UNIT HYDROGRAPH THEORY

Input, I Output, OTransferFunction

Ot = f{It,It-1,It-1, ......, It-n}

RainfallHyetograph

DrainageBasin

DischargeHydrograph

Time

Rai

nfal

l

0

10

20

TimeD

isch

arge

0

10

20 IO

LUMPED HYDROLOGIC MODELS

Before GEO3280!

BLACK BOX

UNIT HYDROGRAPH THEORY

Sherman - 1932Horton - 1933Wisler & Brater - 1949 - “the hydrograph of

surface runoff resulting from a relatively short, intense rain, called a unit storm.”

The runoff hydrograph may be “made up” of runoff that is generated as flow through the soil (Black, 1990).

WHAT IS THE THEORY BEHIND THE UNIT HYDROGRAPH ?

Impervious Parking Lot

OutletContributing A7A6A5A4A3A2A1Time

A1PA1P0-1A2PA2P1-2A3PA3P2-3A4PA4P3-4A5PA5P4-5A6PA6P5-6A7PA7P6-7

A unit of precipitation, P, falling for 1 minute, uniformlyover the entire area.

1mm (0.001m) of precipitation falling for 1 minute, uniformlyover the entire area.

OutletVolumeContributing Areas (m2)

(m3)17032835042443024454Time0.0540.0540-10.2440.2441-20.4300.4302-30.4240.4243-40.3500.3504-50.3280.3285-60.1700.1706-7

Time (mins)

1 2 3 4 5 6 7 8 9 10 11

1 2 3 4 5 6 7 8 9 10 11

Dis

char

ge a

t Out

let (

m3 m

in-1

)

0.0

0.2

0.4

0.6

0.8

1.0

Prec

ipita

tion

(mm

)

01234

56

Outflow Hydrograph in Response to 1mm (0.001m)of Precipitation Falling for 1 minute, Uniformly

over the Entire Area.

OutletVolumeContributing Areas (m2)

(m3)17032835042443024454Time0.0540.0540-10.2440.2441-20.4300.4302-30.4240.4243-40.3500.3504-50.3280.3285-60.1700.1706-7

ASSUMPTIONS:

1. RESPONSE IS ALWAYS DIRECTLY PROPORTIONAL TO THE INPUT.

Time (mins)

1 2 3 4 5 6 7 8 9 10 11

1 2 3 4 5 6 7 8 9 10 11

Prec

ipita

tion

(mm

)

Dis

char

ge a

t Out

let (

m3 m

in-1

)

0.0

0.2

0.4

0.6

0.8

1.0 01234

56

Outflow Hydrograph in Response to 2mm (0.002m)of Precipitation Falling for 1 minute, Uniformly

over the Entire Area.

OutletVolumeContributing Areas (m2)

(m3)17032835042443024454Time0.1080.1080-10.4880.4881-20.8600.8602-30.8480.8483-40.7000.7004-50.6560.6565-60.3400.3406-7

NOTE:

RESPONSE IS ALWAYS PROPORTIONATE TO THE INPUT. DOUBLE INPUT- DOUBLE RESPONSE.

Time (mins)

1 2 3 4 5 6 7 8 9 10 11

1 2 3 4 5 6 7 8 9 10 11

Dis

char

ge a

t Out

let (

m3 m

in-1

)

0.0

0.2

0.4

0.6

0.8

1.0

Prec

ipita

tion

(mm

)01234

56

Outflow Hydrograph in Response to 2.2mm (0.0022m)of Precipitation Falling for 1 minute, Uniformly

over the Entire Area.

OutletVolumeContributing Areas (m2)

(m3)17032835042443024454Time0.1190.1190-10.5370.5371-20.9460.9462-30.9330.9333-40.7700.7704-50.7220.7225-60.3740.3746-7

NOTE:

TWO POINT TWO TIMES THE INPUT- TWO POINT TWO TIMES THE RESPONSE!

ASSUMPTIONS:

1. RESPONSE IS ALWAYS DIRECTLY PROPORTIONAL TO THE INPUT.

2. TIME-BASE OF RESPONSE IS ALWAYSFIXED REGARDLESS OF THE INPUT

Time (mins)

1 2 3 4 5 6 7 8 9 10 11

1 2 3 4 5 6 7 8 9 10 11

Dis

char

ge a

t Out

let (

m3 m

in-1

)

0.0

0.2

0.4

0.6

0.8

1.0 01234

56

Outflow Hydrograph in Response to 1mm (0.002m)of Precipitation Falling for 2 Consecutive Minutes, Uniformly

over the Entire Area.

VolumeContributing Areas (m2)(m3)17032835042443024454Time

0.0540.0540-10.2980.2440.0541-20.6740.4300.2442-30.8540.4240.4303-40.7740.3500.4244-50.6780.3280.3505-60.4980.1700.3286-70.1700.1707-8

NOTE:

TWO CONSECUTIVE INPUTSOF 1mm/min PRODUCESTWO CONSECUTIVE, BUTSUPERIMPOSED RESPONSES,THE TIME BASES OF WHICHARE EQUAL.

Outflow Hydrograph in Response to 0.25mm, 2.1mm and 0.8mmof Precipitation Falling for 3 Consecutive Minutes, Uniformly

over the Entire Area.

VolumeContributing Areas (m2)(m3)17032835042443024454Time

0.0140.0140-10.1740.0610.1131-20.6630.1080.5120.0432-31.2040.1060.9030.1953-41.3220.0880.8900.3444-51.1560.0820.7350.3395-61.0110.0430.6890.2806-70.6190.3570.2627-80.1360.1368-9

Time (minutes)

0 1 2 3 4 5 6 7 8 9 10

Pre

cip

(mm

)

Dis

char

ge a

t Out

let (

m3 m

in-1

)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.401

2

3

NOTE:

THE TIME BASE OF EACHRESPONSE REMAINS AT 7MINUTES, REGARDLESS OF WHETHER THE IMPUT IS 0.22 OR 1 mm/minutes AS IN THIS MORE COMPLEX EVENT

HOW CAN A UNIT HYDROGRAPHBE DERIVED IN REALITY?

Days since April 30, 1969

0 30 60 90 120 150 180 210 240 270 300 330 360

Mea

n D

aily

Dis

char

ge (m

3 s-1

)

0

20

40

60

80

100

120

140

May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr

DAILY DISCHARGE AT ELECTRIONA

1. Select fairly clearly defined single events from available records

Avoid this sort of complex response where it is hard to separate one storm from the next

Days since April 30, 1969.

205 210 215 220 225

Mea

n D

aily

Dis

char

ge (m

3 s-1

)

0

20

40

60

80

100

120

140Dec. 1Nov 25 Dec. 13

2. Extract observed hydrograph

Days since April 30, 1969.

205 210 215 220 225

Mea

n D

aily

Dis

char

ge (m

3 s-1

)

0

20

40

60

80

100

120

140Dec. 1Nov 25 Dec. 13

3. Separate out the “Direct Runoff” response from the groundwater or baseflow.

GROUNDWATER?

Direct RunoffHydrograph

1. Define point at which flood hydrograph rises2. Superimpose Master Recession curve of declining limb.3. Identify “Inflection Point” at which baseflow is dominant process4. Join Time of Rise to Inflection Point with straight line5. Graphically separate Direct Runoff Hydrograph from Base Flow

HYDROGRAPH SEPARATION

Time (T)

0 10 20 30 40 50

Sta

ge (L

)

0

20

40

60

80

100

120

140

160

Time ofRise

InflectionPoint

MasterRecession

Straight Line

DRH

BASE FLOW

Days since April 30, 1969

0 30 60 90 120 150 180 210 240 270 300 330 360

Mea

n D

aily

Dis

char

ge (m

3 s-1

)

0

20

40

60

80

100

120

140

May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr

DAILY DISCHARGE AT ELECTRIONAHOW TO COMPUTE MASTER RECESSION CURVE

CREATING MASTER RECESSION CURVE

Time (T)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Sta

ge (L

)

15

20

25

30

35

40

Master Recession

IndividualRecession

Curves

4. Determine the quantity and temporal distribution of “Effective Rainfall” that gave rise to that Direct Runoff Hydrograph

As is assignments and lectures, this involves consideration of water lost to interception, depression storage and infiltration.

Average StormEffective Precipitation, p

Storm Duration, d

Effective Hydrograph37.6mm over 90 minutes

Time (mins)0 60 120 180 240 300 360 420 480 540 600 660 720 780

Run

off (

mm

)

0

20

40

60

80

100

120

5. Determine the Direct Runoff Hydrograph resulting from an Effective Precipitation of p mm over d hours/minutes in duration

Direct RunoffHydrograph

Effective Hyetograph

HOW DO YOU PROCEED FROM THIS 37.6 mm 90 min UNIT HYDROGRAPH TO ONE OF MORE CONVENTIONAL

DEPTH (e.g. 25mm) AND TIME (e.g. 1 hour) UNITS?

1. Use the Equal Time Base of Response assumption

Time (minutes)

0 60 120 180 240 300 360 420 480 540 600 660 720 780

0 60 120 180 240 300 360 420 480 540 600 660 720 780R

unof

f (m

m)

0

50

100

150

200

250

300 0

2040

Precip.

(mm

)

90mins

90mins

• This S-curve showing the response to a very large number of sequential 37.6mmm in 90 events would be the same no matter what time we actually started the sequence, at time 0, or after 5 minutes, or 60 minutes.

• Imagine that we set such a sequence running at time 0, and a second identical sequence running at time 60 minutes.

• We have two s-curves offset by 60 minutes.• The difference in discharges between the two S-

curves, measured at any time, represents the response of the first S-curve, that the second one hasn’t yet experienced because of the 60 minute shift.

• This difference therefore represents the “missing” rainfall in that 60 minute period, or the 37.6mm for 60 minutes hydrograph.

Time (mins)0 60 120 180 240 300 360 420 480 540 600 660 720 780

0 60 120 180 240 300 360 420 480 540 600 660 720 780R

unof

f (m

m)

0

50

100

150

200

250

0

50

100

150

200

250

OFFSET 37.6mm/90min S-CURVE BY 60 mins.

Time (mins)

0 60 120 180 240 300 360 420 480 540 600 660 720 780

Run

off (

mm

)

0

20

40

60

800 60 120 180 240 300 360 420 480 540 600 660 720 780

0 60 120 180 240 300 360 420 480 540 600 660 720 780

Run

off (

mm

)

0

50

100

150

200

250

0

50

100

150

200

250

Offset 37.6mm-90 min

S-Curves

37.6mm-60 min UnitHydrograph

HOW DO WE CONVERT THIS 37.6mm for 1hr UNIT HYDROGRAPH INTO

SOMETHING MORE STANDARD LIKE A 25mm (1”) for 1 hour UNIT

HYDROGRAPH?

Use the assumption of a proportional response.

Time (mins)0 60 120 180 240 300 360 420 480 540 600 660 720 780

0 60 120 180 240 300 360 420 480 540 600 660 720 780

Run

off (

mm

)

0

20

40

60

80

0

20

40

60

80

Derive 25.4mm-60min U.H.Multiply x-coordinate

by (25.4/37.6)

37.6mm - 60 min U.H.25.4 mm(1")-1hr U.H.

DERIVED 25.4mm (1")-1 hr. UNIT HYDROGRAPH

Time (mins)

0 60 120 180 240 300 360 420 480 540 600 660

0 60 120 180 240 300 360 420 480 540 600 660

Run

off (

mm

)

0

10

20

30

40

50

60

0

10

20

30

40

50

60 HERE IS OUR BASIC “BUILDING BLOCK”, DERIVED WITHOUT HAVING TO GO THROUGH THE EXERCISE OF ISOCHRONES ETC., BUT WHICH CAN BE USED TO FORECAST RESPONSES TO STORM DELIVERING VARYING QUANTITIES OF WATER IN DIVERSE PATTERNS OVER TIME.