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Engineering Hydrology
Instructor Y. DaneshbodSpring 1387
2
Hydrologic Analysis
Change in storage w.r.t. time = inflow - outflow
In the case of a linear reservoir, S = kQ
Transfer function for a linear system (S = kQ).
3
Linear Reservoir
0
dSI QdQdSQdQ
S kQdQQ kdt
dQ dtQ k
− =
− =
=
− =
= −
0 0
10
0 0
00
( )0
1ln( ) ln( ) ( )
1ln( ) ( )
k
Q t
Q t
t t
dQ dtQ k
Q Q t tk
Q t tQ k
Q Q e− −
= −
− = − −
= − −
=
∫ ∫
4
Proportionality and superposition
Linear system (k is constant in S = kQ) Proportionality
If I1 Q1 then C*I1 C*Q1
SuperpositionIf I1 Q1 and I2 Q2, then I1 +I2 Q1 + Q2
5
Impulse response functionImpulse input: an input applied instantaneously (spike) at time Impulse input: an input applied instantaneously (spike) at time ττ and zero and zero everywhere elseeverywhere else
An unit impulse at An unit impulse at ττ produces as produces as unit impulse response function unit impulse response function u(tu(t--ττ) )
Principle of Principle of proportionality and proportionality and superpositionsuperposition
6
Convolution integralFor an unit impulse, the response of the system is given by the unit impulse response function u(t-τ)An impulse of 3 units produces the 3u(t-τ)If I(τ) is the precipitation intensity occurring for a time period of dτ, the response of the system (direct runoff) is I(τ)u(t-τ)dτThe complete response due to the input function I(τ) is given by convolution integral
Response of a linear system is the sum (convolution) of the responses to inputs that have happened in the past.
∫ −=t
dtuItQ0
)()()( τττ
7
Step and pulse inputs
A unit step input is an input that goes from 0 to 1 at time 0 and continues indefinitely thereafterA unit pulse is an input of unit amount occurring in duration Δt and 0 elsewhere.
Precipitation is a series of pulse inputs!Precipitation is a series of pulse inputs!
8
Unit Hydrograph Theory
Direct runoff hydrograph resulting from a unit depth of excess rainfall occurring uniformly on a watershed at a constant rate for a specified duration.Unit pulse response function of a linear hydrologic systemCan be used to derive runoff from any excess rainfall on the watershed.
9
Unit hydrograph assumptions
AssumptionsExcess rainfall has constant intensity during duration (indefinite UH)Excess rainfall is uniformly distributed on watershedBase time of runoff is constantOrdinates of unit hydrograph are proportional to total runoff (linearity)Unit hydrograph represents all characteristics of watershed (lumped parameter) and is time invariant (stationarity)
10
Discrete Convolution
∫ −=t
dtuItQ0
)()()( τττ
∑≤
=+−=
Mn
mmnmn UPQ
11
Continuous
Discrete
Q is flow, P is precipitation and U is unit hydrograph
M is the number of precipitation pulses, n is the number of flow rate intervals
The unit hydrograph has n-M+1 pulses
11
Discrete vs. Continuous
12
Application of convolution to the output from a linear system
n=1, m=1
Q1=P1U1-1+1=P1U1
n=2, m=1,2
Q2=P1U2-1+1+P2U2-2+1=P1U1+P2U1
n=3, m=1,2,3
Q3=P1U3-1+1+P2U3-2+1+P3U3-3+1=P1U3+P2U2+P3U1
n=4,5,…
Qn=P1Un+P2Un-1+P3Un-2
Q6=P1U6+P2U5+P3U4
Q7=0+P2U6+P3U5
Q8=0+0+P3U6
13
Unit Hydrograph Derivation
14
Unit Hydrograph Derivation
15
Unit Hydrograph Derivation
16
Application of UH
Once a UH is derived, it can be used/applied to find direct runoff and stream flow hydrograph from other storm events.
17
Application of UH
Ex. 7.5.1
Given:P1 = 2 in, P2 = 3 in and P3 = 1 in, baseflow = 500 cfs and watershed
area is 7.03 mi2. Given the Unit Hydrograph below, determine the streamflow hydrograph
18
Application of UHThe time interval used in defining the excess rainfall hyetograph ordinates must be the same as that for which the unit hydrograph was specified.
19
7.5.1 solution (cont’d)
20
Unit Hydrograph Derivation
How to obtain 2hr unit hydrograph from 1hr unit hydrograph? How to obtain 0.5hr unit hydrograph from 1hr hydrograph? (S–Curve)
Integer Multiple vs. Real Multiple
21
S-Curve
22
S-Curve
23
Gauged and ungauged watersheds
Gauged watershedsWatersheds where data on precipitation, streamflow, and other variables are available
Ungauged watershedsWatersheds with no data on precipitation, streamflow and other variables.
24
Synthetic UH
Synthetic hydrographs are derived by Relating hydrograph characteristics such as peak flow, base time etc. with watershed characteristics such as area and time of concentration.Using dimensionless unit hydrographBased on watershed storage