Criteria for the choice of flood routing methods

for natural channels with overbank flowby

Roger MOUSSA

Laboratoire d'étude des Interactions Sol - Agrosystème - HydrosystèmeUMR LISAH ENSA-INRA-IRD, Montpellier

Context : Flood events

In the world

More than 50 % of the damages caused by natural disasters are due to floods : on average 20 000 deaths/year.

In France

Over a 160 000 km long river network, an area of 22 000 km² is recognised as very prone to flooding: 7 600 villages and 2 000 000 residents are concerned.

Objectives

Develop a spatially distributed hydrological modelling tool to simulate flood events on natural and farmed catchments.

Spatially distributed hydrological modelling of flood events on natural and farmed catchments

I. Process

representation

III.Integration

into adistributed

model

II.Catchment

segmentation

I. Flood routing

Context

Flood routing in natural channels is governed by the Barré de Saint-Venant (1871) equations.

xLateral inflow q(x,t)>0W

B

Longitudinal section Lateral section

ContextI. Flood routing

Saint-Venant system

Equation of mass conservation

0x

yV

x

Vy

t

y=

∂

∂+

∂

∂+

∂

∂

Terms I II III

Equation of energy conservation

( ) 0SSgx

yg

x

VV

t

Vf

=−+∂

∂+

∂

∂+

∂

∂

Terms IV V VI VII VIII

I. Flood routing I.1. Approximation zones

I.1. Approximation zones of the Saint-Venant system(Moussa and Bocquillon, 1996, J. Hydrol.)

Propagation of a sinusoidal function :- Froude number (F²)- a characteristic time of the hydrograph

0,01

0,1

1

10

100

0,01 0,1 1 10 100 1000

Kinematicwave

(IV, V et VI)Diffusivewave

(IV et V)

Gravity wave(VII et VIII)

CompleteSaint-Venant

system

F²=V²/gyDynamic

wave(VI)

(VIII)

(VII)

T+ = TVS/y

W/B = 1

I. Flood routing I.1. Approximation zones

Approximation zones of the Saint-Venant systemwith overbank flow : example W/B = 20

(Moussa and Bocquillon, 2000, Hydrol. Earth System Sc.)F²=V²gy

T+ = TVS/y0.01

0.1

1

10

100

1000

0.01 0.1 1 10 100 1000

KinematicwaveDiffusive

wave

Gravitywave

CompleteSaint-Venant

system

W

B

In some hydrological models, the flood routing model can be usedoutside its application domain.ex : kinematic wave in KINEROS, TOPOG, TOPMODEL, LISEM, QPBRRM.

I. Flood routing I.2. Numerical schemes

I.2. Numerical resolution of the diffusive wave model

In general :

- resolution using finite differences schemes

- problems of numerical stability and convergence

j

j-1

i-1 i

∆ t

∆xt

x

P

i+1O

Case where celerity and diffusivity can be considered constant :

Hayami (1951) analytical solution.

I. Flood routing I.2. Numerical schemes

I.2. Development of new numerical scheme to resolve the diffusive wave equation in flood routing applications over

treelike channels

j+1

j-1

j

i-1 i

∆t

t

x

P uj

uj-1

uj+1

vj-1

vj

vj+1

O

∆xi-1 ∆xi ∆xi+1

- a scheme that enables the use of variable spatial steps (Moussa and Bocquillon, 1996, Hydrol. Pro.).

- a scheme based on the fractional step method that enables to separate convection and diffusion (Moussa and Bocquillon, 2001, J. Hydrol. Eng.).

I. Flood routing I.2. Numerical schemes

Application : flood prevision on the Loire river(between Grangent and Feurs : distance 43 km)

Loire : Event of 20 - 23 September 1980

0

500

1000

1500

2000

2500

3000

3500

4000

0 24 48 72 96 120Time (hours)

Dis

char

ge (m

3 /s)

InputMeasured outputCalculated output

I. Flood routing I.3. Analytical schemes

I.3. Development of a new scheme for the analytic resolution of the diffusive wave equation (Moussa, 1996, Hydrol. Pro.)

- Taking into account uniformly distributed lateral inputs or outputs.

- Identification of the lateral inflow by input/ouput deconvolution.

Input Output

Lateral inflow ? ? ?

L

I. Flood routing I.3. Analytical schemes

Example of application on the Allier river between Prades (1350 km² catchment) and Vieille Brioude (2269 km² catchment) distant of 42 km.

(data source : EPALA)

Allier : Event of april 25 – may 3 1983

0

250

500

750

1000

0 48 96 144Time (hours)

Dis

char

ge (m

3/s) 0

10

20

30

40

Rainfall

Input

Output

Lateralinflow

192

Rai

nfal

l (m

m/2

h)

I. Flood routing I.4. Errors

I.4. Development of a method to estimate numerical errors (Moussa and Bocquillon, 1996, J. Hydrol.)

Three types of error :

eQ = b/aeT = f/eeOsc = d/a

The analysis enables the choice of the spatial increment ∆x and the time step ∆t that minimise the errors.

I. Flood routing Conclusions

Applications

France- Roujan (1 km²)- Gardon d’Anduze (542 km²)- Allier (2269 km²)- Loire (4978 km²)- Moselle (9400 km²)

Canada The diffusive wave equation resolution scheme is integrated in the Hydrotel model and has been used on several Canadian catchments (de 100 à 10000 km²) (Fortin et al., 1995, RSE; 2001a, 2001b, JHE).

ArgentinaSalado (1500 km²) (MSc. Del Valle Morresi, 2001).

Spatially distributed hydrological modelling of flood events on natural and farmed catchments

I. Process

Representation

III.Integration

into adistributed

model

II.Catchment

segmentation

The study site : Tech, Têt, Agly, Aude, Orb, Héraultand Vidourle catchments (738 to 5346 km²)

II. Catchment segmentation II.1. GIUH

II.1. Identification of a Geomorphological Unit Hydrograph(Moussa, 2003, Hydrol. Pro.)

0 12 24 36 48 60 720

1

2x 10-5

Time (hours)

Dis

char

ge (m

3 /s) Agly

Vidourle

OrbHérault

Têt

Aude

Tech

Comparison of the Unit hydrograph transfer function of seven Mediterranean basins.

This transfer function takes into account : - the channel network structure.- the spatial variability of rainfall.

II. Catchment segmentation II.2. Contribution of hydrological units

0

35

70

P1 (m

m/h

)

RuissellementInfiltration

0

35

70

P2 (m

m/h

)

0

35

70

P3 (m

m/h

)

0

35

70

P4 (m

m/h

)

0

35

70

P5 (m

m/h

)

0

35

70

P6 (m

m/h

)

0

35

70

P7 (m

m/h

)

0

250

500

750

1000

00:00 12:00 00:00 12:00 00:00 12:00

Evénement du 22-24 octobre 1977 (heures:minutes)

Déb

it (m

3 /s)

Sbv1Sbv2Sbv3Sbv4Sbv5Sbv6Sbv7Débit caculé (bassin)Débit mesuré (bassin)

II.2. Identification of the contribution of each sub-catchment.Example : The Gardon d’Anduze(545 km²)(Moussa, 1997, Hydrol. Pro.)

II. Catchment segmentation II.2. Contribution of hydrological units

0

250

500

750

1000

00:00 12:00 00:00 12:00 00:00 12:00

Event of 22-24 october 1977 (hours:minutes)

Dis

char

ge (m

3 /s)

Sbv1Sbv2Sbv3Sbv4Sbv5Sbv6Sbv7Débit caculé (bassin)Débit mesuré (bassin)

Spatially distributed hydrological modelling of flood events on natural and farmed basins

I. Process

representation

III.Integration

into adistributed

model

II.Catchment

segmentation

The MHYDAS spatially distributed hydrological model(Modelling HYDrological AgroSystems)

(Moussa et al., 2002, Hydrol. Pro.)

Groundwater level

Channel routingDiffusive wave

Stream-aquifer interactionEmpirical equation based on Darcy’s law

Hydrological unitHydrological unit

Overland flowKinematic wave equation Rainfall

Infiltration(Green and Ampt)

III. Distributed modelling III.3. Test of scenarios

Testing the channel network management impact

0

250

500

750

00:00 04:00 08:0030 April 1993 (hour : minute)

Dis

char

ge (l

/s)

ReferenceScenario

0

250

500

750

3:00 7:00 11:0031 August 1994 (hour : minute)

Dis

char

ge (l

/s)

Actual ditch network Hypothetical scenario

Conclusions

Development of a spatially distributed hydrological modelling approach (MHYDAS) that is well adapted to natural and farmed catchments based on :

- The analysis of the Saint-Venant equations for flood routing with overbank flow, and the choice of the numerical method that minimises errors.

- Catchment segmentation into hydrological units and calculation of a transfer function based on catchmentgeomorphology.

- Multiscale calibration and validation.

- Human management impact analysis.

ReferencesMoussa R, Bocquillon C. 1994. TraPhyC-BV : A hydrologic information system. Environmental Software 9(4) :

217-226.Moussa R, Bocquillon C. 1996. Algorithms for solving the diffusive wave flood routing equation. Hydrological

Processes 10(1) : 105-124.Moussa R. 1996. Analytical Hayami solution for the diffusive wave flood routing problem with lateral inflow.

Hydrological Processes 10(9) : 1209-1227.Moussa R, Bocquillon C. 1996. Criteria for the choice of flood-routing methods in natural channels. Journal of

Hydrology 186(1-4) : 1-30.Moussa R, Bocquillon C. 1996. Fractal analyses of tree-like channel networks from digital elevation model data.

Journal of Hydrology 187(1-2) : 157-172.Moussa R. 1997. Geomorphological transfer function calculated from digital elevation models for distributed

hydrological modelling. Hydrological Processes 11(5) : 429-449.Moussa R. 1997. Is the drainage network a fractal Sierpinski space? Water Resources Research 33 (10) : 2399-2408.Moussa R, Bocquillon C. 2000. Approximation zones of the Saint-Venant equations for flood routing with overbank

flow. Hydrology and Earth System Sciences 4(2) : 251-261.Moussa R, Bocquillon C. 2001. Fractional-step method solution of diffusive wave equation. Journal of Hydrologic

Engineering 6(1) : 11-19.Moussa R, Voltz M, Andrieux P. 2002. Effects of the spatial organization of agricultural management on the

hydrological behaviour of a farmed catchment during flood events. Hydrological Processes 16 : 393-412.Moussa R. 2003. On morphometric properties of basins, scale effects and hydrologic response.Hydrological Processes 17(1) : 33-58.