Rainfall-Runoff Modeling (2)

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Applied Hydrology

RSLAB-NTU

Lab for Remote Sensing Hydrology and Spatial Modeling

Rainfall-Runoff Modeling (2)

Professor Ke-Sheng ChengDept. of Bioenvironmental Systems EngineeringNational Taiwan University

Lab for Remote Sensing Hydrology and Spatial ModelingRSLAB-NTU

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Peak flow estimation

Rational methodEmpirical methodUnit hydrograph technique

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The concept of Isochrones

Let’s assume that raindrops fall on a spatial point x within the watershed require an amount of travel time t(x) to move from point x to the basin outlet.

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Contour lines of the travel time are termed as isochrones (or runoff isochrones) of the watershed.

The highest value of isochrones represents the time of concentration of the watershed.

Assume the effective rainfall has a constant intensity (in time) and is uniformly distributed over the whole watershed, i.e.,

constant),( ee itxi

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If the effective rainfall duration tr = 2 hours, what is the peak direct runoff at the basin outlet? When will the peak flow occur? When will the direct runoff end?

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The time of peak flow occurrence is dependent on the relative magnitude of Ai’s.

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If the effective rainfall duration tr = 10 hours, what is the peak direct runoff at the basin outlet? When will the peak flow occur? When will the direct runoff end?

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The watershed storage effect is neglected.

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Under the assumptions of constant intensity and uniform spatial distribution for effective rainfall, the peak direct runoff occurs at the time t = tc, if the duration of the effective rainfall is longer than or equal to tc.

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cttr

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However, for storms with duration shorter than tc, the peak flow and its time of occurrence depend on relative magnitude of contributing areas (Ai’s).

cttr

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Rational method for peak flow estimation

Consider a rainfall of constant intensity and very long duration occurring uniformly over a basin. The runoff gradually increases from zero to a constant value as shown in the following figure.

tcoefficien runoff

c

ciAAiQ ep

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Practical application of the rational method for peak flow estimation

Calculating the time of concentration, tc.Determining the average rainfall intensity i

from the IDF curve using tc as the storm duration and a predetermined return period T.

Calculating the runoff coefficient, c.Calculating the peak direct runoff

tcoefficien runoff

c

ciAAiQ epIn engineering practice, it is normally restricted to catchments with basin area not exceeding 15 km2.

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Unit hydrograph of the modified rational method

Original rational method only estimates the peak flow. It does not yield the complete runoff hydrograph.

cttr cttr cttr

Neglecting the watershed storage effect.2tc tc+tr

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Synthetic unit hydrographs For areas where rainfall and runoff data are not

available, unit hydrograph can be developed based on physical characteristics of the watershed. Clark’s IUH (time-area method) SCS unit hydrograph Linear reservoir model (Nash model)

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Clark’s IUH (time-area method) The concept of isochrones

Isochrones are lines of equal travel time. Any point on a given isochrone takes the same time to reach the basin outlet. Therefore, for the following basin isochrone map and assuming constant and uniform effective rainfall, discharge at the basin outlet can be decomposed into individual contributing areas and rainfalls.

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Contributing area and contributing rainfall

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Using the basin isochrone map, the cumulative contributing area curve can be developed. The derivatives or differences of this curve constitute the instantaneous unit hydrograph IUH(t).

Time base of the IUH is the time of concentration of the watershed.

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If the effect of watershed storage is to be considered, the unit hydrograph described above is routed through a hypothetical linear reservoir with a storage coefficient k located at the watershed outlet.

For a linear reservoir with storage coefficient k, we have

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Consider the continuity of the hypothetical reservoir during a time interval t.

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Example. A watershed of 1000-acre drainage area has the following 15-minute time-area curve. The storage coefficient k of the watershed is 30 minutes. Determine the 15-minute unit hydrograph UH(15,t).

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Area between isochrones t = 0 and t = 0.25 hr is 100 acres. Since we are interested in the unit hydrograph UH(15-min, t), the rainfall intensity in the time period (0.25 hr) should be 4 inch/hr. Therefore, the ordinate of UH(15-min, t) at time t = 0.25 hr is:4 inch/hr (1/12 ft/inch) (1/3600 hr/sec) 100 acres

(43560 ft2/acre) = 403.33 cfs

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Synthetic unit hydrographs

Clark’s IUH (time-area method) SCS unit hydrograph Linear reservoir model (Nash model)

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SCS unit hydrograph

Note: tb > tr+tc. The SCS UH takes into account the storage effect.

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Misuse of SCS UH – an example

tc=0.5

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Synthetic unit hydrographs

Clark’s IUH (time-area method) SCS unit hydrograph Linear reservoir model (Nash model)

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Nash’s linear reservoir model Consider a linear reservoir which has the

following characteristics:

)()( tKQtS

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)1()( / kteItQ

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Summary of the Conceptual IUH modelAssumption: Linear Reservoir S(t)=kQ(t)

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Relationship between IUH and the S curve

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43)1(1)1(

1][1][

)(1)]1()1([

)(

)(

Kt

Kt

Kt

Kt

Kt

Kt

Kt

Ktt

Kt

Ktt

Kt

eetA

teAe

teeeA

teeA

ttt

eAeA

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The Nash Linear Reservoir Model

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The above equation represents a gamma density function, and thus the integration over 0 to infinity yields 1 (one unit of rainfall excess).

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Parameter estimation for n and k The IUH of the n-LR system is characterized by

a function which is identical to the gamma density.

=M1

=M2

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The direct runoff can be expressed as a convolution integral of the rainfall excess and IUH, i.e.

If the direct runoff hydrograph (DRH) is divided by the total volume of direct runoff, it can be viewed as a density function. Thus, denotes a density function of a random variable t.

The first moment of this rescaled DRH is

t

dtuitQ0

)()()(

)(/)( tqVtQ DRH

0 001 )()(1)( dtdtuit

VdtttqM

DRHQ

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The second moment of the rescaled DRH is

Similarly, the first and second moments of the rescaled effective rainfall can be respectively expressed as

0 0

2

0

22 )()(1)( dtdtuit

VdttqtM

DRHQ

01 )(1 dttti

VM

ERHI

0

22 )(1 dttitV

MERH

I

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It can be shown that

Thus, given the ERH and DRH, the parameters n and k can obtained from the above equations.

nkMM IQ 11

12

22 2)1( IIQ nkMknnMM

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Calculation of

1IM21 and II MM

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Calculation of 21 and QQ MM

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坡地開發滯留池之水文設計Hydrological Analysis for Detention Pond Design in Hillslope Development

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坡地開發水文設計相關規範

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現行設計規範之檢討

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7957)1020(10)60/15(14.159 43

103502/)60/60(75.5

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逕流量分析 ( 水土保持技術規範第 十七條 ) 洪峰流量之估算，有實測資料時，得採用單位歷線分析；面積在一千公頃以內者，無實測資料時，得採用合理化公式 (Rational Formula) 計算。合理化公式如下：

。公頃：集水區面積小時公釐：降雨強度

無單位：逕流係數

秒立方公尺：洪峰流量式中，

)(),/(

),(

),/(

3601

AIC

Q

CIAQ

p

p

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滯洪設施規劃設計原則 ( 水土保持技術規範第 九十五 條 )

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滯洪量之估算( 水土保持技術規範第 九十六 條 )