Mathematical Modeling of Flooding Due to River Bank Failure
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Advances in Water Resources 59 (2013) 82–94 Contents lists available at SciVerse ScienceDirect Advances in Water Resources journal homepage: www.elsevier.com/locate/advwatres Mathematical modeling of flooding due to river bank failure ⇑ Daniele Pietro Viero a, , Andrea D’Alpaos b, Luca Carniello a, Andrea Defina a a Department ICEA, University of Padova, Via Loredan 20, 35131 Padova, Italy b Department of Geosciences, University of Padova, Via Gradenigo 6, 35131 Padova, Italy article info abstract Article history: Modeling of flooding events resulting from bank overflooding and levee breaching is of relevant social Received 12 February 2013 and environmental interest. Two-dimensional (2D) hydrodynamic models integrating the shallow water Received in revised form 15 May 2013 equations turn out to be very effective tools for the purpose at hand. Many of the available models also Accepted 20 May 2013 use 1D channel elements, fully coupled to the 2D model, to simulate the flow of small channels dissecting Available online 29 May 2013 the urban and rural areas, and 1D elements, referred to as 1D-links, to efficiently model the flow over levees, road and rail embankments, bunds, the flow through control gates, either free or submerged, Keywords: and the operation of other hydraulic structures. In this work we propose a physically-based 1D-link to Breach formation model breach formation and evolution in fluvial levees, and levee failure due to either piping or overtop- Breach evolution Coupled hydrodynamic models ping. The proposed 1D-link is then embedded in a 1D–2D hydrodynamic model, thus accounting for crit- ical feedbacks between breach formation and changes in the hydrodynamic flow field. The breach model also includes the possibility of simulating breach closure, an important feature particularly in the view of hydraulic risk assessment and management of the emergency. The model is applied to five different case studies and the results of the numerical simulations compare favorably with field observations displaying a good agreement in terms of urban and rural flooded areas, water levels within the channel, final breach widths, and water volumes flowed through the breach. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction evacuated, and damages were estimated at more than 300 M€. These failures were ascribed to several factors, among which the The study of flooding events resulting from bank overflooding most relevant seemed to be the poor discharge capacity of the flu- and levee breaching is of relevant social and environmental inter- vial reaches, the bad levee maintenance, and, importantly, the pres- est because flood waves, resulting from levee failure, might cause ence of nutria (Myocastor coypus) holes within the levees body. loss of lives [1] and the destruction of properties and ecosystems The capability of simulating the development, in space and [2]. During severe flood events, many links of the fluvial network time, of a flooding event has, therefore, important implications in lowland areas are commonly characterized by water levels high- on the development of effective intervention plans for the mainte- er than those of the adjacent land, and river cross sections are often nance and management of the emergency. unable to convey the maximum discharges, their banks being pos- Within this framework, two-dimensional (2D) hydrodynamic sibly overflooded. Moreover, in the past, levees have been some- models, integrating the shallow water equations, turn out to be time built by raising the banks ever higher with local material very effective tools for the purpose at hand. unsuitable for the purpose at hand, thus exposing levees to failure A number of 2D models are available in the literature. Indeed, to during flood propagation. Indeed, mainly because of progressively keep a high accuracy and reduce the computational effort, some of less frequent maintenance tasks, critical events are today even these models use 1D elements, coupled to the 2D domain, to more frequent. describe the flow in the network of interconnected channels that As an example, during the last two years, four levee failures oc- dissect the lowland and affect the propagation of the flood wave curred along the main rivers flowing through the Padova district over the initially dry land (e.g., FloodFlow, Infoworks 2D, ISIS2D, (namely, Muson dei Sassi River at Loreggia, Frassine River at Saletto, LISFLOOD, MIKE FLOOD, SOBEK, TUFLOW). and Bacchiglione River at Veggiano and Ponte San Nicolò), North- Most of these models also have other 1D elements, which are re- eastern Italy. During these events three people died, large portions ferred to as 1D-links, that provide an efficient and accurate model- of rural and urban areas were flooded, about 5,000 people were ing of the flow over levees, road and rail embankments, bunds, and the flow through control gates, either free or submerged. These 1D- ⇑ Corresponding author. links also serve to simulate the operation of pumping systems, lock E-mail address: [email protected] (D.P. Viero). gates, and other hydraulic structures (see [3] for a recent review). 0309-1708/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.advwatres.2013.05.011 D.P. Viero et al. / Advances in Water Resources 59 (2013) 82–94 83 A number of numerical models for the simulation of breach ini- where h is the free surface elevation, g is gravity, t denotes time, tiation and growth in fluvial embankments are also available in the q ¼ðqx; qyÞ is the flow rate per unit width, Y the equivalent water literature (see, e.g., [4–6]). However, these models require a refined depth, defined as the volume of water per unit area actually pond- discretization of the domain (i.e., a large number of relatively small ing the bottom, g the local fraction of wetted domain which can be computational elements) and, in the case of small embankments interpreted as an h-dependent storativity coefficient accounting for (i.e., levees rather than earth dams), their coupling to shallow the actual area that can be wetted or dried, J ¼ðJx; JyÞ is energy dis- water models, in order to fully describe the flooding of large areas, sipation per unit length due to bottom shear stress computed using is neither straightforward nor computationally efficient. the Manning formula (see [16]), and Re accounts for the horizontal The aim of the present work is to develop a simplified, robust, dispersion stresses. and physically based 1D-link allowing one to simulate the breach The model uses a semi-implicit staggered finite element meth- initiation and growth by considering both overtopping and piping od based on a mixed Eulerian and Lagrangian Galerkin’s approach processes. Numerically solved, physically-based 1D-links, describ- [21], and the computational domain is discretized with 2D triangu- ing breach evolution, properly match the needs of schematic ap- lar elements and 1D linear elements, the latter being used to de- proach, fast calculation, and applicability to different contexts, scribe the flow in small channels and pipes and to simulate the also in the view of probabilistic analyses aiming at mapping flood- operation of hydraulic structures. ing hazard [7–13]. The use of 1D channel elements in a coupled 1D–2D scheme is An important additional feature of the proposed model is also very effective to account for the presence of small channels dissect- the possibility of simulating breach closure, which is commonly ing the urban and rural area which, in fact, play a crucial role on the performed during the receding phase of a flood event and which propagation of flood waves over initially dry areas [15]. For a de- is of crucial importance in order to accurately predict the evolution tailed description of the hydrodynamic model we refer the reader of the inundation process and the extent of the inundated area. to [16,18,21]. Details on 1D-links equations, and on how 1D-links This feature also allows one to easily reproduce different scenarios, equations are linearized and integrated in the 1D–2D model are gi- depending on when breach closure activities begin and on the rate ven in [22,17], respectively. at which the levee is reconstructed, and to establish efficient plans for the management of the emergency. 2.2. The failing levee model The proposed 1D-link is then coupled to an already developed, widely tested 1D–2D hydrodynamic model [14–20]. The coupling In the last 50 years many numerical models have been devel- is of critical importance, because, as we show, the evolution of oped to simulate breach formation and evolution. Recent reviews the breach strongly depends on the altered driving hydrodynamic can be found, e.g., in [4,6,23,24]. conditions both within the channel and over the adjacent flooded These models can be classified into two major categories: (i) area. parametric models, which use statistically derived regression The proposed model is then tested by simulating breach forma- equations for estimating significant characteristics, such as the tion, growth, and closure for five case studies, namely the breach peak water discharge and the final breach width; and (ii) physi- occurred on January 2009 across the left bank of the Muson dei Sas- cally-based models describing, with a reasonable approximation, si River, north of the city of Padova, and the four breaches occurred the main phases of the actual failure process. Models of the latter on November 2010 along the Bacchiglione River (two of which category use sediment transport formulas to estimate the rate of nearby the city of Padova and two nearby the city of Vicenza). All bank erosion, and are more or less complex, depending on the these cases present different features with regards to the mecha- assumptions made to describe the breach evolution pattern.