CE322 Basic Hydrology
Jorge A. Ramírez
Unit Hydrographs - Example

A. Obtain a Unit Hydrograph for a basin of 315 km2 of area using the rainfall and streamflow data tabulated below.

 Time (h) Observed Hydrograph (m3/s) 0 100 1 100 2 300 3 700 4 1000 5 800 6 600 7 400 8 300 9 200 10 100 11 100
 Time (h) Gross Precipitation (GRH) (cm/h) 0 - 1 0.5 1 - 2 2.5 2 - 3 2.5 3 - 4 0.5
Empirical Unit Hydrograph Derivation
1. Separate the baseflow from the observed streamflow hydrograph in order to obtain the Direct Runoff Hydrograph (DRH).
For this example, use the horizontal line method to separate the baseflow. From observation of the hydrograph data, the streamflow at the start of the rising limb of the hydrograph is 100 m3/s.
2. Compute the volume of Direct Runoff. This volume must be equal to the volume of the Effective Rainfall Hyetograph (ERH).

Thus, for this example:

VDRH = (200+600+900+700+500+300+200+100) m3/s (3600) s = 12'600,000 m3

3. Express VDRH in equivalent units of depth:
VDRH in equivalent units of depth = VDRH/Abasin = 12'600,000 m3/(315000000 m2) = 0.04 m = 4 cm.
4. Obtain a Unit Hydrograph by normalizing the DRH. Normalizing implies dividing the ordinates of the DRH by the VDRH in equivalent units of depth.
 Time (h) Observed Hydrograph (m3/s) Direct Runoff Hydrograph (DRH) (m3/s) Unit Hydrograph (m3/s/cm) 0 100 0 0 1 100 0 0 2 300 200 50 3 700 600 150 4 1000 900 225 5 800 700 175 6 600 500 125 7 400 300 75 8 300 200 50 9 200 100 25 10 100 0 0 11 100 0 0

5. Determine the duration D of the ERH associated with the UH obtained in 4. In order to do this:
1. Determine the volume of losses, VLosses which is equal to the difference between the volume of gross rainfall, VGRH, and the volume of the direct runoff hydrograph, VDRH .

VLosses = VGRH - VDRH = (0.5 + 2.5 + 2.5 +0.5) cm/h 1 h - 4 cm = 2 cm

2. Compute the f-index equal to the ratio of the volume of losses to the rainfall duration, tr. Thus,

f-index = VLosses/tr = 2 cm / 4 h = 0.5 cm/h

3. Determine the ERH by subtracting the infiltration (e.g., f-index) from the GRH:

 Time (h) Effective Precipitation (ERH) (cm/h) 0 - 1 0.0 1 - 2 2.0 2 - 3 2.0 3 - 4 0.0

As observed in the table, the duration of the effective rainfall hyetograph is 2 hours. Thus, D = 2 hours, and the Unit Hydrograph obtained above is a 2-hour Unit Hydrograph. Therefore, it can be used to predict runoff from precipitation events whose effective rainfall hyetographs can be represented as a sequence of uniform intensity (rectangular) pulses each of duration D. This is accomplished by using the principles of superposition and proportionality, encoded in the discrete convolution equation:

where Qn is the nth ordinate of the DRH, Pm is the volume of the mth rainfall pulse expressed in units of equivalent depth (e.g., cm or in), and Un-m+1 is the (n-m+1)th ordinate of the UH, expressed in units of m3/s/cm.

B. Using the UH obtained in A., predict the total streamflow that would be observed as a result of the following ERH:

 Time (h) Effective Precipitation (ERH) (cm/h) 0 - 2 0.5 2 - 4 1.5 4 - 6 2.0 6 - 8 1.0
As observed in the table, the ERH can be decomposed into a sequence of rectangular pulses, each of 2 hours duration. Thus, we can use the 2-hour UH obtained in A.
1. Determine the volume of each ERH pulse, Pm, expressed in units of equivalent depth:
 Time (h) Pm (cm) 0 - 2 1.0 2 - 4 3.0 4 - 6 4.0 6 - 8 2.0
2. Use superposition and proportionality principles:

 1 2 3 4 5 6 7 Time(h) UH (m3/s/cm) P1*UH (m3/s) P2*UH (m3/s) P3*UH (m3/s) P4*UH (m3/s) DRH (m3/s) Total (m3/s) 1 0 0 0 100 2 50 50 50 150 3 150 150 0 150 250 4 225 225 150 375 475 5 175 175 450 0 625 725 6 125 125 675 200 1000 1100 7 75 75 525 600 0 1200 1300 8 50 50 375 900 100 1425 1525 9 25 25 225 700 300 1250 1350 10 0 0 150 500 450 1100 1200 11 75 300 350 725 825 12 0 200 250 450 550 13 100 150 250 350 14 0 100 100 200 15 50 50 150 16 0 0 100

1. Columns 2 - 5: Apply the proportionality principle to scale the UH by the actual volume of the corresponding rectangular pulse, Pm. Observe that the resulting hydrographs are lagged so that their origins coincide with the time of occurrence of the corresponding rainfall pulse.
2. Column 6: Apply the superposition principle to obtain the DRH by summing up Columns 2 - 5.
3. Column 7: Add back the baseflow in order to obtain the Total Streamflow Hydrograph.