hydrograph]   [TR-55 Method]   [Questions]   [Problems]  

  1. Amorocho. J., and G. T. Orlob. (1961). "Nonlinear Analysis of Hydrologic Systems," University of California Water Resources Center, Contribution No. 40, November.

  2. Barnes, H. H. Jr. (1967). "Roughness Characteristics of Natural Channels," U.S. Geological Survey Water Supply Paper No. 1849.

  3. Chow, V. T. (1959). Open-Channel Hydraulics. New York: McGraw-Hill.

  4. Diskin, M. H. (1964). "A Basic Study of the Nonlinearity of Rainfall-Runoff Processes in Watersheds," Ph.D. Diss., University of Illinois, Urbana.

  5. Freeze, R. A., and J. A. Cherry. (1979). Groundwater, Englewood Cliffs, N.J.: Prentice- Hall.

  6. French, R. H. (1986). Open-Channel Hydraulics, New York: McGraw-Hill.

  7. Hall, M. J. (1984). Urban Hydrology. London: Elsevier Applied Science Publishers.

  8. Hawkins, R. H., A. T. Hjelmfelt, and A. W. Zevenbergen. (1985). "Runoff Probability, Storm Depth, and Curve Numbers," Journal of the Irrigation and Drainage Division, ASCE, Vol. 111, No. 4, December, pp. 330-340.

  9. Hjelmfelt, A. T., K. A. Kramer, and R. E. Burwell. (1981). "Curve Numbers as Random Variables," Proceedings, International Symposium on Rainfall-Runoff Modeling, Mississippi State University, (also Water Resources Publications, Littleton, Colorado).

  10. Linsley, R. K. , M. A. Kohler, and L. H. Paulhus. (1962). Hydrology for Engineers, 3d. ed. New York: McGraw-Hill.

  11. McCuen, R. H., W. 1. Rawls, and S. L. Wong. (1984). "SCS Urban Peak Flow Methods," Journal of Hydraulic Engineering, ASCE, Vol. 110, No. 3, March, pp. 290-299.

  12. Newton, D. J. , and J. W. Vineyard. (1967). "Computer-Determined Unit Hydrographs from Floods," Journal of the Hydraulics Division, ASCE, Vol. 93, No. HY5, pp. 219-236.

  13. Rallison, R.E., and R. G. Cronshey. (1979). Discussion of "Runoff Curve Numbers with Varying Soil Moisture," Journal of the Irrigation and Drainage Division ASCE, Vol. l05, No. IR4, pp. 439-441.

  14. Sherman, L. K. (1932). "Streamflow from Rainfall by Unit-Graph Method," Engineering News-Record, Vol. 108, April 7, pp. 501-505.

  15. Singh, K. P. (1962). "A Nonlinear Approach to the Instantaneous Unit Hydrograph," Ph.D. Diss., University of Illinois, Urbana.

  16. Singh, V. P. (1988). Hydrologic Systems. Vol. 1: Rainfall-Runoff Modeling. Englewood Cliffs, N.J.: Prentice-Hall.

  17. Snyder, F. F. (1938). "Synthetic Unit-Graphs," Transactions, American Geophysical Union. Vol. 19, pp. 447-454.

  18. Springer, E. P., B. J. McGurk, R. H. Hawkins, and G. B. Coltharp. (1980). "Curve Numbers from Watershed Data," Proceedings, Symposium on Watershed Management, ASCE, Boise, Idaho, July, pp. 938-950.

  19. Taylor, A. B., and H. E. Schwarz. (1952). "Unit Hydrograph Lag and Peak Flow Related to Basin Characteristics," Transactions. American Geophysical Union, Vol. 33, pp. 235-246.

  20. U.S. Army Corps of Engineers. (1959). "Flood Hydrograph Analysis and Computations," Engineering and Design Manual EM 1110-2-1405, Washington, D.C.

  21. USDA Natural Resources Conservation Service. (1985). NRCS National Engineering Handbook. Section 4: Hydrology, Washington, D.C.

  22. USDA Natural Resources Conservation Service. (1986). "Urban Hydrology for Small Watersheds," Technical Release No. 55 (TR-55), Washington, D.C.

  23. USDA Natural Resources Conservation Service. (1993). "Chapter 4: Storm Rainfall Depth," Part 630, Hydrologic Engineering, Washington, D.C. (part of NRCS National Engineering Handbook, Section 4: Hydrology).

  24. U.S. Forest Service. (1959). Forest and Range Hydrology Handbook, Washington, D.C.

  25. Forest Service. (1959). Handbook on Methods of Hydrologic Analysis, Washington,D.C.

  26. Van Sickle, D. (1969). "Experience with the Evaluation of Urban Effects for Drainage Design," in Effects of Watershed Changes on Streamflow, Proceedings, Water Resources Symposium No. 2, University of Texas, Austin, pp. 229-254.

Suggested Readings

  1. Sherman, L. K. (1932). "Streamflow from Rainfall by Unit-Graph Method," Engineering News-Record. Vol. 108, April 7, pp. 501-505.

  2. Snyder, F. F. (1938). "Synthetic Unit-Graphs," Transactions. American Geophysical Union, Vol. 19, pp. 447-454.

  3. USDA Natural Resources Conservation Service. (1985). SCS National Engineering Handbook, Section 4: Hydrology, Washington, D.C.

  4. USDA Natural Resources Conservation Service. (1986). "Urban Hydrology for Small Watersheds," Technical Release No. 55 (TR-55), Washington, D.C.


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