Comparison between 1D and 2D models to analyze the dam break

Comparison between 1D and 2D models to analyze the dam break waveABSTRACTThe simulation of exceptional events characterized by high hydraulic and hydro-geologic risk is an actual problem forthe scientific community, working in the topic of environmental protection. There have been around 200 notabledam and reservoir failures worldwide in the twentieth century. These failures have caused severe devastation in thevalleys downstream both in terms of lives lost and widespread damage to infrastructure and property.The flood wave caused by a dam failure can result in the loss of human lives and have a severe economic impact.Therefore, significant efforts have been carried out over the years to produce methods for determination of theextent and timing of the flood wave.This work describes and compares two models for solving the dam break problem using the FEM (finite elementmethod). Both models are based on solving the Shallow Water equations. The first model is one-dimensional and thesecond is two-dimensional.Most simulation studies for the dam break problem are currently attempted using one-dimensional models, and weinvestigate here the possibility of moving to two-dimensional simulations solving and comparing the two modelswith a finite element analysis and solver software package called Comsol Multiphysics.Comsol Multiphysics is a modelling package for the simulation of any physical process describable with partialdifferential equations (PDEs). The hydraulic analysis of the dam break problem from a different point of view hasbeen done in this work, that is, using a generic tool not adapted to solve this kind of problems. This fact helps us tounderstand the process entirely because of the necessity of introducing all the conditions and equations. We are ablein this way to control all the details within the model, even though we have other problems in relation with thegeneric nature of the program.A simple example of the collapse of a dam is used to demonstrate the capability of the two models. Theinstantaneous collapse over wet bed has been studied. Some difficulties are encountered simulating dry bed conditionfor the appearance of negative depths or velocity overshoots. The important influence of the bathymetry has beenalso demonstrated during the simulations.Key words: Shallow water equations; Dam-break simulation; Finite elements method; Bathymetry.1. - INTRODUCTION1.1 Literature review1.1.1 GeneralThe first law on dam safety was passed in France inMay 1968. Emergency plans had to be prepared forevery dam higher than 20 m and with a reservoirvolume over 15 Mm 3 . At that time, the numericalsimulation of dam-break problems could beaccomplished only with one-dimensional tools andaccordingly these were used to predict the spreadof the flood wave (Hervouet 2000).Since 1994, however, new regulations havedemanded an updating of existing emergency plansfor large dams. Emergency Response Plans nowhave to be prepared by the local authority afterconsulting the relevant mayors and dam operators.The scope of the Emergency Response Planincludes:1.- the identification of potential risks;2.- a list of protective measures and mechanisms toimplement them;3.- definition of the responsibilities of localgovernment agencies and other organizationsinvolved;4.- dissemination of information to the public;The dam operator is now required to carry out arisk assessment study which provides informationon the area covered by the Emergency ResponsePlan and which distinguishes between:1.- the near safety zone, flooded in less than 15 minafter the dam-break;2.- the remote flooded area. Since 1994, thisextends to the limit at which ‘there is no danger forthe population’.As a consequence of this new law, a risk analysis ofmany dams must be completed. Although many ofthese studies, particularly those relating tohydroelectric power stations in narrow uplandvalleys, could be completed with a one-dimensionalprogram, a small but significant number ofproblems demanded a two-dimensional approach.These include flood waves spreading in largevalleys or those that ultimately reach the sea via aflat coastal plain. In both these cases, widespread,and relatively shallow inundation results in a flowfield that is significantly 1D in nature and which isamenable to simulation using the shallow water orSaint Venant equations. Given the complexity ofdam-break hydraulics, it is also probable that moreand more two-dimensional studies will have to becompleted in the future (Hervouet 2000).1.1.2 Review of numerical methods for dambreak.A brief review of the numerical methods used in1

Ignacio Valcárcel LencinaTRITA LWR Masters Thesisthe dam break problem has been done, although itis not the objective of this work.Numerical simulation models are powerful tools toassess the impact of floods due to dam failure events.Different codes are available for numerical simulationsof floods caused by dam failures.At present, classical methods and central differenceschemes dominate the software products for theshallow-water system of equations. Some years aftertheir adoption for solving problems in gas dynamics,upwind schemes have been successfully used for thesolution of the shallow water equations, with similaradvantages. Historically, upwind schemes weredeveloped specifically for the solution of the Eulerequations, but there is no reason why the techniquesinvolved cannot be applied for the solution of othersystems of conservation laws.First-order and second-order schemes have been usedto solve two-dimensional dam break flows onstructured meshes following central differences andupwind flux difference approaches. More recently,first and second-order upwind schemes onunstructured meshes have been presented for this kindof flow.Besides this, discussion is open as to whether theschemes based on one-dimensional Riemann solversare the most suitable choice for multi-dimensionalcalculations because they seem inadequate forcapturing two-dimensional flow features. Multidimensionalupwinding techniques have demonstratedto give very good resolution for dam break problemsin genuinely two-dimensional domains, but theirimplementation is more complicated.1.1.3 Hydraulic flow models available for dam breakflood simulations.Some flow models available for dam break flowsimulation are listed in table 1 (ICOLD, 1999).1.1.4 FEM, FVM or FDM?Several techniques are available in literature, to solvethe two-dimensional shallow water equations for thesimulation of free-surface flows transients. Theseinclude finite difference methods (FDM), finiteelement methods (FEM) and the finite volumemethods (FVM) (Valiani et al 1999).Table 1. Some flow models for dam-break flood simulations (ICOLD, 1999). (*2D models)AgencyName of models1 USA / National Weather Service DAMBRK (original)2 USA / National Weather Service SMPDBK (Simpified Dambreak)3 BOSS International BOSS DAMBRK4 HAESTED METHODS HAESTED DAMBRK5 Binnie & Partners UKDAMBRK6 Department of Water Affairs and Foresty Pretoria, DWAF-DAMBRK7 USA / COE- Hyrdologic Engineering Center HEC-programs (HEC-RAS)8 Tams LATIS9 (IWHR), PR China DBK 110 (IWHR), PR China *DBK 211 Royal Institute of Technology, Stockholm TVDDAM and *COMSOL12 Cemagref RUBAR 313 Cemagref * RUBAR 2014 Cemagref CASTOR15 Delft Hydraulics SOBEK16 Delft Hydraulics *DELFT 2 D17 Consulting Engineers Reiter Ltd. DYX 1018 ANU-Reiter Ltd. DYNET - ANUFLOOD19 ENEL Centro di Ricerca Idraulica RECAS20 ENEL Centro di Ricerca Idraulica *FLOOD 2D21 ENEL Centro di Ricerca Idraulica STREAM22 Danish Hydraulic Institute MIKE 1123 Danish Hydraulic Institute *MIKE 2124 ETH Zürich FLORIS25 ETH Zürich *2D-MB26 EDF - Ladoratoire National Hydraulique RUPTURE27 EDF - Ladoratoire National Hydraulique *TELEMAC-2D2

Comparison between 1D and 2D models to analyze the dam break waveA finite element method (FEM) discretization is basedupon a piecewise representation of the solution interms of specified basis functions. The computationaldomain is divided up into smaller domains (finiteelements) and the solution in each element isconstructed from the basis functions. The actualequations that are solved are typically obtained byrestating the conservation equation in weak form: thefield variables are written in terms of the basisfunctions, the equation is multiplied by appropriatetest functions, and then integrated over an element.Since the FEM solution is in terms of specific basisfunctions, a great deal more is known about thesolution than for either FDM or FVM. This can be adouble-edged sword, as the choice of basis functionsis very important and boundary conditions may bemore difficult to formulate. Again, a system ofequations is obtained (usually for nodal values) thatmust be solved to obtain a solution.Comparison of the three methods is difficult, primarilydue to the many variations of all three methods. FVMand FDM provide discrete solutions, while FEMprovides a continuous (up to a point) solution. FVMand FDM are generally considered easier to programthan FEM, but opinions vary on this point. FVM aregenerally expected to provide better conservationproperties, but opinions vary on this point also. Theimplementation of the model is based on thecommercial FEM package Comsol Multiphysics,which reduces the programming effort required to aminimum.1.1.5 Numerical problems of the Shallow WaterEquations.The numerical solution of the shallow water equationsfor flood waves propagation over real domains posesthree specific problems, (Valiani et al 1999).The first problem is the simulation of the fronts orabrupt water waves that can be representednumerically as a propagating discontinuity. Thisproblem can be considered as solved since the lateeighties or early nineties.The second problem derives from abrupt changes inbathymetry. As long as the bottom surface remainssufficiently smooth, most numerical techniquesprovide an accurate solution of the flow, but if thebottom surface is very rough the majority of methodsfail.The last problem especially arises when these schemesare applied to study the wave front propagation overdry bed. In fact, during the simulations of this work, aswe said before, some difficulties are encounteredsimulating dry bed condition for the appearance ofnegative depths or velocity overshoots; therefore, ourprincipal focus has been the wet case.To study the wave over dry bed is necessary to find asuitable numerical scheme (which is not the case ofComsol) that can efficiently and accurately simulatethe flow with all its characteristics and features (usingthe two-step Taylor-Galerkin method, just to mentionone).Furthermore, is the hyperbolic character of theshallow water equations that makes finding solutionsto these equations difficult. Hyperbolic equationsadmit discontinuous as well as smooth or classicalsolutions. Even for the case in which the initial data issmooth, the non-linear character combined with thehyperbolic type of the initial of the equations can leadto discontinuous solutions in finite time.The non-linear character of the equations also meansthat analytical solutions to these equations are limitedto only very special cases. Numerical methods must beused to obtain solutions to practical problems.Therefore, numerical techniques which admitdiscontinuities in the solution, secondary shocks,reflections of waves, dry bed and obstructions arenecessary for the solution of the shallow waterequations. The method of characteristics, finitedifferences, finite elements and Godunov-typeschemes can be used to solve the shallow waterequations (Zoppou and Roberts 1991).1.2 Objectives of the study.The main objective of this work is to study andunderstand the hydraulic analysis of the dam-breakproblem through the shallow water equations, solvingthem from a different point of view, which is using ageneric tool based on the FEM. Comparing both onedimensionaland two-dimensional approaches, weunderstand entirely the whole process and we are ableto answer some old questions about the benefit ofusing the 1D or the 2D model.Through simulation of two different models, we wishto achieve the following goals:- study the capacity of a particular general purpose 2Dtool to deal with dam-break simulations- compare simulation results of 1D and 2Dsimulations for comparable conditions- set a “point of reference” for 2D computations.2. - GOVERNING EQUATIONS2.1 Navier-Stokes or Shallow Water Equations.Before describing the shallow water equations, it needsto be specified that these are not the only equationsused to solve the dam-break problem. Therefore,using the approach (Quecedo et al 2004), where theauthors compared two mathematical models forsolving the dam-break problem, the Navier-Stokesequations are also a possible candidate.As in many engineering problems, simulation of dambreak can be done with alternative mathematical andnumerical models with different levels ofapproximation. Quecedo compared two methodsbased on (i) solving the Navier–Stokes equations and(ii) using a depth integrated model which is discretizedusing a Taylor–Galerkin scheme. These mathematicalmodels are based on assumptions such as disregarding3

Ignacio Valcárcel LencinaTRITA LWR Masters Thesisair trapping in the former or vertical accelerations onthe latter, and their range of application depends onthem.The validity of the shallow water approach was basedon comparisons of the model against problems with aknown analytical solution, which has been done bymany researchers over the past years.Concerning comparisons with in situ measurements,Quecedo applied the shalow water approach tosimulate breaking of a tailing dam in East Texas. Allthese comparisons do not concentrate on intervalswhere vertical accelerations are important, like theinitial instants following failure, or the interaction withobstacles like vertical dykes. In these cases, the errorscan be important.Quecedo used a more accurate approach to assess thevalidity of the shallow water approach in these casesfor which no known analytical solutions are available.Based on the results obtained, the authors concludedthat:(i) The shallow water approach predicts (i) faster floodwaves when propagating over dry beds, thus providingsafer predictions of flood wave arrival times, (ii)reasonable estimations of the evolution of water depthfar from the dam.(ii) If failure takes place onto a wet bed, the shallowwater results can be considered reasonable incomparison to those obtained using the Navier–Stokesapproach.(iii) If the free surface curvature is high, the shallowwater approach misspredicts the pressure. Thus, itshould not be used to calculate, for instance, stresseson structures.In spite of the shortcomings of the shallow waterapproach when applied to dam break problems, thecomputational effort required by other methods, suchas the Navier–Stokes Level-Set algorithm presentedherein, in actual problems makes the shallow waterapproach attractive for large computational domains.The Navier–Stokes approach would be used for theanalyses of small areas when knowledge of the threedimensionalstructure of the flow is needed.2.2 One-dimensional Shallow Water equations.Once we have seen that the shallow water equations(from now on “SWE”) are the most suitable todescribe the dam break problem, the next step is todescribe them for the one-dimensional approach.The SWE are frequently used for modelling bothoceanographic and atmospheric fluid flow. Models ofsuch systems lead to the prediction of areas eventuallyaffected by pollution, coastal erosion, and polar icecapmelting.As previously seen in [2.1], comprehensive modellingof the dam break using physical descriptions such asthe Navier-Stokes equations can often be problematic,due to the scale of the modeling domains. The SWE,of which there are a number of representations,provide an easier description of such phenomena.The SWE are obtained from the Navier-Stokesequations, assuming that vertical velocities andaccelerations are negligible and integrating over thewater depth.This 1D model investigates the settling of the dambreakwave over a variable bed as a function of time.A mathematical relation represents the shape of thebed so that is easy to change parameters.The 1D Saint-Venant’s-SWE are the following:∂z∂(zv)+ = 0∂t∂x2∂(zv)∂(zv ) ∂z∂zf+ + g ⋅ z ⋅ + g ⋅ z ⋅∂t∂x∂x∂x22∂ ( zv)∂ ( zv)−υc⋅ + g ⋅ z ⋅ Sf − E ⋅ = 022∂x∂xOr in the compact form:∂z+ ∇ ⋅ ( zv)= 0∂t∂(zv)T+ ∇ ⋅ ( zvv ) + g ⋅ z ⋅∇zS∂t−υ⋅ ∆(zv)+ g ⋅ z ⋅ Sf − E ⋅ ∆(zv)= 0cWhere z is the thickness of the water layer (m), v is thevelocity (ms -1 ), g the gravity constant (m s –2 ), Sf is thefriction term (explained later) and υ c the kinematicviscosity (m 2 s -1 ). The wind effect is not present inthe equations.The dispersion coefficient E has been set to 2 m 2 /s.This value takes into account the turbulence and theheterogeneity of the velocity on the vertical. It is amajor unknown in the case of dam break, but theresults that we will see later, show that ComsolMultiphysics has need of artificial diffusion forstabilizing its solution.The definition of the thickness of the water layer, z, isz s -z f , where z s and z f are the measures in figure 1.(1)(2)(3)(4)4

Comparison between 1D and 2D models to analyze the dam break waveFigure 1. Representative vertical section through the fluiddomain showing the bed of a lake and the water surface.2.3 Two-dimensional Shallow Water equations.Depth averaging of the free surface flow equationsunder the shallow water hypothesis leads to a commonversion of the two dimensional SWE, wich inconservative form is:∂z∂(zu)∂(zv)+ + = 0∂t∂x∂y∂(zu)∂(zu+∂t∂x2) ∂(zuv)+∂y(5)∂z∂zf (6)− f ⋅ zv + g ⋅ z ⋅ + g ⋅ z ⋅∂x∂x22∂ ( zu)∂ ( zu)+ g ⋅ z ⋅ Sfx− E ⋅ − E ⋅ = 022∂x∂y∂(zv)∂(zuv)∂(zv+ +∂t∂x∂y+ g ⋅ z ⋅ Sf2)∂z∂zf (7)+ f ⋅ zu + g ⋅ z ⋅ + g ⋅ z ⋅∂y∂yy22∂ ( zv)∂ ( zv)− E ⋅ − E ⋅ = 022∂x∂yThe equations above are valid under the followingassumptions:1. The fluid is well-mixed vertically with a hydrostaticpressure gradient.2. The density of the fluid is constant.3. We study water waves of long wave lengths.4. The viscosity term is negligible.Where z is the depth of water, zu and zv are unitdischarges along the co-ordinate directions. Inaddition, u and v are the velocities in the x and ydirections respectively, g is the acceleration due togravity and Sf x and Sf y are the friction terms in the xand y directions respectively. The turbulent dissipationterms (E) and Coriolis effect (f) are present in theequations, but not the wind effect, which is notsignificant in the usual valleys where the dam-breakoccurs.2.3.1 The friction term.Many two-dimensional depth-averaged models includeonly friction at the bottom. Specifically, models thatassume vertical channel side-walls and use free-slipboundary conditions do not account for the friction atthe walls. Neglecting this effect, open channel flowwould likely show a marked variation in water depthfrom the one measured experimentally.Evaluation of the friction slope that quantifies theenergy loss at the side-walls of the channel is required.It is common to use a uniform flow law, Manning orChezy formulae, to calculate this term. However, theselaws were developed for one-dimensional flow andmust be extended and properly incorporated to twodimensionalequations.Manning’s formula used in one-dimensional shallowwatermodels is expressed in the formSfn2A⋅ Q ⋅ Q= (8)2⋅ R4 / 3Where n is the Manning coefficient and R = A/P is thehydraulic radius, which depends on the wettedperimeter P. For a rectangular channel P = b + 2zand for an irregular basin P = b.A distinction must be made between an arbitrarychannel cross-section and a cross-section with verticalwalls (fig. 2). Most two-dimensional models assumevertical walls and free-slip wall boundary conditionsfor rectangular channels, while the equations used tocompute the friction term are based on irregular crosssections.To correct this inconsistency, the frictionslope equation has been modified here to reflect thevertical side-wall assumption. Basically, themodification ensures that the entire wetted perimeter(bottom width and side-walls) is accounted for.For an arbitrary cross-section, the wetted perimeter ina cell is equal to the bottom width b (fig. 2), so that thehydraulic radius adopts the form R = z, the waterdepth. In this case, the two-dimensional friction termsare written in the form (Brufau and García-Navarro2000):22n ⋅ u ⋅ u +Sf x 4/ 3zv2= (9)22n ⋅ v ⋅ u +Sf y 4 / 3zv2= (10)5

Ignacio Valcárcel LencinaTRITA LWR Masters ThesisFigure 2. Irregular and rectangular cross sections.3. - MO DEL L I N G THE 1D DAM-BREAKWAVE.3.1 Assumptions.The basic assumption is that a failure and the resultingflood wave can reasonably be modeled using analyticaltechniques. Existing computer models have been usedto recreate actual failures with a close representation toobserved crests. However, unrealistic results can beobtained without a careful analysis.In this model, it has been assumed that the dam failedcompletely and instantaneously.This ‘main’ assumption of instantaneous and completebreach (the breach is the opening formed in the damwhen it fails) was used for reasons of conveniencewhen applying certain mathematical techniques foranalysing dam-break flood waves. The presumptionsare somewhat appropriate for concrete arch-typedams, but they are not suitable for earthen dams andconcrete gravity-type dams.● Earthen dams, which exceedingly outnumber allother types of dams, do not tend to completely fail,nor do they fail instantaneously. Once a developingbreach has been initiated, the discharging water willerode the breach until either the reservoir water isdepleted or the breach resists further erosion. Thefully formed breach in earthen dams tends to have anaverage width (b) in the range (h d

Comparison between 1D and 2D models to analyze the dam break waveTo conclude, the last assumption concerns about thebathymetry profiling. An unreal bathymetry has beengenerated for the 1D model by the function sin(x)/10,which have been developed only downstream of thedam, where it has the real influence.This function creates a group of small elevations of±10 cm downstream of the dam, in order to observethe big influence of the bathymetry profiling when thedam break wave sweep along the valley. In thedenominator of the function there is a 10, it has beenonly the result after some different tests to prove themost suitable profile for the simulations.3.2 Dependent variables.Our dependent variables for this 1D model are z andzv. Where z is the depth of water, and zv is the unitdischarge along the co-ordinate direction.3.3 Constants.In this model, we call ‘constants’ the variables that areindependent of the geometry. Constants are global,that is, they are the same for all geometries andsubdomains. In this 1D model the following constantshave been used:Kinematic viscosity: ν = µ / ρIt is expressed as the dynamic viscosity divided by thedensity of the fluid. Kinematic viscosity is a measureof the resistive flow of a fluid under the influence ofgravity. Kinematic viscosity (Greek symbol nu: ν c ) hasSI units (m 2 s -1 ). The kinematic viscosity of water at20ºC is 10 -6 m 2 /s, and that is the value we assume forour model. This viscosity term is negligible; it has noinfluence although is has been taken intoconsideration. In the 2D model, the viscosity term isneglected and this is not an imprecise assumption.Acceleration due to gravityAt the Earth's surface, denoted g, is approximately 9.8m/s 2 (meters per second squared).Eddy diffusion coefficientIn fluid dynamics, an ‘eddy’ is the swirling of a fluidand the reverse current created when the fluid flowspast an obstacle. The moving fluid creates a spacedevoid of downstream-flowing water on thedownstream side of the object. Fluid behind theobstacle flows into the void creating a swirl of fluid oneach edge of the obstacle, followed by a short reverseflow of fluid behind the obstacle flowing upstream,toward the back of the obstacle. This phenomenon ismost visible behind large emergent rocks in swiftflowingrivers.The ‘eddy diffusivity’ is the exchange coefficient forthe diffusion of a conservative property by eddies in aturbulent flow. Also known as eddy-diffusioncoefficient. In this model, this coefficient is relevant toget the stability in the simulations. Without thiscoefficient, the results have been not satisfactory at all.We need this artificial diffusion for stabilizing thesolution. The dispersion coefficient E has been set to2 m 2 /s.Manning's Roughness CoefficientRoughness coefficients represent the resistance toflood flows in channels and flood plains. Suggestedvalues for Manning's n, tabulated according to factorsthat affect roughness, are found in the literature.Roughness characteristics of natural channels are givenby Barnes (1967). Barnes presents photographs andcross sections of typical rivers and creeks and theirrespective n values. From these guides we fetch atypical value for flood plains, n = 0.025Figure 3. The Constants dialog box.3.4 Scalar expressions.The scalar expressions are used to define scalarexpression variables that are valid on all geometrylevels everywhere in the current geometry. Thefollowing expressions have been defined in the model:The bed profile z f has been defined in this way:z f = (0)*(x=52)This is the analytical expression that has been chosento define the bathymetry in the 1D model; it meansthat there is a flat bed profile the first 52 m (upstreamof the dam) and a ‘sinusoidal’ bed profile downstreamof the dam, that is, the last 48 m of the one hundredmetres-domain.The derivative of z f with respect to the variable x hasbeen introduced here for computational effortreasons. Computing this term apart, the computerdoes not need to do it in the simulation for every finiteelement of the mesh, it takes simply the value which issaved here as another expression. This term appears inthe equations as we saw in [2.2].The initial value of the water layer z 0 has been set inthe same way as the bed profile, that is: z 0 =(10)*(x=50). It is developed in detail in[3.7].The friction term seen in [2.3.1] has been introducedhere with a different expression but it is really thesame. All the expressions as possible have been setapart from the equations for own convenience andcomputational effort reasons.7

Ignacio Valcárcel LencinaTRITA LWR Masters ThesisFigure 4. The Scalar Expressions dialog box.3.5 Geometry.To create the 1D geometry model, a line which is 100m in length has been drawn. This is the first step to doonce the dimensions and the dependant variables havebeen chosen. The figure below show the 1D geometrymodel:Figure 6. Bathymetry profile, z f , used in the model.Figure 7. The initial water surface profile and thebathymetry profileFigure 5. 1D geometry model3.6 Bathymetry profile.As it was written previously, the bed profile z f hasbeen defined in this way:z f = (0)*(x=52)In the figure below we can see the sinusoidal bedprofile only downstream of the dam, the zone ofinterest for us hydrodynamically speaking (notice thedifference in scale between the x and y direction). Thiskind of profile is theoretical, but the purpose here isshowing the influence of the topography of the bed onthe elevation of the water surface. To see thisinfluence, these small elevations of 10 cm have beenintroduced. A higher elevation profile has moreinfluence on the water surface, but it introduces muchmore instabilities in the model.3.7 Initial value.For a time-dependent analysis, initial conditions mustbe provided. An initial value for each dependentvariable is entered. We can also enter initial values forthe first time derivative of the dependent variables.These are used when solving time-dependentproblems containing second time derivatives. In thiscase there are no second time derivatives, then weenter zero in both edit fields.Only one value different from zero in the first editfield is entered, for the dependent variable z. Thisvalue is equal to z 0 because it was defined previously inthe scalar expressions [3.4]. At t=0 the value of theother dependent variable zv is equal to zero.The figure 7 above shows clearly the situation at t = 0,before the dam break occurs.The expression for the initial value is:z 0 = (10)*(x=50).8

Comparison between 1D and 2D models to analyze the dam break waveIn 1D, Comsol Multiphysics always uses regularrefinement, where it divides each element into twoelements of the same shape. The refinement of themesh has gone from 15 to 7680 elements, the finaldivision that has been chosen to define the model withan acceptable accuracy.Figure 8. The initial condition specification page in theSubdomain Settings dialog box.3.8 Meshing.A mesh is a partition of the geometric model intosmall units of simple shape. For 1D geometry thesoftware partitions the subdomains (intervals) intosmaller intervals (or mesh elements). The endpoints ofthe mesh elements are called ‘mesh vertices’.The value in the ‘Maximum element size’ edit fieldspecifies the maximum allowed element size, which bydefault is 1/15th of the size of the geometry.The value in the ‘Maximum element size scalingfactor’ edit field comes into play if we do not explicitlygive a maximum element size. In that case, thesoftware multiplies the default maximum element sizeby this factor; its default value is 1.The value in the ‘Element growth rate’ edit fielddetermines the maximum rate at which the elementsize can grow from a region with small elements to aregion with larger elements. The value must be greateror equal to one. The default value is 1.3, that is,element size can grow by 30% (approximately) fromone element to another.Table 2. Mesh parameters and statistics.ParameterMaximum element size 1Maximum element size scaling factor 1ValueElement growth rate 1.3X-direction scale factor 1.0Number of degrees of freedom 30722Number of elements 76803.9 Subdomain settings.The subdomain settings describe the physics on amodel’s main domain, which is divided intosubdomains. In this model, there is only onesubdomain.The PDE in its general form is used with a timedependent analysis. The PDE in the general form hasthe following aspect in the program:e2∂ u+ d∂t∂u⋅ + ∇ ⋅ Γ∂ta⋅a=2F(11)The next step is to introduce the equations seen in[2.2] adapting them to this general form, that is, the1D-SWE must be introduced in the proper formatthat Comsol Multiphysics requires.Figure 9. The flux vector edit fieldWhen trying to solve a problem that does notconverge, or having solved a problem where thesolution does not seem properly resolved, there isoften a need for re-solving the problem using a finermesh. In these situations it is sometimes easier torefine the existing mesh than to regenerate the meshfrom scratch using finer mesh settings.Figure 10. The source term edit field9

Ignacio Valcárcel LencinaTRITA LWR Masters ThesisFigure 11. The e a mass coefficient edit field.Figure 13. The type of the boundary condition (Dirichlet).Figure 12. The d a damping/mass coefficient edit field.Extending the general form using the coefficients thathave been introduced in the edit fields above; we getthe equations explained in [2.2]. Since the e a masscoefficient is zero, the second time derivatives aredeleted in the general form, therefore the equationbecomes:∂ud a+ ∇ ⋅ Γ = F∂t⋅ (12)3.10 Boundary settings.If the boundary gives a value to the problem then it isa ‘Dirichlet boundary condition’. In this model,Dirichlet boundary conditions are implemented toboth ends, while the physics are described by theequations seen in [2.2].Figure 14. The R coefficient (boundary value).3.11 Time stepping.The next table shows the parameters used in the 1Dmodel. The ‘Times’ edit field contains a vector oftimes at which we want the solution to the timedependentmodel. The syntax 0:0.1:5.4 represents avector of times starting at 0 with steps of 0.1 up to5.4. The relevant time span depends on the model’sdynamics.The absolute and relative tolerances control the errorin each integration step. Roughly stated, the relativeerror is less than the relative tolerance if the solution islarge, and the absolute error is less than the absolutetolerance for the corresponding solution component ifthe solution is small. In particular, there is no accuracyat all when the solution is less than the absolutetolerance.In the ‘Time steps taken by solver’ edit field, for the‘Free’ option selected, the solver chooses its time stepsarbitrarily; in other words, it ignores the ‘Times’ listwhen selecting them. We fetch all these explanationsfrom the ‘Comsol User’s Guide’. The rest of theparameters are the default values.10

Comparison between 1D and 2D models to analyze the dam break waveTable 3. Time stepping parameters.ParameterValueTimes 0:0.1:5.4Relative tolerance 0.01Absolute tolerance 0.0010Times to store in outputTime steps taken by solverManual tuning of step sizeSpecified timesFreeOffInitial time step 0.0010Maximum time step 1.0Maximum BDF order 5Singular mass matrixConsistent initialization of DAEsystemsError estimation strategyMaybeBackward EulerIncludealgebraic3.12 Results.The simulation runs for 5.4 seconds (when the wavereaches the 100 m of the domain). The followingfigures show the water surface and the bed profile atseven output times toward the beginning of thesimulation (only cross sections are shown). Time spansfrom t = 0 to t = 5.4 at steps of 0.9 seconds. The totalcomputation time for the 1D model has been 1m 07swith processor Intel Pentium 4 CPU 2.53 GHz and 1GB RAM.Figure 15b. Water surface (z) at t = 0.9 sFigure 15c. Water surface (z) at t = 2.7 sFigure 15d. Water surface (z) at t = 5.4 sFigure 15a. Water surface (z) at t = 0 sFigure 16. Flow plot at t = 5.4 s11

Ignacio Valcárcel LencinaTRITA LWR Masters ThesisThe simulation clearly shows the influence of the bed’stopography on the elevation of the water surface.Comparing these results with those obtained from themodel with flat bottom topography, the influence ofthe topography is proved.Figure 17d. Water surface at t = 5.4 s. Model with flatbottom topography.The total computation time for this model has been 59s, eight seconds faster than the previous model;obviously the complexity of this model is different.Figure 17a. Water surface at t = 0 s. Model with flat bottomtopography.4. - VALIDATION OF TH E 1D MODELIn general, validation is the process of checking ifsomething satisfies a certain criterion. At this point, itis needed to check if the results are satisfactory or not.It will be done comparing the solution obtained withComsol Multiphysics (CM) with other solutions of thesame problem.The definition sketch of the model that we comparewith is the figure below:Figure 17b. Water surface at t = 2.7 s. Model with flatbottom topography.Figure 17c. Water surface at t = 4 s. Model with flat bottomtopography.Figure 18. Definition sketch of the model for the validation.A wide, horizontal, rectangular and frictionlesschannel is used to simulate the dam-break flow. Attime t = 0 + , the dam is removed instantly. The nexttwo figures below show that the most similar solutionof the model solved with CM is the analytical. In thissolution we observe that the wave front is almostvertical (both upstream and downstream of the dam),which is very close to our solution. The othersolutions are also similar. By comparison with the fivesolutions given by Rahman and Chaudhry (1997), weconclude that the code is satisfactory valid to solve the1D dam-break flow.The analytical dam break solution is given by Ritter in1892. This solution is:2 ⎛ x⎜ + c3 ⎝ t=0⎞⎟⎠u (13)12

Comparison between 1D and 2D models to analyze the dam break wave4 ⎛= ⎜c9g⎝0x ⎞− ⎟2t⎠2h (14)with c0 = g ⋅ h0(15)being h 0 the initial water depth in the reservoir and tthe time.Coriolis parameter takes into account the Corioliseffect caused by the rotation of the Earth. Theexpression to get f is:f = 2ω ⋅sinφ(16)Where ω is the angular velocity vector of the rotatingsystem and φ is the latitude. The assumed value for fhas been 2.4110 -5 . The other values are the same as inthe 1D model.Figure 21. The constants dialog box.Figure 19. Comparison of different solutions given by Rahmanand Chaudhry.5.4 Scalar expressionsIn this case, the same expressions as in the 1D modelhave been used, but obviously in two dimensions.Figure 22. The Scalar Expressions dialog box.Figure 20. Model solved with Comsol Multiphysics.5. - MO DEL L I N G THE 2D DAM-BREAKWAVE5.1 Assumptions.In this 2D model we accept the same assumptions asin the 1D model (section 3.1). The only difference tostand out is the bathymetry profile that will be seenfurther on in [5.6].5.2 Dependent variables.The dependent variables for the 2D model are z, zuand zv. Where z is the depth of water, zu and zv areunit discharges along the co-ordinate directions.5.5 Geometry.A rectangle is drawn to define the geometry of the 2Dmodel (figure 23). This is the domain where theequations are valid. The rectangle has dimensions100x50 m. The long side of the rectangle has the samelength as the line drawn for the 1D domain in order tocompare both models, one of the objectives of thisthesis. The first simulations were run with a 200 mdomain but the computational effort was too high forthe 2D model using such a dense mesh. As a matter offact, the dimensions of the domain are not soimportant because the whole model is defined exactlythe same. The maximum length that the computer wasable to run in 2D has been chosen, with the highernumber of elements in the mesh to be as accurate aspossible.5.3 Constants.The same constants seen previously in the 1D modelexcept the Coriolis parameter f have been used. The13

Ignacio Valcárcel LencinaTRITA LWR Masters ThesisFigure 23. 2D Geometry model.5.6 Bathymetry profile.In the two-dimensional model, it is necessary to definea function with two variables (x,y) to define thebathymetry. For the test case, the geometry andbathymetry illustrated in the figures below have beenimplemented. Since a source of strong non-linearitiesis the real bathymetry, a sin(x)cos(y) generated surfacehas been used for the test case. The function used issin(x)·cos(y)/10 , in order to compare with the 1Dmodel with an equivalent function.It defines a similar surface as the one used for the 1Dmodel with the function sin(x)/10, that is, a group ofsmall elevations of ±10 cm downstream of the dam.This kind of bathymetry defined analytically allows usto check its effect on the water surface easily, justchanging the denominator, for instance.Figure 25. View in perspective of the initial conditions andthe bathymetry.5.7 Initial value.The initial conditions are already well-known. Thelevel of water is 10 m high before the dam break and 3m downstream of the dam. The way to define it isexactly the same as it was done in the 1D model, thatis, z 0 = (10)*(x=50).Figure 26. The initial condition at t = 0.Figure 24. Layout of the bathymetry profile.5.8 Meshing.When trying to solve a problem that does notconverge, or having solved a problem where thesolution does not seem properly resolved, there isoften a need for re-solving the problem using a finermesh. In these situations it is sometimes easier torefine the existing mesh than to regenerate the meshfrom scratch using finer mesh settings. The mesh hasbeen refined two times to solve the model correctly.In 2D, the regular refinement method divides eachelement into four elements of the same shape. Thefirst mesh is composed by 496 elements, the secondhas 1984 elements (496x4) and the mesh used (showedin the figure below) is formed by 7936 elements(1984x4). Since Comsol Multiphysics remeshautomatically, the finest mesh that the computer isable to run has been selected.14

Comparison between 1D and 2D models to analyze the dam break waveMesh curvature factor 0.3Mesh curvature cut off 0.001Resolution of narrow regions 1Resolution of geometry 10x-direction scale factor 1.0y-direction scale factor 1.0Mesh geometry to levelSubdomainFigure 27. Mesh with 7936 elements.Figure 28. Mesh with 496 elements.The consequence of solving the model with a wrongmesh is the inaccurate definition of the whole model.The effect is shown in figure 29, where is present astrange appearance of the initial value. The number ofelements of this mesh has been 496 instead of 7936.In the ‘Mesh Statistics’ (table 5), there are somestatistical data for the current mesh such as thenumber of elements and the minimum element quality.Table 5. Mesh Statistics.Number of degrees of freedom 48171Number of boundary elements 184Number of elements 7936Minimum element quality 0.60365.9 Subdomain settings.The previous discourse about the 1D-subdomainsettings is analogous for the 2D model. The samegeneral equation has been adapted in this case, usingthe two-dimensional expressions, as we can see in thefollowing figures:∂ud a+ ∇ ⋅ Γ = F∂t⋅ (17)Figure 29. Mesh with 496 elementsFigure 30. The flux vector tab.Table 4. Mesh Parameters.ParameterValueMaximum element sizeMaximum element size scaling factor 1Element growth rate 1.315

Ignacio Valcárcel LencinaTRITA LWR Masters ThesisTable 6. The algebraic form of the source term and the fluxvector in Comsol Multiphysics.Subdomain 1Source term (f)Conservativeflux sourceterm (ga){0;'-g*Z*Zx-g*Z*dZfdx+f*ZV-T*(ZU/Z)';'-g*Z*Zy-g*Z*dZfdyf*ZU-T*(ZV/Z)'}{{'ZU';'ZV'};{'((ZU*ZU)/Z)-E*ZUx';'((ZU*ZV)/Z)-E*ZUy'};{'((ZU*ZV)/Z)-E*ZVx';'((ZV*ZV)/Z)-E*ZVy'}}Figure 31. The source term tab.5.10 Boundary settings.In the 2D model, Dirichlet boundary conditions (theboundary gives a value to the problem) areimplemented to both ends (edges 1 and 4 in the figurebelow). The lateral side of the simulation domain isnot interested by the flow and then a simple wallconditionis used (Neumann boundary conditionshave been implemented on the edges 2 and 3). Theseconditions are shown in table 7.Figure 32. The damping/mass coefficient (d a ) tab.Figure 34. The boundary conditions plot.Table 7. The boundary conditions types.Boundary 1, 4 2-3TypeDirichlet boundaryconditionNeumann boundaryconditionR coeff. {0;'-ZU';'-ZV'} {'-Z';'-ZU';'-ZV'}5.11 Time stepping.The parameters given by Comsol Multiphysics arelisted in table 8:Figure 33. The initial value tab.16

Comparison between 1D and 2D models to analyze the dam break waveTable 8. Time stepping parameters for 2D model.ParameterValueTimes 0:0.1:5.4Relative tolerance 0.01Absolute tolerance 0.0010Times to store in outputTime steps taken by solverManual tuning of step sizeSpecified timesFreeOffInitial time step 0.0010Maximum time step 1.0Maximum BDF order 5Singular mass matrixConsistent initialization of DAEsystemsError estimation strategyAllow complex numbersMaybeBackwardEulerIncludealgebraicOffFigure 35c. Water surface in the 2D approach at t = 1.8 s5.12 Results.The following figures show the evolution of the watersurface z from t = 0 s to t = 5.4 s in steps of 0.9seconds.Figure 35d. Water surface in the 2D approach at t = 2.7 sFigure 35a. Water surface in the 2D approach at t = 0 sFigure 35e. Water surface in the 2D approach at t = 3.6 sFigure 35b. Water surface in the 2D approach at t = 0.9 sFigure 35f. Water surface in the 2D approach at t = 4.5 s17

Ignacio Valcárcel LencinaTRITA LWR Masters ThesisFigure 35g. Water surface in the 2D approach at t = 5.4 sFigure 37. Flow plot of arrows at t = 5.4 s of the 2D model.The total computation time for this 2D model hasbeen 7m 30s with a processor Intel Pentium 4 CPU2.53 GHz and 1 GB RAM.A similar model with flat bottom bathymetry has beenimplemented as it was done for the 1D model todemonstrate the big influence of the bathymetry in theresults.Figure 35h. Water surface in the 2D approach at t = 5.4 sFigure 38a. Water surface at t = 0 sFigure 36. Contour plot of depths (t = 5.4 s) of the 2Dmodel.Figure 38b. Water surface at t = 2.7 s18

Comparison between 1D and 2D models to analyze the dam break wavem. The initial water level is 10 m at upstream and 5 mat downstream. The breach is 75 m wide.There is not analytical reference solution for this testcase, but in literature numerical solutions of variousauthors are available. The original example wasdescribed by Fennema and Chaudhry (1990) and thesame example has been reproduced here.Figure 38c. Water surface at t = 5.4 sFigure 38d. Water surface at t = 5.4 sFigure 40. The initial condition of the validation model solvedwith CM.ResultsThe validation model has been solved with a meshdefined by 4088 elements. An interesting view of theresults including this mesh is shown in figure 41.Figure 39. Contour plot of depths (t = 5.4 s) of the 2Dmodel with flat bathymetry.In this model with flat bottom bathymetry, the resultsare completely different. The water surface is uniformand flat. This is due to the bathymetry, it does notintroduce alterations, and thus the aspect is “smooth”.6. - VALIDATION OF TH E 2D MODELThe aim of this test case is the study of the capacity ofthe code to reproduce solutions, with particularattention to the 2D aspect of the flow motion. A200x200 m flat region represents the spatial domain.The bottom is friction less. The upstream flow is equalto 0 m 3 /s and the downstream level is set equal to 5Figure 41. Layout of the simulation at t = 6 s. Detail of themesh used in the model.Figure 42. Profile at t = 3 s.19

Ignacio Valcárcel LencinaTRITA LWR Masters Thesisplot of depths is compared in the following figures,where the similarities are clear.Figure 43. Profile at t = 6 s.Figure 46. Contour plot of depths (t = 6 s) of the validationmodel solved with Comsol.Figure 44. Profile solved by Fennema and Chaudhry at t =6 s.These results are very similar to others found in theliterature for this problem. The following figure showsanother version of the situation 6 seconds after thehypothetical instantaneous dam break through abreach 75 m wide.Figure 47. Contour plot of depths (t = 6 s) given by Valianiet al.Figure 45. Profile at t = 6 s. Realistic behaviour of theproblem.A comparison with the model solved by Valiani et al(1999) has been done in order to validate the code in2D. A priori, observing the results in 2D (which arevery close to the results obtained with the alreadyvalidated 1D model) one guess that this 2D model isvalid also. The results obtained by Valiani et al showsome graphics and figures that here will be used tovalidate the 2D model by comparison. The contourFigure 48. Isovalue z = 5.4545 m. (Comsol result).20

Comparison between 1D and 2D models to analyze the dam break waveFigure 52. Vectors of Velocity plot (t = 6 s) obtained withComsol.Figure 49. Isovalue z = 9.0909 m (Comsol result).Figure 53. Flow plot (t = 6 s) solved with Comsol.Figure 50. Flow plot (t = 6 s) given by Valiani et al.Figure 54. Streamline plot solved with CM at t = 6 s.Figure 51. Vectors of Velocity plot (t = 6 s) given byValiani et al.Therefore, the models (1D and 2D) have beenvalidated by comparison with the results given byRahman & Chaudhry and Valiani et al respectively intwo test cases involving dam break flow. Software isgenerally validated against analytical solutions of theSWE or with laboratory experimental measurements.Both methods are very useful but are not fullysatisfactory. In the first case (our case) the validity ofthe equations is not questioned and in the latter theturbulence is overlooked, not to mention sedimenttransport effects.21

Ignacio Valcárcel LencinaTRITA LWR Masters Thesis7. - DISCUSSION AND CO MPARISON OFBOTH MO DEL S.The results show that the two modelling approachesare close to each other. The most importantparameters to analyze after a dam-break flood (interms of safety) are the maximum free surfaceelevation and the propagation time of the wave. In the1D approach, the maximum surface elevation is a fewcentimetres higher than the elevation obtained withthe 2D, as we see in figures 55 and 56. In these figures,the evolution of the water surface is shown starting att = 0 with steps of 0.1 up to 5.4. Every line representsevery time step. Both figures are practically equal. Themaximum surface elevation is around 6 meters high(with 10 m as initial level at the reservoir).The propagation time is exactly the same for bothmodels. The results show an average velocity of thewave of 33.3 km/h. The wave propagation on a wetbed is faster than on a dry bed. This study focusesonly on the wet case, but we have to take this fact intoconsideration in real applications where the wave goesthrough both wet and dry zones (depending on thereached depths). In addition, the friction coefficientbecomes unimportant in a wet bed case. This is due tothe fact that the depressions that are normally filled bythe flood wave as it passes by are now already full ofwater when the computation begins. In realapplications, the friction coefficient is much moreimportant, especially for the propagation times. Theinfluence of friction on the maximum free surface issmall, even if it is known that the lower the frictioncoefficient, the higher the elevations.This thesis analyses the dam-break wave from atheoretical point of view. A simple example of thecollapse of a water supply reservoir in a narrow valleyhas been used to demonstrate the capability of themodel. In real contexts, the differences between 1Dand 2D models are significant and nowadays they areknown. We have studied the dam-break problem witha generic tool, which is not adapted to solve realapplications on large domains. On the other hand wehave studied in depth the problem and we can ensurethe validity of the two-dimensional approach.Until recently it was considered that two-dimensionaltools could not be applied to large domains, forreasons of computer time and memory. Moreover, thetopographic data available from field survey, generallycross-section geometry profiles, were simple toassimilate into 1D models. However, 1D results mustbe carefully interpreted to work out the final twodimensionalmaps showing flooded areas and arrivaltimes, and this may introduce additional errors into theanalysis. Calibration is also easier in 2D becausephenomena such as the effect of bends or obstaclesare naturally reproduced, whereas they are artificiallytaken into account via empirical head loss correctionsin 1D.Figure 55. Cross section of the evolution of the water surfacewith the 1D approach.Figure 56. Cross section of the evolution of the water surfacewith the 2D approach.In terms of non-theoretical simulations, someconsulted papers conclude that the new twodimensionaltools obtain an improvement on onedimension when there are sharp bends in the river (thebends are not taken into account in 1D simulations)and are necessary in cases where the flood spreads in acoastal zone. They also improve predictions of thewave celerity. Two-dimensional simulations of floodwaves are already possible on domains with a length ofsome 10 s kilometers. Larger domains (100 to 400 km)are also within reach with supercomputers, or withparallel architectures.There are a lot of uncertainties in the modeling andespecially in the one-dimensional flow modeling wherethe user of the model can have a significant effect onthe results by selecting the locations of cross-sectionscarelessly. The debris flow, clogging of bridges andother structures and erosion of flooded areas alsocause uncertainties in the flood simulation and thoseuncertainties must be taken into account.In addition, there are several topics in dam breakhazard analysis that require additional research.According to the RESCDAM Project (2001), thefollowing topics are of primary importance:22

Comparison between 1D and 2D models to analyze the dam break wave• Determining a dam breach formation (flowhydrograph).• Determining roughness coefficients used inflow simulations.• The effect of debris flow and urban areas in adam-break hazard analysis.8.- CON CL USION.One of the main goals of this study was to show thecapacity of a general tool called Comsol Multiphysicsto deal with dam-break simulations. This capacity hasbeen proved even though it has been done using asimple and idealized case. In parallel with the rapiddevelopment of computer technology within the last20 years, dam break and flood simulations havestepped from the use of simplified models to onedimensionalchannel network models and finally toexpensive two-dimensional models. The trend is tominimize the expert work in favor of computerresources.The main improvement expected from a 2Dcomputation was in terms of the accuracy of thepredicted propagation times. After the comparisondone in this work we don’t see this improvementbecause the models used have been theoretical andwithout irregularities, it has not been a realisticapproach. The use of a 1D model needs a lot ofexperience since the cross-sections have to be put atthe right locations. The use of a 2D model is morestraightforward. However, the results of the 1Dapproach are actually considered good because theyare on the safe side and safety is the key concept inthese problems.The 1D approach is even recommended by somedepartments like the Spanish Ministry ofEnvironment, where in its “Technical guide for thedevelopment of emergency action plans for dams” we read: “Ingeneral, it is desirable to use 1D models because they aresufficiently accurate and the 2D models usually are complex tomodel and to get the basic data (topography, directionalroughness, etc.) and force us to assume additional simplificationsto take into account the second dimension.”Thinking in terms of implementation, it is clear thatthe 1D approach is valid because it is conservative andgood enough; however, actually it is possible to thinkabout more and more accurate 2D models, thanks tothe rapid development of the technology. Today’sstate of the art procedure is to develop complexterrain models and superimpose them with floodcomputation parameters.Although a general conclusion of this report could bethat it is necessary to decide between 1D and 2D, it isnot quite that simple. The use of one doesn’t excludethe other. In fact, it is desirable to take both of theminto consideration. The resources for the 1D approach(computers and software) are more accessible,therefore this approach is very useful to getconservative screenings at the beginning. If theresources and data are available, a 2D model isadvisable for all the reasons expounded previously. Inaddition, the results of the performed 1D model canbe a great help to predict the results in the 2D model.In practice, the implementation may be best achievedcombining 1D and 2D models. A framework for adam break analysis is summarized in the Appendix,where the current practice is presented according tothe ICOLD (International Commission On LargeDams) recommendations. As it can be seen in theAppendix, the analysis is based using 1D and 2Dmodels.Typically 1D models are based on finite differencemethods (FDM). Apart from the research, even thelargest energy companies achieve the dam break floodanalysis with this kind of one-dimensional approach. Itis faster, cheaper and the results are fully satisfactory.The dam break studies have been attempted with the1D approach in the past, actually the “1D+2D”approach is becoming more common, and in thefuture, when the resources are completely accessible,the 2D approach could be the primary way to achievethese analyses.23

Ignacio Valcárcel LencinaTRITA LWR Masters ThesisAP PEN DI X: DEV EL OPM E N T OF RESCU EACTIONS BASED ON DAM-BREA K FLOODANALYSIS (PRES EN T PROCED URE) .The previous part of this work is a description of twomodels developed with a solver software packagecalled Comsol Multiphysics. Once the validation ofthese models has been done, the first purpose shouldbe the application in real contexts. This Appendix hasbeen achieved to explain this practical approach in ageneral way, not using concrete software applied to aspecific dam.The enormous complexity of the dam break problemdoesn’t make possible to introduce simplifications inthe beginning. They are completely dependent of thespecific case analyzed. In addition, the constantevolution of the mathematical models makes verydifficult to recommend any specific model for ageneral use. Nevertheless, as a help for the selection ofthe model is recommended the study of the ICOLD(International Commission On Large Dams) about“Dam Break Flood Analysis” carried out by theSubcommittee of “Analysis of dam-break flooding andrelated parameters normally assumed” where 27existing models are described and evaluated. Actuallyonly a few are widespread and accessiblecommercially.The RESCDAM Project is a Grant Agreement (NoSubv 99/52623) signed between the EuropeanCommission and the Finnish Environment Institute. Itis actually the last complete study about the dam breakproblem, (after the study of the ICOLD) and it couldbe a sort of “guide” for the engineers involved withinthis task. Because dams and especially consequences ofa dam failure are quite similar in different countries,results and activities of the RESCDAM project shouldbenefit EU and associated countries.The main purpose of the RESCDAM project was todevelop emergency action planning for dams. A dambreak hazard (flood) analysis is a necessary aid for this.A risk assessment (analysis) was also included in theproject work. The pilot project of RESCDAM was theembankment dam of the Kyrkösjärvi reservoir locatedin Seinäjoki in Western Finland.According to the RESCDAM Project, a completeproject work should be divided into three subprojects:A) Risk Assessment (Analysis).B) Dam Break Hazard Analysis (DBHA).C) Emergency/Rescue Action Planning.Following the RESCDAM project steps, the actualprocedure to analyze entirely a specific dam-breakproblem is developed from now on. The second point(DBHA) is the principal focus of this ninth section ofthe thesis (numerical flow models, 1D and 2D). Sincethe comparison between one and two dimensionalmodels has been developed in terms of flood analysis,the first and third point are not the main purpose ofthis section; it has been developed in this appendix toexpound a complete dam break approach.This Appendix will focus on the scientific approachesthat have produced today’s engineering practices.After a very long period when the art of damconstruction remained empirical, but when there werenevertheless some remarkable inventions, dambuilding gradually took on a scientific basis, usingmathematical tools and techniques. Today, dambuilding calls for scientific and technical knowledge invaried disciplines.Each of these disciplines makes use of less specialisedtools provided by mathematics (continuum mechanics,materials strength), physics and chemistry. In terms ofsafety, it is in fact “rare” or “extreme” events that areof interest, that is, accidents. The usual approach istherefore to specify the design flood and designearthquake at the design stage.A) RISK ASSESSMENTConcept and bases of risk analysis for damsRisk assessment is the process of deciding whetherexisting risks are tolerable and present risk controlmeasures are adequate and if not, whether alternativerisk control measures are required. Risk assessmentincorporates, as inputs, the outputs from the riskanalysis and risk evaluation phases.The primary steps of a risk assessment consist of thefollowing (ICOLD, 1999):- Risk analysis.- Risk evaluation.- Risk management.Risk analysis.The risk analysis of a dam is the use of availableinformation to estimate the risk to individuals orpopulations, property or the environment, fromhazards. A risk analysis generally contains thefollowing steps:• scope definition• hazard identification• risk estimation.A risk analysis involves the de-aggregation ordecomposition of the dam system and sources of riskinto their fundamental parts.The scope and expectations of the risk analysis shouldbe defined at the outset. Definition of the scopeshould include (ICOLD, 1999):- description of the issues that initiated the riskanalysis,- description of the dam system to be analyzed,- identification of the point in time to which the studyrelates,24

Comparison between 1D and 2D models to analyze the dam break wave - Appendix- formulation of the objective of the risk analysisbased on the main concerns identified,- identification of the decisions to be made, thedecision-making criteria and the decision-makers,- description of the proposed risk analysis processincluding statement of the assumptions andconstraints governing the analysis.Risk analysis for dam systems can be categorized bythe nature of the hazards leading to the consequencesof concern. The hazards in question may be groupedinto four general categories, namely:- hazards due to natural conditions (normal loads,earthquakes, floods, debris, winds, etc.);- operational hazards (spillway reliability, operatorerror, etc.);- internal hazards (aging, alkali-aggregate reaction,internal erosion, metal fatigue, etc.); and- social hazards (war, sabotage, etc.).Failures are often categorized as static, hydrologic orseismic. Some of the various factors, are presented inthe figure below:Figure 57. Some factors to cause dam failures.Risk estimation entails the assignment of probabilitiesto the events and responses identified under riskidentification. The assessment of appropriateprobability estimates is one of the most difficult tasksof the entire process. The outcome of this step is acalculation of the risk of failure.Risk evaluation.Risk evaluation is the process of examining andjudging the significance of risk. The risk evaluationstage is the point at which values (societal, regulatory,legal and owners) and judgements enter the decisionprocess, explicitly or implicitly, by includingconsideration of the importance of the estimated risksand the associated social, environmental, andeconomic consequences, in order to identify a range ofalternatives for managing the risks.Risk mitigation is a selective application of appropriatetechniques and management principles to reduceeither likelihood of an occurrence or its consequences,or both (ICOLD, 1999).Risk acceptance is an informed decision to accept thelikelihood and the consequences of a particular risk(ICOLD, 1999). Risk acceptance is coupled to the riskreduction evaluation and is a basis for deciding whatresidual risk will be accepted for the affectedcommunity and structures.Risk managementRisk management is the systematic application ofmanagement policies, procedures and practices to thetask of identifying, analyzing, assessing, treating andmonitoring risk (ICOLD, 1999). Having received therisk information, and knowing the risk valuationcriteria, a decision-maker must come to a decision. Anumber of good practice steps to be taken to aid thedecision process are given by the ICOLD. These stepsshould include e.g. consultations with stakeholders andcommunity, insurance issues, legal defensibility ofdecisions, risk information to decision-maker, and tothe public, clarification of the role of decision-makerand documentation of the decision and its rationalimposes a discipline that is helpful to sound decisionmaking.Causes and probabilities of dam incidentsThe data, which have become available in the last 10 -20 years, provide a foundation for the ideasrecommended to be used in risk analysis. This studyaddresses risks associated with the construction,operation, and failure of dams, excluding indirectenvironmental or resettlement problems. The purposeof the study is to identify the main real risks associatedwith each type and height of dam, for allcircumstances. It may be of help in several respects:- in extensive risk analysis of very large dams, tosubstantiate reliably the probabilities chosen in eventtrees;- in simplified risk analysis of smaller dams, to focuslow-cost risk analysis on a few main risks;- in identifying possibilities for reducing these risksthrough low-cost structural or non-structuralmeasures.When studying the possibilities of risk assessment thereported failure circumstances can be classified asfollows:• Exceptional circumstances: floods, upstreamdam-break wave, earthquakes, and war. Riskassessment for exceptional circumstances islogically based on the probability and impactof such circumstances.• Normal circumstances: first filling, aging(after two years operation) and internalerosion, and construction. For dams innormal circumstances the safety is essentiallybased on human behaviour, and riskassessment should pay a great deal ofattention to this point.B) DAM BREAK HAZARD ANALYSISIntroductionDam break modelling consist of first, prediction of theoutflow hydrograph due to dam breach and latter,routing the hydrograph through the downstream valley25

Ignacio Valcárcel LencinaTRITA LWR Masters Thesisto get the maximum water level and discharge alongwith the time of travel at different locations of theriver downstream of the dam.Dam break hazard analysis (DBHA) provideinformation on the consequences of a possible dambreak for risk estimation and rescue planningpurposes. To determine the flow through a dambreach and to simulate flood propagation in thedownstream valley numerical models are used inDBHA.Any output of a dam break hazard analysis is related tothe accuracy and amount of input data. We cannotexpect to have accurate results if our data is poor. Therequired data is characterized as (ICOLD 1999):• reservoir data (including catchmenthydrology and river flood conditions)• structure data• topographic data• sediment and debris data.Reservoir data includes stage-area and stage-volumerelationships and the bathymetric data for thereservoir. Catchment hydrology data may for examplebe reservoir inflow and river flow data includingnatural flood observations. Structural data consists ofdata for dams and other structures. Topographic datais needed for the whole area which may be flooded bya dam break flood. Terrain model technology providesexcellent tools for handling of topography data.The results of DBHA should provide information foremergency action planning, land use planning andother purposes in a form that is appropriate to the enduser. A Geographical Information System (GIS)provides efficient tools to transfer the information indigital form and to analyse the results. By using GISflood inundation maps and other related informationcan be produced in different scales depending on thescale of the base map which is used in thepresentation. In a project, maps are produced forflood inundation (max inundation and the inundation0,5 h, 1 h, 2 h and 3 h after the initiation of thefailure), for flow velocity, for damage parameter (waterdepth times flow velocity) and for flow depth. Thescale used for printed maps is generally 1:20000 butthe scale 1:10000 is also used.Water level hydrographs and tables for certainlocations may be important for emergency actionplanning. From hydrographs the arrival time of floodwave and the speed of the rising of water level can beobtained.Hydrological analysisThe watershed model is a semi-distributed model withthe sub-basins. Each sub-basin has separateprecipitation, temperature and potential evaporation asinput. The model has a flood area-model whichsimulates water exchange between river and floodplains. The flood-area-model is simulated with shortertime steps than the main model. At every reach ofriver with embankments and weir, the model calculatesthe water level in the river and discharge through theweir over the embankment into the flood plain. Thispart of the model is important during the floodseason.This simple hydraulic model is verified against theresults from a complete hydraulic model in order tokeep up the accuracy of the simulation. The watershedmodel is based on a conceptual distributed runoffmodel, water balance model for lake, river routingmodel and flood area models.The input variables for the model are dailyprecipitation, temperature and potential evaporation.As input for a dam break simulation three floodsituations should be simulated. Floods with a returnperiod of 20, 100 and 10000 years create or determinewith the operational hydrological catchment model.The method used to determine a 10000 year flood isbased on precipitation with a return period of 10000years.Determination of the breach hydrographsDetermination of flow through a dam breach has lotof uncertainties. A numerical model for erosion of anembankment dam can be used for the determinationof the discharge hydrograph. The results of theerosion model are compared with those of othermethods. In the studies the breach is normallyassumed to happen at three locations with twohydrological conditions which are HQ 1/100 flood andmean flow (MQ).One-dimensional flow modellingA usual 1D model is based on four point implicitdifference scheme and uses the solution algorithmDAMBRK model.The 1D flow model for the DBHA normally covers abig area mainly downstream of the reservoir. We talkabout kilometres to cover the whole domain. Thecross-sections used in the model are taken from aterrain model. We talk about cross-sections, reachesand junctions in the 1D model. Some of the reachesare fictive channels (flood plains, connecting channelsetc).In the 1D model the breach is normally assumed tohappen at three locations with two hydrologicalconditions which are HQ 1/100 flood (with a returnperiod of 100) and mean flow (MQ). The breachhydrographs which are used in the simulations aredefined by using the breach erosion model. A constantroughness coefficient (Manning n = 0,060) could beused in all simulation cases.Two-dimensional flow modellingIn a 2D simulation a finite element model is normallyused. The finite element mesh has thousand ofelements depending on the computer capacity to dealwith it. The mesh size ranges from approximately 5 mto 50 m. The river beds, the bridges, the roads, and therailways have been highly refined to accurately modelflood propagation. In regular areas such as roads,26

Comparison between 1D and 2D models to analyze the dam break wave - Appendixreservoir dykes, and river beds, regular grid applicationis used. For other features such as railways,embankments, and constraint lines are imposed. Allthese features are quite visible on the mesh.Urban areas and floating debrisIt is necessary to develope methods to calculate floodwave propagation in urban area. Some models use thegeometry approach and houses are taken into accountin model geometry. Other models use the porosityapproach.In the first model, the geometry is represented by agrid consisting of rectangles (10x10 metres forinstance). Generally, the calculated water levels arehigher in the cases where buildings where taken in toaccount (maximum difference about 0,5 m). Thepropagation of a flood wave is generally a bit slowerwhen the buildings are taken into account. However,the effect on propagation speed is not very significant.The effect of buildings on damage parameter is notvery significant and can not be seen as very importantfor the planning of emergency actions. However, insome specific cases the effect can be important.Analyses of the resultsThe following analyses can be simulated during theproject, for instance:RUN 1Base flow (HQ 1/100 ) + breach hydrograph for maxreservoir level, roughness varies between Strickler 15(Manning's n = 0,06666) and 40 (Manning's n =0,025).RUN 2Base flow and breach hydrograph as in RUN 1,roughness is constant for the entire modelling areawith Strickler 15 (Manning's n = 0,06666).RUN 3MQ base flow + breach hydrograph, constantManning n varying in the entire modelling areaaccording to land formation and vegetation.RUN 4HQ 1/100 base flow + breach hydrograph, otherconditions as in RUN 3.RUN 5Conditions of RUN 3 modified according to thechoice of modelling buildings (porosity or geometry).RUN 6Conditions of RUN 4 modified according to thechoice of modelling buildings (porosity or geometry).The breach location is assumed to be the same for allthese analyses.The reason behind using different models to simulatethe same case is not to compare the computationalalgorithms. The reason is to get an idea how much theresults differ depending on the models.The modeller uses his own judgement for choosingthe friction factors for different areas. The water levelcomparison is done on different locations downstreamof the dam. The progression of the dam break flood isalso compared on maps.In the case of very complicated topography, the use ofa two-dimensional model seems to be morereasonable. The use of a 1-dimensional model needs alot of experience since the cross-sections have to beput at the right locations.Summary .To determine flow hydrographs through the dambreach opening is crucial to the results of a DBHA.There are a lot of uncertainties in the determination ofa breach hydrograph and sensitivity analyses have tobe used to ensure the results.A one or two-dimensional model can be used in thecalculations. The experience shows that with carefulmodelling and accurate data the results of differentmodelling approaches may be relatively close eachother. However, there are a lot of uncertainties in themodelling.Several methods can be used to model the flow inurban areas (porosity or geometry approach). Theconcrete results of a dam break hazard analysis foremergency action planning consist of inundationmaps, water depth maps, damage parameter (flowvelocity times water depth) maps as well as water leveland velocity hydrographs and tables. The results ofDBHA should be presented in such a way that theycan be used efficiently in the emergency/rescue actionplanning and risk analysis. The use of theGeographical Information System (GIS) is essentialfor that purpose.C) EMERGENCY ACTION PLANNINGIntroductionThe main purpose of the RESCDAM project was todevelop emergency action planning in the event of adam failure. The topics of the emergency actionplanning are approached from the point of view ofrescue authorities.Dam safety is an issue for a few people only, such asdam owners or dam holders, the rescue andenvironmental authorities. Only a few of these peoplehave been faced with a serious dam failure or damemergency situation. In case of an extensive accident,situation management is almost impossible withoutadvance planning. Lack of experience is also aproblem when considering the emergency action planof a dam. It is difficult to prepare an emergency actionplan for an event that can only be imagined.Organizing the warning and evacuation of thepopulation and rescue operations in a dam failuresituation.Dam failure is an extremely demanding accidentsituation to everyone involved. A strong water flow,varying water depth, floating debris, damaged road27

Ignacio Valcárcel LencinaTRITA LWR Masters Thesisnetwork and power supply etc. makes the situationespecially difficult for rescue operations. Thus thepopulation in the danger area must be evacuatedbefore the arrival of a flood wave.The essential part of warning the population is to getthe population to take the action necessary. Peoplemust be convinced of the reality of the danger andthey must be advised to act according to theinstructions of rescue authorities. A mere warning isoften not enough. People in danger need instructionson how to save themselves from a threateningsituation.Definitely the most important factor affecting thewarning, evacuation and rescuing of the localpopulation is the time period between dam failure andnotification of the failure and commencement of therescue operation. If a dam failure alarm is given late,the conditions for making emergency repairs to thedam are poor and the probability of human casualtiesincreases.At the same time the benefit from the public alarmsystem installed to the dam danger area is lost. Thenotification speed of the dam failure has beenobserved to have a direct correlation to the loss ofhuman life (Douglas et al. 1999, Lemperiere 1999). Thesafety monitoring of the dam must correspond to therisk factors of the dam. The greater the failure risk ofthe dam, the greater the demands on dam safetymonitoring functions.The credibility of following warning methods andachieving the necessary action was assessed:• public warning signals (sirens)• emergency bulletin on the radio• loudspeaker vehicles of rescue units• telephone and mobile phone (group calls)• warning people indoors from house tohouse.Public warning signals are best suited to urban areas,because an alarm can be given quickly and a largepopulation can be reached. The working state ofequipment is generally secured using backup batteries,to make sure the system is independent from anexternal power supply. Spoken messages andinstructions can also be given using modern electronicequipment.One problem with public warning signals is thecredibility factor – do people believe that the danger isreal. In addition to a public alarm signal, attentionmust be paid to the visibility of the warning.The success of a warning operation is partlydependent on the fact that the population is aware ofliving in a dam danger area. If a lot of people, whosesafety the rescue services cannot guarantee in case ofan emergency, are living in a dam danger area, anindependent initiative is also required for escaping thedanger area. The population must be aware of thepossible personal danger so that they can protectthemselves and act on their own in case of anemergency. The necessary information can bedistributed through an advance information bulletin.This bulletin has a special purpose due to the nature ofa dam failure. In case of a chemical or radiationemergency, the population may be able to protectthemselves in their homes, but in case of a dam failurethe situation is different. Some buildings fill with waterand collapse due to structural strain of masses ofwater, whereas some buildings remain intact. Thebulletin is very important for the credibility of thewarnings and decisions about action taken in a damfailure situation. The bulletin decreases the need for anindividual to find further assurance of the threateningdanger and gives a prepared operational model forescaping the danger area.The purpose of warning and evacuation is to evacuateareas that will be flooded according to the inundationmaps. This applies in particular to buildings withoutupper storeys. Rooms below ground must always beevacuated.Rescue authorities carry out evacuation in dangerareas. Taxis, local buses and ambulances are used asevacuation vehicles in the evacuation organised by therescue authorities. If people living in the danger arealeave on their own, they go on foot, use their ownvehicles or other means of transportation. Whenwarning and evacuating people, they are instructed totake personal medication with them.Vehicles used for evacuation are alerted to specifiedmeeting places by the emergency response centre.There drivers are given their tasks as well asinstructions and materials required to execute thetasks. Subsequently vehicles are sent to the evacuationareas to execute their evacuation tasks. Evacuationunits execute the evacuation under the supervision ofthe personnel of rescue services. Vehicles are not toproceed with the evacuation on their own, sincedrivers may have problems managing anxious peopleon their own.Evacuated people are taken to evacuation centres.These centres are organised by the social and healthservices using voluntary rescue service personnel. Thelocation of the evacuation centre must be planned sothat a dam failure does not have an effect on its waterand heat supply, distribution of electricity and sewagesystem.The following supply services are organised in theevacuation centres:• first aid• crisis bulletin for evacuated people, theirfamilies and people who have lost theirhomes• psychosocial help for evacuated populationand people who have lost family members orfriends• catering and clothing for evacuatedpopulation28

Comparison between 1D and 2D models to analyze the dam break wave - Appendix• temporary accommodation.One of the golden rules of rescue operations is”control the greatest threat first”. The starting point ofa rescue operation must be the warning andevacuation of the population so that an actual rescue isunnecessary. In the emergency action plan there mustbe an assumption that people may have to be rescuedfrom the water or from buildings surrounded by water.SummaryThe important part of preparing an emergency actionplan for a dam must be in organizing the warning,alerting and evacuation activities. The consequences ofa dam failure as well as the conditions following theemergency are very difficult for rescue operations, andevacuation before the arrival of flood waters must bethe main line of approach. In addition to thedevelopment of warning and evacuation procedures,an automatic dam monitoring system to provide anearly warning of a potentially serious incident or a damfailure should be developed further.Assembly rooms, hospitals, maintenance institutionsand corrective institutions should not be built in thedam danger area, because the evacuation of suchbuildings is very difficult in an emergency situation.Buildings in the danger area should be built in such amanner that dam failure will not endanger peopleliving in the buildings.When preparing for a dam failure, it is important toconsider human behaviour in a crisis situation. Studiesof this topic show that people do not always believe inthe reality of warnings. Home is generally considered asanctuary and leaving home is no easy step. In theplanning and implementation of rescue operations, thecompliance with warning and evacuation instructionsmust always be ensured using vehicles with aloudspeaker system moving in the danger area andrescue units going from house to house.Compliance with warnings and instructions given byauthorities can be made easier by issuing a safetybulletin to people in the danger area in advance. Thiskind of advance information has generally been amatter of recommendation, as opposed to aregulation. Advance information does play animportant part in the success of rescue operations andas such it should be a routine requirement.The hazard assessment of a dam together withinundation maps for the flood area subsequent to adam failure are only the first steps for an emergencyaction plan prepared by rescue services. The needs ofrescue services must be noted when presenting theflood information and inundation maps. Theavailability of maps in digital and printout form shouldbe further developed.Besides the duty of competence placed on engineers,there is also a vital on-going duty to inform andeducate the public. At a higher level, those withpolitical power, who represent the public, will have torespond to this new educated view. Any society, richor poor, can maintain and even improve the safety ofits structures. It is the engineer's duty to see that thestakes are clear.29

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