An Investigation of Preferential Flow Path in Case of Urban Flooding,

Using 2D Mathematical Hydrodynamical Modeling

GHIłESCU MARIE-ALICE

Department of Water Developments & Land Improvement Department (CHIF)

DAVID IOAN

Department of Hydraulics, Sanitary Engineering & Water Management Department (HISGA)

Faculty of Hydrotechnical Engineering, “Politehnica” University from Timişoara

George Enescu Street 1/A, 300022 Timişoara

ROMANIA

alice.ghitescu@hidro.upt.ro http://www.hidro.upt.ro

Ioan.David@gmx.net http://www.hidro.upt.ro

Abstract: - This study has concentrated on benchmarking model code against one another, by simulating a flood through an idealized city caused by a dam-break, assessing the sensitivity of simulation to model parameters and the

discretization of domain’s topography. Two hydrodynamic models approaches (2D SOBEK and 2D BreZo) were applied to simulate a flow through an idealized city. A comparison has been done between the mathematical models

results and physical model experiments. Experiments were conducted involving the configuration: a square city layout of 5 × 5 buildings aligned with the approach flow direction [2].The hydrodynamic models BreZo and SOBEK results showed that the simulated model behavior was close to the physical behavior regarding the flood wave propagation

through the urban district.

Key-Words: - Urban flooding, dam-break, 2D hydrodynamic modelling, numerical model

1 Introduction Experimentally urban floods were studied mainly for steady flow conditions, either subcritical or supercritical. It has been investigated that during the flood, in transient

flow conditions the severe transient phase is followed by a slow transient which is in some aspects close to steady

flow conditions. Dam-break flow experiments are often used to simulate floods in urban areas.

In case of urban flooding a major importance has the severe phase which involves specific features such as high velocities and transcritical flows, wave impact on

structures, or higher water rise following wave reflection against a structure [7].

Fig.1.Experimental set-up and channel dimensions in

(m): a) plan view, b) cross section [7]

The case study was chosen from a Hydraulic Laboratory experiment presented in the paper of Sandra Soares-

Frazao, “Dam-break flow through an idealized city”. In

this research the dam-break flow experiment is presented in an idealized city layout: A square city layout of 5×5 buildings aligned with the flow direction

Fig.1. Data were recorded by means of water level gauges and digital-imaging technique.

2 Problem Formulation The main characteristics of the models and the problem which is aimed to be solved are presented in the following subsections.

2.1 Input Data

The experiments were conducted at the Hydraulic Laboratory of the Civil and Environmental Engineering Department of the Université Catholique de Louvain,

Belgium. The test channel was 36m long and 3.6m wide, with a horizontal bed and a trapezoidal tail water cross

section (Fig.1). A gate was located between two impervious abutment blocks to simulate a breach. The initial reservoir water level was 0.40 m; the

downstream reach was wetted with a thin layer of 0.011m water. The reason for this initial wetting was the

imperfect tightness of the gate and the impossibility to completely dry the channel bed before conducting an experiment. The location of gauging points is shown in

Fig.2.

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ISSN: 1792-6114 281 ISBN: 978-960-474-243-1

Fig.2. Idealized city layout and positions of the gauging

points (not at scale) [7]

The Manning friction coefficient for the channel was assessed to 0.010sm-1/3 by steady-flow experiments without the blocks and the gate. The ratio between

building and street widths was chosen from aerial views of Brussels, Belgium.

The configuration of the urban district is symmetrical. The “city” consisted of impervious wooden locks of 0.3mx0.3m, while the “streets” were 0.10m wide.

Each test run was filmed at a rate of 40 frames per second during 10 s. The surface velocity field was

recorded using digital-imaging technique. A total of 22 experimental runs resulted in the surface-velocity data were done, in addition to the 16 runs made for the water

level measurements at the 64 gauging stations. Observations indicated that the flow rises at the city

front before entering the streets. A hydraulic jump forms at the impact section with the water level locally higher

than without the presence of the buildings, and is illustrated in Fig.3.

Fig.3.The upstream hydraulic jump of the urban district

in a symmetrical layout [7]

Due to the physical experiments, the geometrical characteristics (width, heights. slopes and cross-

sections), the hydraulic characteristics (water levels, steady flow type) and physical characteristics (roughness, material types) of the system were taken

into consideration.

2.2 Numerical Models For this study were used two programs for flooding simulations, BreZo2D and SOBEK2D.

BreZo solves the shallow-water equations by using a Godunov-type finite volume algorithm optimized for

wetting and drying applications involving natural topography and runs on an unstructured grid of

triangular cells. (source http://sanders.eng.uci.edu/brezo.html). SOBEK is an integrated software package, developed by

WL | Delft Hydraulics for river, urban or water management. 2D hydrodynamic prediction package

known as Delft-FLS (Delft-Flooding System), it is based upon the complete de Saint Venant Equations, thus including transient flow phenomena and backwater

profiles.

2.2.1 Governing Equations for Mathematical

Model create with BreZo Code [9] The flow model is based on two-dimensional 2D shallow-water equations, which are depth-integrated

equations of motion consisting of a fluid continuity equation and two momentum equations, one for each horizontal coordinate direction. For consistency with the

finite-volume method, which involves the direct discretization of the governing equations in integral

form Anderson 1995, the shallow-water equations are written as follows:

∫ ∫∫ Ω∂ ΩΩΩ=−+Ω

∂∂

SdGxFdyUdt

)( (1)

Where: - Ω represents a 2D domain (e.g., a

computational cell) with a boundary Ω∂ - U is vector of flow variables,

ThvhuhU )(= The flux terms F, G and the source term S are defined as it follows:

+=

huv

ghhu

hv

F 22

2

1 (2)

+

=22

2

1ghhv

huv

hv

G (3)

+−∂∂

−

+−∂∂

−=

22

22

0

vuvcy

zgh

vuucx

zghS

D

D (4)

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Where: - h is the flow depth (m);

- g is the gravity acceleration (m/s2); - u is vertically averaged velocities in the x

direction (m/s); - v is vertically averaged velocities in the y

direction (m/s);

- z is bed elevation (m); - cD is the drag coefficient that is typically

treated as a constant or computed using the Manning

formula 3/12 / hgncD = ;

- n is the Manning bed roughness factor.

2.2.2 Governing Equations for Mathematical

Model create with SOBEK software The 2D shallow water equations, de Saint Venant equations expressed in their non-conservative form per

unit width of channel are integrated. The basic assumption applied in the 2D shallow water equation is that the water is much smaller than the horizontal length

scales, vertical velocities being neglected. The equations are concentrated on the x-y plane, due to the fact that

maximum number of flow dimensions is two. The continuity equation of two dimensional flow is:

Continuity equation:

0)()(=

∂∂

+∂

∂+

∂∂

y

vh

x

uh

t

ζ (5)

Momentum equation:

02

=++∂∂

+∂∂

+∂∂

+∂∂

uauhC

uug

xg

x

uv

x

uu

t

u ζ (6)

02

=++∂∂

+∂∂

+∂∂

+∂∂

vavhC

vvg

xg

x

vv

x

vu

t

v ζ (7)

Where: ζ is the water level (with respect to the reference level) ζ = h +zb

h is the local water depth (m) zb is the local bottom level (m)

u is the velocity in x- direction (m/s) v is the velocity in y- direction (m/s)

t is the time (s) x is the distance (m) cf is the friction coefficient

V is the velocity (m/s)

2

1

22 )( vuV += C is Chezy coefficient (m ½ /s)

a is the wall friction coefficient(1/m)

3 Models Construction Construction of 2D hydrodynamic model in BreZo

The grid files were generated by Triangle, a 2D mesh generator created by Jonathan Shewchuk. The buildings blocs have been considered as holes for this model.

Fig.4. illustrates the idealized city layout plan view.

Fig.4. Plan view of the channel dimensions in (m) (Not to scale)

Fig.5. illustrates the positions and the type of the boundary markers.

Fig.5. Positioning of boundary markers and types

The resulted cell (triangle) size of the mesh is

A=0.01m2, with an approximate resolution of 0.001m and with the minimum permitted angle of 30o. Fig.6.

shows the overview of the resulted mesh.

Fig.6. Triangular mesh generated after using Triangle: a) Channel plan view; b) Mesh detail in the ideal city (dimensions in m); c) Upstream boundary (reservoir)

mesh detail (dimensions in m)

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The value of the inflow discharge has been set at 0m3/s and the outflow water level at 0.1m.

An .ic (initial condition) file was used to set up the initial water level of the reservoir, for this purpose, the

entire upstream area has been considered separately. The roughness type and value, were set as a uniform Manning’s n=0.011.

Construction of 2D hydrodynamic model in SOBEK In the figure Fig.7.a) is presented the DEM of the

channel, including the urban district, without the grid. In detail of the “city” illustrated in the Fig.7.b) the dimensions and the measurement stations number can be

observed.

Fig.7. Digital Elevation Model (DEM) of the channel with the urban district: a) Channel with the urban district

plan view; b) DEM detail in the ideal city (not to scale)

The grid cell size of the rectangular mesh has been set at 0.05m, taking into account a street width of 0.1m.

In the whole channel an initial water level have been set at h=0.011m and a uniform roughness value of Manning’s n = 0.011m.

3.1 Time Step and Simulation Period The simulation period taken into consideration, due to the available data for further comparison, was 30 seconds. The time step was chosen taking into account

the importance of maintaining the stability of the model. BreZo is conditionally stable. Due to these

considerations, the time step value in BreZo was fixed to 0.1second and in SOBEK to 1second.

3.2 Boundary Conditions Data The boundary conditions data have been provided from

the reference paper [7]. The upstream boundary condition has been set as an initial reservoir with water level h=0.4m (above datum)

and the downstream boundary condition has been set as

free flow, also as a water level h=0.01m (above datum). An initial water level h=0.011m has been also set in the

downstream channel area, as in the physical model experiment. All the conditions mentioned were set for

both of the 2D hydrodynamic models, in BreZo and in SOBEK.

4 RESULTS Numerical results obtained with the two two-

dimensional finite-volume models are compared herein with the experimental results.

The simulation in BreZo model was performed on unstructured triangular mesh; while the mesh grid of SOBEK model is rectangular.

Measurement stations have been fixed in the urban district in order to collect data. In Fig.8 and Fig.9, the

location of these points is presented for both of the hydraulic models, SOBEK and respectively BreZo.

Fig.8. Location of Station points in SOBEK

Fig.9. Location of Station points in BreZo

The results measurements are taken at time t and time t1, which are corresponding with T=5s and respectively

T=10s in the physical experiments measurements. Time

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t is equal with the time when the water enters in the city and the hydraulic jump occurs at the impact section.

Time t1 represents the time when the supercritical situation is obtained, when the hydraulic jump is still

present but the street water levels have significantly increased.

Fig.10. Water surface profiles results along the central longitudinal street: a) numerical at time T=t s; b)

experimental at time T=5s

Fig.10a),b) and Fig.11a),b) (a) the physical experiments and b) the models measurements) are showing the water-

surface profile computed in the street close to the channel axis (y=0.2m) from the physical experiments

and from the two hydraulic models results.

Fig.11. Water surface profiles results along the central longitudinal street: a) Numerical at time T=t1s; b)

Experimental at time T=10s

In the physical model, the profiles at two times T=5s and T=10s, are compared with the corresponding times t and

t1 of the hydraulic models. It can be simply observed that the flow rises at the city front before entering the

streets, and a hydraulic jump forms. The water level without the presence of the buildings is higher than in the urban district.

A discrepancy in the tendency of the flow, between the experiments and models, arises after entering into the

city, and could be due to the way in which the boundary condition has been set different upstream. The figures Fig.10 a) and Fig.11 a) are showing the

relations between distance and depths for particular times in SOBEK and BreZo models. The water depths

calculated thus are higher than the values measured in the physical model. After analyzing the flow for to both

time frames, the flow in the streets has evolved from supercritical to sub-critical. In Fig.12 a), b), c) the comparison of velocity vectors

between the two hydraulic models and the experiment is made.

Fig.12. Comparison of vectors: a) Experimental (T=5 s), b) BreZo model (T=t s), c) SOBEK model (T=t s)

Both hydraulic models are graphically presenting the

direction and the magnitude of the velocity vectors, which can be observed in Fig.12b) and Fig.12c).

In BreZo the water depths and velocities are calculated in the nodes of the triangular grid and those points are located at the side of the street as showing in Fig.9.

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These values of the vectors are not totally comparable with the measured values because the mesh size is

bigger in the streets of the “ideal city’. But they show the same tendency as the measured velocity vectors.

In SOBEK the vectors and the values of the velocities are calculated in the centre of the rectangular grid cell, and because of the rectangular shape the cells

distribution in on the streets is better. Due to that it shows more accurate the velocities distribution and the

behavior of the flow path. The differences between the measured and calculated values of the water surface profiles can occur due to 3

facts: - The calculation points in each diagram do not

have exactly the same coordinates as the measured ones; - The influence of roughness of the buildings,

which in both models has not been taken into account; - The hydraulic models use different numerical

methods to solve the 2D equations.

5 Conclusion Two model approaches (2D SOBEK and 2D BreZo) were applied to simulate a flow through an idealized city

caused by a dam-break. The hydrodynamics BreZo and SOBEK gave a good estimation of flood flow path through the urban area.

The models provide velocity vector maps, water level and water depth maps which can be used to track the

water movement through a city. The 2D models represent a better approach of the physical phenomena. They consider the topographical

variations of the terrain and roughness while calculating the depths and velocities in two directions, which

represents the key factor to define the preferential flow path due an extreme event. Despite of the required amount of computational time,

the 2D hydrodynamic models are computational intensive, and can be used as an important tool for the

decision maker support in order to reduce the flood risk. Apart from data problems for the Ideal City Case the SOBEK2D model suffered itself from technical flaws.

As an example the upstream water levels in the reservoir could not been set up by the use of an initial water level

node. That is why a water level time series had to be created.

The differences between the comparisons are due to the construction of the models, but the values obtained have the same order of magnitude as the measured ones. The

overall agreement between the numerical models and physical experiments is good, so these 2D models can be

used in estimating the flow path. It is recommended to use the 1D model approach in urban flooding modelling in order to find the flow path

in urban area, by building reaches over the streets connected to the main river in order to propagate the

flood wave through the city. Also a two-dimensional model could be used to validate 1D network approaches,

which might appear as a good compromise to limit the computational effort. The aim of this study is to contribute to better

understanding of urban flooding phenomena and to bring some useful inputs for decision making and

planning of flood events. References:

[1] X1. Abbott, M.B., Computational Hydraulics,

Pitman, London, 1979. [2] X2. Bradford, S.F., and Sanders, B.F., Finite-volume

model for shallow-water flooding of arbitrary topography, International Journal of Hydraulic

Engineering, Vol.128, No.3. 2002, pp. 289–298

[3] X3. Douben, N., Flood Management, UNESCO-IHE Delft, The Netherlands, 2002.

[4] X4. Ghitescu, M.A., An Investigation of Inundation

Modelling for Urban Areas, MSc Thesis, UNESCO-IHE, Delft, 2009.

[5] X5. Horritt M.S., Bates P.D., Evaluation of 1D and 2D numerical models for predicting river flood

inundation, Journal of Hydrology, Vol. 268, 2002, pp. 87–99.

[6] X6. Sanders, B.F., and Bradford, S.F., Impact of

limiters on accuracy of high-resolution flow and transport models, Journal of Mechanical

Engineering, 2006, Vol.132, No.1, pp. 87–98. [7] X7. Soares-Frazão, S. Zech, Y. Dam-break flow

thorough an idealized city, Journal of Hydraulic

Engineering, 2008, Vol.46, No.5, pp. 648-658. [8] X8. Soares-Frazão, S. Zech, Y. (2002). “Dam-break

in channels with 90 bend”. Journal of Hydraulic

Engineering 128(11), 956–968. [9] X9. Sanders, B.F., and Bradford, S.F., Unstructured

grid finite volume algorithm for shallow-water flow and transport with wetting and drying, Journal of

Hydraulic Engineering 2006, Vol.132 (4), pp.371-384.

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