Dam-break Modeling in Alpine Valleys

J. Mt. Sci. (2014) 11(6): 1429-1441

e-mail: [email protected]

http://jms.imde.ac.cn DOI: 10.1007/s11629-014-3042-0

Dam-break Modeling in Alpine Valleys

Marco PILOTTI1*, Andrea MARANZONI2, Luca MILANESI1, Massimo TOMIROTTI1, Giulia VALERIO1 1 DICATAM, Università degli Studi di Brescia, Via Branze 43, 25123 - Brescia, Italy 2 DICATeA, Parco Area delle Scienze 181/A, 43124 - Parma, Italy *Corresponding author, e-mail: [email protected]; 2nd author, e-mail: [email protected]; 3rd author e-mail: [email protected]; 4th author, e-mail: [email protected]; 5th author, e-mail: [email protected] Citation: Pilotti M, Maranzoni A, Milanesi L, et al. (2014) Dam-break modeling in alpine valleys. Journal of Mountain Science 11(6). DOI: 10.1007/s11629-014-3042-0

© Science Press and Institute of Mountain Hazards and Environment, CAS and Springer-Verlag Berlin Heidelberg 2014

Abstract: Dam-break analysis is of great importance in mountain environment, especially where reservoirs are located upstream of densely populated areas and hydraulic hazard should be assessed for land planning purposes. Accordingly, there is a need to identify suitable operative tools which may differ from the ones used in flat flood-prone areas. This paper shows the results provided by a 1D and a 2D model based on the Shallow Water Equations (SWE) for dam-break wave propagation in alpine regions. The 1D model takes advantage of a topographic toolkit that includes an algorithm for pre-processing the Digital Elevation Model (DEM) and of a novel criterion for the automatic cross-section space refinement. The 2D model is FLO-2D, a commercial software widely used for flood routing in mountain areas. In order to verify the predictive effectiveness of these numerical models, the test case of the Cancano dam-break has been recovered from the historical study of De Marchi (1945), which provides a unique laboratory data set concerning the consequences of the potential collapse of the former Cancano dam (Northern Italy). The measured discharge hydrograph at the dam also provides the data to test a simplified method recently proposed for the characterization of the hydrograph following a sudden dam-break. Keywords: Dam-break modeling; Shallow water Received: 1 March 2014 Accepted: 16 June 2014

equations; Alpine valleys; Case study; Hydraulic hazard

Introduction The risk assessment of dam failure is a topic that deserves great attention since, although its probability of occurrence is rather small, the related consequences might be catastrophic. The problem could become even worse in the future because of the intensification of extreme meteorological events, the general increase of exposure, and the ageing of the structures. The growth of exposure is related to the increased number of new reservoirs built over the last decades in developing countries and to the expansion of strongly populated areas in developed nations. Therefore, in many countries, technical guidelines concerning dam safety require hydraulic studies aimed to estimate hazard levels in dambreak flood prone areas. This requires the use of suitable mathematical tools capable of effectively modeling the dynamics of very rapid floods. In alpine valleys, dam-break wave propagation may be very complex because of the interaction of the impulsive nature of the process with the strong irregularity of natural topographies. This particular

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feature suggests the adoption of purposely devised computational tools. Despite the hypothesis of spatially gradually varied flow is often violated in practical applications, especially in mountain environment, SWE are usually accepted in the literature and used in the engineering practice dealing with flood propagation problems (e.g., Soares Frazão et al. 2003; Capart et al. 2003; Aureli et al. 2008a; Petaccia et al. 2008; Pilotti et al. 2011). Indeed, SWE conveniently combine computational efficiency and satisfactory reconstruction of the physical phenomenon. This paper compares a one-dimensional (1D) and a two-dimensional (2D) shallow water model for dam-break wave propagation in alpine regions. The 1D de Saint-Venant equations are solved using a finite volume numerical scheme based on the approach proposed by Capart et al. (2003). It takes advantage of a topographic toolkit that includes an algorithm for pre-processing the Digital Elevation Model (DEM) and a novel criterion for the automatic cross-section refinement. The 2D model is based on the commercial software FLO-2D, widely used for hydraulic hazard mapping in mountain areas, mostly for debris-flow and mudflow. Recently its performance in granular debris-flow propagation was comparatively tested by Wu et al. (2013). Accordingly, it is interesting to explore its performances also when dam-break wave propagation is concerned. Although 2D models are gaining increasing popularity and provide a more accurate description of the hydrodynamic process, the 1D schematization is still widely applied whenever the topography of the valley and the discharge involved justify this assumption. A priori, the main reasons are the greater practitioners’ acquaintance with 1D modelling, the more limited computational burden and implementation effort, and the need of a sufficiently detailed DEM for 2D studies, which may be difficult to obtain in some cases. Moreover, 1D models could be still competitive against 2D models with respect to the capability of predicting the flood dynamics and the inundation extent in rivers and streams (Horritt and Bates 2002). In very wide domains, in order to limit the computational time, 1D and 2D approaches can be sinergically coupled (Fernández-Nieto et al. 2010): the former can be used for the wave propagation in channel-like stretches, the second for the flood

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propagation in flood plains where the onedimensionality of the flow, the uniformity of velocity, and horizontality of water level across the cross-section are violated. Accordingly, a comparative analysis of the performances provided by 1D and 2D modelling is useful, although rather rare in literature (e.g., Horritt and Bates 2002; Tayefi et al. 2007). Pilotti et al. (2011) showed that for an accurate description of very steep and irregular natural bathymetries by means of a 1D approach, it could be necessary to have closely spaced cross-sections, which would be impossible to acquire by conventional field survey procedures. In this direction, the increasing availability of detailed DEMs has fostered the development of efficient geographical information systems (GIS) able to analyse, process, and manage massive quantities of spatial data. Many algorithms for the automatic extraction of drainage networks from digital elevation models have been proposed in literature (e.g., Pilotti et al. 1996), almost exclusively for space-distributed hydrological and environmental applications (e.g., Pilotti and Bacchi 1997). In this paper these methods are used to deal with a hydraulic problem and an algorithm that performs a space adaptive sampling along the drainage network obtained from a DEM is presented. Consequently, the cross-sections (and the associated geometric information) were extracted with an adaptive spatial resolution. Commonly, the reliability of the results obtained using SWE is firstly verified on the basis of theoretical test cases with analytical solution that usually are characterized by simple geometry and mostly by frictionless bottom (Ritter 1892; Stoker 1948; Thacker 1981; McDonald et al. 1997; Liska and Wendroff 1998; Liska and Wendroff 1999). However, the numerical modeling of flood wave propagation in mountain regions demands that the chosen mathematical and numerical tool simultaneously deals with several stumbling blocks (e.g. wet/dry boundaries, non prismatic geometry, non uniform bed slope, transcritical flows, shocks) that in the simplified literature test cases are dealt with separately. For this reason, laboratory measures and field data concerning real events are valuable information for validating numerical modeling, even if they do not provide the exactness

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of analytical solutions. Data collected during laboratory investigations usually concern idealized situations and schematic geometries (e.g., Soares Frazão 2007; Soares Frazão and Zech 2007; Aureli et al. 2008b). Very few are the experimental data from physical models reproducing a real topography. In particular, the most known physical model is probably the one built by Electricité de France in 1964 to reproduce the Malpasset dambreak (e.g., Goutal 1999; Hervouet and Petitjean 1999; Valiani et al. 2002), where maximum water depths were measured at 14 gauges. Other experimental data widely used in literature (Soares Frazão and Testa 1999; Brufau et al. 2002; Caleffi et al. 2003) concern the flood wave propagating in the physical model built by ENEL–HYDRO (Italy) reproducing a portion of the Toce River valley (located in the Northern Italian Alps) and including water depth hydrographs measured at 33 selected points. On the other hand, documented historical dam-break events are rather rare. Among these, the well-known catastrophic events of Malpasset (Goutal 1999), Tous (Alcrudo and Mulet 2007), St. Francis (Begnudelli and Sanders 2007), Gleno (Pilotti et al. 2011), and Lawn Lake dam (Jarrett and Costa 1986) can be mentioned. In general, the available information is rather limited, scattered and uncertain, but nonetheless it can be useful to define dam-break test cases and verify the capability of numerical models to reproduce the aspects of the flood event that are most relevant for hazard classification (maximum water levels and velocity, flood extent, and timing of the flood wave). This paper shows for the first time the application of the numerical models to the case study examined by De Marchi (1945) and concerning the propagation along the Adda River (Northern Italy) of the flood wave consequent to the potential collapse of the first Cancano dam as a possible war target during World War II. The resulting report of De Marchi is very interesting, since it synergistically mixes theoretical, experimental, and numerical considerations. In particular, the measured discharge hydrographs reported in that paper can be used to assess the performances of SWE and the reliability of numerical results in mountain applications. Furthermore, the measured discharge hydrograph at the dam provides the opportunity to validate the simplified method proposed by Pilotti et al. (2010)

for the characterization of following a sudden dam-break.

1

the

hydrograph

1D Modeling

One-dimensional flood propagation modeling requires a proper treatment of the source term, especially in the presence of natural complex bathymetries. In fact, in the classical formulation of the SWE (e.g. Toro 2001), the geometrical source terms involving the spatial gradients of bottom elevation and of cross-sectional width cannot be easily evaluated from the topographical information, usually based on a limited number of cross-sections. In order to cope with these difficulties, Capart et al. (2003) proposed an approximate evaluation of the pressure force acting on the surface of the wetted boundary that allows transferring the geometric source terms (that account for non-prismaticity and bed slope) within the momentum flux, enhancing mass and momentum conservation properties of the numerical model in the simulations over real bathymetries. As shown by Capart et al. (2003) and Chen et al. (2007), the consequent reformulation of the governing equations does not introduce remarkable inaccuracies, even in the presence of hydraulic jumps or shock waves. The numerical integration of the modified SWE was performed by the first-order algorithm of Capart et al. (2003), which makes use of the PFP (Pavia Flux Predictor) upwind method proposed by Braschi and Gallati (1992) for the numerical flux prediction. A detailed description of this numerical model can be found in Pilotti et al. (2011). However, despite its robustness and suitability to unsteady river flow modelling over natural topography, this scheme requires a very fine computational mesh in the presence of a steep water surface slope combined with a steep bathymetry, typical of mountain streams (Capart et al. 2004). This kind of spatial discretization can increase the computational effort. 1.1 DEM pre-processing and cross-section extraction In river flow computations, an extensive numerical DEM pre-processing is usually needed

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to extract the geometryy of the modeelled system.. If a oach is adopted, a topograp phic 1D appro representattion of the valley v by crosss-sections must m be provideed. This can be done ussing the chan nnel and hillslop pe drainage networks ob btained from m the informativee content off the sourcee DEM. Herre, a specific GIS S tool perforrming this ta ask was used (for details, seee Pilotti et al. 1996). First, the pre-pro ocessing algo orithm remo oves depressions (where flo ow-lines con nverge) and flat regions (w where data points aree at the same elevation), that actu ually would d prevent the identification of the connected basin drain nage network. Using thee enhanced topograp phic information n, a space filling drain nage networrk is then derivved followiing the stteepest desccent directions. Whilst th he space filling f drain nage network reeproduces alll the flow paths p inside the basin, in riiver flow mo odelling the channel c netw work (that is a subset s of thee overall dra ainage network) must be id dentified. With W this aim m, the algorithm here employed filters the space filling drain nage network eiither accordiing to the method m based d on the fixed threshold t co ontributing area or to the method ba ased on thee slope dep pendent crittical

pport area. According tto the first method, a sup dra ainage line is i considereed as a cha annel if the upsstream drain ned area A iss larger than n a suitable threeshold At. Due D to the d dependence of the bed sheear stress on the locall gradient, the second metthod takes into accoun nt also the local slope thro ough the con ndition: A>A AtSα, S and α being the loca al slope and an exponentt, respectivelyy. Figures 1a a-1d show ffour differeent channel netw works deriveed for an up pper reach of o the Adda Riv ver valley, where w the Caancano dam is located. Botth drainage and channell networks are a encoded usin ng a tree-lik ke pointers sstructure, th hat logically rep produces thee network topology allowing a an efficient manip pulation of iits informatiive content. Afteer that the lo ocation of th he upstream section has beeen selected on o the chan nnel network k, the code extrracts the hyd draulic path d down to the basin b outlet alon ng with thee informatio on on local elevations, slop pes, and diistances. Th he river rea ach is then disccretized acccording to o the orig ginal DEM reso olution Δl: accordinglly, the rea ach length betw ween two co onsecutive po oints is Δl or o 20.5 Δl. At eacch point along the ch hannel the local flow direection is id dentified draawing the secant line

Figure 1 Different D chan nnel networks obtained from m the DEM off the alpine wa atershed consiidered in the paper p using the slope dependent d algo orithm (A>At Sα, S being thee local bottom m slope). (a) At=0.1 km2, α = =0; (b) At=0.1 km2, α=1.7; (c) At=0.2 km2, α=2; (d)) At=0.5 km2, α=1.7. The white w area in (d d) is the locattion of Cancan no reservoir, whereas w the black spot is the outlet off the watersheed at Ponte Ceepina.

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connecting the second point upstream and the second point downstream. However, the automatic recognition of the flow direction is not straightforward when the stream bends or meanders. Finally, the DEM surface is sampled orthogonally to the local flow direction by bilinear interpolation in order to extract cross-section profiles and derive the geometric information required for the 1D hydrodynamic simulation. 1.2 Spatial discretization

(a)

(b)

Figure 2 Transcritical steady flow through a sinusoidal contraction-expansion: (a) comparison between numerical results obtained by different mesh sizes; (b) plan view of the channel and distribution of the εa parameter for different space discretizations.

Although the use of 1D SWE is a standard in literature, both in steady and unsteady flows, the problem of the definition of appropriate criteria for cross-section selection and mesh refinement has apparently been overlooked in the past and few literature contributions deal with it (e.g., Samuels 1990; Castellarin et al. 2009). This topic is here briefly analyzed and a novel method for automatic cross-section space refinement is suggested. As an example, let us consider the simple case of a steady flow through a sequence of a contraction-expansion in a frictionless, horizontal, 60 m long, rectangular channel where the width b varies according to the equation:

   x − 30  for x − 30 ≤ 15  5 − 1.5 1 + cos  2π  . b( x) =  30     5 otherwise (1) The inflow discharge is set to 20 and a transmissive boundary condition is assigned downstream. This test problem was numerically solved for different mesh sizes ∆x using the numerical model described above. The results shown in Figure 2a emphasize a considerable dependence on cross-section spacing: a value of ∆x greater than 10 m implies maximum error on stage

h and discharge Q higher than 20%. We propose to evaluate the acceptability of the space discretization on the basis of the comparison, at a fixed stage h , between the uniform flow stagedischarge relationships of two consecutive crosssections spaced ∆x. Accordingly, a relative deviation

ε(x ,h ,Δx ) =

Q(x + Δx ,h ) − Q(x ,h ) Q(x ,h ) (2)

is defined. In the discussed test case, introducing a first order accurate approximation, this coefficient reads:

  Δx .  (3) This quantity can be averaged in each crosssection between 0 and H, being H a reasonably expected maximum value for the local water depth during the flood: ε( x, h , Δx ) =

1 db( x )  5 2  − 3 dx  b( x ) b( x ) + 2h

m3/s

1 ε a ( x , Δx ) = H

H

 ε( x, h , Δx) dh . (4) 0

The maximum value of ε a ( x , Δ x ) along the discretized channel can be assumed as an indicator of the appropriateness of the selected space discretization (see Figure 2b).

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Real situations are by far more complex than the one shown in this example, since steep slopes and continuous changes in cross-section shape could occur in addition to changes of width. However, the indicator εa takes into account also these effects, providing a simple criterion for the automatic local refinement of the cross-sections along the channel. Therefore, after the selection of the channel from the DEM and the automatic extraction of the cross-sections (along with geometric quantities and uniform flow stagedischarge relationships), the local value εa (x, ∆x) is computed. According to this vector, cross-sections are locally and automatically refined where necessary, obtaining an improved and adaptive 1D mesh.

2

form, so that the numerical scheme is not ideal to accurately reproduce shock-waves or hydraulic jumps.

3

The Cancano Dam Test Case

The effectiveness of the operative tools described in the previous sections was verified on the basis of the Cancano dam-break test case recovered from the historical paper of De Marchi (1945). In 1943, after the bombing of several dams in the Ruhr region, the possibility that the Cancano dam (see Figure 3 and Figure 4) could become a

2D Modeling

The 2D numerical model used in this paper is the commercial code FLO-2D (2009). This choice was suggested by the compliance of this software with the requirements of FEMA about flood hazard studies, with particular reference to steep mountain alluvial fans, where mud and debris flows can be expected. With reference to this type of events, FLO-2D was compared in the past with other 2D codes for debris flow simulation (Rickenmann et al. 2006; Wu et al. 2013). However, FLO-2D is a general code suitable also for flood propagation. Accordingly, verification of its effectiveness in clear water dam-break cases can be of interest. FLO-2D solves the full 2D SWE on a Cartesian grid using an explicit, central, finite difference numerical scheme along eight potential flow directions. Each velocity computation is essentially one-dimensional and is performed independently for each direction. The Courant-Friedrichs-Lewy condition and the stability criterion developed by Ponce and Theurer (1982) for nonlinear equations are applied, as well as a check on the percentage time change in the flow depth within each computational cell. If any of these stability criteria is not met, the time step is reduced and the computations are repeated, so that the method can require lengthy computer runs to simulate steep rising flood waves. According to O’Brien et al. (1993), the solved equations are not in conservative

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Figure 3 Location of the Cancano dam in the Adda River valley (Northern Italian Alps).

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Figure 4 Historical picture and topography of the former Cancano dam viewed from downstream and in plan (from ANIDEL 1953).

war target was considered. The dam was a concrete gravity structure and was built in the period between 1924 and 1929 for hydroelectric purposes. It was 60 m high and 265 m long, and retained 24 × 106 m3 of water. Nowadays it is submerged into the new Cancano reservoir that was built in the 50’s with a total storage capacity of about 124 × 106 m3. De Marchi was commissioned to evaluate the expected hydraulic consequences of the possible collapse of the Cancano dam and the resulting study provides an outstanding example of coupling between numerical and experimental analysis. According to the fact that “it is practically impossible to apply the usual equations of Hydraulics to the mostly irregular, sometimes extremely irregular, bottom of mountain valleys” (translation from De Marchi 1945), for the first rugged reach downstream of the reservoir an experimental approach was adopted. Actually, along the 8 km reach from the Cancano dam down to the village of Premadio (Section 23 in Figure 5), the Adda River valley has a mean bed slope of 7.5% with long stretches characterized by bed slope greater than 20%. From Section 23 to Ponte Cepina (see Figure 5) the valley is more regular and mean bed slope reduces to about 1%. In this stretch De Marchi applied also a simplified numerical approach based on SWE. A 1:500 scale physical model reproducing the bathymetry of the 16 km long valley stretch from the Cancano dam to Ponte Cepina was built according to the Froude similitude. As one can deduce from the original paper, the Cancano lake was reproduced as a horizontal rectangular-shaped

box with prototypal dimension of approximately 1220 × 704 m; a gap of 265 m along its shortest side represented the Cancano dam. The sudden and complete removal of the plate covering the gap allowed to simulate the collapse of the dam. Both total and partial collapses of the dam were considered. In both cases, discharge hydrographs were measured at three locations: a section just downstream of the dam, at Section 23 and at Ponte Cepina. Each discharge hydrograph was obtained by graphic derivation from the filling time series recorded by using a floating device in a calibrated tank. For each collapse scenario, several runs were carried out in the same test conditions to verify the consistency of the measuring procedure. The final hydrographs were obtained by averaging. Furthermore, the delimitation of the flooded areas between Section 23 and Ponte Cepina was

Figure 5 Longitudinal profile of the thalweg of the Adda River between the former Cancano dam and Ponte Cepina.

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accomplished. These data provide a unique benchmark for validating numerical models for flood routing in mountain environment.

mass conservation law can be used (Pilotti et al. 2013),

Q(t ) =

3.1 Reconstruction of the hydrograph at the dam De Marchi’s experimental data also allows the validation of a procedure recently proposed by Pilotti et al. (2010) for the computation of the hydrograph following a partial dam-break. However, this procedure is not suitable to the reconstruction of the whole hydrograph in the case of a horizontal reservoir like the one built by De Marchi; accordingly, here it will be used to compute the peak discharge only. Considering the overall volume stored in the reservoir model and its area, the initial water depth during the tests can be computed as 0.056 m, that corresponds to h0=27.9 m in the prototype. For this depth and for the total collapse of the dam, the Ritter formula provides a peak discharge of 36327 m3/s. Due to the relative dimension of the dam with respect to the length of the rectangularshaped box where it was opened, the sudden and complete removal of the dam has a similar effect of a partial dam-break with a breach ratio b/B=265/740= 0.376. Pilotti et al. (2010) have shown that, whenever a partial dam-break occurs, the widening of the negative wave front upstream of the breach causes an amplification of the peak discharge with respect to the Ritter value. Pilotti et al. (2010) provides an amplification factor for the Ritter discharge as a function of the breach ratio (see Eq. 12 of the cited paper), that in the case of rectangular cross-section can be rewritten as:

Qp Bh0

b

 27  B   = k (b / B ) gh0  8 

(5)

where k(b/B) can be interpolated by the tabulated values in the cited paper and is equal to 0.2412 in this case. Accordingly, the computed peak discharge Qp is 49690 m3/s, in excellent agreement with the value of approximately 50000 m3/s measured by De Marchi. De Marchi timed the peak discharge immediately after the removal of the dam and measured a monotonically decreasing hydrograph. In order to interpret its falling limb, the curve obtained by coupling a weir flow with the

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Qp  Qp  1 + 2V t   

(6) 3

where V is the water volume initially stored in the reservoir. Figure 6 shows an excellent match between the experimental and theoretical hydrographs.

Figure 6 Comparison between measured and calculated discharge hydrograph at the dam section.

4

Results and Discussion

The dam-break wave following the total collapse of Cancano dam was simulated using the two numerical models presented in Sections 1 and 2. Only a 20 × 20 m resolution DEM is available for this part of the valley and it was used as a basis for the description of the computational domain. We completed the DEM by including the valley stretch (now submerged by the reservoir) between the old and the present Cancano dam. To this purpose, elevation contour lines obtained from historical official topographic maps were used. Unfortunately, the original cross-sections used by De Marchi are not available and the cross-section profiles for the 1D model were extracted from the DEM using the semi-automatic procedure described in Section 1. In both models, the discharge hydrograph measured at the dam cross-section (see Figure 6) was imposed as upstream boundary condition, whereas a free outflow condition was applied at the outlet. Moreover, a dry bed condition was initially assumed along the modeled reach. For a proper

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numerical treatment of wetting and drying fronts, a 1 mm and 3 cm water depth threshold was imposed in the 1D and 2D models, respectively, accepting a maximum final mass error less than 1% with respect to the total volume initially stored in the reservoir. The Courant-Friedrichs-Lewy number was set at 0.8. De Marchi made the concrete surface of the physical model artificially rough to reproduce the natural irregularity of the upper part of the valley and the effect of vegetation. Although an estimation of the roughness of the physical model is missing in the historical paper, the morphology of the valley upstream of Section 23 (very steep and winding) suggests the adoption of a Manning’s coefficient n between 0.1 and 0.067 m−1/3s, as confirmed in literature studies concerning mountain creeks (e.g. Jarrett 1984). Similarly, visual inspection of the downstream reach between Section 23 and Ponte Cepina may justify values between 0.05 and 0.033 m−1/3s. This choice is supported by the fact that a similar value (n = 0.05 m−1/3s) was adopted by De Marchi for the numerical modeling of the dam-break propagation downstream of Ponte Cepina. Operating with the 1D model, first a sensitivity analysis on mesh resolution was accomplished. A non uniform mesh (N = 125 cross-sections characterized by a mean spacing Δxm of about 130 m) was initially adopted, doubling the number of cross-sections used in the experimental study. Then spatial resolution was gradually increased obtaining three refined meshes with N = 667 (Δxm=24.4 m), 1332 (Δxm=12.2 m) and 4987 crosssections (Δxm=3.3 m) respectively. Figure 7 shows the discharge hydrographs computed at the locations indicated in Figure 5 (Section 23 and Ponte Cepina) using different meshes. The Manning’s coefficient was set at n= 0.067 m−1/3s for the reach upstream of Section 23 and at n= 0.04 m−1/3s for the downstream stretch of the valley. The numerical scheme achieves convergence for Δxm =12.2 m. Consequently, the corresponding mesh was chosen for the calculations. Furthermore, flood arrival time is sensitive to mesh resolution: in particular, as one could expect, mesh refinement induces a delay of the arrival time of the wetting front. Actually, a coarse mesh smooths bed irregularities and, consequently, induces a faster propagation of the dam-break wave. On the other

hand, peak discharge seems much less affected by grid size. The sensitivity of numerical results with respect to n was also analyzed. In Figure 8 the gray bands represent the envelope of the discharge hydrographs predicted by the 1D model for values of the Manning coefficient within the ranges mentioned above. One can notice that this parameter has a stronger influence on the timing of the flood than on the peak discharge value. As expected, the wetting front arrival time decreases with the reduction of the Manning coefficient. Moreover, the effect of roughness on the arrival time is less evident in the upper reach of the valley

Figure 7 Sensitivity of the 1D model to spatial resolution: computed discharge hydrographs at two selected cross-sections based on n=0.067m−1/3s upstream of Section 23 and n=0.04m−1/3s downstream.

Figure 8 Sensitivity of the 1D model to roughness coefficient: envelope of the computed discharge time series at two selected cross-sections based on the mesh characterized by Δxm=12.2 m.

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aracterized byy very steep p bed slopes.. On that is cha the whole, the speed of o the flood wave w seems well reproduced d by the num merical modeel. Figure 8 also highlights that the 1D model systematiccally overestima ates both th he peak disccharge and the steepness of the rising limb off the discha arge hydrograph hs. This behavior is prob bably due to o the difficulty of the sch heme in capturing c t twodimensiona al expansion ns and rela ated attenua ation effects at lateral l conflluences. Thee falling lim mb is less influen nced by rou ughness and d shows a good g agreement with the experimental e l one. Figurre 9 shows the hydrographs h s calculated by b FLO-2D. The results sho ow a minor sensitivity to t the Mann ning coefficient in the samee range explo ored with thee 1D model. In both b section ns the peak discharge d is well reproduced d, although there t is a dellay in the arrrival time of thee wave and the t computeed rising lim mb is much steep per than the observed o onee.

Add da River and d its tributaary upstream m of Section 23, where stro ong two-dim mensional efffects occur. Oth her discrepa ancies can be observ ved in the floo odplain dow wnstream o of Section 23, where floo oded areas are a considerrably overesstimated, as onee could expect. As one caan see in Fig gure 10, the 2D simulation n reproduces more faiithfully the obsserved inun ndation exttent. The agreement betw ween observved and comp puted floodeed areas can be quantified by using the performance index sug ggested by Ho orritt and Baates (2002):

P =100 ⋅

Asim ∩ Aobs Asim ∪ Aobs

(7)

wheere A makees reference to the sim mulated and obsserved extension of flood ded areas. The T P index ran nges from 100 (perfect ag greement) do own to 0 in

Figure 9 Sensitivity of o the FLO-2D model to the c Ca ase (a) n= 0.10 m−1/3s in the Manning coefficient. reach upstrream of Section 23 and n= = 0.04 m−1/3s for the downsttream part of the t domain. Case C (b) n= 0.0 067 m−1/3s and n= n 0.03 m−1/3s, s respectivelyy.

Finallyy, the calculated inunda ation maps were w compared with the experimental e l results of De Marchi (F Figure 10). The T inunda ated areas were w delimited on o the basis of o the envelo ope of maxim mum water depth hs calculated d by the 1D numerical n mo odel according to the assu umption of horizontalityy of water level in the cross-sections. Th his mapping was repeated fo or different roughness values, v show wing that the shorelines off flooded arreas are alm most insensitive to the Ma anning coeff fficient. The 1D model doees not faith hfully match h the obserrved pattern, esspecially at the t confluen nce between the

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Figu ure 10 Co omparison b between calcculated and mea asured flooded d areas betweeen Section 23 and Ponte Cepina.

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the case of complete uncorrelation between predicted and observed inundated areas. In the case of the simulation accomplished by the 1D model, P is 63.8 while using FLO-2D the P index rises up to 73.4. In considering these values, one should take into account that De Marchi represented this part of the domain by using 32 cross-sections: accordingly, one can expect slight variations of the bathymetry which could affect flood propagation in the shallowest parts of the flow field. From the computational point of view, the run of the 1D code lasts approximately 0.7 hour. The same simulation accomplished with FLO-2D (calibrated to obtain the results here presented) takes approximately 72 hours using i7-core PC with 4 GB RAM. Other 2D solvers can certainly be more efficient that the one currently implemented in FLO2D but there is a clear disproportion between the computational time required by the two approaches. Finally, one can note that although a long stretch of the river downstream of Section 23 is not strictly suitable to a 1D simulation, also in this area 1D results are conservative without being too unrealistic. Given that the peak time is of minor interest in such a fast-evolving process, the predicted peak discharge and extent of flooded areas are sufficiently close to the measured ones to retain engineering interest. However, a 1D approach only provides the average velocity and depth cross-sections and, accordingly, is not suitable to the use of physically-based criterions for hazard mapping (Xia et al. 2014; Milanesi et al. 2014) that require a local description of the flow field.

5

Conclusions

The study of dam break in alpine areas involves several stumbling blocks and can be tackled using 1D and 2D models for flood propagation. In this paper we compared the results provided by a 1D code (Pilotti et al. 2011) and the 2D code FLO-2D. The 1D approach is computationally less demanding and simpler to implement and is acceptable whenever the extent of the discharge and the narrowness of the valley justify the underlying hypothesis. In wide domains, this approach could be effectively used to route the flood wave as far as floodplain areas, where it tends

to overestimate the extent of flooding. Accordingly, whilst the use of a 2D approach throughout the domain would imply a considerable increase of the computational time (non compatible with a realtime application in the examined case), a 2D description of the flood limited to these areas could greatly improve the computational efficiency. Accordingly, the 1D modelling of wave propagation in channel-like stretches can provide a boundary condition for 2D flood propagation in flood plains. When a 1D approach is used in steep and irregular alpine valleys, the problem of crosssections extraction could not be solved using traditional surveying techniques. In the considered case, the distance between two consecutive crosssections cannot exceed 20 m on average and the convergence of the solution is obtained for an average grid spacing of approximately 12 m. In order to solve this problem, in this paper we outline a novel procedure to derive cross-sections by automatic sampling of the DEM channel network. This can be derived by the analysis of the steepest terrain directions, as usually done in hydrologic rainfall-runoff modeling. The introduction of a control criterion based on the comparison of consecutive normal flow depths allowed an automated mesh refinement where required. In this way, an accurate and efficient 1D reconstruction of the topography was performed. The Cancano test case, which was recovered from the historical paper of De Marchi, is an extraordinary benchmark for the validation of numerical SWE solvers in alpine valleys. The three measured hydrographs and the measured extent of flooding documented in the original paper were used to compare the results provided by the two mathematical models and to show that SWE provide a powerful prediction tool even in the mountain environment, where the underlying hypotheses are far from being rigorously satisfied. The measured hydrograph at the dam was also used to verify the predictive effectiveness of a simple theoretical method to evaluate the dambreak peak discharge.

Acknowledgements We thank the anonymous reviewers for their valuable suggestions that contributed to the

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improvement of the paper. The study has been developed within the European Project Kulturisk

(Grant agreement 265280).

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