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