Development of unit hydrographs: Direct method

• For the direct method, the stream must be gaged.

• The rainfall-runoff records should be screened to identify storms suitable for hydrograph analysis.

• The selected storms should be of uniform rainfall intensity temporally and spatially.

• This latter requirement is not met as catchment size increases above 250-1000 km².

• Catchment lag is a measure of the time elapsed between the occurrence of unit rainfall and the occurrence of unit runoff.

• Catchment lag is a global measure of catchment response, encompassing hydraulic length, catchment gradient, drainage density, and drainage patterns.

• There are many definitions of catchment lag, lag time, or simply lag.

• The T₂ lag is the most common: the time elapsed from the centroid of effective rainfall to the peak of runoff.

• Runoff volume is equal to 1 unit of effective rainfall depth.

• Short lags feature high peaks and short time bases; long lags result in low peaks and long time bases.

• Catchment lag is empirically related to catchment characteristics:

  tl = C [L Lc / S1/2]N

• in which t_l = catchment lag; L = catchment length; L_c = length to catchment centroid; S = weighted measure of catchment slope, and C and N are empirical parameters.

  Methodology

• Storms suitable for unit hydrograph analysis should be about the same duration.

• The duration should be about 10 to 30% of the catchment lag.

• This implies that runoff is of the subconcentrated type.

• Subconcentrated flow is a characteristic of midsize catchments.

• Not all catchments concentrate in nature.

• Midsize catchments are too large to concentrate flow; calculation shifts to the hyetograph and the maximum rainfall intensity.

• Several individual storms are analyzed for consistency.

• The following steps are applied to each storm:
  
  
  1. Separation of the measured hydrograph into direct runoff hydrograph (DRH) and baseflow (BF).

  2. Integration of the DRH to calculate the direct runoff volume (DRV).

  3. Dividing the DRV by the catchment area to determine direct runoff depth (DRD).

  4. Division of DRH ordinates by DRD to determine the unit hydrograph (UH) ordinates.

  5. Estimation of the UH duration.

• The catchment unit hydrographs is obtained by averaging ordinates for all storms.

• Minor adjustments may be necessary to ensure that the volume under the UH is equal to 1 unit of runoff.

  Hydrograph separation

• Only the direct runoff component is used in the computation of the unit hydrograph.

• It is necessary to separate the measured hydrograph into direct runoff and baseflow.

• Procedures for baseflow separation are arbitrary and empirical.

• Baseflow separation: Figure 5-7.

• Example of the development of a unit hydrograph by the direct method: Example 5-2.

Development of unit hydrographs: Indirect method

• In the absence of stream gaging, unit hydrographs are derived by synthetic means.

• A synthetic unit hydrograph is derived based on an established formula, derived empirically.

• Principle of synthetic unit hydrograph theory: Since the volume of the unit hydrograph is known (1 unit of volume), an assumption of hydrograph shape leads to the peak flow.

• If a triangular shape is assumed, the volume is equal to:  

  V = [Qp Tbt ) / 2 = A (1)

• In which V= unit hydrograph volume; Q_p = peak flow; T_bt= time base of the triangular unit hydrograph; A = catchment area.

• Solving for peak flow:

  Qp = 2 A / Tbt

• Synthetic unit hydrographs related time base to catchment lag.

• In turn, catchment lag is related to catchment shape, length, and slope.

• Synthetic unit hydrographs
  
  
  • Snyder (U.S. Army Corps of Engineers)

  • NRCS (ex-SCS)

  • TVA

  • Clark (Chapter 10, Section 2).

Snyder's synthetic unit hydrograph

• The analysis of a large number of hydrographs from catchments in the Appalachian region led Snyder to the following formula for lag:

  tl = Ct (L Lc)0.3

• In which C_t = a coefficient accounting for slope and associated catchment storage.

• Snyder's formula for peak flow is:

  Qp = Cp (A / tl)

• with C_p defined as follows:

  Cp = 2 / (Tbt / tl)

• C_p is an empirical coefficient relating triangular time base to lag.

• C_p values are in the range 0.56-0.69.

• These are associated with T_bt / t_l ratios values in the range 3.57-2.90.

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

  Qp = 2.78 Cp (A / tl)

• in which Q_p for 1 cm in m³/s, A in km², t_l in hours.

• In U.S. customary units, Snyder's formula for peak flow is:

  Qp = 645 Cp (A / tl)

• in which Q_p for 1 inch in cfs, A in mi², t_l in hours.

• The unit hydrograph duration is estimated as:

  tr = (2/11) tl

• The time-to-peak is equal to one-half of the storm duration plus the lag:

  tp = (12/11) tl

• When calculating the actual time base of the unit hydrograph, Snyder included interflow as part of direct runoff.

• This results in a time base longer than that corresponding only to direct runoff.

• Snyder's formula for actual time base is:

  Tb = 72 + 3 tl

• in which T_b is in hours and t_l in hours.

• For a 24-hr lag, this formula gives T_b/t_l = 6, which is reasonable for large lag.

• For a 6-hr lag, this formula gives T_b/t_l = 90/6 = 15, which is too long a time base.

• For short lags, experience shows that T_b/t_p = 5.

• This translates into T_b/t_l = 5.45 instead of 15.

• The Snyder method gives peak flow, time-to-peak, and time base.

• Additional formulas for W₅₀ (page 176) and W₇₅ (page 177) help estimate the shape of the unit hydrograph.  

   

• It can be shown that the maximum value of C_p is 11/12.

• Triangular time base cannot be less than twice the time-to-peak.

• In the limit, when diffusion is absent, T_bt / t_p = 2.

• Since t_p / t_l = 12/11.

• Then C_p = 2 / (T_bt / t_l) = 2 / [(T_bt / t_p) (t_p / t_l)] = t_l / t_p = 11/12.

• Since C_t increases with catchment storage, and C_p decreases with catchment storage, the ratio C_t/C_p can be directly related to catchment storage.

• The reciprocal ratio C_p/C_t can be directly related to extent of urban development, since the latter is inversely related to catchment storage.

NRCS synthetic unit hydrograph

• This method differs from Snyder's in that it uses a constant ratio of triangular time base to time-to-peak equal to 8/3.

• This implies that Snyder's C_p = 0.6875.

• The method also uses a constant ratio of actual time base to time-to-peak equal to 5.

• To calculate lag, the NRCS uses two methods:
  
  
  1. the curve number method, and

  2. the velocity method.

• The curve number method is limited to areas less than 8 km².

• In the curve number method, the lag is:

  tl = L0.8 (2540 - 22.86 CN)0.7 / (14104 CN0.7Y0.3)

• In which L catchment length, in m, CN = curve number, and Y = average catchment land surface slope in m/m.

• In the velocity method, mean velocities are estimated along the main stream, and time of concentration determined.

• The lag is estimated as follows:

  tl = (6/10) tc

• The time-to-peak is estimated as follows:

  tp = 5 tr

• The time-to-peak is defined as:

  tp = (tr/2) + tl

• This leads to:

  tp = (10/9) tl

  [Compare to (12/11) t₁ for Snyder's formula].

• Then:

  tr = (2/9) tl

  [Compare to (2/11) t₁ for Snyder's formula].

• Then:

  tr = (2/15) tc

• The main assumption of the NRCS unit hydrograph is:

  Tbt/tp = 8/3

• This leads to:

  Qp = (3/4) (A / tp)

• In SI units, the NRCS formula for peak flow is:

  Qp = 2.08 (A / tp)

• in which Q_p for 1 cm in m³/s, A in km², t_l in hours.

• In U.S. customary units, the NRCS formula for peak flow is:

  Qp = 484 (A / tp)

• in which Q_p for 1 inch in cfs, A in mi², t_l in hours.

• The time-to-peak is estimated as:

  tp = (1/2) tr + (6/10) tc

• The dimensionless unit hydrograph is used to calculate the hydrograph ordinates.

• The NRCS method provides a hydrograph shape, and leads to results which are more reproducible than Snyder's.

• By fixing the ratio T_bt/t_p = 8/3 and t_l/t_c = 6/10, the method is rendered inflexible.

• However, it is widely used because of its simplicity.

• Because diffusion is fixed to an average value, NRCS should be limited to catchments in the lower end of the size spectrum (2.5-250 km²).

• Because diffusion is variable, Snyder's method can be used for larger catchments (250 to 5000 km²).

• By relaxing the ratio T_bt/t_p to values greater than 8/3, the method is rendered more flexible.

• When T_bt/t_p = 8/3, then the reciprocal p = 3/8.

• In general, this leads to:

  Qp = 2p (A / tp)

• This equation is applicable in areas of flat relief, where more diffusion is desired, over and above what the NRCS method can provide.

• The method has been used in Florida.

Change in unit hydrograph duration

• A unit hydrograph is valid only for a given (effective) storm duration.

• The storm hyetograph is defined in terms of a specified constant time interval.

• For convolution, the storm duration has to be the same as the time interval of the storm hyetograph.

• If an X-hr unit hydrograph is going to be used with a Y-hr-defined storm hyetograph, it is necessary to convert the X-hr unit hydrograph into a Y-hr unit hydrograph.

• Once an X-hr unit hydrograph is known, any other Y-hr unit hydrograph can be derived from it.

• There are two methods:
  
  
  1. superposition

  2. S-hydrograph.

• The superposition method converts an X-hr UH into an nX-hr UH, in which n is an integer such as 2 or 3.

• The S-hydrograph method converts an X-hr UH into a Y-hr UH, regardless of the ratio X/Y (for example, 3/2 or 2/3).

Superposition method

• This method converts an X-hr UH into an nX-hr UH, in which n is an integer.

• The procedure consists of lagging n X-hr UH 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-hr UH.

• The procedure is illustrated by Example 5-5.

S-hydrograph method

• This method converts an X-hr UH into an Y-hr UH, regardless of the ratio X/Y.

• The procedure consists of the following steps:
  
  
  1. Determine the X-hr S-hydrograph accumulating the X-hr unit hydrograph ordinates at intervals equal to the duration.

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

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

  4. Multiply the resulting hydrograph ordinates by X/Y to obtain the Y-hr UH.

• The procedure is illustrated by Example 5-6.

Convolution and composite or flood hydrographs

• The UH is convoluted with the effective storm hyetograph to derive a composite or flood hydrograph, to be used for design.

• Convolution is based on the principles of linearity and superposition.

• The procedure is illustrated by Example 5-7.

Go to Chapter 5C.