Basic Hydrology: Rainfall-Runoff – I

Rainfall-Runoff • Rainfall-runoff (modeling) is a tool used to

estimate the discharge produced by a rainfall event over a project area for hydraulic design.

• Unlike flood frequency the rainfall probabilities are expressed as a combination of frequency (same idea as AEP), depth, and duration.

• The inclusion of depth and duration reflects that different “storms” can produce the same total depth, but deliver that depth over much different times Consider a slow gentle rain for a long time versus a fast hard rain very rapidly

Rainfall-Runoff

Loss Model(s)Precipitation

Excess PrecipitationRouting/Storage

Discharge

Depth-Duration-Frequency• Depth of rainfall is the accumulated depth (in a

gage) over some time interval.

• Duration is that time interval.

• Frequency is the probability (like AEP) of observing the depth over the given duration.

• These relationships for a location are conveyed in families of DDF Curves analogous to a flood frequency curve, but different curves for different AEP

Depth-Duration-Frequency• DDF curve

e.q. 12 hour, 100-year (AEP=1%), depth is 70 millimeters

FrequencyAEP; ARI

Duration

Depth

Intensity-Duration-Frequency• An alternate form of DDF is to present the

magnitude as an intensity (a rate).

• Intensity is the ratio of an accumulated depth to some averaging time, usually the duration.

• Intensity is NOT the instantaneous rainfall rate.

Depth-Duration-Frequency• Intensity (average rate)

e.q. 12 hour, 100-year (AEP=1%), depth is 70 millimetersaverage intensity is 70mm/12hr = 5.8 mm/hr

Duration

Depth

Intensity-Duration-Frequency• IDF curves

e.q. 20 minute hour, 5-year (AEP=20%), intensity is 5.5 inches/hr

FrequencyAEP; ARI

Intensity

DDF and IDF

• Depth is average intensity multiplied by the time of accumulation.

• Depth is also the integral of the instantaneous intensity over the duration

Instantaneous rate

Depth, Intensity, and Duration

Slope of this line isAverage intensity for 24 hours

Slope of tangent line isinstant intensity for hour 9

Depth, Intensity, and Duration

• Conversion from Depth-Duration to Intensity-Duration is obtained by the ratio of depth to duration.

• Conversion from Intensity-Duration to Depth-Duration is obtained by multiplication.

using same duration!

DDF Data Sources• DDF from various sources.

• NWS TP40 (online)• NWS HY35 (online)• Texas DDF Atlas (online)

• All similar – have maps of depth for given duration and AEP (ARI)

DDF Data Sources

Harris County 3 hour, 5-year (AEP=20%) depth = 3.6 inches

Location

ARI (AEP = 1/5 = 20%)Duration = 3 hour

Duration (hours)

Inte

nsity

(inc

hes

per h

our)

Intensity-Duration Curve

0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 250

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Frequency matching – P(flood)=P(rainfall)

• The principal assumption of rainfall-runoff analysis is that a particular discharge event is produced by a rainfall event of the same probability

• It is only an assumption; there are situations where there is no reason to expect the probabilities to be equivalent

• Although an assumption, it is the only way (currently) to proceed with rainfall-runoff modeling

Rational equation• The rational equation is a model that estimates

peak discharge for small drainage areas as the product of the contributing area, a rainfall intensity for a time of duration called the time of concentration, and a rational runoff coefficient.

• The duration is obtained from consideration of the drainage area’s flow paths, slopes and such – this duration is called the “Time of Concentration”

• The runoff coefficients are tabulated and associated with descriptive properties of the drainage area.

Rational equation• The HDM and HDS-2 (as well as many other

references) contain guidance on computing the time of concentration.

• The HDM and HDS-2 (as well as many other references) contain guidance on selecting appropriate runoff coefficients.

• The rational equation produces estimates of peak discharge only – it does not produce a hydrograph

• The rational equation has limited applicability – the limits are listed in the HDM

Hydrographs• A hydrograph is a plot of discharge versus time at a

location.

• When rainfall that produces runoff falls on a watershed, the runoff signal will eventually be observed in the hydrograph, usually as some kind peak a certain time after the rainfall begins

Hydrographs

Time

Flow rate

• Flow rate, Q, as a function of time.

• Values depend upon:• Rainfall depth• Watershed

characteristics

Rising limb

PeakRecession

limb

Rainfall-Runoff and Hydrographs

• Precipitation– DDF/IDF

Transformation

Direct runoff hydrograph shape• Hydrograph shape depends on watershed characteristics.• What will the hydrograph look like for each of these

watersheds, assuming a long duration storm?

Direct runoff hydrograph shape• Hydrograph shape depends on watershed characteristics.• Which of these shapes corresponds to the previous

watershed shapes?

? ?

Unit-hydrograph based methods• Unit hydrographs (UH) are another method to

estimate discharge caused by rainfall• Unlike the rational equation, the UH method

generates a hydrograph estimate – both peak discharge and volumes can be reasonably examined

Runoff Volume

• Runoff volume = “area under the curve” of a hydrograph

• For example• Sum of flows =2502 cfs• Volume = 2502 cfs * 3 hr =

27 million ft3 = 620 ac ft• Divide by watershed area to

compute an equivalent uniform depth (EUD)

• The runoff volume is expressed in “watershed inches”

0

50

100

150

200

250

300

0 3 6 9 121518212427303336394245485154576063666972Time (hr)

Flow

(cfs

)

…

Add areas under curve

Unit-hydrograph based methods

• A unit hydrograph is the graph of outflow from a watershed from an input of 1 unit of EXCESS rainfall in a given time period.

• The EXCESS is the result of abstractions (storage, loss) from the incoming precipitation signal.

• The EXCESS volume is equal to the runoff volume (expressed in watershed inches)

• If there were no losses or storage, the volume of raw rain into the watershed would equal the volume of runoff produced by the watershed

Unit Hydrographs

Unit Hydrographs

Excess as series of pulses

Inte

nsity

Time

Time

Inte

nsity

If pulses went immediately to the outlet, would expect direct hydrograph to have same shape. Q(t) = i(t)A

Run

off

Time

But pulses are assumed uniform over whole area – close to outlet arrive sooner than far from outlet– Hence there is time re-distribution

Unit Hydrographs

Excess as series of pulses

Inte

nsity

Time

Time

Inte

nsity

If pulses went immediately to the outlet, would expect direct hydrograph to have same shape. Q(t) = i(t)A

Run

off

Time

But pulses are assumed uniform over whole area – close to outlet arrive sooner than far from outlet– Hence there is time re-distribution

Unit Hydrographs

Excess as series of pulses

Inte

nsity

Time

Time

Inte

nsity

The unit hydrograph is the “function” that maps the time-distribution of pulses of excess precipitation to the time-distribution of direct runoff.

Run

off

Time

Unit-Hydrograph Assumptions

• Direct runoff duration (time) is same for all uniform-intensity storms of same duration (time).

• Two excess hyetographs of the same duration (time) will produce direct runoff hydrographs of the same duration (time) but with runoff rates proportional to the volumes (depth) of the excess hyetographs.

• The time distribution of direct runoff from a given storm duration is independent of concurrent runoff form prior storms.

Unit-Hydrograph Assumptions

• Unit hydrographs have a particular time base – a UH response for a 1-hour storm is different than a UH response for a 3-hour storm.

• The HDM describes several different UH models for design application• Involve estimation of a characteristic

response time that has various names: basin lag, time to peak, etc.

• Involves some characterization of hydrograph shape.

Summary

• Rainfall input described by DDF or IDF curves• Intensity is an average rate over the duration• DDF values are obtained from NWS or similar

sources – they are mapped to locations• DDF curves can be constructed for a location by

combining values for different durations and AEPs

• Rainfall-runoff analysis ASSUMES the X-probability rainfall event produces the X-probability discharge event

Summary• Rational equation estimates peak discharge• Rational equation has limited applications• Hydrograph based methods produce entire

hydrographs