Dr. M. Perumal - SWAT · 2012. 8. 13. · Dr. M. Perumal, Professor & Head, Dept. of Hydrology,...

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___________________________________________________________________ Dr. M. Perumal, Professor & Head, Dept. of Hydrology, I.I.T. Roorkee, India

_________________________________________________________________ Dr. M. Perumal, Professor & Head, Dept. of Hydrology, I.I.T. Roorkee, India

Dr. M. Perumal Professor & Head

Department of Hydrology Indian Institute of Technology Roorkee

INDIA Co-authors: Dr. B. Sahoo & Dr. C.M. Rao

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Linearity of channel routing scheme in SWAT model using classical Muskingum (CM) method

Frameworks of variable parameter Muskingum-Cunge (VPMC) & Muskingum-Cunge-Todini (MCT) routing methods available in literature

Development of variable parameter McCarthy-Muskingum (VPMM) routing method

Performance evaluation criteria Numerical Application

Trapezoidal, Rectangular & Triangular channel sections

Performance of CM, VPMC, MCT & VPMM methods Conclusions

Presentation Outline


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Classification of River Flood Waves

Bulk waves

Dynamic waves

Gravity waves

Diffusion wave

Kinematic wave

Old classification by Ferrick (1985) :

Bulk waves

Dynamic waves

Gravity waves

Diffusion wave

ACD wave

New suggested classification :

VPMM method

• Kinematic wave (KW) is a special case of ACD wave

• Works in transition range of DW & KW

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VPMM- directly derived from Saint-Venant Equations. Can be used for routing flood waves in semi-infinite

rigid bed prismatic channels having any cross-sectional shape. Allows simultaneous computations of stage and

discharge hydrographs at any ungauged river site. During steady flow in the channel reach, there exists

one-to-one discharge - depth relationship; which is not valid in case of unsteady flows. During unsteady flow in the channel reach, the

discharge observed at any section has its corresponding normal depth at the upstream section.

Development of VPMM Method

Hypothesis used:


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___________________________________________________________________ Dr. M. Perumal, Professor & Head, Dept. of Hydrology, I.I.T. Roorkee, India 8/10/2012 5


Definition Sketch of VPMM Method

Δx

Qd

Qu

1

Q3 QM

L Δx/2

yu

y3 yd

M

3 2

yM

3

2 o M Mo

QLS B c

=

xL ∆−= 5.0θ

Mo

xKV∆

=

Routing parameters


6


Framework of VPMM Method (Perumal & Ponce, 2012)

C1 C2 C3 :

1,1 ++ ∆= jMoj VxK( )xcBSQ jMojMoojj ∆−= ++++ 1,1,1,31 25.0θ

11111,3 )1( +++++ −+= jjjjj QIQ θθ

1 2

j

j+1

Δx

Δt M

Numerical scheme of VPMM method

( ) ( )( )( )

1 11 1

1 1 1 1 1 1

2. . 12. . 2. .. . .

2. . 1 2. . 1 2. . 1j jj j j j

j j j jj j j j j j

t Kt K t KO I I O

t K t K t K

θθ θ

θ θ θ+ +

+ ++ + + + + +

−∆ + −∆ − ∆ += + +∆ + − ∆ + − ∆ + −

For derivation details of VPMM

method, click here


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Framework of VPMC Method (Ponce and Chaganti, 1994)

:

13211 ++ ++= jjjj ICICOCO

( ) tKtKC∆+−

∆+−=

5.015.0

1 θθ

( ) tKtKC∆+−

∆+=

5.015.0

2 θθ ( )

( ) tKtKC

∆+−∆−−

=5.015.01

3 θθ

cxK ∆= ( )xBcSQ o ∆−= 25.0θIn CM method, the routing parameters remain constant at every routing time step so that C1+C2+C3=1, making it mass conservative.


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Framework of MCT Method (Todini, 2007)

:

jj

j

jj

jjj

j

j

jj

jjj

jj

jjj OC

C

DC

DCI

C

C

DC

DCI

DC

DCO *

*1

*1

*1

**

*

*1

*1

*1

**

1*1

*1

*1

*1

1 1

1

1

1

1

1 +

++

+

+++

++

+++ ++

+−+

++

−++

++

++−=

)/)(/(* xtcC jjj ∆∆= β

)/)(/( 11*

1 xtcC jjj ∆∆= +++ β

jjj vc /=β

111 / +++ = jjj vcβ

)/(* xcBSQD jojjj ∆= β

)/( 111*

1 xcBSQD jojjj ∆= ++++ β


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Comparative evaluation of CM, VPMC, MCT & VPMM methods with respect to benchmark solutions of the Saint-Venant equations.

Evaluated these methods by routing a given hypothetical flood wave in uniform prismatic Trapezoidal, Rectangular and Triangular cross-section channel reaches of different configurations.

Performance Evaluation


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Performance Evaluation Criteria

( ) ( ) 10012

1

2

1×

−−−= ∑∑

==

N

ioioi

N

icioiq QQQQη

( ) 1001 ×−= popcper qqq( ) 1001 ×−= qpoqpcqper ttt

1 11 100

N N

ci ii i

EVOL Q I= =

= − × ∑ ∑

*Negative indicates–

underestimation;

*Positive indicates– over estimation

Nash and Sutcliffe (1970)


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Numerical Application

Inflow discharge hydrographs corresponding to the input stage hydrographs in the form of

Pearson type III distribution are used to generate random sizes of discharge hydrographs.

# Total reach length= 40 km # ∆x = 1 km; ∆x= 5 min. # Total no. of runs = 9847 (for each of the Saint-Venant, VPMC & VPMM methods)

Parameters Values Gamma, γ 1.05, 1.15, 1.25, 1.50 Channel bed slope, So 0.002, 0.001, 0.0008, 0.0005, 0.0004, 0.0002, 0.0001 Manning’s roughness, n 0.01, 0.02, 0.03, 0.04, 0.05 Initial discharge, bQ (m

3/s) 100.0 Peak stage, py (m) 5.0, 8.0, 10.0, 12.0, 15.0

Time-to-peak stage, pt (h) 5.0, 10.0, 15.0, 20.0

Channel bottom width, b (m) 100.0 Channel side slope, z 0.0, 1.0, 3.0, 5.0

( ) ( )

−

−

−+=

−

11

exp)(,0)1/(1

γ

γp

pbpb

ttttytyyty

Upstream boundary condition:


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Evaluation of CM, VPMC, MCT & VPMM Methods

Nash-Sutcliffe Efficiency

CM always performs with

η≥50% for all the cases except five

runs η≥95% for all the runs except 115 (1.17%) cases

50

60

70

80

90

100

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

(1/S o)(∂y /∂x )max

η (%

)

(a) CM method


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CM always performs with


runs η≥95% for all the runs except 115 (1.17%) cases

50

60

70

80

90

100

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

(1/S o)(∂y /∂x )max

η (%

)

(b) VPMC method


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MCT always performs with


runs

50

60

70

80

90

100

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

(1/S o)(∂y /∂x )max

η (%

)

(c) MCT method


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VPMM always performs with

η≥90% for all the cases except nine

runs

50

60

70

80

90

100

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

(1/S o)(∂y /∂x )max

η (%

)

(d) VPMM method


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-40

-30

-20

-10

0

10

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

(1/S o)(∂y /∂x )max

EV

OL

(%)

(a) CM method

-40

-30

-20

-10

0

10

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

(1/S o)(∂y /∂x )max

EV

OL

(%)

(b) VPMC method

-40

-30

-20

-10

0

10

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

(1/S o)(∂y /∂x )max

EV

OL

(%

)

(c) MCT method

-40

-30

-20

-10

0

10

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

(1/S o)(∂y /∂x )max

EV

OL

(%)

(d) VPMM method

Error in Volume


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Error in Peak Discharge

-50

-40

-30

-20

-10

0

10

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

(1/S o)(∂y /∂x )max

qper

(%

)

(a) CM method

-50

-40

-30

-20

-10

0

10

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

(1/S o)(∂y /∂x )max

qper

(%

)

(b) VPMC method

-50

-40

-30

-20

-10

0

10

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

(1/S o)(∂y /∂x )max

qper

(%)

(c) MCT method

-50

-40

-30

-20

-10

0

10

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

(1/S o)(∂y /∂x )max

qper

(%

)

(d) VPMM method


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Evaluation of CM, VPMC, MCT & VPMM Methods Error in Time-to-peak Discharge

-30

-20

-10

0

10

20

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

(1/S o)(∂y /∂x )max

tpqer

(%

)

(a) CM method

-30

-20

-10

0

10

20

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

(1/S o)(∂y /∂x )maxtpqe

r (%

)

(b) VPMC method

-30

-20

-10

0

10

20

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

(1/S o)(∂y /∂x )max

tpqer

(%)

(c) MCT method

-30

-20

-10

0

10

20

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

(1/S o)(∂y /∂x )max

tpqe

r (%

)

(d) VPMM method


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Conclusions VPMM method consistently performs better than the MC & VPMC method.

VPMM, MCT & MC methods are fully volume conservative, whereas the VPMC method has the tendency to always loose mass.

Due to the nonlinearity dynamics of river flood convection, the MC method should not be used in SWAT model.

Since the VPMM method has a sound physical basis, it may be preferred over the other existing methods for its incorporation in the SWAT model for dealing with various field problems.


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21



0Q Ax t

∂ ∂+ =

∂ ∂

01

fy v v vS Sx g x g t∂ ∂ ∂

= − − −∂ ∂ ∂

So= bed slope; Sf = friction slope; ∂y/∂x = water surface slope; (v/g)(∂v/∂x) = convective acceleration; (1/g)(∂v/∂t) = local acceleration. Magnitudes of various terms in eqn. (2) are usually small in comparison with So [Henderson, 1966; NERC, 1975].

Continuity equation:

Momentum eqn.

(1)

(2)

Mathematical Formulation of VPMM Method

Back to Slide 6

Saint-Venant Equations:


22


• Unsteady flow relationship:

• Using equations (1), (2) (4), (5), & (6) friction slope:

•Celerity of the flood wave (Henderson,1966; NERC,1975):

• Where Froude number: 1 22v dA dyF

gA

=

221 41 1

9f o o

y PdR dyS S FS x dA dy

∂ = − − ∂

…Mathematical Formulation of VPMM Method Important steps of derivation:

Back to Slide 6

• Considering ∂y/∂x≠0 & ∂2y/∂x2→0:

vdydAdyPdR

dAdQc

+==

//

321

( )xyBc

xAv

xQ

∂∂

=∂

∂=

∂∂

( ) 0/)(, SSyQxyyQ f=∂∂ (3)

(4)

(5)

•Velocity by the Manning’s friction law: 2/3 1/21

fv R Sn= (6)

(7)

(8) To Eq.

(14)


23


• Using Eq. (7) in (3), unsteady discharge:

• Similarly, unsteady velocity:

• Hydraulic continuity Eq. (1):

• Using Eqs. (9) & (10) in (11):

• Normal discharge in Eq. (9) can be rewritten as:

….Mathematical Formulation of VPMM Method

Back to Slide 6

( )21

22

0 94111,

−

∂∂

−=

∂∂

=dydR

BPF

xy

SyQ

xyyQQ o (9)

(10) ( )21

22

0 94111,

−

∂∂

−=

∂∂

=dydR

BPF

xy

Syv

xyyvv o

0=

∂∂

+

∂∂

xQ

vQ

t (11)

(12) 00

0 =∂∂

+

∂∂

xQ

vQ

t

( )21

22

0 94111

−

−

∂∂

−==

dydR

BPF

xy

SQyQQ oo (13)


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Where:

• Which is in the form of:

•Using binomial series expansion in Eq. (13) with (1/S0)(∂y/∂x)

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•Using Eq. (18) in continuity Eq. (12):

• Expressing Eq. (19) at the centre point ‘M’ of the finite difference grid:


Back to Slide 6

(19)

(20)

00

0

0

=∂∂

+

∂∂

+

∂∂

xQ

vxQ

ca

Q

t

Finite-difference grid representation of the VPMM

method computational scheme

1 2

j

j+1

Δx

Δt M

0211 1

,0

0

0

1,0

1

0

01

=

∂∂

+∂∂

+

∂∂

+

−∂∂

+

∆

+

+

++

j

M

j

Mj

M

j

M

jM

jM

j

M

jM

xQ

xQ

v

xQ

ca

Q

v

xQ

ca

Q

t


26


• Eq. (20) can be reorganized as:

• Which can be written as:


Back to Slide 6

(21)

(22)

+−

+∆=

∆++

∆−

∆−

∆++

∆−

∆

+++

+

+

++

+

+

+

+

22

1211

21

1211

21

11

11

10

0

0

0

,0

11

1

0

011

0

01

,0

ji

ji

ji

ji

ji

j

M

ji

j

Mj

M

ji

j

M

ji

j

Mj

M

QQQQt

Qxc

aQxc

av

x

Qxc

aQxc

av

x

[ ] [ ]t

QQQQ

QQKQQKj

ij

ij

ij

i

ji

jji

jjji

jji

jj

∆

+−

+=

−+−−+

+++

+

+++

++++

22

)1()1(

11

11

11

11111 θθθθ


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• Where:

• Eq. (22) can also be written in the form of the classical Muskingum routing equation as:


Back to Slide 6

(23)

(24)

1,0

1+

+ ∆= jM

j

vxK

122

,00

,01

0

01

941

21

21

++

+

−

∆=

∆−=

j

MM

MM

Mj

M

j

dydR

BPF

xcBSQ

xcaθ

21

3

2

=

M

MMM gA

BQF

ji

ji

ji

ji QCQCQCQ 132

11

11 +

+++ ++=

• Where:

( )1 1

11 1

2. .2. . 1

j j

j j

t KC

t Kθθ

+ +

+ +

∆ −=∆ + − ( )2 1 1

2. .2. . 1

j j

j j

t KC

t Kθ

θ+ +

∆ +=∆ + −

( )( )3 1 1

2. . 1

2. . 1j j

j j

t KC

t K

θ

θ+ +

−∆ + −=∆ + −


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•The reach storage at any computational time level can be given by:

• Corresponding downstream stage hydrograph can be computed using Eq. (4) as:

• Where:


Back to Slide 6

[ ])( 11111111 ++++++++ −+= jijijjijj QQQKS θ

11

1 ++

+ ji

j QKPrism storage = Wedge storage = )( 11

111 ++

+++ − jij

ijj QQK θ

1

11111

1)(

+

+++++

+

−+= j

MM

jM

jij

Mj

i cBQQ

yy

This proves that the classical Muskingum method advocated by McCarthy (1938) has the physical basis


A Simplified Channel Routing Scheme Suitable for Adoption in SWAT Model�Slide Number 2Slide Number 3Slide Number 4Definition Sketch of VPMM Method Slide Number 6Slide Number 7Slide Number 8Slide Number 9Performance Evaluation CriteriaNumerical ApplicationEvaluation of CM, VPMC, MCT & VPMM MethodsEvaluation of CM, VPMC, MCT & VPMM MethodsEvaluation of CM, VPMC, MCT & VPMM MethodsEvaluation of CM, VPMC, MCT & VPMM MethodsEvaluation of CM, VPMC, MCT & VPMM MethodsEvaluation of CM, VPMC, MCT & VPMM MethodsEvaluation of CM, VPMC, MCT & VPMM MethodsConclusionsSlide Number 20Slide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Slide Number 26Slide Number 27Slide Number 28

Dr. M. Perumal - SWAT · 2012. 8. 13. · Dr. M. Perumal, Professor & Head, Dept. of Hydrology,...

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