Modèle eectif par une approche de Saint-Venant pour les écoulements complexes lors d'inondations urbaines

Présentation des résultats majeurs de la thèse

Validation

Le code Flood1D2D a été validé à l'aide de diérents cas tests. La capacité du modèle 1D2D à modéliser des écoulements très transitoires a été évaluée sur l'expérience de rupture de barrage

Introduction

Les inondations représentent le premier risque naturel sur chaque continent et ont causé en Europe environ 100 milliards d'euros de dommage entre 1986 et 2006 (Moel et al. (2009)). Dans le contexte du changement climatique et de ses eets potentiels sur les régimes hydrologiques, une urbanisation sans précédent dans les plaines inondables a accru la vulnérabilité sociétale. L'amélioration de la prévision des inondations et de la ré-analyse combinée à des analyses d'incertitudes, notamment en zone urbaine, est ainsi une priorité pour l'aide à la décision en matière de protection civile et pour les politiques d'assurance.

Des données ables mesurées lors d'inondations urbaines qui sont nécessaires pour améliorer la compréhension des phénomènes d'inondation urbaine, élaborer de nouveaux modèles ou calibrer les modèles existants sont rares et complexes à obtenir. Les jeux de données obtenus sur des plateformes expérimentales d'étude des inondations urbaines sont donc précieuses voire essentielles (voir par exemple Mignot et al. (2006); Schubert and Sanders (2012); Finaud-Guyot et al. (2018)). Par ailleurs, il apparaît que les interactions entre des écoulements fortement énergétiques et les obstacles contrôlent l'hydrodynamique 3D (comme cela apparaît dans les carrefours Mignot et al. (2006)) ce qui implique qu'une représentation réaliste de ces phénomènes peut nécessiter la résolution d'un modèle 3D complet basé sur les équations de Navier-Stokes. Les débits injectés dans le modèle, qui peuvent provenir de données in-situ (impliquant généralement les incertitudes sur les courbes de tarage employées) ou de simulation hydro-météorologique impliquant des incertitudes météorologiques et sur le modèle hydrologique employé, constituent également une source connue d'incertitudes notamment dans le cas d'écoulements complexes en zone urbaine.

Les cartes d'inondation, de hauteurs de submersion et de vitesse d'écoulement sont généralement produites à partir de modèles 'shallow water 2D' correspondant à une moyenne sur la hauteur d'eau des équations de Navier-Stokes. En conséquence, la paramétrisation des phénomènes de dissipation d'énergie dans un modèle simplié (tel qu'un modèle '2D shallow water') peut dépendre de l'échelle de modélisation considérée pour la représentation des pertes de charge localisées ou du frottement pariétal sur des micro ou macro rugosités dans des géométries complexes (Guinot (2012)). Cependant si la taille du domaine augmente de quelques blocs à l'échelle de tout un quartier voire une ville, le coût de simulations 2D reste élevé. Face à cela, des couplages 1D2D (Finaud-Guyot (2009); Finaud-Guyot et al. (2010a)) et des méthodes macroscopiques (Guinot and Soares-Frazão (2006); Sanders et al. (2008); Guinot (2012); Guinot et al. (2017a)) apparaissent comme des compromis avantageux pour réduire le coût de calcul des modélisations en utilisant la précision correcte du 2D au niveau des intersections hydrauliques telles que les carrefours et des modélisations 1D dans les portions d'écoulement plus simples comme les rues. III Dans le contexte de la modélisation eective des inondations urbaines et en vue de proposer un nouveau modèle 1D2D, les questions suivantes sont étudiées dans cette thèse :

1. Quelles sont les zones d'écoulement clé et quelles caractéristiques doivent être reproduites par un modèle simplié 1D2D sur un réseau maillé ? 2. Comment paramétriser un tel modèle par exemple dans des rues où des structures d'écoulement 3D telles que des recirculations peuvent apparaître à l'aval des carrefours ? 3. Quelles sont les sensibilités paramétriques des modèles 2D SW et 1D2D dans le cas de crues urbaines dans des réseaux maillés ? 4. Quelle complexité de modélisation et quelle paramétrisation doivent être employées pour permettre une représentation eective des écoulements lors d'inondations urbaines sur de grands domaines avec un coût de calcul raisonnable en vue d'applications opérationnelles ?

Ce travail de thèse est proposé en vue d'améliorer la compréhension des mécanismes d'écoulement en jeu lors de crues urbaines et leur modélisation équivalente à un coût de calcul réduit -un nouveau code C/C++ Flood1D2D est présenté. Le manuscrit est organisé de la manière suivante : la section 1 présente une étude bibliographique ; les équations et leur discrétisation sont présentées en section 2 ; la section 3 illustre la validation sur des cas synthétiques du nouveau modèle Flood1D2D proposé dans ces travaux ; la sensibilité du modèle SW 2D à ses paramètres d'entrée est étudié avec des méthodes de décomposition de la variance au chapitre 4 ; l'application du nouveau modèle Flood1D2D sur sur plusieurs congurations hydrauliques complexes incluant un réseau des rues et carrefours est détaillée au chapitre 5.

Présentation de Flood1D2D

Le modèle Flood1D2D proposé dans cette thèse vise à modéliser les inondations urbaines en se basant sur le caractère parcimonieux des modélisations 1D et incluant des zooms 2D dans les zones à plus fortes complexités de l'écoulement tels que les zones de conuences / déuences.

2.1

Couplage 1D2D

Les rues sont donc modélisées par une approche 1D et les carrefours par une approche 2D (voir gure 1). La distinction 1D -2D n'est toutefois pas liée à une quelconque contrainte sur le jeu d'équations résolues. Dans les deux cas, les équations 2D de Saint-Venant impliquant la hauteur d'eau et les deux composantes horizontales de la vitesse sont employées sans aucune orientation privilégiée de la vitesse dans les mailles 1D. La distinction réside dans le fait que les rues sont modélisées avec une unique maille sur la largeur.

Les équations résolues sont les suivantes :

∂U ∂t + ∂F ∂x + ∂G ∂y = S (1a) U =   φh φq φr   F =     φq φ q 2 h + 1 2 gφh 2 φ qr h     G =     φr φ qr h φ r 2 h + 1 2 gφh 2     S =      0 gφh (S 0,x -S f,x ) + 1 2
gh 2 ∂φ ∂x gφh (S 0,y -S f,y ) + 1 2 gh 2 ∂φ ∂y

     (1b)
IV x h 4/3 (S f,y = v √ u 2 +v 2 K 2 y h 4/3 ) sont la pente du fond et la pente de la ligne des frottements dans la direction x (respectivement y) ; z b est la cote du fond et φ est la porosité qui traduit le fait que la surface en plan d'un élément de calcul n'est pas totalement disponible pour l'eau.

L'équation (1a) est discrétisée sur chaque maille (1D ou 2D) par une méthode aux volumes nis :

U n+1 i = U n i - t φ i A i k∈N (i) w k P k F n k + tS (U n i ) (2)
où U n i représente l'état hydraulique du système dans la maille i à l'instant n, ∆t le pas de temps de calcul, φ i A i la surface en plan de la maille i disponible pour l'eau, N (i) l'ensemble des interfaces délimitant la maille i ; w k la longueur de l'interface k, P k la matrice de passage permettant de passer du référentiel global au référentiel attaché à chaque interface et F n k = F k (U n ) le ux de U à travers l'interface k.

En considérant les diérents types d'interfaces pouvant exister du fait de la topologie du maillage considéré (voir gure 1), on a N (i) = N 1D,W (i) ∪ N 1D,BC (i) ∪ N 1D1D (i) ∪ N 2D,BC (i) ∪ N 2D2D (i) ∪ N 1D2D (i) qui regroupe les interfaces 1D correspondant à des murs, à des conditions limites ou entre deux mailles 1D ; les interfaces 2D correspondant à des conditions limites ou entre deux mailles 2D ; et les interfaces entre mailles 1D et 2D. Les ux F n k entre deux mailles (pour les ensembles N 1D1D ∪ N 2D2D ∪ N 1D2D ) sont calculés en utilisant un solveur de Riemann u (x) est la vitesse longitudinale ; y s (x) est la taille de la zone de recirculation comptée à partir du mur ; L s est la longueur totale de la zone de recirculation (où y s (x) > 0 et L s = max x y s (x), ∀x ∈ [0, L s ]) ; r 0 est la distance depuis le bord du carrefour jusqu'à l'endroit où la taille de la zone de recirculation est maximale et vaut B s .

(HLLC Soares-Frazão and Guinot (2007) ou PorAS Finaud-Guyot et al. (2010b)) an de garantir le caractère well-balanced du schéma numérique. Le solveur de Riemann calcule le ux F n k à partir des grandeurs dans les mailles gauche et droite de l'interface k. Les ux F n k à la limite d'une maille (pour les ensembles N 1D,W ∪ N 1D,BC ∪ N 2D,BC ) sont calculés de manière similaire en utilisant une estimation des variables hydrauliques au niveau de la limite à partir des spécications de l'utilisateur et des invariants de Riemann le long de la caractéristique entre la maille et la limite (Araud (2012)).

2.2

Prise en compte des recirculations dans les mailles 1D

La bibliographie (par exemple Best and Reid (1984); Gurram et al. (1997); Mignot et al. (2006)) et les expériences menées sur le pilote d'inondation urbaine d'Icube montrent la présence de zones de recirculation dans les biefs à l'aval des carrefours qui contractent l'écoulement et peuvent de ce fait jouer un eet signicatif sur la répartition des débits dans un ou plusieurs carrefours (voir gure 2). Une topologie de maillage 1D permet de prendre en compte ces zones de recirculations. Il convient donc de prendre en compte leur eet sur l'hydrodynamique modélisée et qui n'est pas représenté dans les équations résolues (1a).

Dans les rues (modélisées par des mailles 1D), une paramétrisation complémentaire est donc introduite. Sur ces mailles, on considère que la veine d'écoulement eective est contractée par la présence de la zone de recirculation apparaissant à l'aval du carrefour. Les mailles 1D, dénies pour représenter l'ensemble de la largeur de la rue, contiennent donc deux domaines Ω 1D = Ω u ∪ Ω v avec Ω u la zone d'écoulement ineectif située dans la recirculation pour laquelle il n'y a en moyenne pas d'écoulement et Ω v la veine d'écoulement par laquelle tout le débit circulant das la rue transite (voir gure 3).

Le partage de la maille 1D est basé sur la fonction ω qui prend en compte les eets de la recirculation. ω mesure la proportion de la largeur de la rue disponible pour l'écoulement : ω = L v /L où L = L u + L v représente la largeur totale de la maille 1D, L u (respectivement L v ) est la largeur de la recirculation (respectivement veine d'écoulement). On peut démontrer que ω = Ω v /Ω 1D . Cette nouvelle dénition de ω est similaire à la notion de porosité dénie dans Lhomme (2006). La porosité des mailles 2D est donc prise égale à 1 et pour toutes les mailles, elle est conservée constante dans le temps (et xée a priori par le modélisateur). Cela suppose que le régime permanent dans un réseau maillé d'écoulement à surface libre ne dépend pas du régime VI Figure 3 Représentation schématique de la prise en compte des recirculations dans les mailles 1D transitoire ayant conduit à cet état. Cette hypothèse, qui devra être étudiée de manière plus exhaustive sur le pilote d'inondation urbaine d'Icube, permet de négliger les eets de couplage entre l'hydrodynamique et la porosité.

De par la topologie du maillage, les ux F n k dont le calcul doit prendre en compte la nouvelle paramétrisation ω sont calculés à travers les interfaces dans l'ensemble

N uv = N 1D,W ∪ N 1D1D ∪ N 1D2D = N 1D,W u ∪ N 1D,W v ∪ N 1D1D,u ∪ N 1D1D,v ∪ N 1D2D,u ∪ N 1D2D,v .
Pour ces ensembles d'interface, on écrit :

k∈Nuv w k P k F n k = k∈Nuv P k w k,u F n k,u + w k,v F n k,v (3)
où w k,u (respectivement w k,v ) est la longueur de l'interface k le long de la zone de recirculation (respectivement la veine d'écoulement) telle que

w k = w k,u + w k,v = (1 -φ k ) w k + φ k w k et F n k,u
(respectivement F n k,v ) est le ux à travers w k,u (respectivement w k,v ).

Table 1 Type Ensemble Domaine

U L U R φ k Gauche Droit 1D 2D 1D/1D N 1D1D,u Ω 1D,u Ω 1D,u h 0 0 T h 0 0 T φ G + φ R 2 1D/1D N 1D1D,v Ω 1D,v Ω 1D,v h q/φ r/φ T h q/φ r/φ T φ G + φ R 2 1D/BC N 1D,W u Ω 1D,u Mur h 0 0 T BC 1 1D/BC N 1D,W v Ω 1D,v Mur h q/φ r/φ T BC 1 1D/2D N 1D2D,u Ω 1D,u Ω 2D h 0 0 T h q r T φ 1D 1D/2D N 1D2D,v Ω 1D,v Ω 2D h q/φ r/φ
La géométrie de l'expérience est composée d'un réservoir alimentant un canal avec un angle de 90 • terminé par une chute d'eau (voir gure 4). Au début de l'expérience, la porte du réservoir est ôtée permettant l'écoulement de l'eau du réservoir (initialement 0.25m au dessus du fond du canal). L'expérience mesure les niveaux d'eau en diérents endroit du canal jusqu'à la vidange totale du réservoir. La conguration expérimentale est modélisée avec Flood1D2D. Les paramètres expérimentaux sont repris et la valeur du coecient de Manning employée par les auteurs est conservée (n = 0.011m -1/3 .s). Le maillage employé est 1D dans les parties rectilignes et 2D dans le réservoir, dans et immédiatement à l'aval du coude (voir gure 4). Les prols de hauteur d'eau obtenus à diérents instants avec Flood1D2D (en utilisant le maillage 1D2D ou seulement un maillage 2D n) est comparés à l'expérimental (voir gure 5). Les résultats montrent une bonne adéquation entre l'expérimental et la modélisation 1D2D ou 2D. Les caractéristiques hydrodynamiques de l'écoulement (hauteur et position des fronts, vitesse de déplacement) sont bien représentées par le modèle 1D2D.

Propagation d'incertitudes dans les modélisations d'inondations urbaines

Le but de la modélisation hydraulique est de prévoir le comportement d'écoulements à surface libre sur un domaine donné, l'évolution spatio-temporelle des hauteurs d'eau et débits y est simulée. Selon la quantité de données disponibles et les objectifs de modélisation, diérentes complexités et paramétrisations hydrauliques peuvent être choisies. Ainsi, l'incertitude dans les sorties de modèles peut provenir de plusieurs sources : le type de données et erreurs d'observation, les incertitudes structurelles de modélisation ou encore les incertitudes paramétriques.

La modélisation hydraulique est utilisée pour prédire l'hydrodynamique des crues, comme ici dans le contexte de réseaux urbains maillés. Cette partie s'intéresse à l'analyse de sensibilité d'un modèle SW 2D. Le but est d'analyser comment la répartition des écoulements et la distribution des hauteurs d'eau peuvent être inuencées par la structure du modèle, les conditions aux limites et les paramètres. Les sources d'incertitude en modélisation hydraulique incluent typiquement les conditions aux limites amont (ex : débits entrants) et aval (ex : hauteur, courbe de tarage), rugosité, topographie... Dans cette partie, plusieurs sources d'incertitude sont étudiées an d'évaluer leur inuence sur la variance des sorties du modèle à l'aide d'une méthode d'analyse de sensibilité VIII Figure 5 Comparison des résultats issus de Flood1D2D (avec un maillage 2D ou 1D2D) avec les expériences réalisées par Soares-Frazão and Zech (2002). Ligne d'eau le long du prol A-B-C (voir gure 4). La zone 2D du maillage 1D2D est localisée entre les deux barres verticales noires.

IX globale (GSA).

3.1

Modèlisation 2D SW Des simulation 2D en régime permanent sont réalisées sur le pilote inondation Icube, représentant un réseau urbain (5m*5m), en vue de reproduire des structures d'écoulement complexes (cf. Arrault et al. (2016)). Les conditions aux limites sont basées sur un jeu de données expérimental. Des débits et hauteurs d'eau constants sont xés aux entrées et sorties du domaine, à fond plat horizontal, ainsi qu'un coecient de frottement uniforme (K 2D = 100m 1/3 .s -1 ) puisqu'il est simplement calibré sur la distribution des débits mesurée en sortie. Notons qu'une analyse de convergence des résultats par rapport à la nesse du maillage a été réalisée pour des mailles de taille variant de 0.01 à 0.0025m -la taille de maille retenue est de 0.005m. Les analyses de sensibilité sont réalisées pour un débit entrant total de 80m 3 .h -1 , correspondant à un fort débit de crue d'occurrence rare dans la réalité, répartit de manière diérente entre les faces ouest et nord (50-50%, 80-20%). Sur une face le débit est répartit proportionnellement à la largeur de rue.

La distribution spatiale des hauteur d'eau et nombres de Froude est présentée sur la gure (6). La hauteur d'eau décroît des faces amont vers les faces aval, du coin nord-ouest au coin sud-est. L'écoulement est partout sous-critique sauf dans la rue 1 à l'aval du dernier carrefour entre les rues 1 et G, où une zone d'écoulement supercritique apparaît. La répartition de débit asymétrique dans le cas 80-20% a tendance a augmenter la hauteur d'eau au coin sud-est et réduire la région d'écoulement supercritique. Il est intéressant de noter que la répartition de débit simulée dans les deux congurations (50-50% et 80-20%) est très similaire et correspond bien aux mesures expérimentales. Ces résultats mettent en avant l'eet ltrant de la géométrie urbaine sur l'hydrogramme entrant comme démontré dans Finaud-Guyot et al. (2018).

GSA sur les hauteurs d'eau

La sensibilité d'un modèle 2D à ses quatre paramètres principaux, Q west et Q north les débits d'entrée sur la face ouest et nord, la hauteur d'eau en sortie h out et un coecient de frottement uniforme K, est testée à l'aide d'une méthode d'analyse de sensibilité globale (GSA). Elle permet d'évaluer la contribution des diérents paramètres testés sur la variance des sorties du modèle (hauteurs d'eau et débits de sortie). Les gammes de variations des paramètres sont déterminés à partir des incertitudes expérimentales (Q cal north ± 5%, Q cal west ± 5% ; h cal out ± 10% et K cal ± 50%). Un ensemble de N s = 2000 jeux de paramètres permet une bonne estimation a des indices de Sobol d'ordre 1 dans cette conguration avec la méthode de Ratto et al. (2007).

Les cartes de sensibilité de la hauteur d'eau modélisée aux paramètres sont présentées sur la gure (6) avec les cartes de nombre de Froude F r locaux pour 2 répartitions de débit entrant. La somme des indices de Sobol ΣS i est proche de 1 pour les zones d'écoulement sub-critiques (correspondant à un nombre de Froude F r < 1) indiquant que la variance du modèle est bien expliquée par les paramètres testés. ΣS i est inférieure à 1 pour la zone où apparaît une zone d'écoulement supercritique pour certains jeux de paramètres.

Les cartes de sensibilité de la hauteur d'eau aux quatre paramètres sont présentées sur les gures 6b, c, d, e. Pour la conguration de débits entrants équi-répartis, la rugosité K et la hauteur d'eau aval h out sont les paramètres les plus importants expliquant la variance des hauteurs d'eau. La sensibilité à K et h out évolue globalement dans la direction nord-ouest vers sud-est excepté dans le cercle correspondant aux écoulements supercritiques : décroissance (resp. croissance) de la sensibilité à K (resp. h out ). La sensibilité au débit entrant décroît le long de cette direction eective d'écoulement, elle est quasiment nulle à l'aval. X Les histogrammes présentés sur la gure 7 montrent la sensibilité de la hauteur d'eau aux paramètres testés pour quelques intersections hydrauliques -carrefours. Elles sont présentées selon la distance hydraulique à partir du coin nord-ouest et décrivent quantitativement la décroissance de l'eet des contrôles amonts Q west , Q north , K.

La gure 6 présente les cartes de sensibilité pour la conguration 80-20% des débits entrant. Les tendances des sensibilités spatiales sont similaires à la conguration 50-50%. Pour la conguration 80-20%, ces cartes montrent que la sensibilité des hauteurs d'eau à Q west et quasiment deux fois plus grande que celle à Q north . L'indice de Sobol de Q north décroît vers l'aval alors que celui de Q west augmente et explique 2% de la variance des hauteurs d'eau pour les rues 1 à 4. Dans ce cas l'inuence du contrôle aval (h out ) est plus faible pour la région située près des faces nord et ouest. La région d'écoulement supercritique est également réduite puisqu'un débit moins grand provient de la face nord. 4 Modélisation de congurations urbaines avec La modélisation des écoulements à l'échelle d'un carrefour avec le modèle Flood1D2D a été étudiée sur la conguration expérimentale proposée par Nanía et al. (2004). Le carrefour étudié expérimentalement est maillée avec l'approche 1D2D proposée : le carrefour est 2D, les rues sont 1D (voir gure 8). Des conditions aux limites symétriques (Q in,x = Q in,y = 0.075m 3 .s -1 et hauteur d'eau aval h out = 0.1m) permettent de garantir une symétrie des écoulements dans les rues de sorties et seules les résultats d'une rue sont donc présentés. La simulation est menée en régime permanent et confrontée à une simulation avec le logiciel Telemac 2D qui sert de référence (voir gure 9).

Flood1D2D est paramétrisé avec une porosité dans les rues aval :

B (s) = as 2 + bs + c (4)
où B est la largeur de la recirculation, s est l'abscisse le long de la rue aval (s = 0 est positionné arbitrairement à l'amont de la rue et orienté dans le sens de l'écoulement). a, b et c sont trois coecients calculés tels que B (0) = B (L s ) = 0 et B (L s /2) = B s avec L s la longueur de la recirculation et B s la largeur maximale de la recirculation. La modélisation de référence (Telemac2D) permet de déterminer les valeurs de L s = 3.3m et B s = 0.52m. La modélisation est réalisée avec Flood1D2D avec un coecient de Strickler uniforme K = 100m 1/3 .s -1 et B s = {0; 0.2; 0.26; 0.3; 0.52}.

Les résultats de modélisation 1D2D sont comparés à la référence pour déterminer dans quelle mesure le modèle proposé peut reproduire les caractéristiques 2D de l'écoulement (voir gure 10). La référence montre que le niveau d'eau passe par un minimum dans la rue aval (x ≈ 5.15m) qui est lié à la présence d'une zone de recirculation (visible à partir de la carte des vecteurs vitesses -non présentées ici). La modélisation d'une telle conguration en utilisant une approche 1D2D paramétrisée uniquement avec un coecient de frottement (B s = 0m) ne permet pas de reproduire ce minimum dans le niveau d'eau modélisé. Cette constatation peut être généralisée : l'analyse des termes sources de l'équation SW modélisée montre que ce type de prol des hauteurs d'eau ne peut être obtenu même avec un coecient de frottement non-uniforme. Pour une porosité B s = {0.2; 0.26; 0.3}, le prol de référence est relativement bien reproduit. Les écarts entre la modélisation 1D2D et la référence dépendent de la fonction de paramétrisation employée. Pour une porosité B s = 0.52m (correspondant à la valeur calculée à partir des résultats de référence), la contraction de la veine d'écoulement est susamment importante pour que l'écoulement modélisé XI Figure 6 Cartes de sensibilités paramétriques 2D pour un débit entrant total de 80 m 3 /h : gauche, débit entrant équi-réparti (50-50%) entre faces nord et ouest, droite conguration 80-20%. Les cercles repèrent les zones d'écoulement supercritique (Fr > 1). Comme illustré précédemment, la modélisation de la ligne d'eau d'une rue en 1D avec une paramétrisation prenant en compte seulement la friction ne permet pas de rendre compte de l'accélération locale de l'écoulement où la veine d'eau est contractée par la zone de recirculation. L'ajout d'une nouvelle paramétrisation prenant en compte la réduction de la veine d'eau permet de palier ces dicultés et montre une inuence sur la répartition des débits à l'aval d'une intersection hydraulique. Cette section montre les eets d'une distribution de porosité sur l'écoulement modélisé à l'échelle de plusieurs carrefours.

XII

Présentation de la conguration étudiée

La réponse de Flood1D2D est étudiée sur une sous partie du pilote d'inondation urbaine d'Icube an d'utiliser toute la complexité d'un jeu de données expérimental tout en limitant les coûts de calcul et le nombre de paramètres. La sous partie choisie comprend 4 carrefours connectés par des rues de largeurs et orientations variables, au centre du dispositif expérimental (voir gure 11). La largeur d'une rue large (F et 4) est 12.5cm et 5cm pour les plus petites. Un carrefour est identié par le nom des deux rues le formant (ex. 'E3'). Une sous partie d'une rue est nommée en utilisant les noms des carrefours amont et aval (ex. 'E3E4').

Les données expérimentales en régime permanent (Araud (2012); Finaud-Guyot et al. ( 2018)) permettent de connaître le débit et les hauteurs d'eau dans chaque rue, ces dernières étant mesurées de chaque côté de la rue. Le réseau est modélisé en utilisant un modèle 1D2D sur un maillage non structuré avec un pas d'espace x = 0.025m (gure 11). Les rues sont modélisées avec des mailles 1D et les carrefours en 2D. Les conditions aux limites consistent en des débits amonts pour les rues E, F, 3 et 4 (correspondant aux valeurs expérimentales) et des courbes de tarage expérimentales pour chaque condition limite aval. Le solveur de Riemann PorAS est utilisé XIV Cong 1 : Surface de réponse à K street et K cross Le modèle est paramétrisé avec deux coecients de Strickler uniformes : K street dans les rues et K cross dans les carrefours. Chaque paramètre est pris parmi les valeurs [25,50,75,100,125,150] et toutes les combinaisons sont testées.

Les débits simulés dans les rues de sorties sont comparés aux valeurs expérimentales sur la gure 12. Pour chaque rue, la première barre correspond à la mesure expérimentale. Quelque soit la paramétrisation du modèle, chaque débit simulé pour les rues E et F (respectivement 3 et 4) sous-(respectivement sur-)estime la valeur expérimentale. K street semble imposer un débit moyen dans la rue (noter par exemple le saut de débit pour la rue F quand K street varie de 25 à 50). K cross a un impact plus réduit sur la répartition des débits de sorties avec une inuence toujours similaire autour de la valeur imposée par K street .

Il apparaît donc que Flood1D2D paramétrisé uniquement avec un coecient de Strickler est incapable de reproduire la ligne d'eau mesurée et la répartition expérimentale des débits.

Cong 2 : Surface de réponse à K unif , φ large et φ narrow Le modèle est paramétrisé avec un coecient de Strickler uniforme K = 75m 1/3 .s -1 dans la mesure où la sensibilité à K street et XV Les débits simulés dans les rues de sorties et dans les rues intermédiaires sont comparés aux valeurs expérimentales 13 et 14. Pour chaque rue, la première barre correspond aux valeurs expérimentales. Pour chaque rue, on obtient des simulations sur-et sous-estimant la valeur expérimentale (contrairement à ce qui était obtenu dans le cas de la conguration 1 avec seulement K street et K cross ). Malgré l'échantillonnage grossier de l'espace des paramètres (en raison du coût de chaque simulation), la surface de réponse est pratiquement centrée autour de la valeur expérimentale ce qui laisse supposer qu'il existe au moins un jeu de paramètres permettant de retrouver les valeurs expérimentales.

Conclusions et perspectives

Cette thèse propose une approche eective basée sur les équations Shallow water modiées pour inclure une paramétrisation adaptée à la modélisation des inondations urbaines à l'échelle locale et du quartier. Ce travail de thèse est basé sur un nouveau code C/C++ Flood1D2D et sur les jeux de données expérimentales du pilote inondation Icube à Strasbourg. Une attention particulière est portée sur les zones de recirculations à l'aval des carrefours, leur prise en compte eective sur la modélisation des écoulements à échelle du quartier. Le jeu d'équations est présenté ainsi que leur discrétisation et l'algorithme de résolution. La validation du code sur de nombreux cas tests est réalisée.

XVII

Acknowledgment

This PhD thesis is performed in the hydraulic lab of INSA-Strasbourg under the direction of Abdallah Ghenaim and Abdelali Terfous. In this acknowledgment, I want to express my gratitude to all the people that provide help to nish this thesis. I thank my supervisors, Abdallah Ghenaim and Abdelali Terfous, who provide this opportunity that I can study in French. In the past three and half years, they have given me a lot of help both in study and life, which is really appreciated.

I want to thank Pierre-André Garambois and Pascal Finaud-Guyot, who teach me how to make research and encourage me when I depressed in the study, The support they have given to me is cherished and I have really enjoyed the time when working with you. I think words is not enough to express my thanks to your guys. Without the help, I can't nish the thesis like this. I want to thank two reviewers of the thesis, Hélène Roux and Vincent Guinot, for accepting review my work, and also for the advices provided. I want to thank Mrs Sandra Soares-Frazao for the validation data provided in the thesis.

I also want to thank all the members in the laboratory for their support during all my PhD career, and the help from calculation center of University of Strasbourg. and underground networks resulting in complex ow path depending on the topology of a city [START_REF] Paquier | From hydraulic modelling to urban ood risk[END_REF]. Flood eects can be exacerbated by existing paved streets and roads, which increase the speed of owing water. The submersion levels and ow velocities constitute a hazard to both the population and infrastructure related to ood nature and urban networks conguration. Therefore it is of paramount importance to improve risk assessment based on ow physics understanding and predictive models both at the local and city scales. However real time forecasts of urban ood hydrodynamics, with an accurate and robust estimation of the spatio-temporal evolution of water levels and ow velocities, and their associated uncertainties, remains a real scientic challenge.

The predictive performances of a model generally depend on its complexity and the mist to the real (nonlinear) physical processes complexity but also of the availability of constraining data. Nevertheless, reliable ood measurements, that are required to improve (urban) ood mechanism understanding, elaborate models and calibrate them are dicult to perform. That is why urban ood datasets obtained on densely instrumented laboratory scale experimental devices are precious (e.g. Mignot et al. (2006); Schubert and Sanders (2012); Finaud-Guyot et al.

( 2010a)) and macroscopic (Guinot and Soares-Frazão (2006); Sanders et al. (2008); Guinot 1 (2012a); Guinot et al. (2017a)) methods appear to be promising approaches to reduce ood modeling computational costs but taking advantage of the relative accuracy of 2D on hydraulic intersections such as crossroads and of cheaper 1D models on street channels. Such simplied models, aiming at preserving essential ow features for an eective ood modeling with sucient accuracy on key hydraulic variables, would also be parsimonious in terms of parameterization.

In the context of urban ood eective modeling and in view to propose an eective 1D2D model, the following questions arise and are tackled in this PhD:

1. What are the key local ow zones and features that need to be reproduced by a simplied 1D2D model of a branched network?

2. How to parameterize such a model for example in streets where 3D ow features such as recirculation zones downstream of crossroads can appear?

3. What are the sensitivity of 2D SW or 1D2D models to their parameters for urban ood ows in a branched network?

Outline of the thesis

In this PhD, a 1D2D model, Flood1D2D, is proposed with a new parameterisation adapted to the main ow features of a urban district conguration -ICube urban ood experimental rig.

An isotropic porosity is implemented in Flood1D2D in order to be able to take into account the ow contractions due to the eect of recirculations. Sensitivity analysis of a state of the art 2D SW model and of the proposed 1D2D model to their input parameters are performed using global sensitivity analysis (GSA). Input parameters are sampled in ranges dened from typical uncertainties observed in experiments.

This PhD thesis is organized in 5 chapters plus the present general introduction and a conclusion. Chapter 1 presents a bibliography synthesis on urban oods at local and urban district scales using experimental and numerical approaches. Some shallow water models applied to urban ood are presented including 1D2D couplings and sub-grid methods like porosity function; some sensitivity analysis are also presented. Chapter 2 presents the derivation of the shallow water equations (depth-averaged) with porosity and their discretization with a nite volume method, but also the proposed numerical method and parameterization implemented in Flood1D2D. The Chapter 2

Literature review

This PhD deals with urban ood modeling at the street and city scale. This chapter presents a bibliography synthesis of research works related to urban oods, including numerical and experimental investigations along with sensitivity analysis. It begins with a general description of the continental water cycle and the hydrological contextualization of urban oods. Then, major hydrodynamic features involved in urban ood ows are presented, and especially local scale hydraulic intersections. Next, the numerical methods commonly used for urban ood modeling are introduced including a review of works with dierent paradigms (1D, 2D and 3D), with 1D2D couplings and sub-grid scale methods. Sensitivity analysis (mainly variance based global sensitivity analysis) is introduced and its application in hydrology and hydrodynamic is also briey reviewed. Finally, some problematics are formulated and manuscript objectives are presented.

2.1 Water cycle and ooding

Water cycle

The great water cycle driven by the thermal energy of the sun describes the uxes of water within the so-called hydrosphere on earth including oceans and continents. The quantity of water on earth (around 1, 386, 000, 000 km 3 ) is nearly constant since about 4 billion years (https://water.usgs.gov/edu/earthhowmuch.html, access on April 03, 2018) whereas its partitioning into the oceans, the atmosphere, the cryosphere and the continental surface and underground water is variable in time and space. The continental water cycle, also known as the hydrological cycle, describes the movement of water on, above and below continental surfaces as illustrated in gure 2.1. Hydrology is the scientic eld, at the conuence of meteorology and uid mechanics, focused on the continental water cycle. The main physical processes involved in water repartition are evaporation, condensation, precipitation, inltration, surface runo, subsurface ow, groundwater ow. These vertical and lateral hydrometeorological processes are non linear and coupled, characterized by various relaxation times.

Soil atmosphere exchanges, including evaporation, evapotranspiration, precipitations, snow and ice melt represent the main drivers, also called forcing of surface runo and groundwater ows. In hydrology, rainfall-runo transformation is generally studied at the topographic basin scale whose properties (topography, soil occupation and nature, bedrock, etc) inuence the spatiotemporal variability of ow processes along with the variability of the forcings. It is of particular interest to improve the understanding and predictive performances of models of those hydrological processes at various spatial and temporal scales for: Improving our understanding of the water cycle on earth and reach its closure in terms of quantitative water balance between ocean, continents, atmosphere and cryosphere.

Building sustainable water management and engineering techniques in a context of climate change [START_REF] Pachauri | Climate change 2014: synthesis report. Contribution of Working Groups I, II and III to the fth assessment report of the Intergovernmental Panel on Climate Change[END_REF]).

Water management includes many problematics involving dierent spatio-temporal scales. Hydrological extremes such as oods may occur at short time scales and their predictivity represent an important challenge. Indeed, weather related disasters seem to become increasingly frequent and especially storm-oods which represent nowadays 47% of weather-related disasters worldwide over the period 1995-2015 [START_REF] Pachauri | Climate change 2014: synthesis report. Contribution of Working Groups I, II and III to the fth assessment report of the Intergovernmental Panel on Climate Change[END_REF]; [START_REF] Wahlstrom | The human cost of weather-related disasters 19952015[END_REF]).

Depending on storm and catchment properties various ood types can occur as detailed in the following section.

Flood types

General classication

A ood is a natural hazard that can occur in many places worldwide due to the combination of several factors that can include: precipitation, soil (and/or underground networks) saturation, ocean storm surge, dam/breach breaking, debris chocked rivers. It is commonly dened as the overow of water over land and built-up areas that are usually dry. Flooding can be classied as follows:

Coastal oods generally occurring along the coastline where land can be submerged by seawater because of a storm surge due to a hurricane or tsunami and/or overwhelming rainfall.

River (or uvial) ooding occurring after long (oceanic) rainfall events of generally moderate intensity on large catchments (area greater than 500km 2 for instance). The river or stream is unable to convey the excedent of runo drained by its upstream catchment and slowly oods its major bed (oodplains). The submersion duration can reach several days or even such as for the ood in Paris in 1910 which lasted 10 days, with a measured water level of 8.62m at Paris-Austerlitz gauging station where the water level is usually around 2.5m.

Floods due the groundwater level rise due to a long rainfall period over the relling area(s) of an aquifer. This kind of events were badly known until the events in the Somme catchment (France) in 2000 and 2001. The plains in Picardie (Northern France) have know submersion levels of about 15m at some places.

Flash oods that are extreme and quick catchment responses typically smaller than 24h with high peak discharge often produced by severe localized thunderstorms on catchment headwaters (area smaller than 500km 2 for instance) [START_REF] Garambois | Characterization of process-oriented hydrologic model behavior with temporal sensitivity analysis for ash oods in mediterranean catchments[END_REF][START_REF] Garambois | Parameter regionalization for a process-oriented distributed model dedicated to ash oods[END_REF]; [START_REF] Douinot | Modelling errors calculation adapted to rainfallruno model user expectations and discharge data uncertainties[END_REF]). Various denitions of ash oods are given in the literature but ve criteria can be used [START_REF] Garambois | Elaboration de méthodologies de transfert à des bassins versants non jaugés[END_REF]): ashiness of the hydrological response, river slope, Froude number, debris transport, ood eects/damages. Due to high ow power those oods can signicantly modify a river geometry and are consequently highly morphogenic. Let us cite some catastrophic oods in the south of France: the Ouvèze River in 1992, the Aude River in 1999, the 2002 oods in the Gard region and the 2010 and 2016 oods in the Var department.

Urban ooding consisting in signicant submersion of densely populated and often highly impermeable areas. It can be caused by the combination of various factors including upstream runo and river ooding, storm surge or tsunami, rainfall localized on the urban area and the saturation of the capacity of drainage systems such as storm sewers (Mignot et al. (2006); [START_REF] Sattar | Case study: 17th street canal breach closure procedures[END_REF]).

Urban oods

Urban ooding is dened as the inundation of land or property in a more or less densely built and populated environment. Such hydrological events can be due to various combinations of hydrometeorological factors as presented above and also site specic features such as topography, soil occupation and properties and antecedent wetness. Urban oods are generally caused by rainfalls overwhelming the inltration capacity of a (urban) catchment and the storage and conveyance capacities of drainage systems, such as storm sewers as in gure 2.2.

Physical phenomena involved in urban hydrology

Urban hydrology is a science investigating the hydrological cycle (see gure 2.3) and its changes, water regime and quality within the urbanized landscape and its impacted zones . The hydrological process in urbanized catchments consist in [START_REF] Fletcher | Understanding, management and modelling of urban hydrology and its consequences for receiving waters: A state of the art[END_REF]; Salvadore et al.

( Surface runo process: the urbanization is accompanied with the replacement from natural/cultivated soil to impervious surface, which has profound impacts on urban hydrology.

)): 2015
The volume of surface runo can increase dramatically due to impervious surfaces where inltration hence water storage is impossible.

Subsurface ow processes: subsurface ows in urban sewers and water supply networks, underground networks [START_REF] Ishigaki | Hydraulic model tests of inundation in urban area with underground space[END_REF]; [START_REF] Noh | Ensemble urban ood simulation in comparison with laboratory-scale experiments: Impact of interaction models for manhole, sewer pipe, and surface ow[END_REF]).

The accumulation of overland ow in urban areas caused by excedent rainfall, due to imperious surface and inappropriately designed drainage systems can be the main reasons for urban ooding.

Urban ood ows physics and risk to people

The increase of ood damages is particularly marked in urbanized areas representing an increasing percentage of the world population. Both an increasing urbanization and the potential rise of high to extreme rainfall events should lead to more frequent urban ooding. In urban areas, ood eects can be exacerbated by existing paved streets and roads, which increase the speed of owing water. The ood ow in urbanized areas constitutes a hazard to both the population and infrastructure. Important features of this hazard are the variability of water depth and velocities (Mignot et al. (2006); Yu and Lane (2006a)).

In urban area ood ow paths are quite complex and related to the topology of the city.

From a hydraulic point of view, the structure of the city formed of building blocks is organized on the basis of networks [START_REF] Paquier | From hydraulic modelling to urban ood risk[END_REF]): the hydrographic network including the rivers, underground sewer network and street networks (Finaud-Guyot et al. (2018)). During urban ooding, the streets can become the main ow paths while remaining the only means for emergency services and evacuation of the population.

The hydrodynamics of urban ood ows can be viewed at two dierent scales of interest: Therefore it is of paramount importance to improve risk assessment based on ow physics and prediction tools ecient both at the local and the city scale. A bibliography synthesis of local phenomena at hydraulic intersections is presented in the next section followed by a synthesis on ow studies at the district and city scales.

Hydraulic intersections at the local scale

Open channel intersections are commonly encountered in natural rivers (deltas, oodplains) and anthropized channels (irrigation networks and sewerage, urban street networks). In the literature mainly three types of hydraulic intersections are documented:

Three branch conuence ow: two upstream reaches feeding one downstream reach;

Three branch dividing ow: one upstream reach feeding two downstream reaches;

Four branch intersection ow: two upstream reaches feeding two downstream reaches.

The basic intersections are three branch intersections including conuence ow or bifurcation ow. Both types of hydraulic intersections involve modications in the mass and momentum repartition:

Dividing ow The continuity equation imposes that upstream discharge is splitted into two downstream discharges. The momentum is also splitted into the downstream branches but energy losses can appear due to basal friction, turbulence dissipation (e.g. ow in zones with complex hydrodynamics such as recirculation or vortex).

Junction The mass continuity imposes that the downstream discharge is the sum of the two upstream ones. Similarly the downstream momentum is the sum of the two upstream minus the energy losses mentioned above.

Due to ow physics and the variability of ow regimes in each branch, there is not the same number of hydraulic unknowns (depth, discharges) for a conuence and a dividing ow. In the case of the four branches ow, the hydraulic problem can even be more complex. [START_REF] Ming | Experimental and numerical study of open channel intersections[END_REF] among others, considers that a four branch intersections (with two inlet and outlets) can be seen as the combination of a conuence ow and a bifurcation ow. Street intersections when city is ooded are the typical examples of free surface ows at hydraulic intersections. The local and global ow features of those intersections have been thoroughly reviewed by [START_REF] Ming | Experimental and numerical study of open channel intersections[END_REF]. In this thesis, we make a more detailed review on local ow feature and especially the separation zone.

Three branch conuences

General introduction

The general ow pattern is summarized in gure 2.4 based on the experiment of [START_REF] Weber | Experiments on ow at a 90 • open-channel junction[END_REF] and numerical modeling by [START_REF] Huang | Three-dimensional numerical study of ows in openchannel junctions[END_REF]. A three branch conuence ow is generally characterized, depending on ow regimes, by complex ow patterns. The general features are:

A shear plane which delimitates the separation between the two upstream incoming ows (identied as mixing layer in gure 2.4). A mixing layer exists between main channel ow and tributary inow. The velocity dierence between them causes shear forces and coherent structures into the mixing layer [START_REF] Thanh | Numerical simulation of ow at an open-channel conuence using depth-averaged 2d models with eects of secondary currents[END_REF]; [START_REF] Mignot | Mixing layer in open-channel junction ows[END_REF]).

A separation zone that occurs due to the tributary branch momentum causing the ow to detach at the downstream corner of the crossroad (Best and Reid (1984)). The separation zone is characterized by low ow pressure, which causes the separation streamline to deect towards the branch-side bank further downstream, forming a three-dimensional separation zone [START_REF] Hsu | Flow at 90 • equal-width open-channel junction[END_REF]). A recirculation zone may appear into the separation zone with ow circulation.

A ow contraction due to the recirculation. The combined ow accelerates as it passes through this part of the channel, where water height decreases and velocity increases for subcritical ow.

A stagnation zone that is characterized by a low velocity close to the upstream corner between two incoming ows. The dimensions of the stagnation zone are inuenced by the incoming discharge ratios.

Among others, [START_REF] Schindfessel | How dierent cross-sectional shapes inuence the separation zone of an open-channel conuence[END_REF] propose a schematic representation of a recirculation zone as presented in gure 2.5. Two methods are used to delineate the dimensions of a separation zone, which is based on the 2D ow eld. The rst method is named as the isovel method proposed that the isoline of zero longitudinal velocity is used to delineate the separation zone [START_REF] Qing-Yuan | Experimental study on characteristics of separation zone in conuence zones in rivers[END_REF]). The second method is referred as the zero-discharge method that a zero discharge in the direction of the main ow dened the longitudinal extent of the separation ow [START_REF] Schindfessel | How dierent cross-sectional shapes inuence the separation zone of an open-channel conuence[END_REF]) (see gure 2.5): where u (x) is the longitudinal velocity; y s (x) is the distance of the recirculation limit to the wall. L s is dened as the length of separation zone (for which y s (x) > 0) and B s is dened as the maximum value of y s (x) for all x between 0 and L s . The length r 0 corresponds to the distance from the downstream corner to the x-coordinate where B s is found.

Flow features on separation zone

A rst comprehensive study of open channel conuence ows is proposed in [START_REF] Taylor | Flow characteristics at rectangular open-channel junctions[END_REF]. The experiment proposed consists in a rectangular channel of uniform width with junction angles of Several studies have been conducted on three branch conuence ow [START_REF] Webber | An investigation of ow behaviour at the junction of rectangular channels[END_REF]; [START_REF] Modi | Conformal mapping for channel junction ow[END_REF]; Best and Reid (1984); [START_REF] Hager | Discussion of separation zone at open-channel junctions by James L. Best and Ian Reid (november, 1984)[END_REF][START_REF] Hager | Transitional ow in channel junctions[END_REF]; [START_REF] Mamedov | Hydraulic calculation of a conuence[END_REF]; Gurram et al. (1997); [START_REF] Hsu | Flow at 90 • equal-width open-channel junction[END_REF]; [START_REF] Weber | Experiments on ow at a 90 • open-channel junction[END_REF]; [START_REF] Huang | Three-dimensional numerical study of ows in openchannel junctions[END_REF]; [START_REF] Schindfessel | How dierent cross-sectional shapes inuence the separation zone of an open-channel conuence[END_REF]). The research is focused on local ow patterns like ow separation, secondary current and mixing layers, global ow features like upstream and downstream depth ratio and energy loss at a ow crossroad based on momentum and energy balance. [START_REF] Webber | An investigation of ow behaviour at the junction of rectangular channels[END_REF] perform experimental studies on three branch conuence ow with varied angle 30 The recirculation zone grows at the expense of the channel width (ow vein) occupied by the combined streams. For combined ows a relation is presented by Best and Reid (1984) for a 90 • angle to determine B s and L s and writes:

B s = B × (0.506 + 0.205 ln q)

(2.2a)

L s = B × (2.587 + 1.025 ln q) (2.2b)
where B is channel width, q = Q L Q d

, Q L and Q d are lateral and downstream discharges. Best and Reid (1984) also compared his relationship to the dataset of [START_REF] Modi | Conformal mapping for channel junction ow[END_REF].

It shows that predicted width of the separation zone is overestimated and the fraction of the channel occupied by the ow vein is therefore underestimated. [START_REF] Hager | Transitional ow in channel junctions[END_REF] expressed the upstream to downstream water depth ratio and the contraction coecient as functions of the junction angle and the discharge ratio, assuming critical ow conditions at the maximum ow constriction downstream of the junction. To quantify the dimensions of separation zone, an empirical relationship is proposed by [START_REF] Mamedov | Hydraulic calculation of a conuence[END_REF] which relates the length and width of the separation zone to the conuence angle, the momentum ratio, the inow velocity ratio, and a factor depending on the inow conditions. Gurram et al. (1997) provide a detailed study on subcritical three-branch junction ows using an experimental device with angles of 30

B s = B × 1 2 × F d - 2 3 2 + 0.45 × q 1/2 × δ 90 • (2.3a) L s = B × 3.8 × sin 3 δ × 1 - 1 2 × F d × q 1/2 for F d < 1 (2.3b) L s = B × 0.26 × 1 + 3 × δ 90 • × q 1/2 for F d = 1 (2.3c)
where B is channel width, F d is the downstream Froude number, q = Q L Q d

, Q L and Q d are lateral and downstream discharges and δ is junction angle.

Y s = [sin (90 • X s )] 2/3 (2.4)
where Y s = y s (x)/B s and X s = x/(L s /2), leading to a symmetrical prole. y s (x) and x are dened in gure 2.5.

Hsu et al. (1998) conducted T-branch (90

• angle) conuence for subcritical ows in the upstream main and branch channels, with Froude numbers lower than 0.45. The maximum width and length of the separation zone were determined by averaging three plane observations of dye traces that were injected near the water surface at the downstream corner. Shape index and contraction coecient at maximum ow constriction µ are obtained and compared with Best and Reid (1984) and [START_REF] Hager | Discussion of separation zone at open-channel junctions by James L. Best and Ian Reid (november, 1984)[END_REF] (see gure 2.6). Shape index keeps constant as 0.17 with the variation of discharge ratio q compared with 0.19 provided by Best and Reid (1984). µ increases when discharge ratio q changes from 0.1 to 1. [START_REF] Weber | Experiments on ow at a 90 • open-channel junction[END_REF] conducted experiments on a T-branch conuence. Separation zones with 3D ow features are observed. Recirculation inside the separation zone is highlighted as reverse ows are identied. The separation zone is larger near the surface than at the bottom, both in length and width. As presented by Best and Reid (1984); Gurram et al. (1997), the separation zone grows as the discharge ratio q = Q L Q d increases. However, the growth of the separation zone has a limitation observed by [START_REF] Weber | Experiments on ow at a 90 • open-channel junction[END_REF]. Depending on q a reection of the lateral inow on the opposite wall can appear and reduce the length of the recirculation zone. [START_REF] Weber | Experiments on ow at a 90 • open-channel junction[END_REF] also show that a smaller inow angle at the bottom resulted in a lower lateral momentum, which caused a narrower separation zone near the bottom than at the surface.

Due to a negative v-component of the ow (crossow horizontal velocity component), the ow near the bottom is the rst portion of the junction ow that enters the separation zone along the junction-adjacent wall. The ow is reected o by the wall opposite to the junction and comes to the bottom from the surface. It is shown that the upper portion of the separation zone can et al. (1999) be lled by the vertical ow, but not lateral ow. When ow near the bottom approaches the junction adjacent wall, it is redirected like the surface ow at the junction-opposite wall. Some of this is in the downstream direction but the bed ow is also deected upward into the zone of separation [START_REF] Weber | Experiments on ow at a 90 • open-channel junction[END_REF]).

Except experiments conducted on three-branch crossroad ows, 3D numerical approaches are also adopted to delineate the separation zone.

Three branch bifurcations

General introduction

Three branch bifurcations can be seen in irrigation networks or in the separation of a river around an island with one incoming ow and two unknown downstream discharges. The main dierence between conuence ow and bifurcation ow is that discharge distribution at crossroad is unknown. Mass conservation is kept for the whole domain, therefore only an extra variable is introduced to the problem. The general ow patterns of three branches bifurcation in subcritical ow conditions is summarized in gure 2.7 by [START_REF] Neary | Three-dimensional ow structure at open-channel diversions[END_REF]; [START_REF] Neary | Three-dimensional numerical model of lateralintake inows[END_REF]:

A separation zone highlighted as zone A in gure 2.7 occurs in the branch channel just downstream of the crossroad for the transformation of momentum from main channel to the branch. The size of the separation zone is related to the discharge ratio Q L Q total between lateral discharge in the branch and total inow discharge. As previously presented, this separation zone can cause a reduction of the eective ow vein and thus locally increase the ow velocity up to a potential transition from subcritical to supercritical ow.

Flow contraction: as previously presented, due to the existence of a separation zone at branch channel, can cause the reduction of eective ow vein and increase the ow velocity, which can cause the transition from subcritical to supercritical ow.

Separation zone in the main channel (zone B in gure 2.7): several studies have identied this separation zone in the main channel downstream of a crossroad [START_REF] Neary | Three-dimensional ow structure at open-channel diversions[END_REF]; [START_REF] Neary | Three-dimensional numerical model of lateralintake inows[END_REF]). Its dimensions mainly depended on the discharge ratio

Q L Q total .
Dividing stream surface: the dividing stream surface is composed of stream lines connected to the downstream corner representing the splitting of the inlet discharge [START_REF] Neary | Three-dimensional ow structure at open-channel diversions[END_REF]; [START_REF] Momplot | Typology of the ow structures in dividing open channel ows[END_REF]).

For three branch bifurcation ows, the research is mainly focused on the prediction of discharge distribution at a crossroad. As presented by [START_REF] Taylor | Flow characteristics at rectangular open-channel junctions[END_REF], this problem can't be solved only using momentum balance to provide the solution. Therefore, an extra equation is needed to close the system and provide the full solution like energy equation, analogously to other ow phenomena and dimensional analysis.

Flow features on separation zone in the branch channel [START_REF] Kasthuri | Discussion of separation zone at open-channel junctions by James L. Best and Ian Reid (november, 1984)[END_REF], who discuss Best and Reid (1984), provide an empirical relationship for the maximum separation width B s and length L s based on the ratio between the lateral and total inlet discharge for three branch dividing ow with right angle:

B s = B × 0.504 × Q L Q 2 -0.893 × Q L Q + 0.861 (2.5a) L s = B × 6.49 × Q L Q 2 -8.44 × Q L Q + 4.45 (2.5b)
where Q L and Q are lateral and total inlet discharges. This formula is more deeply studied in section 6.2.

The ow structure at a 90 • open channel bifurcation ow is investigated experimentally by [START_REF] Neary | Three-dimensional ow structure at open-channel diversions[END_REF]. Experimental data on velocity shows that the ow at an open channel bifurcation is three-dimensional, which is similar to river bend ows. Results show that roughness of the main channel bed and the ratio of diversion ow velocity to main ow velocity key parameters to determine the three-dimensional ow features, like ow dividing and secondary currents. The ow separation is characterized by low velocities, eddies, and recirculation. The separation zone is larger at the surface than at the bottom. [START_REF] Neary | Three-dimensional numerical model of lateralintake inows[END_REF] B at maximum width-contracted section in the separation region is almost linearly related to q, the branch to main channel upstream discharge ratio (see gure 2.8), which resulted in a small eective width in the separation zone at branch channel when q is small. (e.g., kε or kω models). In complex hydraulic zones, this assumption may be invalid.

Four branch intersections

Among the various congurations of inlet and outlet street in four branch intersections (Araud (2012)), the more deeply studied is the conguration with two inlets and two outlets shown as in gure 2.9. As suggested by [START_REF] Ming | Experimental and numerical study of open channel intersections[END_REF], such conguration can be seen as a combination of conuence ow and bifurcation ow, leading to a more complex problem. The general ow patterns are similar to the three branch intersections with shear plane and separation zone in two outlet channels. Nanía et al. (2004) studied experimentally supercritical ows in a four branch intersection with slope. It has been found that the ow distribution as well as the ow pattern can be predicted by a non-dimensional parameter, the inow power ratio at the entrance of the crossroad. The inow power is dened as γ

× Q × z + U 2 2g
, where γ the specic weight of ow, Q the discharge, z the water surface elevation, U the velocity and g the acceleration of gravity. Two types of ow patterns had been identied according to the presence and location of hydraulic jumps.

The third type of ow pattern with two oblique jumps is observed and dened by Mignot et al. (2008b) in the intersection as illustrated in gure 2.10. Theoretical explanations to three ow types are also provided.

Except supercritical ow mentioned above, subcritical four-branch junction ow is studied experimentally by [START_REF] Rivière | Flooding ows in city crossroads: experiments and 1-d modelling[END_REF][START_REF] Rivière | Subcritical open channel ows in four branch intersections[END_REF]; [START_REF] Nanía | Experimental study of subcritical dividing ow in an equal-width, four-branch junction[END_REF]. The intersection and channels are all horizontal and ow depths are controlled by vertical weirs at the downstream end of outlet et al. (2013). They point out that the existence of a critical section in the lateral outow channel is not sucient to isolate ow in the intersection from the downstream control in the lateral channel. [START_REF] Li | Flow division at a channel crossing with subcritical or supercritical ow[END_REF] conducted 3D simulations at a channel crossing in subcritical or supercritical ow conditions. Using numerical models validated on T-branch junction ow, the dimensions of recirculation zone are quantied. Results show that under the identical outlet conditions, the contraction coecient µ = B-Bs B is only dependent on the input ow ratio Qyi Qxi (Q xi and Q yi are the inlet discharges in x and y channels). For asymmetrical outlet conditions, a higher value of the ratio of the inlet ow rates Qyi Qxi leads to a larger recirculation zone in the x-channel and smaller recirculation zone in the y-channel. A larger exit weir height always leads to a smaller recirculation in both the xand y-channels.

Conclusion on hydraulic intersections

In this section, scientic research concerning hydraulic intersections at local scale has been reviewed. Several points can be raised:

For three branch conuence ows: the separation zone downstream of a crossroad has been extensively studied both experimentally and numerically. 2D ow velocity elds could be used to quantify the dimensions of separation zone for dierent ow conditions. 3D experimental data provided by [START_REF] Weber | Experiments on ow at a 90 • open-channel junction[END_REF] gives insight into the mechanism of the separation zone interacting with the main ow. 3D simulation appears to be a appropriate tool to provide 3D ow features of separation zone. The studies are mainly focused on ows congured with equal channel width without slope under steady state.

For three branch bifurcation ow: the separation zone at bifurcation ow has been studied experimentally and numerically. The researches are mainly focused on subcritical ows.

The studies are mainly focused on ows congured with equal channel width in subcritical ow regime without slope under steady state.

For four branch intersections: few studies have been conducted on four branch intersections mainly on supercritical ows. The function of recirculation zone on ow patterns (water height and discharge distribution) still needs to be explored. The studies are mainly focused on ows with two inlets and two outlets in equal channel width under steady state, whereas no studies have been performed with one inlet and three outlets or one outlet and three inlets.

Hydrodynamic investigations at district/city scale

This section presents research works related to the study of hydrodynamic at the district/city scale explored using experimental or numerical approaches.

An experimental model of Kyoto city is constructed by [START_REF] Ishigaki | Hydraulic model tests of inundation in urban area with underground space[END_REF] to study the interaction between surface ows and underground spaces like the subway network. The water height and velocity at dierent locations are measured. The measurements consist in water depth distribution over the whole domain and velocity around the breach. Results show that in the tested congurations, about 50% of total inow discharge is owing into the underground space. Assessment of the 2D models for ash ood simulation from local scale to eld scale is conducted based on two experimental and two eld datasets [START_REF] Abderrezzak | Modelling ash ood propagation in urban areas using a two-dimensional numerical model[END_REF]). Sensitivity analysis of uncertainty sources on crossroad obstacles and sewer networks is studied by [START_REF] Paquier | From hydraulic modelling to urban ood risk[END_REF].

A model of Toce valley with idealized urban area has been constructed during the project IMPACT [START_REF] Zech | Dam-break ow experiments and real-case data. A database from the European IMPACT research[END_REF]; [START_REF] Testa | Flash ood ow experiment in a simplied urban district[END_REF]). The urban area is made of cubic concrete blocks (aligned or staggered) to represent the buildings (see gure 2.12). The timevaried water depth forced by an inow hydrograph is recorded at ten locations using probes into At the downstream ends of the streets, the Froude numbers correspond to supercritical ows.

An analysis of the discharge distribution at the city scale is proposed. At the scale of the dis- A local experimental study consisting in a rainfall simulator, ground surface, and sewer pipe system is conducted by [START_REF] Noh | Ensemble urban ood simulation in comparison with laboratory-scale experiments: Impact of interaction models for manhole, sewer pipe, and surface ow[END_REF]. Based on the Froude similarity law, the scale is assumed to be 1/20 between the experimental rig and the real-world. An ensemble method is proposed to consider the interaction eect among a manhole, a sewer pipe and surface ow for urban ood modeling. Numerical results are compared with experimental data and parameters uncertainty to model output is analyzed. Results show that ensemble simulation using interaction models based on the weir and orice formulas can reproduce experimental data with high accuracy both for steady and unsteady conditions. At the same time, the identiability of model parameters is also detected.

As a conclusion, several experimental and numerical models have been proposed to simulate urban oods from transient dambreak due to a ash ood to quasi steady state for long-duration ood events. The models sizes range from several building to small European city with variable street widths and angles. However, detailed data of sucient quality about real-world city ooding still lacks for numerical model improvement and validation.

Numerical models applied to urban oods

Numerical modeling is nowadays widely used in uid mechanics and especially hydrodynamic modeling in the eld of water management. Hydrodynamic modeling can be used in view to better understand and forecast urban ood ows to protect property and people's life. Flood inundation models have been reviewed by [START_REF] Teng | Flood inundation modelling: A review of methods, recent advances and uncertainty analysis[END_REF] who list their advantages and limitations (see table 2.1).

Hydrodynamic models are mathematical models that attempt to replicate uid motion by solving the continuity and momentum equations. Depending on their modeling paradigm and especially their spatial representation of a ooded domain for instance, models can be dimensionally grouped into 1D, 2D and 3D models:

1D shallow water models (SW) are very ecient in terms of computation, but their hypothesis do not allow to simulate complex ow features (2D, or even 3D) as discussed in the present work.

2D shallow water models (SW), widely used in urban ood modeling, are a good compromise between 1D and 3D (relatively limited computational resources but correct ow simulation accuracy).

3D simulation generally based on Reynolds averaged Navier-Stokes equations with a turbulent closure, are generally applied at the local scale like the crossroad scale [START_REF] Bradbrook | Investigation of controls on secondary circulation in a simple conuence geometry using a three-dimensional numerical model[END_REF]; [START_REF] Neary | Three-dimensional numerical model of lateralintake inows[END_REF]; [START_REF] Huang | Three-dimensional numerical study of ows in openchannel junctions[END_REF]; [START_REF] Ramamurthy | Numerical and experimental study of dividing openchannel ows[END_REF]; Li andZeng (2009, 2010)). 3D models are too expensive for systematic simulations at the district/city scale and hard to calibrate -wall function and turbulence model parameters. 3D models are not used in this thesis.

Review on 1D2D coupling

Integrated 1D2D models consisting in mixing 1D and 2D SW models when needed have been presented in the literature [START_REF] Ghostine | Comparison between a coupled 1d-2d model and a fully 2d model for supercritical ow simulation in crossroads[END_REF]). Using such a method, less computation points are required in simple ow zones leading to a reduced computation time and memory.

The 1D quasi 2D model as extensions of existing 1D models was rst proposed by [START_REF] Cunge | Two dimensional modeling of ood plains[END_REF]. 1D shallow water equations are solved on the looped channel ow (using Preissmann scheme) and the domains are linked using a storage cell algorithm based on mass conservation.

However, the ability of 1D-quasi 2D schemes in determining the front wave advance and recession process over a oodplain is always limited [START_REF] Ghostine | Comparison between a coupled 1d-2d model and a fully 2d model for supercritical ow simulation in crossroads[END_REF]). The 1D2D coupling between sewer and street networks is also conducted by dierent authors [START_REF] Seyoum | Coupled 1d and noninertia 2d ood inundation model for simulation of urban ooding[END_REF][START_REF] Fan | A coupled 1d-2d hydrodynamic model for urban ood inundation[END_REF]).

For river ow modeling with lateral ood plains, Finaud-Guyot et al. (2010a, 2011) developed a new 1D2D coupling methodology. The mass and momentum exchanges between the 1D and 2D models are taken into account for momentum transfer whereas generally neglected.

Based on nite volume method, a new 1D2D coupling method is proposed by [START_REF] Bladé | Integration of 1d and 2d nite volume schemes for computations of water ow in natural channels[END_REF] between river and oodplain. The connection between 1D and 2D mesh is divided into: Table 2.1: Some well known software/models that are capable of modeling ood inundation taken from [START_REF] Teng | Flood inundation modelling: A review of methods, recent advances and uncertainty analysis[END_REF] connection in ow direction and lateral connections. For 1D2D integration, a ux-based method (FB) is proposed using numerical uxes for connection between 1D and 2D. The results are compared with Source Term-based (STB) connection which no momentum transfer is considered.

The ability of new ux-based method to predict velocity eld is proved, compared to ne 2D results.

For supercritical ow simulation at four-branch crossroads, a comparison between a coupled 1D2D model and a fully 2D model is conducted by [START_REF] Ghostine | Comparison between a coupled 1d-2d model and a fully 2d model for supercritical ow simulation in crossroads[END_REF]. The 1D Saint-Venant equations are applied in the two upstream branches up to a distance of 2B (channel width) from the junction to ensure that the ow is one-dimensional in this area. In the other areas, the 2D Saint-Venant equations are applied for the ow is two-dimensional. The coupling between 1D and 2D model is conducted using ux computation between the interface using Roe's solver. The averaged variables are used on 2D side for ux computation.

Except classical domain decomposition approach, a superposition approach for the coupling between 1D and 2D models is proposed by [START_REF] Gejadze | On a 2d zoom for the 1d shallow water model: Coupling and data assimilation[END_REF]; [START_REF] Marin | Superposition of local zoom models and simultaneous calibration for 1d2d shallow water ows[END_REF]. The coupling term in 1D equations are source terms derived from the 3D Navier-Stokes

Review on shallow water (SW) models with macroscopic porosity approaches

Among the possibilities explored in the literature to tailor shallow water models for complex geometries such as those encountered in urban areas, macroscopic approaches involving a parametrisation of sub-grid features have been proposed. Such approaches adapted to an operational context. The idea is to solve SW equations on a grid adapted to data availability, while preserving the predictive capacity of such model. This can lead to a reduced computational cost. However an additional parameterisation is required at the sub-grid scale with the so-called porosity function.

Depending on the complexity considered, the porosity function can be either isotropic [START_REF] Dena | Two-dimensional shallow ow equations for partially dry areas[END_REF]; [START_REF] Hervouet | Modelling urban areas in dam-break ood-wave numerical simulations[END_REF]; Guinot and Soares-Frazão (2006); Lhomme (2006)

Origin of the porosity-based approach

The concept of the porosity is a measure of the void (i.e. "empty") spaces in a material as a fraction of the volume of voids over the total volume thus comprised between 0 and 1 originally.

The porosity is initially introduced in the porous media eld (see for instance [START_REF] Bachmat | Macroscopic modelling of transport phenomena in porous media. 1: The continuum approach[END_REF]). The approach substitutes, on a representative elementary volume (REV), the heterogeneous medium constituted by the uid and the solid matrix by a ctitious homogeneous medium with parameters that have to be globally representative of the pore scale variables (see Figure 2.14).

Isotropic porosity method

To our best knowledge, [START_REF] Dena | Two-dimensional shallow ow equations for partially dry areas[END_REF] is the rst to propose to modify the shallow water equations with porosity (introduced as a h-dependent storativity coecient ) to statistically account for partially wet and very irregular domain (such as bottom irregularity). Assuming a hydrostatic approximation, the three-dimensional Reynolds-averaged equations are rst averaged over a representative elementary domain including porosity and then integrated over the depth. The proposed model is shown to signicantly improve the hydraulic results in modeling wetting-drying problems.

Following that, [START_REF] Hervouet | Modelling urban areas in dam-break ood-wave numerical simulations[END_REF] modify the shallow water equations with time-varied porosity to represent a local density of buildings for urban ood modeling. In these modied equations the porosity accounts for the presence of buildings, structures, etc. that restrict the area available to water ow. The new set of equations are used to model various test cases and the gap between reference and numerical results is claimed to be due to the miss-t of the porosity estimation. Guinot and Soares-Frazão (2006); Lhomme (2006) present a detailed derivation of the shallow water equations with porosity for urban ood modeling using an isotropic porosity function and complementary head loss parametrisation to take into account the urban singularities eect. Guinot (2012a) also proposed a multiple porosity shallow water model for preferential ow. As highlighted on gure 2.17, such model involves several sub-domains with various characteristics:

Ω b , Ω s and Ω m dened as for the dual porosity model for isotropic ow , Ω {1,...,M } corresponding to M domains with mobile water for which the ow is oriented in a given predened direction and Ω i corresponding to mobile water with no preferential direction connecting two anisotropic mobile water zones. Several test cases highlight the ability of this proposed model to accurately reproduce the ow characteristics without detailed accounting of the urbanized area in the mesh. 2017)). [START_REF] Bruwier | Shallow-water models with anisotropic porosity and merging for ood modelling on cartesian grids[END_REF] thus propose a new method for computing the conveyance porosity that leads to results with a similar accuracy than using the classical conveyance porosity computation method but without requiring to calibrate complementary drag force term. Finally the numerical results using the anisotropic porosity shallow water model are compared using various parameter values.

Comparison of the dierent porosity methods

Schubert and Sanders (2012) present the rst eld-scale application of the anisotropic IP method and oer insight into its practical utility. Four dierent methods to account for the eect of buildings on the ow are proposed: setting the limit of the building as impervious boundary conditions (BH), increasing the topography of the cell including building (BB), increasing the friction coecient without modifying the topography (BR) or using porosity approach (BP). All those approaches are compared to data available (Baldwin Hills, CA urban dam break scenario).

BR and BB methods are shown to be easy to implement. However, the rst method fails to reproduce accurately the ow features close to buildings as they are not represented and the second one can leads to computationally expensive models due to rened mesh close to buildings.

BH method allows for the same level of details than the BB method with coarser mesh but the model is dicult to prepare. Finally BP method is shown to be the best compromise between accuracy and computational cost. The conveyance porosity thus appears to be insucient and should be associated with a ow direction indicator. Finally, Guinot (2017b) details research direction both for improving the DIP model but also to design experiments able to validate the shallow water porosity model. Guinot (2017c) presents a consistency analysis and a determination of the wave propagation properties of both the IP and DIP model. The sensitivity of the porosity models to the mesh is highlighted and criteria are proposed to reduce the mesh dependence.

Conclusion

In this section, numerical models on urban ood modeling have been reviewed, including integration between 1D and 2D models and sub-grid method to account for sub-grid features (buildings and blocks). 1D2D coupling methods reveal to be a good compromise between simple 1D and relatively costly 2D. Porosity models are a promising approach for urban ood modeling at rather large scale -e.g. city scale. Porosity functions are used to account for the inuence of street/buildings on urban ood propagation, whereas its application to account for sub-scale ow features still needs to be studied.

2.5 Numerical model sensitivity analysis 2.5.1 Uncertainty analysis (UA) and sensitivity analysis (SA)

The goal of a numerical model is generally to predict the behavior of a (physical) system. The chosen and aordable complexity of a model depends on the modeling goals and the data availability. For a given model it is of prior interest to understand and quantify the links between its outputs and inputs. The uncertainty in model outputs may be related to dierent sources: data type and observation errors, model structural uncertainties, parametric uncertainties.

Therefore it is important to systematically assess the uncertainty associated with a model and its sensitivity to input parameters. Let us distinguish two denitions:

Uncertainty quantication (UQ) is the forward propagation of uncertainty to predict the overall uncertainty in model outputs.

Sensitivity analysis (SA) is the study of how the uncertainty in model output can be apportioned to dierent sources of uncertainty in model inputs [START_REF] Saltelli | Global sensitivity analysis: the primer[END_REF][START_REF] Pianosi | Sensitivity analysis of environmental models: A systematic review with practical workow[END_REF]). Sensitivity analysis can be based on various techniques including the analysis of local gradients or of a whole region of a parameter space (cf. section 5.1.1). Variance-based methods are a class of probabilistic approaches which quantify the input and output uncertainties as probability distributions, and decompose the output variance into parts attributable to input variables and combinations of variables [START_REF] Saltelli | Global sensitivity analysis: the primer[END_REF]).

A review of application of sensitivity analysis on environmental models has been provided by [START_REF] Pianosi | Sensitivity analysis of environmental models: A systematic review with practical workow[END_REF], which summarizes and classies the concepts and methods in sensitivity analysis (SA) and provides a general workow (cf. gure 2.19). 2016)) where uncertainty sources can be model structure and parameterization (e.g. basal friction), initial and boundary conditions (bathymetry and source terms). A full shallow water model including inertia terms may be required to capture small scale features and rapidly varying ows over rural oodplains [START_REF] Neal | How much physical complexity is needed to model ood inundation?[END_REF]), along with appropriate numerical methods for shock capture and wet dry front treatments [START_REF] Monnier | Inverse algorithms for 2d shallow water equations in presence of wet dry fronts: Application to ood plain dynamics[END_REF]). As a matter of facts, the interaction of high energy ows with obstacles triggers 3D ow processes such as crossroads junctions (Mignot et al. (2006)) -which for a realistic representation generally require to solve the Navier-Stokes equations in 3D. Consequently, ow resistance parameterization for a 2D SW model, which is an eective representation of 3D (turbulent) free surface ow structures, may depend on its scale for localized head losses in complex geometries (Guinot (2012a)) or parietal friction on smooth or macro roughness [START_REF] Cassan | Velocity proles in a real vegetated channel[END_REF][START_REF] Cassan | A semi-analytical model for the hydraulic resistance due to macro-roughnesses of varying shapes and densities[END_REF]). Another boundary condition that is a well known source of uncertainty is inow discharge especially in complex urban ood congurations -which can be provided by in situ measurements (involving rating and hydrological model uncertainties [START_REF] Douinot | Modelling errors calculation adapted to rainfallruno model user expectations and discharge data uncertainties[END_REF]).

Sensitivity analysis applied to shallow water models

Flood simulation accuracy is not necessarily improved by increasing model complexity and resolution [START_REF] Dottori | Detailed data is welcome, but with a pinch of salt: Accuracy, precision, and uncertainty in ood inundation modeling[END_REF]) but uncertainty sources may also vary (even interact) as the amount of data required to constrain the model. Sensitivity analysis, that assesses how the uncertainty in the output of a model can be apportioned to dierent sources of uncertainty in the model input [START_REF] Saltelli | Global sensitivity analysis: the primer[END_REF]), has become a popular tool in environmental modeling [START_REF] Pianosi | Sensitivity analysis of environmental models: A systematic review with practical workow[END_REF]). SA is used in catchment hydrology to explore high dimensional parameter spaces, assess parameter identiability and understand uncertainty sources [START_REF] Hornberger | Approach to the preliminary analysis of environmental systems[END_REF]; [START_REF] Beven | The future of distributed models: model calibration and uncertainty prediction[END_REF]; [START_REF] Pushpalatha | A downward structural sensitivity analysis of hydrological models to improve low-ow simulation[END_REF]; [START_REF] Garambois | Characterization of process-oriented hydrologic model behavior with temporal sensitivity analysis for ash oods in mediterranean catchments[END_REF][START_REF] Garambois | Parameter regionalization for a process-oriented distributed model dedicated to ash oods[END_REF] among others). Temporal variations of distributed rainfall runo and simplied hydraulic model parameters sensitivities are analyzed and ranked in the case of Mediterranean ash oods [START_REF] Garambois | Characterization of process-oriented hydrologic model behavior with temporal sensitivity analysis for ash oods in mediterranean catchments[END_REF] or large scale conceptual model of the Amazon river [START_REF] Emery | Temporal variance-based sensitivity analysis of the river-routing component of the large-scale hydrological model isbatrip: Application on the amazon basin[END_REF]). Temporal sensitivities of simulated ood response highlight phases and locations of higher sensitivities to runo production parameters and/or runo routing depending on rainfall forcing variabilities and drainage network shape among other parameters.

Among the few SA studies in the eld of river hydraulic modeling, [START_REF] Roux | Sensitivity analysis and predictive uncertainty using inundation observations for parameter estimation in open-channel inverse problem[END_REF] propose a generalized sensitivity analysis of a 1D Saint Venant model with simplied geometry adapted to remote sensing and ungauged rivers, in the case of ood scenario on the Lèze River,

France. The authors show the importance of downstream ow depth in controlling ood extent for a 1.5 km reach of a small river (around 100m bankfull width). Guinot and Cappelaere (2009a) extent to topography and model resolution is limited whereas interactions of those parameters with others (hydrographs and roughness) increases at the end of the ood given their inuence on oodplain ow pathways in this case. A methodology for quantifying the parametric uncertainty due to friction in urban ooding simulations is proposed by [START_REF] Bellos | Quantifying roughness coecient uncertainty in urban ooding simulations through a simplied methodology[END_REF]. Due to the fact that quantication of uncertainty in urban ood modeling is often computationally expensive they propose a method based on a model surrogate obtained with polynomial chaos expansion.

They investigate the sensitivity due to roughness parameter.

Variance based methods will be applied to the various SW models used in this PhD.

Bibliography synthesis and scientic questions

From the bibliography synthesis above on urban ood modeling involving either classical or porosity approaches and local features on hydraulic intersections, or the few studies at the city scale, several scientic questions arise:

Is it possible to propose an eective model for urban oods at a reduced complexity (coherent with operational costs) and computational cost?

Which modeling paradigm could be used? Given a modeling paradigm which parameterization is aordable (with respect to calibration problems) and eective for describing main physical processes involved in urban oods?

What would be the predictive capacity of such a simplied model compared to a reference hydrodynamic model and/or experimental datasets?

What would be the sensitivity of such a simplied model output with respect to its inputs?

In the light of these questions, this thesis aims to develop a new eective model based on SW equations adapted to simulate the main operational features of urban ood ows at the local scale and the district scale. This work is based on ICube urban ood experimental rig presented above (see gure 2.13) and aims at studying ood ows in a real-like street network. A particular attention is paid to recirculation areas downstream of crossroads, their local eective modeling and their eect on the ow pattern at the district scale. This PhD work is based on the development of a new C code Flood1D2D. The next chapter presents the ow governing equations and their discretization.

Chapter 3

Material and methods

This PhD aims to propose a coupled and eective 1D2D approach adapted to simulate urban ood ows with a reasonable computational cost. Based on the conclusions of the bibliography study and observations made on ICube urban ood experimental rig, this new model aims at:

taking advantage of the relative accuracy of a 2D shallow water model for describing the main aspects of ood ows patterns;

taking advantage of low computational cost of 1D in streets, assuming that an accurate representation of velocity eld in 2D in streets is already simplied with respect to the real physics and unnecessary for operation forecasts of water levels and discharge distribution in urban area.

This chapter presents the derivation of the 2D shallow water equations from 3D Navier-Stokes that are assumed to be valid in urban ood ows patterns. Then 2D shallow water equations with porosity (isotropic-and constant in time) are derived (cf. Guinot and Soares-Frazão (2006); Lhomme (2006); Finaud-Guyot ( 2009)) as they are implemented in Flood1D2D and validated in this PhD (cf. chapter 4). Finally a new numerical approach is introduced for coupling 1D-like shallow water model for streets and 2D shallow water model for crossroads with the help of a porosity-like parameter (cf. section 3.3). It will be shown in chapter 5 that such a model is able to simulate realistic discharge and water surface patterns, including water surface shape in ow veins contracted by 3D recirculations in streets downstream of some crossroads.

3.1 Derivation of the governing equations

3D Navier-Stokes equations

The 3D Navier-Stokes equations are derived using a mass and momentum balance on a control volume.

Derivation of the continuity equation

First of all, we consider a xed control volume of volume V and surface S as in gure 3.1. A small area dS of the surface has a outward normal unit vector -→ n . Let us dene the uid velocity -→ U , and its density ρ. The innitesimal mass ux q through this small area is:

q = ρ - → U . - → n dS (3.1)
Q = S ρ - → U . - → n dS = V - → ∇ • ρ - → U dV (3.2)
Knowing that the mass ux Q through the surface S is just equal to the mass change in the control volume V ∂ρ ∂t dV , which lead to:

V - → ∇ • ρ - → U dV = - V ∂ρ ∂t dV V - → ∇ • ρ - → U dV + V ∂ρ ∂t dV = 0 V - → ∇ • ρ - → U + ∂ρ ∂t dV = 0 (3.3)
Because the equation must hold for any small arbitrary control volume anywhere in the uid, the previous integrand is also equal to zero. So we get the local form of continuity equation:

- → ∇ • ρ - → U + ∂ρ ∂t = 0 (3.4)
Expansion of the second term yields:

∂ρ ∂t + - → U • - → ∇ (ρ) + ρ - → ∇ • - → U = 0 (3.5)
Introducing the material derivative dened as (3.5) gives:

D Dt ≡ ∂ ∂t + - → U • - → ∇ in equation
Dρ Dt + ρ - → ∇ • - → U = 0 (3.6)
D ρ - → U Dt = ∂ ∂t V ρ - → U dV + S ρ - → U - → U . - → n dS (3.7)
This change in the momentum, according to Newton's second law, is equal to the sum of forces -→ F applied to the uid inside this control volume:

D ρ - → U Dt = - → F (3.8)
The forces acting on the uid in the control volume in the -→ x i direction -→ F i are either the volumic integral of the body forces ρf i per unit volume or the surface integral of the surface forces n j σ ij per unit area giving:

- → F i = V ρf i dV + S n j σ ij dS (3.9)
where n j is the jth component of the unit normal vector -→ n that points outward from the control volume and σ ij is the component of stress tensor σ. By substituting equations (3.7) and (3.9) into equation (3.8), we obtain the integral form of the momentum equation in the i direction,

where

- → U = (u 1 , u 2 , u 3 ): ∂ ∂t V ρu i dV + S ρu i ( - → U . - → n )dS = V ρf i dV + S n j σ ij dS (3.10)
Applying the divergence theorem to the surface integral terms of equation (3.10) yields:

V ∂ ∂t (ρu i ) + - → ∇ • ρu i - → U -ρf i - ∂σ ij ∂x j dV = 0 (3.11)
Because this integral holds for any arbitrary small control volume, the integrand must be zero, and therefore:

∂ ∂t (ρu i ) + - → ∇ • ρu i - → U = ρf i + ∂σ ij ∂x j (3.12)
Expanding the left-hand side of equation (3.12) and using the continuity equation we get:

∂ ∂t (ρu i ) + - → ∇ • ρu i - → U = u i ∂ρ ∂t + - → ∇ • ρ - → U + ρ ∂u i ∂t + - → U • - → ∇ (u i ) = Dρu i Dt (3.13)
that is the uid acceleration given by the material derivative of velocity Dui

Dt = ∂ui ∂t + - → U • - → ∇ (u i ).
So after substituting this expression of the acceleration term into equation (3.12), we nd that the local form of the momentum equation is:

Dρ - → U Dt • - → x i = ρf i + ∂σ ij ∂x j (3.14)
The stress tensor σ is often decomposed into two terms of interest: the volumetric stress tensors which tends to change the volume of the body (contribution of pressure force) and the stress deviator tensor which determines body deformation and movement (composed of shear stresses). The stress tensor components as represented on an innitesimal cuboid uid element (gure 3.2) can be summarized in matrix form as follows:

σ = -   p 0 0 0 p 0 0 0 p   +   σ 11 + p τ 12 τ 13 τ 21 σ 22 + p τ 23 τ 31 τ 32 σ 33 + p   = -pI + τ (3.15)
Remark : the decomposition σ = -pI + τ leads to the general form of the Navier-Stokes equations where uid properties must be specied for a practical use:

Dρ - → U Dt = ρ - → f - - → ∇ (p) + - → ∇.τ (3.16)
The moment equilibrium results in the symmetry of the stress vector components so that σ ij = σ ji and for a Newtonian uid the stress is proportional to the rate of deformation:

τ ij = τ ji = µ ∂u j ∂x i + ∂u i ∂x j (3.17)
where µ is the dynamic viscosity of the uid. As proposed by Stokes in 1845 [START_REF] Stokes | On the theories of internal friction of uids in motion[END_REF]): τ ii = σ ii + p can be combined by:

τ ii = λ - → ∇ • - → U + 2µ ∂u i ∂x i (3.18)
where λ is the second viscosity coecient. Stokes made the commonly used hypothesis that

λ = - 2 3 µ (3.19)
The stress components are related to the velocity eld by

σ ij = -p - 2 3 µ ∂u k ∂x k δ ij + µ ∂u i ∂x j + ∂u j ∂x i (3.20)
the dummy variable k is summed from 1 to 3, and δ ij is the Kronecker delta function dened by:

δ ij ≡ 1 i = j 0 i = j (3.21)
Substituting equation (3.20) into equation (3.14), we obtain the Navier-Stokes equations:

Dρu i Dt = ρf i - ∂ ∂x i p + 2 3 µ - → ∇ • - → U + ∂ ∂x j µ ∂u i ∂x j + ∂u j ∂x i (3.22)
Assuming a constant viscosity coecient µ, equation (3.22) becomes:

Dρu i Dt = ρf i - ∂p ∂x i -µ ∂ ∂x i 2 3 - → ∇ • - → U + µ ∂ 2 u i ∂x 2 j + µ ∂u 2 j ∂x i ∂x j = ρf i - ∂p ∂x i + µ ∂ ∂x i ∂u j ∂x j - 2 3 - → ∇ • - → U + ∂ 2 u i ∂x 2 j (3.23) For - → ∇ • - → U = ∂ui ∂xi and ∂ 2 ui ∂x 2 j = u i , equation (3.23) yields: Dρu i Dt = ρf i - ∂p ∂x i + µ 1 3 - → ∇ - → ∇ • - → U + u i (3.24)
Then, by assuming an incompressible uid (for which continuity equation becomes

- → ∇ • - → U = 0)
, and considering the gravity -→ g as the only body force we nd that equation (3.24) reduces to:

ρ ∂ - → U ∂t + - → U • ∇ - → U = ρ - → g - - → ∇p + µ - → U (3.25)

Reynolds-averaged Navier-Stokes equations and Boussinesq approximation

The Navier-Stokes equations of motion, expressed in tensor notation for an incompressible Newtonian uid write:

- → ∇. - → U = 0 (3.26a) ρ ∂ - → U ∂t + - → U • ∇ - → U = ρ - → g - - → ∇p + µ - → U (3.26b)
Next, assuming each instantaneous quantity can be split into time-averaged and uctuating components, like u i = u i + u i , p = p + p , yields a time-averaged equation:

∂u i ∂x i = 0 (3.27a) ∂u i ∂t + u j ∂u i ∂x j + u j ∂u i ∂x j = f i - 1 ρ ∂p ∂x i + ν ∂ 2 u i ∂x j ∂x j (3.27b)
Let's introduce S ij the mean strain-rate tensor, which is:

S ij = 1 2 ∂u i ∂x j + ∂u j ∂x i (3.28)
The momentum equation can then be written as:

∂u i ∂t + u j ∂u i ∂x j = f i - 1 ρ ∂p ∂x i + ∂ ∂x j 2νS ij -u i u j (3.29)
The quantity u i u j is known as the Reynolds stress tensor which is symmetric (u i u j = u j u i ) and thus has only six components to determine.

The Boussinesq hypothesis is commonly used to relate the Reynolds stresses to the mean velocity gradients:

-

u i u j = 2ν t S ij - 2 3 kδ ij (3.30)
where k = 1 2 u i u i is the turbulent kinetic and ν t is the kinetic eddy viscosity assumed as an isotropic quantity. Equation (3.29) becomes:

∂u i ∂t + u j ∂u i ∂x j = f i - 1 ρ ∂p ∂x i + ∂ ∂x j [2νS ij + 2ν t S ij - 2 3 kδ ij ] = f i - ∂ ∂x i p ρ + 2 3 k + ∂ ∂x j 2 (ν + ν t ) S ij (3.31)
Therefore, Reynolds-averaged Navier-Stokes equations using Boussinesq approximation become:

∂u i ∂x i = 0 (3.32a) ∂u i ∂t + u j ∂u i ∂x j = f i - ∂ ∂x i p ρ + 2 3 k + ∂ ∂x j (ν + ν t ) d (3.32b)
where d = 2S ij .

Three-dimensional Navier-Stokes equations

Let consider a three-dimensional ow on a tilted plane making an angle γ with the horizontal. By choosing the Ox axes to be parallel to the tilted plane, one denotes by ( -→ e x , -→ e y , -→ e z ) the unit vectors of the (x, y, z) plane. The gravity vector then reads -→ g = g sin γ -→ e xg cos γ -→ e z . One chooses the axes origin at the bottom so the z = 0 corresponds to the tilted plane.

A turbulent ow of an incompressible and Newtonian uid can be described by the incompressible Reynolds-averaged Navier-Stokes equations with turbulent viscosity (equation (3.32)).

For the sake of simplicity, the notation u i , p, S ij and d are used for the Reynolds means rather than u i , p, S ij and d. One assumes that the turbulent viscosity ν t is constant in the fully turbulent ow for z ≥ z lim (x, y, t) where z lim is the top of a bottom boundary layer [START_REF] Southard | Introduction to Fluid Motions, Sediment Transport and Current-Generated Sedimentary Structures[END_REF]). One supposes that ν is negligible in front of ν t in the upper layer, which allows to write the Navier-Stokes equations under the form:

- → ∇. - → U = 0 (3.33a) ∂ - → U ∂t + - → U • ∇ - → U = - → g - 1 ρ - → ∇p t + ν t - → U (3.33b)
where p t = p + 2 3 ρk is the turbulent pressure and

u i = ∂d ∂xj = ∂2Sij ∂xj = ∂ ∂xj ∂ui ∂xj + ∂uj ∂xi .

Flow boundary conditions

Assuming the free surface equation can read F (x, y, z, t) = zh(x, y, t) = 0 which excludes deformations of the breaking type [START_REF] Thual | Hydrodynamique de l'environnement[END_REF]). The normal -→ n =

∂h ∂t + u 1 ∂h ∂x + u 2 ∂h ∂y = u 3 for z = h (x, y, t) (3.34)
The dynamic boundary condition on the free surface expresses the continuity of the surface forces. If one assumes that the liquid (water) is in contact with a perfect uid (air) of constant pressure p a , it reads:

σ t • - → n = -p a • - → n for z = h (x, y, t) (3.35)
where σ t (x, y, z, t) is the turbulent stress tensor.

σ t = -p t I + 2ρν t d, with d (x, y, z, t) = 2S ij (3.36)
The settlement of the boundary conditions at z = z lim have to match with the bottom boundary layer. A very simple model consists in considering that z lim is very small and to impose the boundary conditions such as:

u 3 = 0 and - → e x • σ t • - → e z = τ * x for z = 0, (3.37a) - → e y • σ t • - → e z = τ * y for z = 0, (3.37b)
where τ * is the bottom shear stress applied by the uid on the wall. The stress τ * can then be modeled as a function of the velocity, of the height as well and other parameters such as the molecular viscosity ν of the uid or the characteristic height k s of the bottom roughness.

Derivation of the Saint-Venant equations

The Saint-Venant equations are obtained by integrating in the vertical direction the turbulent Navier-Stokes equations, assuming that the depth is small in front of the variation scale of the phenomena in the direction of the ow (Guinot (2012b)). The following sets of hypothesis can be found in Guinot (2012b) and [START_REF] Thual | Hydrodynamique de l'environnement[END_REF]:

The shallow water approximations apply so that vertical acceleration is negligible, resulting in a vertical pressure distribution that is hydrostatic;

The channel bed is stable, so that the bed elevation do not change with time;

The water is assumed to be incompressible in the range of ordinary pressure and water levels. The density is constant;

The ow is fully turbulent. The head loss due to bottom friction is proportional to the square of the velocity;

The diusion of momentum due to turbulence and viscosity, the Coriolis eect and shear stress due to the wind are neglected.

Neglecting the turbulence eect, k = 0 and thus p t = p + 2 3 ρk = p. Since the pressure prole is hydrostatic, p t writes:

p t (x, y, z, t) = p a -ρg cos γ [z -h (x, y, t)] (3.38)
The remaining boundary conditions read:

∂h ∂t + u 1 ∂h ∂x + u 2 ∂h ∂y = u 3 and ∂u 1 ∂z = 0 for z = h (3.39a) w = 0, ρν t ∂u 1 ∂z = τ * x and ρν t ∂u 2 ∂z = τ * y for z = 0 (3.39b)
One denes the uid layer longitudinal velocity u (x, y, t) =

1 h(x,y,t) h(x,y,t) 0 u 1 (x, y, t) dz and v (x, y, t) = 1 h(x,y,t) h(x,y,t) 0 u 2 (x, y, t) dz.
The integration along -→ e y of the continuity equation (3.33a)

∂ ∂x h 0 u 1 dz + ∂ ∂y h 0 u 2 dz -u 1,h ∂h ∂x -u 2,h ∂h ∂y + u 3,h -u 3 (x, y, 0, t) = 0 (3.41)
Using the boundary condition equation (3.39) to eliminate last terms u 3 (x, y, 0, t) = 0 and

∂h ∂t = -u 1,h ∂h ∂x -u 2,h ∂h ∂y + u 3,h , equation (3.41) becomes: ∂ ∂x h 0 u 1 dz + ∂ ∂y h 0 u 2 dz + ∂h ∂t = 0 (3.42)
Using the denition of uid layer longitudinal velocity u (x, y, t) , v (x, y, t), we get:

∂ ∂x (hu) + ∂ ∂y (hv) + ∂h ∂t = 0 (3.43)
Substituting equation (3.38) into the momentum equations (3.33b) in x and y direction, we get:

∂u 1 ∂t + u 1 ∂u 1 ∂x + u 2 ∂u 1 ∂y + u 3 ∂u 1 ∂z = -g cos γ x ∂h ∂x + g sin γ x + ν t ∂ 2 u 1 ∂z 2 (3.44a) ∂u 2 ∂t + u 1 ∂u 2 ∂x + u 2 ∂u 2 ∂y + u 3 ∂u 2 ∂z = -g cos γ y ∂h ∂y + gsinγ y + ν t ∂ 2 u 2 ∂z 2 (3.44b)
By using the continuity equation (3.33a) in the momentum equations (3.44a) and (3.44b), one obtains:

∂u 1 ∂t + ∂u 2 1 ∂x + ∂u 1 u 2 ∂y + ∂u 1 u 3 ∂z = -g cos γ x ∂h ∂x + g sin γ x + ν t ∂ 2 u 1 ∂z 2 (3.45a) ∂u 2 ∂t + ∂u 1 u 2 ∂x + ∂u 2 2 ∂y + ∂u 2 u 3 ∂z = -g cos γ y ∂h ∂y + g sin γ y + ν t ∂ 2 u 2 ∂z 2 (3.45b)
Integrating momentum equations (3.45) along the direction -→ e z , and using the Leibnitz formula, one then obtains:

∂ ∂t h 0 u 1 dz-u 1 ∂h ∂t z=h + ∂ ∂x h 0 u 2 1 dz-u 2 1 ∂h ∂x z=h + ∂ ∂y h 0 u 1 u 2 dz-u 1 u 2 ∂h ∂y z=h +[u 1 u 3 ] z=h z=0 = -gh cos γ x ∂h ∂x + gh sin γ x + ν t ∂u 1 ∂z z=h z=0 (3.46a) ∂ ∂t h 0 u 2 dz-u 2 ∂h ∂t z=h + ∂ ∂x h 0 u 1 u 2 dz-u 1 u 2 ∂h ∂x z=h + ∂ ∂y h 0 u 2 2 dz-u
∂uh ∂t + ∂ ∂x h 0 u 2 1 dz + ∂ ∂y h 0 u 1 u 2 dz + gh cos γ x ∂h ∂x = gh sin γ x - τ * x ρ (3.47a) ∂vh ∂t + ∂ ∂x h 0 u 1 u 2 dz + ∂ ∂y h 0 u 2 2 dz + gh cos γ y ∂h ∂y = gh sin γ y - τ * y ρ (3.47b)
One seeks to obtain a model that only involves the u (x, y, t), v (x, y, t) and h (x, y, t) elds.

Thus it only remains to express terms like ∂ ∂x h 0 u 2 1 dz and τ *

x as functions of these elds. A rst modeling consists in writing, through a dimensional analysis, that

h 0 u 2 1 dz = αu 2 h h 0 u 1 u 2 dz = αuvh h 0 u 2 2 dz = αv 2 h (3.48)
Since the ow is turbulent, u 1 (z) and u 2 (z) are nearly constant along the depth and one can assume α = 1.

For the modeling of the bottom shear τ * , one denes the dimensionless quantity C f (h, u, v)

by the dimensional relation:

τ * x = 1 2 C f (h, u, v) ρu u 2 + v 2 (3.49a) τ * y = 1 2 C f (h, u, v) ρv u 2 + v 2 (3.49b)
The Chezy coecient C f in open channel hydraulics is often expressed with the Manning-Strickler formula:

C f (h) = 2g K 2 h 1/3 (3.50)
where K is the Strickler coecient (m 1/3 .s -1 ).

Eventually, the Saint-Venant equations read:

∂h ∂t + u ∂h ∂x + v ∂h ∂y = -h ∂u ∂x + ∂v ∂y (3.51a) ∂u ∂t + u ∂u ∂x + v ∂u ∂y + g cos γ x ∂h ∂x = g sin γ x - u √ u 2 + v 2 K 2 x h 4/3 (3.51b) ∂v ∂t + u ∂v ∂x + v ∂v ∂y + g cos γ y ∂h ∂y = g sin γ y - v √ u 2 + v 2 K 2 y h 4/3 (3.51c)
where K x = K y = K in all the following congurations. In equation (3.51), g cos γ x ∂h ∂x and g cos γ y ∂h ∂y represent the pressure gradient, g sin γ x and g sin γ y the projection of the gravity in the x and y direction.

For the continuity equation (3.51a), it can be rewritten as:

∂h ∂t + ∂hu ∂x + ∂hv ∂y = 0 (3.52)
For momentum equation (3.51b) in x direction, it can be rewritten as:

∂hu ∂t + ∂hu 2 ∂x + ∂huv ∂y -u ∂h ∂t + ∂hu ∂x + ∂hv ∂y +gh cos γ x ∂h ∂x = gh sin γ x - u √ u 2 + v 2 K 2 x h 4/3
∂hu ∂t + ∂hu 2 ∂x + ∂huv ∂y + ∂ ∂x 1 2 gh 2 = gh - ∂z b ∂x - u √ u 2 + v 2 K 2 x h 4/3 ∂hu ∂t + ∂ ∂x hu 2 + 1 2 gh 2 + ∂huv ∂y = gh (S 0,x -S f,x ) (3.54) where S 0,x = -∂z b ∂x and S f,x = u √ u 2 +v 2 K 2 x h 4/3 .
For momentum equation in y direction a similar form is obtained:

∂hv ∂t + ∂ ∂x (huv) + ∂ ∂y hv 2 + 1 2 gh 2 = gh (S 0,y -S f,y ) (3.55) where S 0,y = -∂z b ∂y and S f,y = v √ u 2 +v 2 K 2 y h 4/3 .
The vector formulation the equations are written in conservative form as:

∂U ∂t + ∂F ∂x + ∂G ∂y = S (3.56) U =   h q r   , F =     q q 2 h + gh 2 /2 qr h     , G =     r qr h r 2 h + gh 2 /2     , S =   0 gh (S 0,x -S f,x ) gh (S 0,y -S f,y )  
where the unit discharge component q = hu and r = hv. The most common denition of porosity is a volumetric porosity φ, that is a fraction of a control volume occupied by voids. This fraction can itself be splitted into two parts: a storage porosity φ s and an anisotropic conveyance porosity formed by φ x and φ y along the two directions of the plane (Lhomme ( 2006)) (see gure 3.4). The storage porosity can be dened as the fraction of the plan view area available to the ow. On the other hand, the conveyance porosity is the fraction of the cross section available to the ow, which is directionally dependent on the plane orientation.

In this section, only dierential formulation using isotropic and constant porosity is derived following Guinot and Soares-Frazão (2006); Lhomme (2006); [START_REF] Lhomme | A general approximate-state Riemann solver for hyperbolic systems of conservation laws with source terms[END_REF] since it is implemented in the C/C++ code: Flood1D2D. Such parameterization is either used to account for the impact of sub-grid scale features: geometrical obstruction or ow features such as recirculations.

Derivation of shallow water equations with isotropic porosity in

dierential formulation (Guinot and Soares-Frazão ( 2006))

The equations are derived by calculating a mass and momentum balance over a rectangular innitesimal volume dΩ of horizontal and vertical dimensions δx and δy as illustrated in gure 3.5. The derivation of SW equations with a constant isotropic porosity φ follows a similar procedure as 3D Navier-Stokes equations proposed in section 3.1.1.

The continuity equation The volume Ω (t) of water contained in the control volume is given by:

Ω (t) = y0+δy y0 x0+δx x0 φ (x, y) h (x, y, t) dxdy (3.57)
where (x 0 , y 0 ) are the coordinates of the lower left corner of the control volume presented in gure 3.5. The volumic uxes F V,{N,W,E,S} (where subscripts N,W,E,S stands respectively for

F V,S = x0+δx x0 (φhv) (x, y 0 ) dx (3.58c) F V,N = x0+δx x0 (φhv) (x, y 0 + δy) dx (3.58d)
Using mass conservation, the continuity equation can be written as:

Ω (t + dt) -Ω (t) = t+dt t (F V,W -F V,E + F V,S -F V,N ) dt t+dt t ∂Ω ∂t dt = t+dt t (F V,W -F V,E + F V,S -F V,N ) dt ∂Ω ∂t = F V,W -F V,E + F V,S -F V,N (3.59)
that can be written as:

y0+δy y0 x0+δx x0 ∂ (φh) ∂t (x, y) dxdy + y0+δy y0 [(φhu) (x 0 + δx, y) -(φhu) (x 0 , y)] dy + x0+δx x0 [(φhv) (x, y 0 + δy) -(φhv) (x, y 0 )] dx = 0 (3.60)
When both δx and δy tend to 0:

(φhu) (x 0 + δx, y) -(φhu) (x 0 , y) ≈ δx→0 δx ∂ ∂x (φhu) (3.61a) (φhv) (x, y 0 + δy) -(φhv) (x, y 0 ) ≈ δy→0 δy ∂ ∂y (φhv) (3.61b)
we can get the nal continuity equation with porosity:

∂ ∂t (φh) + ∂ ∂x (φhu) + ∂ ∂y (φhv) = 0 (3.62)
The momentum equation The total x-momentum in the control volume is given by:

M x = ρ y0+δy y0 x0+δx x0 (φuh) (x, y, t) dxdy (3.63)
The x-momentum uxes across the various edges of the control volume are given by:

F M,W = ρ y0+δy y0 φhu 2 (x 0 , y) dy (3.64a) F M,E = ρ y0+δy y0 φhu 2 (x 0 + δx, y) dy (3.64b) F M,S = ρ x0+δx x0 (φhuv) (x, y 0 ) dx (3.64c) F M,N = ρ x0+δx x0 (φhuv) (x, y 0 + δy) dx (3.64d)
The external forces exerted in the x-direction on the water in the control volume are the following:

The pressure force P W is exerted from left to right on the western side of the control volume such as:

P W = ρg 2 y0+δy y0 φh 2 (x 0 , y) dy (3.65)
The pressure force P E exerted from right to left on the eastern side of the control volume writes:

P E = - ρg 2 y0+δy y0 φh 2 (x 0 + δx, y) dy (3.66)
The reaction W x exerted by the walls on the water body owing to the variation of the porosity in the x-direction is given by:

W x = ρg 2 y0+δy y0 x0+δx x0 ∂φ ∂x h 2 dxdy (3.67)
The x-reaction B x of the bottom to the weight of the water body per unit surface is the product of the local bottom pressure, the bottom slope in the x-direction and the porosity:

B x = -ρg y0+δy y0 x0+δx x0 (φh) (x, y) ∂z b ∂x dxdy (3.68)
The resistance R x due to friction is accounted for by a classical Strickler law. The friction force is exerted only at the points occupied by the water:

R x = -ρgh y0+δy y0 x0+δx x0 u 2 + v 2 1/2 K 2 x h 4/3 φu (x, y) dxdy (3.69)
The momentum balance can be written as:

M x (t + dt) -M x (t) = t+δt t (F M,W -F M,E + F M,S -F M,N + P W -P E + W x + B x + R x ) dt (3.70)
It is rst noted that in the limit of vanishing δx and δy, the dierences between the uxes and the forces at opposite edges of the control volume can be simplied as follows:

F M,W -F M,E ≈ δx→0 -ρ y0+δy y0 x0+δx x0 ∂ ∂x φu 2 h dxdy (3.71a) F M,S -F M,N ≈ δy→0 -ρ x0+δx x0 y0+δy y0 ∂ ∂x (φuvh) dydx (3.71b) P W -P E ≈ δx→0 - ρg 2 y0+δy y0 x0+δx x0 ∂ ∂x φh 2 dxdy (3.71c)
We get the nal equation in x direction:

∂ ∂t (φuh) + ∂ ∂x φu 2 h + g 2 φh 2 + ∂ ∂y (φuvh) = g h 2 2 ∂φ ∂x -gφh ∂z b ∂x -gh u 2 + v 2 1/2 K 2 x h 4/3 φu (3.72)
A similar reasoning leads to the following equation for the y-momentum:

∂ ∂t (φvh) + ∂ ∂x (φuvh) + ∂ ∂y φv 2 h + g 2 φh 2 = g h 2 2 ∂φ ∂y -gφh ∂z b ∂y -gh u 2 + v 2 1/2 K 2 y h 4/3 φv (3.73)
The vector form of equations is

∂ ∂t (U φ ) + ∂ ∂x (F φ ) + ∂ ∂y (G φ ) = S φ (3.74) with U φ =   φh φq φr   , F φ =     φq φ q 2 h + 1 2 gφh 2 φ qr h     , G φ =     φr φ qr h φ r 2 h + 1 2 gφh 2     , S φ =      0 gφh (S 0,x -S f,x ) + gh 2 ∂φ ∂x gφh (S 0,y -S f,y ) + gh 2 ∂φ ∂y     
where q = hu and r = hv are the unit discharge components. S 0,x = -∂z b ∂x (respectively,

S 0,y = -∂z b ∂y ) and S f,x = u √ u 2 +v 2 K 2 x h 4/3 (respectively, S f,y = v √ u 2 +v 2 K 2 y h 4/3
) are the bottom slope and head slope at x (respectively, y) direction.

The equation of (3.74) is implemented in Flood1D2D using a nite volume approach (see section 3.2.2) and an explicit time discretization.

1D2D coupling model

The numerical model Flood1D2D is developed to achieve eective and computationally aordable ood simulations in complex branched urban networks. It aims, with a parsimonious parameterization, to predict realistic ood ow patterns at the city scale with adapted complexities at the local scale. It is based on a full 2D shallow water model at crossroads and a much cheaper 1D eective approach in streets remaining physically coherent and suciently informative for operational purpose (i.e. providing the averaged water depth and velocity on the cross-section of the street).

Topology of the coupled 1D2D model

The modeling paradigm, based on the shallow water equations consists in 2D cells in the crossroad and eective 1D cells including the full street width (see gure 3.6). As introduced in Finaud-Guyot ( 2009) and unlike classical 1D2D approaches (see section 2.4.1), the distinction between 1D and 2D cells is not linked to any particular constraint on the set of equations. The equations on both 1D and 2D cells involves the water depth and the two components of the horizontal velocity without any assumption on the velocity orientation.

The only constraint for the junction between 1D and 2D cells is that a 1D edge (understand the edge of a 1D cell) should correspond several 2D edges (understand edges of a 2D cell) (see gure 3.7).

Particular case of the 1D2D width variation

The junction between 1D and 2D cells can correspond to a variation of the channel width (being either an enlargment or a narrowing). As illustrated on gures 3.8a and 3.8b, the connection of a large domain to a narrow one introduces particular boundary on the larger domain (illustrated by Γ 2D,BC-C on gure 3.8a and Γ 1D2D on gure 3.8b).

In the case of a large 2D domain connected to a narrow 1D domain, such boundary can easily be handled as they corresponds to 2D cell edges (see gure 3.8a). In the opposite case of a narrow 2D domain connected to a large 1D domain, the handling of those boundaries would require that the rst 1D cell has 6 dierent edges: one 1D1D: Γ 1D,1D , one 1D2D: Γ 1D,2D , two classical BC: Γ 1D,BC (as the other 1D cells) and two '1D2D' BC: Γ 1D,2D (see gure 3.8b).

Such hexaedral cells cannot be handled by the meshing software (Aquaveo Surface-Water Modeling System 11.2.5) and would require a particular implementation in Flood1D2D to handle cells with six interfaces. Instead, two dierent solutions have been designed for this particular problem: deleting the nodes of the 1D cell corresponding to the channel corner (see gure 3.8c) or moving the junction between the two parts of the mesh of the width variation position (see gure 3.8d).

Finite volume method

The ow is modeled using the shallow water equations with porosity (section 3.1.3) over the whole domain, applied to both 1D eective and 2D cells. The porosity can be imposed to 1 when relevant to come back to the classical shallow water equations without porosity. The numerical tool Flood1D2D has been implemented to solve the governing 2D shallow water equations using a nite volume approach. The set of equations considered can be written in conservative form as:

∂U ∂t + ∂F ∂x + ∂G ∂y = S (3.75)
where U is the vector of the conserved variables, F (respectively G) is the ux in the x (respectively y) direction and S is the source term. For the sake of simplicity U φ , F φ , G φ and S φ (dened in equation (3.74)) are written as U, F, G and S.

The nite volume method consists in solving the integral of equation ( 3.75) on a given domain

Ω ⊂ R 2 : Ω ∂U ∂t + ∂F ∂x + ∂G ∂y dΩ = Ω SdΩ (3.76)
F (respectively G ) being the x (respectively y) component of the ux vector

- → H = -→ H 1 -→ H 2 -→ H 3 T -→ H i = F i - → x + G i - → y ∀i ∈ [1..3], ∂F ∂x + ∂G ∂y thus represents the divergence of - → H: Ω ( ∂U ∂t + div - → H)dΩ = Ω SdΩ (3.77)
By additivity of the integral we get the sum of the integrals on each derivatives:

Ω ∂U ∂t dΩ + Ω div - → HdΩ = Ω SdΩ (3.78)
Therefore, by applying the Green Ostrogradsky's theorem, the integral of the divergence of the ux vector (div -→ H) on the volume Ω can be transformed into the integral of ux vector through Γ the closed border of Ω:

Ω ∂U ∂t dΩ + Γ - → H. - → n dΓ = Ω SdΩ (3.79)
where -→ n is the normal outward unit vector to Γ. The integration volume Ω being assumed to be constant in time, equation (3.79) can be written as:

∂ ∂t Ω UdΩ + Γ - → H. - → n dΓ = Ω SdΩ (3.80)
Introducing the integral denition of the average: f = 1 Ω Ω f dΩ in equation (3.80) leads to:

∂ΩU ∂t + Γ - → H. - → n dΓ = ΩS (3.81)
where S is the average source term S in the cell.

Discretisation

Equation (3.81) is discretized on each computational cell Ω i ⊂ D (see gure 3.9). The sequence

(Ω i ) 1≤i≤N dening a mesh of the computational domain D such as Ω i is a part of D, Ω i ∩ Ω j = ∅, ∀i =j∈[1.
.N], and ∪ N i=1 Ω i = D. Applying this framework allows to write equation (3.81) on the cell Ω i as:

∂Ω i U i ∂t + Γi - → H. - → n dΓ = Ω i S (U i ) (3.82)
where U i the vector of the conserved variables in cell i.

The closed border of cell Ω i is Γ i that corresponds to a set of edges N (i). Using a classical Euler explicit discretisation of equation (3.82) leads to: where n is the timestep, N (i) represents all the interfaces of cell Ω i , A i the area of cell i and w k the length of edge k between Ω i and Ω j . N (i) the numerical ensemble corresponding to Γ i the mathematical ensemble.

U n+1 i = U n i - t A i k∈N (i) P k F n k w k + tS (U n i ) (3.83)
The matrix P k expresses the rotation from the global coordinate system (x, y) to the local coordinate system (ξ, ψ) attached to each interface as illustrated in gure 3.9 for interface k between cells Ω i and Ω j . The rotation leaves the water depth invariant but acts on the unit discharges. The expression of P k is therefore: between the cells i and j (pointing outward from i to j) (see gure 3.9).

P k =    1 0 0 0 n (x) k -n (y) k 0 n (y) k n (x) k    (3.

Time splitting

The source term S (U n i ) of equation (3.83) can be splitted to make appear the eect of the three source terms written in equation (3.74):

U n+1 i = U n i - t A i k∈N (i) P k F n k w k + t [S 0 (U n i ) + S φ (U n i ) + S f (U n i )] (3.85)
where S 0 (U n i ), S φ (U n i ) and S f (U n i ) correspond respectively to the average contribution of bottom source term, the porosity variation source term and the friction eect on cell Ω i .

In order to simplify the computation of the variation of U i on the time step and to allow the

Step 1 -Well balanced computation of the hyperbolic part

The rst computational step does not account the eect of the friction:

U n+h i = U n i - t A i k∈N (i) P k F n k w k + t [S 0 (U n i ) + S φ (U n i )] (3.86)
where U n+h i stands for the estimation of U i including the eect of the hyperbolic uxes and bottom and porosity source terms on one timestep.

Considering the dierent types of interfaces that may exist due to the topology of the 1D2D considered mesh (see gure 3.6),

N (i) = N 1D,W (i) ∪ N 1D,BC (i) ∪ N 2D,BC (i) ∪ N 1D1D (i) ∪ N 2D2D (i) ∪ N 1D2D (i).
Note that depending on the considered cell Ω i some sets of interfaces might be empty: for a 2D cell that does not share any interface with a boundary or with a 1D cell, all the sets are empty excepts N 2D2D (i). In the particular case of a 1D cell coupled to several 2D cells, the junction interface of the 1D cell is accounted as multiple edges corresponding to the edges of each 2D cell.

The uxes F n k between two cells (for the set of edges N 1D1D (i) The uxes F n k between one cell Ω i and a boundary (for the set of edges N 1D,W (i)∪N 1D,BC (i)∪ N 2D,BC (i)) are computed using an estimation of h BC and q BC at the boundary. The user dened boundary condition allows to determine either h BC , q BC or a relationship between them and the Riemann invariants conservation along the characteristic from cell Ω i to the boundary allows to provide the second equation that close the system (Araud (2012)).

∪ N 2D2D (i) ∪ N 1D2D (i))

Step 2 -Friction eect computation

U n+1 i = U n+h i + tS f (U n i ) (3.87)
The friction term is solved using an ordinary dierential equation in the form (Finaud-Guyot ( 2009)):

df dt = -αf (3.88)
where f = q or f = r. This equation is equal to:

df f = -αdt (3.89)
The analytical solution for this equation can be obtained by integrating equation (3.89) over the timestep:

f n+1 f n 1 f df = t n+1 t n -αdt [lnf ] f n+1 f n = -α t lnf n+1 -lnf n = -α t ln f n+1 f n = -α t f n+1 f n = exp (-α t) f n+1 = f n exp (-α t) (3.90)
For the 2D mesh, we can get:

α = g q 2 + r 2 K 2 h 7/3 (3.91)
Equation ( 3.91) can be used into equation (3.90), the variables vector U n,h can be transformed as:

h n+1 = h n,h (3.92a)
q n+1 = q n exp -g (q n ) 2 + (r n ) 2 K 2 x (h n ) 7/3 t (3.92b) r n+1 = r n exp -g (q n ) 2 + (r n ) 2 K 2 y (h n ) 7/3 t (3.92c)

New hydrodynamic reconstruction

The proposed 1D2D shallow water coupling between 1D streets and 2D crossroads achieved with a cut-cell technique allows a suciently informative modeling for operational purpose.

For open channel ows downstream of a crossroad, complex ow features are not nely accounted by such a 1D approach parameterized with roughness only (cf. chapter 6 section 6.5.2). In the streets, a complementary parsimonious parameterization of the shallow water equations is thus introduced.

In what follows, each computational cell representing a part of either a street or a crossroad, the full area of the modeled domain is available for water. The porosity φ in each computational cell, representing the ratio of the cell area available for water to the total area of the cell (see section 3.3.1), is thus equal to one. Ω u : an ineective ow area located in a recirculation zone on which there is on average no ux in the street direction;

Ω v : an eective ow vein.

The cell splitting is based on the function ω used to account for recirculation eects. The variable ω measures the proportion of the cell width that is not occupied by the recirculation:

ω = L v L (3.93)
where L = L u + L v represents the full width of the 1D cell and L u (respectively L v ) is the width of the recirculation (respectively ow vein). If ω = 1 the full street width is available for ow and no recirculation is taken into account. The variable ω is also the area ratio of the ow vein to the full cell: ω = Ω v /Ω 1D . Indeed, the integration of L v on the 1D cell along the street axis s is the area of the ow vein on the cell:

Ω v = s L v (s) ds = s ωL (s) ds = ω s L (s) ds = ωΩ 1D (3.94)
Such a denition of ω, corresponding to the eective ow width with respect to the full cell width, is similar to a porosity as it can be dened in Lhomme (2006). In the following, the notation φ has been chosen instead of ω to simplify the implementation of Flood1D2D. Moreover, the parameter is also dened for the 2D cells but taken equal to 1.

In the present study, the parametrisation φ is constant in time even if all the hydraulic variables can be variable in time. This assumes that the steady-state conguration in a complex

Modied topology of the edge -hyperbolic part

Compared to the classical 1D2D model, the overall computational algorithm (see section 3.2.4) and the computation of the friction eect (see section 3.2.4.2) remains unchanged. In this section, only the computation of the hyperbolic part of the equation is thus presented.

Finally the computational domain still consists in D = Ω 1D ∪ Ω 2D and thus the discretisation equation (3.86) remains unchanged with a computation of the uxes for k in N

(i) = N 1D,W (i) ∪ N 1D,BC (i) ∪ N 2D,BC (i) ∪ N 1D1D (i) ∪ N 2D2D (i) ∪ N 1D2D (i): U n+h i = U n i - t A i { k∈N 1D,BC (i) P k F n k w k + k∈N 2D,BC (i)∪N 2D2D (i) P k F n k w k } + k∈N 1D,W (i)∪N 1D1D (i)∪N 1D2D (i) P k F n k w k + t [S 0 (U n i ) + S φ (U n i )] (3.95)
The ux computation for N 1D,BC (i) ∪ N 2D,BC (i) ∪ N 2D2D (i) remains unchanged. However, the ux computation for the edges in N uv = N 1D,W (i) ∪ N 1D1D (i) ∪ N 1D2D (i) is modied to take into account the eect of the recirculation on the ow:

k∈Nuv P k F n k w k = k∈Nuv P k F n k,u w k,u + F n k,v w k,v (3.96)
where w k,u (respectively w k,v ) is the length of interface k along the recirculation zone (respectively the ow vein) such as

w k = w k,u + w k,v and F n k,u (respectively F n k,v ) is the ux through w k,u (respectively w k,v ).
Introducing the porosity denition leads to:

w k,v = φ k w k w k,u = (1 -φ k ) w k (3.97)
where φ k is the porosity at the interface. Detailing each subset of N uv corresponding to each dierent computation method leads to

N uv = N 1D,W u ∪N 1D,W v ∪N 1D1D,u ∪N 1D1D,v ∪N 1D2D,uu ∪ N 1D2D,u ∪ N 1D2D,v ∪ N 1D2D,vv (see gure 3.11): N 1D,W u
is the edges subset of the wall boundary along the recirculation zone. The ux through this type of interface is computed using the Riemann invariants between the 1D cell and the boundary condition set to q BC = 0. The edge being fully along a recirculation zone, the hydraulic variables in the 1D cell U 1D are take equal to h 0 0

T . N 1D,W v
is the edges subset of the wall boundary along the ow vein. The ux through this type of interface is computed using the Riemann invariants between the 1D cell and the boundary condition set to q BC = 0. The edge being fully along a ow vein, the hydraulic variables in the 1D cell U 1D are take equal to h q r T .

N 1D1D,u is the subset of the edges between two recirculation zones in 1D cells. The ux through this type of interface is computed using either the HLLC or the PorAS Riemann solver parameterized with U 1D = h 0 0 T . Even with only a water surface elevation variation across the interface, such Riemann solver computes a mass ux through the interface that is not coherent with the physical hypothesis that considers a nul discharge across a recirculation. The computed mass ux is thus ignored and taken equal to 0.

N 1D1D,v is the subset of the edges between two ow veins in a 1D cells. The ux through this type of interface is computed using either the HLLC or the PorAS Riemann solver

parameterized with U 1D = h q /φ r /φ T .
N 1D2D,uu is the subset of edges between a 1D and a 2D cell that are fully along the recirculation zone. The ux through this type of interface is computed using either the HLLC or the PorAS Riemann solver parameterized with U 1D = h 0 0 T and U 2D = h 0 0 T . As for N 1D1D,u , the computed mass ux is ignored and taken equal to 0. N 1D2D,u is the subset of the edges between a 1D and a 2D cells that are neither fully along a recirculation zone nor the ow vein. The ux through this type of interface is computed using either the HLLC or the PorAS Riemann solver parameterized with U 1D = h 0 0 T and U 2D = h 0 0 T . As for N 1D1D,u , the computed mass ux is ignored and taken equal to 0.

N U 1D U 2D φ k N 1D,W u h 0 0 T 0 N 1D,W v h q r T 1 N 1D1D,u h 0 0 T φ L +φ R 2 N 1D1D,v h q /φ r /φ T φ L +φ R 2 N 1D2D,uu h 0 0 T h 0 0 T 0 N 1D2D,u h 0 0 T h 0 0 T φ 1D N 1D2D,v h q /φ 1D r /φ 1D T h q /φ k r /φ k T φ 1D N 1D2D,vv h q r T h q r T 1 N 1D2D,v
is the subset of edges between a 1D and a 2D cells that are neither fully along a recirculation zone nor the ow vein. The ux through this type of interface is computed using either the HLLC or the PorAS Riemann solver parameterized with

U 1D = h q /φ 1D r /φ 1D T and U 2D = h q /φ k r /φ k T .
N 1D2D,vv is the subset of the edges between a 1D and a 2D cells that are fully along the ow vein. The ux through this type of interface is computed using either the HLLC or the PorAS Riemann solver parameterized with U 1D = h q r T and U 2D = h q r T .

For every computation using a Riemann solver, the bottom elevation value is classically parameterized i.e. the value for z bL (respectively z bR ) is the bottom elevation in the left (respectively right) cell attached to the considered interface and the porosity is taken equal to 1 in every conguration.

Once the uxes are computed, the quantities exchanged between cells depend on the length of the interfaces (either w k , w k,u or w k,v ) and thus on the porosity φ k at the interface (see table 3.1): Prescribed unit discharge at the upstream boundary (x = 0m) 0m 2 .s -1 h

φ k =          0 for k ∈ N 1D,W u ∪ N 1D2D,uu 1 for k ∈ N 1D,W v ∪ N 1D2D,vv φ L +φ R 2 for k ∈ N 1D1D,u ∪ N 1D1D,v φ 1D for k ∈ N 1D2D,u ∪ N 1D2D,v (3
Prescribed water depth at the downstream boundary (x = 40m) 0m z ini Initial water surface elevation over the domain

0.75m t Simulation time 1000s

The simulation is performed on a structured mesh of rectangular cells (∆x = 0.1m in the longitudinal direction of the channel and ∆y = 1m in the transversal direction). The initial water surface elevation is set to z ini = 0.75m (with no velocity) and the computation is run during 1000s, time at which the results are presented (see gure 4.2). Numerical results show that for both Riemann solver (HLLC and PorAS), the water surface elevation remains constant and equal to the initial water depth with no velocity after 1000s. This is coherent with the expected result and conrms the ability of the code to deal with wetting and drying front. This test case aims at validating the ability of the code to deal with variable porosity in steady state. A permanent ow is thus simulated in a rectangular channel without friction nor slope and with a spatially varying porosity with a parabolic prole:

φ(x) = ax 2 + bx + c for x ∈ [x m -l 0 ; x m + l 0 ] φ 0 elsewhere (4.1a) a = φ 0 -φ m l 2 0 , b = -2ax m , c = φ 0 + a(x 2 m -l 2 0 ) (4.1b)
where the length of the porosity variation is 2l 0 , x m is the position of the middle of the porosity variation, φ 0 (respectively φ m ) is the porosity outside (respectively in the middle) of the variation As highlighted by Finaud-Guyot ( 2009), the reference solution for this test case is obtained by solving the steady-state momentum conservation equation in a non-prismatic rectangular channel derived from the shallow water equations with porosity and no bottom source term:

φ dh dx (1 -Fr 2 ) = hFr 2 dφ dx (4.2)
Equation (4.2) is discretized and solved using an Euler method on a very ne grid (∆x = 0.002m) from downstream to upstream (respectively upstream to downstream) for subcritical (respectively supercritical) ow regime. The reference for test case T02a (subcritical ow) is established using q upstream = 1m 2 .s -1 and h downstream = 1m. For the transcritical case (T02b), the discharge is the same as for T02a and no water depth is needed at the boundary since the ow is controlled by a critical depth (such as Fr = 1) that appears at x m . For both hydraulic regimes, the domain length is l = 100m with a uniform mesh of regular cells of 0.1m×1m. For both hydraulic regimes, the simulations are performed from an initial water surface elevation set to 1m and initial velocity in x direction is set to 0m.s -1 . The computational parameters are given in table 4.2. The simulations are carried out until steady ow conditions are reached over the computational domain (after barely 3000s).

The simulation results for subcritical and transcritical ow regimes are presented in gures 4.4 and 4.5. Numerical results show a good agreement between the computed hydraulic variables (h, Fr, q and Q) using either the HLLC or the PorAS Riemann solvers and the reference whatever the hydraulic ow regime. However, in both congurations, the total discharge Q = φ × ∆y × q computed with the PorAS Riemann solver is constant and variable with the HLLC solver. As the test case corresponds to a steady-state, the total discharge computed by the HLLC Riemann solver is wrong. This is due to the fact that the total discharge is computed by a Riemann solver at the interface between two cells i.e. based on the geometrical features (bottom elevation and porosity) of both cells whereas the computed (and plotted) values are in the middle of each cell.

One-dimensional ow over a bump (T03)

This test case aims at validating the ability of the code to deal with variable bottom elevation in steady state. A permanent ow is thus simulated in a rectangular channel with a spatially The ability of the code to deal with variable bottom is tested for two dierent hydraulic regimes: a subcritical ow (T03a) and a transcritical ow (T03b).

As highlighted by Finaud-Guyot (2009), the reference solution for this test case is obtained by solving the steady-state momentum conservation equation in rectangular channel derived from the shallow water equations with a bottom source term:

dh dx (1 -Fr 2 ) = - dz b dx (4.3) Equation (4.
3) is discretized and solved using an Euler method on a very ne grid (∆x = 0.05m) from downstream to upstream (respectively upstream to downstream) for subcritical (respectively supercritcal) ow regime. The reference for test case T03a (subcritical ow) is established using q upstream = 4.42m 2 .s -1 and h downstream = 2m. For the transcritical case (T03b), the discharge is q upstream = 1.53m 2 .s -1 and no water depth is necessary at the boundary since the ow is controlled by a critical depth (such as Fr = 1) that appears at x m (middle of the bump, see gure 4.6).

For both hydraulic regimes, the domain length is l = 25m with a mesh of uniform cell

0.1m

Prescribed unit discharge at the upstream boundary (x = 0m) q upstream for subcritical ow 4.42m 2 .s -1 for transcritical ow 1.53m 2 .s -1 h downstream Prescribed water depth at the downstream boundary (x = 100m) 2m 0.1m × 1m. The simulation is carried out from an initial velocity in x direction set to 0m.s -1

(for both congurations: T03a and T03b) and an initial constant water surface elevation set to z ini = 2m (respectively z ini = 0.1m) for test case T03a (respectively T03b). The computational parameters are given in table 4.3. The simulation is carried out until steady ow conditions are reached over the computational domain (after barely 1000s).

The simulation results for subcritical and transcritical ow are presented in gures 4.7 and 4.8. Numerical results show a good agreement between the computed hydraulic variables (h, z, Fr and q) using either the HLLC or the PorAS Riemann solver and the reference whatever the hydraulic ow regime. However, in both congurations, the total discharge Q = ∆y × q (Q = 4.42m 3 .s -1 for test case T03a and Q = 1.53m 3 .s -1 for test case T03b) computed with the PorAS Riemann solver is constant over the whole domain but variable with the HLLC solver.

As this test case corresponds to a steady-state ow, the total discharge computed by the HLLC Riemann solver is wrong. This is due to the fact that the total discharge is computed by a Riemann solver at the interface between two cells i.e. based on the geometrical features (bottom elevation and porosity) of both cells whereas the computed (and plotted) values are in the middle of each cell as explained for test case T02. As highlighted by Finaud-Guyot ( 2009), the reference solution of this test case is obtained by solving the steady-state momentum conservation equation in a non-prismatic rectangular channel derived from the shallow water equations with porosity and a bottom source term:

φ(1 -Fr 2 ) dh dx = hFr 2 dφ dx -φ dz b dx (4.4)
Equation (4.4) is discretized and solved using an Euler method on a very ne grid (∆x = 0.002m) from downstream to upstream for subcritical ow regime. The reference for test case T04 is established using q upstream = 1m 2 .s -1 and h downstream = 1m. The domain length is l = 40m with a structured mesh of uniform cell size: 1m × 1m (T04a), 0.1m × 1m (T04b) and 0.01m × 1m (T04c) to test the convergence. The computational timestep is reduced accordingly to the space step ∆x as Flood1D2D uses an adaptive timestep algorithm. For all cases, the simulation is carried out from initial water surface elevation z ini = 1m and an initial velocity in x direction is set to 0m.s -1 . The computational parameters are given in table 4.4. The simulation is carried out until steady ow conditions are reached over the computational domain (after barely 1000s).

The simulation results based on the dierent decreasing cell sizes are presented in gures 4.9, 4.10 and 4.11. For every test case, the computed hydraulic variables (h, q) using the PorAS Riemann solver are closer from the reference than using the HLLC Riemann solver whatever the space discretisation step ∆x. As for the previous test cases (and for similar reasons), the total computed discharge Q = φ × q for the HLLC results are not constant (see gure 4.12). Logically, as the cell size decreases, the hydraulic variables (whatever the Riemann solver) get closer to the reference (see gure 4.12).

A convergence study is conducted based on the results of test case T04a, T04b and T04c.

Let dene the error f on the computed variable f :

f = 1 N N i=1 (f i,sim -f i,ref ) 2 (4.5)
where N is the total number of cell, f i,sim is the computed value of the variable f in cell i Prescribed unit discharge at the upstream boundary (x = 0m) 1m 2 .s -1 h downstream Prescribed water depth at the downstream boundary (x = 40m) 1m presents the evolution of h and q when the spatial discretization step

x is changed for the results computed using the HLLC Riemann solver. Over the range of variation for

x, h and q decrease with x conrming that Flood1D2D converges to the reference solution.

All the previous test cases (T01 to T04) have allowed to validate the correct implementation of both HLLC and PorAS Riemann solvers in steady-state congurations. The latter allowing to avoid any miss-evaluation of the total discharge Q = φ × q in the computational cell (even if both solvers correctly compute the uxes through the mesh interfaces). The following test cases are designed to assess the validity of the Riemann solvers in transient congurations. For both hydraulic regimes, the domain is l = 200m long with a structured mesh of uniform cell size 0.1m × 1m. The dam is located at x 0 = 100m in the middle of the channel and initially separates two zones with no velocity and a constant water level: h 0,L = 10m and h 0,R = 5m (respectively h 0,R = 1m) for test T05a (respectively T05b). The dam disappears instantaneously at t = 0s. The comparison between the reference and the computational results is done at t = 8s such as the eect of the dambreak does not reach the boundary of the domain. The computational parameters are summarized in table 4.5. The reference solution for those test cases can be obtained by solving analytically the shallow water equation (Guinot and Soares-Frazão (2006)). The simulation results for subcritical and transcritical ows are presented in gures 4.14.

Numerical results show a good agreement between the computed hydraulic variables (h, q and Fr) and the reference using either the HLLC or the PorAS Riemann solver whatever the hydraulic ow regime. In both cases, a slight numerical diusion can be identied at locations where the variable value changes. Even if not presented here, the numerical diusion can be reduced by decreasing accordingly the computational space and time step. This test case highlights the ability of both Riemann solvers to correctly localize both rarefaction and shock waves.

One dimensional velocity dambreak without porosity (T06)

The previous dambreak test case (T05) creates whatever the initial conguration of the water depth (expect h 0,L = h 0,R for which there is no test case anymore) both a rarefaction and a shock wave. The test case T06 is also a dambreak problem but the initial discontinuity is applied on the velocity allowing to obtain a solution with two rarefaction waves. In this test case, there is no variation of the porosity nor the bottom elevation and no friction is taken into account.

For this test case, two dierent hydraulic regimes are tested: subcritical (T06a) and transcritical (T06b).

For both hydraulic regimes, the domain is l = 200m long with a structured mesh of uniform cell size 0.1m × 1m. The dam is located at x 0 = 100m in the middle of the channel and initially separates two zones with a constant water level h 0,L = h 0,R = 1m and a constant velocity u 0,L = 0m.s -1 and u 0,R = 3m.s -1 (respectively u 0,R = 5m.s -1 ) for test T06a (respectively T06b). The dam disappears instantaneously at t = 0s. The comparison between the reference and the computational results is done at t = 8s such as the eect of the dambreak does not reach the boundary of the domain. The computational parameters are summarized in table 4.6. The reference solution of those test cases can be obtained by solving analytically the shallow water equation (Guinot and Soares-Frazão (2006)).

The simulation results with subcritical and transcritical ow are presented in gures 4.15 and 4.16. Numerical results show a good agreement between the computed hydraulic variables (h, q, u and Fr) and the reference using either the HLLC or the PorAS Riemann solver whatever the hydraulic ow regime. As for the case T05, a slight numerical diusion appears close to the wave location.

One-dimensional dambreak with variable porosity (T07)

The two previous test cases (T05 and T06) highlight the ability of Flood1D2D to correctly solve the shallow water equations (3.56) in highly transient congurations without porosity. This test case aims at validating the ability of the code to deal with transient dambreak problem with a porosity varying linearly from 0 to 1 over a at bottom with no friction. As presented in Guinot and Soares-Frazão (2006), this one-dimensional dambreak with variable porosity is equivalent to a classical circular dambreak with uniform porosity. The reference is computed as a classical dambreak over a grid with uniform cell width x = 10 -3 m. The domain is l = 100m long with a structured mesh of uniform cell size 0.1m × 1m. The dam is located at x 0 = 50m with an initial water depth of h 0,L = 10m on the left-hand side of the dam and h 0,R = 1m on the right hand side; the initial velocity in x direction is set to 0m.s -1 . The dam disappears instantaneously at t = 0s. The comparison between the reference and the computational results is done at t = 4s such as the eect of the dambreak does not reach the boundary of the domain. The computational parameters are given in table 4.7.

The simulation results are presented in gure 4.17. Numerical results show a good agreement between the computed hydraulic variables (h, q, u and Fr) and the reference using either the HLLC or the PorAS Riemann solver.

One-dimensional dambreak with porosity discontinuity (T08)

This test case aims at validating the ability of the code to deal with transient dambreak problem with a porosity discontinuity, which is located at the same place that the initial water height The domain length is l = 100m long with a structured mesh of uniform cell size 1m × 1m for T08a, 0.1m × 1m for T08b and 0.01m × 1m for T08c. The dam is located at x 0 = 50m in the middle of the channel and initially separates two zones with a constant porosity φ L = 1 and φ R = 0.1, water level h 0,L = 10m and h 0,R = 1m and the initial velocity in x direction is set to 0m.s -1 . The dam disappears instantaneously at t = 0s. The comparison between the reference and the computational results is done at t = 4s such as the eect of the dambreak does not reach the boundary of the domain. The computational parameters are summarized in table 4.8. The reference solutions of those test cases can be obtained by solving analytically the shallow water equation with porosity (Guinot and Soares-Frazão (2006)).

For test case T08, the simulation results are presented in gures 4.18. For each mesh, numerical results show a good agreement between the computed hydraulic variables (h and q) and the reference using either the HLLC or the PorAS Riemann solver. However, a slight dierence appears between the computed velocity for x > 50m and the reference. The gap remains constant as the computational space and time step are decreased accordingly (see gure 4.19 with the velocity), which may be accounted by the Riemann solver adopted. The friction computation is validated on two test cases: a comparison of the computed water surface prole with a classical backwater curve (T10) and a comparison between the converged water depth and the theoretical normal depth in both sub-and supercritical congurations.

Backwater curve comparison (T10)

This test case aims to check the ability of Flood1D2D to correctly take into account the friction eect. A steady-state ow is thus simulated in a rectangular channel of width B = 5m over a at bottom (S 0 = 0m.m -1 ) and compared to the classical backwater curve:

dh dx = S o -S f 1 -Fr 2 (4.7) S f = Q KBh 5/3 2 Fr = Q Bh √ gh
Note that the expression of S f in equation ( 4.7) is written with the assumption that the hydraulic radius R h is equal to the water depth h as it is implemented in Flood1D2D. Equation (4.7) is discretized and solved using an Euler method on a very ne grid (∆x = 0.001m) from downstream to upstream for subcritical ow regime:

h i-1 = h i -∆x S 0 -S f (h i ) 1 -Fr 2 (h i ) (4.8)
The reference for test case T10 is established using q upstream = 0.05m 2 .s -1 and h downstream = 0.1m. The domain length is l = 100m long with a grid of uniform cell width 0.1m × 1m. The simulation is carried out from an initial water level z ini = 0.1m and no initial velocity. The friction coecient K is set to 25m 1/3 .s -1 . The computational parameters are summarized in table 4.10. The simulation is carried out until steady ow conditions are reached over the computational domain.

0.1m q upstream

Prescribed unit discharge at the upstream boundary (x = 0m) 0.05m 2 .s -1 h downstream Prescribed water depth at the downstream boundary (x = 100m) 0.1m

K Strickler coecient 25m 1/3 .s -1

Normal depth computation (T11)

This test case aims to check the ability of Flood1D2D to correctly take into account the friction eect and the bottom slope. A steady-state ow is thus simulated in a rectangular channel of width B = 5m with bottom slope S 0 = 0.001m.m -1 (respectively S 0 = 0.005m.m -1 ) for test case T11a (respectively T11b) and compared to the classical backwater curve obtained using equation 4.7. Equation (4.7) is discretized and solved using an Euler method on a very ne grid (∆x = 0.001m) from downstream to upstream for subcritical ow regime (equation (4.8)). The reference for test case T11a (respectively T11b) is established using q upstream = 2m 2 .s -1 and h downstream = 1m (respectively h downstream = 0.7m).

For test case T11a (respectively T11b), the domain length is l = 3000m (respectively l = 500m) long with a grid of uniform cell width 3m×5m (respectively 0.5m×5m) . The simulation is carried out from an initial water level z ini = 1m (respectively z ini = 0.5m ) and no initial velocity. The friction coecient K is set to 25m 1/3 .s -1 (respectively K = 90m 1/3 .s -1 ). The computational parameters are summarized in table 4.11 (respectively table 4.12). The simulations are carried out until steady ow conditions are reached over the computational domain.

The simulation results are presented in gures 4.22 and 4.23. Numerical results show a good agreement between the computed hydraulic variables (h, and Fr) using the PorAS Riemann solver and the reference.

Cut cell technique

A novelty of the developed model Flood1D2D is the use of a cut cell technique. Several test cases are developed to show that this new functionality allows to simulate correctly specic hydraulic problems of ow contraction/expansions while being insensitive to mesh type change between 1D-like and 2D domains.

0.5m q upstream

Prescribed unit discharge at the upstream boundary (x = 0m) 2m 2 .s -1 h downstream Prescribed water depth at the downstream boundary (x = 539m) 0.7m This test case aims to check that the computation at the junction between a 1D and a 2D parts of the mesh does not introduce unexpected uxes (in case of implementation error). The domain represents a three-branch rectangular channel junction: 3 channels of 30m long and 10m large joining at 90°in a junction measuring 10m × 10m (see gure 4.24, top). The channels are considered as 1D parts of the mesh ( x 1D = 10m) and the junction is 2D ( x 2D = 5m). Water at rest is simulated considering a variable bottom elevation (see gure 4.24, bottom left) and porosity (see gure 4.24, bottom right).

K Strickler coecient 90m 1/3 .s -1
The simulation is carried out from an initial water elevation z ini = 10m and no velocity. At each channel end, a water elevation z BC = 10m is set for the boundary conditions (see gure 4.24, top). The computational parameters are given in table 4.13. The simulation is carried out during 1000s, time at which the computed water depth and water surface elevation are presented (see gure 4.25, top). After 1000s, the initial water surface elevation is preserved and the water remain stagnant (in every cell: u = 0m.s -1 and v = 0m.s -1 , see gure 4.25, bottom). Those results highlight that the cut-cell technique implemented allows to preserve the well-balancing property of Flood1D2D.

Steady-state ow in a variable width rectangular channel (T13)

This test case aims to check that the model at the junction between a 1D and a 2D parts of the mesh correctly estimates the ux between the two parts of the mesh. The domain is a rectangular channel with a length l = 400m with 5 reaches of various widths (see table 4.14). The channel is meshed with 1D and 2D parts such as reproducing all possible congurations between 1D and 2D zones that may be encountered (see gure 4.26). Particularly, two dierent meshing for the junction between 1D and 2D parts of the mesh have be created: at x = 50m the junction corresponds exactly with the width variation and at x = 250m, the 2D part overlaps the width variation. Connecting a unique cell (1D part) to a larger 2D mesh does not require any meshing trick (at x=100m) whereas connecting multiple 2D cells to a larger unique cell requires a meshing trick (cf. zoom 3, see chapter 3 section 3.2.1 for details). The simulation is carried out over a at bottom without friction with an initial water surface elevation z ini = 1m and the initial velocity in x and y direction is set to 0m.s -1 . The test case is established using q upstream = 1m 2 .s -1 and h downstream = 1m. The computational parameters Prescribed unit discharge at the upstream boundary (x = 0m) 1m 2 .s -1 h downstream Prescribed water depth at the downstream boundary

(x = 400m) 1m Q Total discharge 10m 3 .s -1
are given in table 4.14. The simulation is carried out until steady ow conditions are reached over the computational domain (after barely 1500s).

The discharges at the junction between the 1D and the 2D parts and the water depth computed are presented in gure 4.27. Figure 4.27c shows the total discharge computed at the boundaries of the domain and at the junction (see gure 4.26, top) between each submesh part.

The computed discharge diers from the reference (total discharge Q) from less than 0.5%; this is explained by the simulation that is not fully converged. However, it highlights that Flood1D2D is able to correctly estimate the ux between the subparts of the mesh. Figures 4.27a and 4.27b plot the water depth along the channel. The water level remains barely at on each reach of constant width even if the reach is subdivided into a 1D and a 2D mesh part. The water level changes are due to the width variations of the channel.

Steady-state ow in a rectangular channel with various 1D2D junction orientation (T14)

This test case aims to check that the orientation of a junction between a 1D and a 2D part of the mesh has no impact on the ux computation. A steady state ow is thus simulated in a rectangular channel without friction nor slope.

For this simulation, the domain length is l = 225m with a uniform width B = 10m. The mesh involves both 1D and 2D parts with oblique edges between cells to check if ux orientation remains unaected through those edges (see gure 4.28). The average dimensions of the cells are: 2m × 10m for the 1D part and 2m × 2m for 2D part of the mesh.

The simulation is carried out over a at bottom without friction with an initial water surface elevation z ini = 5m and the initial velocity in x and y direction is set to 0m.s -1 . The test case is established using q upstream = 10m 2 .s -1 and h downstream = 5m. The computational parameters are summarized in table 4.15. The simulation is carried out until steady ow conditions are reached over the computational domain (after barely 10000s).

The simulation results are presented in gure 4.29. The water depth prole is constant and equal to the prescribed downstream water depth which is coherent with a computation without 4.29c and 4.29d). The total discharge computed at the boundaries of the domain and at the junctions between the 1D and 2D part of the mesh (see gure 4.29e) is constant over the whole domain highlighting that the orientation of the junction has no impact on the ux computation. The velocity vector maps (see gures 4.29a and 4.29b) show that all velocity vectors are aligned with the channel direction; this highlights that any orientation of computational edge with respect to the ux direction allows to produce good results.

Description of the experiment

The experimental geometry is composed of a rectangular channel of width B = 0.495m with a 90 • bend. The canal is connected to a reservoir of dimensions 2.44m × 2.39m that is located 4m upstream of the bend and is then ended after 3m with a free fall. The channel is initially empty and the initial water level in the reservoir is z ini = 0.25m above the bottom of the channel (gure 4.30, left side). At t = 0s, a ap gate is opened and the water ows out of the reservoir and reaches the bend, then changes direction abruptly. The wave reection on the opposite wall of the bend (x = 6.805m) causes the increase of water height in the bend. Water then propagates to upstream and downstream of the bend. During the experiment, the water level in the reservoir decreases until it reaches the level of the channel bottom. The manning roughness is 0.011m -1/3 .s as described by the authors. The hydraulic congurations of the experiment is modeled using Flood1D2D either using only a classical structured/unstructured 2D mesh or a 1D2D coupling mesh (see gure 4.30, right side). In both cases, the reservoir is included in the mesh and the initial water surface elevation is set to z ini = 0.25m. At the downstream end of the channel, the boundary condition is set as a free outlet.

As stated by the authors, the convergence of mesh size is checked successfully to 0.1m for square grids and the simulation was conducted using grid size 0.4m. In this section, the convergence of mesh size is also checked using a 2D unstructured triangular mesh and results are shown in gure 4.31. Computed water height varied with time is compared with experimental results, where the convergence is reached at grid size x = 0.025m.

Sensitivity analysis of water height to mesh type and size

Before validating the ability of the code to deal with 1D2D coupling in transient state, the sensitivity to mesh type and size is conducted. Two types of mesh including triangular and quadrilateral elements are tested using grid size x = 0.025m. The parameters used are the same as stated in section 4.4.4.1 with results presented in gure 4.32. Results show that the Except mesh type, the sensitivity to mesh size is also tested. Fine mesh with grid size x = 0.025m and coarse mesh with x = 0.1m ( x = 0.025m around bend) are tested. Same parameter congurations are used as stated in section 4.4.4.1. Results are presented in gure 4.33 which shows that coarse mesh with rened mesh at bend is enough to reproduce the ow feature (water height) compared with ne mesh and selected for further use.

Results

The 2D and 1D2D simulated water depth and ow lines along the channel centerline are presented at dierent time steps. The results from Soares-Frazão and Zech (2002) are also used for the comparisons presented in gures 4.31 to 4.35.

Conclusions

In this chapter, the code Flood1D2D presented in chapter 3 is validated against various test cases.

Introduction

Flooding is one of the rst natural hazard on each continent, and it caused in Europe around 100 billion euros of damage between 1986 and 2006 (Moel et al. (2009)). In the context of climate change and its possible eects on rainfalls regimes and extremes [START_REF] Pachauri | Climate change 2014: synthesis report. Contribution of Working Groups I, II and III to the fth assessment report of the Intergovernmental Panel on Climate Change[END_REF]), an unprecedented urbanization of oodplains increases the vulnerability of human societies (e.g. [START_REF] Werners | Water resources in Europe in the context of vulnerability[END_REF]). That is why improving the accuracy of ood inundation forecasts and reanalysis combined with uncertainty quantication, especially in urban areas, has becomes a priority for decision making in civil protection or the insurance industry.

The predictive performances of a model generally depend on model complexity and data availability (e.g. [START_REF] Grayson | Spatial patterns in catchment hydrology: observations and modelling[END_REF] for hydrological models). Flood inundation maps are commonly generated with a 2D shallow water (SW) model consisting in depth averaged Navier-Stokes equations (e.g. [START_REF] Hervouet | Hydrodynamics of free surface ows: modelling with the nite element method[END_REF]; Arrault et al. (2016); [START_REF] Monnier | Inverse algorithms for 2d shallow water equations in presence of wet dry fronts: Application to ood plain dynamics[END_REF] where uncertainty sources can be model structure and parameterization (e.g. basal friction), initial and boundary conditions (bathymetry and source terms). A full shallow water model including inertia terms may be required to capture small scale features and rapidly varying ows over rural oodplains (e.g. [START_REF] Neal | How much physical complexity is needed to model ood inundation?[END_REF]), along with appropriate numerical methods for shock capture and wet dry front treatments (e.g. [START_REF] Monnier | Inverse algorithms for 2d shallow water equations in presence of wet dry fronts: Application to ood plain dynamics[END_REF]). As a matter of facts, the interaction of high energy ows with obstacles triggers 3D ow processes such as crossroads junctions (Mignot et al. (2006)) -which for a realistic representation generally require to solve the Navier-Stokes equations in 3D. Consequently, ow resistance parameterization for a 2D SW model, which is an eective representation of 3D (turbulent) free surface ow structures, may depend on its scale for localized head losses in complex geometries (e.g. Guinot (2012a)) or parietal friction on smooth or macro roughness (e.g. [START_REF] Cassan | Velocity proles in a real vegetated channel[END_REF][START_REF] Cassan | A semi-analytical model for the hydraulic resistance due to macro-roughnesses of varying shapes and densities[END_REF]). Another boundary condition that is a well known source of uncertainty is inow discharge especially in complex urban ood congurations -which can be provided by in situ measurements (involving rating curve uncertainty [START_REF] Delrieu | The catastrophic ash-ood event of 89 September 2002 in the Gard Region, France: a rst case study for the CévennesVivarais Mediterranean hydrometeorological observatory[END_REF]; [START_REF] Paris | Stage-discharge rating curves based on satellite altimetry and modeled discharge in the amazon basin[END_REF]; [START_REF] Savage | Quantifying the importance of spatial resolution and other factors through global sensitivity analysis of a ood inundation model[END_REF]) or simulated by a hydro-meteorological chain involving meteorological model uncertainties [START_REF] Vié | Cloud-resolving ensemble simulations of mediterranean heavy precipitating events: uncertainty on initial conditions and lateral boundary conditions[END_REF] and hydrological model uncertainties (e.g. [START_REF] Douinot | Modelling errors calculation adapted to rainfallruno model user expectations and discharge data uncertainties[END_REF]).

Given its importance in ood modeling and forecasting, sensitivity analysis (SA) of 1D Saint Venant and 2D SW models to their parameters and boundary conditions are addressed in the present contribution. The study focuses on the case of urban ood ows in a complex streets network, based on a ne and unique laboratory scale experiment while in situ ood data are still rare (Finaud-Guyot et al. (2018)). Remark this streets network is composed of several 4 branch crossroads creating complex patterns of free surface ows conuences and deuences. Flood simulation accuracy is not necessarily improved by increasing model complexity and resolution (cf. [START_REF] Dottori | Detailed data is welcome, but with a pinch of salt: Accuracy, precision, and uncertainty in ood inundation modeling[END_REF]) but uncertainty sources may also vary (even interact) as the amount of data required to constrain the model. Sensitivity analysis, that assesses how the of water height and discharge to input parameters are analyzed on two main streets and the whole experimental rig using respectively 1D and 2D models.

The paper is organized as follows. Section 2 describes the GSA method, experimental device and numerical models used to investigate urban ood ows controls. Spatial patterns of parameter sensitivity are presented in section 3 both for 1D and 2D models. Section 4 presents a detailed discussion around sensitivity patterns in various congurations including the eect of bathymetric slope. Conclusions and perspectives are presented in section 5.

Methods and models

Experimental rig and numerical models

This study is based on the ICube urban ood experimental rig designed and built at the ICube laboratory in Strasbourg (France) (Araud (2012);Finaud-Guyot et al. (2018)). It is supposed to be representative of typical urban geometries at the scale 1 /200 with street widths of 12.5cm and 5cm and various angles. It is composed of 64 impermeable blocks on a 5m × 5m horizontal plan.

The streets network is composed of seven streets in the north-south direction crossed by seven streets in the west-east direction (see gure 5.1 -c). The inow discharge of each street of the north and west faces is supplied by a volumetric pump, a rectangular weir controls water levels at the outlet of each street and outlet discharges are measured using calibrated weirs relationships (Finaud-Guyot et al. (2018)). The studied congurations correspond to steady state experiments and the corresponding boundary conditions are presented in Appendix (table A.1).

1D and 2D steady models are used for ow simulations under experimental conditions. 1D modeling is performed for widest streets C and 4 using 1D Saint Venant equations which are solved with a nite volume approach (e.g. Guinot (2012b); [START_REF] Brisset | Identiability and assimilation of sparse altimetry data in 1d saint-venant river models[END_REF]) using a classical HLLC Riemann solver (Guinot and Soares-Frazão (2006)). 2D modeling is performed for the whole experimental rig using Telemac2D [START_REF] Hervouet | Telemac modelling system: an overview[END_REF]) which solves the 2D shallow water equations coupled to a classical kε closure turbulence model using a nite volume approach on unstructured triangular mesh.

2D model is run with upstream discharge and downstream water depth determined from experimental data. Roughness is calibrated with a trial and error procedure in order to best match outlet discharges and measured ow lines (cf. gure 5.1 -d and section 5. 1.3.2.1). 1D model is run with experimental data for the lateral discharge and downstream water depth, and its roughness is calibrated in order to best match measured ow lines (cf. gure 5.4 -top and section 5.1.3.1.1).

Background on model sensitivity analysis with variance decomposition methods

Sensitivity analysis is dened as the study of how the uncertainty in the output of a model can be apportioned to dierent sources of uncertainty in the inputs [START_REF] Saltelli | Global sensitivity analysis: the primer[END_REF]; [START_REF] Pianosi | Sensitivity analysis of environmental models: A systematic review with practical workow[END_REF]). It includes local sensitivity analysis that evaluates the eect on model outputs of a small perturbation of the considered input parameters around a specic value (Guinot and Cappelaere (2009a,b); [START_REF] Delenne | Sensitivity of the 1d shallow water equations with source terms: Solution method for discontinuous ows[END_REF][START_REF] Monnier | Inverse algorithms for 2d shallow water equations in presence of wet dry fronts: Application to ood plain dynamics[END_REF]); and global sensitivity analysis that explore model behavior over a whole parameter space but that is often more computationally expensive (e.g. [START_REF] Pianosi | Sensitivity analysis of environmental models: A systematic review with practical workow[END_REF]).

For a generic model f , let Ω k ∈ k denote the set of all possible values that the model parameters can take. Let X ∈ Ω k be a possible value of the k model parameters normalized by their variation range. We denote by Y = f (X) = f (X 1 , X 2 , ..., X k ) the relationship that links the model inputs to the model output. Following Sobol ' (1990); [START_REF] Sobol | Sensitivity estimates for nonlinear mathematical models[END_REF]; [START_REF] Saltelli | Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index[END_REF]

V (Y ) = i V i + i j>i V ij + . . . + V 1,2,...,k (5.1) where V (Y ) is the total variance, V i = V ar Xi (E X∼i (Y |X i )) is the rst-order variance caused by parameter X i , V ij = V ar Xij (E X∼ij (Y |X i , X j )) -V i -V j
is the covariance caused by X i and X j (second-order variance), and higher order terms show the variance contribution from multiple parameters.

The sensitivity index of rst-order eect of X i on Y can be dened as:

S i1,•••ip = V i1•••ip V (5.2)
where S i is the Sobol index, always between 0 and 1. The sum of all Sobol indices is equal to 1: downstream. This is a reasonable pattern for a subcritical ow. 5.1.3.1.3 GSA on ten parameters with experimental uncertainties This section investigates the sensitivity of 1D simulated ow lines to boundary conditions and source terms.

k i=1 S i + k i<j S ij + • • • + S 12•••k = 1 (5.3)
Sensitivities are presented for the main streets C and 4 of the urban ood rig, conveying around

(Q A ) Lateral street 1 (A) inow discharge 0.171 l/s 0.014 l/s ±5% Q 2 (Q B ) Lateral street 2 (B) inow discharge 0.198 l/s 0.155 l/s ±5% Q 3 (Q C ) Lateral street 3 (C) inow discharge 0.093 l/s -0.107 l/s ±5% Q 4 (Q D ) Lateral street 4 (D) inow discharge 0.407 l/s 0.261 l/s ±5% Q 5 (Q E ) Lateral street 5 (E) inow discharge 0.237 l/s 0.077 l/s ±5% Q 6 (Q F ) Lateral street 6 (F) inow discharge 0.296 l/s 0.380 l/s ±5% Q 7 (Q G ) Lateral street 7 (G) inow discharge -0.319 l/s -0.553 l/s ±5% K uniform roughness coecient 21.5 m 1/3 .s -1
21 m show that the sum of all three parameters is close to 1 highlighting no interactions eects between those three parameters. Modeled water depth variance is mainly explained by roughness and downstream water depth for the whole streets length. Upstream of street C or 4, roughness explains more than 95% of model response and its inuence steadily decreases with increasing ow distance. The opposite pattern is obtained for downstream water depth sensitivity with S hout greater than 95% downstream. Flow in both streets C and 4 is subcritical everywhere and such a pattern is logical. There are no interaction eects between those three parameters along street C and 4, and water depth variance is perfectly explained by their rst-order Sobol indices.

In magnitude, the inuence of inlet discharge can be neglected for the tested ranges compared to water height and friction coecient. For Sobol index of 10 parameters, S Qi < 0.002, ∀i∈ [START_REF] Chen | Variance based uncertainty quantication for 1d and 2d hydraulic modeling: an experimental urban ood case[END_REF]2,3,4,5,6,7; A, B, C, D, E, F, G]), over experimental uncertainty ranges (cf. table 5.1) Figure 5.5: Sobol indices of seven lateral discharges with minus signs for ow loss.

The sensitivity of output water depth to a given lateral discharge is maximal at its inow point (crossroad). Interestingly the inuence of a lateral mass ux propagates to the downstream outlet but also upstream in subcritical ow conditions as a downstream control. After the last crossroad, the Sobol indices of lateral discharges stay constant and their values are proportional to lateral discharge values which can be correlated to the prescribed water depth at the downstream end of the street: S Qi = αQ 2 i ∀i ∈ [START_REF] Chen | Variance based uncertainty quantication for 1d and 2d hydraulic modeling: an experimental urban ood case[END_REF]2,3,4,5,6,7; A, B, C, D, E, F, G] where S Qi stands for the Sobol index at the downstream end to the lateral discharge of street i and α is a constant factor. 5.1.3.1.6 GSA on distributed roughness The spatial pattern of water depth sensitivity to a uniform roughness has been studied in section 5. 1.3.1.3 and 5.1.3.1.4. Now the inuence of a distributed roughness pattern for streets (K s ) and crossroads (K c ) is considered. The variation ranges of two roughness are both set to 50% around their calibrated values (cf. section 5. 1.3.1.1); the ranges for the other parameters are the same. The results show that no interaction eect exists between those two input parameters since the sum of rst order S i s is close to 1 in each street (gure 5.6). For both streets, K s has an increasing inuence on water depth variance with increasing ow distance contrarily to crossroad roughness sensitivity that decreases.

Interestingly, the sensitivity prole between two crossroads (identied by the vertical dashed lines) evolve from linear upstream to parabolic downstream. This might be explained because the friction term (involving the roughness K) is proportional to v 2 and as the water decrease along the street, the velocity increase. Remark also that the magnitude of the sensitivity jump at the crossroads is correlated to the magnitude of the lateral injection.

Interestingly, crossroads roughness plays a larger role (up to 55%) for street 4 than for street C (only up to 34%). This demonstrates that a best eort modeling of the same domain (street C or street 4) would not be achieved using the same parameter (K c for street 4 and K s for street C). This highlights that the importance of the roughness parameter is not related to the area of the modeled domain on which the parameter apply, the crossroads representing 15% of it. inlets and outlets along with a spatially uniform friction coecient, since it is only calibrated with measured outlet discharge distribution. Remark that contrarily to 1D model calibrated on measured water depth proles given known discharge, the 2D model is only calibrated against outlet discharges to reproduce the correct ow pattern within the urban district.

Note that a mesh convergence analysis has been performed for cell sizes ranging between 0.01 and 0.0025m, a grid with 0.005m cells is nally selected. A total inow discharge of 80 m 3 /h, corresponding to rare and high ood ow, is used for steady state simulations with dierent inow discharge partitions between west and north face: 1) 50% west and 50% north; 2) 80% west and 20% north. Within a face, the discharge is divided proportionally to the street width.

The spatial distributions of simulated water height and Froude number are presented in gure 5.7 for the two inow repartitions. Water height decreases from west-north corner to south-east corner in both congurations. The ow in the device is subcritical everywhere except in street 1 after the last crossroad between street 1 and G, where supercritical ow occurs. The unequal ow repartition with increased inow at 80% on west face increases the water height at south-west corner and reduces the region of supercritical ow in street 1 close to the outlet. Interestingly, the outlet street discharges modeled in both conguration (50-50% and 80-20%) are very similar and correspond to the experimental measurements (gure 5.1). This highlights the ltering eect of the experimental rig geometry on inow discharge as demonstrated in Finaud-Guyot et al.

(2018). 5.1.3.2.2 GSA on water height Based on previous investigations with 1D model (cf. section 5. 1.3.1.3), four main input parameters are tested using GSA, namely west-face discharge Q west , north-face discharge Q north , outlet water height h out and uniform friction coecient K.

Again their variation ranges are determined based on experimental uncertainties (Q cal north ± 5%, Q cal west ± 5%; h cal out ± 10% and K cal ± 50%). Interestingly, the sum ΣS i is signicantly below 1 in the part of street 1 in which a supercritical ow zone and a hydraulic jump appear for some parameter sets. A detailed investigation of this particular point is made in section 5.1.4.1.

2D maps of rst order sensitivities of the water depth to the four tested parameters are presented in gure 5.7b, c, d, e. For the 50-50% hydraulic conguration, roughness and downstream water depth are the most important parameters explaining output water depth variance; as found for the 1D model (cf. section 5. 1.3.1.3). Water depth sensitivity to downstream depth h out and roughness K evolves globally in the direction from the north-west corner to the south-east corner except in the circle corresponding to supercritical ows; both patterns evolve as for the 1D GSA: decrease (resp. increase) in the direction of the ow for the sensitivity to K (resp. h out ). The sensitivity to upstream discharge decreases along eective ow direction, i.e. along the diagonal from north-west to south-east corner; and is almost null in the downstream part of the streets.

Interestingly, for the 50-50% conguration the maps show:

the maximal sensitivity to both upstream discharge (that has the same nominal value and the same range of variation) is almost two time bigger for Q west that for Q north ;

the water depth is more sensitive to upstream discharge in 2D than in 1D. This might be explained by the fact that discharge is more free on the 2D domain than on the 1D. As the modeling is performed in steady state, the water surface prole between two crossroads is described by a classical backwater curve that is function of the discharge (between other parameters). Indeed in 1D, the discharge is totally imposed by the upstream value and the lateral inputs on one street whereas in 2D, the ow repartition between street remains free (driven by the modeled crossroad behavior and mass exchanges between streets).

Histograms presented on gure 5.8 show the sensitivity of water depth to the tested parameters for several precise locations consisting mainly in crossroads within the streets network. They are ordered by increasing hydraulic distance from the upstream north-west corner. It depicts clearly the decrease of water depth sensitivity to upstream controls: Q west , Q north , K.

Figure 5.7 shows sensitivity patterns for a 80-20% inow discharge partition between west and north faces. Spatial Sobol index patterns for this asymmetrical inow are similar to the ones for the 50-50% conguration. However, the patterns are oriented from the north-west to the south-east corner; i.e. oriented in the direction of the global ow. As expected, the Sobol index of north-face discharge decreases, whereas west-face discharge index increases and explains more than 25% of water depth variance for streets 1 to 4. In that case, the inuence of downstream control (h out ) is lower for the region close to the west and north faces. The supercritical ow region downstream of street 1 is also reduced for less discharge passing by from the north-face.

Discussion

Metamodel error on S i estimation with hydraulic shocks

As previously shown in section 5.1.3.2.2, the sum of the rst order Sobol indices is signicantly below 1 in the supercritical region (illustrated by the circles on gure 5.7). Figure 5.9 presents the S i proles along the downstream part of the street 1; gure 5.9 -bottom right shows the range of Froude numbers obtained on the parameter sample, which denes the shock zone where supercritical ows (Fr > 1) and hydraulic jumps appear for some parameter sets. Figure 5.9top right presents all the sampled water surface prole; it highlights that the range of variation of the water depth is almost 3 times larger in the shock zone compared to outside. Downstream of the shock zone (x > 0.9), the variance of the water depth is totally explained by h out (cf. gure 5.9 -top left); for x between 0.15 and 0.9, the sensitivity to h out decreases from downstream to upstream. It can be correlated with the fact that downstream water depth represents a hydraulic control only if the ow is uvial while the most upstream point is characterized by the highest number of torrential ows over the N s ows simulated. The spatial variation of water depth sensitivity to the roughness coecient K is barely the negative of the sensitivity pattern to h out . As presented in section 5.1.3.2.2, the sensitivity of the water depth to the upstream discharge (both Q north and Q west ) is almost null along the whole street for the tested ranges.

One cannot ignore that a sum of the rst order S i lower than one claims for interaction eect between the tested parameters or for metamodel error on Sobol index (e.g. [START_REF] Janon | Analyse de sensibilité et réduction de dimension[END_REF]). The quality of the S i estimation with the metamodel is assessed by the correlation coecient r 2 (cf. gure 5.9). It appears that outside of the shock zone, the r 2 is almost equal to 1 and decreases to 0.7 in the shock zone for which the water depth vary of almost 50%, with respect to 10% outside, around the nominal one (gure 5.9). Moreover, note that metamodel standard deviation is quite narrow as shown for each S i prole on gure 5.9 that remain hydraulically coherent.

Sensitivity of outlet discharge

In the context of urban ood forecasting, spatial distributions of water height are of great interest but also discharge distribution, at the downstream end of the 14 streets here. The Sobol indices of outlet discharges to 4 main ow controls (Q

cal north ± 5%, Q cal west ± 5%; h cal out ± 10%, K cal ± 50%)
in gure 5.10 for dierent inow discharge partitions (50-50% and 80-20%); Sobol indices are also calculated for subdistricts as detailed hereafter. ltered out by those larger streets. This is conrmed by the same sensitivities to h out for those streets on the 80-20% ow repartition (cf. gure 5.10, bottom). Moreover, concerning street 1 a supercritical ow zone occurs (cf. gures 5.7 and 5.9) hence the ow in street 1 is totally controlled by upstream conditions. Therefore the output discharge of street 1 is not controlled by its downstream water depth but those of other streets. It may be explained by the fact that, applying a stronger downstream control on all streets (i.e. increasing h out ) reduces the discharge owing out streets 2 to G thus increases relatively the discharge of street 1.

The outlets of street 2 and 3 (resp. street A, B, D and E) are located close to the inlet on north face (resp. west face), which may be the reason for higher sensitivities to Q north (resp. Q west ) for those streets (cf. gure 5.10a). The inuence of Q north (resp. Q west ) on outlet discharge decreases from street 2 to 3 (resp. street A to E) as their distance to north-face inlet (resp. west-face inlet) increases, whereas the inuence of west-face discharge (resp. north-face discharge) increases. Interestingly for streets A to E, the sensitivity of the outlet discharge to Q west is divided by more than 2 between streets A-B and D-E. This may be related to the inuence of the main street C between those two groups of streets.

h out explains around 30% of the outlet discharges of streets A-B and 60% for streets D-E.

The rst being located closer from the inlet hence more sensitive to inow discharge than global outlet water height. Moreover ow is conveyed/crossing big streets before reaching streets D-E.

For big streets C, F and 4, the inuence of west-face discharge and north-face discharge is equal which makes sense given their central locations within the district geometry. Interestingly, outlet where north-face discharge decreases respectively. The Sobol index of outlet water height and friction coecient is not very sensitive to this change except for streets 4, A, B, C that are less (resp. more) sensitive to h out (resp. K); street F has an opposite trend. The increase of the sensitivity to K may be related to the increase of the velocity for the concerned streets.

As a conclusion of this section, big streets acts as global ow pattern separators into several sub-district, which can block the inuence of upstream inlet discharge and increase the inuence of downstream outlet water height.

Sensitivity to slope

The inuence of bottom slope on the simulated free surface ows in the urban district geometry is investigated here. A constant slope in the north-south direction I NS = 2% is simulated (the increased elevation of inlets on north face is 0.1 m). The total inlet discharge is 80m 3 /h with 50-50% and 80-20% discharge partitions at west and north-face. The outlet water height is obtained from 3D model simulations with interface tracking performed using InterFoam and validated on the measured outlet discharge in the at conguration -not presented here. The four input parameters tested previously are investigated here with a bottom slope: west-face discharge Q cal west ± 5%, north-face discharge Q cal norh ± 5%, uniform friction coecient K cal ± 50% and the downstream water height h out with a range of ±10% for all streets and congurations except for street 1, 2, 3 and 4 with the 80-20% inow discharge partition because reverse ow appears at the outlet; the range is thus modied to [-10%, -1%].

Figure 5.11 presents the results (maps of water depth and Froude number) of the calibrated model with slope for both congurations and the rst order Sobol index. The Froude number maps show that by adding a slope in north-south direction, more supercritical ows and hydraulic jumps are generated in streets of north-south direction than in the horizontal case. As presented in section 5.1.3.2.2, the sum of the rst order Sobol index is equal to 1 for the whole subcritical ow and smaller than 1 where supercritical ows appear as shown by Froude distribution. The GSA remains consistent for the whole domain as shown in section 5.1.4.1. A comparison of Sobol index maps with and without slope (cf. gure 5.11 and 5.7) shows that the sensitivity patterns are barely the same except in the torrential zones. In the conguration with slope, the inuence of Q west and Q north (resp. h out ) on the water depth is almost multiplied (resp. divided) by 2. As in the conguration without slope, the sensitivity to roughness K becomes non negligible in the torrential zone due to the increase of the velocity.

Histograms presented on gure 5.12 show the sensitivity of water depth to the tested parameters for several precise locations consisting mainly in crossroads within the streets network. They are ordered by increasing hydraulic distance from the upstream north-west corner. It depicts clearly the decrease of water depth sensitivity to upstream controls: Q west , Q north , K. However their inuence is lower than previously without slope (gure 5.8) because of the occurrence of several super critical ow zones characterized by a higher sensitivity to K even downstream and acting as a hydraulic deconnection. Figure 5.13 illustrates the inuence of four input parameters on 14 outlet discharges in 50-50% inow discharge partition with slope. Compared with the results presented in gure 5.10, the sum of rst Sobol indices is reduced between 0.8 and 1 for all street except for streets 6 and 7 for where it reduces to almost 0.7. This can be explain by possible parameter interactions but also the occurrence of more supercritical ow zones and hydraulic jumps. Moreover streets 5 to 7 outlets are almost disconnected from upstream (due to torrential ow in streets F and G illustrated in gure 5.11) and corroborates a sensitivity of outlet discharges mainly explained by K and h out . The outlet discharges for the streets 5 to 7 and G are also more sensitive to the roughness coecient K in the conguration with slope which can be related to the increase of the velocity in the southern streets. The outlet discharge for the streets 2 to 4, B, D and E are signicantly less depending on Q west and Q north (with a similar S Qwest /S Q north ratio for streets B, D and E) and conversely more depending on h out . This can be related to north-south slope that deviate the ow in the south and south-east outlets. No signicant change can be observed for street 1 and A. The slope has an opposite eect for the main streets C and F outlet sensitivity patterns: the sensitivity to h out decreases (resp. increases) for street C (resp. F). Street C becomes more sensitive to Q west whereas street F becomes insensitive to both Q west and Q north . Those trends might be explained by more torrential ows appearing in street F than in street C, leading to a more important deconnection from upstream.

Conclusions

This study proposes a framework for performing global sensitivity analysis (GSA) in the case of complex free surface ood-ow modeling. It quanties the inuence of the uncertainty of 1D and 2D Shallow water model input parameters on simulated urban ood ows using GSA. For our case study based on the urban ood experimental rig of ICube laboratory Strasbourg dierent General sensitivity patterns of 1D SW equations have been obtained with parameters sampled on equal ranges. The results show that in 1D conguration, the closer from the downstream boundary condition on water height, the higher the Sobol index as predicted by hydraulic theory for subcritical ow. Interestingly, the sensitivity to friction and upstream discharge are very similar. At a 'far enough' upstream end of the domain, the sensitivity to the downstream water depth is null and both roughness coecient and upstream discharge explain 50% of the water depth variance.

1D Sensitivity patterns corresponding to our experimental uncertainties are similar to previous ones with a lower inuence of the upstream discharge that is less uncertain. Water depth variance is mostly explained by friction and downstream water depth showing opposite spatial variations. The sensitivity analysis to the lateral discharges highlight that the main (resp. smaller) streets of the experimental rig explain 30 to 45% (resp. 0 to 15%) of the water depth variance with a higher sensitivity close to the crossroads at which the ow is injected. Sensitivity to distributed (street and crossroad) friction coecient has shown that the water depth variance is not proportional to the domain size on which a friction coecient (either crossroad and street) is applied and that similar parameterization for dierent streets produce variables sensitivity responses. This highlights the need of identifying the strongest hydraulic controls during a calibration process.

Maps of Sobol indices are provided for 2D SW model with two dierent upstream discharge partitions. The results are in agreement with the ndings in 1D but in dierent proportions: the water depth variance is explained at almost 70% by the downstream water depth, to less than 10% by the friction coecient and to less than 25% by the upstream discharge. This highlights that the friction coecient plays a dierent role in the energy dissipation at 2D scale. Moreover it quanties the ltering eect of the topography on the discharge distribution through the whole district as suggested by Finaud-Guyot et al. ( 2018) which is conrmed by the downstream discharge sensitivity to upstream inow that is barely null.

Concerning the street outlet discharge, dierent sensitivity patterns are obtained with a variance explained at 100% by the downstream water depth, or almost equally explained by the downstream water depth and the upstream faces discharge; the friction coecient being only involved in some particular streets. The key ndings is that larger streets acts as global ow pattern separators into several sub-district, which can block the inuence of upstream inlet discharge and increase the inuence of downstream outlet water height.

Sensitivity maps are also presented for hydraulic regime changes and ows with a positive bottom slope triggering more supercritical ow in the district. The global sensitivity patterns remains similar but with an increased role of the friction coecient, a water depth distribution that is less downstream dependent (especially in the sub-district that is closer of the inlets), and an asymmetrical sensitivity to the upstream discharge (with an increased sensitivity to the north discharge).

This work highlights and somehow quanties the nonlinearities of these complex free surface ow patterns and dependencies. Subsequent works could investigate unsteady aspects of urban ood ows for other geometries, ood magnitude and source terms such as street/underground networks or building interactions. The sensitivity of 1D/2D coupled models could also be investigated along with other parametrizations of ow resistance. Note that the results presented involve a signicant number of model runs -on the order of 10 4 2D model ( 4• 10 5 cells) evaluations were needed corresponding 35 days of calculation on 256 cores. New methods for computing Sobol indices at a lower cost could also be of interest.

Variance based sensitivity analysis of 2D unsteady ood ow hydraulic model

Many ood hazard studies and predictions are performed with steady state models, including ood hazard mapping for dierent return period. However unsteady models (see e.

Calibration of the rating curve

Unsteady simulation on the whole ICube urban ood experimental rig illustrated in gure 5.1, considered without bottom slope, is conducted with a time-varied inow hydrograph (see gure 5.14) dened for each street on north and west inlet faces and rating curves for each street on east and south outlet faces. The lattest being calibrated with respect to the steady experiment with total inow discharge 80 m 3 .h -1 (respectively 0.02222 m 3 .s -1 as presented in gure 5.14).

Currently, the ICube urban ood experimental rig is not equipped to measure water depth in transient conguration at the downstream end of the streets. No experimental values can then be used to provide downstream boundary conditions for the modeling. A stage discharge relationship has then been used [START_REF] Bos | Discharge measurement structures[END_REF]; [START_REF] Gharahjeh | Weir velocity formulation for sharp-crested rectangular weirs[END_REF]):

Q = 2 3 C d B 2gh 3/2 (5.4)
where Q is the total discharge, B is the weir width (channel width in this conguration), h is the water depth above weir crest elevation and C d is the discharge coecient. The stage-discharge relationship is calibrated such as a nil water depth in the street corresponds to a nil discharge.

This implies that the weir height is assumed to be nil in the numerical modeling. The discharge coecient C d is calibrated for each street using the steady state experimental value: the water depth and associated discharge measured in the downstream part of each street. The calibrated discharge coecient is listed in table 5.2. Full 2D depth-averaged shallow water equations are solved and discretized on a triangular mesh grid with a resolution of 0.005m. The mesh convergence analysis has been conducted as mentioned above (see section 5.1.3.2.1). Finite volume method and HLLC Riemann solver are used to calculate the uxes between neighboring cells. The initial water depth over the whole domain is set to 1.5mm before ash ood simulation starts.

Q(t = 0) = Q(t = 350) = 0 m 3 .h

Analysis of the direct run

The resulting spatial and temporal patterns of water depth and Froude number are presented in gures 5.15 and 5.16.

Figure 5.15 shows spatially distributed water height maps every 20s from 0s (starting of the ash ood) to 350s (end of the ash ood). The water height increases and decreases at rising and falling limb following the inow hydrograph variations and spreads from north-west corner to east-south corner along the diagonal. The water height reaches the maximum at steady state of total inow discharge 80m 3 .h -1 from 100s to 150s which has been analyzed in the steady simulation of section 5.1. Remark that for all snapshots of this unsteady simulation the water depth is higher upstream than downstream. Figure 5.16 presents the time-varied Froude number on the whole device. More supercritical zones can be observed for this unsteady ash ood case. One reason can be the outlet weir height set to 0m instead of weir height of 0.05m in previous studies in steady state. Interestingly, maps of the distributed Froude number are very similar during rising limb (0s100s) and steady state (100s150s) and only decreases at the end of the falling limb (300s350s), when the areas of supercritical ow reduce. The supercritical zones are located:

Around outlets for t > 40s;

Inside the urban ood device and mostly during ood rising limb. The resulting hydraulic jumps downstream of those supercritical ow zones when a uvial zone exists may aect the spread of the ash ood within the geometry.

As a matter of facts, the torrential zones located in downstream ends of each street tend to impose a critical depth for upstream uvial ow lines. This may explain the general decrease of water depth along the apparent ow direction (NW-SE diagonal).

The time-varied outlet discharges of the 14 streets and their ratios to total discharge Qi Q total,out (Q total,out , total outlet discharges) are presented in gure 5.17. The temporal variations of outlet discharges follow the same trend than inow hydrograph with a delay of about 10s, which is the propagation time needed to cross the whole device. The three biggest streets (street C, F and 4 having a width 0.125m) have the biggest outlet discharges compared to the small streets with a width 0.05m. To estimate if those biggest outlet discharges are related to the particular role of those streets or mainly due to their larger width, the discharges of larger streets have been computed and divided by 0.125m 0.05m = 2.5 to allow a simple comparison with The ratios between each outlet discharge Q i and the total discharge Q total,out are presented in gure 5.17 right. For all the streets, the relative discharge is constant between 0s and 5s. This corresponds to the contained water depth within the domain that ow out by the downstream weir. After 60s (dash line in gure 5.17 right), the relative discharges are barely constant. Before (respectively after) 350s, the biggest streets (4, C and F) represent almost 48% (respectively 42%) of the total discharge. Focusing on the trends of variation of the relative discharge around t = 350s, several groups of streets can be identied:

Streets [START_REF] Chen | Variance based uncertainty quantication for 1d and 2d hydraulic modeling: an experimental urban ood case[END_REF]2,3) and(A, B) that have a reduction of the relative discharge; Streets (D, E) that have a barely constant relative discharge; Streets (5,6,7) and (G) that have an increase of the relative discharge.

Interestingly, these groups of streets are the same than the one proposed by Finaud-Guyot et al.

(2018) based on steady state experiment. This highlights the role of the streets network geometry to redistribute the discharge between streets.

It appears that during the falling limb of the ood, the streets that become more important are further from the inlet (NW corner).

Global sensitivity analysis

Global sensitivity analysis (GSA) described in section 5.1.2 is applied to unsteady ash ood to determine the inuence of uncertainty sources on unsteady model outputs. Two input parameters are analyzed: inow discharge and uniform roughness K. The sampling range for inow discharge is set to Q cal i,in ± 25% (see gure 5.14 for Q cal i,in ), hence maximum total inow discharge Q cal in,total ranges from 60m 3 .h -1 to 100m 3 .h -1 . The sampling range for uniform roughness is set to K cal ± 50%, which corresponds to roughness ranging between 50m 1/3 .s -1 and 150m

Sensitivity analysis on spatially varied water height

The inuence of uncertain inow discharge and uniform roughness on the spatially distributed water height during a ash ood is studied. Sobol indices maps for inow discharge, uniform roughness and their sum are presented in gure 5.18 for 9 dierent times equally sampling the three hydrograph phases (rising limb, steady peak state and falling limb).

Figure 5.18 left shows the sum of all Sobol indices S sum that are close to 1 except at some locations where hydraulic jump and interaction eects exist. Such zones are reduced with increasing simulation time since hydraulic jumps tend to vanish throughout the device as explained in section 5.2.1 (see for instance, the map at t = 20s and t = 40s, street 4 between t = 160s and t = 220s and street F between t = 220s and t = 280s). Again this shows water depth variance pattern is mainly explained at rst order by the two parameters investigated (inow discharge and uniform roughness), except at hydraulic jump locations as shown for steady state in previous section (see section 5.1.4.1). (t=350s) where inow discharge has decreased to 0m 3 .h -1 but water still propagates, the inow discharge and roughness has similar eect on water height except for supercritical zones.

The Sobol index of uniform roughness has an opposite trend compared to inow discharge as illustrated in gure 5.18 right. In supercritical ow zones where no hydraulic jump exists, the Sobol index of roughness is close to 1 and roughness has dominant control instead of inow discharge for the increase of velocity. For the supercritical zones followed by a hydraulic jump inside the device, interaction eect exist and the function of roughness on the water height decreases as explained in section 5.1.4.1.

Sensitivity analysis on outlet discharge

Another output of interest is outlet discharge for the 14 outlet streets. Considering the Sobol estimation is approximately correct in terms of shape and R 2 and S istd illustrated in gure 5.21 and sum always lower than 1.0577. Time-varied Sobol indices of outlet street discharges to the tested parameters (inow discharge and uniform roughness) are presented in gure 5.20 at every 20s from 20s to 350s.

First of all, the sum of Sobol indices, for each outlet street, is close to 1 after 60s until to the end of the ash ood. Note that the sum of Sobol indices is lower than one during submersion but not during ow recession phase. This may suggest higher order interactions between the tested parameters. This could either be order two or more interactions between parameters, which could give insight for modifying model parameterization along with improving the accuracy of Sobol (see e.g. [START_REF] Janon | Analyse de sensibilité et réduction de dimension[END_REF]). The inuence of total inow discharge variation explains more than 80% of the outow discharge variance of each outlet street except for the rst point (t = 20s) for which the outlet discharge is also inuenced by the drainage of the initial water depth (see section 5.2.3). Sobol index of streets 5 and 7 show that those streets are less sensitive to Q in (cf. gure 5.20 middle). This may either be due to their particular geometry (broken street at crossroads E5 and E7) or high Froude numbers. The other parameter considered is uniform roughness which explains on the order of 10% or less of the outow discharge variance of each outlet street. Interestingly, the sensitivity to roughness is higher for streets 5 and 7(cf. gure 5.20 right).

Conclusions on temporal sensitivities

This temporal sensitivity analysis investigated the inuence of two input parameters (inow discharge and uniform roughness) on 2D shallow water model response. The case of a synthetic ash ood has been simulated on ICube urban ood experimental rig geometry. The results show that:

1. Water spreads from north-west corner to the south-east corner along the diagonal and the variation of water depth is comparable to the inow hydrograph shape; 2. Discharge distribution into the device reaches stable state at 60s during submersion ow phases and changes at the end of ash ood (t=350s); Their locations are presented in gure 5.22, which can be classied as: 1) blockage in small streets (blockage 1, 2, 3, 4, 5, 8 and 9, width 0.05m; 2) blockage in big street C (blockage 6, width 0.125m); 3) half-blockage in big street 4 (blockage 7, width 0.125m). The reference is the steady state simulation studied in section 5.1 without blockage.

To assess the inuence of the tested blockage, we dene f as:

f = 1 N N i=1 f sim i -f ref i f ref i (5.5)
where N is the number of cells; subscript f is either h (for the water depth), Fr (for the Froude number) or Q (for the outlet discharge); f sim i stands for the simulated value of f in cell i and f ref i

is the reference of this same variable in the same cell. f is thus a measure of the relative variation on f induced by a blockage. Blockage 1, 2, 3, 4, 5, 8 and9) has limited inuence on distributed water height whatever the street's location and direction. Simulated Froude number is sensitive to full blockage of narrow streets only if the street is located rather upstream or downstream as illustrated in gure 5.23 (right side).

Conclusions

This chapter proposed global sensitivity analysis (GSA) in the case of complex free surface ood- The sensitivity of ow features to the location of street blockage is conducted. Undoubtedly, the variation of ow features and outlet discharge caused by blockage in big streets (width 0.125m) is much bigger than small streets (0.05m). However, the role of small street is correlated with the location and direction of the street. This study can be used to identify the key small streets to the whole feature and discharge distribution into the device.

The presented sensitivity analysis can be the guidelines on model calibration and data collection before urban ood simulation. The friction drag force is an essential physical eect that is accounted in the shallow water equations (see section 3.1.2). In this work, a classical Manning-Strickler parameterisation is used to compute the friction slope S f (equation 3.54) involving the computation of the hydraulic radius. In classical 1D approaches, the hydraulic radius R h involves the computation of the full wetted perimeter whereas in 2D, the hydraulic radius is classically approximated by the water depth in the computational cell to simplify the implementation. The friction slope S f thus writes S f,1D in 1D or S f,2D in 2D:

S f,1D = u 2 K 2 R 4/3 h R h = S χ (6.1a) S f,2D =          u √ u 2 + v 2 K 2 x R 4/3 h along Ox v √ u 2 + v 2 K 2 y R 4/3 h along Oy R h = h (6.1b)
where K is the Strickler coecient (K x (respectively K y ) being the Strickler coecient in the x (respectively y) direction), u (respectively v) is the depth-average velocity in the x (respectively y) direction.

The 1D cells in Flood1D2D represent the full street width (as it would be in a classical 1D model) but the classical 2D shallow water equations are computed on it (equations (3.56)). It thus seems logical in the context of 1D2D hydraulic modeling to wonder how to compute the hydraulic radius.

Approximations of hydraulic radius

For the implementation in Flood1D2D, the friction eect in the so-called 2D cells is computed using the classical R h = h assumption for the 2D modeling tools. In the case of the 1D cells, the question of the hydraulic radius R h computation remains. Indeed, the hydraulic radius R h = S/χ traduces the fact that the friction eect is induced by the contact length between the ow and the soil (represented by the wetted perimeter χ) that imposes a nil velocity and that this eect impacts the whole wetted cross section S.

In the particular case of a 1D cell, a part of the total wetted cross-section is assumed to be a recirculation for which the ow velocity is nil in average and thus for which the friction eect can be accounted to be also nil in average. The friction eect on the whole wetted cross section thus reduces to the part on the ow vein with three zones of contact between the ow and a nil velocity (see gure 6.1): the bottom, the solid wall and the interface between the ow vein and the recirculation zone.

Three dierent approaches have been proposed to estimate the wetted perimeter χ and thus the hydraulic radius R h . Considering 1D cells with recirculation zone (thus a porosity φ a priori Figure 6.1: Scheme of the considered cross-section with recirculation smaller than 1), χ (h, φ) and R h (h, φ) can be written:

χ SW = φB R h,SW (φ, h, B) = φBh φB = h (6.2a) χ wall = φB + h R h,wall (φ, h, B) = φBh φB + h (6.2b) χ f ull = φB + 2h R h,f ull (φ, h, B) = φBh φB + 2h (6.2c)
where subscripts SW stands for shallow water as the water depth is neglected in front of the eective width φB of the ow vein, wall indicates that the friction is only accounted on the bottom and the side wall of the ow vein and f ull corresponds to a full and classical computation of the wetted perimeter.

Introducing the denition of the discharge Q = u × φ × B × h in equation (6.1a) leads to:

S f (φ, h, B) = Q 2 φ 2 B 2 h 2 K 2 R 4/3 h (φ, h, B) (6.3)
A comparison of the dierent estimation of the hydraulic radius is thus performed hereafter and the sensitivity of the friction slope to hydraulic approximations is explored.

Sensitivity of the S f computation to porosity

This section aims to highlight the modication of the friction slope due to the introduction of the recirculation eect through the porosity parameter. For a chosen hydraulic radius formulation, the friction slope S f,{SW,wall,f ull} (φ, h, B) is adimensionalised by the friction slope S f,{SW,wall,f ull} (φ = 1, h, B) computed for the same hydraulic radius formulation with φ = 1: Introducing equation ( 6.2) for the each formulation of the hydraulic radius into equation (6.4) leads to:

S f,{SW,wall,f ull} (φ, h, B) S f,{SW,wall,f ull} (φ = 1, h, B) = Q 2 φ 2 B 2 h 2 K 2 R 4/3 h,{SW,wall,f ull} (φ, h, B) B 2 h 2 K 2 R 4/3 h,{SW,wall,f ull} (φ = 1, h, B) Q 2 = 1 φ 2 R h,{SW,wall,f ull} (φ = 1, h, B) R h,
S f,SW (φ, h, B) S f,SW (φ = 1, h, B) = 1 φ 2 (6.5a) S f,wall (φ, h, B) S f,wall (φ = 1, h, B) = 1 φ 2 φB + h φB + φh 4/3 = 1 φ 7/3 φB + h B + h 4/3 (6.5b) S f,f ull (φ, h, B) S f,f ull (φ = 1, h, B) = 1 φ 2 φB + 2h φB + 2φh 4/3 = 1 φ 7/3 φB + 2h B + 2h
4/3 (6.5c) Figure 6.2 presents equation (6.5) for dierent hydraulic radius formulations as a function of porosity φ. For every hydraulic radius formulation R h,{SW,wall,f ull} , the adimensionalised friction slope decreases to 1 as the porosity get closer to 1. For the smaller porosity (φ = 0.1), the friction slope with the porosity is, at most, 100 times bigger than the friction slope without porosity (φ=1). As highlighted by gure 6.2a and equation (6.5a), the adimensionalised friction slope for the shallow water estimation of the hydraulic radius R h,SW does not depend on h/B. For the wall (respectively full) estimation of the hydraulic radius R h,wall (respectively R h,f ull ), the adimensionalised friction slope decreases when h/B increases from 0.01, 0.1, 1 to 10 (whatever the porosity).

As highlighted by Araud (2012);Finaud-Guyot et al. (2018), the typical water depth in the ICube urban ood experimental rig is in the range 1cm ≤ h XP ≤ 15cm, for street width B XP in the range 5cm ≤ B XP ≤ 12.5cm. The scaling of the experimental rig being λ h = B XP /B Real = 1/200 in the horizontal direction and λ v = h XP /h Real = 1/20 in the vertical direction, the corresponding water depth and velocity in real world congurations are respectively 0.2m ≤ h Real ≤ 3m and 5m ≤ B Real ≤ 25m. The ratio h/B is thus in the range 0.2 ≤ h XP /B XP ≤ 1.2 (respectively 0.02 ≤ h Real /B Real ≤ 0.12) for the experimental (respectively real) conguration. Let us consider that a signicant eect of the porosity is dened by a friction slope computed with recirculation (φ < 1) 10% bigger than the friction slope computed without recirculation. In the real congurations (thus considering h/B = 0.1), gure 6.2 shows the recirculation introduces a signicant eect for φ<0.95 (whatever the estimation chosen for the computation of the hydraulic radius). Similar reasoning for the experimental conguration (h/B = 1) shows that the recirculation introduces a signicant eect from a similar threshold φ<0.95.

For a given friction slope formulation (either the shallow water, the wall or the full), the friction slope appears to be signicantly modied when introducing a recirculation (by decreasing the porosity). The recirculation reducing the size of the ow vein for a given discharge, the velocity is thus increased leading as highlighted here to a higher friction eect. However, this does not account for the potential eect of the recirculation on the discharge distribution in a street network that should be investigated more deeply (see section 6.6). Indeed, the parametrisation (using either the Strickler coecient K or the porosity φ) is dened at the computational cell scale. However, those parameters eects on the hydrodynamic are much global (see for instance Guinot and Cappelaere (2009b)), especially in branched network for which the owpaths are not easy to dene.

Sensitivity of the S f computation to hydraulic radius

In this section, the sensitivity of the S f computation to hydraulic radius formulation is assessed for similar geometrical and hydraulic parameterisation. For a chosen hydraulic radius formulation, the friction slope S f,{SW,wall} is adimensionalised by the friction slope S f,f ull computed using the full and classical computation of the wetted perimeter: ), the adimensionalised friction slope decreases as the porosity get closer to 1. The adimensionalised friction slope for the shallow water estimation of the hydraulic radius R h,SW is always bigger than the one for the wall estimation of the hydraulic radius R h,wall . As the geometrical ratio h/B decreases the dierence between the adimensionalised friction slopes (whatever the porosity) reduces.

S f,{SW,wall} S f,f ull = Q 2 φ 2 B 2 h 2 K 2 R 4/3 h,{SW,wall} φ 2 B 2 h 2 K 2 R 4/3 h,f ull Q 2 = R h,f ull R h,{SW,wall} 4/3 (6.6)
As explained above, the ratio h/B is in the range 0.2 ≤ h XP /B XP ≤ 1.2 (respectively 0.02 ≤ h Real /B Real ≤ 0.12) for the experimental (respectively real) conguration. In the real congurations (thus considering h/B = 0.1), gure 6.3 shows that the friction slope computed with the shallow water (respectively wall) is 4.36 (respectively 1.72) times bigger than using the full estimation for the smaller considered porosity (φ = 0.1); 1.566 (respectively 1.23) times bigger for φ = 0.5 and only 1.275 (respectively 1.123) times bigger without recirculation (φ = 1). Similar reasoning for the experimental conguration (h/B = 1) shows that the friction slope computed with the shallow water (respectively wall) is 57.94 (respectively 2.37) times For a given geometrical and hydraulic conguration, the friction slope appears to be signicantly inuenced by the friction computation formulation (either shallow water or wall compared to full). Interestingly, the friction slope is up to 60 times bigger than the one with a full computation for the experimental conguration (h/B = 1). This ratio has the same order of magnitude than taking into account the recirculation eect (as highlighted in section 6.1.2). In the real-world conguration, this ratio decreases to 4 which is smaller than the inuence of the recirculation.

The choice of the friction slope formulation is thus much more important for the experimental conguration than for modeling real-world domain. Moreover and as highlighted in section 3. 1.2, this study only focuses on the friction slope but shallow water ows are known to be inuenced by several other parameters (like porosity, initial and boundary condition et al.).

Response surface of existing three branches recirculation laws

As presented in section 2.2, only few studies have been performed on four branch intersections especially on a recirculation zone downstream of a crossroad. [START_REF] Ming | Experimental and numerical study of open channel intersections[END_REF] proposed in his thesis that a four branch intersection can be seen as a combination of a three branch conuence and bifurcation. It seems feasible to study the recirculation zone in four branch intersections from the results on three branch intersections. Therefore, to delineate the dimensions of separation zone in four branch intersections, results on three branch intersections quantifying the dimensions of recirculation zone are reviewed, compared and analyzed in this section. Figure 6.4 illustrates the recirculation zone in three branch intersections. y s (x) delineates the shape of recirculation zone. L s is the length of the recirculation zone, and B s the maximum value of y s for all x between 0 and L s . The length r 0 corresponds to the distance from the downstream corner to the x-coordinate of the maximum contraction. At maximum contraction point, u s is the longitudinal velocity outside the recirculation zone and h s is the averaged water height perpendicular to the ow direction.

Three empirical relationships quantifying the dimensions of the recirculation zone are rewritten here and adapted for comparison: Best and Reid (1984) L s B = 2.587 + 1.025 ln q (6.8b)

where B is the channel width, q = Q L /Q d is the discharge ratio with Q L and Q d the lateral and downstream discharges. Those empirical relationships are determined with a downstream Froude number F d in the range 0.1 to 0.3 and q in the range 0.2 to 1. q = 1 corresponds to a 90 • bend and q = 0 would correspond to a straight channel. The recirculation zone is thus expected to be maximal for q = 1.

B s B = 1 2 × F d - 2 3 2 + 0.45q 1/2 (6.9a) L s B = 3.8 × 1 -1 2 F d × q 1/2 for F d < 1 1.04q 1/2 for F d = 1 (6.9b)
where B is the channel width; q = Q L /Q d is the discharge ratio, Q L and Q d are lateral and downstream discharges; F d is the downstream Froude number, which varies from 0.1 to 0.99 for subcritical ow regime. Qualitative analysis of equations (6.9a) and (6.9b) shows that, as for equations (6.8a) and (6.8b), the maximal recirculation zone might appear for q = 1. Gurram et al. (1997) also suggest to compute the shape of the recirculation zone:

Y s = [sin (90 • X s )] 2/3
where Y s = f (B s ) and X s = f (L s ). 6.10b) where q = Q L /Q (in range 0.2 to 1), Q L and Q being the lateral and total inlet discharges. Despite q has not the same denition as q, it appears to be quite similar as (i) for q = 0 it corresponds to the straight channel conguration (with thus no recirculation) and q = 1 corresponds to a 90 • bend (with a priori the maximal recirculation zone).

B s B = 0.504 × (q ) 2 -0.893 × (q ) + 0.861 (6.10a) L s B = 6.49 × (q ) 2 -8.44 × (q ) + 4.45 (
The comparison of the three relationships for the computation of the ratio Bs B is presented in gure 6.5b. It shows that when the discharge ratio q increases, the ratio Bs B increases for T-conuence (corresponding to equations (6.8a) and (6.9a)) and decreases for T-bifurcation (equation (6.10a)).

For T-branch conuence ow, downstream Froude number has small inuence on Bs B and limited dierence is observed between Best and Reid (1984) and Gurram et al. (1997). Interestingly

for T-bifurcation ow, the recirculation appears to be maximal for q → 0 (corresponding to no discharge in the lateral branch). That can be understood by reversing the reasoning: to have all 6.11) This equation is plotted on gure 6.5a and interestingly, the shape is very similar to the one observed for equations (6.8a) and (6.9a). On the basis of the relations (6.8a), (6.9a) and ( 6.11), it appears that Bs B evolves in the range 0.18 to 0.54 as q changes from 0.2 to 1.

B s B = 1 - B s B Kasthuri (
Figure 6.6a presents the variations of the ratio Ls B for the dierent above-mentioned formu- lations in function of q (respectively q ). It shows that when the discharge ratio q increases, the ratio Ls B increases for T-conuence (corresponding to equations (6.8b) and (6.9b)). Interestingly, the curve corresponding to the formulation proposed in [START_REF] Kasthuri | Discussion of separation zone at open-channel junctions by James L. Best and Ian Reid (november, 1984)[END_REF] seems dierent from what was published (see gure 6.6b). Indeed, for q = 1, equation (6.10b) gives L s /B = 6.49 × (1) 2 -8.44 × (1) + 4.45 = 2.5 which is quite dierent from the value of 2 plotted by [START_REF] Kasthuri | Discussion of separation zone at open-channel junctions by James L. Best and Ian Reid (november, 1984)[END_REF]. It also gives a minimal length of the recirculation for q = 0.65 of 170% of the channel width. The experimental setup and dimensions are illustrated in gure 6.7a.

A reference simulation is performed with a ne mesh and a 2D SW model accounting for turbulence. The goal is to simulate a realistic ow pattern recirculation zones just downstream of the crossroad for this four branch crossroad ow. In the x and y directions, symmetrical boundaries are used so that analysis can be focused on one direction only. The inlet discharges are set to Q in,x = Q in,y = 0.075m 3 .s -1 and outlet water height is h out,x = h out,y = 0.1m so that Fr outlet = 0.5 ensuring uvial ow conditions over the whole domain -inlet Froude number is uvial and there is no hydraulic jumps. The Strickler coecient is set to K = 100m 1/3 .s -1 in order to be consistent with the study in chapter 5. Initial water surface elevation is set to 0.1m and velocity in x and y directions are set to 0m.s -1 . The modeling is performed using Telemac2D with a nite element method with a kε turbulence model. Grid size is set to 0.06m (ensuring convergence) using an unstructured triangular mesh (see gure 6.7b). The parameters are listed in table 6.1.

Simulated results water height and velocity eld are presented respectively in gures 6.8a and 6.8b. Water height along the line A-B-C-D in x direction is also presented (see gure 6.8c). This water height prole along line A-B and C-D is the averaged water height perpendicular to the street direction. Between B and C, the plotted values correspond to the computed water depth values at the cell corner being in a 0.03m buer around the segment BC.

Water height decreases from inlet to outlet and a sharp decrease occurs across the crossroad (see gure 6.8c, between B-C). After the crossroad, averaged water height rst decreases and then increases along the ow direction, which corresponds to the eect of the recirculation zone contracting the ow vein hence increasing ow speed. Indeed, the shape of the recirculation

9.81m.s -2 Q in,x Inlet discharge in x direction 0.075m 3 .s -1 Q in,y
Inlet discharge in y direction 0.075m

Characterization of the recirculation zone

Three methods are presented in this section to quantify the dimensions of recirculation zone.

Q in,x Inlet discharge at x direction 0.012m 3 .s -1 0.075m 3 .s -1 ±72.5% Q in,y
Inlet discharge at y direction 0.012m 3 .s -1 0.075m 3 .s -1 ±72.5% w out,x

Outlet weir height at x direction 0.02m 0.08m ±60% w out,y

Outlet weir height at y direction 0.02m 0.08m ±60%

the computed value in a circle of radius 0.033B having its center in (r 0 , 0.167B). Indeed, close to a distance of 0.167B to the street side, the longitudinal velocity is barely constant (see gure 6.8d). B s is thus computed as:

B s = B - Q u s × h s (6.14)
Using this method, the obtained B s = 0.5747m. In this section, B s = 0.52m using the zerovelocity method used as reference. The dierence can be explained because the estimation of u s is not exactly the averaged velocity outside the recirculation zone. A correction factor C is thus introduced: B s,c = CB s where B s is estimated using equation (6.14), C = 0.9 and B s,c is the nal width of the recirculation area.

For the sake of clarity, B s,c will also be denoted as B s in the following sections. In the following sections, the dimensions B s and L s of the recirculation will be estimated using the approximated zero-velocity method .

GSA of a full 2D SW model applied to crossroad modeling

Global sensitivity analysis (GSA) used to quantify the inuence of input parameters on model outputs has been studied at the district scale in chapter 5. In this section, GSA is applied at a more local scale on a four branch crossroad. The sensitivity of the hydraulic variables and recirculation dimensions to input parameters is assessed using GSA as described in section 5.1.2.

Variation range of input parameters

The geometry of the four branch crossroad has been presented in gure and an unstructured 2D triangular mesh is used for simulation (see gure 6.7b). Boundary conditions and their variation ranges are based on the experiment provided by [START_REF] Nanía | Experimental study of subcritical dividing ow in an equal-width, four-branch junction[END_REF] (see table 6.2). Q in,x and Q in,y are modied in the range 0.012m 3 .s -1 to 0.075m 3 .s -1 corresponding to a perturbation of 72.5%. Experimentally, the downstream water height is controlled by weirs of variable height (w out,x and w out,y ) in the range 2cm -8cm corresponding to a perturbation of 60%. For the sake of simplicity, the simulation is performed with imposed downstream water height perturbed of 60% around the nominal value of 5cm. Friction coecient K is set to 100m 1/3 .s -1 with a variation range of 50% as done in chapter 5. The parameters for the nominal simulation are listed in table 6.3.

GSA on the four branch crossroad ow

With the conguration of the four branch crossroad described in section 6.4.1, the global sensitivity analysis (GSA) is conducted to assess the inuence of the input parameters on the outputs of interest (namely spatially varied water height and outlet discharge). Preliminary tests using the nominal values and dierent perturbations of the outlet water heights h out,x and h out,y result

Q in,x Inlet discharge in x direction 0.0435m 3 .s -1 ±72.5% Q in,y Inlet discharge in y direction 0.0435m 3 .s -1 ±72.5% h out Outlet water height 0.05m ±60% K Strickler coecient 100m 1/3 .s -1 ±50%
in highly variable hydrodynamics. Indeed, incoming water by the outlet streets with the highest imposed water depth is observed. Computing the sensitivity with such a variable response surface would require an important number of runs. To avoid too expensive computations, only four input parameters are tested: both inlet discharges in x and y directions (Q in,x and Q in,y ), the friction coecient K and h out the outlet water height for both x and y directions.

The nominal values for all the tested parameters and their variation ranges are listed in table 6.4. 2000 Monte Carlo simulations are conducted with input parameter sets randomly sampled following the chosen variation ranges. Figure 6.9 presents the Sobol index of the simulated water height to the tested parameters.

Maps of the water height variance to the inlet discharges Q in.x and Q in,y are symmetrical with main eects in the inlet channels and at the crossroad. Then they decrease from crossroad to outlet as explained by Chen et al. (2018). Interestingly, the upstream discharge Q in.x explains 35% of the variance of the street west (corresponding to the street in which the discharge is injected) and up to 55% for the lateral upstream street (north street). This shows that the discharge of west street creates a backwater control on street north as the ow regime is subcritical on the whole computed domain. A local peak of sensitivity can also be identied where the ow coming from the west street and deviated by the north discharge meets the upstream part of the left side of the south street. Similar trends are visible for Q in,y .

Concerning outlet water height (see gure 6.9), they have biggest inuence at outlet and decreases from downstream to upstream. However, their inuence is limited at channels downstream of the crossroad and has barely no eect upstream of and into the crossroad. Friction has similar eect as inlet discharge and also decreases from upstream inlet to downstream outlet.

Interestingly, friction has big inuence at recirculation zone both in x and y direction just after the crossroad. This may be explained by the contraction eect caused by the recirculation zone, which caused the increase of ow velocity.

The sum of all Sobol indices is greater than 0.9 indicating no signicant interaction eects between the tested parameters for this is completely subcritical ow.

Sobol indices of outlet discharges are presented in gure 6.10. Results show that the sum of In what follows, the parameterization of the code is presented, including the calibration of porosity function φ and roughness. The application of the code on a four branch crossroad is presented with and without porosity.

Parameterisation of the code Flood1D2D

The code Flood1D2D is expected to reproduce the ow patterns both in streets and crossroad.

As Flood1D2D is parameterized using both a Strickler coecient in all cells and a porosity in the 1D cells, this parameterization has to be dened prior to the computation. The Strickler coecient can be classically estimated based on the wall materials. The denition of the porosity represents the proportion of the street width available for the ow vein (see section 3.3.1):

φ (s) = 1 -B recirc (s) /B, B recirc (s) being the width of the recirculation at s where s is the curvilinear axis along the street (s = 0 being located by convention at the junction between the upstream crossroad and the street). Three dierent approaches are proposed to estimate B recirc (s):

The recirculation shape proposed by Gurram et al. (1997):

B recirc (s) B s = sin 90 • s L s /2 2/3 (6.15)
where B s is the maximum width and L s is the total length of recirculation zone.

An hypothetical shape assumed to be a second order polynomial: For every formulation, the recirculation shape can be dened using only two parameters: B s and L s . For similar B s = 0.26m and L s = 3.3m, the three proposed shapes are plotted in gure 6.13. One approach would have been to estimate B s and L s using Best and Reid (1984); Gurram et al. (1997); [START_REF] Kasthuri | Discussion of separation zone at open-channel junctions by James L. Best and Ian Reid (november, 1984)[END_REF] empirical models. However, those models are proposed for three branch junction ows with a discharge ratio parameter q smaller than unity for every experiment that is greater than 1 for the four branch crossroad of [START_REF] Nanía | Experimental study of subcritical dividing ow in an equal-width, four-branch junction[END_REF].

B recirc (s) = as 2 + bs + c (6.16) such as B recirc (0) = B recirc (L s ) = 0 and B recirc (L s /2) = B s hence a = - 4B s L 2
The Telemac2D results have been preferred to estimate those parameters.

Results comparison using Flood1D2D with and without porosity

In this section, Flood1D2D results using various parameterizations are compared on the conguration used in section 6.3.1. The four branch crossroad presented in gure 6.7a is used to show the inuence of the porosity parameterization. The crossroad is meshed using 2D cells and the four streets are meshed using 1D cells (see gure 6.14). In the x and y directions, symmetrical boundary conditions are used with inlet discharges Q in,x = Q in,y = 0.075m 3 .s -1 and outlet Figure 6.13: Distribution of porosity φ after crossroad both in x and y directions Figure 6.14: 1D2D coupling mesh on four branch crossroad (blue is 1D and red is 2D mesh).

Right is a zoom on crossroad.

water height h out,x = h out,y = 0.1m, which listed in table 6.1 as section 6. Water depth proles along the line A-B-C-D in x direction are compared with the 2D reference prole. A uniform roughness K = 100m 1/3 .s -1 is applied both on 1D and 2D cells. Water depth results are presented in gure 6.16. The water depth proles using both the Gurram et al. (1997) and the polynomial relationships are very similar along the whole A-D prole. Downstream of the crossroad (x > 3.5m), all the 1D eective water depth proles present similar trends including a ow prole contraction as the 2D prole (with a local minimum at x ≈ 5.1m) with more or less good agreement depending on the ow zones. Piecewise relationship produces a prole that is closer from the reference, especially after the maximum contraction point compared with the two others. However, at street inlet just downstream the crossroad, the water depth prole is still lower than the 2D prole. The energy loss modeled in 2D is not fully accounted by the 1D model parameterized with a uniform roughness especially downstream of the crossroad. Therefore, distributed roughness is considered in what follows to improve the performance of the 1D2D model.

Inuence of non-uniform Strickler coecient

To account for 2D energy loss caused by 2D ow feature in 1D2D model and improve the agreement between the ow proles, a distributed roughness is used in 1D2D modeling. The whole domain has been divided into several sections: At crossroad, which is meshed using 2D grid, the Strickler coecient is set to K = 100m 1/3 .s -1 without modication;

For the the street portion located upstream of the crossroad, and the one downstream of the maximum contraction point, where uniform ow is dominating, the Strickler coecient is set to K = 100m 1/3 .s -1 without modication; Figure 6.16: Comparison of simulated water height between Telemac2D and Flood1D2D using φ = 1, trigonometric, polynomial and piecewise relationships Figure 6.17: Comparison of simulated water height between Telemac2D and Flood1D2D using uniform and distributed roughness At section just downstream of the crossroad but before the maximum contraction point, which is characterized by 2D streamlines curvature an eective modeling is obtained by increasing the Strickler coecient to K = 60m 1/3 .s -1 . Figure 6.17 shows the calculated water height using distributed roughness compared with uniform roughness. Result show that the curve tting has been improved especially at inlet channel using distributed roughness to account for the extra energy loss caused by 2D ow features.

Conclusion

The proposed model uses an eective 1D approach with a porosity able to account for ow contraction, recall acceleration and water depth decrease, due to a recirculation zone. Simulating such a hydraulic behavior may be impossible to perform with friction (energy dissipation) only for which water depth steadily decreases from upstream to downstream. The previous calibration of the porosity shape and of a spatially variable roughness shows that Flood1D2D is able to reproduce a 2D reference. However there may exist multiple sets of those parameters that could produce comparable ow lines. In the following section, the sensitivity of water height and discharge distribution to recirculation dimensions is assessed.

GSA of Flood1D2D using porosity parameterization

In this section, global sensitivity analysis (GSA) is applied to all the parameters of Flood1D2D (roughness and recirculation zone dimensions) to assess their inuence on water height and discharge distribution. In Flood1D2D, the recirculation zones are parameterised with the porosity function φ (s) = 1 -B recirc (s) /B where B recirc (s) is the width of the recirculation at a distance s of the crossroad.

The piecewise polynomial/linear relationship is chosen to reproduce the recirculation shape.

As discussed at section 6.5.2, porosity function φ implemented into the code Flood1D2D to account for the eect of the recirculation φ min = 1 -B s,φ B = 1 -Bs 2B and the length of the recirculation L s = 3r 0 .

The GSA is used to estimate the sensitivity of water depth and discharge to Q in , h out , K and the recirculation dimensions (B s and r 0 ), in the four branch crossroad conguration described in section 6.3. The variation ranges for Q in , h out , K are equal to the one used for the 2D GSA (see section 6.4) and the one determined from 2000 Monte Carlo simulations described in section 6.4 for B s and r 0 .

GSA on main controls and recirculation dimensions

In this section, 8 parameters are tested on the four branch crossroad ow, including four main control parameters (Q in,x , Q in,y , h out and K) and four parameters on recirculation dimensions (B s,x , r 0,x , B s,y and r 0,y ). The nominal parameters values and their variation ranges are listed in table 6.6, 5000 runs are performed in order to sample extensive the parameter space.

Sobol indices of outlet discharges to all input parameters are presented in gure 6.18. Results show that the sum of all Sobol indices is close to 1, indicating very few interaction eect between parameters in explaining discharge partition caused by the crossroad. For outlet discharges, inlet discharges (Q in,x and Q in,y ) have dominant inuence compared with the other parameters that can thus be neglected. For outlet discharge at x (respectively y) direction, Q in,x (respectively Q in,y ) has dominant inuence. Compared to a full 2D model (see gure 6.10), the inuence of the outlet water height on discharge distribution is signicantly reduced for Flood1D2D. The indices is around 0.8 and less than 1. The friction coecient explains at most 10% of the water height variance and reaches 16% where the ow vein has a reduced width (thus corresponding to a higher velocity zone). The sensitivity of the water height to the height at the downstream boundary decreases from 70% at the downstream end of the streets to less than 10% in the crossroad and upstream. The water height is sensitive to the west upstream discharge Q in,x .

This sensitivity is (i) smaller than 10% in the south street, (ii) between 45% and 10% from the upstream to the downstream end of the east street (iii) equal to 35-40% in the west street and (iv) equal to 45-55% in the north street. The northern part of the crossroad is sensitive to Q in,x at barely 55% whereas the sensitivity in the south west corner decreases to few percent. The Figure 6.20 shows the Sobol index of water height to recirculation dimensions (B s,x , r 0,x , B s,y and r 0,y ). B s,x the cross sectional recirculation width (respectively B s,y ) has an eect on the water height only at the north east corner of the crossroad (respectively south west) and in the outlet channel at x direction (respectively y direction). In both of the downstream streets, the sensitivity to B s,x (respectively B s,y ) appears to be minimal at the nominal value of r 0,x (respectively r 0,y ). The variance of the water height to r 0,x (respectively r 0,y ) is nil except in the east (respectively south) street. The maximum of sensitivity is located around the nominal value of r 0 . Compared with main control parameters (Q in,x , Q in,y , h out and K), recirculation parameters have limited inuence on water height. However the inuence of ow contractions in each streets has a clear eect on water height upstream, within the crossroad and with a maximal sensitivity at the angle from which the recirculation, hence the contraction is represented.

GSA on recirculation dimensions and roughness

In this section, the main control parameters (Q in,x , Q in,y , h out ) are kept constant to their nominal values in order to exhibit the inuence of recirculation dimensions and friction on the four branch crossroad ow features. Five parameters are tested including B s,x , r 0,x , B s,y and r 0,y and roughness K. The boundary conditions are listed in table 6.3. The nominal values of recirculation dimensions and their variation ranges for sampling are listed in table 6.7. The Sobol index on water height and outlet discharges are presented in gures 6.21 and 6.22.

Figure 6.21 illustrates the sensitivity of the water height to the ve input parameters and their sum. The sum of all Sobol indices is close to 1 except at the zones after the crossroad as explained at section 6.6.1. B s,x (respectively B s,y ) has eect on the inlet channels and on the east (respectively south) face of the crossroad. For r 0,x and r 0,y , they only have inuence at porosity variation zone. For friction coecient, it has main eect at inlet channels before the crossroad and decreases from upstream to downstream as explained by Chen et al. (2018). In the downstream streets, friction coecient is higher in the low porosity zones for which the velocity is increased. the discharge and the water at the downstream end of the last streets. PorAS Riemann solver is used for ux computation. In every conguration, the simulation is run during 1000s to ensure steady state ow over the whole domain.

This geometrical conguration is used to highlight the sensitivity of Flood1D2D to three The porosity in the streets 'E3E4' and 'E3F3' are weighted in order to inuence the ow repartition at the 'E3' crossroad.

Characterization in K street /K cross

The model is parameterized using two uniform values of the Strickler coecient: K street in streets and K cross in crossroads. Each parameter is taken in the set of values [25,50,75,100,125,150] and all combinations are tested, that is 36 in total. The relative variation of the water depth in cell i h i is dened as:

h i = 100 × max h k i -min h k i h exp (x i , y i ) (6.20)
where h exp (x i , y i ) represents the experimental water depth at the coordinate of the gravity center of cell i, min h k i and max h k i are the minimal and the maximal water depth in cell i computed in simulation k. h i is presented in gure 6.24 and is smaller than 6% showing that the water depth is barely not inuenced by the Strickler coecient as highlighted in section 6.6.

The street outlet discharge is compared to the experimental value on gure 6.25. For each street, the rst bar corresponds to the experimental value. Whatever the parameterization of the model, each discharge owing out from streets E and F (respectively 3 and 4) is systematically under-(respectively over-) estimated. Interestingly, K street seems to impose an averaged street discharge value (look for instance the shift in the outlet discharge for street F when K street varies from 25 to 50). K cross has a smaller impact on outow discharge repartition (with always the same trend around the value imposed by K street ).

It appears that Flood1D2D parameterized only with a Strickler coecient is unable to correctly reproduce the water depth prole (see section 6.5.2) and the downstream discharge repartition. The model is parameterized using a uniform value of the roughness coecient K = 75m 1/3 .s -1 since few sensitivity to both K street and K cross is found in the previous section on outlet discharges. A distributed porosity is dened for all streets downstream of crossroads following a parabolic prole: φ(s) = as 2 + bs + c f or 0 ≤ s ≤ 0.5 with φ(0) = φ (1/2) = 1 and φ(1/4) = φ r 1 elsewhere (6.21) where s is the adimensional curvilinear abscissa (0 (respectively 1) at the upstream (respectively downstream) end of each street). The model is parameterized, by adjusting a, b and c, with two values of φ r,l in large streets (F and 4) and φ r,n in narrow streets (E and 3). In order to sample Flood1D2D response surface, all the combinations of values of φ r,l and φ r,n in [0.25, 0.50, 0.75, 1.00] are tested, that is 16 in total. The relative variation of water depth is presented in gure 6.26, which ranges from 0 to 100.

The important variations of the water depth are localized in streets E4F4, F3G3 and F4G4.

Hydraulic jumps are localized in those streets and move at a small velocity, which claims for quasi-steady state results. Increased simulations would have been required but would have leads to unaordable computational costs.

The outlet street discharges and those in the four streets between crossroads are compared to the experimental values on gure 6.27 and 6.28. For each street, the rst bar corresponds to the experimental value. For each street and all parameter sets tested, there are both under and In this section, the model is parameterized using a uniform roughness K = 75m 1/3 .s -1 and distributed porosity as described in section 6.7.3: φ large is applied to large streets, φ narrow The relative variation of water depth is presented in gure 6.29, which ranges from 0 to 100. The outlet street discharges and those in the four streets between crossroads are compared to the experimental values on gure 6.30 and 6.31. For each street, the rst bar corresponds to the experimental value. Figure 6.30 shows that the ratio parameter r allows to slightly improve the discharge repartition between streets E3F3 and E3E4. Interestingly, the outlet discharge of each street is almost kept constant with varied street discharges combining dierent φ large and φ narrow (whatever the ratio parameter r). It highlights that the mechanism of discharge redistribution into the street networks. The discharge distribution at the district scale is Q in dependent but results in a comparable outlet discharge partition.

The issues related to the calibration of spatially distributed K and φ parameters should be further studied, especially equinality problems in the light of their inuence on discharge repartition at large scale in branched networks.

Conclusions

In this chapter, the application of Flood1D2D at local and district scale has been studied for urban ood modeling. Result of this chapter shows the ability of Flood1D2D using parameter porosity φ in reproducing 2D ow lines and the function of recirculation zone on water height and discharge distribution at local and district scales.

Firstly, the sensitivity of friction slope calculation to recirculation eect parameterization is assessed and provide advice for choosing dierent hydraulic radius formulations. The friction slope formulation is signicantly modied when recirculation eect is introduced.

Then, Flood1D2D is applied at local scale, a four branch crossroad. Simulated ow lines are compared with 2D simulation results (Telemac2D). Result shows the ability of Flood1D2D to account for the recirculation eect using porosity φ, which can't be explained using roughness only. The function of recirculation zone on water height and discharge distribution is also assessed 161 The new parameterization proposed for an eective urban ood modeling in 1D2D, accounting for the ow contraction due to recirculation zones, with a porosity like approach is studied in chapter 5. An isotropic porosity φ, implemented into the code Flood1D2D and it is parameterized from recirculation dimensions B s (maximum recirculation width) and L s (recirculation length) estimate from classical 2D simulations, assuming a simple geometrical shape. This new parameterization leads to correct 1D eective simulated ow lines in the contracted ow veins, which is impossible with a roughness only. The eect of recirculation zone parameters on the water height and discharge distribution simulated with Flood1D2D is assessed using GSA on a four branched 163 crossroad. B s is shown to have a bigger inuence on discharge distribution compared to L s and K. Finally, Flood1D2D is also applied at the district scale, for a branched network composed of several crossroads taken from ICube experimental dataset. The few parameter sets tested show the ability of the new parameterization, using both roughness and porosity, to inuence both water depth at the local scale and the discharge distribution at the global scale.

Perspectives

The purpose of this thesis was to develop a parsimonious 1D2D coupling method adapted to urban district conguration, taking into account recirculation eects on urban ood ow patterns. The application of the code, at a larger scale, for example at the scale of the full ICube urban ood rig should be investigated along with the question of spatialized parameters calibration on existing datasets. Moreover, further studies could tackle the problem of unsteady urban ood ows and the necessary parameterization complexities by paying attention to potential equinality problems. The mechanism of discharge distribution, including transient phenomena, at the crossroad scale would also be interesting for further research as the generalization at the district scale.

Modèle effectif par une approche de Saint-Venant pour les écoulements complexes lors d'inondations urbaines

Résumé

Les inondations représentent le premier danger naturel sur chaque continent, avec des préoccupations particulières centrées sur les inondations urbaines pour l'urbanisation sans précédent, qui augmente la vulnérabilité des sociétés humaines aux inondations. Pour les difficultés d'accès aux données en temps réel, la méthode numérique semble être une approche puissante pour la prévision des inondations et l'atténuation des dommages. Dans cette thèse, un nouveau logiciel Flood1D2D pour la modélisation des crues urbaines a été proposé en utilisant le maillage de couplage 1D2D, avec 1D dans la rue et 2D appliqué au carrefour. Le logiciel est conçu sur la base de l'installation située dans ICube Strasbourg et validée à l'aide de cas de tests synthétiques. La nouveauté du modèle est de rendre compte de l'effet de recirculation en utilisant la fonction de porosité, et de reproduire la convexité dans les profils de profondeur le long de la rue en aval de chaque carrefour où une zone de recirculation apparaî t. Une analyse de sensibilité basée sur la variance est également réalisée pour les modèles classiques d'eau peu profonde 1D et 2D pour l'étalonnage de modèles et la collecte de données.

Mots clés : Inondation urbaine, 1D / 2D, porosité, analyse de sensibilité, équations de Saint-Venant

Résumé en anglais

Flooding represents the first natural hazard on each continent, with special concerns are focused on urban flooding for the unprecedented urbanization, which increases the vulnerability of human societies to floods. For the accessing difficulties of real-time data, numerical method seems to be a powerful approach for flood forecasting and damage mitigation. In this thesis, a new software Flood1D2D for urban flood modeling has been proposed using 1D2D coupling mesh, with 1D on the street and 2D applied on the crossroad. The software is designed based on the rig located in ICube Strasbourg and validated using synthetic test cases. The novelty of the model is to account for the recirculation effect using porosity function, and it can reproduce the convexity in water depth profiles along the street downstream each crossroad where a recirculation zone appears, which can't be accounted using roughness only. Variance based sensitivity analysis is also performed for classical 1D and 2D shallow water models for model calibration and data collections.

Keywords : Urban flood, 1D/2D, porosity, sensitivity analysis, Saint-Venant equations

Figure 1

 1 Figure 1 Topologie du maillage 1D2D considéré

Figure 2

 2 Figure 2 Dénition des dimensions de la zone de séparation d'après Schindfessel et al. (2017) :

Figure 4

 4 Figure 4 Gauche : schéma du dispositif expérimental (dimensions en m d'après Soares-Frazão and Zech (2002)) ; Droite : maillage 1D2D utilisé pour la modélisation avec Flood1D2D

  Flood1D2D

Figure 7

 7 Figure 7 Décomposition des sensibilités locales des hauteurs d'eau aux quatres paramèters testés -expérience sans pente. Numérotation des points en fonction de la distance hydraulqiue au coin nord-ouest.

Figure 8

 8 Figure 8 Représentation du maillage 1D2D employé avec un zoom sur le carrefour (1D en bleu et 2D en rouge).

4 . 2

 42 Application de Flood1D2D à l'échelle d'un réseau de carrefours

Figure 10

 10 Figure 10 Comparaison entre les hauteurs d'eau de référence (Telemac2D) et celles simulées avec Flood1D2D pour diérentes paramétrisations de la porosité

Figure 11

 11 Figure 11 Dispositif expérimental et illustration de la zone modélisée (gauche) et maillage 1D2D (droite).

Figure 12

 12 Figure 12 Sensibilité du débit dans les rues aval à K street et K cross

Figure 14

 14 Figure 14 Sensibilité du débit dans les rues aval à φ r,n et φ r,l

I 1

 1 'm really grateful to China Scholarship Council for the nancial supporting in the past three and half years, also I'd like to thank INSA de Strasbourg for providing me a wonderful research place. At last not the least, I want to express my gratitude to my family, especially my father, my mother, my old sister and my brother in law, for their seless support on my study career. XXI 7 General conclusions and perspectives 7.1 General conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Context and problematics Flooding represents the rst natural hazard on each continent, and it caused in Europe around 100 billion euros of damage between 1986 and 2006 (e.g. Moel et al. (2009)). In the context of climate change and its possible eects on rainfalls regimes and extremes (Pachauri et al. (2014)), an unprecedented urbanization of oodplains increases the vulnerability of human societies to oods. They are generally caused by intense rainfalls which can lead to a temporary saturation of the urbanized catchment and drainage systems. Such oods can propagate through streets

  ), among others). Fine 3D hydrodynamic models, based on Navier-Stokes equations, can also be used in order to investigate hardly measurable complex turbulent ow patterns such as those involved at the local scale of hydraulic intersections (e.g.[START_REF] Momplot | Typology of the ow structures in dividing open channel ows[END_REF]) or even ow repartition at the scale of branched networks(Finaud-Guyot et al. (2018)). Such calculations, generally requiring a turbulent closure and wall function parameterization, yet remain very computationally expensive and ood models dedicated to operational use are commonly depth averaged (2D or 1D shallow water -SW-equations). Inundation maps, submersion levels and ow velocities are generally performed with a 2D SW model which is a good compromise in terms of realism and computational eciency between 3D and cross sectional averaged 1D. However if the domain size increases from building blocks to a whole district or city, 2D simulations remain very computationally demanding. Facing this, 1D2D coupling(Finaud-Guyot (2009); Finaud-Guyot et al. (

  model is validated in chapter 3 on reference ow cases including synthetic test cases and experimental datasets with various combinations of porosity, bottom elevation, friction and ux computation between 1D and 2D cells. Chapter 4 presents the application of GSA of classical shallow water models applied at the urban district scale using datasets of the ICube urban ood experimental rig. Chapter 5 tests the ability of Flood1D2D in reproducing local ow features at the street scale and their inuence on discharge repartition at the block scale. The inuence of input parameters like recirculation dimensions on local scale ow patterns is also quantied using GSA.

Figure 2 . 1 :

 21 Figure 2.1: Schematic view of the great water cycle with an emphasis on the continental one (source: https://water.usgs.gov/edu/watercycle.html, access on April 03, 2018).

Figure 2 . 2 :

 22 Figure 2.2: Pictures of urban ooding during severe ood

Figure 2 . 3 :

 23 Figure 2.3: Hydrological processes about physical phenomena involved in an urbanized catchment taken from Salvadore et al. (2015)

Figure 2 . 4 :Figure 2 . 5 :

 2425 Figure 2.4: Illustration of a general conceptual model for an open channel conuence (Creëlle et al. (2016))

Figure 2 . 6 :

 26 Figure 2.6: Shape index and maximum contraction coecient µ on separation zone: left, shape index of separation zone; right, contraction coecient at maximum ow constriction taken from Hsu et al. (1998)

Figure 2 .

 2 Figure 2.7: Illustration of a general concept model for a dividing open channel ow from Neary

  [START_REF] Huang | Three-dimensional numerical study of ows in openchannel junctions[END_REF] simulated T-conuence ow using 3D Reynolds-averaged Navier-Stokes equations (cf. next chapter, equation (3.32)) to reproduce the features of separation zone observed by Weber et al. (2001). Schindfessel et al. (2017) used validated large-eddy simulation (LES) to study the ow in conuences with four dierent cross-sectional shapes. Results show that the dimensions of the separation zone are signicantly inuenced by the cross-sectional shapes.

  developed a threedimensional (3D) numerical model for predicting turbulent ows through lateral intakes with rough walls. Reynolds-averaged Navier-Stokes equations are solved with the isotropic turbulence closure model of kω. The complexity of ow in the branch is increased with the increase of bed roughness, which induced the increased number of eddies and the width of the separation zone. Hsu et al. (2002) conducted an experiment for a subcritical open channel dividing ow over a horizontal bed equipped with equal width in angle 90 • . Based on experimental observations, the contraction coecient µ = Bs

  [START_REF] Ramamurthy | Numerical and experimental study of dividing openchannel ows[END_REF] study T-branch dividing ows experimentally and numerically using 3D Navier-Stokes (NS) equations with kω turbulence model. A qualitatively correct

Figure 2 . 8 :

 28 Figure 2.8: Contraction coecient in branch channel taken from Hsu et al. (2002)

Figure 2 .

 2 Figure 2.9: Illustration of a general conceptual model for a four-branch intersection from Rivière et al. (2011)

Figure 2 .

 2 Figure 2.11: Richelieu area of the city of Nimes with S 1 to S 11 the outlets from the area (Mignot et al. (2006))

Figure 2 .

 2 Figure 2.12: Upstream part of the physical model where the model city has been implemented (aligned city layout) (Testa et al. (2007))

  equations. The coupling terms in the 2D equations are the incoming characteristics at open boundaries. The coupling algorithm is based on an optimal control process. The superposition and coupling between 1D and 2D equations with non-at topographies is solved by[START_REF] Fernandez-Nieto | Coupling superposed 1d and 2d shallowwater models: Source terms in nite volume schemes[END_REF].

  ; Soares-Frazão et al. (2008); Guinot (2012a); Velickovic et al. (2017)) or anisotropic (Sanders et al. (2008); Guinot (2012a); Chen et al. (2012a,b); Lhomme (2006); Özgen et al. (2016a,b); Guinot et al. (2017a); Bruwier (2017); Bruwier et al. (2017)).

Figure 2 .

 2 Figure2.14: REV in a 3D porous medium and denition of bounding surfaces A f f (surface between uid and uid), A f s (surface between uid and solid) and A ss (surface between solid and solid) taken from[START_REF] Velickovic | Macroscopic modeling of urban ood by a porosity approach[END_REF]

Figure 2 .

 2 Figure 2.15: Left: Schematic urban geometry involving buildings (Ω b ), ow vein (Ω m ) and stagnant water (Ω s ). Right: Equivalent representation in a computational cell. Illustration from Guinot (2012a)

Figure 2 .Figure 2 .

 22 Figure 2.16: Schematic representation of the computational cell highlighting the non-isotropic conveyance porosity dened as the void portion of each edge. Illustration from Sanders et al.(2008)

  [START_REF] Kim | Urban ood modeling with porous shallow-water equations: a case study of model errors in the presence of anisotropic porosity[END_REF] studied the errors associated with porosity models applied in urban ood modeling. Three dierent types of errors are identied: structural error due to the limitations of the employed equations, scale error produced by relatively too coarse grid, and porosity error corresponding to inadequation between the porosity equations employed and modeled sub-grid scale obstructions. Porosity errors are shown to be greater than scale errors but less than structural errors. Isotropic porosity model compared to anisotropic models leads to bigger error on the computed water depth and velocity. The laters being barely as accurate that the ne grid shallow water models. Guinot (2017b) proposes a comparison of three types of porosity models: Single porosity model (SP), Integral porosity model (IP) and Dual Integral porosity model (DIP) against various periodic computational domains. The DIP model is shown to produce the most accurate results compared to both the SP and the IP model. However, all the tested porosity models are shown to reproduce the ux closure when the wave propagates in direction not aligned with the streets.

Figure 2 .

 2 Figure 2.18: Schematic illustration of uncertainty quantication (UQ) and sensitivity analysis (SA) taken from Vitturi et al. (2016). Please note that N s refers to the number of dierent input parameter sets (number of simulations performed), sampled here on a latin hypercube

Figure 2 .

 2 Figure 2.19: Workow for the application of sensitivity analysis (SA), choices to be made and recommended practice for their revision made by[START_REF] Pianosi | Sensitivity analysis of environmental models: A systematic review with practical workow[END_REF]

Flood

  inundation maps are commonly generated with a 2D shallow water (SW) model consisting in depth averaged Navier-Stokes equations ( Hervouet (2007); Arrault et al. (2016); Monnier et al. (

  curve uncertainty Delrieu et al. (2005); Paris et al. (2016); Savage et al. (2016)) or simulated by a hydro-meteorological chain involving meteorological model uncertainties[START_REF] Vié | Cloud-resolving ensemble simulations of mediterranean heavy precipitating events: uncertainty on initial conditions and lateral boundary conditions[END_REF]

  propose local sensitivity equations for the 2D steady state SW equations without shocks and provide some guidelines for model calibration and validation. These sensitivity equations are also derived and implemented for 1D Saint Venant model[START_REF] Delenne | Sensitivity of the 1d shallow water equations with source terms: Solution method for discontinuous ows[END_REF]). Local sensitivities derived with the adjoint method (2D SW DassFlow model), and involving a cost function, are presented in the case of a high resolution model of a ood on a 2km reach of the Lèze River[START_REF] Monnier | Inverse algorithms for 2d shallow water equations in presence of wet dry fronts: Application to ood plain dynamics[END_REF]). The authors nd higher sensitivities of water depth to bathymetry and roughness downstream (subcritical regime) of the observation points in the oodplain and in the main channel, roughness sensitivities are higher in the main channel.Recently variance based SA has been applied to 2D hydraulic models in ooding conditions[START_REF] Abily | Spatial global sensitivity analysis of high resolution classied topographic data use in 2d urban ood modelling[END_REF];[START_REF] Savage | Quantifying the importance of spatial resolution and other factors through global sensitivity analysis of a ood inundation model[END_REF]).[START_REF] Abily | Spatial global sensitivity analysis of high resolution classied topographic data use in 2d urban ood modelling[END_REF] present a spatial SA approach of a 2D SW model based on high resolution digital elevation model (DEM). Sensitivity maps of simulated water depth to uncertain parameters including topography are presented for the last 5km of the Var valley, France -November 1994 ood. For a 50km² rural oodplain in Sicily,[START_REF] Savage | Quantifying the importance of spatial resolution and other factors through global sensitivity analysis of a ood inundation model[END_REF] highlight the sensitivity of simulated ood extent to inow discharge during ood rising limb then the channel friction parameter during ood peak and the oodplain friction parameter during recession. First order sensitivity of both maximal water depth and ood

Figure 3 . 1 :

 31 Figure 3.1: Finite control volume xed in space fromAnderson and Wendt (1995)

Figure 3 . 2 :

 32 Figure 3.2: Innitesimally small volume of moving uid. Only the forces in the x direction are shown. Model used for the derivation of the x component of the momentum equation fromAnderson and Wendt (1995)

Figure 3 . 3 :

 33 Figure 3.3: Tilted channel with a at bottom adapted from Thual (2010)

  to the surface is proportional to the vector -→ ∇F = ∂F ∂x -→ e x + ∂F ∂y -→ e y + ∂F ∂z -→ e z . The kinematic boundary condition dF dt = 0 then reads:

  (3.53) Using continuity equation(3.52) and setting γ x → 0 (cos γ x → 1 and sin γ x = tan γ x = -∂z b ∂x ), equation(3.53) can be rewritten as:

Figure 3 . 4 :

 34 Figure 3.4: Scheme of control volume with dierent representation of porosity (Lhomme (2006))

Figure 3 . 5 :

 35 Figure 3.5: Schematic of control volume dΩ

Figure 3 . 6 :Figure 3

 363 Figure 3.6: Topology of the 1D2D coupling model: Γ 1D,W , interfaces of the 1D cell corresponding to walls; Γ 1D,BC , interfaces of the 1D cell corresponding to BC; Γ 2D,BC , interfaces of the 2D cell corresponding to hydrodynamic BC; Γ 1D1D , interfaces between two 1D cells; Γ 2D2D , interfaces between two 2D cells; Γ 1D2D , interfaces between 1D and 2D cells

Figure 3 . 9 :

 39 Figure 3.9: Conceptual view of the nite volume discretization: two elements Ω i and Ω j of D

  along x and y directions of the normal unit vector -→ n

  use of the numerical method adapted to each part of equation(3.85), a classical time splitting algorithm is used[START_REF] Maccormack | Computational eciency achieved by time splitting of nite dierence operators[END_REF]). The rst step is devoted to the computation of the hyperbolic part of equation(3.74) including the bottom and porosity source terms in order to achieve a well-balanced computation of the ux (F) n k . The second step focuses on the friction eect computation.

  are computed using a Riemann solver(either HLLC or PorAS (Finaud-Guyot et al. (2010b))). Such choice allows to preserve the well-balanced condition due to the bottom and porosity source term eect S 0 (U n i ) and S φ (U n i ) across interface k. The parameters of the Riemann solver (being the values of U, φ and z b in the left and right cells attached to interface k) are thus directly taken in the corresponding cells of the mesh: Ω i and Ω j .

3. 3 . 1

 31 Physical considerations on ow downstream crossroadsThis parameterization takes advantage of empirical ndings on the eects of 3D ow structures on local and global ow patterns in urban geometries. Modeling and experimental observations of ow hydrodynamic within complex branched networks highlight the presence of recirculation areas downstream of crossroad and their eects on global ow repartition (e.g.Mignot et al. (2008a,b); Finaud-Guyot et al. (2018), see details in section 2.2).

Figure 3 .

 3 Figure 3.10: Sketch of the recirculation zone on the 1D cell

Figure 3 .

 3 Figure 3.11: Sub-edges of 1D2D mesh for the eect of recirculation zone. For the sake of readability, a gap has been introduced between Ω 1D and Ω 2D

  .98) φ L (respectively φ R ) being the porosity in the cell on the left (respectively right) side of interface k and φ 1D being the porosity of the 1D cell attached to interface k.3.4 Conclusions/modeling methodologyIn this chapter, a new physical model based on 2D shallow water equations including porosity has been proposed. The original nite volume discretization, using a cut-cell technique between 1D and 2D interfaces, has been implemented into the code Flood1D2D. In the next chapter, Flood1D2D model will be validated on several ow cases where reference solutions, either analytical or experimental, are available.The numerical approach presented in chapter 3 has been implemented in a new program Flood1D2D developed during this PhD. This numerical tool is based on a classical nite volume approach (see section 3.2.2) and the proposed cut cell technique (see section 3.3) to solve the 2D shallow water equations with porosity (see equation(3.74)). As highlighted in section 3.3, the computation requires: (i) the computation of the ux through the interface between 2 cells while preserving the well-balancing property for both the bottom slope and the porosity variations;(ii) the computation of the friction eect; and (iii) a cut-cell technique at the junction between streets and crossroads.First, the eciency of the Riemann solvers are assessed on classical academic test cases, essentially to detect any mistake in their implementation. Next, the source terms computation corresponding to the friction eect is validated. Finally, the proposed cut-cell technique abilities are assessed on academic cases including porosity in view of urban ood modeling.4.2 Flux computationFlood1D2D being based on a nite volume approach, the hydraulic computations are based on a Riemann solver (cf. section 3.2.3) at each interface between the cells of the mesh. The code uses two classical Riemann solvers (HLLC(Guinot and Soares-Frazão, 2006) and PorAS(Finaud- Guyot et al. (2010b))) adapted to conguration of both bottom and porosity variations for computing the uxes through interfaces while preserving the well-balancing condition. The test cases presented hereafter aim to validate the correct implementation of those Riemann solvers in all possible congurations: 1D or 2D, steady-state or transient regime, various hydraulic regimes (sub, super or transcritical), with either smooth or abrupt variation of the porosity and the bottom elevation (one of them or both).

4. 2 . 1

 21 Water at rest with variable porosity and bottom elevation (T01)This test case aims at validating the ability of the code to compute wetting and drying front in case of stagnant water in a partly immerged domain including a variable porosity. The modeled domain is thus a rectangular channel with no friction and a spatially varied porosity and bottom elevation along the channel. A linear variation of both the porosity and the bottom elevation starts at x 0 = 10m over l 0 = 20m (see gure4.1 and table 4.1). An uniform water level (with no velocity) is set as an initial condition and is supposed to remain unchanged during the simulation.

Figure 4 . 1 :

 41 Figure 4.1: Test case T01 -Bottom and porosity variation along the channel

Figure 4 . 2 :

 42 Figure 4.2: Test case T01 -Comparison of Flood1D2D results (using either the HLLC or the PorAS Riemann solver) with the reference. Top left: water surface elevation; top right: water depth; bottom left: velocity; bottom right: unit discharge. Note that the computed water surface elevation (for both Riemann solvers) varies for x > 20m because the elevation returned by the code is the maximal value between the bottom and the free surface elevation (i.e. the bottom elevation when the water depth is nil). Only 1 point every 10 are plotted in space for the sake of clarity

Figure 4 . 3 :

 43 Figure 4.3: Diagram of variables denition for test case T02, top view along ow distance from Finaud-Guyot (2009)

Table 4 . 2 :

 42 Parameters for the test case with variable porosity (Initial water level over the domain 1m q upstream Prescribed unit discharge at the upstream boundary (x = 0m) 1m 2 .s -1 Prescribed water depth at the downstream boundary (x = 100m) h downstream for subcritical ow (T02a) 1m for transcritical ow (T02b) 0.1m

Figure 4 . 4 :

 44 Figure 4.4: Comparison of Flood1D2D results (using either the HLLC or the PorAS Riemann solver) with the semi-analytical reference for the test case T02a (subcritical ow regime). Only 1 point every 15 are plotted for the sake of clarity

Figure 4 . 5 :

 45 Figure 4.5: Comparison of Flood1D2D results (using either the HLLC or the PorAS Riemann solver) with the semi-analytical reference for the test case T02b (transcritical ow regime). Only 1 point every 15 are plotted for the sake of clarity

Figure 4 . 6 :

 46 Figure 4.6: Geometry denition for test case T03, side view along ow distance

Figure 4 . 7 :

 47 Figure 4.7: Comparison of Flood1D2D results (using either the HLLC or the PorAS Riemann solver) with the semi-analytical reference for the test case T03a (subcritical ow regime over a bump). Only 1 point every 5 are plotted in space for the sake of clarity

Figure 4 . 8 :

 48 Figure 4.8: Comparison of Flood1D2D results (using either the HLLC or the PorAS Riemann solver) with the semi-analytical reference for the test case T03b (transcritical ow regime over a bump). Only 1 point every 5 are plotted for the sake of clarity

Figure 4 . 9 :

 49 Figure 4.9: First test case for convergence analysis (T04a): x = 1m. Comparison of Flood1D2D results (using either the HLLC or the PorAS Riemann solver) with the semi-analytical reference (subcritical ow regime with porosity and bottom bump)

Figure 4

 4

Figure 4 .

 4 Figure 4.10: Second test case for convergence analysis (T04b):

Figure 4 .

 4 Figure 4.11: Third test case for convergence analysis (T04c):

Figure 4 .

 4 Figure 4.12: Test case for convergence analysis on total discharge (T04a, T04b and T04c).Comparison of Flood1D2D results (using either the HLLC or the PorAS Riemann solver) with the semi-analytical reference (subcritical ow regime with porosity and bottom bump). Top left, T04a; top right, T04b; bottom, T04c. For the sake of clarity, only 1 point every 10 (respectively 100) are plotted for T04b (respectively T04c)

Table 4 . 5 :

 45 Parameters for the one-dimensional dambreak problem (T05) Initial water depth on the left-hand side of the dam 10m Initial water depth on the right-hand side of the dam h

Figure 4 .

 4 Figure 4.14: Comparison of Flood1D2D results (using either the HLLC or the PorAS Riemann solver) with the analytical reference for the test case T05a (subcritical ow regime, left side) and T05b (transcritical ow regime, right side). Only 1 point every 50 are plotted for the sake of clarity

Figure 4 .

 4 Figure 4.15: Comparison of Flood1D2D results (using either the HLLC or the PorAS Riemann solver) with the semi-analytical reference for the test case T06a (subcritical ow regime). Only 1 point every 40 are plotted for the sake of clarity

Figure 4 .Figure 4 .

 44 Figure 4.16: Comparison of Flood1D2D results (using either the HLLC or the PorAS Riemann solver) with the semi-analytical reference for the test case T06b (transcritical ow regime). Only 1 point every 40 are plotted for the sake of clarity

Figure 4 .Figure 4 .

 44 Figure 4.18: Comparison of Flood1D2D results (using either the HLLC or the PorAS Riemann solver) with the analytical reference for the test case T08 (transcritical ow regime): top, x = 1m; middle, x = 0.1m; bottom, x = 0.01m. For the sake of clarity, only 1 point every 2 (respectively, 15 and 150) are plotted for test case T08a (respectively, T08b and T08c)

Figure 4 .

 4 Figure 4.21: Comparison of Flood1D2D results (using either the HLLC or the PorAS Riemann solver) for the test case T10 (subcritical ow regime with friction). Only 1 point every 20 are plotted for the sake of clarity

Figure 4 .

 4 Figure 4.22: Comparison of Flood1D2D results (using the PorAS Riemann solver) for the test case T11a (subcritical ow regime with friction). Only 1 point every 15 are plotted for the sake of clarity

Figure 4 .

 4 Figure 4.23: Comparison of Flood1D2D results (using the PorAS Riemann solver) for the test case T11b (supercritical ow regime with friction). Only 1 point every 15 are plotted

Figure 4 .

 4 Figure 4.24: Parameterization of test case T12: top, 1D2D mesh, blue is 1D mesh and red is 2D; bottom left, spatially varied bottom elevation z 0 ; bottom right, spatially varied porosity φ

Figure 4 .

 4 Figure 4.26: Map of the full domain with 1D and 2D mesh (blue is a 1D mesh and red is a 2D mesh) for test case T13: Zoom 1, zoom on mesh junction (x = 300m) -1D to 2D (same width); Zoom 2, zoom on mesh junction (x = 50m) -2D to narrower 1D; Zoom 3, zoom on mesh junction (x = 250m) -2D to narrower 1D; Zoom 4, zoom on mesh junction (x = 100m) -1D to larger 2D; Zoom 5, zoom on mesh junction (x = 350m) -2D to larger 1D

  (a) The map of water depth -h (b) Water depth along the channel from upstream to downstream (c) Fluxes at boundary conditions and interfaces

Figure 4 .Figure 4 .

 44 Figure 4.27: Flood1D2D results with varied channel width for the test case T13 (1D2D coupling)

Figure 4 .

 4 Figure 4.29: Comparison of Flood1D2D results (using the PorAS Riemann solver) for test case T13

4. 4 . 4

 44 Dambreak in channels with 90 • bend (T15) This test case aims to demonstrate the ability of the 1D2D coupling method implemented in Flood1D2D to correctly model a highly transient hydrodynamics. Flood1D2D is thus compared to an experimental dambreak in open channel with 90 • bend (Soares-Frazão and Zech (2002)).

Figure 4 .

 4 Figure 4.30: Channel with 90 • bend Top view left: dimensions in m (Soares-Frazão and Zech (2002)); Right 1D2D mesh used with Flood1D2D

Figure 4 .

 4 Figure 4.31: Comparison of Flood1D2D results using dierent grid sizes (dx = 0.1m, dx = 0.05m, dx = 0.025m and dx = 0.01m) with the experiment data from Soares-Frazão and Zech (2002): water height along the line A-B-C as illustrated in gure 4.30

Figure 4 .

 4 Figure 4.34 illustrates spatially varied water heights at dierent time steps: 3s, 5s, 7s and 14s.The simulated 2D results provided by Soares-Frazão andZech (2002) and results calculated by the code Flood1D2D using 2D and 1D2D coupling mesh are compared. The results show that the code Flood1D2D can reproduce the water depth measurements of Soares-Frazão andZech (2002)

Figure 4 .

 4 Figure 4.32: Comparison of Flood1D2D results (2D triangular mesh ( x = 0.025m) or 2D quadrilateral mesh ( x = 0.025m)) with the experiment provided by Soares-Frazão and Zech (2002): water height along the line A-B-C as illustrated in gure 4.30

  Several classical 1D and 2D test cases provided bySoares-Frazão and Zech (2002);Guinot and Soares-Frazão (2006); Finaud-Guyot et al. (2010b) are used with and without porosity. Numerical results are compared with analytical and experimental results. Several new test cases are also designed to test the 1D2D coupling functionalities and results are compared with full 2D results and experimental results when available. The ability of the code is fairly well demonstrated especially concerning: Open channel ows with and without porosity; Open channel ows in non at bottom with and without friction eect; Transient dambreak problem with and without porosity in 1D and 2D scale;

Figure 4 .

 4 Figure 4.33: Comparison of Flood1D2D results (2D ne ( x = 0.025m) or coarse ( x = 0.1m except in bend where is x = 0.025m) triangular mesh) with the experiment provided by Soares-Frazão and Zech (2002): water height along the line A-B-C as illustrated in gure 4.30

Figure 4 .

 4 Figure 4.34: Comparison of Flood1D2D results (2D water height): left, simulated water height provided by Soares-Frazão and Zech (2002); middle, 2D water height in coarse mesh ( x = 0.1m except in bend which is x = 0.025m); right, 1D2D water height in coarse mesh (see gure 4.30, right side) ( x = 0.1m except in bend which is x = 0.025m)

Figure 4 .

 4 Figure 4.35: Comparison of Flood1D2D results (2D or 1D2D mesh) with the experiment provided by Soares-Frazão and Zech (2002): water height along the line A-B-C as illustrated in gure 4.30. 2D mesh is located into two vertical lines

3 .

 3 In the third section, the sensitivity of model output to dierent street blockage locations is assessed. Dierent locations of blockage are described and their inuence to discharge distribution and local ow features are studied. 5.1 Chen, S., Garambois, P.-A., Finaud-Guyot, P., Dellinger, G., Mose, R., Terfous, A., Ghenaim, A., 2018. Variance based uncertainty quantication for 1d and 2d hydraulic modeling: an experimental urban ood case. Under review for Environmental Modelling & Software.

Figure 5 . 1 :

 51 Figure 5.1: (a) Experimental rig, (b) modeled (2D SW) velocity eld for Q = 80m 3 /h equidistributed between upstream west and north faces, (c) naming of the streets and (d) modeled output discharge comparison with experiment.

Figure 5 . 2 :

 52 Figure 5.2: Main steps of the methodology used for performing the Global Sensitivity Analysis (GSA) of 1D or 2D hydraulic models. Step 0 consists in choosing the parameters to test and dening their physical bounds hence dening parameter space. Each parameter is sampled randomly in step 1. Next, hydrodynamic model is run for each parameter set in step 2 and sensitivity indices are calculated for all those model realizations in step 3.

Figure 5 . 3 :

 53 Figure 5.3: Sobol index of water depth for 1D model estimated for parameters sampled on equal ranges of amplitudes various amplitudes ranging from ±5% to ±60% each parameter for street C.

  The spatial variations of Sobol indices of ten input parameters are presented in gure5.4 with parametric uncertainty ranges corresponding to experimental ones. In that case, the sum of rst order indices S i∈[1..10] is very close to one indicating very small higher order interactions between parameters. The results for both street C and 4 show that main ow controls over the tested uncertainty ranges are roughness, downstream water depth and upstream discharge. The remaining sensitivities of the seven lateral discharges are very small compared to the three other parameters. Therefore, for the sampled ranges those ten input parameters are studied separately in subsequent sections: three main parameters and seven lateral discharges.5.1.3.1.4 GSA on the main three parameters The spatial variation of Sobol indices of the main three parameters (upstream discharge, downstream water depth and friction) sampled in experimental uncertainty ranges are presented in gure 5.4. Results for both street C and 4

5. 1

 1 .3.1.5 GSA on lateral discharges We consider here the case where uncertainty sources are only the lateral discharges. The spatial patterns of Sobol indices are presented in gure 5.5. Their sum stays close to one along street C or 4 and again shows no interaction eects.

Figure 5 . 4 :

 54 Figure 5.4: Top: Comparison of the computed and measured water height proles and Sobol indices of ten (Middle) and three (Bottom) main input parameters. The results are plotted for streets C (Left) and 4 (Right).

5. 1 . 3 . 2

 132 Sensitivity analysis in 2D conguration 5.1.3.2.1 2D SW model set up and calibration 2D steady state SW simulations are performed for the whole experimental rig in order to better reproduce complex ow structures (cf. Arrault et al. (2016); Tran et al. (2016)). The boundary conditions are based on the experimental data-set listed in Appendix table A.1. Constant discharges and water heights are prescribed at

Figure 5 . 6 :

 56 Figure 5.6: Sobol indices of distributed roughness (For street C, K s = 26.5 m 1/3 .s -1 and K c = 9.5 m 1/3 .s -1 ; For street 4, K s = 28.5 m 1/3 .s -1 and K c = 9 m 1/3 .s -1 ).

  A sample size N s = 2000 ensures a good convergence of Sobol indices estimation in this 2D SW model conguration (cf. section 5.1.3.1.1). Sensitivity pattern of water depth is presented through 2D maps of Sobol indices along with their sum for the two inow discharge partitions (gure 5.7). The sum of all Sobol indices is close to 1 for subcritical ow zones (corresponding to Froude number below in gure 5.7) meaning water depth variance is perfectly explained by the rst-order Sobol indices of the tested parameters.

Figure 5 .

 5 Figure 5.7: 2D maps of Sobol indices for 80 m 3 /h: Left, inow discharge partition at west and north-face 50-50%; Right, inow discharge partition at west and north-face 80-20%. The circles highlight supercritical ow zones (Fr > 1).

Figure 5 . 8 :

 58 Figure 5.8: Decomposition of local sensitivity of water depth to the tested parameters, experiment without slope. Numbering of points made in function of hydraulic distance to the upstream north-west corner.

Figure 5 .

 5 Figure 5.10, top left illustrates the Sobol indices for 14 outlets in south and east face with equal repartition of inow discharges. For all streets, the sum of all Sobol indices is close to1, which means no interaction eect exist between the tested parameters controlling the ow distribution at outlets. The outlet discharges in street 1, 5, 6, 7 and G are nearly totally controlled by downstream water height. We retrieve a total dependence between outlet water depth and discharge, in other words a rating curve relationship for those streets. The outlets of street 5, 6, 7 and G are located at the south-east corner of the device, which is far from the inlet at west and north face, and separated by three big streets C, F and 4. Therefore, the inuence of upstream discharge distribution on west and north faces can not propagate and is

Figure 5 .

 5 Figure 5.10: Sobol indices of outlet discharges obtained with the 2D SW model (for each test, ΣS i ∈ [0.97; 1]).

  discharge are barely not sensitive to uniform roughness for those steady congurations. However, about 20% of the outlet discharge variance of street F is explained by roughness, which may be linked to its particular orientation north-west to south-east.As observed byFinaud-Guyot et al. (2018), the 14 outlet discharges can be divided into several sub-districts using big streets, which are sub-district composed of street 1 (for supercritical ow), sub-district 23, sub-district 567, sub-district AB, sub-district DE, sub-district G and sub-district 4CF composed of big streets. The Sobol indices of sub-districts are presented in gure 5.10 for 50-50% and 80-20% inow repartition. When west-face discharge partition changes from 50% to 80% illustrated in gure 5.10, results show that the inuence of west-face discharge increases at sub-district 23, sub-district AB, sub-district DE, and big street sub-district 4CF,

Figure 5 .

 5 Figure 5.11: 2D maps of Sobol indices for 80 m 3 /h with slope: Left, equal 50-50% inow discharge partition between west and north-face; Right, 80-20% inow discharge partition between west and north-face

Figure 5 .

 5 Figure 5.13: Sobol indices of outlet discharges obtained with the 2D SW model with slope

  g.Mignot et al. (2006);Schubert and Sanders (2012);[START_REF] Skinner | Simulating tidal and storm surge hydraulics with a simple 2d inertia based model, in the Humber Estuary, UK[END_REF];Arrault et al. (2016); Guinot et al. (2017a)) simulate spatio-temporal hydrodynamic patterns and consequently the variations of ood hazard throughout the passage of a ood wave. In this part, GSA of an unsteady urban ood model is conducted. The uncertainty of inow hydrograph dened hereafter and uniform roughness are considered. First the input factors and especially the hydrograph shape are dened, then the resulting calibrated model outputs are analyzed (subsection 5.2.1). Next, Sobol variation ranges are dened and Monte Carlo simulations with sampling size 200 are conducted to check time-varied Sobol index to dierent uncertainty sources (inow discharge and uniform roughness).

  -1 and Q(t) = 80 m 3 .h -1 , t ∈ [100; 150], hence dening a rising limb of 100s a steady state of 50s and a falling limb of 200s. The inow discharge is equally distributed at west and north face and proportional to the channel width for each street. The maximum inow discharge of 80 m 3 .h -1 corresponds to a typical ood discharge tested experimentally in steady state (cf. chapter 5 section 5.1).

Figure 5 .

 5 Figure 5.14: Inow hydrograph on west and north face of each street

Figure 5 1 F 1 Figure 5 .

 5115 Figure 5.15: Time-varied spatially distributed water height

Figure 5 .

 5 Figure 5.17: Time-varied outlet discharge (left side) and ratio Qi Q total,out (right side) of 14 streets

Figure 5 .

 5 Figure 5.18 middle shows the Sobol index of inow discharge, which extends and decreases

Figure 5 .

 5 Figure 5.18: 2D maps of time-varied Sobol indices calculated on distributed water height: left, the sum of all Sobol indices; middle, Sobol index of inow discharge; right, Sobol index of roughness

Figure 5 .

 5 Figure 5.19: 2D maps of time-varied Sobol indices calculated on distributed water height: left, R 2 ; middle, S istd of Q in ; right, S istd of K

Figure 5 .

 5 Figure 5.20: Time-varied Sobol indices on outlet discharge (left, east-face; right, south-face): top, the sum of all Sobol indices; middle, Sobol index of inow discharge; bottom, Sobol index of roughness

Figure 5 .

 5 Figure 5.23 illustrates h and Fr on 9 blockages compared with normal setting. Results show that water height is more sensitive to blockage 6, whereas Froude number is less sensitive to blockage 3, 4 and 5 compared with other settings. Figure5.24 presents the evolution of Q for dierent blockage. Compared with other settings, outlet discharge is more sensitive to blockage 6 and 7.The sensitivity analysis conducted above shows that full blockage (Blockage 6) on the wider street has bigger eect on both water height and Froude number distribution and outlet discharge.

  ow steady and unsteady modeling which based on the urban ood experimental rig of ICube laboratory Strasbourg. The inuence of uncertainty sources of 1D and 2D Shallow water model input parameters on steady and unsteady model outputs is assessed in the case of urban ood ows using GSA with presented spatial and temporal sensitivity indices. The sensitivity of model output to street blockage is also assessed: Spatial Sobol index of steady 1D or 2D SW models has provided the guidelines for the calibration on input parameters before simulation to improve the model output accuracy. Input parameters studied are inow discharge, outlet water height and uniform and distributed roughness based on the experimental ranges. The sensitivity to equal range is also tested to obtain the general sensitivity patterns. The sensitivity to bottom slope is conducted for special cases. Temporal and spatial Sobol index in 2D scale is provided to study the development of input parameter sensitivity along the time. Inow discharge obtained from classical ash ood and uniform roughness are studied. The mechanism of discharge distribution into the device varied with time is studied. The Sobol index of inow discharge spreads and disappears on the whole device as the same trend of inow hydrograph of a ash ood.

Figure 5 1

 51 Figure 5.23: on water height and Froude number for 9 blockages: left, water height; right, Froude number

Figure 6 . 2 :

 62 Figure 6.2: Evolution of the friction slope S f due to the porosity parameter φ. For the sake of clarity, the porosity φ starts at φ = 0.1 as the plotted formulas are in 1 /φ 2 leading to vertical asymptote and innite value of the relative friction slope for φ = 0.

Figure 6 . 3 :

 63 Figure 6.3: Inuence of the friction slope S f formulation for dierent aspect ratio h/B. Top left: h/B = 0.01, top right: h/B = 0.1, bottom left: h/B = 1, bottom right: h/B = 10. For the sake of clarity, the porosity φ starts at φ = 0.1 as the plotted formulas are in 1 /φ 2 leading to vertical asymptote and innite value of the relative friction slope for φ = 0. Note that for h/B = 10 (bottom right), line χ = B + h is hidden by the line χ = B + 2h

Figure 6 . 4 :

 64 Figure 6.4: Denition of the dimensions of the recirculation zone adapted from Schindfessel et al.(2017)

Figure 6 . 5 :

 65 Figure 6.5: Comparison of three empirical relationships quantifying the maximum recirculation width

  For T-branch conuence ow, F d has hydrodynamic analysis of a four branch crossroad In chapter 5, 1D and 2D steady and transient hydrodynamics have been analyzed in the congurations of the ICube urban ood experimental rig. In this section, similar analysis is conducted at the local scale on a four branch crossroad. The ow patterns on a four branch crossroad are analyzed, and especially the dimensions of the recirculation zone downstream of the crossroad.

Figure 6 . 6 :

 66 Figure 6.6: Left, comparison of three empirical relationships quantifying the total recirculation length; Right, dimensionless length of separation zone taken from Kasthuri and Pundarikanthan(1987)

Figure 6

 6 Figure 6.7: Left: illustration of a four branch crossroad with dimensions: B s,x and L s,x (respectively B s,y and L s,y ) are the maximum width and total length of the recirculation zone in x (respectively y) direction downstream of the crossroad (Nanía et al. (2011)); right, unstructured triangular mesh on a four branch crossroad

  Figure 6.8: Telemac2D results on a four branch crossroad

Figure 6 .

 6 Figure 6.9: Sobol indices of spatially distributed water height simulated with Telemac2D to input parameters on a four branch crossroad ow

as 2 +

 2 bs + c for 0 ≤ s ≤ r 0 ds + e for r 0 ≤ s ≤ L s (6.17) such as B recirc (0) = B recirc (L s ) = 0, B recirc (r 0 ) = B s and L s = 3r 0 hence a = -

3 .

 3 Initial water surface elevation is 0.1m and velocity in x and y directions are set to 0m.s -1 . PorAS Riemann solver is used for the ux computation. The simulation is carried out until steady ow conditions are reached over the computed domain.Simulated 1D2D results are compared with 2D results to test in which extent it can reproduce the 2D ow features (water height). For symmetrical boundary conditions, discharge distribution at crossroad is not considered and water height is only presented along the x direction (line A-B-C-D see gure 6.8a).

Figure 6 .

 6 Figure 6.15: Comparison of simulated water height between Telemac2D and Flood1D2D using dierent B s,φ

Figure 6 .

 6 Figure 6.18: Sobol indices of outlet discharges simulated with Flood1D2D to the main controls including recirculation dimensions

Figure 6 .

 6 Figure 6.19 illustrates the Sobol index of water height to the main control parameters. The sum of all Sobol indices shows that no interaction eects exist at the inlet channels and into crossroad. However, at the zone where porosity varies after the crossroad, the sum of all Sobol

Figure 6 .

 6 Figure 6.19: Sobol indices of spatially distributed water height simulated with Flood1D2D on a four branch crossroad (main controls considered in the full parameter set are Q in,x , Q in,y , h out and K)

Figure 6 .

 6 Figure 6.20: Sobol indices of spatially varied water height simulated with Flood1D2D to recirculation dimensions (considered in full parameter sets composed of B s,x , r 0,x , B s,y and r 0,y ): top left, sensitivity of the water depth to the maximum recirculation width in x direction; top right, to the recirculation radius in x direction; bottom left, to the maximum recirculation width in y direction; bottom right, to the recirculation radius in y direction

Figure 6 .

 6 Figure 6.22 presents Sobol indices of input parameters on outlet discharges. Results show that recirculation width has main eect on discharge distribution at crossroad compared with length of recirculation and roughness.

Figure 6 .

 6 Figure 6.21: Sobol indices of spatially distributed water height simulated with Flood1D2D to recirculation dimensions on a four branch crossroad.

Figure 6 .

 6 Figure 6.22: Sobol indices of outlet discharges simulated with Flood1D2D to recirculation dimensions on a four branch crossroad.

Figure 6 .

 6 Figure 6.23: Experimental rig and illustration of the modeled zone (left side) and 1D2D mesh (right side). The white dot in the south east crossroad is a plotting artefact.

Figure 6 .

 6 Figure 6.24: Relative variation of water depth to a parameterization with K street and K cross

Figure 6 .

 6 Figure 6.26: Relative variation of water depth to a parameterization with φ r,n and φ r,l

  over estimations of street outlet discharge which was not possible with roughness only. Despite the sparse sampling of this wide parameter space, the spread of model response quite centered around discharge measurements suggests that there may exist parameter sets reproducing those values. It clearly shows the ability of the proposed model and parameterization to model various non monotonous ow patterns and potentially realistic local discharge and global discharge distributions.

Figure 6 .

 6 Figure 6.28: Outlet discharge sensitivity to a parameterization with φ r,n and φ r,l

  to the street E and to subpart F3G3 of street 3. Subpart E3F3 of street 3 is parameterised with r × φ narrow . The set of values for the dierent parameters are [0.6, 0.75, 0.9] for φ large , [0.7, 0.8, 0.9, 1] for φ narrow and [0.3, 0.475, 0.65, 0.825, 1] for r. Choosing r ≤ 1 allows to impose a porosity in street E3F3 smaller than in street E3E4. It is expected that it would increase the discharge in street E3E4 as gure 6.27 highlights that the discharge is overestimated (respectively underestimated) in street E3F3 (respectively E3E4). In order to sample Flood1D2D response surface, all the combinations of values of φ large , φ narrow and Ratio are tested, that is 60 in total.

Figure 6 .

 6 Figure 6.29: Relative variation of water depth to a parameterization with φ narrow and Ratio

Figure 6 .

 6 Figure 6.30: Street discharge sensitivity to a parameterization with φ narrow and Ratio

Figure 6 .

 6 Figure 6.31: Outlet discharge sensitivity to a parameterization with φ narrow and Ratio

  • , 60 • and 90 • using equal width on horizontal plan. Theoretical ow B s and length L s of recirculation zone based on the ratio between lateral and downstream discharge with angles of 15 • , 45 • , 70 • and 90 • . The results show that the width and length

	patterns are derived using free-streamline theory and conformal mapping to dene a theoretical
	ow pattern throughout the junction and compared with experiment. The location of stagnation
	point at the upstream corner of the channel junction and delineation of separation zone are also
	presented. No attempt of quantifying the dimensions of separation zone is conducted. Modi
	et al. (1981) uses a conformal mapping technique to predict the geometry of the separation zone.
	Best and Reid (1984) conduct experimental studies on separation zones in the case of an
	open channel junction of equal width and an empirical relationship is derived for the maximum
	of the ow separation zone at open-channel conuences increases with both junction angle and
	the contribution of the tributary to the total discharge. The shape of the separation zone
	remains more or less constant. The ratio between maximum width and length is	Bs Ls = 0.19.

width

  • , 60 • and 90 • and uniform width. Then, an empirical relationship is provided for maximum separation width B s and length L s ,

Table 3 .

 3 1: Sub-edges of 1D2D coupling with the consideration of recirculation eect

Table 4 .

 4 10: Parameters for one-dimensional ow with friction (T10)

	Symbol	Meaning	Value
	g	Gravitational acceleration	9.81m.s -2
	l B	Domain length Channel width	100m × 1m 5m
	S 0	Bottom slope	0m.m -1
	x	Cell size	0.1m
	y	Cell width	1m
	z ini	Initial water level

Table 4 .

 4 11: Parameters for one-dimensional ow with friction and bottom slope (T11a)

	Symbol	Meaning	Value
	g	Gravitational acceleration	9.81m.s -2
	l B	Domain length Channel width	3000m × 5m 5m
	S 0	Bottom slope	0.001m.m -1
	x	Cell size	3m
	y	Cell width	5m
	z ini q upstream	Initial water level Prescribed unit discharge at the upstream boundary (x = 0m)	1m 2m 2 .s -1
	h downstream Prescribed water depth at the downstream boundary (x = 3234m)	1m
	K	Strickler coecient	25m 1/3 .s -1

Table 4 .

 4 12: Parameters for one-dimensional ow with friction and bottom slope (T11b)

	Symbol	Meaning	Value
	g	Gravitational acceleration	9.81m.s -2
	l B	Domain length Channel width	500m × 5m 5m
	S 0	Bottom slope	0.005m.m -1
	x	Cell size	0.5m
	y	Cell width	5m
	z ini	Initial water level

Table 4 .

 4

		13: Parameters for 1D2D coupling cases (T12)
	Symbol	Meaning	Value
	g	Gravitational acceleration	9.81m.s -2
	l S 0,x S 0,y x 1D	Domain length Slope in x direction Slope in y direction Cell size for 1D mesh	70m × 40m 0.1m.m -1 -0.1m.m -1 10m
	x 2D	Cell size for 2D mesh	5m
	z ini u ini v ini	Initial water level over the domain Initial velocity at x direction Initial velocity at y direction	10m 0m.s -1 0m.s -1
	z BC	Water surface elevation at boundary conditions	10m
	t	Computation time	1000s

Table 4 .

 4 14: Parameters for 1D2D coupling case with varied width (T13)

	Symbol	Meaning	Value
	g	Gravitational acceleration	9.81m.s -2
	l B 1	Domain length Width of the channel from 0m to 50m	10m × 1m 10m
	B 2	Width of the channel from 50m to 100m	4m
	B 3	Width of the channel from 100m to 250m	10m
	B 4	Width of the channel from 250m to 350m	4m
	B 5	Width of the channel from 350m to 400m	10m
	x 1D	Cell size for 1D mesh	0.25m
	x 2D	Cell size for 2D mesh	0.25m
	z ini u ini v ini	Initial water surface elevation over the domain Initial velocity at x direction Initial velocity at y direction	1m 0m.s -1 0m.s -1
	q upstream

Table 5 .

 5

		1: Parameters used and associated perturbation ranges for the 1D street modeling
	Symbol	Meaning	Value (street C)	Value (street 4)	Range
	Q in h out Q 1	Street C (4) inow discharge Street C (4) outow water height	2.67 l/s 0.083 m	3.01 l/s 0.078 m	±5% ±10%

  1/3 .s -1 ±50% L

	x	Domain length Cell size	5 m × 0.125 m 0.02 m	5 m × 0.125 m 0.02 m

30% of the total ow (cf. gure 5.1). Ten input parameters are considered including upstream inlet discharge Q in , outlet water height h out , uniform roughness K and seven lateral discharges corresponding to mass exchange at crossroads Q {1,...,7;A,...,G} (cf. table 5.1). The variation range of each input parameter around its calibrated value (. cal ) is chosen corresponding to the experi- mental uncertainties. The ranges considered here are: Q cal in ± 5%, h cal out ± 10%, K cal ± 50% and Q cal {1,...,7;A,...,G} ± 5%; note that each lateral discharge is perturbed independently.

Table 5 .

 5

	2: Calibrated discharge coecient of rating curve
	Street	C d	Street	C d
	1	2.8	A	1.2
	2	1.2	B	1.2
	3	1.65	C	1.15
	4	1.05	D	1.2
	5	1.8	E	1.2
	6	1.2	F	1.25
	7	1.75	G	1.2
			114

  1/3 .s -1 . The outlet BC, rating curve, is not considered in this study. The sampling size is set to 200 given the computational cost of 1.5h on 32 processors of a forward run for each parameter set. The convergence of sampling size is reached based on the Sobol indices of R 2 and S istd presented in gure 5.19. The standard deviation of the water depth Sobol index to Q in and K is very low on the whole domain. R 2 ∼ 1 on the main part of the domain except where the ow hydraulic jump appears. In what follows, we present the Sobol maps of sensitivity of water depth to

  established for T-conuence:

	B s B	= 0.506 + 0.205 ln q	(6.8a)

Table 6 .

 6

	1: Parameters used on a four branch crossroad modeling
	Symbol	Meaning	Value
	g	Gravitational acceleration

Table 6 .

 6 2: Variation range of boundary conditions provided by[START_REF] Nanía | Experimental study of subcritical dividing ow in an equal-width, four-branch junction[END_REF]

	Symbol	Meaning	Min	Max	Variation range

Table 6 .

 6 3: Nominal parameters of the four branch crossroad modeling

	Symbol	Meaning	Value
	g Q in,x Q in,y	Gravitational acceleration Inlet discharge in x direction Inlet discharge in y direction	9.81m.s -2 0.0435m 3 .s -1 0.0435m 3 .s -1
	h out,x	Outlet water height in x direction	0.05m
	h out,y	Outlet water height in y direction	0.05m
	K	Strickler coecient	100m 1/3 .s -1
	x	Cell size	0.06m
	h ini	Initial water level over the domain	0.1m

Table 6 .

 6 4: Parameters used and associated perturbation range for the four branch crossroad ow

	Symbol	Meaning	Value	Range

Table 6 .

 6 6: Parameters used and associated perturbation range for the four branch crossroad ow Inlet discharge in x direction 0.0435m 3 .s -1 ±72.5% Q in,yInlet discharge in y direction 0.0435m 3 .s -1 ±72.5% h out,xOutlet water height in x direction 0.05m ±60% h out,yOutlet water height in y direction 0.05m ±60% K

	Symbol	Meaning	Value	Range
	Q in,x
	B s,x r 0,x B s,y r 0,y	friction coecient Maximum recirculation width in x direction Recirculation radius in x direction Maximum recirculation width in y direction Recirculation radius in y direction	100m 1/3 .s -1 0.225m 1.8m 0.225m 1.8m	±50% ±68.6% ±41.7% ±68.6% ±41.7%

Table 6 .

 6 7: Parameters used and associated perturbation range for the four branch crossroad ow

	Symbol	Meaning	Value	Range
	B s,x r 0,x B s,y r 0,y K	Maximum recirculation width at x direction Recirculation radius at x direction Maximum recirculation width at y direction Recirculation radius at y direction Strickler coecient	0.225m 1.8m 0.225m 1.8m 100m 1/3 .s -1	±68.6% ±41.7% ±68.6% ±41.7% ±50%

compared with the analytical reference on grid size: left, comparison on velocity; right, zoomed space on rectangular shape 4.2.9 Two-dimensional dambreak with variable porosity (T09)

All the previous test cases highlight the eciency of both the HLLC and PorAS Riemann solver in one-dimensional congurations. This test case aims at validating the ability of the code to deal with a two-dimensional dambreak problem with a variable porosity over a at bottom without friction. As shown in Guinot and Soares-Frazão (2006), the circular dambreak problem with a porosity proportional to the inverse of the distance to the center of the dam has the same analytical solution as a classical one-dimensional dambreak with a constant porosity.

The domain length is 200m × 200m wide and the center of the circular dam (having a radius of R = 50m) is located in the middle of the domain in (x 0 , y 0 ). The dam initially separates two zones with no velocity and a constant water level: h (r ≤ R) = 10m and h (r > R) = 1m. The porosity is dened as: φ (x, y) = 1 r where r = (xx 0 ) 2 + (yy 0 )

The dam disappears instantaneously at t = 0s. The comparison between the reference and the computational results is done at t = 4s such as the eect of the dambreak does not reach the boundary of the domain. The computational parameters are given in table 4.9.

The computation is done using an unstructured mesh with triangular cells of typical dimension ∆x = 1m. Figure 4.20 presents both the water depth and the unit-discharge prole along the radius of the dam (either aligned with the x or y direction). Numerical results show a good agreement between the computed hydraulic variables using either the HLLC or the PorAS Riemann solver and the reference. Moreover the proles are very similar whatever the orientation chosen (either x or y) highlighting that Flood1D2D does not introduce anisotropy when solving this dambreak problem.

Friction handling

The algorithm used in Flood1D2D being based on the time splitting method (see section 3.2.4), the computation of the hyperbolic uxes is done rst, then the friction eect is taken into account.

The Riemann solver is only used for the hyperbolic ux computation and since both HLLC and uncertainty in the output of a model can be apportioned to dierent sources of uncertainty in the model input [START_REF] Saltelli | Global sensitivity analysis: the primer[END_REF]), has become a popular tool in environmental modeling [START_REF] Pianosi | Sensitivity analysis of environmental models: A systematic review with practical workow[END_REF]). SA is used in catchment hydrology to explore high dimensional parameter spaces, assess parameter identiability and understand uncertainty sources [START_REF] Hornberger | Approach to the preliminary analysis of environmental systems[END_REF]; [START_REF] Beven | The future of distributed models: model calibration and uncertainty prediction[END_REF]; [START_REF] Pushpalatha | A downward structural sensitivity analysis of hydrological models to improve low-ow simulation[END_REF]; [START_REF] Garambois | Characterization of process-oriented hydrologic model behavior with temporal sensitivity analysis for ash oods in mediterranean catchments[END_REF][START_REF] Garambois | Parameter regionalization for a process-oriented distributed model dedicated to ash oods[END_REF] among others). Temporal variations of distributed rainfall runo and simplied hydraulic model parameters sensitivities are analyzed and ranked in the case of Mediterranean ash oods [START_REF] Garambois | Characterization of process-oriented hydrologic model behavior with temporal sensitivity analysis for ash oods in mediterranean catchments[END_REF] or large scale conceptual model of the Amazon river [START_REF] Emery | Temporal variance-based sensitivity analysis of the river-routing component of the large-scale hydrological model isbatrip: Application on the amazon basin[END_REF]). Temporal sensitivities of simulated ood response highlight phases and locations of higher sensitivities to runo production parameters and/or runo routing depending on rainfall forcing variabilities and drainage network shape among other parameters.

Among the few SA studies in the eld of river hydraulic modeling, [START_REF] Roux | Sensitivity analysis and predictive uncertainty using inundation observations for parameter estimation in open-channel inverse problem[END_REF] propose a generalized sensitivity analysis of a 1D Saint Venant model with simplied geometry adapted to remote sensing and ungauged rivers, in the case of ood scenario on the Lèze River,

France. The authors show the importance of downstream ow depth in controlling ood extent for a 1.5 km reach of a small river (around 100m bankfull width). Guinot and Cappelaere (2009a) propose local sensitivity equations for the 2D steady state SW equations without shocks and provide some guidelines for model calibration and validation. These sensitivity equations are also derived and implemented for 1D Saint Venant model [START_REF] Delenne | Sensitivity of the 1d shallow water equations with source terms: Solution method for discontinuous ows[END_REF]). Local sensitivities derived with the adjoint method (2D SW DassFlow model), and involving a cost function, are presented in the case of a high resolution model of a ood on a 2km reach of the Lèze River [START_REF] Monnier | Inverse algorithms for 2d shallow water equations in presence of wet dry fronts: Application to ood plain dynamics[END_REF]). The authors nd higher sensitivities of water depth to bathymetry and roughness downstream (subcritical regime) of the observation points in the oodplain and in the main channel, roughness sensitivities are higher in the main channel. Recently variance based SA has been applied to 2D hydraulic models in ooding conditions [START_REF] Abily | Spatial global sensitivity analysis of high resolution classied topographic data use in 2d urban ood modelling[END_REF];

Results

This section details sensitivity analysis of 1D and 2D SW models outputs. The sensitivities to their parameters are investigated with the GSA method presented above (cf. section 5.1.2.1) for 1D hydraulic models of the two main streets of the experimental rig and for full 2D models .

The spatially distributed variances of model outputs are decomposed with respect to dierent hydraulic parameters. Each input parameter vector the small number of parameters a trial and error procedure is used. The calibrated roughness and 1D model inputs with their variation ranges are presented in table 5.1. For uniform roughness, the Strickler coecient found for street C is K cal C = 21.5m 1/3 .s -1 and for street 4 K cal 4 = 21m 1/3 .s -1 . The calibration results for distributed roughness consist in a roughness value for streets: K cal Cs = 26.5m 1/3 .s -1 for street C and K cal 4s = 28.5m 1/3 .s -1 for street 4 and in a value for crossroads: K cal Cc = 9.5m 1/3 .s -1 for street C and K cal 4c = 9m 1/3 .s -1 for street 4. Recall that ows are characterized by complex 3D hydrodynamic structures, but this contribution focuses on comparing the sensitivities computed for 1D and 2D modeling approaches in a branched hydraulic network.

Chapter 6

Model applications

The numerical model Flood1D2D using cut-cell technique to account for the eect of recirculation zones downstream of crossroads has been derived in chapter 3 and validated in chapter 4 for various hydraulic congurations. In this chapter, the Flood1D2D model is applied on urban ood congurations using 1D2D coupling to reproduce ow patterns expecting performances comparable to those of a classical 2D model.

In view to tailor 1D2D shallow water model (Flood1D2D ) to urban ood ow simulations the following points are investigated in this chapter:

In the rst section, the impact of dierent modeling options of the friction term is assessed. The modication of the friction slope due to the introduction of the recirculation eect through the porosity parameter is highlighted and the sensitivity of the S f computation to hydraulic radius formulation is assessed for similar geometrical and hydraulic parameterisation;

In the second section, the response surfaces of existing laws for predicting recirculation shape downstream of three branch intersections computed and compared, to give idea of the reasonable porosity on four branch intersections;

In the third section, real like recirculation dimensions are estimated, with various methods, from 2D modeling (full 2D shallow water with turbulence) on a four branch crossroad conguration taken from an experimental dataset of [START_REF] Nanía | Experimental study of subcritical dividing ow in an equal-width, four-branch junction[END_REF];

In the fourth section, the sensitivity of the simulated recirculation zone dimensions to input parameters of the full 2D SW model is investigated with GSA applied on a four branch crossroad;

In the fth section, Flood1D2D is parameterized and tested on the four branch crossroad conguration of [START_REF] Nanía | Experimental study of subcritical dividing ow in an equal-width, four-branch junction[END_REF]. The inuence of the porosity parameterisation and roughness on ow lines is tested;

In the sixth section, the sensitivity of Flood1D2D on parameterisation is investigated with GSA;

Finally, the model is applied to a district for reproducing experimental congurations of the ICube urban ood experimental rig.

Zero-discharge method

The rst method is referred to as the zero-discharge method [START_REF] Schindfessel | How dierent cross-sectional shapes inuence the separation zone of an open-channel conuence[END_REF]). In the recirculation area, the discharge is assumed to be nil in a cross-section (i) in the normal direction to the street axis; (ii) starting from the bank attached to separation zone and ending at y s the limit between the recirculation and the ow vein (see gure 6.4). The width of the recirculation is thus dened as: ys(x) 0 u (x) dy = 0 (6.12)

where u (x) represents the average on the water depth of the longitudinal velocity and y s (x) is the distance to the bank attached to separation zone at which the recirculation ends.

Using this method, the determination of L s (the total length of the recirculation zone) and B s (the maximum value of y s for all x between 0 and L s ) requires thus to compute y s along the downstream street (at least until L s ) and then to detect the value of B s . This method appears to be quite time consuming especially in view of the determination of B s and L s in lots of modeling.

Zero-velocity method

The zero-velocity method is a simplied approach, compared to the zero-discharge method (see section 6.3.2.1), to determine B s and L s . The maximum contraction prole (for which the recirculation width is B s ) is located at the minimal water depth on the longitudinal prole (see gure 6.8c). The value of r 0 can thus be obtained from the downstream corner to the maximum contraction point featured by localized minimum water height, which is named as recirculation radius. As illustrated in gure 6.8c, r 0 = 1.65m. Assuming symmetrical recirculation shape (as proposed by Gurram et al. (1997)) leads to L s = 2r 0 = 3.3m.

For a given cross-section, the center of the recirculation is assumed to correspond to the point where the longitudinal velocity is nil. At maximum contraction point, velocity is depicted along the direction perpendicular to the ow as in gure 6.8d (line F-G, x = 5.15m). Velocity u decreases from point F to G, and stays almost constant outside the recirculation zone and then decreases when entering the recirculation zone reaching a symmetrical prole with negative velocities on half of this recirculation width. The center of the recirculation is thus located (assuming u = 0m.s -1 ) for y 0 = 1.24m. This leads to the following estimation of B s :

This methodology is less time-consuming than the zero-discharge method (see section 6.3.2.1), as it only requires to compute the longitudinal water depth prole and one transversal velocity prole to determine both B s and L s .

Approximated zero-velocity method

Finally, the approximated zero-velocity method is proposed to allow the computation of both B s and L s with smaller computational cost than with the zero-velocity method . L s is computed as for the zero-velocity method . To avoid the determination and the analysis of the longitudinal velocity along the transversal cross-section, the mass equation is written for the maximal contraction prole:

where Q is the total discharge, u s the longitudinal velocity outside the recirculation zone and h s is the averaged water height perpendicular to the ow direction. h s is estimated from the longitudinal water depth prole as it is the water depth in r 0 . u s is computed as the average of

B s,x

Maximum recirculation width at x direction 0.16m 0.86m ±68.6% L s,x Recirculation length at x direction 2.1m 5.1m ±41.7% B s,y

Maximum recirculation width at y direction 0.16m 0.86m ±68.6% L s,y Recirculation length at y direction 2.1m 5.1m ±41.7%

all Sobol indices is close to 1 again indicating no signicant interaction eect between the tested parameters both Q out,x and Q out,y . Q in,x and Q in,y have similar eects on discharge distribution downstream of the crossroad for this symmetrical boundary conditions. Compared with the other parameters, friction has limited eect on discharge distribution at crossroad, which is in contrast with its inuence on water height.

Variation range of the recirculation dimensions

The 2000 Monte Carlo simulations are used in this section to obtain the variation range of the recirculation dimensions, namely B s and L s . As detailed at section 6.

Inuence of the minimal porosity value

In this section, the inuence of porosity parameterization in reproducing contracted ow proles is tested. 1D2D simulations with and without porosity are performed using uniform roughness.

The Strickler coecient is set to 100m 1/3 .s -1 for both 1D and 2D cells. B s = 0.52m and r 0 = 1.65m are obtained from 2D simulation results using the approximated zero-velocity method presented in section 6. 3.2.3. Polynomial relationship (6.16) is used to reproduce the recirculation shape and L s = 2r 0 = 3.3m. φ min is calculated by B s :

Computed water depth proles for variations B s are compared with the reference (Telemac2D) on gure 6.15. For any positive B s , results show that downstream of the crossroad, because of the recirculation, the ow vein is contracted, hence accelerated, and the free surface presents an inection for x ∈ [3.5; 5.15] as in the reference.

When B s = 0.52m, the ow accelerates to supercritical ow after the crossroad and a hydraulic jump occurs at the end of the recirculation zone. The hydraulic jump is not modelised in the reference obtained with Telemac.. The above dierence can be explained by an overestimation of the ow contraction by calculating B s from a 2D simulation. Indeed, the size in plane of a real 3D recirculation may be variable on the vertical and even in 2D scale it may not be fully isolated from the ow. However the porosity represents a zone that is totally separated from the main stream with no ux exchange between them. Therefore, B s calculated from 2D simulation may be overestimated compared to the one necessary to perform an eective modeling of the ow contraction in 1D.

By decreasing B s,φ (0.3m, 0.26m, 0.2m, and 0m) the calculated proles presented in gure 6.15 remain subcritical. Compared to the reference, B s,φ = 0.26m is the value for which the eective ow is better reproduced at the maximum contraction point, which is the half of the value calculated from 2D results B s,φ = Bs 2 . Therefore, B s,φ = 0.26m is used in 1D model to reproduce the 2D ow lines and φ min is calculated by B s,φ : 6.19) When B s,φ = 0m, corresponding to no porosity and friction only, the ow line modeled with Flood1D2D is linear, hence not reproducing ow contraction behavior. Note that even with a distributed friction coecient such contracted ow proles cannot be reproduced using a 1D approach as the friction coecient is embedded in the source term S f of the SW equation, only able to dissipate energy. Indeed this term is of the opposite sign of the velocity which remains monotonous in a prismatic reach as water depth prole.

The hydraulic jump is triggered by the parameter values creating high ow velocity, namely a ow contracting at B s = 0.52 combined to a low roughness value K = 100m 1/3 .s -1 . Moreover, the roughness value used in Flood1D2D in 1D mesh does not account for the head loss due to turbulence modeled in Telemac2D. This explains why no hydraulic jump appears in Telemac2D results.

Inuence of the porosity shape

In order to test the inuence of the porosity shape on reproducing water depth prole, three relationships proposed at section 6.5.1 are tested: Gurram et al. (1997) relationship (6.15), polynomial relationship (6.16) and piecewise relationship (6.17) as plotted in gure 6.13.

Appendix