A Rapid Model for Delimiting Flooded Areas

Abstract

Analyzing hydraulic risk involves simulating a high number of scenarios from which to draw statistical information about flood extent and depth in relation to variations in the defence conditions and the entity of the event. With the aim of keeping the computational times of simulations within reasonable values while maintaining a sufficient reliability of the results, rapid models enabling flooded areas to be delimited using a DEM have been introduced into the scientific literature. These models, called Rapid Flood Spreading Models (RFSMs), are based on highly simplifying hypotheses that are analyzed and discussed in this paper thus arriving to a new formulation. A comparison of the results produced by this new formulation versus those of other RFSMs in a test case largely characterized by flat land shows it to be superior and reliable, while maintaining the computational times to within a few seconds. The new formulation thus lends itself to being used to support hydraulic risk assessment and management procedures requiring a reliable simulation of numerous flood scenarios.

1 Introduction

Floods are natural events that can give rise to severe economic losses and in some cases also the loss of human lives: between 1998 and the 2009, Europe was hit by over 213 major floods, in countries such as Austria, Italy, Germany, the Czech Republic, the United Kingdom and Hungary (European Agency 2005; CRED 2009; European Agency 2010). The frequency with which these episodes have occurred over the past two decades, together with the huge damage caused, have induced the European Community to reflect upon the growing hydraulic risk and design a legislative framework for the definition of flood risk management plans (Directive 2007/60/EC). Drawing up these plans is a very complex undertaking, as it involves numerous aspects tied to land management (see, for instance, Correia et al. 1998; Costa et al. 2004; Kundzewicz et al. 2010). An indispensable step in this case is to simulate a flood phenomenon starting from its trigger point or points. Knowledge of the flood’s characteristics in terms of timing, extent, speed and depth represents the starting point for assessing the possible damage and quantifying, accordingly, the hydraulic risk connected to different land and/or crisis event management policies. In other words, the ability to assess and compare flood risk management plans depends on the possibility of creating different flood scenarios, each connected to particular surrounding conditions arising from the operational choices made. The number of simulations to be carried out in this context is certainly high (Voortman et al. 2003; Dawson et al. 2005; Gouldby et al. 2008; Harvey et al. 2009; Lhomme et al. 2009; Harvey et al. 2012) and the computational speed of individual simulations thus takes on particular importance. Two-dimensional hydraulic models such as MIKE21 (DHI Water & Environment 2007), SOBEK (WL | Delft Hydraulics 2005) and TELEMAC-2D (EDF-DRD 2002), just to mention the best known commercially available ones, enable in-depth, detailed flood analysis, but have the disadvantage of requiring very long computational times (even days) when the surfaces concerned become very large and the degree of descriptive detail required remains high.

In response to the need to speed up simulations, some researchers have turned towards simplified approaches based on diffusive formulations (Bascià and Tucciarelli 2004; Moussa and Boucquillon 2009; Aricò et al. 2011), or on coupling a 1D model (used to study propagation in the riverbed) with a 2D model (used to represent the submersion wave). The 2D model in this coupling enables shallow water equations to be solved using a numerical diagram in which the cells making up the calculation grid represent areas of accumulation (Cunge et al. 1980). In this context, the increasingly wide use of applications developed in a GIS environment with the aim of storing and processing topographical data has led to the formulation of raster-based models in which the calculation grid exactly coincides with the original DEM. Therefore, every cell of the DEM represents an area of accumulation, whose capacity is defined through appropriate geometric relationships directly generated in a GIS environment. The mass flows between cells are evaluated relying on simple formulations (based, for example, on the Manning formula for uniform flow) (Bates and De Roo 2000; Horrit and Bates 2002).

Another simplified approach envisages the use of methods based on cellular automata, which by their nature enable implementation of the parallel type of computing code (Dottori and Todini 2010, 2011). However, the results obtained to date with the latter approach are not yet comparable to those achieved by integrating, in various ways, the classic hydraulic equations of mass and momentum balance.

In any event, though all of these simplified models can achieve good levels of computational efficiency, they are not yet sufficient when a very large number of scenarios needs to be analyzed, as in the case of hydraulic risk analysis (Voortman et al. 2003; Dawson et al. 2005; Gouldby et al. 2008; Harvey et al. 2009; Lhomme et al. 2009; Harvey et al. 2012).

These limitations are due to the intrinsic character of the models mentioned up to now, which take into consideration the complete temporal evolution of the hydraulic process.

One line of research that aims to overcome such limitations is based on the development of rapid conceptual models for assessing the extent of a flooded area for fixed incoming volumes and fixed conditions of the riverbank defence system or in any case of the flood trigger point (Gouldby et al. 2008; Lhomme et al. 2009). In these models, the temporal evolution of the flood phenomenon is wholly disregarded and thus kinetic energy, head gradients and friction forces are considered to affect the final result less significantly: attention is focused only on delimiting the maximum flooded area and associated depths when water is still everywhere.

This simplification constitutes an intrinsic limit of these models, making them inadequate for applications that require detailed knowledge of the timing and velocity fields associated with flood phenomena. It should be observed, however, that defining the perimeter of geographical areas that could potentially be affected by floods according to different inundation scenarios, with an indication of water depth, represents sufficient information for the purpose of characterizing risk as required by European Directive 2007/60/EC.

In all of these simplified models, the ground is schematized as a system of storage reservoirs, whose hydraulic behaviour is governed solely by the law of conservation of mass. The distinction among the various approaches lies in the way in which the various zones (or storage reservoirs) interact. In particular, the RFSM (Rapid Flood Simulation Model) of Gouldby et al. 2008 is based on a unidirectional spilling, that is, a reservoir can feed only one of its neighbours because of the lower crest existing between the reservoir considered and those around it. The degree of schematization is clearly very great and a comparison between the final extent of the flooded areas resulting from this approach and the extent deduced using two-dimensional hydraulic models shows significant divergences in some cases (Gouldby et al. 2008). For this reason, Lhomme et al. 2009 introduce some variants in order to better represent the effect that the shape of the different reservoirs and roughness of the ground may have on the final result; they moreover assume that the volume exceeding the capacity of a generic accumulation zone is equally distributed among its neighbours.

This paper presents a further evolution of the approach described by Gouldby et al. 2008 and Lhomme et al. 2009. In particular, we propose an alternative to the modes of exchange among neighbouring reservoirs in order to render it multidirectional but not uniform, so as to better simulate the real situation, in which the interaction between one zone and adjacent ones is highly complex. Moreover, we propose subdividing the volume of water spread over the land in subsequent phases to better mimic the effect that temporal dynamics have on how a flood spreads.

Section 2 first describes the studies by Gouldby et al. 2008 and of Lhomme et al. 2009, which are seen here as starting points, and then the proposed method. The numerical application presented in section 3 makes reference to an area of flat land in order to provide a worse-case scenario of the problems connected to the spreading of water: we consider both the case of a single flood trigger point and the case in which there are a number of trigger points. Section 4 presents the conclusions of the study.

2 The Rapid Model Developed

Since the rapid model described in this paper represents an evolution of the RFSM (Gouldby et al. 2008), subsequently modified by Lhomme et al. (2009), we think it appropriate to start off by describing the RFSM and the modification of Lhomme et al. (2009) and then present the further modifications we propose.

2.1 The RFSM

As mentioned previously, the RFSM (Rapid Flood Spreading Model) is a rapid model for defining flooded areas and the corresponding water depths. The input of this model is represented by volumes of water: they are input into the flood-prone area at one or more points representing the places where the flood originates. These volumes in turn summarize the entire incoming flood wave in the area considered.

The model is structured in two logical blocks: the first comprises a set of operations serving to define the calculation grid and generate and organize the data necessary for subsequent processing; the second is represented by the actual flood calculation.

The definition of the calculation grid is based on a schematization of the area of interest into a system of accumulation zones (AZs), which represent the natural depressions in the ground where water flows toward during a flood. From an operational viewpoint, the calculation grid can be automatically defined based on the DEM of the area of interest using specific applications developed in the ArcGIS™ environment. These applications enable the following parameters to be calculated for each AZ: minimum elevation within the AZ (accumulation point), level-volume curve, identification of adjacent AZs, identification of the cells of the DEM which define the border or line of separation between the accumulation zone considered and the adjacent accumulation zones (surrounding cells). Among the cells surrounding each adjacent AZ, the ones of interest are those with the lowest elevation (communication point): generally speaking, there is one communication point for each line of separation and one line of separation for each AZ adjacent to the one considered (see Fig. 1).

Fig. 1

Schematic representation of the division of the calculation domain into accumulation zones (AZs) with the associated boundary cells, communication points and accumulation points

The data generated, duly organized into tables, together with the AZ or AZs of the incoming flood volume V, represent the input for the “simulation”, whose output is a definition of the maximum extent of the flooded area and the corresponding depths. In the “simulation”, the RSFM completely disregards the temporal evolution of the flood phenomenon and describes the behaviour of the previously defined system of accumulation zones through a sequence of “hydrostatic conditions” governed solely by the law of conservation of mass. Each “hydrostatic condition” represents a computational step in which the volume V (input to the model), initially discharged into the AZs from where the flood phenomenon originates, is divided among neighbouring AZs on the basis of exchange mechanisms whereby the accumulation zones are filled and the volumes exceeding their capacity spill over into adjacent accumulation zones. The filling/exchange process ends when the volume stored in each of the AZs involved does not exceed its corresponding storage capacity.

The steps the calculation algorithm of the RSFM is based on are described in detail below. For the sake of simplicity, a single flood trigger point is assumed (Gouldby et al. 2008):

• 1. The incoming volume V is totally ascribed to the AZ adjacent to the place where the flood is assumed to have been triggered (for example, the AZ considered may be adjacent to a zone where the river bank collapses): this AZ will be hereafter identified as AZ₁.

• 2. The communication points (CPs) between AZ₁ and the adjacent AZ are identified and the one having the lowest elevation (CP_1min) is selected. Based on the level-volume curves previously defined, the following are then calculated:

  • the volume V_AZ₁ stored in AZ₁, setting in the accumulation zone a level equal to CP_1min;

  • the level h₁ inside AZ₁, corresponding to the volume discharged into that zone;

• 3. The condition which determines the exchange of volumes of water between AZ₁ and the contiguous AZs is the following:

  • If h ₁ < CP_1min no exchange or spilling occurs and the procedure comes to an end;

  • If h ₁ > CP_1min it means that the volume discharged into AZ₁ exceeds its storage capacity and the excess part must thus be transferred to the neighbouring AZs.

    In the latter case, the excess volume V _exc is calculated as the difference between the volume V discharged into AZ₁ and the stored volume V_AZ₁.

• 4. The mechanism by which the exchange takes place is of a unidirectional type: the excess volume V _exc spills completely into the adjacent AZ (hereafter identified as AZ₂) which communicates with AZ₁ through CP_1min. If a number of lines of separation have a communication point with an elevation equal to CP_1min, the excess volume is divided into equal parts among the corresponding adjacent AZs.

• 5. Step 2 is repeated for AZ₂ and CP_2min, the corresponding stored volume V_AZ₂ and level h ₂ are identified, in that order. If the latter exceeds CP_2min, it is determined whether a merger has occurred between adjacent AZs, based on the following conditions:

  • if CP_2min = CP_1min, AZ₂ and AZ₁ are merged into a single AZ₁₊₂, for which the operations described in step 2 are repeated again.

  • If CP_2min ≠ CP_1min, no merger occurs.

• 6. Steps 3 and 4 are repeated.

• 7. Step 5 (and thus steps 3 and 4) is repeated for as long as the volume that spills into the last flooded AZ (whether a single one or resulting from a merger) exceeds the corresponding storage capacity.

• 8. The levels in the flooded AZs (single or resulting from a merger) are determined by exploiting the respective level-volume curves and the extent of the flooded area is defined accordingly.

The exchange mechanism described above is greatly simplified, as it is based solely on the topography of the ground and is unable to distribute the volumes of water among several contiguous zones. In order to overcome these limits and render the exchange mechanism more realistic, Lhomme et al. (2009) propose an alternative procedure (hereafter identified as RFSM_unif) which attempts to represent the effect that the shape of the different AZs and roughness of the ground may have on the final result.

The idea of involving the shape of the generic AZ in the water exchange mechanism was conceived with the aim of reproducing the dynamic effects tied to the velocity with which the AZ is filled. In this specific case, according to Lhomme et al. (2009), the smaller the AZ and thus the faster the filling takes place, the more likely it is that the water volumes will spill in multiple directions into adjacent AZs.

The rapidity of the filling phenomenon is indirectly estimated based on the shape of the AZ, imagined as a cone, and using the following formula (Lhomme et al. 2009):

$$ MSTol= Ktol\cdot \frac{q_2-{q}_1}{V_2} $$

(1)

where:

MSTol :

fictitious depth to be added to the “current” water level in the AZ

Ktol :

a constant whose value is set equal to 1,400

q ₂ :

elevation of the lowest communication point

q ₁ :

elevation of the accumulation point of the AZ

V ₂ :

volume accumulated in the AZ when the water level reaches q ₂ .

The volume being equal, the contribution of MSTol becomes increasingly significant as the width of the cone decreases (narrow AZ). It should be observed that since MSTol is added to the water level in the AZ, its effect, according to Lhomme et al. (2009), is to accelerate the discharge of water from the AZ considered.

Another physical phenomenon that is introduced into the exchange mechanism proposed by Lhomme et al. in 2009 in view of the relevance it takes on in the dynamics of flood processes is the friction tied to the roughness of the ground: the larger the flooded area, the more significant its effect on fluid motion (water movement). Obviously, friction has a “global” effect of slowing and hindering flood expansion.

In the RFSM_unif this slowing effect is represented by raising the communication points by a quantity equal to:

$$ {S}_f={C}_f\cdot {A}_{wet} $$

(2)

where:

S _f :

fictitious increase in the elevation of the communication points

C _f :

a constant whose value is set equal to 10⁻⁹

A _wet :

flooded area within the AZ considered following a generic calculation step

A graphic representation of the MSTol and S _f is given in Fig. 2 (see also Fig. 7 in Lhomme et al. 2009).

Fig. 2

Representation of the physical meaning of MSTol and S _f (after Lhomme et al. 2009)

From an operational standpoint, the procedure proposed by Lhomme et al. (2009) does not modify the structure of the calculation algorithm, but it does introduce a new criterion for establishing whether the water volumes spill towards adjacent AZs and redefines the manner in which these volumes are distributed.

In the case considered, in step 3 of the calculation algorithm the condition that determines the exchange of volumes of water between a fixed AZ and the neighbouring AZs becomes:

• If h ₁ + MSTol < CP_1min + S _f no exchange or spilling takes place and the procedure comes to an end;

• If h ₁ + MSTol > CP_1min + S _f it means that the input volume exceeds the storage capacity of AZ₁ and the excess part must spill over into the neighbouring AZs.

The mechanism of exchange of the excess volume is now represented using a multiple spilling approach and the operations included in step 4 are modified as follows: it is determined for which of the i-th CPs identified in step 2 the condition h ₁ + MSTol > CP_i + S _f is met and the excess volume is consequently divided into equal parts among the corresponding adjacent AZs.

2.2 The Method Proposed

Although the procedure proposed by Lhomme et al. (2009) introduces a multidirectional distribution of the volumes of water, dividing them equally among the receiving AZs represents a very rough schematization of the real physical process. Another aspect which limits the validity of the proposed modifications, reducing their generalizability, is the value of the two constants (Ktol and C _f ), defined on the basis of a trial-and-error procedure relying on an exiguous number of case studies. As noted earlier, these constants are connected to an attempt to better represent the effect that temporal dynamics have on the spatial extent of a flood.

In order to overcome these limits we propose a version called RFSM_w (w = weighted), which represents a further evolution of the approaches described by Gouldby et al. 2008 (RFSM) and Lhomme et al. 2009 (RFSM_unif).

In particular, two modifications have been introduced in the RFSM_w. With the first modification we maintain the criterion introduced by Gouldby et al. 2008 for determining the exchange of volumes of water between contiguous AZs (see step 3 of the RFSM calculation algorithm), thus eliminating the dependency on the constants Ktol and C _f . However, we introduce a new method for distributing the water volumes which maintains the multidirectional nature of the exchange/spilling but not the uniformity, in order to better represent a real situation, which involves a highly complex interaction between one accumulation zone and its neighbours. With the second modification we imitate the temporal dynamics of a flood by subdividing the volume V in cell AZ₁ (and thus in all subsequent cells) into r parts: in other words, the various AZs are filled in successive steps, that is, a little at a time. This second modification is in contrast with the one proposed by Lhomme et al. (2009), where the effect of the dynamics of the flood event is accounted for by considering the shape of the AZs and raising the communication points according to the previously described steps.

2.2.1 First Modification

Underlying this modification is the hypothesis that the connections between adjacent accumulation zones AZ can be represented through outflowing weir crests. In this manner, the volume exceeding the storage capacity of one zone can be proportionally distributed among adjacent accumulation zones based on weights derived from equations which regulate weir flow.

Weir flow is commonly described by means of an equation of the following type (Chow 1959):

$$ Q={C}_q\cdot \varOmega (h)\cdot \sqrt{2\cdot g\cdot h} $$

(3)

where:

Q :

outflowing discharge

C _q :

coefficient of discharge

Ω(h) :

wet area above the weir

h :

depth relative to the point of the crest having the lowest elevation

One assumption made here is to consider C _q constant, i.e. more complicated dependencies on hydraulic or geometric parameters of the problem are excluded: given the substantial simplification inherent in the RFSM_w., characterizing the coefficient C _q with the shape of the boundary line between adjacent accumulation zones seems completely pointless.

Given these basic elements, once we have focused on an AZ, the volume (V _exc ) exceeding its storage capacity (relative to the lowest crest) is divided among the receiving n AZs according to quantities V _i , derivable from the following system:

$$ \left\{\begin{array}{l}{V}_{exc}={\displaystyle \sum_{i=1}^n{V}_i}\hfill \\ {}{V}_i={Q}_iT=\left({C}_q\cdot {\varOmega}_i\left({h}_i\right)\cdot \sqrt{2\cdot g\cdot {h}_i}\right)\cdot T\kern0.5em \mathrm{con}\ i=1,\dots, n\hfill \end{array}\right. $$

(4)

where T represents the emptying time (a single value). The system (4) leads to the following solutions:

$$ {V}_i={p}_i\cdot {V}_{exc}\kern1.25em i=1,\dots, n\kern1em \mathrm{with}\kern1em {p}_i=\frac{\varOmega_i\left({h}_i\right)\sqrt{h_i}}{{\displaystyle \sum_{k=1}^n{\varOmega}_k\left({h}_k\right)\sqrt{h_k}}} $$

(5)

It is important to observe that V _exc is distributed among the n adjacent receiving AZs in proportion to weights p _i , which are a function of the hydraulic parameters involved (the receiving AZs are the ones for which h _i is positive: if h _i is negative it means that the water level in the AZ considered is lower than the communication point with its neighbour AZ_i and hence there is no spilling in that direction). In this case the weights (and thus the distributed volumes) are directly proportional to the head relative to the point of the crest having the lowest elevation and the area whose base corresponds to the line of separation between the two AZs considered (see Eq. 5).

2.2.2 Second Modification

The RFSM and RFSM_unif assume an instantaneous filling of the AZ inundated on each occasion. The instant flow of the entire volume into the accumulation zone results in a very high water level within the zone itself and thus a simultaneous triggering of flow over all weir crests.

This phenomenon does not reflect the actual physics of the problem, since in reality the inundated accumulation zone fills gradually, with a velocity that is a function of the inflow hydrogram and ground morphology. Consequently, the flow over the various weir crests occurs at successive points in time. The velocity with which the accumulation zone fills will thus play a decisive role in determining the mechanisms of exchange of incoming water volumes, since it influences the number of weir crests water will flow over, which in turn determine how the volumes are distributed among adjacent accumulation zones.

In order to simulate the effect of these dynamics, the model provides for a gradual flow of the volume entering the AZ inundated on each occasion. The gradualness is simulated by dividing the total volume V into r equal parts, which are introduced into the accumulation zone in an equal number of the steps. More precisely, a spreading simulation is carried out for every r-th fraction of the volume V: the simulations following the first obviously assume an area that is already partly flooded, so that the triggering of the various weir crests occurs at increasingly higher levels and further directions of spreading are identified.

3 Case Studies

3.1 Area of Investigation and Description of Case Studies

The RFSM, RFSM_unif and RFSM_w were applied to a flat area of land situated in the municipality of Argenta in the province of Ferrara. The altimetry of the study area, which extends over 95 km², is described by means of a 30 × 30 m DEM (105,703 square cells). The calculation domain is made up of 925 accumulation zones (AZs) (see Fig. 3), automatically generated by the ArcHydro extension of ArcGIS™ software, together with the information necessary for subsequent hydraulic simulations.

Fig. 3

Calculation domain: division of the study area into accumulation zones (AZ) and identification of the AZs where the flooding starts: AZ 918 – test case 1; AZ 673 – test case 2; AZ 918 + 673 + 933 – test case 3

This information, organized in tabular form, comprises:

• 1. for each accumulation zone: identification code, minimum elevation (accumulation point), level-volume curve, boundary lines and identification code of the adjacent accumulation zones;

• 2. for each boundary line: identification code, minimum elevation (communication point), level-area curve (useful for quantifying the law Ω(h) – see Eq. 5) and identification code of the adjacent accumulation zones separated by it.

With reference to the calculation domain generated, three synthetic test cases were constructed, two of which characterized by a single flood trigger point and one characterized by multiple (three) trigger points (see Fig. 3). The points of discharge of the water volumes into the area of interest were chosen in such a way as to initially flood the accumulation zones which lie adjacent to the river bank line and vary in size (918: large; 673: medium; see Fig. 3 and Table 1). More precisely, Fig. 3 shows, in particular, that the large AZ (code 918) is surrounded by other medium-large AZs, all of which are at more or less at the same elevation as the most depressed zone, whereas the medium-sized zone (code 673) is surrounded by medium-small AZs which slope downward toward the most depressed zone of the study area. The combination between the inlet AZ and the surrounding ones (within a range of varying extent), in addition to the general altimetry of the flood-affected zone, influences the behaviour of the various rapid models analyzed here as regards the characterization of the direction or directions of spreading, as will be evident from the results shown further below.

Table 1 Case studies: areas of the initially flooded accumulation zones and inflow hydrograms. For the first two test cases the duration of the hydrogram was assumed to be 10,000 s, whereas for the third test case the duration was assumed to be 21,500 s

The flooded areas and depths used as a basis for comparing and evaluating the different rapid models were produced using the two-dimensional hydraulic model FLO-2D™ (O’Brien 2007), assuming, as the initial condition, a constant hydrogram (duration 10,000 s for the first two test cases and 21,500 s for the third test case: see Table 1) and a Manning coefficient of 0.04 m^−1/3 s for each cell of the calculation DEM.

The three rapid models were applied to the three case studies. The second modification proposed in the RFSM_W, previously described (volume V subdivided into r parts), was also applied to the RFSM_unif assuming for both models r = 1, 12, 60. This variant was not considered for the RFSM, since the latter is characterized by a purely unidirectional spilling into the adjacent AZ and is therefore not sensitive to the subdivision of the volume V into r fractions.

3.2 Performance Indicators

Four performance indicators were introduced for the purpose of comparing the RFSM, RFSM_unif and RFSM_w and to provide an estimate of the quality of the results produced by them (extent of the flooded area and relative depth). These indicators take into account areas and water depths, which are determined making reference to the cells of the DEM from which the AZs were derived, using appropriate procedures. This means that the wet areas are the set of the (wet) cells of the DEM present in the AZs where there is water, whereas the water depths are given by the difference between the water level (always equipotential) in the generic AZ (simple or compound) and the elevations of the (wet) cells.

These indicators are:

• 1. a fit indicator, defined as (Lhomme et al. 2009):

  $$ Fit=\frac{B}{B+C+D}\cdot 100 $$

  (6)

  where:

  B :

  wet area when using both rapid and FLO-2D models

  C :

  wet area when using rapid model but dry area when using FLO-2D model

  D :

  wet area when using FLO-2D model but dry area when using rapid model

  This indicator expresses the relation, in percentage form, between the intersection of the wet areas obtained with the two models (rapid model and FLO-2D) and their combination or union. The closer the value of the Fit indicator is to 100 %, the better the match between the reference wet area (D) and the simulated wet area (C) and, consequently, the better the estimate obtained with the rapid model.

• 2. Bias indicator, defined as (Lhomme et al. 2009):

  $$ Bias=\left(\frac{B+C}{B+D}-1\right)\cdot 100 $$

  (7)

  where: B, C and D take on the previously described meaning.

  This indicator quantifies the relative percentage error with respect to the final extent of the flooded area. Positive values of the Bias indicator indicate an overestimation of the extent compared to the expected value, whereas negative values indicate an underestimation. The closer the value of the Bias indicator is to zero, the smaller the error and, consequently, the better the estimate obtained with the rapid model.

• 3. mean (m) of the deviations (errors) between the depths obtained with the rapid model and the FLO-2D model, calculated with reference to the flooded DEM cells in both models:

  $$ m=\frac{1}{n_{cell}}{\displaystyle \sum_{i=1}^{n_{cell}}\left({y}_{sim,i}-{y}_{FLO2D,i}\right)} $$

  (8)

  where:

  y _sim,i :

  depth associated with the i-th cell wet in the rapid model

  y _FLO2D,i :

  depth associated with the i-th cell wet in the FLO-2D model

  n _cell :

  total number of cells flooded simultaneously in both models.

  Positive values of m indicate that, on average, the simulated depths overestimate the expected ones, whereas negative values indicate, on average, an underestimation.

• 4. root-mean-square error (RMSE) between the depths obtained with the rapid model and FLO-2D model, calculated with reference to the flooded cells in both models:

  $$ RMSE=\sqrt{\frac{{\displaystyle \sum_{i=1}^{n_{cell}}{\left({y}_{sim,i}-{y}_{FLO2D,i}\right)}^2}}{n_{cell}}} $$

  (9)

where: y _sim,i , y _FLO2D,i and n _cell take on the previously specified meaning.

This indicator enables us to evaluate, on average, the modulus of the error committed by the rapid model in estimating the depth. The closer the RMSE is to zero, the better the estimate provided by the rapid model.

The performance indicators were calculated considering only the flooded cells characterized by depths greater than or equal to 10 cm. This choice is based on the intrinsic aim of rapid models, developed to provide a fast, efficient tool to support the creation of flood risk management plans. In such a context, depths of less than 10 cm appear to be of little significance from an engineering viewpoint and thus need not be taken into consideration.

3.3 Application to the Test Cases

3.3.1 Test Case 1 (Code of Inlet AZ: 918)

Figure 4 compares the flooded areas produced by the rapid models RFSM (Fig. 4–a), RFSM_unif (Fig. 4–b,c,d) and RFSM_w (Fig. 4–e,f,g) with those produced by the two-dimensional hydraulic model FLO-2D (reference model). With regard to the RFSM_unif and RFSM_w, which envisage a multidirectional water exchange mechanism, 3 different flood maps are presented; these were obtained considering no subdivision of the volume V, i.e. r = 1 (Fig. 4–b for the RFSM_unif and Fig. 4–e for the RFSM_w), a division into 12 equal parts (r = 12, Fig. 4–c for the RFSM_unif and Fig. 4–f for the RFSM_w) and a division into 60 equal parts (r = 60, Fig. 4–d for the RFSM_unif and Fig. 4–g for the RFSM_w).

Fig. 4

Test case 1 – initial accumulation zone AZ 918. Comparison between the extent of the flooded areas with the FLO-2D and each of the following rapid models: RFSM, a); RFSM_unif, r = 1, b); RFSM_unif, r = 12, c); RFSM_unif, r = 60 d); RFSM_w r = 1, e); RFSM_w r = 12, f); RFSM_w, r = 60, g)

Table 2 presents the results of the simulations carried out in terms of extent of the flooded area, mean and mean square error of the depths.

Table 2 Test case 1: extent of the flooded areas, mean and mean square error of the depths obtained with the model FLO-2D and the rapid models RFSM, RFSM_unif and RFSM_w

An analysis of the maps shown in Fig. 4 and the values of the performance indicators presented in Table 3 enables us to affirm that the RFSM_unif and RFSM_w perform distinctly better when the volume V is subdivided into a large number of parts (r = 60), both in terms of representing the extent of the flooded areas and in terms of the depths associated with them. In the case considered here, dividing the incoming volume V into many fractions enables us to better reproduce what occurs in the FLO-2D model, where the flow into AZ₁ (initial accumulation zone) and - in cascade fashion - into the neighbouring and subsequent (medium-large) ones causes the level in these AZs to grow slowly. The fact that the RFSM_w works better than the RFSM_unif can be explained by observing that in this situation, where the levels in the context of the FLO-2D model grow more or less slowly, the outflow across the lowest crest becomes preponderant compared to that across the other crests, hence the importance of dividing the volume V _exc in a non-uniform manner among the various accumulation zones adjacent to the one considered each time.

Table 3 Test case 1: Performance indicators associated with the rapid models RFSM, RFSM_unif and RFSM_w

The latter observation also explains why the RFSM works well in this case: in fact, in the RFSM the spilling is only unidirectional and guided by the communication point at the lowest elevation; therefore, in this specific case the water mass moves largely in the right “direction”.

3.3.2 Test Case 2 (Code of Inlet AZ: 673)

Figure 5 shows the seven flood maps produced by the rapid models (RFSM, Fig. 5–a, RFSM_unif, Fig. 5–b, r = 1, Fig. 5–c, r = 12, Fig. 5–d, r = 60, RFSM_w, Fig. 5–e, r = 1 Fig. 5–f, r = 12 Fig. 5–g, r = 60) together with those produced by the FLO-2D model (reference model). Table 4 presents the results of the simulations performed in terms of extent of the flooded area, mean and mean square error of the depths.

Fig. 5

Test case 2 – initial accumulation zone AZ 673. Comparison between the extent of the flooded areas with the FLO-2D model and each of the following rapid models: model RFSM, a); RFSM_unif, r = 1, b); RFSM_unif, r = 12, c); RFSM_unif, r = 60 d); RFSM_w r = 1, e); RFSM_w, r = 12, f); RFSM_w, r = 60, g)

Table 4 Test case 2: extent of the flooded areas, mean of the depths and mean square error of the depths obtained with the FLO-2D model and with the rapid models RFSM, RFSM_unif and RFSM_w

Unlike in the previous test case, here the effect tied to the distribution of the water volumes among a number of adjacent AZs is very evident. The values of the performance indicators calculated for the models which envisage a multidirectional water exchange mechanism (RFSM_unif and RFSM_w) are distinctly better than those obtained with the unidirectional model (RFSM), irrespective of whether the incoming volume V is divided into fractions or not (see Table 5). Practically speaking, when the inlet AZ and neighbouring AZs lying within a given range are of modest size relative to the incoming volume V and under real conditions (here represented by the FLO-2D model) fill more rapidly, a unidirectional distribution is conceptually inadequate, since spilling towards the contiguous AZs occurs in all directions. The fact that the RFSM_w performs better than the RFSM_unif demonstrates that a multiple spilling approach guided by weights that take into account the different elevations of the cells involved in the exchange is more realistic than a uniform one.

Table 5 Test case 2: Performance indicators associated with the rapid models RFSM, RFSM_unif and RFSM_w

Briefly summarizing, these two test cases show that if the combination of cell size, elevation of the communication points and general altimetry of the study area leads to the identification of a territory in which there is a prevalent direction of spreading (guided by the lowest communication points), the RFSM can compete with the other two versions RFSM_unif and RFSM_w. If, on the other hand, the combination of the above-mentioned parameters does not lead to the identification of a territory with a prevalent direction of spreading, the RFSM will inevitably give inferior results compared to the other two rapid models and the concept of multiple spilling becomes dominant. If we then compare the RFSM_unif and RFSM_w, the results shown here demonstrate that the latter performs systematically better, thus supporting the idea that the multiple spilling approach based on weir flow behaviour is more realistic than a uniform distribution. Moreover, dividing the volume V into a number of parts enables a better representation of the sequence in which the different crests of the accumulation zones are triggered (or activated).

3.3.3 Test Case 3 (Flood with Multiple Trigger Points)

In the last test case we assume a multiple triggering of the flood from the 2 accumulation zones AZs considered individually in test cases 1 and 2, plus a third (AZ code - 933) (see Fig. 3): in this case, therefore, all the conditions considered separately in test cases 1 and 2 will be addressed together. Figure 6 shows the seven flood maps produced by the rapid models (RFSM, Fig. 6–a), RFSM_unif (Fig. 6–b, r = 1; Fig. 6–c, r = 12; Fig. 6–d, r = 60) and RFSM_w (Fig. 6–e, r = 1; Fig. 6–f, r = 12; Fig. 6–g, r = 60) and the FLO-2D model (reference model). Table 6 presents the results of the simulations carried out in terms of extent of the flooded area, mean and mean square error of the depths.

Fig. 6

Test case 3 – initial accumulation zones AZ 918 + 673 + 933. Comparison between extent of the flooded areas with the FLO-2D model and each of the following rapid models: RFSM, a); RFSM_unif, r = 1, b); RFSM_unif, r = 12, c); RFSM_unif, r = 60 d); RFSM_w r = 1, e); RFSM_w, r = 12, f); RFSM_w, r = 60, g)

Table 6 Test case 3: extent of the flooded areas, mean of the depths and mean square error of the depths obtained with FLO-2D the model and the rapid models RFSM, RFSM_unif and RFSM_w

The rapid model which best matches the expected result (produced by the FLO-2D model) is the RFSM_w, with r = 60 (see Table 7). This means that the combination of “non-uniform distribution” of volumes based on considerations of a hydraulic character (weir-type exchange along the lines of contact) and “subdivision of the incoming volume V” is the approach which, among the rapid models, best represents the complexity of the phenomenon considered.

Table 7 Test case 3: Performance indicators associated with the rapid models RFSM, RFSM_unif and RFSM_w

Although the unidirectional model RFSM provides a good result in terms of Fit and RMSE (see Table 7), it should be observed that the corresponding Bias and m indicators respectively show a non-negligible underestimation of the flooded area (−12.1 %) and a consequent overestimation of the depth (0.05 m). This is due to the fact that the RFSM, given its structure, does not envisage the possibility of distributing the excess water volumes in different directions and this results in the flooding of smaller areas, generally located in the most depressed zones of the calculation domain, with depths that are greater on average than the expected ones (see Table 6).

It should also be stressed that when the dynamics of the simulated phenomenon are complicated, the intrinsic limits tied to the schematization of the rapid models emerge to a greater degree. Therefore, their performances in this test case, though good, are in general inferior to the performances obtained in the previous two cases.

Comparing the results obtained in the three test cases enables us to draw some general conclusions:

• 1. The multidirectional rapid models (RFSM_unif and RFSM_w) give better results if associated with a division of the volume V into a large number of fractions (high r). The explanation for this fact lies in the dynamics associated with the real mechanism by which the accumulation zones are filled and the consequent gradual triggering over time of outflow over the various weir crests.

• 2. The RFSM_w systematically performs better than the RFSM_unif as shown by the indicators. This is due to the fact that a distribution of excess volumes based on hydraulic considerations (weir equation, i.e. weir-type exchange) is more realistic than a simple uniform distribution. To this we may add the fact that the RFSM_unif makes use of two “crest” values (see Eqs. (1) and (2)), which are in turn a function of the two coefficients Ktol and C _f , estimated by Lhomme et al. (2009) with reference to a limited number of cases and hence of reduced general validity. What is more, estimating these two coefficients time after time on the site considered would eliminate all of the advantages of being able to use a rapid model, since the calibration would require a systematic use of a complete two-dimensional hydraulic model with extremely long computational times.

• 3. The performance of the RFSM is comparable to that of the other two rapid models when the flood water clearly spreads in a dominant direction in the area considered. In this case the unidirectional technique is acceptable, since it predicts the movement of the mass of water towards more depressed zones with sufficient accuracy. In other cases, disregarding spread in different directions becomes an excessively rough approximation and the results deviate from what was expected.

3.4 Computational Times and Sensitivity Analysis

The above-described rapid models were implemented in the MATLAB™ environment and the simulations for the three test cases were conducted on an Intel Pentium Dual Processor E2160 (1.80GHz) CP with 2.00 GB of RAM. The reference simulations, obtained using the FLO-2D two-dimensional hydraulic model, were run on an Intel Xeon Processor E5345 (2.33GHz) CP with 2.47 GB of RAM.

The computational times associated with the rapid models RFSM, RFSM_unif and RFSM_w and the FLO-2D hydraulic model are compared in Table 8.

Table 8 Computational times, in seconds, for the rapid models RFSM, RFSM_unif and RFSM_w and the two-dimensional reference hydraulic model FLO-2D

The RFSM is the most efficient of all in this respect, since the computational times for the three test cases do not exceed 15 s. The RFSM_w gives a slightly inferior performance, with average simulation times close to 35 s. Among the rapid models the RFSM_unif has the longest computational times, around 60 s on average. Clearly, the quantification of the “crests” in Eqs. (1) and (2) and their use in the process of spreading the water volume V has an effect of slowing down the calculations, which thus take longer than those produced in the RFSM_w.

The decline in computational efficiency observed for the RFSM_{unif and} RFSM_w compared to the RFSM is tied to the greater complexity of the calculation algorithm, which envisages a multidirectional water exchange mechanism and thus entails the execution of a larger number of instructions and cycles. It should be observed, however, that despite this decline, the computational performance of the rapid multidirectional models, the RFSM_w in particular, remains high.

A comparison between the computational times of the rapid models and those of the FLO-2D two-dimensional hydraulic model clearly justify the use of the former.

All previous numerical examples and computational time evaluations were performed referring to a DEM with a grid size of 30 m while simulations with FLO-2D were based on a Manning coefficient of 0.040 m^−1/3 s. Two questions are now considered. The first question is: is the good performance of the rapid models (and in particular of the RFSM_w) affected by the Manning coefficients used in the FLO-2D simulations? More precisely, would we have observed the same level of performance if a larger Manning coefficient was used for the FLO-2D simulations? The second questions: what is the effect of the grid size on the evaluation of the performance of the rapid models?

To reply to the first question a simulation with FLO-2D was performed setting the Manning coefficient equal to 0.080 m^−1/3 s and considering the accumulation zone AZ 673 as that from where the inundation starts (see test case 2). Making reference to the inundation map obtained at the end of the simulation, i.e. when the water is still everywhere, a comparison was made with that obtained when Manning coefficient was set equal to 0.040 m^−1/3 s and the 4 indicators defined with Eqs. (6), (7), (8) and (9) were calculated. Incidentally, the simulation now considered lasted 260 h, i.e. a time of the same order of magnitude as that requested by the simulation with n = 0.040 m^−1/3 s (about 350 h). The qualitative comparison of the two maps (not reported here) shows that the two inundated areas are almost coincident and the four indicators are then equal to Fit≅96.00 %; Bias≅0 %; m≅0 m; RMSE≅0.05 m. This result allows us to conclude that the performance of the rapid models observed in the previous section is not affected by the roughness coefficient used to perform the hydraulic simulations. This is not surprising: indeed the roughness coefficient can affect the spatial-temporal evolution of the inundation but not the final situation which is static since water is still everywhere.

To reply to the second question, a further simulation with FLO-2D was performed using the same Manning coefficient used in all the test cases described in the previous section but adopting a grid size of 100 m. The accumulation zone from which the inundation starts is still that used in test case 2, i.e. AZ 673. The same grid size was used when the RFSM_w with r = 60 was applied. For this case as well the four indicators defined by Eqs. (6), (7), (8) and (9) were calculated and reported in Table 9.

Table 9 Comparison between FLO-2D and RFSMw (r = 60) when a DEM with grid size of 100 m is used. The inundation starts from the accumulation zone AZ 673 and the Manning coefficient used is 0.040 m^−1/3 s

The analysis of the values reported in Table 9 shows that the indicators based on the water depths are almost the same as those reported in the last line of Table 5 while the other two indicators (Fit and Bias) are clearly worse. This entails two considerations: (a) the use of a coarse DEM produces results which are more imprecise since territory, especially in such a flat area as that considered herein, is poorly characterised; (b) a poor description of a very flat territory makes the concept of Accumulation Zones less significant. As a consequence, the performance of the RFSM_w (r = 60) deteriorates with respect to the case with grid size 30 m. At the same time it may be worth noting that the computational time for FLO-2D was of 3.8 h while for RFSM_w was of 10 s. In other words, the computational time for FLO-2D was certainly much less than the about 360 h requested for the simulation when the grid size was 30 m and Manning coefficient equal to 0.040 m^− 1/3s but still much higher than the few seconds requested by the RFSM_w we propose. We can thus observe that, while the proposed rapid model can work in few seconds even using a detailed map (as shown in Table 8) (hence no reason for using a coarse DEM), a significant reduction in the computational time can be obtained, with reference the FLO-2D model, by using a coarse grid at the cost of a poor characterization of the territory: however, even in this case, the computational times remain of the order of magnitude of many hours.

4 Conclusions

This paper presents an evolution of the RFSM (Rapid Flood Spreading Model) proposed by Gouldby et al. (2008) and revised by Lhomme et al. (2009). Two modifications are proposed: (1) multidirectional exchanges among (or multiple spilling towards) accumulation zones compatible with a weir-type behaviour at the lines of communication, (2) a subdivision of the inflow volume in such a way as to simulate the temporal sequence of triggering at different communication points. The first modification is in contrast with the one previously proposed by Lhomme et al. (2009), who had indeed introduced the idea of multiple spilling (absent from the original version of the RFSM), but assumed a division into equal parts among the various accumulation zones involved. The second modification aims to render the representation of the phenomenon of a mass of water spreading in the area of interest a little less “static” without however introducing any “timing” into the flood’s expansion.

The rapid model developed - the RFSM_w - was compared with the previous formulations (RFSM, Gouldby et al. (2008) and RFSM_unif, Lhomme et al. 2009) in an application to three test cases, in which the basis for comparison was the extent of the flooded areas and the depths obtained with the FLO-2D two-dimensional hydraulic model.

The results showed that when the proposed model RFSM_w is applied together with a division of the incoming volume into a sufficiently large number of fractions, it systematically outperforms the other rapid models at the cost of a slight increase in computational times compared to the original version of the RFSM, though it is in any case more efficient than the RFSM_unif. In particular, the rapid model RFSM_w is more reliable than the RFSM whenever the direction of spreading is not well defined (as in the case of very flat areas) and is also more reliable than the RFSM_unif when the communication points between neighbouring AZs are at different elevations. Dividing the volume V into multiple parts (a step included in both the RFSM_unif and the RFSM_w) improves the rapid model’s performance and the larger the number of fractions, the better the performance.

A sensitivity analysis also showed that all the results presented in this paper are not affected by the roughness coefficient used for the hydraulic simulation performed, in our case, with the FLO-2D software. Furthermore it was observed that a coarse DEM is unsuitable for applying the proposed rapid model: in fact its computational time remains of the order of magnitude of a few seconds as in the case of fine DEM, but its precision gets worse, especially in very flat areas.

Overall, the RFSM_w has the necessary features to be used as a support for hydraulic risk assessment and management procedures which require a reliable simulation of numerous flood scenarios.