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Lecture 6b_Unit Hydrograph lecture presentation
- 1. Department of WR WaterInstitute (WI) Dar es Salaam, Tanzania WRUO7212: Hydrological Analysis and Design Lecture 6b – Unit Hydrograph
- 2. Unit Hydrograph • Thetheory of the unit hydrograph was introduced by Sherman in 1932. The method is based on the assumption that the physical characteristics within a river basin (such as slope, size, drainage network, etc.) do not change significantly, and consequently there should be a great similarity in the shape of the hydrographs resulting from similar high intensity rainfalls • The unit hydrograph is defined as the runoff of a catchment resulting from a unit depth of effective rainfall (e.g. 1 cm) falling uniformly in space and time during a period T (minute, hour, day). It should be noted that the intensity of the rainfall during this period T is equal to 1/T in order to obtain unit depth. The requirement of an effective precipitation falling uniformly in space limits the application of the unit hydrograph theory to catchments smaller than 100 - 500 km2 , since for larger basins the assumption of a uniform distribution of the rainfall is hardly ever valid.
- 3. Unit Hydrograph • EffectiveRainfall is that part of rainfall that becomes prompt surface runoff i.e. The portion of rainfall that reaches a river. • The other part of rainfall contributes to initial losses such as soil moisture deficit, interception, evaporation, depression storage, etc. • The portion of effective rainfall increases and that of lost rainfall decreases with increasing rainfall duration • The purpose of establishing a Unit Hydrograph which establishes a rainfall-runoff relationships for a given basin is to give a quick prediction of expected flows from a storm that has already occurred. to predict future floods from data of probabilities of rainfall.
- 4. Unit Hydrograph • Thespecific period of time for the excess rainfall T is known as the ‘unit storm period’. • For small to medium sized drainage basins there is a certain unit storm period for which the shape of the hydrograph is not significantly affected by changes in the time distribution of the excess rainfall over this unit storm period • This means that equal depths of excess rainfall with different time-intensity patterns produce hydrographs of direct runoff which are the same when the duration of this excess rainfall is equal to or shorter than the unit storm period.
- 5. Example of aunit hydrograph Effective rainfall, Pe and the unit hydrograph, DUH (Distribution Unit Hydrograph) are expressed in the same units: cm/d. The unit hydrograph has a length of 4 days. The memory of the rainfall-runoff system is 3 days, since 3 days after the rain has stopped, the last rainfall excess comes to runoff. If the ordinates of the DUH are expressed in the same unit as the rainfall excess, they should sum to one. This will not be so if the ordinates convert unites from e.g. cm/d to m3 /s.
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- 7. • Assumption 1: Assumption1: The excess rainfall has a constant intensity constant intensity within the effective duration effective duration – the storms selected for analysis should be of short duration, since these will most likely produce an intense and nearly constant excess rainfall rate, yielding a well-defined single-peaked hydrograph of short time base • Assumption 2: Assumption 2: The excess rainfall is uniformly distributed uniformly distributed throughout the drainage area – the unit hydrograph may become inapplicable when the drainage area is too large to be covered by a nearly uniform distribution of rainfall 7 Unit Hydrograph Assumptions Assumptions
- 8. • Assumption 3: Assumption3: The base time base time of the DRH (the duration of direct runoff) resulting from an excess rainfall of given duration is constant is constant – the base time of DRH is generally uncertain but depends on the method of baseflow separation . The base time is usually short if the direct runoff is considered to include the surface runoff only and long if the direct runoff also includes subsurface runoff. 8 Unit Hydrograph Assumptions… Assumptions…
- 9. • Assumption 4: Assumption4: The ordinates of all DRH’s of a common base time are directly proportional to the total amount of direct runoff represented by each hydrograph – the principles of superposition superposition and proportionality proportionality are assumed so that the ordinates Qn of the DRH may be computed by discrete convolution equation. • Principle of Proportionality: Principle of Proportionality: If a solution f(Q) is multiplied by a constant c, the resulting cf(Q) is also a solution • Principle of Additivity or Superposition: Principle of Additivity or Superposition: If two solutions f1(Q) and f2(Q) of the equation are added, the resulting function f (Q) = f1(Q)+f2(Q) is also a solution of the equation 9 Unit Hydrograph Assumptions… Assumptions…
- 10. • Assumption 5: Assumption5: For a given watershed, the hydrograph resulting from a given excess rainfall reflects the unchanging characteristics of the watershed – the unit hydrograph is considered unique unique for a given watershed and invariable invariable with respect to time (principle of time invariance principle of time invariance) – This principles of time invariant, superposition and proportionality are fundamental to the unit hydrograph model – Unit hydrographs are applicable only when channel conditions remain unchanged and watersheds do not have appreciable storage • This condition is violated when the drainage area contains reservoirs or when floods overflow into the floodplain thereby producing considerable storage Unit Hydrograph Assumptions… Assumptions…
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- 12. 12 Resultant DR hydrograph Effectiverainfall unit hydrographs Unit Hydrograph Theoretical Derivation Theoretical Derivation Unit duration
- 14. The set ofequations for discrete time convolution n M n m n m 1 m 1 Q P U 1 1 1 Q PU 2 2 1 1 2 Q P U PU 3 3 1 2 2 1 3 Q P U P U PU M M 1 M 1 2 1 M Q P U P U ..... PU M 1 M 2 2 M 1 M 1 Q 0 P U ..... P U PU N 1 M N M M 1 N M 1 Q 0 0 ..... 0 0 ..... P U P U N 1 M 1 N M 1 Q 0 0 ..... 0 0 ..... 0 P U n = 1, 2, …,N 2 3 1 2 1 n n n n U P U P U P Q For n > m, there continue to be just three terms Note: The notation n ≤ M as the upper limit of the summation shows that the terms are summed for m = 1, 2, …, n for n ≤ M, but for n > M, the summation is limited to m = 1, 2, …, M 14 Unit Hydrograph Derivation… Derivation… Discrete Convolution Equation
- 15. Example 1: Find thehalf-hour unit hydrograph using the excess rainfall excess rainfall hyetograph hyetograph (ERH) and direct runoff hydrograph direct runoff hydrograph (DRH) given in the table. Solution: The ERH and DRH in table have M=3 and N=11 pulses, respectively. Hence, the number of pulses in the unit hydrograph is N-M+1=11- 3+1=9. Substituting the ordinates of the ERH and DRH into the equations yields a set of 11 simultaneous equations. Time (1/2hr) Excess Rainfall (in) Direct Runoff (cfs) 1 1.06 428 2 1.93 1923 3 1.81 5297 4 9131 5 10625 6 7834 7 3921 8 1846 9 1402 10 830 11 313 15 Unit Hydrograph Derivation… Derivation…
- 16. 1 1 1 Q428 U 404 cfs/in P 1.06 3 3 1 2 2 3 1 Q P U P U 5,297 1.81x404 1.93x1 ,079 U 2,343 cfs/in P 1.06 and similarly for the remaining ordinates 4 9,131 1.81x1 ,079 1.93x2,343 U 2,506cfs/in 1.06 5 10,625 1.81x2,343 1.93x2,506 U 1 ,460 cfs/in 1.06 6 7,834 1.81x2,506 1.93x1 ,460 U 453 cfs/in 1.06 n 1,079cfs/i 1.06 1.93x404 1,928 P U P Q U 1 1 2 2 2 16 Unit Hydrograph Derivation… Derivation…
- 17. n 1 23 4 5 6 7 8 9 Un (cfs/in) 404 1,079 2,343 2,506 1,460 453 381 274 173 7 3,921 1.81x1 ,460 1.93x453 U 381cfs/in 1.06 8 1 ,846 1.81x453 1.93x381 U 274 cfs/in 1.06 9 1 ,402 1.81x381 1.93x274 U 173 cfs/in 1.06 Unit hydrograph 17 Unit Hydrograph Derivation… Derivation…
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- 19. • Once theunit hydrograph has been determined, it may be applied to direct runoff and streamflow hydrograph. • Procedures: • A rainfall hyetograph is selected. • The abstractions are estimated. • The excess rainfall is calculated. • The time interval time interval used in defining the excess rainfall hyetograph excess rainfall hyetograph ordinates must be the same must be the same as that for which the unit hydrograph unit hydrograph was specified. • The discrete convolution equation may then be used to yield the direct runoff hydrograph. • By adding an estimated baseflow to the direct runoff hydrograph, the streamflow hydrograph is obtained. 19 Unit Hydrograph Unit Hydrograph Applications – Applications – Total Runoff Estimation Total Runoff Estimation
- 20. Example 2: Example 2: Calculatethe streamflow hydrograph for a storm of 6 in excess rainfall, with 2 in the first half-hour, 3 in in the second half-hour and 1 in the third half-hour. Use the half-hour unit hydrograph computed in Example 1 and assume the baseflow is constant at 500 cfs throughput the flood. Check that the total depth of direct runoff is equal to the total excess precipitation. (Watershed are = 7.03 mi2 ) 1 1 1 Q PU 2.00x404 808cfs 2 2 1 1 2 Q P U PU 3.00x404 2.00x1 ,079 1 ,212 2,158 3,370cfs 3 3 1 2 2 1 3 Q P U P U PU 1.00x404 3.00x1 ,079 2.00x2,343 404 3,237 4,686 8,327cfs Solution: Solution: 20 Time interval = ½ hour Unit Hydrograph Unit Hydrograph Applications – Applications – Total Runoff Estimation… Total Runoff Estimation…
- 21. Time (1/2 hr) Excess Precipitation (in) Unit HydrographOrdinates (cfs/in) Direct Runoff (cfs) Stream flow (cfs) 1 2 3 4 5 6 7 8 9 404 1079 2343 2506 1460 453 381 274 173 N=1 2 3 4 5 6 7 8 9 10 11 2.00 3.00 1.00 808 1212 404 2158 3237 1079 4686 7029 2343 5012 7518 2506 2920 4380 1460 906 1359 453 762 1143 381 548 822 274 346 519 173 808 3370 8327 13120 12781 7792 3581 2144 1549 793 173 1308 3870 8827 13620 13281 8292 4081 2644 2049 1293 673 Calculation of the direct runoff hydrograph and streamflow hydrograph Baseflow = 500 cfs 21 Unit Hydrograph Unit Hydrograph Applications – Applications – Total Runoff Estimation… Total Runoff Estimation…
- 22. The total directrunoff volume is N d n n 1 3 7 3 V Q t 54,438x0.5 cfs.h ft .h 3,600s 54,438x0.5 x s 1s 9.80x10 ft The corresponding depth of direct runoff is found by dividing by the watershed area A=7.03 mi2 =7.03 x (5,2802 ) ft2 =1.96x108 ft2 7 d d 8 V 9.80x10 r ft 0.50 ft 6.00 in A 1.96x10 22 Unit Hydrograph Unit Hydrograph Applications – Applications – Total Runoff Estimation… Total Runoff Estimation…
- 23. Limitations of theUnit Hydrograph theory 1. Space invariance of effective rainfall (ER) ER seldom occurs uniformly over a drainage basin (watershed) Storm movement involves variability of precipitation Can be minimized by dividing the drainage basin into sub-watersheds 2. Time invariance of effective rainfall (ER) ER does not occurs uniformly, even for a 5 minutes duration… However, the effect of rainfall variability diminishes for small watersheds Infiltration varies as well over the surface of the watershed 3. Validity of the linearity hypothesis The principles of proportionality and superposition assume linearity of the watershed, which is not physically true All watershed are non-linear in nature: some more non-linear, some less The resulting hydrograph is only an approximation, which is satisfactory for many practical applications