A GENERAL DIMENSIONLESS UNIT HYDROGRAPH

Professor of Civil and Environmental Engineering

San Diego State University

San Diego, California

May 15, 2009

ABSTRACT:   A general dimensionless unit hydrograph (GDUH) based on the cascade of linear reservoirs is formulated and calculated online. The GDUH is shown to be solely a function of the Courant number and the number of linear reservoirs. Since the GDUH is independent of the basin drainage area and the unit hydrograph duration, it is applicable on a global basis. Each GDUH is related only to the basin's prevailing runoff diffusion properties. The model's two-parameter feature provides increased flexibility for simulating a wide range of diffusion effects.

1.  INTRODUCTION

A unit hydrograph (UH) is a hydrograph for a unit depth of effective rainfall, applicable to a given basin and for a given duration (Sherman, 1932). In hydrologic practice, the unit hydrograph is convoluted with the effective storm hyetograph to obtain the composite flood hydrograph. For proper convolution, the UH duration should be the same as the unit interval of the storm hyetograph. If this is not the case, the S-hydrograph [see online version] may be used to change the UH duration to match the unit interval (Ponce, 1989).

The shape of the unit hydrograph, as characterized by its peak and time lag, is a measure of the amount of runoff diffusion prevailing in the basin (Ponce, 1989). Steeper basins will have little runoff diffusion, while milder basins may have a significant amount. Diffusion acts to spread the flows in time and space. Less diffusion means sharp hydrographs with little attenuation; more diffusion means substantially attenuated hydrographs. Therefore, less/more diffusion means a shorter/longer time lag.

2.  LINEAR RESERVOIRS

A linear reservoir provides a certain amount of runoff diffusion. In a discrete solution, the amount of diffusion is characterized by the Courant number C, defined as follows (Ponce, 1989):

 C = Δt /K (1)

in which Δt (hr) is the time interval and K (hr) is the reservoir storage constant.

In unit hydrograph theory, for proper convolution, the time interval Δt (hr) has to match the unit hydrograph duration tr (hr). Therefore:

 C = tr /K (2)

For hydrograph attenuation to occur, i.e., positive runoff diffusion (Ponce, 1989):

 K ≥ tr / 2 = Δt / 2 (3)

i.e.,

 C ≤ 2 (4)

Larger values of K, compared to tr, provide a greater amount of runoff diffusion. Typical values of C are in the range 0.1 ≤ C ≤ 2.

The cascade of linear reservoirs (CLR) is a hydrologic model that uses a number of linear reservoirs in series in order to provide a wide range of possibilities for runoff diffusion and associated unit hydrograph peak flow attenuation and lag effects. In a CLR, the outflow from the first reservoir is the inflow to the second; the outflow from the second is the inflow to the third; and so on (Ponce, 1989). Typical values of the number of reservoirs N are in the range 1 ≤ N ≤ 10. Once C and N are defined, the CLR method provides a unique amount of runoff diffusion, i.e., a unique unit hydrograph peak and associated time lag.

An online version of the CLR unit hydrograph is given in online_uh_cascade.

4.  DIMENSIONLESS UNIT HYDROGRAPH

The dimensionless unit hydrograph (DUH) has dimensionless time t* in its abscissa and dimensionless discharge Q* in its ordinate. The dimensionless time is:

 t* = t / tr (5)

in which t = time (hr). The dimensionless discharge is:

 Q* = Q / Qmax (6)

in which Q = discharge, in m3/s; and Qmax = maximum discharge, i.e., the discharge attained in the absence of runoff diffusion, in m3/s. According to the runoff concentration principle (Ponce, 1989):

 Qmax = i A (7)

in which i = effective rainfall intensity, in m/s; and A = basin drainage area, in m2. Then:

 Q* = Q / (i A) (8)

For 1 cm of unit hydrograph rainfall in tr hr (in SI units):

 i = 0.01 (m) / [3,600 (s/hr) × tr (hr) ] (9)

Therefore:

 Q* = 0.36 Q (tr / A) (10)

in which tr is in hr and A in in km2. An online version of the CLR DUH is given in online_dimensionless_uh_cascade.

5.  GENERAL DIMENSIONLESS UNIT HYDROGRAPH

A general dimensionless unit hydrograph (GDUH) may be generated using the CLR method for a basin of drainage area A and unit hydrograph duration tr (i.e., the storm hyetograph's unit interval). The resulting set of Q* vs t* unit hydrograph values are solely a function of C and N, and independent of either A or tr. Thus, for a given set of C and N, there is a unique GDUH, of general, i.e., global applicability.

In practice, the set of C and N is chosen such that the runoff diffusion properties of the basin are properly represented in the GDUH. This requires careful consideration and the analysis of measured rainfall-runoff events. Steeper basins will require a large C and a small N; conversely, milder basins will require a small C and a large N. The recommended practical range of parameters is:  0.1 ≤ C ≤ 2; 1 ≤ N ≤ 10. Within this range, C = 2 and N = 1 provides zero diffusion; on the other hand, C = 0.1 and N = 10 provides a very significant amount of diffusion. The case of zero diffusion is equivalent to the assumption of runoff concentration only, which is inherent in the rational method (Ponce, 1989).

Once the GDUH is chosen, the ordinates of the unit hydrograph may be calculated from Eq. 10:

 Q = Q* A / (0.36 tr) = 2.7778 Q* A / tr (11)

and the abscissa from Eq. 5:

 t = t* tr (12)

The unit hydrograph thus calculated may be convoluted with the effective storm hyetograph to determine the composite flood hydrograph (Ponce, 1989).

An online version of the GDUH as a function of C and N is given in online_general_uh_cascade.

An online version of a GDUH series as a function of C in the range 0.1 ≤ C ≤ 2.0 and N in the range 1 ≤ N ≤ 10 is given in online_series_uh_cascade.

An online version of a GDUH series for C equal to 2, 1.5, 1.0, 0.5, 0.2, and 0.1, and N in the range 1 ≤ N ≤ 10 is given in online_all_series_uh_cascade.

6.  PROCEDURE

The procedure to derive a unit hydrograph based on the GDUH is the following:

1. Determine the basin drainage area A (km2).

2. Determine the unit interval tr (hr) of the effective storm hyetograph.

3. Run online_all_series_uh_cascade to choose among a pair of six predetermined C values between 0.1 ≤ C ≤ 2.0, and N between 1 ≤ N ≤ 10, applicable to the basin under consideration.

4. If necessary, run online_series_uh_cascade to calculate the dimensionless unit hydrograph for any C between 0.1 ≤ C ≤ 2, and 1 ≤ N ≤ 10.

5. If necessary, run online_general_uh_cascade to calculate the applicable dimensionless unit hydrograph for a given pair of C and N.

Either the convolution or the CLR methods may be used to calculate the composite flood hydrograph.

1. Convolution method:

• Using the GDUH, calculate the unit hydrograph discharges using Eq. 11, and the corresponding times using Eq. 12.

• Given the effective storm hyetograph and the unit hydrograph ordinates calculated in a, use online_convolution to calculate the composite flood hydrograph. Use curve number CN = 100 to specify an effective storm hyetograph.

2. Cascade of linear reservoirs (CLR) method:

• Calculate the reservoir storage constant K = Δt / C = tr / C.

• Given the effective storm hyetograph, and the parameters K and N, use online_routing08 to calculate the composite flood hydrograph.

The composite flood hydrographs calculated by both methods are shown to be essentially the same.

The GDUH described herein has the following significant advantages:

1. The procedure is global, applicable anywhere, and solely a function of C and N.

2. Unlike other established unit hydrograph procedures such as the NRCS unit hydrograph (Ponce, 1989), the GDUH is a two-parameter model; therefore, it is able to simulate a wider range of runoff diffusion effects.

A disadvantage of the GDUH is that while the parameters C and N may be readily related to the basin's general topographic relief, considerable experience and/or measured rainfall/runoff data may be required for a proper estimation of these parameters.

8.  CONCLUSIONS

A general dimensionless unit hydrograph (GDUH) based on the cascade of linear reservoirs is formulated and calculated online. The GDUH is shown to be solely a function of the Courant number C and the number of linear reservoirs in series N. Since the GDUH is independent of the basin drainage area and the unit hydrograph duration, it is applicable on a global basis. Each GDUH is related only to the basin's prevailing runoff diffusion properties. The latter are a function of the general topographic relief, with more relief, less diffusion, and less relief, more diffusion. Indeed, the model's two-parameter feature provides increased flexibility for simulating a wide range of runoff diffusion effects.

Research into the estimation of C and N on the basis of basin relief, geomorphology and related ecology will lead to an improved prediction of flood hydrographs, for both design and forecasting applications.

REFERENCES

 Ponce, V. M., 1989. Engineering Hydrology: Principles and Practices. Prentice Hall, Upper Saddle River, New Jersey. Sherman, L. K., 1932. Streamflow from rainfall by unit-graph method. Engineering News-Record, Vol. 108, April 7, 501-505.

NOTATION

The following symbols are used in this document:

A = basin drainage area (m2 or km2);

C = Courant number, dimensionless, Eq. 1 or Eq. 2;

CN = (NRCS runoff) curve number;

i = effective rainfall intensity (cm/hr);

K = (linear) reservoir storage constant (hr);

N = number of linear reservoirs in series;

Q = unit hydrograph discharge (m3/s);

Qmax = maximum discharge, in the absence of runoff diffusion (m3/s), Eq. 7;

Q* = dimensionless discharge, Eq. 6 or Eq. 8;

t = time (hr);

tr = unit hydrograph duration (hr);

t* = dimensionless time, Eq. 5; and

Δt = unit interval of the storm hyetograph (hr).

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