Advances in Water Resources 59 (2013) 82–94

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Advances in Water Resources

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Mathematical modeling of flooding due to river bank failure

0309-1708/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.advwatres.2013.05.011

⇑ Corresponding author.E-mail address: daniele.viero@unipd.it (D.P. Viero).

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, Italyb Department of Geosciences, University of Padova, Via Gradenigo 6, 35131 Padova, Italy

a r t i c l e i n f o

Article history:Received 12 February 2013Received in revised form 15 May 2013Accepted 20 May 2013Available online 29 May 2013

Keywords:Breach formationBreach evolutionCoupled hydrodynamic models

a b s t r a c t

Modeling of flooding events resulting from bank overflooding and levee breaching is of relevant socialand environmental interest. Two-dimensional (2D) hydrodynamic models integrating the shallow waterequations turn out to be very effective tools for the purpose at hand. Many of the available models alsouse 1D channel elements, fully coupled to the 2D model, to simulate the flow of small channels dissectingthe urban and rural areas, and 1D elements, referred to as 1D-links, to efficiently model the flow overlevees, road and rail embankments, bunds, the flow through control gates, either free or submerged,and the operation of other hydraulic structures. In this work we propose a physically-based 1D-link tomodel breach formation and evolution in fluvial levees, and levee failure due to either piping or overtop-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 modelalso includes the possibility of simulating breach closure, an important feature particularly in the view ofhydraulic risk assessment and management of the emergency. The model is applied to five different casestudies and the results of the numerical simulations compare favorably with field observations displayinga good agreement in terms of urban and rural flooded areas, water levels within the channel, final breachwidths, and water volumes flowed through the breach.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

The study of flooding events resulting from bank overfloodingand levee breaching is of relevant social and environmental inter-est because flood waves, resulting from levee failure, might causeloss of lives [1] and the destruction of properties and ecosystems[2]. During severe flood events, many links of the fluvial networkin lowland areas are commonly characterized by water levels high-er than those of the adjacent land, and river cross sections are oftenunable to convey the maximum discharges, their banks being pos-sibly overflooded. Moreover, in the past, levees have been some-time built by raising the banks ever higher with local materialunsuitable for the purpose at hand, thus exposing levees to failureduring flood propagation. Indeed, mainly because of progressivelyless frequent maintenance tasks, critical events are today evenmore frequent.

As an example, during the last two years, four levee failures oc-curred along the main rivers flowing through the Padova district(namely, Muson dei Sassi River at Loreggia, Frassine River at Saletto,and Bacchiglione River at Veggiano and Ponte San Nicolò), North-eastern Italy. During these events three people died, large portionsof rural and urban areas were flooded, about 5,000 people were

evacuated, and damages were estimated at more than 300 M€.These failures were ascribed to several factors, among which themost relevant seemed to be the poor discharge capacity of the flu-vial reaches, the bad levee maintenance, and, importantly, the pres-ence of nutria (Myocastor coypus) holes within the levees body.

The capability of simulating the development, in space andtime, of a flooding event has, therefore, important implicationson the development of effective intervention plans for the mainte-nance and management of the emergency.

Within this framework, two-dimensional (2D) hydrodynamicmodels, integrating the shallow water equations, turn out to bevery effective tools for the purpose at hand.

A number of 2D models are available in the literature. Indeed, tokeep a high accuracy and reduce the computational effort, some ofthese models use 1D elements, coupled to the 2D domain, todescribe the flow in the network of interconnected channels thatdissect the lowland and affect the propagation of the flood waveover the initially dry land (e.g., FloodFlow, Infoworks 2D, ISIS2D,LISFLOOD, MIKE FLOOD, SOBEK, TUFLOW).

Most of these models also have other 1D elements, which are re-ferred to as 1D-links, that provide an efficient and accurate model-ing of the flow over levees, road and rail embankments, bunds, andthe flow through control gates, either free or submerged. These 1D-links also serve to simulate the operation of pumping systems, lockgates, and other hydraulic structures (see [3] for a recent review).

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-tiation and growth in fluvial embankments are also available in theliterature (see, e.g., [4–6]). However, these models require a refineddiscretization of the domain (i.e., a large number of relatively smallcomputational elements) and, in the case of small embankments(i.e., levees rather than earth dams), their coupling to shallowwater models, in order to fully describe the flooding of large areas,is neither straightforward nor computationally efficient.

The aim of the present work is to develop a simplified, robust,and physically based 1D-link allowing one to simulate the breachinitiation and growth by considering both overtopping and pipingprocesses. Numerically solved, physically-based 1D-links, describ-ing breach evolution, properly match the needs of schematic ap-proach, fast calculation, and applicability to different contexts,also in the view of probabilistic analyses aiming at mapping flood-ing hazard [7–13].

An important additional feature of the proposed model is alsothe possibility of simulating breach closure, which is commonlyperformed during the receding phase of a flood event and whichis of crucial importance in order to accurately predict the evolutionof the inundation process and the extent of the inundated area.This feature also allows one to easily reproduce different scenarios,depending on when breach closure activities begin and on the rateat which the levee is reconstructed, and to establish efficient plansfor the management of the emergency.

The proposed 1D-link is then coupled to an already developed,widely tested 1D–2D hydrodynamic model [14–20]. The couplingis of critical importance, because, as we show, the evolution ofthe breach strongly depends on the altered driving hydrodynamicconditions both within the channel and over the adjacent floodedarea.

The proposed model is then tested by simulating breach forma-tion, growth, and closure for five case studies, namely the breachoccurred on January 2009 across the left bank of the Muson dei Sas-si River, north of the city of Padova, and the four breaches occurredon November 2010 along the Bacchiglione River (two of whichnearby the city of Padova and two nearby the city of Vicenza). Allthese cases present different features with regards to the mecha-nisms of breach initiation and evolution (i.e., by piping or overtop-ping), the topography of the flooded areas, the final breach widthand the water volumes flowed out through the breach, the presenceor absence of activities aimed at repairing the breach.

The paper is organized as follows. In Section 2, for the sake ofcompleteness, we briefly recall the 1D–2D hydrodynamic modeland describe the newly developed breach module. We then presentthe five case studies and discuss the results of the simulations, interms of urban and rural flooded areas, water volumes and breachsize evolution, that are compared to the available surveyed data(Sections 3 and 4). Finally, a set of conclusions closes the paper.

2. Model description

2.1. The hydrodynamic model

A hydrodynamic numerical model, named 2DEF [14–16,18], ischosen as the basis framework on which the proposed 1D-link,describing breach formation, growth and closure, is embedded in.2DEF is a coupled 1D–2D model that solves the full shallow waterequations, modified to deal with flooding and drying processesover very irregular topography [16]

rhþ 1g

ddt

qY

� �þ J�r � Re ¼ 0 ð1Þ

gðhÞ @h@tþrq ¼ 0 ð2Þ

where h is the free surface elevation, g is gravity, t denotes time,q ¼ ðqx; qyÞ is the flow rate per unit width, Y the equivalent waterdepth, defined as the volume of water per unit area actually pond-ing the bottom, g the local fraction of wetted domain which can beinterpreted as an h-dependent storativity coefficient accounting forthe actual area that can be wetted or dried, J ¼ ðJx; JyÞ is energy dis-sipation per unit length due to bottom shear stress computed usingthe Manning formula (see [16]), and Re accounts for the horizontaldispersion stresses.

The model uses a semi-implicit staggered finite element meth-od based on a mixed Eulerian and Lagrangian Galerkin’s approach[21], and the computational domain is discretized with 2D triangu-lar elements and 1D linear elements, the latter being used to de-scribe the flow in small channels and pipes and to simulate theoperation of hydraulic structures.

The use of 1D channel elements in a coupled 1D–2D scheme isvery effective to account for the presence of small channels dissect-ing the urban and rural area which, in fact, play a crucial role on thepropagation of flood waves over initially dry areas [15]. For a de-tailed description of the hydrodynamic model we refer the readerto [16,18,21]. Details on 1D-links equations, and on how 1D-linksequations are linearized and integrated in the 1D–2D model are gi-ven in [22,17], respectively.

2.2. The failing levee model

In the last 50 years many numerical models have been devel-oped to simulate breach formation and evolution. Recent reviewscan be found, e.g., in [4,6,23,24].

These models can be classified into two major categories: (i)parametric models, which use statistically derived regressionequations for estimating significant characteristics, such as thepeak water discharge and the final breach width; and (ii) physi-cally-based models describing, with a reasonable approximation,the main phases of the actual failure process. Models of the lattercategory use sediment transport formulas to estimate the rate ofbank erosion, and are more or less complex, depending on theassumptions made to describe the breach evolution pattern.

An additional classification is based on the mechanisms thatlead to the embankment failure: (i) overtopping of the levee crestresulting in channel incision over the crest and over the down-stream face, driven by shear forces; (ii) piping or internal erosion,driven by seepage forces, resulting in the removal of fine sedimentsalong a path between the upstream and downstream levee faces;and (iii) foundation and structural defects, including differentialsettlement, sliding and slope instability, high uplift pressure anduncontrolled foundation seepage.

The most frequent mechanisms for riverine levee failures aresurface erosion due to overtopping and internal erosion by piping[24]. An internal erosion process can result in the formation andgradual enlargement of an earlier small pipe through the levee un-til the collapse of the crest. As an alternative, it can lead to a low-ering of the levee crest due to the widespread removal of fines. Inboth cases, an overtopping like breach evolution process canfollow.

The proposed model describes the above failure mechanisms,i.e., overtopping and piping, whereas at present, the less frequentcrest lowering mechanism due to removal of fine sediments isnot included.

Accordingly, the model comprises two sub-modules: one de-scribes the breach formation and evolution due to overtopping,the other describes the enlargement of an earlier pipe throughthe embankment. In the latter case, the levee crest is assumed tocollapse when the pipe size exceeds a threshold value and thebreach that forms in the levee is further evolved using the firstsub-module.

84 D.P. Viero et al. / Advances in Water Resources 59 (2013) 82–94

To describe the whole breaching process triggered by overflowevents, a schematic approach was implemented, which further im-proves the approach proposed by Visser [25] (see also [26]). Theevolution process starts from a small initial breach at the top ofthe levee for which a trapezoidal cross section is assumed (seeFig. 1), with B0 the bottom width, d0 the depth relative to theundamaged levee crest, and b1 the sides slope angle.

In the first stage, the steepening of the slope angle of the outerface of the levee, from the initial value b0 to the critical value b1,occurs (Fig. 1(a)). The outer face is then eroded at constant angleb1, with unvarying breach width and bottom elevation, until thecrest vanishes (Fig. 1(b)). During these two stages the flow overthe outer slope is assumed uniform and supercritical. The outerslope is then gradually eroded and moved back toward the riverat constant angle b1, yielding a lowering of the top of the levee(Fig. 1(c) on the left), a widening of the breach (Fig. 1(c) on theright), and, progressively, increasing the water discharge throughthe breach. Finally, when the base of the dike is reached, the breachprogressively enlarges (Fig. 1(d)). In the last two stages, the poten-tial increase in the external water level may turn the flow fromcritical to subcritical, slowing down the lateral growth rate of thebreach as a consequence of the reduction in flow velocities. Insome cases, the flux direction can be inverted with water flowingback into the river, essentially because of the lowering of the innerwater level. As suggested by Visser [25], the erosion rate is as-sessed by using Bagnold–Visser and Van Rijn formulas for stagesa–d, respectively.

The original model proposed by Visser [25] did not consideredthe possible presence of a grass cover protecting the downstreamface of the embankment and the erosion process was assumed tostart as the water initiated to overflow. In our model the stabilizingeffect of grass is accounted for following Fread’s [27] approach: thevelocity of the overtopping flow along the grassed downstreamface is computed at each time step by the Manning equation withn0 ¼ aqb, where q is the specific flow rate, a and b are fitting

d

β1

c

β0β1

a

β1

b

Fig. 1. Schematic illustration of different stages of the breach evolution process by ovesteepening of the slope angle of the external face of the levee. Stage b: erosion of the exteelevation. Stage c: erosion of the external face at constant angle b1 yielding a lowering oconstant angle b1.

coefficients suitable to mathematically reproduce the graphicalcurves given in [28]. This velocity is then compared to the maxi-mum permissible velocity (VMP) for grass-lined channels (see[28]). The erosion process is initiated only when the thresholdvelocity (VMP) is exceeded.

In addition, the present model allows one to select the numberof sides of the breach affected by the erosion process (one-sidedbreach or centered breach).

As to the schematization of the piping process, many modelsproposed in the literature [27,29,30,31] consider the flow througha pipe with a rectangular cross section at the beginning of the ero-sion process. The flow through the pipe is then determined usingthe orifice flow equation, and the erosion rate at the pipe perimeteris determined using a sediment transport formula. At each timestep, the pipe is uniformly enlarged according to the volume oferoded sediments until the collapse of the top part of the levee oc-curs, when the weight of the vault of the pipe exceeds soil resis-tance. The collapsed material is assumed to be washeddownstream instantaneously or, in some cases, is transportedalong the breach channel at the current rate of sediment transportbefore further erosion occurs. Once the top part of the levee is col-lapsed, the overtopping flow module is used to simulate the evolu-tion of the originated breach.

In the present model the pipe cross section is assumed to be cir-cular as suggested by Bonelli and Brivois [32]. The model furtherassumes an initially straight pipe, with radius R0 (i.e., dR0=dx ¼ 0)and neglects the influence of sediment concentration on flowvelocity and shear stress at the wall. Under these restrictions, theradius remains uniform along the pipe during the erosion process.

Lachouette et al. [33] performed a numerical investigation on theevolution of an initially straight pipe under constant input and out-put pressure conditions. They found that sediment concentrationcan have significant effects at the very beginning of the erosion pro-cess, resulting in a greater enlargement of the hole close to the out-let. However, they also showed that this spatial non-uniformity

β1

B0

β1

d0

rtopping: longitudinal sections (left panels); cross sections (right panels). Stage a:rnal face of the levee at constant angle b1, with unvarying breach width and bottomf the top of the levee and a widening of the breach. Stage d: breach enlargement at

D.P. Viero et al. / Advances in Water Resources 59 (2013) 82–94 85

rapidly decreases as the erosion process develops thus becomingrapidly negligible. Lachouette et al. [33] therefore concluded thatthe dilute flow assumption, which considerably simplifies thedescription, is suitable for modeling piping erosion flows.

The erosion rate @R=@t is estimated via the Partheniades for-mula for cohesive materials as

@R@t¼

kerqs

s� scð Þ s > sc

0 s 6 sc

(ð3Þ

where R is the radius of the pipe, ker is the coefficient of soil erosion,qs is the soil density, sc is the critical (threshold) shear stress forerosion.

The shear stress at the interface is given by s ¼ c R=2ð ÞJ, with cthe specific weight of water and J ¼ ðHin � HoutÞ=L the energy lossper unit length. The actual inner (Hin) and outer (Hout) drivingwater levels used to compute J are provided, at each time step,by the hydrodynamic module.

Eq. (3) can therefore be rewritten as

@R@t¼ R 1� sc

s

� �1

ters > sc

0 s 6 sc

(ð4Þ

where ter ¼ 2qs= ckerJð Þ is a suitable time scale. In the numericalmodel, Eq. (4) is discretized by a finite difference method, and itis solved using the same time step as for the 1D–2D model.

Given an initial pipe radius R0 and a critical shear stress sc , to beset up by the end-user, the erosion process starts when the actualshear stress s exceeds the threshold value sc .

The vault of the pipe is assumed to collapse when the holediameter exceeds a prescribed threshold value (about 0.5–0.75times the height of the levee). When the vault collapses, a transi-tion from piping to overtopping occurs. The breach cross sectionis consequently reshaped to trapezoidal, and the model estimatesa reliable breach width, B0, at the bottom of the new trapezoidal

a

c

b

N

0 5scale (km)

Fig. 2. (a) Loreggia case study, north of the city of Padova (Italy). The red polygon identifienetwork. (b) Color coded topography of the computational domain. The main roads (blackthe smaller channels (blue thin lines, represented as 1D elements). (c) Aerial view of the faJanuary 21st 2009). The light blue arrow indicates flow direction in the main stream; th

cross section, based on the radius of the hole at collapse, Rmax,and on the maximum angle of breach side slopes, b1, as

B0 ¼ 2Rmax1

cos b1� 1

tan b1

� �ð5Þ

The collapsed material is assumed to be washed downstreaminstantaneously, and the overtopping flow module is then used tosimulate the breach evolution, starting from stage d (Fig. 1(d)).

The proposed breach model also allows one to simulate breachclosure. It is worth pointing out that this is an important feature ofthe model, particularly in the view of hydraulic risk assessment. Aprompt or a delayed breach closure determines, in fact, rather dif-ferent extensions of the flooded area. The possibility of recon-structing different scenarios, in which the breach closure is moreor less delayed and it is advanced at different rates, is thereforeof critical importance in the management of the emergency.

To our knowledge, a few studies dealt with emergency closureof levee breaches, mainly focusing on optimum techniques forthe closure procedures [34,35]. However, the consequences of re-pair activities on the volume of discharged water and flooding ex-tents were not considered.

In our model, the breach closure process is simulated by gradu-ally reducing the width of the breach cross section, while keepingthe slope-side angle unchanged. The time at which breach closurestarts and the rate at which breach width is reduced can be arbi-trarily prescribed.

In the following Sections, the proposed model is applied to fivecase studies and model results are compared with available fielddata. A sensitivity analysis is also presented and discussed.

3. The Muson dei Sassi River case study: the January 2009 floodevent

The Muson dei Sassi River is the main tributary of the middlereach of the Brenta River (Fig. 2(a)) and it is characterized by rap-

20.0

22.5

25.0

27.5

30.0

botto

m e

leva

tion

(m a

sl)

N

0 21scale (km)

s the limits of the computational domain,whereas the blue lines portray the channellines) are also shown, together with the Muson dei Sassi River (blue thick line) and

ilure (breach L) occurred across the left levee of the Muson dei Sassi River at 3 am ofe red arrow indicates the direction of the flow through the breach.

86 D.P. Viero et al. / Advances in Water Resources 59 (2013) 82–94

idly varying water levels. On January 21st 2009, a failure of the leftlevee of the Muson dei Sassi River occurred at about 3 am, causingthe flooding of rural and urban areas (approximately 1 km2) atLoreggia and involving about 200 buildings and 170 families(Fig. 2(c)). To analyze the formation and evolution of the breach(hereafter referred to as breach L) and the related inundation, acomputational mesh was set up (Fig. 2(b)) which reproduces thecourse of the Muson dei Sassi River and the adjacent areas fromCastelfranco Veneto to the confluence with the Brenta River closeto Vigodarzere (Fig. 2(a)). The unstructured triangular grid, consist-ing of about 15,000 nodes and 25,600 triangular elements, wascoupled with 550 1D channel elements and about 1,260 1D-links,one of them being deputed to simulate the levee breach.

The topography of the computational domain was recon-structed using technical maps and cross-sectional and geotechnicalsurveys. As a result of a calibration procedure, Manning coeffi-cients were assumed equal to 0.04 sm�1/3 for the main river andthe minor channel network, 0.067 sm�1/3 for the rural areas, and0.033 sm�1/3 for roads.

The flow hydrograph for the Muson dei Sassi River at Castelfr-anco Veneto (see Fig. 2(a)), which was imposed as upstreamboundary condition, was provided by the Regional Agency for Envi-ronmental Protection of Veneto (ARPAV); water levels measured at

Table 1Piping model: Case A: ker ¼ 0:003 s=m; sc ¼ 13 Pa. Effect of the initial pipe radius R0

on: the time of the levee collapse (tC ), the peak value of the flow rate (Q max), the totalvolume of water (Vt) through the breach, and the final width of the breach (Bb).

R0 ðmÞ 0.03 0.05 0.10tC 26 h 510 25 h 170 24 h 480

Qmax (m3/s) 18.1 23.1 24.8Vt (106m3) 0.789 0.954 1.030Bb (m) 18.5 19.9 20.3

τ(Pa)

0

20

40

60

80

100

0 5 15 20 30

τc=13 Pa

R0=5.0 cm

10 25 t (h)

R0=3.0 cm

R0=10.0 cm

a

(mBb

(m)

0

15

20

25

20 25 35 40 50

R0=5.0 cm

30 45 t (h)

R0=3.0 cm

R0=10.0 cm

10

5

c

Peak of water level

Fig. 3. Piping model: Case A. Results of sensitivity analysis with respect to the initial valradius, R; (c) breach L width at the bottom, Bb; and (d) flow rate through the breach L,

the confluence with the Brenta River were imposed as downstreamboundary condition.

3.1. Model testing and sensitivity analysis

A sensitivity analysis was carried out in order to verify the pip-ing model and to determine its sensitivity to input parameters. Theimpact of the initial pipe radius R0 (Case A, Table 1, and Fig. 3), ofthe critical shear stress sc (Case B, Table 2), and of the coefficient ofsoil erosion ker (Case C, Table 3) were investigated. We focused onthe role exerted by the above parameters on critical quantities forthe flood and breach formation, namely, the instant at which thelevee collapses (tC), the peak value of the flow rate through thebreach (Qmax), the total volume of water through the breach (Vt),and the final width of the breach (Bb). The results of our analysesare shown in Fig. 3 and are summarized in Tables 1–3.

Fig. 3(a) shows the influence of the initial radius, R0, on the tem-poral evolution of the radial shear stress, s: when larger initial R0

values are assumed, the critical shear stress (sc) is reached earlierin time but the maximum shear stress is relatively small. On thecontrary, smaller initial R0 values determine quite a larger shearstress, hence a larger erosion rate, which, however, starts later intime.

According to the pipe model, the piping process begins whenthe shear stress, s, becomes greater then sc , which can occur onlyduring the rising phase of the flood. The dynamics of the pipingprocess can be analyzed by observing that as the pipe radius in-creases, the energy at the outflow of the pipe Hout increases andfriction slope J ¼ ðHin � HoutÞ=L decreases together with the radialshear stress s ¼ c R=2ð ÞJ. A smaller pipe, therefore, produces a high-er peak value of s that occurs close to the peak of inner water level(see the vertical dash-dotted line in Fig. 3(a)). On the contrarywider pipe produces lower peak value of s that occurs before thepeak of inner water level is reached.

R(m)

0

1.0

1.5

0 5 15 20 30

Rmax=1.14 m

R0=5.0 cm

10 25 t (h)

R0=3.0 cm

R0=10.0 cm0.5

b

Q3/s)

0

30

40

50

15 20 30 35 45

R0=5.0 cm

25 40 t (h)

R0=3.0 cm

20

10

Qupstream

R0=10.0 cm

d

ue imposed to pipe radius, R0. Time evolution of: (a) Radial shear stress, s; (b) pipeQ

Table 2Piping model: Case B: ker=0.003 s/m, R0=0.05 m. Effect of the critical shear stress sc

on: the time of the levee collapse (tC ), the peak value of the flow rate (Qmax), the totalvolume of water (Vt) through the breach, and the final width of the breach (Bb).

sc (Pa) 7.0 13.0 17.0tC 23 h 220 25 h 170 27 h 130

Qmax (m3/s) 29.0 23.1 17.3Vt (106 m3) 1.164 0.954 0.796Bb (m) 21.4 19.9 18.2

Table 3Piping model: Case C: sc=13 Pa, R0=0.05 m. Impact of soil erosion coefficient ker on:the time of the levee collapse (tC ), the peak value of the flow rate (Q max), the totalvolume of water (Vt) through the breach, and the final width of the breach (Bb).

ker (s/m) 0.002 0.003 0.005tC 28 h 300 25 h 170 23 h 170

Qmax (m3/s) 14.9 23.1 29.1Vt (106 m3) 0.737 0.954 1.127Bb (m) 17.2 19.9 21.4

D.P. Viero et al. / Advances in Water Resources 59 (2013) 82–94 87

Fig. 3(b) shows the influence of the initial radius, R0, on the tem-poral evolution of the pipe radius, R. As the pipe radius becomeslarger than a critical value, Rmax (2Rmax=75% of the levee height),the collapse of the levee occurs. Interestingly, model results showthat, while the initial value of the pipe radius, R0, strongly influ-ences the time at which erosion starts (Fig. 3(b)), it does not cru-cially affect the time of levee collapse tC (see Table 1 and Fig. 3(b)).

It also emerges that the maximum breach width, as well as itstemporal evolution, is only slightly affected by the initial radiusR0 (Table 1 and Fig. 3(c)). The final width of the breach is mainlyrelated to the magnitude and the persistence of water-level differ-ence ðHin � HoutÞ across the levee, which controls the flow velocityand, consequently, the sediment transport rate. Accordingly, thegrowth rate of the outer water level plays a leading role in deter-mining the final breach width. For this reason, very large breaches

0.00

0.50

1.00

wat

er d

epth

(m

)

Surveyed flooded areas

0 40200scale (m)

N

a

Fig. 4. Loreggia case study (breach L). (a) Comparison between surveyed (magenta) and cand (c) water discharge (Q) 2 km upstream of the breach (blue line), 2 km downstream (denote the results obtained when formation of the breach along the main channel is no

can be produced if topographical conditions of the rural areas adja-cent to the river promote a rapid expansion of the flow down-stream of the breach, thus preventing outer water levels toincrease.

A larger value of the initial pipe radius, R0, is also shown to in-crease the discharge through the breach and to anticipate the peakof the flow rate through the breach Qmax (Table 1 and Fig. 3(d)).

Variations in the critical shear stress for erosion, sc , mainly af-fect the initiation of the piping process, which is the most difficultaspect of the breach formation to be modeled, since it is related tothe presence of preferential hydraulic paths across the levee,through which sediments can be removed. In many practical cases,these preferential hydraulic paths are related to the presence ofnutria burrows, which are conceptualized in the model by assum-ing an initial value for R0. Once the piping process is initiated, itsevolution is not strongly affected by the value assumed for sc

(Table 2).Finally, Table 3 shows the model sensitivity to the soil erosion

coefficient ker . This parameter has a strong impact on the breachsize and on the water volume and rate through the breach. Severalfield and laboratory methods are available to assess model param-eters sc and ker [36–38].

3.2. Model results

Fig. 4 shows the results of the simulations for the Muson deiSassi case study (breach L), in terms of flooded areas (a), watersurface elevations within the river (b), and water discharges at spe-cific locations (c). The values assumed for the parameters of thebreach formation model are provided in Table 4. In particular,the initial value of the pipe radius, R0, was set equal to 5 cm,whereas values of the parameters related to terrain characteristics(ker and sc) were derived from the literature for the case of a siltyterrain [36,37].

The final width of the breach observed at the end of the floodevent was of about 20 m, and the water volume flowed out throughthe breach was estimated to be about 106 m3. Both these values are

0

23

27

29

20 Jan 12:00 21 Jan 22 Jan12:00

25

26

24

WSupstreamWSbreachWSdownstream

30

40

50

20

10

0

Qupstream

QbreachQdownstream

20 Jan 12:00 21 Jan 22 Jan12:00

b

c

Piping

Overtopping

Stages:

Wat

er s

urfa

ce e

leva

tion

(m a

sl)

28

Wat

er d

isch

arge

(m

3 /s)

alculated (color coded water depths) flooded areas. (b) Water surface elevation (WS)black line) and through the breach (red line). For comparison purpose, dashed linest considered.

Table 4Loreggia case study (breach L): input parameters for the breach model.

Initial radius of the pipe R0 0.05 mCoefficient of soil erosion ker 0.003 s/mCritical shear stress sc 13 PaUndamaged levee top width W0 4.2 mUndamaged levee elevation Z0 29.0 m aslLevee inner slope angle a 35�Levee outer slope angle b 30�50th Percentile of terrain particle diameter D50 0.1 mm90th Percentile of terrain particle diameter D90 0.5 mmPorosity n 0.35Terrain internal friction angle u 40�Soil density qs 1700 kg/m3

Critical slope angle b1 60�No. of erodible sides ns 2

88 D.P. Viero et al. / Advances in Water Resources 59 (2013) 82–94

nicely met by the model which predicts a final breach width ofabout 22 m and a water volume of 0.95 �106 m3. Moreover a goodagreement between surveyed and computed flooded areas wasfound (Fig. 4(a)), the minor disagreements being possibly due tothe poor resolution of the terrain elevation model.

The time evolution of water surface elevations within the Mu-son dei Sassi River close to, and downstream of, the breach sug-gests the necessity of coupling hydrodynamic and breachformation models. Breach formation, in fact, induces relevantreductions in both water levels and flow rates along the mainreach. The maximum discharge flowing through the river is equalto 47 m3/s, whereas the maximum discharge through the breachis equal to 23 m3/s (about 47% of maximum river discharge). This

scale (km)0

N

10

b

a

Fig. 5. Geographic location of the Bacchiglione River case studies in the Veneto regionchannels (blue lines), minor 1D channels (light blue lines), and main levees (red lines) ar(b) the Veggiano case study (breach V); and (c) the Ponte San Nicoló case study (South

is shown in Fig. 4(b) and (c) in which we compare water surfaceelevation and water discharge computed by considering (solidlines) and neglecting (dashed lines) the formation of the breach.

4. The Bacchiglione River case studies: the November 2010 floodevent

During the flood event occurred in November 1–3, 2010 fourbreaches formed along the Bacchiglione River. Three of them wereproduced by overtopping: at Due Ponti di Caldogno (breach C1) andBoschi di Caldogno (breach C2), north of the city of Vicenza(Fig. 5(a)), and at Veggiano (breach V), near the confluence of theTesina Padovano and the Bacchiglione river (Fig. 5(b)). The breachoccurred at Ponte San Nicolò (breach P), south of the city of Padova(Fig. 5(c)), was instead triggered by a piping process.

In order to analyze the flood event, the related breach formationand evolution, and the inundation of areas adjacent to the Bacchi-glione River, a computational mesh was set up that reproduces thecourse of the Bacchiglione River, its main tributaries and the adja-cent rural and urban areas from north of Vicenza to the Adriatic sea(see Fig. 5). The most significant channels and hydraulic structureswere also schematized in order to correctly reproduce the hydro-dynamic field. The resulting computational grid (shaded area inFig. 5) consists of about 65,000 nodes, 115,000 triangular elements,2,000 1D channel elements, and 6,000 1D-links.

The topography of the computational domain was recon-structed on the basis of technical maps, LiDAR [39], and cross-sec-tional and geotechnical surveys. An extensive calibration

20

c

(NE Italy) upon Esri Street Map. Computational mesh (shaded areas), main rivere shown. Insets: (a) the Caldogno case study, (North of Vicenza, breaches C1 and C2);of Padova, breach P)

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Fig. 6. Caldogno case studies, north of the city of Vicenza (breaches C1 and C2). (a) Comparison between: surveyed (magenta) and computed (color coded water depths)flooded areas. (b) Measured and computed water surface elevation at Ponte Marchese by considering (red solid line) or neglecting (red dashed line) breaches formation. (c)Aerial view of the failures occurred across the right levee of the Timonchio River; light blue arrows denote the direction of the main stream, red arrows indicate flow directionthrough the breaches.

D.P. Viero et al. / Advances in Water Resources 59 (2013) 82–94 89

procedure against measured data was carried out allowing themodel to correctly reproduce the maximum water levels as wellas the propagation speed of the flood wave along the BacchiglioneRiver, by matching computed and observed peak arrival times atdifferent sections. Manning coefficients were assumed in the rangebetween 0.030 and 0.067 sm�1/3 for the course of the main rivers,and equal to 0.04 sm�1/3 for the 1D channels, 0.067 sm�1/3 for therural areas, and 0.033 sm�1/3 for roads.

Flow hydrographs, which were imposed as upstream boundaryconditions for all the rivers described in the computational do-main, were provided by the Alto Adriatico River Basin Authority.As downstream boundary condition the tidal levels measured atthe Adriatic sea were imposed.

1 For interpretation of color in Figs. 2, 4, 5, 6, 7 and 8, the reader is referred to theweb version of this paper.

4.1. The two breaches developed at Caldogno (Vicenza)

Both the breaches which occurred at Caldogno, north of Vicenza(Fig. 5(a)), were triggered by levee overtopping. One of thebreaches (breach C1) occurred at Due Ponti, along the TimonchioRiver which is a tributary of the Bacchiglione River. The breachC1 started developing on November 1st at about 7.30 am andreached a maximum width of about 90 m. The other breach(breach C2) occurred along the Timonchio River, about 1 km up-stream of the breach C1, at Boschi di Caldogno. This breach starteddeveloping on November 1st at about 8.30 am and reached a max-imum width of about 50 m.

It is important to note that two interventions were carried outin order to reduce the effects of flooding related to the formationof these two breaches. On November 2nd at 12 pm the breach C1

started to be repaired: the activities lasted the whole day and partof the following one. Moreover, in the early morning of November3rd, an artificial cut of the levee was made upstream of Ponte Mar-chese (see Fig. 6(a)) in order to favor the draining of the areasflooded after the formation of the two breaches. However, boththese interventions were carried out after the flow peak arrivaltime and therefore did not provide valuable reductions neither in

the total discharged water volume nor in the extent of the floodedarea.

In order to achieve good simulation results, the initial breachdepth d0 and the grass cover maximum velocity VMP were cali-brated in the range of 0–0.5 m and of 0.6–1.2 m/s, respectively.

The results of the simulations, for the set of parameters listed inTable 5, are shown in Fig. 6 in terms of flooded areas (a) and watersurface elevations within the river (b), whereas Fig. 6(c) shows twoaerial views of the failures captured during the event. Model re-sults nicely match both the time of the levee collapse for the twobreaches (see Fig. 6(b)) and their maximum width: the final mod-eled width of the breach C1 was, in fact, 92 m quite close to theabout 90 m estimated for the actual breach; the final modeledwidth of the breach C2 was 54 m to be compared with the about50 m measured for the actual one. A good agreement was alsofound between surveyed and modeled flooded areas (Fig. 6(a)).An official estimation of the total water volume dischargedthrough the breaches was not available, whereas the modelprovides an estimate of about 17:6 � 106 m3 equally shared be-tween the two breaches. Measured water surface elevations withinthe Bacchiglione River were also available at Ponte Marchese,downstream of the two breaches and of the artificial cut of the le-vee (Fig. 6(b)). It clearly emerges that modeled water levels com-pare favorably with measured ones at the Ponte Marchesesection, and that the model is furthermore capable to correctlyreproduce the increase in water levels due to the artificial cut ofthe levee, carried out to promote the draining of the flooded areas.The effect of the breaches is clearly visible by the comparison of thewater levels computed by neglecting the formation of the breaches(red1 dashed line in Fig. 6(b)). In this case, the maximum dischargeflowing through the Bacchiglione river was equal to 330 m3/s,whereas the peak discharge through the breaches was equal to

Table 5Caldogno case studies (breaches C1 and C2): input parameters for the breach model.

Initial breach width B0 2.0 mInitial breach depth d0 0.2 mGrass cover maximum velocity VMP 0.8 m/sGrass cover fitting coefficient a 0.085Grass cover fitting coefficient b �0.22Undamaged levee top width W0 3.5 mC1 undamaged levee elevation Z0 49.5 m aslC2 undamaged levee elevation Z0 51.8 m aslLevee inner slope angle a 30�Levee outer slope angle b 30�50th Percentile of terrain particle diameter D50 0.3 mm90th Percentile of terrain particle diameter D90 10 mmPorosity n 0.35Terrain internal friction angle u 40�Soil density qs 1700 kg/m3

Critical slope angle b1 60�No. of erodible sides ns 2

90 D.P. Viero et al. / Advances in Water Resources 59 (2013) 82–94

240 m3/s (about 75% of maximum river discharge) thus importantlyinfluencing water levels within the river.

4.2. The Veggiano breach

The Veggiano levee failure (breach V) was also triggered byovertopping. The right levee of the Tesina Padovano River failednearby the connection with Bacchiglione River (Fig. 7). The over-topping started in the first hours of November the 2nd, while theerosion process started only at about 6 am, because of the grasscover protecting the downstream face of the levee. The maximumwidth of the breach, of about 35 m, was reached in few hours. Allmodel parameters used in the simulation are summarized in Ta-ble 6. The initial breach depth d0 and the grass cover maximumvelocity VMP were calibrated in the range of 0–0.2 m and of 0.6–1.2 m/s, respectively.

Fig. 7 shows the results of the simulation in terms of floodedareas (Fig. 7(a)) and water surface elevations at the Ponte dei Ped-agni cross section within the Tesina Padovano River and at the con-

a

m a

sl

b

c

0 21scale (km)

N

Surveyed flooded area0.0

2.0

4.0

wat

er d

epth

(m

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Fig. 7. Veggiano case study (breach V). (a) Comparison between surveyed (magenta)computed water surface elevation at Ponte dei Pedagni and at the confluence with th(dashed lines) the breach formation. (c) Aerial view of the failures of the right embankmered arrow denotes the flow through the breach.

fluence with the Bacchiglione River (Fig. 7(b)). Fig. 7(c) shows anaerial view of the failure captured during the event.

The maximum width of the breach computed by the model was33 m which nicely matches the observed one of about 35 m. A goodagreement was also found by comparing surveyed and computedflooded areas (Fig. 7(a)). Official estimation of the total water vol-ume passed through the breach was not available, while the modelestimated a value was about 3:1 � 106 m3. Fig. 7(b) shows that themodel correctly reproduced the time evolution of water levelsmeasured at Ponte dei Pedagni cross section, nearby Veggiano.The breach formation, in this case, did not affect water levels with-in the main course of the river, as clearly shown by the comparisonof the results obtained with the numerical model by either consid-ering or neglecting the breach formation (solid and dashed lines inFig. 7(b)). In this case, in fact, the maximum discharge flowingthrough the Bacchiglione river was equal to 660 m3/s, whereasthe peak discharge through the breaches was equal to 140 m3/s(about 21% of maximum river discharge) with minor influence onwater levels within the river as compared to the previous cases.It is finally worth noting that the Tesina Padovano contributionto the Bacchiglione flood wave was not very important, especiallyclose to the peak of the discharge when the maximum water levelsalong the final reach of Tesina Padovano River were mainly drivenby water levels at the confluence. This clearly emerges fromFig. 7(b), where the computed water levels within the main courseof the Bacchiglione River (green line) almost overlap those at Pontedei Pedagni, located about 6 km upstream of the confluence alongthe Tesina Padovano river (red line).

4.3. The Ponte San Nicolò breach

South of the center of Padova the Bacchiglione River flows in theeast–west direction. This part of the river is actually an artificialchannel (Scaricatore channel, see Fig. 5(c)) which was dug duringthe nineteenth century in order to prevent the flooding of the city.At the end of the Scaricatore channel (Voltabarozzo hydraulicnode) two sets of sluice gates (Fig. 5(c)) control the flow: a first

1 Nov14

17

18

15

16

19

2 Nov 3 Nov 4 Nov 5 Nov

Water surface elevations

Ponte dei Pedagni (modeled)Ponte dei Pedagni (measured)

Confluence (modeled)

without breach formation

and calculated (color map) flooded areas. (b) Comparison between measured ande Bacchiglione River in two different cases: considering (solid lines) or neglectingnt of the Tesina Padovano River: light blue arrow denotes the main stream direction,

Table 6Veggiano case study (breach V): input parameters for the breach model.

Initial breach width B0 2.0 mInitial breach depth d0 0.01 mGrass cover maximum velocity VMP 1.0 m/sGrass cover fitting coefficient a 0.085Grass cover fitting coefficient b �0.22Undamaged levee top width W0 3.5 mUndamaged levee elevation Z0 19.7 m aslLevee inner slope angle a 29�Levee outer slope angle b 28�50th Percentile of terrain particle diameter D50 0.3 mm90th Percentile of terrain particle diameter D90 1.0 mmPorosity n 0.3Terrain internal friction angle u 40�Soil density qs 1700 kg/m3

Critical slope angle b1 60�No. of erodible sides ns 2

D.P. Viero et al. / Advances in Water Resources 59 (2013) 82–94 91

set of four gates regulates the flow toward the lower part of theBacchiglione River, while a second set of two gates allows thediversion of part of the discharges flowing along the BacchiglioneRiver toward the Brenta River through the San Gregorio–Piovegochannel. The lower course of the Bacchiglione River, downstream

c

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ba

Fig. 8. (a) Aerial view of the failure (breach P) of the right bank of Bacchiglione River (9denotes the main stream direction, red arrow denotes the flow through the breach. (bBacchiglione River at Bovolenta in two different cases: considering (solid red line) or negsurveyed (magenta) and computed (color map) flooded areas. (d) Breach width time evowitnesses. (e) Water surface levels inside and outside the river; the inside water level wabreach formation. Green shaded area denotes piping stage, red shaded area denotes the pclosure phase. The period of time during which submerged flow occurs is also highlight

of the Voltabarozzo node, flows southward toward the town ofBovolenta where it receives the inflow of the Cagnola channel.

During the flood event of November 2010 a breach formed(breach P) across the right levee of the Bacchiglione River down-stream of Ponte San Nicolò (see Fig. 5(c)), thus leading to the flood-ing of an area of about 21 km2. The Ponte San Nicolò breachdeveloped across the right levee of the Bacchiglione River next toa culvert structure (Fig. 8(a)). Because of the presence of the culvertthe breach developed only downstream, the upstream side beingprotected by the structure. As reported by witnesses, the breachformation started around midnight between the 1st and the 2ndof November. At about 3 am the water flowing through a breachwide about 10 m caused the black out of the electrical equipmentof an adjacent rubbish dump. The breach width increased rapidlyand reached approximately 30 m at 6 am and the maximum widthof nearly 40 m at 9 am, when activities aiming at stopping thebreach enlargement and at repairing the levee began. The floodedareas pertain to the Patriarcati reclamation district, bounded by theScaricatore channel (North), the Bacchiglione River (East) and theCagnola channel (South), the three rivers being characterized bythe presence of man-made embankments. The whole area iscrossed by a dense network of minor channels with many hydrau-

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am of 2nd November 2010), nearby a culvert structure (red circle); light blue arrow) Comparison between measured and modeled water surface elevation within thelecting (dashed red line) breach closure during the event. (c) Comparison betweenlution; red circles denote the width of the breach at different time, as reported bys computed both considering (blue solid line) and neglecting (blue dashed line) the

hase of breach enlargement by overtopping and blue shaded area denotes the breached.

Table 7Ponte San Nicolò case study (breach P): input parameters for the breach model.

Initial radius of the pipe R0 0.25 mCoefficient of soil erosion ker 0.003 s/mCritical shear stress sc 5.0 PaUndamaged levee top width W0 5.0 mUndamaged levee elevation Z0 10.2 m aslLevee inner slope angle a 25�Levee outer slope angle b 11�50th Percentile of terrain particle diameter D50 0.1 mm90th Percentile of terrain particle diameter D90 0.5 mmPorosity n 0.3Terrain internal friction angle u 40�Density of the soil qs 1700 kg/m3

Critical value slope angle b1 60�No. of erodible sides ns 1

92 D.P. Viero et al. / Advances in Water Resources 59 (2013) 82–94

lic regulation structures for flood protection and irrigation. Amongthese, seven pump stations are present for a total discharge ofabout 50 m3/s.

The opening of each of the two sets of sluice-gates at the Volta-boarozzo hydraulic node was recorded during the flood event, thusenabling us to correctly reproduce, in particular, the dischargeflowing along the lower course of the Bacchiglione River, wherethe breach formed. The sluice gates were modeled using 1D-linkswhich simulate the flow through this type of hydraulic structuresby considering both the case of free flow and the case of sub-merged flow. In order to correctly account for the Cagnola channelflow contribution, measured water levels were imposed at the up-stream cross section of this channel. Measured tidal levels, at theBrenta–Bacchiglione outflow in the Adriatic sea were also imposedas downstream boundary conditions.

The water flowing out of the breach P mainly propagated south-ward reaching the urban area of Bovolenta (Fig. 5(c)). During thedraining operations after the flood event, which lasted 6 days, awater volume of about 15 � 106 m3 was pumped by the seven pumpstations into the Cagnola channel and the Bacchiglione River. Thetotal volume flowed out through the breach, after subtracting therainfall volume over the basin, was estimated to be about13� 14 � 106 m3.

Table 7 provides the values of the input parameters of thebreach model. The critical shear stress sc and the coefficient of soilerosion ker were calibrated in the range of 5–15 Pa and of0.002–0.005 m/s, respectively. In particular the critical shear stresssc ¼ 5 Pa was chosen among the smallest values suitable for siltyterrain (see [36,37]) to better reproduce the very fast formationof the breach.

The levee breach closure started at 9 am of November 2nd, dur-ing the flood event, and lasted approximately one day (blue shadedarea in Fig. 8). The intervention was simulated with the model byassuming a reduction of the breach width, linear in time, at the rate@B=@t=-1.5 m/h.

Model results show a good agreement with measured and esti-mated data in terms of water levels within the Bacchiglione Riverat Bovolenta (red solid line and dots in Fig. 8(b)), breach growthrate, and maximum width (Fig. 8(d)).

The computed outflowed water volume of 14 � 106 m3 was thenconsistent with the estimated one (13� 14 � 106 m3). A quite goodagreement was also found between surveyed and computedflooded areas (Fig. 8(c)): uncertainty in defining the rural areasactually flooded and possible failures of the minor channel net-work can reliably explain local discrepancies.

The time evolution of water levels nearby the breach (blue andmagenta solid lines in Fig. 8(e)) confirms the effectiveness of thecoupled numerical model. Due to the levee failure, in fact, waterlevels within the river significantly reduced (more than 1 m) with

respect to a hypothetical event without levee failure (dashed blueline in Fig. 8(e)). Moreover, water levels outside of the river in-creased and produced submerged flow through the breach for morethen 10 h (Fig. 8(e)), thus affecting both the breach evolution andthe water discharge through the breach. This behavior further con-firms the necessity of coupling hydrodynamic and breach models.

In order to show the importance of including in the model thepossibility of simulating the levee breach closure, a further simula-tion was run in which the breach was allowed to enlarge duringthe whole flood event and a final breach width of nearly 60 meterswas computed. In this case, computed water surface elevationswithin the Bacchiglione River at Bovolenta did not meet the mea-sured ones (see red dashed line in Fig. 8(b)), and the computed out-flowed water volume of approximately 35 � 106 m3 is much largerthan the estimated one.

5. Conclusions

We have developed a 1D-link model to simulate breach forma-tion and evolution in fluvial levees, either triggered by piping orovertopping. The breach formation model is then coupled with a1D–2D hydrodynamic model and tested by considering five casestudies. The results from our simulations highlight model effec-tiveness in reproducing both the levee failure and the subsequentflooding processes. Model results are, in fact, consistent with fieldobservations displaying a good agreement in terms of floodedareas, water levels within the channel, final breach widths, andwater volumes flowed out through the breach. The model accountsfor critical feedbacks between breach formation and changes in thehydrodynamic flow field on the basis of a limited number ofparameters, thus providing means to improve our understandingof the related processes and our capability of simulating possibleemergency scenarios. The main conclusions of this paper can besummarized as follows.

Our analysis emphasizes that the coupling between breach-for-mation and hydrodynamic models is of critical importance,because the evolution of the breach is strongly controlled by thetemporal variability of the hydrodynamic conditions, both withinthe channel and over the adjacent flooded areas, during levee col-lapse. Water discharge through the breach, in fact, can often becomparable with the discharge conveyed by the river, thus signif-icantly affecting water levels both within the river and over theadjacent areas. Water levels within the river play a key role indetermining the flow rate through the breach and, consequently,the breach growth rate, while the raising of the outer water levelis the main factor limiting the water volume flowed out throughthe breach and the breach final width, via tailwater effects.

The evolution in time of the breach and, as a consequence, thepeak discharge and total volume of water through the breach, areinfluenced by possible tailwater effects, by the shape and the dura-tion of the flood wave, and by the possible repair of the collapsedlevee during the event, as suggested by the model applications tothe case studies described above. In particular, when consideringthe upper-middle course of rivers, usually the elevation of adjacentunchanneled areas is steep enough to prevent significant tailwatereffects, whereas flood waves are characterized by short durationwhich typically determines breach enlargement to stop very rap-idly. Due to the rapid evolution of the process, in these cases rep-aration activities usually take place at the end of the flood eventand therefore do not significantly affect the inundation process.On the contrary, in the lower course of rivers, flood waves are char-acterized by long duration and adjacent areas are generally flat.Breach enlargement can therefore be counteracted by tailwater ef-fects and a prompt reparation can be performed during the floodevent providing beneficial effects in reducing the extent of flooded

D.P. Viero et al. / Advances in Water Resources 59 (2013) 82–94 93

areas. It is therefore worth emphasizing that the capability of themodel of reproducing, besides of the tailwater effects, also the con-sequences of reparation activities is crucial to correctly simulateactual events in which these type of intervention were performed,as shown by model results. Moreover the possibility of simulatingthe effects of human intervention for levee reparation can also pro-vide a useful tool for civil protection purposes.

The sensitivity analysis presented in Section 3.1 shows that theparameters controlling breach initiation has a non-negligible im-pact on the early evolution of the breach. However, we also showthat, once the breach is initiated, these parameters (such as the ini-tial radius of the pipe) and the geotechnical characteristics of thelevees slightly affect its evolution (e.g. grow rate, final width, totaldischarge through the breach) when compared with the role ofhydraulic driving conditions.

The high level of uncertainty concerning the localization of thebreach, as well as the description of the processes driving the ini-tiation phase of the breaching process, is known to be an open task[23,24]. In the case of overtopping, the locations of potential leveebreaches can be predicted with some confidence since the hydro-dynamic model reliably predicts the most overflowing-prone le-vees. In this case, however, the breach initiation is largelyaffected by the threshold velocity VMP which controls the inceptionof the erosion process of the outer levee slope when protected by agrass cover.

In the case of piping, an a priori localization of the breach israther unfeasible. Moreover, the early stage of the process is af-fected by the possible presence of preferential paths such as ani-mals’ burrows or discontinuity surfaces due to the constructionmethodology or to the presence of hydraulic structures acrossthe levee.

Accordingly, the model can profitably be used to construct plau-sible inundation scenarios. In addition, a predictive analysis ofinundation scenarios, in the statistical sense, could be performedby generating an ensemble of model runs based on a probabilisticsampling of the possible levee failure locations and mechanisms, assuggested by, e.g., [8,11,13].

Acknowledgments

This work was carried out within the 2008 University of Padovaproject ‘‘Analysis of the stability of fluvial levees through mathe-matical modeling and non-invasive geophysical investigations’’(CPDA088893/08), that is gratefully acknowledged. A.D. thanks‘‘Thetis spa’’ for financial support, L.C. thanks ‘‘Fondazione Vajont’’for financial support.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.advwatres.2013.05.011. These data include Google maps of the most importantareas described in this article.

References

[1] Jonkman SN, Vrijling JK. Loss of life due to floods. J Flood Risk Manage2008;1(1):43–56. http://dx.doi.org/10.1111/j.1753-318X.2008.00006.x.

[2] Messner F, Penning-Rowsell E, Green C, Meyer V, Tunstall S, van der Veen A.Evaluating flood damages: guidance and recommendations on principles andmethods. FLOODsite rep. T09-06-01. FLOODsite Consortium, Wallingford, UK;2007. Available at: <www.floodsite.net>.

[3] Bladé E, Gómez-Valentín M, Dolz J, Aragón-Hernández JL, Corestein G,Sánchez-Juny M. Integration of 1D and 2D finite volume schemes forcomputations of water flow in natural channels. Adv Water Resour2012;42:17–29. http://dx.doi.org/10.1016/j.advwatres.2012.03.021.

[4] Wahl TL. Prediction of embankment dam breach parameters: a literaturereview and needs assessment. Dam safety research report DSO-98-004, USDept. of the Interior, Bureau of Reclamation, Denver, Colorado; July 1998.

[5] Faeh R. Numerical modeling of breach erosion of river embankments. J HydrolEng 2007;133(9):1000–9. http://dx.doi.org/10.1061/(ASCE)0733-9429(2007)133:9(1000).

[6] Morris MW, Kortenhaus A, Visser PJ, Hassan MAAM. Breaching processes: Astate of the art review. FLOODsite rep. T06-06-03, FLOODsite Consortium,Wallingford, UK; 2009. Available at: <www.floodsite.net>.

[7] Bates PD, Horritt MS, Aronica G, Beven K. Bayesian updating of floodinundation likelihoods conditioned on flood extent data. Hydrol Process2004;18:3347–70. http://dx.doi.org/10.1002/hyp.1499.

[8] Dawson R, Hall J, Sayers P, Bates P, Rosu C. Sampling-based flood risk analysisfor fluvial dike systems. Stoch Environ Res Risk Assess 2005;19:388–402.http://dx.doi.org/10.1007/s00477-005-0010-9.

[9] Di Baldassarre G, Castellarin A, Montanari A, Brath A. Probability-weightedhazard maps for comparing different flood risk management strategies: a casestudy. Nat Hazards 2009;50:479–96. http://dx.doi.org/10.1007/s11069-009-9355-6.

[10] Di Baldassarre G, Schumann G, Bates PD, Freer JE, Beven KJ. Floodplainmapping: a critical discussion of deterministic and probabilistic approaches.Hydrol Sci J 2010;55(3):364–76. http://dx.doi.org/10.1080/02626661003683389.

[11] Vorogushyn S, Merz B, Lindenschmidt KE, Apel H. A new methodology for floodhazard assessment considering dike breaches. Water Resour Res2010;46:W08541. http://dx.doi.org/10.1029/2009WR008475.

[12] Neal J, Keef C, Bates P, Beven K, Leedal D. Probabilistic flood risk mappingincluding spatial dependence. Hydrol Process 2013;27:1349–63. http://dx.doi.org/10.1002/hyp.9572.

[13] Mazzoleni M, Bacchi B, Barontini S, Di Baldassarre G, Pilotti M, Ranzi R.Flooding hazard mapping in floodplain areas affected by piping breaches in thePo river Italy. J Hydrol Eng 2013. http://dx.doi.org/10.1061/(ASCE)HE.1943-5584.0000840.

[14] Defina A, D’Alpaos L, Matticchio B. A new set of equations for very shallowwater and partially dry areas suitable to 2D numerical models. In: Molinaro P,Natale L, editors. Modelling flood propagation over initially dry areas. NewYork: American Society of Civil Engineers; 1994. p. 72–81.

[15] D’Alpaos L, Defina A, Matticchio B. A coupled 2D and 1D finite element modelfor simulating tidal flow in the Venice channel network. In: Proceedings of theninth international conference on finite elements in fluids, Venezia; 15–21October 1995. p. 1397–406.

[16] Defina A. Two dimensional shallow flow equations for partially dry areas.Water Resour Res, 0043-1397 2000;36(11):3251–64. http://dx.doi.org/10.1029/2000WR900167.

[17] Martini P, Carniello L, Avanzi C. Two dimensional modelling of flood flows andsuspended sediment transport: the case of Brenta River, Veneto (Italy). NatHazard Earth Sys 2004;4(1):165–81. http://dx.doi.org/10.5194/nhess-4-165-2004.

[18] D’Alpaos L, Defina A. Mathematical modeling of tidal hydrodynamics inshallow lagoons: a review of open issues and applications to the Venicelagoon. Comput Geosci 2007;33:476–96. http://dx.doi.org/10.1016/j.cageo.2006.07.009.

[19] Rinaldo A, Nicotina L, Alessi Celegon E, Beraldin F, Botter G, Carniello L, et al.Sea level rise, hydrologic runoff, and the flooding of Venice. Water Resour Res2008;44:W12434. http://dx.doi.org/10.1029/2008WR007195.

[20] Carniello L, D’Alpaos A, Defina A. Modeling wind waves and tidal flows inshallow micro-tidal basins. Estuar Coast Shelf Sci 2011. http://dx.doi.org/10.1016/j.ecss.2011.01.001.

[21] Defina A. Numerical experiments on bar growth. Water Resour Res2003;39(4):ESG21–ESG212. http://dx.doi.org/10.1029/2002WR001455.

[22] Defina A, Susin FM. Multiple states in open channel flow. In: Brocchini M,Trivellato F, editors. Vorticity and turbulence effects in fluid structuresinteractions-advances in fluid mechanics. Southampton: Wessex Institute ofTechnology Press; 2006. p. 105–30.

[23] Morris MW, Kortenhaus A, Visser PJ. Modeling breach initiation and growth.FLOODsite rep. T06-08-02, FLOODsite Consortium, Wallingford, UK; 2009.Available at: <www.floodsite.net>.

[24] ASCE/EWRI Task Committee on Dam/Levee Breaching. Earthen embankmentbreaching. J Hydraul Eng 2011;137(12):1549–64. http://dx.doi.org/10.1061/(ASCE)HY.1943-7900.0000498.

[25] Visser PJ. Breach growth in sand-dikes. Ph.D. thesis, Delft University ofTechnology, Delft, the Netherlands; 1998.

[26] Visser PJ, Zhu Y, Vrijling JK. Breaching of dikes. Proceedings of the 30thinternational conference on coastal engineering, San Diego, USA; 2006. p.2893–905.

[27] Fread DL. BREACH: an erosion model for earthen dam failures. Silver Spring,Maryland, USA: Model description and User Manual. NOAA; 1988.

[28] Chow VT. Open-channel hydraulics. New York: McGraw-Hill Book Co; 1959.[29] Paquier A, Nogues P, Herledan R. Model of piping in order to compute

dambreak wave. In: Morris M, editor. Concerted action on dambreakmodelling (CADAM Meeting). Germany: Munich; 1998.

[30] Mohamed MAA, Samuels PG, Morris MW, Ghataora GS. Improving theaccuracy of prediction of breach formation through embankment dams andflood embankments. International conference on fluvial hydraulics (Riverflow2002), Louvain-la-Neuve, Belgium; September 2002.

94 D.P. Viero et al. / Advances in Water Resources 59 (2013) 82–94

[31] Wu W, Kang Y, Wang SSY. An earthen embankment breach model. Proceedingof the 33rd Congress of IAHR 2009. Vancouver, Canada.

[32] Bonelli S, Brivois O. The scaling law in the hole erosion test with a constantpressure drop. Int J Numer Anal Methods Geomech 2008;32:1573–95. http://dx.doi.org/10.1002/nag.683.

[33] Lachouette D, Golay F, Bonelli S. One-dimensional modeling of piping flowerosion. CR Mecanique 2008;336:731–6. http://dx.doi.org/10.1016/j.crme.2008.06.007.

[34] Sattar AMA, Kassem AA, Chaudhry MH. Case study: 17th street canal breachclosure procedures. J Hydrol Eng 2008;134(11):1547–58. http://dx.doi.org/10.1061/(ASCE)0733-9429(2008)134:11(1547).

[35] Chaudhry MH, Elkholy M, Riahi-Nezhad C. Investigations on levee breachclosure procedures. Dep. of Civil and Environmental Engineering: TechnicalReport, University of South Carolina; 2010.

[36] Wan CF, Fell R. Investigation of rate of erosion of soils in embankment dams. JGeotech Geoenviron 2004;130(4):373–80. http://dx.doi.org/10.1061/(ASCE)1090-0241(2004)130:4(373).

[37] Wan CF, Fell R. Laboratory tests on the rate of piping erosion of soils inembankment dams. Geotech Test J 2004;27(3):295–303. http://dx.doi.org/10.1520/GTJ11903.

[38] Hanson GJ, Cook KR. Apparatus, test procedures and analytical methods tomeasure soil erodibility in situ. Appl Eng Agric 2004;20(4):455–62.

[39] Bates PD, Marks KJ, Horritt MS. Optimal use of high-resolution topographicdata in flood inundation models. Hydrol Process 2003;17:537–57. http://dx.doi.org/10.1002/hyp.1113.