INDIAN INSTITUTE OF TECHNOLOGY ROORKEE
EFFECT OF SPATIAL AND TEMPORAL DISCRETIZATIONS ON THE SIMULATIONS USING CONSTANT-PARAMETER AND VARIABLE-PARAMETER MUSKINGUM METHODS
Muthiah Perumal and C. Madhusudana RaoMuthiah Perumal and C. Madhusudana RaoProfessor, Department of Hydrology, Indian Institute of Technology Roorkee
Assist. Professor, Department of Civil Engineering, National Institute of Technology Jamshedpur
Introduction
• Flood wave movement is a nonlinear process
• Many models used for catchment runoff simulations still
employ linear theory based models for runoff simulations
• FLOOD ROUTING METHODS
• CONSTANT-PARAMETER MUSKINGUM METHOD (MRM)
(as used in the SWAT model based on the Muskingum-Cunge (Cunge1969) approach)
• VARIABLE-PARAMETER MUSKINGUM METHOD (VPMM)
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In focus
Objective is:• To explore the impact of spatial and temporal
The focus of this study is concerned only with one of the componentprocesses, namely, flood routing in main channels using the Muskingummethod.
• To explore the impact of spatial and temporaldiscretizations on the routing simulations of theconstant-parameter and the variable-parameterMuskingum methods
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CONSTANT-PARAMETER MUSKINGUM METHOD
1 0 0 0 kLK
Frame work of Muskingum Routing Method (MRM)
2 1 2 2 1 3 1Q C I C I C Q
(as used in the SWAT model based on the Muskingum-Cunge (Cunge1969) approach)
1
2
2 1
K tC
K t
CoefficientsRouting Equation
Parameters
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3k c
QC V
A
k
KC
2 2 1K t K To avoid the numerical instability and negative outflowcomputation, the following condition is recommended in theSWAT model
2
2
2 1
K tC
K t
3
2 1
2 1
K tC
K t
Parameters
is weighing factor (0-0.5);
respectively are the channel length (km) and celerity of the flood wave (m/s)
where
is the channel flow velocity (m/s).
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Qpave = (Qpinf + qpout)/2
Qref = (Qpave + Qb)/2
The constant parameter-Muskingum method by estimating its two constantparameters using a reference discharge.
Yref (from Newtorn Raphson method)
If < 2k , the Muskingum coefficient Co = - VeGenerally negative values of coefficients are avoided
For best results, the should be so chosen that k > >2k t
t
t
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The Variable Parameter McCarthy-Muskingum (VPMM) method proposed by Perumal and Price [2013]
directly derived from the Saint-Venant Equations.
for routing flood waves in semi-infinite rigid bed prismatic channelshaving any cross-sectional shape/follows either Mannings friction law.
VARIABLE-PARAMETER MUSKINGUM METHOD (VPMM)
Frame Work of the VPMM Method
during steady flow in the channel reach there exist discharge anddepth relationship.
during unsteady flow in the channel reach, the discharge observed atany section has its corresponding normal depth at upstream section.
Allows the Simultaneous computations of stage hydrographcorresponding to a given inflow/routed hydrograph.
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0Q A
x t
0
1f
y v v vS S
x g x g t
Saint-Venant equations
Continuity equation
Momentum equation
(1)
(2)
Frame Work of VPMM Method
0fS Sx g x g t
Sf = friction slope;So= bed 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].
Momentum equation (2)
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Parameter estimation using the VPMM method
M t
∆t/2
3
j+1
∆t
I
1
QM
yu
y
M
3
2
Fig. 1 Definition Sketch of the VPMM Routing Reach
Fig. 2 Numerical grid adopted for VPMM application in
synchronization with Fig.1
x ∆x
L
2
1
∆x/2 j
∆t
Δx
O
Q3
QM
L Δx/2
yM
y3 yd
8
, 1d j tQ The estimated downstream discharges at time
, 1 , 1
1
, 1 , 1
2. .
2. . 1
j t j t
j t j t
t KC
t K
2. .t K
Muskingum Routing Equation
1 2 , 3 ,, 1 , 1 u j t d j td j t u j tQ CQ C Q CQ
Where
1j t
Coefficients
Parameter estimation using the VPMM method
, 1u j tQ
,u j tQ
,d j tQ
t
discharges at time
, ,
2
, 1 , 1
2. .
2. . 1
j t j t
j t j t
t KC
t K
, ,
3
, 1 , 1
2. . 1
2. . 1
j t j t
j t j t
t KC
t K
The observed upstream discharges at time
The estimated downstream discharges at time
The observed upstream discharges at time
The routing time interval
1j t
1j t
j t
j t
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Routing parameters
, 1, 1
j tM o j t
xK
V
2
241
PdR dyQ F
3, 1
, 1, 1 , 1
1
2 2. . . .j t
j to M j t Mo j t
Q
S B c x
( By neglecting inertial terms)
Parameter estimation using the VPMM method
23, 1
, 1
, 1, 1 , 1
41
91
2 2. . . .
j t
M j t
j to M j t Mo j t
PdR dyQ F
dA dy
S B c x
, ,3, 1 , 1 , 13,
,, 1 2 2
j t j tj t j t j tj t
Mo j tMo j t
Q I I O OQx tV V
The governing finite difference equation of the VPMM routing method
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3
dQ P dRc v
dA B dy
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Numerical Experiments
Test Run No. Space step(km)
Time step(h)
1 40 (Single Reach) 1.00
Twelve numerical test runs have been conducted in this study by changing space (∆x)and routing time (∆t) intervals to explore the effect of spatial and temporaldiscretizations on the routing simulations of the constant-parameter and the variable-parameter Muskingum methods.In this study, a uniform prismatic trapezoidal channel with a bed slope, So = 0.0002, Manning’s roughness coefficient, n = 0.04 and bed width of 50m contained by dykes with a slope ratio (1 V : 5 Horizontal (z)) has been used.
1 40 (Single Reach) 1.002 40 (Single Reach) 3.003 40 (Single Reach) 6.004 40 (Single Reach) 24.005 4 (10 Sub-reaches) 1.006 4 (10 Sub-reaches) 3.007 4 (10 Sub-reaches) 6.008 4 (10 Sub-reaches) 24.009 1 (40 Sub-reaches) 1.00
10 1 (40 Sub-reaches) 3.0011 1 (40 Sub-reaches) 6.0012 1 (40 Sub-reaches) 24.00 11
b
z y
Trapezoidal channel
**Flow characteristics
Qb = 100 m3/sec
Qp = 3000 m3/sec
Pearson type-III distribution expressed as
1 11
( ) ( ) exp1p
b p bp
t ttQ t Q Q Q
t
Inflow Hydrograph and Benchmark solution details
The benchmark solutions were obtained by HEC-RASTp = 192 h (8 days),
Shape factor 1.05
The benchmark solutions were obtained by HEC-RASmodel (USACE, 2010) by routing the inflowhydrograph for a reach length of 40km in prismaticTrapezoidal channel using the Space step = 1000mand Time step = 300sec
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Results and DiscussionsTable 1 Summary of performance criteria showing reproduction of pertinentcharacteristics of the HEC-RAS results by the VPMM method for routing in Trapezoidalchannel reaches using spatial and temporal discretization on the simulations.
Test Run No.
Space step
(km)
Time step
(h)
µq
(%)µy
(%)Discharge Routing Stage Computation
EVOL
(%)
ηq(%)
qper(%)
tpqer(h)
ηy(%)
yper(%)
tpyer(h)
1 40 1.00 0.63 0.26 -0.00029 100.00 0.02 0.00 100.00 0.28 2.002 40 3.00 0.64 0.26 -0.00086 94.95 -4.30 3.00 93.36 -1.14 6.00
3 40 6.00 0.64 0.2 -0.00262 75.82 -13.73 4.00 66.62 -4.00 6.00
4 40 24.00 1.36 0.86 -0.06547 -22.23 -48.67 3.00 -69.74 -13.96 5.00
5 4 1.00 0.63 0.26 -0.00031 100.00 0.02 0.00 100.00 0.15 2.006 4 3.00 0.64 0.26 -0.00088 94.66 -4.45 3.00 92.86 -2.26 6.007 4 6.00 0.64 0.2 -0.00171 73.56 -15.24 4.00 64.50 -8.22 6.00
8 4 24.00 1.36 0.86 -0.00572 -9.89 -61.89 5.00 -51.24 -37.75 7.00
9 1 1.00 0.63 0.26 -0.00030 100.00 0.02 0.00 100.00 0.14 2.0010 1 3.00 0.64 0.26 -0.00089 94.66 -4.46 3.00 92.85 -2.35 6.0011 1 6.00 0.64 0.2 -0.00168 73.55 -15.28 4.00 64.62 -8.49 6.00
12 1 24.00 1.36 0.86 -0.00572 -9.11 -62.10 5.00 -48.35 -38.49 7.0013
0
500
1000
1500
2000
2500
3000
0 96 192 288 384 480
Dis
char
ge (m
3 /se
c)
Time (h)
Trapezoidal ChannelSpace step = 40 km
Time step = 1.00 hour
Inflow
HEC-RAS
VPMM
Figure 1. Typical simulated discharge hydrograph of the VPMM method for a space step of 40km and a time step of 1.00hour in a
trapezoidal channel.
2
4
6
8
10
12
14
16
0 96 192 288 384 480
Stag
e (m
)
Time (h)
Trapezoidal ChannelSpace step = 40 km
Time step =1.00 hour
Input stage
HEC-RAS
VPMM
Figure 2. Typical computed stage hydrographs of the VPMM method for a space step of 40km and a time step of 1.00hour in a
trapezoidal channel.
3000
16
-500
0
500
1000
1500
2000
2500
3000
0 10 20 30 40 50 60
Dis
char
ge (m
3 /se
c)
Time (Days)
Trapezoidal ChannelSpace step = 40 kmTime step = 24 hour
Inflow
HEC-RAS
VPMM
Figure 3. Typical simulated discharge hydrograph of the VPMM method for a space step of 40km and a time step of 24.00hour in a
trapezoidal channel.
-2
0
2
4
6
8
10
12
14
16
0 10 20 30 40 50 60
Stag
e (m
)
Time (Days)
Trapezoidal ChannelSpace step = 40 kmTime step = 24 hour
Input stage
HEC-RAS
VPMM
Figure 4. Typical computed stage hydrographs of the VPMM method for a space step of 40km and a time step of 24.00hour in a
trapezoidal channel.14
0
500
1000
1500
2000
2500
3000
0 96 192 288 384 480
Dis
char
ge (m
3 /se
c)
Time (h)
Trapezoidal ChannelSpace step = 4 km
Time step = 1.00 hour
Inflow
HEC-RAS
VPMM
Figure 5. Typical simulated discharge hydrograph of the VPMM method for a space step of 4km and a time step of 1.00hour in a
trapezoidal channel.
2
4
6
8
10
12
14
16
0 96 192 288 384 480
Stag
e (m
)
Time (h)
Trapezoidal ChannelSpace step = 4 km
Time step =1.00 hour
Input stage
HEC-RAS
VPMM
Figure 6. Typical computed stage hydrographs of the VPMM method for a space step of 4km and a time step of 1.00hour in a
trapezoidal channel.
0
500
1000
1500
2000
2500
3000
0 8 16 24 32 40
Dis
char
ge (m
3 /se
c)
Time (Days)
Trapezoidal ChannelSpace step = 4 km
Time step = 24.00 hour
Inflow
HEC-RAS
VPMM
Figure 7. Typical simulated discharge hydrograph of the VPMM method for a space step of 4km and a time step of 24.00hour in a
trapezoidal channel.
2
4
6
8
10
12
14
16
0 8 16 24 32 40
Stag
e (m
)
Time (Days)
Trapezoidal ChannelSpace step = 4 km
Time step = 24.00 hour
Input stage
HEC-RAS
VPMM
Figure 8. Typical computed stage hydrographs of the VPMM method for a space step of 4km and a time step of 24.00hour in a
trapezoidal channel.15
0
500
1000
1500
2000
2500
3000
0 96 192 288 384 480
Dis
char
ge (m
3 /se
c)
Time (h)
Trapezoidal ChannelSpace step = 1 km
Time step = 1.00 hour
Inflow
HEC-RAS
VPMM
Figure 9. Typical simulated discharge hydrograph of the VPMM method for a space step of 1km and a time step of 1.00hour in a
trapezoidal channel.
2
4
6
8
10
12
14
16
0 96 192 288 384 480
Stag
e (m
)
Time (h)
Trapezoidal ChannelSpace step = 1 km
Time step =1.00 hour
Input stage
HEC-RAS
VPMM
Figure 10. Typical computed stage hydrographs of the VPMM method for a space step of 1km and a time step of 1.00hour in a
trapezoidal channel.trapezoidal channel. trapezoidal channel.
0
500
1000
1500
2000
2500
3000
0 8 16 24 32 40
Dis
char
ge (m
3 /se
c)
Time (Days)
Trapezoidal ChannelSpace step = 1 km
Time step = 24.00 hours
Inflow
HEC-RAS
VPMM
Figure 11. Typical simulated discharge hydrograph of the VPMM method for a space step of 1km and a time step of 24.00hour in a
trapezoidal channel.
2
4
6
8
10
12
14
16
0 8 16 24 32 40
Stag
e (m
)
Time (Days)
Trapezoidal ChannelSpace step = 1 km
Time step = 24.00 hours
Input stage
HEC-RAS
VPMM
Figure 12. Typical computed stage hydrographs of the VPMM method for a space step of 1km and a time step of 24.00hour in a
trapezoidal channel. 16
500
1000
1500
2000
2500
3000
Dis
cha
rge
(m3/s
ec)
Trapezoidal ChannelSpace step = 40km
Time step = 24.00hour
Inflow
Outflow
Observed (HEC-RAS)
0
0 4 8 12 16 20Time (Days)
Figure 13. Typical computed discharge hydrographs of the constant-parameter method for a space step of 40km and a time step
of 24.00hour in a trapezoidal channel using reference discharge
Conclusions
• A preliminary investigation carried out with the considered objectiveshows that the routing solution obtained using a longer routing timeinterval induces significant numerical diffusion of the routed hydrographleading to over attenuation of the inflow flood peak and, thereby,resulting in poor reproduction of the benchmark solution.
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