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Modelling flash flood propagation in urban areas using a two-dimensional numerical model

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Abstract

This paper reports on the numerical modelling of flash flood propagation in urban areas after an excessive rainfall event or dam/dyke break wave. A two-dimensional (2-D) depth-averaged shallow-water model is used, with a refined grid of quadrilaterals and triangles for representing the urban area topography. The 2-D shallow-water equations are solved using the explicit second-order scheme that is adapted from MUSCL approach. Four applications are described to demonstrate the potential benefits and limits of 2-D modelling: (i) laboratory experimental dam-break wave in the presence of an isolated building; (ii) flash flood over a physical model of the urbanized Toce river valley in Italy; (iii) flash flood in October 1988 at the city of Nîmes (France) and (iv) dam-break flood in October 1982 at the town of Sumacárcel (Spain). Computed flow depths and velocities compare well with recorded data, although for the experimental study on dam-break wave some discrepancies are observed around buildings, where the flow is strongly 3-D in character. The numerical simulations show that the flow depths and flood wave celerity are significantly affected by the presence of buildings in comparison with the original floodplain. Further, this study confirms the importance of topography and roughness coefficient for flood propagation simulation.

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Abbreviations

A :

Cell area

C r :

Courant number

c :

Average wave celerity

d i,j :

Distance between the midpoint of the edge (i, j) and one of the adjacent edges

E :

x-Component of flux vector (Eq. 2)

\( \bar{E}_{\text{rr}} \) :

Average relative error

\( {\mathbf{F}} \) :

Flux vector \( = \left[ {{\mathbf{E}}({\mathbf{U}}),{\mathbf{G}}({\mathbf{U}})} \right] \)

g :

Gravitational acceleration

\( {\mathbf{G}} \) :

y-Component of flux vector (Eq. 2)

H :

Flow depth

I :

Cell index

i, j :

Index for the edge between the cells i and j

K s :

Strickler coefficient for flow resistance calculations (Eq. 3a, b)

l :

Edge length

n :

Time index

N i :

Set of neighbour cells of a cell

\( {\mathbf{n}} \) :

Edge outside normal unit vector

\( {\mathbf{P}}_{{}} \) :

Transformation matrix (Eq. 7)

RMSE:

Root mean square error

\( {\mathbf{S}} \) :

Source term vector \( = \;{\mathbf{S}}_{{\mathbf{0}}} + {\mathbf{S}}_{{\mathbf{f}}} \)

\( {\mathbf{S}}_{0} \) :

Bottom slope vector

\( {\mathbf{S}}_{\text{f}} \) :

Energy losses vector

t :

Time

u :

Flow velocity in the x-direction

\( {\mathbf{U}} \) :

Vector of conservative variables = [h, hu, hv]T

\( {\mathbf{U}}^{{\;{\text{L}}}} \), \( {\mathbf{U}}^{{\;{\text{R}}}} \) :

Values of \( {\mathbf{U}} \) at the left- and right-hand sides of an edge, respectively

\( {\mathbf{U}}_{x} \), \( {\mathbf{U}}_{y} \) :

Slopes of \( {\mathbf{U}} \) over a cell in the x-and y-directions, respectively

α (x), α (y) :

x- and y-Components of the normal unit vector \( {\mathbf{n}} \)

v :

Flow velocity in the y-direction

x, y :

Cartesian co-ordinates

z b :

Bed elevation

z w :

Water surface elevation

z *w :

Average water surface elevation over a cell

Δt :

Computational time step

η, ξ :

Local coordinates

∂ :

Partial derivative

∇.:

Divergence operator

|:

Posing that

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Acknowledgements

The authors wish to acknowledge the financial support offered by the French National Research Agency (ANR) for Research Contract ANR-05-PGCU-004, “RIVES.” Dr. Sandra Soares-Frazão and Professor Yves Zech (Université Catholique de Louvain), Guido Testa and David Zuccalà (CESI, Milan), Professor Francisco Alcrudo and Jonatan Mulet (Universidad de Zaragoza) are gratefully acknowledged for the work concerning the availability of experimental and field data. Finally, the authors would like to thank the guest editor (G. Iovine) and three anonymous reviewers for their detailed review and improvement of the English language of the original manuscript.

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Authors and Affiliations

  1. Cemagref, UR HHLY, 3bis quai Chauveau-CP220, F-69336, Lyon, France

    Kamal El Kadi Abderrezzak & André Paquier

  2. LEGI, Laboratory of Geophysical and Industrial Fluid Flows, 1025 rue de la Piscine BP 53, 38041, Grenoble Cedex 9, France

    Emmanuel Mignot

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  1. Kamal El Kadi Abderrezzak
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  2. André Paquier
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Correspondence to Kamal El Kadi Abderrezzak.

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El Kadi Abderrezzak, K., Paquier, A. & Mignot, E. Modelling flash flood propagation in urban areas using a two-dimensional numerical model. Nat Hazards 50, 433–460 (2009). https://doi.org/10.1007/s11069-008-9300-0

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