CE322 Basic Hydrology
Jorge A. Ramírez

Unit Hydrographs

 

 

1. Obtain a Unit Hydrograph for a basin of 282.6 km² of area using the rainfall and streamflow data tabulated below.

 

 

         In this process: use the horizontal straight-line method to separate baseflow.

 

 

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 150 m³/s. Thus, use 150 m³/s as the baseflow.

 

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:

V_DRH = (200+650+1050+750+600+400+200+75) m³/s (3600) s = 14'130,000 m³

 

3.             Express V_DRH in equivalent units of depth:

 

V_DRH in equivalent units of depth = V_DRH/A_basin = 14'130,000 m³/(282600000 m²) = 0.05 m = 5 cm.

 

4.             Obtain a Unit Hydrograph by normalizing the DRH. Normalizing implies dividing the ordinates of the DRH by the V_DRH in equivalent units of depth.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5.             Determine the duration D of the ERH associated with the UH obtained in 4. In order to do this:

a)             Determine the volume of losses, V_Losses which is equal to the difference between the volume of gross rainfall, V_GRH, and the volume of the direct runoff hydrograph, V_DRH .

V_Losses = V_GRH - V_DRH = (0.25 + 2.75 + 2.75 +0.25) cm/h 1 h - 5 cm = 1 cm

 

b)             Compute the f-index equal to the ratio of the volume of losses to the rainfall duration, t_r. Thus,

 

f-index = V_Losses/t_r = 1 cm / 4 h = 0.25 cm/h

 

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

 

 

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 Q_n is the n^th ordinate of the DRH, P_m is the volume of the m^th rainfall pulse expressed in units of equivalent depth (e.g., cm or in), and U_n-m+1 is the (n-m+1)^th ordinate of the UH, expressed in units of m³/s/cm.

 

2. For the basin of Problem 1, predict the total streamflow hydrograph that would be observed as a result of a storm whose effective rainfall is tabulated below. Use the same value of baseflow as for Problem 1.

 

 

 

         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 Problem 1. To do so:

 

1.             Determine the volume of each ERH pulse, P_m, expressed in units of equivalent depth:

 

 

2.             Use superposition and proportionality principles:

 

 

a)             Columns 2 - 5: Apply the proportionality principle to scale the UH by the actual volume of the corresponding rectangular pulse, P_m. Observe that the resulting hydrographs are lagged so that their origins coincide with the time of occurrence of the corresponding rainfall pulse.

 

b)            Column 6: Apply the superposition principle to obtain the DRH by summing up Columns 2 - 5.

 

c)             Column 7: Add back the baseflow in order to obtain the Total Streamflow Hydrograph.

 

 

3. Use the 1-h Unit Hydrograph tabulated below to predict the total streamflow hydrograph that would be observed as a result of a storm whose effective rainfall is also tabulated below. Use the same value of baseflow as for Problem 1. Obtain the solution using the S-Hydrograph method.

 

 

 

The ERH is composed of 3 rectangular pulses of 3-hour duration each. Thus, we need a 3-h Unit Hydrograph. Since the only available hydrograph is a 1-h Unit Hydrograph, we need to develop a new 3-h UH using the S-Hydrograph method, as illustrated below. Observe that the S-hydrograph is obtained in 2 different ways: a) in tabular form as the superposition of an infinite sequence of 1-h UH's each lagged by 1-h; and b) as the cumulative volume of UH per unit time.

 

 

 

 

 

3.             Determine the volume of each ERH pulse, P_m, expressed in units of equivalent depth:

 

 

4.             Use superposition and proportionality principles:

 

 

 

 

a)      Columns 2 - 4: Apply the proportionality principle to scale the UH by the actual volume of the corresponding rectangular pulse, P_m. Observe that the resulting hydrographs are lagged so that their origins coincide with the time of occurrence of the corresponding rainfall pulse.

 

b)      Column 5: Apply the superposition principle to obtain the DRH by summing up Columns 2 - 4.

 

c)      Column 6: Add back the baseflow in order to obtain the Total Streamflow Hydrograph.

 

4. The table below presents the recession limb of a total streamflow hydrograph from a basin. Use the tabulated data to obtain the groundwater recession constant k for this basin.

 

 

 

 

 

 

 

Assuming that the basin responds as a linear reservoir, the recession limb of the hydrograph is described by the following:

 

 

where k is the recession constant of the system. Observe that this equation is linear in the semi-log domain:

 

Therefore, the recession constant k can be estimated as the negative of the slope of a least-squares fit to the pairs ((t-t_o), lnQ(t)). This is accomplished below.

 

 

 

 

 

However, given that there are several distinct storages in a basin, the recession limb of hydrographs includes contributions from all of those storages. Thus, the procedure outlined above can be used sequentially to obtain the corresponding recession constants for each one of the storages (e.g., groundwater storage, subsurface storage). The existence of the different storages is easily observable in the semi-log domain as shown below.

 

         In the graph below, observe that the slowest portion of the recession starts at time t = 14 h. Thus, we can use the streamflow data for t > 14 h to estimate the groundwater recession constant. Using least squares on ((t-t_o), lnQ(t)), t > 14 h, the recession constant is obtained as k = 0.5.