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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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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
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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___________________________________________________________________
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
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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Dr. M. Perumal, Professor & Head, Dept. of Hydrology, I.I.T.
Roorkee, India
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
_________________________________________________________________
Dr. M. Perumal, Professor & Head, Dept. of Hydrology, I.I.T.
Roorkee, India
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
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Dr. M. Perumal, Professor & Head, Dept. of Hydrology, I.I.T.
Roorkee, India
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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Dr. M. Perumal, Professor & Head, Dept. of Hydrology, I.I.T.
Roorkee, India
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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Dr. M. Perumal, Professor & Head, Dept. of Hydrology, I.I.T.
Roorkee, India
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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Dr. M. Perumal, Professor & Head, Dept. of Hydrology, I.I.T.
Roorkee, India 8/10/2012 9
_________________________________________________________________
Dr. M. Perumal, Professor & Head, Dept. of Hydrology, I.I.T.
Roorkee, India
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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Dr. M. Perumal, Professor & Head, Dept. of Hydrology, I.I.T.
Roorkee, India 8/10/2012 10
_________________________________________________________________
Dr. M. Perumal, Professor & Head, Dept. of Hydrology, I.I.T.
Roorkee, India
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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Dr. M. Perumal, Professor & Head, Dept. of Hydrology, I.I.T.
Roorkee, India
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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Dr. M. Perumal, Professor & Head, Dept. of Hydrology, I.I.T.
Roorkee, India
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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Dr. M. Perumal, Professor & Head, Dept. of Hydrology, I.I.T.
Roorkee, India
Evaluation of CM, VPMC, MCT & VPMM Methods
Nash-Sutcliffe Efficiency
CM always performs with
η≥90% 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 (%
)
(b) VPMC method
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Dr. M. Perumal, Professor & Head, Dept. of Hydrology, I.I.T.
Roorkee, India
Evaluation of CM, VPMC, MCT & VPMM Methods
Nash-Sutcliffe Efficiency
MCT always performs with
η≥80% for all the cases except five
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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___________________________________________________________________
Dr. M. Perumal, Professor & Head, Dept. of Hydrology, I.I.T.
Roorkee, India
Evaluation of CM, VPMC, MCT & VPMM Methods
Nash-Sutcliffe Efficiency
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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Dr. M. Perumal, Professor & Head, Dept. of Hydrology, I.I.T.
Roorkee, India
Evaluation of CM, VPMC, MCT & VPMM Methods
-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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Dr. M. Perumal, Professor & Head, Dept. of Hydrology, I.I.T.
Roorkee, India
Evaluation of CM, VPMC, MCT & VPMM Methods
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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___________________________________________________________________
Dr. M. Perumal, Professor & Head, Dept. of Hydrology, I.I.T.
Roorkee, India
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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Dr. M. Perumal, Professor & Head, Dept. of Hydrology, I.I.T.
Roorkee, India
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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___________________________________________________________________
Dr. M. Perumal, Professor & Head, Dept. of Hydrology, I.I.T.
Roorkee, India
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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
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:
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___________________________________________________________________
Dr. M. Perumal, Professor & Head, Dept. of Hydrology, I.I.T.
Roorkee, India
• 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)
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___________________________________________________________________
Dr. M. Perumal, Professor & Head, Dept. of Hydrology, I.I.T.
Roorkee, India
• 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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___________________________________________________________________
Dr. M. Perumal, Professor & Head, Dept. of Hydrology, I.I.T.
Roorkee, India
Where:
• Which is in the form of:
•Using binomial series expansion in Eq. (13) with
(1/S0)(∂y/∂x)
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___________________________________________________________________
Dr. M. Perumal, Professor & Head, Dept. of Hydrology, I.I.T.
Roorkee, India
•Using Eq. (18) in continuity Eq. (12):
• Expressing Eq. (19) at the centre point ‘M’ of the finite
difference grid:
….Mathematical Formulation of VPMM Method
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
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___________________________________________________________________
Dr. M. Perumal, Professor & Head, Dept. of Hydrology, I.I.T.
Roorkee, India
• Eq. (20) can be reorganized as:
• Which can be written as:
….Mathematical Formulation of VPMM Method
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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___________________________________________________________________
Dr. M. Perumal, Professor & Head, Dept. of Hydrology, I.I.T.
Roorkee, India
• Where:
• Eq. (22) can also be written in the form of the classical
Muskingum routing equation as:
….Mathematical Formulation of VPMM Method
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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___________________________________________________________________
Dr. M. Perumal, Professor & Head, Dept. of Hydrology, I.I.T.
Roorkee, India
•The reach storage at any computational time level can be given
by:
• Corresponding downstream stage hydrograph can be computed
using Eq. (4) as:
• Where:
….Mathematical Formulation of VPMM Method
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
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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