CIVE22 BASIC HYDROLOGY Jorge A. Ramírez Homework No 7

Transcription

1 Hydrologic Science and Engineering Civil and Environmental Engineering Department Fort Collins, CO (970) CIVE22 BASIC HYDROLOGY Jorge A. Ramírez Homework No 7 1. Obtain a Unit Hydrograph for a basin of km 2 of area using the rainfall and streamflow data tabulated below. Use the horizontal straight-line method to separate baseflow. Observed Hydrograph (m 3 /s) Gross Precipitation (GRH) (cm/h) 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 m 3 /s. Thus, use m 3 /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).

2 V DRH = t Q DRH ( t) dt i Q DRH i Δt Thus, for this example: V DRH =( )m 3 /s(3600)s= m 3 3. Express V DRH in equivalent units of depth: V DRH in equivalent units of depth = V DRH /A basin = m 3 /( m 2 ) = m = 3.5 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. Observed Hydrograph (m 3 /s) Direct Runoff Hydrograph (DRH) (m 3 /s) Unit Hydrograph (m 3 /s/cm) Discharge (m3/s) Observed Hydrograph (m3/s) Direct Runoff Hydrograph (m3/s) Unit Hydrograph (m3/ s/cm) 5. Determine the duration D of the ERH associated with the UH obtained in 4. In order to do this: 2 Jorge A. Ramírez

3 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 = ( ) cm/h 1 h 3.5 cm = 2 cm b) Compute the φ-index equal to the ratio of the volume of losses to the rainfall duration, t r. Thus, φ-index = V Losses /t r = 2 cm / 3 h = 0.67 cm/h c) Determine the ERH by subtracting the infiltration (e.g., φ-index) from the GRH: Effective Precipitation (ERH) (cm/h) As observed in the table, the duration of the effective rainfall hyetograph is 1 hour. Thus, t = 1 hour, and the Unit Hydrograph obtained above is a 1-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 t. This is accomplished by using the principles of superposition and proportionality, encoded in the discrete convolution equation: Q n = n m=1 P m U n m+1 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 m 3 /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. Effective Precipitation (ERH) (cm/h) Jorge A. Ramírez

4 A - The ERH can be decomposed into a sequence of 8 rectangular pulses, each of 1 hour duration. Thus, we can use the 1-hour UH obtained in Problem 1. In order to predict the resulting total streamflow hydrograph, proceed as follows: 1. Determine the volume of each ERH pulse, P m, expressed in units of equivalent depth: P m (cm) Use superposition and proportionality principles: Unit P1*UH P2*UH P3*UH P4*UH P5*UH P6*UH P7*UH P8*UH DRH Hydrograph (m 3 /s) (m 3 /s) (m 3 /s) (m 3 /s) (m 3 /s) (m 3 /s) (m 3 /s) (m 3 /s) (m 3 /s) (m 3 /s/cm) a) Columns 3-10: 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 11: Apply the superposition principle to obtain the DRH by summing up Columns 2-5. c) Column 12: Add back the baseflow in order to obtain the Total Streamflow Hydrograph. d) 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. e) Column 6: Apply the superposition principle to obtain the DRH by summing up Columns 2-5. f) Column 7: Add back the baseflow in order to obtain the Total Streamflow Hydrograph. OH (m 3 /s) 4 Jorge A. Ramírez

5 B Observe that the ERH can also be decomposed into a sequence of 4 rectangular pulses, each of 2 hours duration. Thus, we could also use a 2-hour UH, which could be obtained with the S-hydrograph procedure. 3. 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. (min) Discharge (m 3 /s) Jorge A. Ramírez

6 Assuming that the basin responds as a linear reservoir, the recession limb of the hydrograph is described by the following: Q(t) = Q(t o )e (t t o )/k where k is the recession constant of the system. Observe that this equation is linear in the semi-log domain: lnq(t) = (t t o ) / k + lnq(t o ) Therefore, the recession constant k can be estimated as the inverse of the negative of the slope of a leastsquares fit to the pairs ((t-t o ), lnq(t)). This is accomplished below. (min) Discharge ln(q(t) Jorge A. Ramírez

7 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 = 21 min. Thus, we can use the streamflow data for t >= 21 min to estimate the groundwater recession constant. Using least squares on ((t-t o ), lnq(t)), t >= 21 min, the recession constant is obtained as k 1 = 5 min. In order to estimate the recession constant of the sub-surface storage, k 2, subtract the contribution from the groundwater storage from the total streamflow hydrograph and then follow the same procedure as 7 Jorge A. Ramírez

8 above for the resulting hydrograph. Doing so leads to the following value for the recession constant of the subsurface storage, k 2 = 3.33 min. 8 Jorge A. Ramírez