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

5

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

5

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)

7

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

9

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

21

3

dQ P dRc v

dA B dy

10

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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