# Probabilistic Flood Hazard Assessment for Multiple Flood and Levee Breaching Scenarios: A Case Study of Etobicoke Creek, Canada

Florence Mainguenaud, Usman T Khan, Laurent Peyras, Claudio Carvajal, Bruno Beullac, Jitendra Sharma

<figure><img src="image_1.png" /></figure>

## To cite this version:

Florence Mainguenaud, Usman T Khan, Laurent Peyras, Claudio Carvajal, Bruno Beullac, et al. Probabilistic Flood Hazard Assessment for Multiple Flood and Levee Breaching Scenarios: A Case Study of Etobicoke Creek, Canada. Journal of Flood Risk Management, 2025, 18 (1), pp.e70002. 10.1111/jfr3.70002. hal-05228180

HAL Id: hal-05228180

https://hal.inrae.fr/hal-05228180v1

Submitted on 28 Aug 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_3.png" /></figure>

ORIGINAL ARTICLE

OPEN ACCESS

# Probabilistic Flood Hazard Assessment for Multiple Flood and Levee Breaching Scenarios: A Case Study of Etobicoke Creek, Canada

Florence Mainguenaud¹,² ID | Usman T. Khan² | Laurent Peyras¹ | Claudio Carvajal¹ ID | Bruno Beullac¹ | Jitendra Sharma²

¹INRAE, RECOVER, Aix Marseille Université, Aix-en-Provence, France | ²Lassonde School of Engineering, York University, Toronto, Ontario, Canada

**Correspondence:** Claudio Carvajal (claudio.carvajal@inrae.fr)

**Received:** 28 December 2023 | **Revised:** 11 September 2024 | **Accepted:** 5 November 2024

**Funding:** Funding was provided by INRAE, York University, and Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant (DG).

**Keywords:** breach failure | flood risk analysis | hydraulic modelling | integrated flood risk management

## ABSTRACT

Flood hazard assessment is crucial to mitigate the risks associated with flooding. Integrating levee failure scenarios into these assessments should improve the evaluation of flood risks and enhance the resilience of communities and infrastructure. This research presents a probabilistic flood hazard approach to assess levee failure and its impact on flood hazard. Our method includes a comprehensive assessment of backward erosion and overflowing failure mechanisms, integrated within a 1D/2D hydraulic model that simulates flood propagation and levee breaching. We calculate the cumulative probability of flood depth and velocity considering various scenarios, taking into account levee failure breaching for various failure mechanisms and several flood intensities. We apply the method to a residential area along Etobicoke Creek in Ontario, Canada. The results highlight which levee segment has the most impact on flood hazard, emphasizing the importance of incorporating levee failure scenarios in flood hazard assessments. The cumulative probability curve provides a more holistic result in locating the most hazardous areas rather than considering one return period or one failure mechanism. It can be expended to every location of the protected area, allowing for the creation of a probabilistic map for a desired probability.

## 1 | Introduction

Flood hazard assessment is the first step in flood risk analysis (Oliver et al. 2018; Rincón et al. 2022) and includes modelling the extent of flood propagation, as described in the review of flood conceptual models by Teng et al. (2017). We focus on fluvial flood assessments, which require a hydraulic model of the river system and existing or projected hydraulic structures. Hydraulic structures (e.g., levees, dams, channels and detention basins) impact the water flow, hence modifying the hydraulic structures within a watershed often goes hand in hand with the recalibration of the hydraulic model. Building flood defense structures is a typical response to protect populations and properties against

rising water levels, making fluvial levees a common feature in urban areas (Di Baldassarre et al. 2013). Building levees can also trigger the levee effect: the construction of a levee can increase risk instead of reducing it. Better protection offered by levees may lead to lower flood frequency, thus increasing confidence among property owners, who may raise the value of their property and, by so doing, raise the exposure, increasing the overall risk (Tobin 1995; Ferdous et al. 2019).

By simulating a flood, hydraulic models can create a flood hazard map based on either data from a historical event or a specific return period scenario (Winsemius et al. 2013). Usually, several flood events are considered, resulting in

<footnote>This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.</footnote>

<footnote>© 2025 The Author(s). Journal of Flood Risk Management published by Chartered Institution of Water and Environmental Management and John Wiley & Sons Ltd.</footnote>
multiple flood extents, each depending on the intensity and duration of the modelled flood event and on hydraulic structure breaching. The flood extent is an essential component of decision-making (e.g., emergency plans, flood mitigation strategies, urban resilience improvements). Therefore, the choice of which flood events and breaching scenario to model must be carefully made to remain understandable for city officials or other decision-makers.

In this context, Maranzoni, D'Oria, and Rizzo (2023) provide a comprehensive review of quantitative flood hazard assessment methods, categorizing them between heuristic, conceptual or empirical approaches. They highlight that flood hazard indicators include the type of flood, depth, velocity, duration, arrival time, return period, flow discharge, sediment transportation, breaching scenarios, flowing debris and people behavior (e.g., evacuation time, traffic modelling, building occupancy). For each flood assessment exercise, we must choose which of these flood indicators to focus on, depending on the objective of the study. Merz, Thieken, and Gocht (2007) identify water depth and velocity as the most influential flood indicators; hence, they are the most commonly used in the literature and thus are the ones we will consider in this study.

Flood assessment maps can be either deterministic or probabilistic, as presented by D'Oria, Maranzoni, and Mazzoleni (2019). Deterministic flood mapping is calibrated using historical flood events and simulates flood propagation of various magnitudes (Di Baldassarre et al. 2010; Mazzorana et al. 2011). The uncertainties associated with deterministic flood assessments are due to model inputs and parameters and can be assessed using sensitivity analysis (Hall et al. 2005). Ferrari et al. (2020) is an interesting example of deterministic flood mapping and presents a methodology to numerically simulate levee-breach-induced flooding in areas protected by fluvial levees. Di Baldassarre et al. (2010) compare deterministic and probabilistic flood mapping approaches, recommending probabilistic approaches as a deterministic approach cannot offer a range of possible flood events and provide an incomplete understanding of the flood hazard. On the other hand, probabilistic flood mapping integrates flood hazard assessment uncertainties and various scenarios, as shown by Maranzoni, D'Oria, and Rizzo (2023). In the literature, Vorogushyn et al. (2010) introduce the Inundation Hazard Assessment Model (IHAM), which considers overtopping, piping and microinstability levee breaches within a probabilistic framework. D'Oria, Maranzoni, and Mazzoleni (2019) consider backward erosion fragility functions to characterize levee breaches combined with a 1D-2D hydraulic model but do not consider multiple breach locations. Maranzoni, D'Oria, and Mazzoleni (2022) do consider multiple breaches occurring throughout a flood event but for one type of failure mechanism. The main advantage of a probabilistic approach is its suitability to assess multiple breaching scenarios. Following these approaches, we propose a method that can account for multiple flood events and multiple breaching scenarios and combine them together in a probabilistic way to generate easy-to-use maps.

An important gap in the literature is the integration of (i) flood propagation models of rivers, (ii) different failure mechanisms of levee breaching, (iii) flood propagation resulting from the breach and (iv) probabilistic spatial flood hazard assessment

obtained by aggregating different scenarios. The following paragraphs summarize each of these four components.

River flood propagation models include 1D, 2D, 3D and 1D-2D models. In a 1D model, calculations are made for each section and are suited for river channels and floodplains modelled with storage areas. However, it needs significant postprocessing to generate flood maps. Beilicci and Beilicci (2012) compare the Muskingum method and DUFLOW 1D models, and Meire et al. (2010) present the interaction of the model STRIVE (STReam RIVer Ecosystem) with modelled storage areas. 2D models solve flow equations across a predefined mesh on ground elevation from a Digital Terrain Model (DTM) and are used for modelling rainfall and complex floodplains. However, the run time can be long, and the in-channel hydraulics lack precision. Li et al. (2013) propose a 2D model based on the finite proximate method. Lavoie and Mahdi (2017) compare 2D models SRH-2D and Hydro_AS-2D. Morales-Hernández et al. (2013) propose a conservative 1D-2D coupled model for solving the shallow water equations (SWE). Mitsopoulos et al. (2022) recommend a 1D-2D coupled model with SWE for an accurate prediction of water depths and flow velocities. Therefore, we use a 1D model for the river and a 2D model for the levee-protected area. The flood propagation integrates levee breaching, and the levee reliability should be assessed accordingly.

Levee reliability can be estimated for different failure mechanisms, such as internal erosion, overtopping or overflowing and slope instability. CIRIA, Ministry of Ecology, and USACE (2013) present earthen levees' main internal erosion mechanisms: concentrated leak erosion, backward erosion, suffusion and contact erosion. Concentrated leak erosion in levees occurs mainly at the interface between levees through its structure, and it may be triggered by animal burrows connecting the waterside and land side, as presented by Ceccato, Malvestio, and Simonini (2022). Backward erosion failure estimation consists in using a fragility function that compares a hydraulic gradient to a critical gradient value using Terzaghi (1943) or Sellmeijer et al. (2011) criteria, for example. For overtopping or overflowing, Aguilar-Lopez et al. (2018) propose a probabilistic wave overtopping model to quantify erosion failure by wave overtopping. Vuik, Van Balen, and van Vuren (2017) estimate overflow and wave overtopping levee failure probabilities for levees and sand dunes with the first-order reliability method (FORM) and Monte Carlo simulations. Mainguenaud et al. (2023) estimate overflow-induced failure by using expert judgement, as described by Hathout et al. (2020) and Rongen, Morales-Nápoles, and Kok (2022). Concerning slope instability, Mainguenaud et al. (2023) estimate slope instability, taking into account backward erosion and overflowing failure probabilities for a levee. For flood hazard assessments, among potential levee failure mechanisms, backward erosion is the mechanism the most taken into consideration to assess levee reliability (e.g., Mazzoleni et al. 2014; D'Oria, Maranzoni, and Mazzoleni 2019). A definite gap in the literature is the lack of inclusion of several failure mechanisms to assess levee reliability. The evaluation of several levee failure mechanisms would lead to more accurate predictions of flood hazard and flood extents, and thus, more effective flood mitigation strategies.

Flood propagation software including breaching models (e.g., HEC-RAS, TELEMAC) and their application to levees are
already present in the literature. Mazzoleni et al. (2014) model a flood hazard with levee breaching due to backward erosion in the Po River in Italy, where they study the impact of breach location, size and the presence of obstacles on flood hazard. There are two important choices to make in a flood model: how to model levee breaching and which propagation equation to use. Firstly, levee breaching equations (e.g., weir flow equation, Manning's equation) are critical components in integrated flood models, capturing the complex processes involved in the failure of levees. They are used in combination to breach parameter predictive equations (e.g., Froehlich equations, Von Thun and Gillette equations, Wahl's method). Rifai et al. (2018) warn about the limitations of fluvial levee breach models that are transposed from dam studies, rely on empirical formulas and are tested for dam breach cases like Mohamed et al. (2002). Secondly, Yilmaz et al. (2023) compare SWE and diffusion wave equations (DWE) for assessing flood hazard with dam breach- ing and conclude that depth, velocity and arrival time should be determined using SWE, while flood extent can be accu- rately determined by DWE. Similarly, Costabile, Costanzo, and Macchione (2017) present the limitations of the DWE for 2D flood propagation and conclude that diffusive-type flood mod- els should be avoided when simulating urban areas due to the poor results around buildings. SWE models have limitations, such as not considering groundwater (Doble et al. 2012); it is nonetheless the preferred model for flood hazard assessment considering levee breaching (Mazzoleni et al. 2014; Maranzoni, D'Oria, and Rizzo 2023).

Spatialized flood hazard assessment consists of assessing the flood indicators in the areas impacted by the flood. The European Environment Agency (2007) prompted the creation of large-scale flood maps and one way to do so is to juxtapose a mosaic of local flood maps. Scaling up comes with its own challenges; however, a good example is presented by Dottori

et al. (2016), who struggle with managing river channel con- fluences and accounting for the impacts of climate change on flood hazard. Tufano et al. (2023) challenged combining flood hazard maps of different flood events by overlaying them, mak- ing visually appealing and comprehensible flood hazard maps destined for stakeholders, city officials, or insurance companies. However, another limitation is the quality of individual flood hazard maps and which flood indicator or scenario they cover (Maranzoni, D'Oria, and Rizzo 2023).

This article aims to provide a probabilistic assessment of flood hazards in areas protected by levees that consider different levee breaching scenarios. The study uses a 1D-2D hydraulic model that outputs flood depth and velocity. We choose HEC-RAS for its ability to simulate 1D-2D hydrodynamic systems and its in- tegrated levee breaching functionality and implement the SWE to propagate flooding in the levee-protected area. Concerning the failure modes of the levees, we consider backward erosion and overflowing failures, along with a “no failure” scenario for comparison. We apply our method to the Etobicoke Creek levee (located near Brampton City in the Ontario region, Canada, as shown in Figure 1), which is well-documented thanks to data provided by local authorities.

Finally, the objective of this study is to provide a probabilis- tic flood assessment method that incorporates levee failure for several return periods. To this end, we estimate levee failure probabilities for overflowing and backward erosion failure mech- anisms using fragility curves. Then, we model flood propagation and simulate extensive flood events and breaching scenarios. For each simulation, we obtain two maps: one showing the water depth and the other showing the flow velocity. We compute and associate a scenario probability to each map. We apply this method to a case study located in Ontario, Canada, where a levee protects a residential area (details of the case study are presented

<figure><img src="image_4.png" /><figcaption>**FIGURE 1** | Location of study area along the Etobicoke Creek in Brampton City, Peel region in Ontario, Canada.</figcaption></figure>
in Section 3). To analyze the results, we selected five locations in this levee-protected area (Figure 2) for which we plot the cumulative probabilities against water depth and flow velocity. This selection is meant for a better explanation and understanding of the method; however, this method can be extended to every location of the levee-protected area. This work will contribute to creating a probabilistic integrated method to assess flood risk in levee-protected areas.

# 2 | Materials and Methods

## 2.1 | Probabilistic Assessment of Levee Failure

We use probabilistic methods presented by Mainguenaud et al. (2023) to estimate backward erosion failure probabilities and overflowing failure probabilities. These methods are summarized in Sections 2.1.1 and 2.1.2.

### 2.1.1 | Backward Erosion Probability Assessment

Among the different mechanisms of internal erosion, backward erosion is commonly used as the design criteria for levees. In this study, we consider backward erosion initiation, while not accounting for the progression of erosion. The resulting probability of failure corresponds to the probability of erosion initiation, which is a safe approximation, as erosion does not always lead to failure. Backward erosion analysis can be performed by comparing hydraulic gradients to a critical gradient.

We compared local hydraulic gradients obtained from a finite element model based on a seepage analysis to the critical gradient defined by Terzaghi (1943). Using the seepage module from the software GeoStudio, we computed hydraulic gradients through the levee and foundation assuming steady-state as a conservative approximation. The soil model parameters used for the seepage analysis are presented in the case study (Section 3.3).

<figure><img src="image_5.png" /><figcaption>**FIGURE 2** | Points of interest location in the protected area (referred to in the article as a, b, c, d, e for more clarity). The flood extent shown in the figure corresponds to a 350-year return period for an overtopping levee breach located near point b.</figcaption></figure>
We selected the maximum gradient located on the area centered
on the foot of the levee, where backward erosion started.

To integrate uncertainty, we considered the critical gradient as a normally distributed random variable and defined the distribution parameters from expert judgement and literature to fit the case study (Mainguenaud et al. 2023). Therefore, the probability of initiation of backward erosion is estimated as the probability that the critical gradient is less than the hydraulic gradient. Applying this approach, we estimated the backward erosion probability of each levee segment for the Etobicoke Creek levee and presented the corresponding fragility curve in Figure 3.

**2.1.2 | Overflowing Probability Assessment**

As described by Visser et al. (1995), overflowing is a time-
dependent mechanism that can be split into three phases: the
beginning of the overflowing and erosion, the progression of the
erosion at various speeds and the collapse of the structure. The
complexity of the mechanisms involved and the lack of simple
methods for estimating erosion make these phases difficult to
assess. Once overflow starts, the stability of the levee is not guar-
anteed, and one of the relevant parameters to consider is the du-
ration of the overflow.

To estimate overflowing failure probability, we rely on expert
judgement (Peyras et al. 2012; Mainguenaud et al. 2023). As de-
scribed by Hartford and Baecher 2004, structured expert judge-
ment involves obtaining judgements and opinions of experts
following several steps: selecting experts, defining the problem
and analyzing expert judgement. We constitute a panel of four
specialist engineers with knowledge of the geotechnical and hy-
draulic fields, including at least one who is specialized in risk
analysis. Each expert independently assesses qualitatively the
annual overflow failure for each flood scenario knowing water
level, soil type, geometry, mechanical parameters and overflow
duration. A group discussion follows, where misunderstandings

of questions and biases are addressed and the experts select to-
gether in a consensual way the final qualitative scale of failure.
Then, we use Mainguenaud et al. (2023)'s method to transpose
qualitative expert judgement into quantitative probabilities
that rely on the table from Vick and Stewart (1996) and Peyras
et al. (2010) that converts qualitative expert judgment into prob-
abilities. As a result, we obtain overflowing failure probabilities
for each levee segment of our case study and we present the cor-
responding fragility curve in Figure 4.

2.2 | Hydraulic Model

2.2.1 | Flood Propagation Model

We use HEC-RAS's 1D-2D model to compute the river water surface for a subcritical flow regime in the 1D area and SWE in the 2D area (Chen et al. 2012). We define the 1D area by cross-sections providing the geometry and water elevation in the channel. A hydrograph defines water flow rate inputs for the upstream boundary of the 1D area and the water flow propagates between cross-sections using Saint-Venant Equations following the conservation of mass and the conservation of momentum. In the 2D area, we use SWE, which assumes incompressible flow, negligible vertical velocity and vertical derivative terms. The model receives as input an unsteady flow represented with simplified synthetic triangular hydrographs (Maione, Mignosa, and Tomirotti 2003). By using synthetic hydrographs, we assume that a simplified approach can recreate a realistic scenario. It is the most conservative approach for engineering design as we allow the flow to reach its peak value quickly and with a heavy tail. Actual hydrographs do not necessarily mimic a worst-case scenario. Synthetic hydrographs are presented in Figure 5: we simulated a 24-h flood event for flows corresponding to return periods of 100, 200, 350, 500, 750 and 1000 years, where the peak flow values were computed using flood frequency analysis. For further analysis, we extracted the maximum depth and maximum velocity of every pixel.

<figure><img src="image_6.png" /><figcaption>**FIGURE 3** | Backward erosion fragility curves for the levee along Etobicoke Creek. BXSi refers to the breached levee segment.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>**FIGURE 4** | Overflowing failure fragility curves for the levee along Etobicoke Creek. BXSi refers to the breached levee segment.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**FIGURE 5** | Synthetic flow hydrographs of Etobicoke Creek for station 02HC017 in Ontario, Canada, developed using flood frequency analysis.</figcaption></figure>

### 2.2.2 | Levee Breaching Model

HEC-RAS has two built-in breaching opening mechanism options: (i) piping or (ii) overtopping (Hydrologic Engineering Center 2023). The piping breach option (i) opens from the bottom of the levee, and the erosion will begin on the protected side of the levee as a result of seepage flow carrying soil particles. As the pipe grows larger, the material will detach and fall into the flowing water, resulting in an open breach. The overtopping breach erosion option (ii) starts on the downstream side of the levee and is deemed satisfactory to model the overflow failure mechanism. We use both options in this article.

Paquier and Goutal (2016) mention that the breaching start time and final breach dimensions greatly influence the water discharge in the protected area. The faster a breach opens, the more water flow goes through the breach during the flood event,

resulting in higher depths or velocity values in the flooded area. Therefore, in addition to an opening process, the final geometry of the breach must be defined carefully.

HEC-RAS includes three models that can be used to define the final geometry of the breach: (i) the parametric breach model, (ii) the “simplified physical” breaching method which requires an input of velocity versus erosion rates and breach widening relationships and (iii) the dam/levee (DL) breach model developed by Wu (2016). Considering available data, we chose to use the first model, parametric breach, for which we calibrated the final breach width, slopes and time of breach opening using Froehlich's (1995) empirical equations provided in Table 1. In general, backward erosion breaches are deep with steep slopes, whereas overtopping breaches tend to have wider bottom sections and gentler slopes than breaches due to backward erosion.
<table><caption>**TABLE 1** | Froehlich (1995) empirical breach parameter equations.</caption><thead><tr><th>Breach final width</th><th>Breach slopes</th><th>Breach opening time</th></tr></thead><tbody><tr><td>B=0.1803K<sub>o</sub>(V<sub>w</sub>)<sup>0.32</sup>(h<sub>b</sub>)<sup>0.19</sup>, K<sub>o</sub>=1.4 for overtopping,<br>1 otherwise, V<sub>w</sub> is the reservoir volume at the time<br>of failure, h<sub>b</sub> is the breach final height</td><td>Z=1.4 for overtopping<br>Z=0.9 for other<br>failure modes</td><td>t<sub>f</sub>=0.00254(V<sub>w</sub>)<sup>0.53</sup>(h<sub>b</sub>)<sup>-0.90</sup><br>V<sub>w</sub> is the reservoir volume at the time<br>of failure, h<sub>b</sub> is the breach final height</td></tr></tbody></table>

<figure><img src="image_9.png" /><figcaption>**FIGURE 6** | Schematics of the probabilistic flood hazard method: Estimating levee reliability and simulating flood scenarios to generate probabilistic flood maps.</figcaption></figure>

We did not consider simultaneous or successive breach failures, as we assumed that when a breach occurs, it decreases some of the pressure forces on the structure and should be less likely to breach on nearby downstream segments. The consideration of multiple breaches may be relevant for long levees of several kilometers.

## 2.3 | Cumulative Probability of Flood Depth and Velocity for Each Scenario

This section describes the method to obtain a cumulative flood exceedance probability for depth and velocity in various locations of the flooded area (Figure 2). Each step of the method is presented in Figure 6, including an assessment of the levee reliability, parameters of the hydraulic model on which the flood simulation relies and the data processing leading to flood hazard maps. Computing scenario probabilities is based on a statistical analysis of the results of a series of hydraulic simulations of different flood scenarios. These flood scenarios are defined by the return period of the flood, the breach mode (such as overtopping, piping, or no breach) and the levee segment that could be affected by a breach.

The general process involves the following four steps. Firstly, for a given flood return period (T), we compute the probability of the breach ($P_{breach}$). The probability of overtopping and the probability of piping are derived from the fragility curves presented in Section 2.1. The no-breach probability is obtained from the complement of the overtopping and piping probabilities.

Secondly, we use Equation 1 to compute the probability of each flood scenario, conditional on the flood considered ($P_{scenario|flood}$) by weighting the probability of the breach mode ($P_{Breach}$) by the

number of levee segments considered, where the breach can be located ($n_{segment}$) and by the number of failure mechanisms considered ($n_{failuremechanism}$). Each failure scenario considers a single breach, but its location may vary from one scenario to another. Therefore, we assume that failure mechanisms are independent. For each flood of the return period (T), the sum of the probabilities of the scenarios conditional on the flood considered is equal to 1.

$$P_{scenario|flood} = \frac{P_{breach}}{n_{segment} \cdot n_{failuremechanism}} \quad (1)$$

In reality, the scenarios are not independent and are often linked. However, the probability obtained using Equation 1 maximizes the probability assessment and is thus a conservative estimate of the actual probability.

Thirdly, we compute the scenario probability of each flood scenario ($P_{scenario}$) by multiplying the conditional probability from Equation 1 ($P_{scenario|flood}$) by the probability of the flood ($P_{flood}$), simplifying by assuming that $P_{flood} = 1/T$. The probability of each flood scenario ($P_{scenario}$) is thus obtained using Equation 2.

$$P_{scenario} = P_{scenario|flood} \cdot P_{flood} = \frac{P_{scenario|flood}}{T} \quad (2)$$

Where T corresponds to the return period in years.

Finally, for each flood scenario considered, we have its probability of occurrence ($P_{scenario}$) and the results of the maximum depth and velocity of the flooded area. This allows us to obtain a cumulative flood exceedance probability for depth and velocity for different points in the flooded area.
# 3 | Case Study

## 3.1 | Location of the Case Study

We apply the probabilistic approach presented in Section 2 to the case study, located in Meadowland Park in Brampton, Ontario, Canada, focusing on a levee along Etobicoke Creek (Figure 1). The Etobicoke Creek flows North to South, and the west side is protected by the natural elevation of the terrain. The levee is approximately 0.5 km long and 2 m high, situated about 80 m from Etobicoke Creek. On the protected side (to the east), there are predominantly residential areas, with commercial areas to the south. City officials raised concerns about continual flooding of the residential neighborhood despite the presence of the levee. We applied the proposed probabilistic flood assessment method to this protected area.

## 3.2 | Hydrology Considerations

Meadowland Park is located in the Etobicoke Creek watershed, which drains 211 km² with a mean river slope of 0.005. The watershed main land uses are 63% urban, 22% rural and 15% natural land cover, while our study area consists of mostly urban residential areas (Toronto Region Conservation 2010).

We base our unsteady flow propagation model on the overall type and shape of hydraulic events from the Water Office historical database, which provides water levels and flow values from the 02HC017 flow gauge station located 2 km upstream of

the study area, as shown in Figures 1 and 2 (Water Office 2023). Using flood frequency analysis, we determined flows for return periods of 100, 200, 350, 500, 750 and 1000 years (selected to assess the reliability of the levee), and they range from 85 to 530 m³.s⁻¹. We used simplified, synthetic and triangular hydrographs for a 24-h duration to simulate flood events associated with return periods, as shown in Figure 5.

## 3.3 | Geotechnical Considerations

The levee is located approximately 80 m away from the river stream's steep vegetated banks, creating a flood mitigation floodplain. This levee is approximately 60 years old with a homogeneous silty embankment, 2 m high throughout the 500 m of length. The levee slopes are fairly consistent throughout the levee length and consist of 3:1 (horizontal/vertical) for the water side and 2:1 for the protected area side of the levee, as shown in Figure 7. The asphalt trail on top of the levee crest follows the terrain's gentle slope (downstream) and is open for both pedestrians and light utility vehicles. This levee has been designed to withstand a 100-year flood return period without overtopping.

The earth fill of the levee is considered homogeneous (Terraprobe 2023); however, as topography and geometry change over the length of the levee, those criteria prevailed for the segmentation shown in Figure 8a (5 homogeneous segments *i*, each associated with a cross-section labelled BXSi in Figures 3, 4 and 9). The levee crest elevation follows the natural slope of the river;

<figure><img src="image_10.png" /><figcaption>**FIGURE 7** | Schematic of the levee cross-section for the levee segment 3 on the left and the creek on the right side. A sample 100-yr water level is provided for reference.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**FIGURE 8** | (a) Levee segments 1–5, (b) photography of the levee oriented toward downstream and located at the beginning of segment 3.</figcaption></figure>
thus, we considered an average crest elevation for each segment. The curvature of the levee, visible in Figure 8b, was a decisive factor in segmentation. We assumed that the failure probability is constant through each levee segment.

Terraprobe performed a geotechnical assessment for the Toronto and Region Conservation Authority (TRCA) to assess the geotechnical parameters of the case study area. The geotechnical assessment indicated a silty soil (gravel 10%, sand 41%, silt 31%, clay 18%) globally homogeneous, 0.3 m³.m⁻³ water content, 10⁻⁸ m.s⁻¹ levee permeability, 10⁻⁶ m.s⁻¹ foundation permeability, finished soil settlement and an estimated earth fill cohesion of 15 kPa and an angle of internal friction of 28°. These data were used in GeoStudio software for the seepage analysis.

As described in Section 2.1.1., we estimated backward erosion failure probabilities by computing a critical hydraulic gradient with Terzaghi's equation. To compute a critical gradient, we used values based on expert judgement: porosity $n \in [0.3; 0.45]$, the solid particle density $\gamma_s \in [2600; 2800]$ kg.m⁻³ and water density $\gamma_w = 1000$ kg.m⁻³. With these elements, we assumed a critical gradient following a normal distribution N (1.06; 0.27).

## 3.4 | Hydraulic Considerations

The TRCA provided a steady flow hydraulic model of the Etobicoke Creek in HEC-RAS, which we adapted to run unsteady flow simulations with synthetic hydrographs, a 1D river and a 2D area for the flood to propagate behind the levee. While the levee was designed to withstand a 100-year return period event, consistent flooding in the area and the age of the levee have raised interest in assessing the performance of the levee.

The High-Resolution Digital Elevation Model derived from LiDAR data and satellite imagery provides a Digital Terrain Model (DTM) of 1m spatial resolution. For the case study, we use the York 2019 survey sourced from National Resource Canada (2019).

We assumed a final depth breach corresponding to the levee crest height. We decided to maintain a constant breach opening time for every simulation, enabling the comparison of depths or velocity between simulations.

## 3.5 | Points of Interest in the Protected Area for the Probabilistic Flood Assessment

For the flood probability assessment, we consider five locations of interest from the protected area to analyze the flood results. Figure 2 shows each point location:

* Point A has been chosen in the residential neighborhood,

* Points B and C stand behind the levee, respectively, at segments 3 and 4,

* Point D is located on the parking lot, and point E is on the main road downstream of the levee.

# 4 | Results

## 4.1 | Influence of Breach Location, Failure Mechanism and Terrain on the Flood Exceedance Probabilities of Flood Depth and Flow Velocity

Following the steps described in Section 2.3, we compute each scenario probability and the corresponding cumulative flood exceedance probability. In Figure 9, the graphs contain 11 lines, each associated with a flood scenario labelled with the following: failed cross-section (BXSi) and breaching mechanism (O for overtopping, P for piping, NB for no breach). Each line is composed of 4–5 points depending on the flood intensity (return period from 100 to 1000 years).

We can distinguish between failure mechanisms based on the order of magnitude of their associated probability. Backward erosion probabilities range between $10^{-7}$ and $10^{-5}$, overtopping probabilities range between $10^{-4}$ and $10^{-3}$ and the no-breach scenario probabilities range between $10^{-3}$ and $10^{-2}$. This demonstrates that the levee has a high reliability.

The breach location is not a relevant parameter for overtopping in this case study as the tuple (depth, overtopping failure probability) overlaps for all breach locations (Figure 9). The backward erosion exceedance probability decreases from levee segments 3, 4, 2, 1–5 (Figure 9, graphs show solid lines with dotted points in the same order). Levee segments 3 and 4 are located at the levee curvature and are the most likely to fail.

In general, higher values of depth, velocity and flood exceedance probabilities are achieved close to the levee. We observe the impact of storage areas on depth in Figure 9, where points B and C (located close to the levee) show higher depth values than points A or E (located far from the levee). However, point D (located far from the levee where water can accumulate) shows a similar depth range as points located close to the levee. Point B (located in front of breach 3) shows a peak in velocity exceedance probabilities for the breaching scenario of the levee segment 3 (BXS3). A similar pattern was expected for point C. Instead, point C shows high-velocity probabilities for all breach locations. This behavior is due to the bottleneck location: the water previously restricted by the terrain is given free range to a flat slightly inclined area. This confirms that, as expected, topography plays a decisive factor in flood exceedance probabilities.

The simulation results show that the depth probability decreases following a similar trend regardless of the proximity to the levee and that higher values of depth, velocity and flood exceedance probabilities are achieved close to the levee except when the global trend in terrain conditions changes. This method enables us to identify which levee segments are most critical for flood risk management and can help determine which segments should be the focus of improved levee reliability and safety of the protected area. In this case study, the most critical levee segments are segments 3 and 4.
## 4.2 | Probabilistic Assessment of Floods in the Levee-Protected Area

Following the steps described in Section 2.3, we aggregate conditional failure probabilities of every simulation into a single cumulative curve, as shown in Figure 10. Out of the selection of locations presented in Figure 2, the ranking of depth hazard levels from the highest to the lowest are C, B, D, A and E (see Figure 10, depth graph), whereas the velocity hazard level ranks from the highest to the lowest as C, B, E, A and D (see Figure 10, velocity graph).

We can observe the impact of flat terrain on flow velocity in (Figure 9b), where we observe a decrease in velocity as the distance to the levee increases. With the exception of point E (located on the road area), where fewer obstacles to slow water propagation result in similar probabilities to locations close to the levee.

The curve provides a probability for a given depth or velocity value. For example, based on Jonkman and Penning-Rowsell (2008), the stability of an individual may be compromised for a depth of 0.3 m depth and a velocity of 1 m.s⁻¹. The probability for this

<figure><img src="image_12.png" /><figcaption>**FIGURE 9** | Cumulative flood exceedance probabilities for depth and velocity for 5 points of interest (A–E).</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>FIGURE 9 | (Continued)</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>FIGURE 10 | Aggregated cumulative flood exceedance probability of (a) water depths and (b) velocities of various locations within the flooded area. For point locations, refer Figure 2 or see the detailed description in Section 3.5.</figcaption></figure>
velocity value ranges between 0 and $9.10^{-2}$ for the point selection available in Figure 10.

We can also focus on the residential area (Figure 10, point A), where we observe a maximal depth of 2.35 m and a velocity of $<0.75 \text{ m.s}^{-1}$. Our method provides relevant support to help choose which level of hazard one is willing to accept and adopt appropriate mitigation measures.

Considering depth and velocity probabilities is necessary as both provide a different insight into risk analysis. In particular, velocity probabilities should be considered near levee breaches because their values are high.

# 5 | Discussion

## 5.1 | Comparison With Other Probabilistic Flood Hazard Assessment Methods

The probabilistic flood hazard model of D'Oria, Maranzoni, and Mazzoleni (2019) integrates piping levee breaching to a 1D-2D hydraulic model, and they propose a probabilistic flood hazard map considering the uncertainty of breach location and failure time. There are 3 key points of their method that differ from ours. Firstly, they consider a single flood event for various time steps (i.e., not necessarily peak flow). This choice induces a noticeable difference with scenario probabilities. In our method, we weigh scenario probabilities by the number of breaching scenarios considered in a flood event and we consider multiple flood events.

Secondly, they selected locations along the levee linear that are a priori more likely to fail due to the piping mechanism based on fragility curves. In contrast, we consider the possibility of multiple failure mechanisms for each breach location. Thirdly, they assess piping failure using an estimate of the grade line of the seepage flow, taking into consideration the variation of porosity of the levee and assuming instantaneous breaching. Instead, we use finite-element seepage analysis and compare hydraulic gradients located at the toe of the levee and used Froehlich 1995, parameters to simulate the breach opening. Finally, we both neglect multiple breach scenarios during a flood event and plot the maximum depth on flood hazard maps.

Multiple levee failures are realistic occurrences though their possibility is usually overlooked in flood risk management (Jonkman, Kok, and Vrijling 2008). Therefore, Ceccato, Malvestio, and Simonini (2022) propose an extension of D'Oria, Maranzoni, and Mazzoleni (2019)'s method that includes multiple dependent breaches throughout a flood event. The key points of Ceccato, Malvestio, and Simonini's (2022) method include having breaches open sequentially from upstream to downstream during a flood event at the peak water level (dependent levee breaches) and representing a probabilistic qualitative level hazard based on a combination of depth and velocity thresholds. This method differs from ours because we consider independent breaches, some of which occur in the same location but involve different failure mechanisms. As a result, we calculate the scenario probability differently.

## 5.2 | Limits and Advantages of Our Method

One limitation of our method is its reliance on the representativeness of the selected flood scenarios. The set of flood scenarios should cover a range as wide as possible to improve the resulting probabilistic map and the precision of its flood extent. Our range of flood scenarios extends from 100 to 1000 years, and we assume that levee failure is impossible for events with a return period shorter than 100 years. Therefore, we do not consider scenarios with a return period below 100 years, which limits the highest observed probability to $1.10^{-2}$. Another limitation of our method is the lack of quantified uncertainty. To address this, we could perform a sensitivity analysis and vary the parameters and methods to evaluate how they impact levee failure probabilities.

Our new probabilistic method computes flood exceedance probabilities and its representation of flood hazard is able to distinguish between failure mechanisms based on their probability. Our quantification of levee failure mechanisms highlights which critical failure mechanism to focus on and which levee segment shows the highest risk of failure. By aggregating flood hazard maps from various scenarios (differentiated by levee breach location, failure mechanism and flood intensity), we generate a flood hazard curve that quantifies both depth and velocity for cumulative flood exceedance probabilities. Our approach enables the computation of detailed flood hazard maps for inclusion in flood risk assessments. Key aspects of our method include a 2D model of the area of interest, the evaluation of levee breach scenarios at multiple locations and for various failure mechanisms and the consideration of multiple flood events.

# 6 | Conclusions

We presented a probabilistic flood hazard assessment method for levee-protected areas. Our method includes (i) river hydraulics, (ii) levee reliability for different failure modes and for several flood events, (iii) flood propagation with levee failure scenarios and (iv) aggregated probabilistic flood hazard mapping. Our approach consists of a levee reliability study for homogeneous levee segments, considering various levee failure mechanisms triggered by different flood intensities, propagating flood in the river and through the levee breach and statistically analyzing depth and velocity located in the protected area. We propose a new approach to probabilistic flood hazard mapping for flood hazard assessments considering several levee breaching scenarios and return periods.

The case study showcases the advantages of our proposed method, such as the ability to

* Identify the levee segment the most impactful on flood hazard and the most impacting failure mechanism on the reliability of a given levee segment,

* Locate the most hazardous areas when considering depth and velocity probabilities,

* Discriminate against the most impactful failure mechanisms for flooding in the levee-protected area and

* Provide a more holistic result rather than considering one return period or one failure mechanism.
In future works, we will explore the potential of dynamic flood mapping and capitalize on probabilistic flood hazards to propose a probabilistic flood risk map.

## Acknowledgements

We acknowledge the Toronto and Region Conservation Authority (TRCA), which provided a hydraulic model of Etobicoke Creek and a geotechnical assessment of the study area. Funding was provided by INRAE, York University, and Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant (DG). Everett Snieder contributed to the figures.

## Data Availability Statement

The data that support the findings of this study are openly available in Real-Time Hydrometric Data Graph for Etobicoke Creek, Greater Toronto Area at https://wateroffice.ec.gc.ca/report/real_time_e.html? stn=02HC017, station number: 02HC017.

## References

Aguilar-Lopez, J. P., J. J. Warmink, A. Bomers, R. M. J. Schielen, and S. J. M. H. Hulscher. 2018. "Failure of Grass Covered Flood Defences with Roads on Top Due to Wave Overtopping: A Probabilistic Assessment Method." *Journal of Marine Science and Engineering* 6, no. 3: 74. https:// doi.org/10.3390/jmse6030074.

Beilicci, E., and R. Beilicci. 2012. "A Comparative Analysis of Two Flood Wave Propagation Models." *International Multidisciplinary Scientific GeoConference: SGEM* 3: 523.

Ceccato, F., S. Malvestio, and P. Simonini. 2022. "Effect of Animal Burrows on the Vulnerability of Levees to Concentrated Erosion." *Watermark* 14, no. 18: 2777. https://doi.org/10.3390/w14182777.

Chen, Y., Z. Wang, Z. Liu, and D. Zhu. 2012. "1D-2D Coupled Numerical Model for Shallow-Water Flows." *Journal of Hydraulic Engineering* 138, no. 2: 122-132.

CIRIA, Ministry of Ecology, and USACE. 2013. *The International Levee Handbook*. London, UK: CIRIA. ISBN: 978-0-86017-734-0. https://www.ciria.org/ItemDetail?iProductCode=C731F&Category= FREEPUBS.

Costabile, P., C. Costanzo, and F. Macchione. 2017. "Performances and Limitations of the Diffusive Approximation of the 2-d Shallow Water Equations for Flood Simulation in Urban and Rural Areas." *Applied Numerical Mathematics* 116: 141-156. https://doi.org/10.1016/j.apnum. 2016.07.003.

D'Oria, M., A. Maranzoni, and M. Mazzoleni. 2019. "Probabilistic Assessment of Flood Hazard due to Levee Breaches Using Fragility Functions." *Water Resources Research* 55, no. 11: 8740-8764. https://doi. org/10.1029/2019WR025369.

Di Baldassarre, G., A. Viglione, G. Carr, L. Kuil, J. L. Salinas, and G. Blöschl. 2013. "Socio-Hydrology: Conceptualising Human-Flood Interactions." *Natural Hazards and Earth System Sciences* 17: 3295- 3303. https://doi.org/10.5194/hess-17-3295-2013.

Di Baldassarre, G., G. Schumann, P. D. Bates, J. E. Freer, and K. J. Beven. 2010. "Flood-Plain Mapping: A Critical Discussion of Deterministic and Probabilistic Approaches." *Hydrological Sciences Journal* 55, no. 3: 364- 376. https://doi.org/10.1080/02626661003683389.

Doble, R. C., R. S. Crosbie, B. D. Smerdon, L. Peeters, and F. J. Cook. 2012. "Groundwater Recharge From Overbank Floods." *Water Resources Research* 48: W09522. https://doi.org/10.1029/2011WR011441.

Dottori, F., P. Salamon, A. Bianchi, L. Alfieri, F. A. Hirpa, and L. Feyen. 2016. "Development and Evaluation of a Framework for Global Flood

Hazard Mapping." *Advances in Water Resources* 94: 87-102. https://doi. org/10.1016/j.advwatres.2016.05.002.

European Environment Agency. 2007. "Directive 2007/60/EC of the European Parliament and of the Council on the Assessment and Management of Flood Risks." Technical Report. http://eur- lex.europa. eu/legalcontent/EN/TXT/?uri=CELEX:32007L0060.

Ferdous, M. R., A. Wesselink, L. Brandimarte, G. Di Baldassarre, and M. M. Rahman. 2019. "The Levee Effect Along the Jamuna River in Bangladesh." *Water International* 44, no. 5: 496-519. https://doi.org/10. 1080/02508060.2019.1619048.

Ferrari, A., S. Dazzi, R. Vacondio, and P. Mignosa. 2020. "Enhancing the Resilience to Flooding Induced by Levee Breaches in Lowland Areas: A Methodology Based on Numerical Modelling." *Natural Hazards and Earth System Sciences* 20: 59-72. https://doi.org/10.5194/ nhess-20-59-2020.

Froehlich, D. C. 1995. "Peak Outflow From Breached Embankment Dam." *Journal of Water Resources Planning and Management* 121, no. 1: 90-97.

Hall, J. W., S. Tarantola, P. D. Bates, and M. S. Horritt. 2005. "Distributed Sensitivity Analysis of Flood Inundation Model Calibration." *Journal of Hydraulic Engineering* 131, no. 2: 117-126. https://doi.org/10.1061/ (asce)0733-9429(2005)131:2(117).

Hartford, D., and G. B. Baecher. 2004. *Risk and Uncertainty in Dam Safety*. Thomas Telford: CEA Technologies. ISBN 07277-3270-6.

Hathout, M., L. Peyras, C. Carvajal, Y. Diab, and M. Vuillet. 2020. "Development of an Approach to Evaluate the Failure Probabilities of River Levees Based on Expert Judgement: Application to a Case Study." *Journal of Flood Risk Management* 13, no. 2: e12603. https://doi.org/10. 1111/jfr3.12603.

Hydrologic Engineering Center. 2023. "HEC-RAS Technical Reference Manual. Version 1.5." https://wwwhec.usace.army.mil/confluence/ rasdocs/ras1dtechref/latest/performing-a-dam-break-study-with-hec- ras/estimating-dam-breach-parameters/estimating-breach-parameters.

Jonkman, S. N., and E. Penning-Rowsell. 2008. "Human Instability in Flood Flows." *Journal of the American Water Resources Association* (JAWRA) 44, no. 5: 1208-1218. https://doi.org/10.1111/j.1752-1688. 2008.00217.x.

Jonkman, S. N., M. Kok, and J. K. Vrijling. 2008. "Flood Risk Assessment in The Netherlands: A Case Study for Dike Ring South Holland." *Risk Analysis* 28, no. 5: 1357-1374. https://doi.org/10.1111/j.1539-6924.2008. 01103.x.

Lavoie, B., and T. F. Mahdi. 2017. "Comparison of Two-Dimensional Flood Propagation Models: SRH-2D and Hydro_AS-2D." *Natural Hazards* 86: 1207-1222. https://doi.org/10.1007/s11069-016-2737-7.

Li, T. R., Z. Y. Bai, X. C. Peng, and T. He. 2013. "Finite Proximate Method for Simulating Flood Propagation in Complicated River Channel and Flood Detention Area." *Applied Mechanics and Materials* 300: 814-820. https://doi.org/10.4028/www.scientific.net/AMM.300-301.814.

Mainguenaud, F., L. Peyras, U. T. Khan, C. Carvajal, J. Sharma, and B. Beullac. 2023. "A Probabilistic Approach to Levee Reliability Based on Sliding, Backward Erosion and Overflowing Mechanisms: Application to an Inspired Canadian Case Study." *Journal of Flood Risk Management* 16, no. 4: e12921. https://doi.org/10.1111/jfr3.12921.

Maione, U., P. Mignosa, and M. Tomirotti. 2003. "Regional Estimation of Synthetic Design Hydrographs." *International Journal of River Basin Management* 1, no. 2: 151-163. https://doi.org/10.1080/15715124.2003. 9635202.

Maranzoni, A., M. D'Oria, and C. Rizzo. 2023. "Quantitative Flood Hazard Assessment Methods: A Review." *Journal of Flood Risk Management* 16, no. 1: e12855. https://doi.org/10.1111/jfr3.12855.

Maranzoni, A., M. D'Oria, and M. Mazzoleni. 2022. "Probabilistic Flood Hazard Mapping Considering Multiple Levee Breaches." *Water
*Resources Research* 58: e2021WR030874. https://doi.org/10.1029/2021W R030874.

Mazzoleni, M., B. Bacchi, S. Barontini, G. Di Baldassarre, M. Pilotti, and R. Ranzi. 2014. "Flooding Hazard Mapping in Floodplain Areas Affected by Piping Breaches in the Po River, Italy." *Journal of Hydrologic Engineering* 19, no. 4: 717–731. https://doi.org/10.1061/(ASCE)HE.1943- 5584.0000840.

Mazzorana, B., F. Comiti, C. Volcan, and C. Scherer. 2011. "Determining Flood Hazard Patterns Through a Combined Stochastic-Deterministic Approach." *Natural Hazards* 59: 301–316. https://doi.org/10.1007/s1106 9-011-9755-2.

Meire, D., L. De Doncker, F. Declercq, K. Buis, P. Troch, and R. Verhoeven. 2010. "Modelling River-Floodplain Interaction During Flood Propagation." *Natural Hazards* 55: 111–121. https://doi.org/10. 1007/s11069-010-9554-1.

Merz, B., A. Thieken, and M. Gocht. 2007. "Flood Risk Mapping at the Local Scale: Concepts and Challenges." In *Flood Risk Management in Europe. Advances in Natural and Technological Hazards Research*, 25, edited by S. Begum, M. J. F. Stive, and J. W. Hall, 231–251. Dordrecht: Springer. https://doi.org/10.1007/978-1-4020-4200-3_13.

Mitsopoulos, G., E. Panagiotatou, V. Sant, et al. 2022. "Optimizing the Performance of Coupled 1D/2D Hydrodynamic Models for Early Warning of Flash Floods." *Watermark* 14, no. 15: 2356. https://doi.org/ 10.3390/w14152356.

Mohamed, M. A., P. G. Samuels, M. W. Morris, and G. S. Ghataora. 2002. "Improving the Accuracy of Prediction of Breach Formation Through Embankment Dams and Flood Embankments." In *International Conference on Fluvial Hydraulics*, 2002. Louvainla-Neuve, Belgium: River Flow.

Morales-Hernández, M., P. García-Navarro, J. Burguete, and P. Brufau. 2013. "A Conservative Strategy to Couple 1D and 2D Models for Shallow Water Flow Simulation." *Computers & Fluids* 81: 26–44. https://doi.org/ 10.1016/j.compfluid.2013.04.001.

National Resource Canada. 2019. "High Resolution Digital Elevation Model From 2019, Project York_2019 for utm17 Coordinate System in a 1 m Spatial Resolution." https://search.open.canada.ca/openmap/95778 2bf-847c-4644-a757-e383c0057995.

Oliver, J., X. S. Qin, O. Larsen, M. Meadows, and M. Fielding. 2018. "Probabilistic Flood Risk Analysis Considering Morphological Dynamics and Dike Failure." *Natural Hazards* 91: 287–307. https://doi. org/10.1007/s11069-017-3126-6.

Paquier, A., and N. Goutal. 2016. "Dam and Levee Failures: An Overview of Flood Wave Propagation Modeling." *La Houille Blanche* 1: 5–12. https://doi.org/10.1051/lhb/2016001.

Peyras, L., C. Carvajal, H. Félix, et al. 2012. "Probability-Based Assessment of Dam Safety Using Combined Risk Analysis and Reliability Methods." *Application to Hazards Studies European Journal of Environmental and Civil Engineering* 16, no. 7: 795–817.

Peyras, L., D. Boissier, C. Carvajal, et al. 2010. "Analyse de risques et fiabilité des barrages. Application aux barrages-poids en béton. Éditions universitaires européennes."

Rifai, I., K. El Kadi Abderrezzak, S. Erpicum, et al. 2018. "Floodplain Backwater Effect on Overtopping Induced Fluvial Dike Failure." *Water Resources Research* 54, no. 11: 9060–9073. https://doi.org/10.1029/ 2017WR022492.

Rincón, D., J. F. Velandia, I. Tsanis, and U. T. Khan. 2022. "Stochastic Flood Risk Assessment Under Climate Change Scenarios for Toronto, Canada Using CAPRA." *Watermark* 14, no. 2: 227. https://doi.org/10. 3390/w14020227.

Rongen, G., O. Morales-Nápoles, and M. Kok. 2022. "Expert Judgment-Based Reliability Analysis of the Dutch Flood Defense System."

*Reliability Engineering & System Safety* 224: 108535. https://doi.org/10. 1016/j.ress.2022.108535.

Sellmeijer, H., J. L. de la Cruz, V. M. van Beek, and H. Knoeff. 2011. "Fine-Tuning of the Backward Erosion Piping Model Through Small-Scale, Medium-Scale and IJkdijk Experiments." *European Journal of Environmental and Civil Engineering* 15, no. 8: 1139–1154. https://doi. org/10.1080/19648189.2011.9714845.

Teng, J., A. J. Jakeman, J. Vaze, B. F. Croke, D. Dutta, and S. Kim. 2017. "Flood Inundation Modelling: A Review of Methods, Recent Advances and Uncertainty Analysis." *Environmental Modelling & Software* 90: 201–216. https://doi.org/10.1016/j.envsoft.2017.01.006.

Terraprobe. 2023. "Geotechnical Assessment Etobicoke Dike." Technical Report n°1-22-0506, March 3rd.

Terzaghi, K. 1943. *Theoretical Soil Mechanics*. New York: John Wiley & Son.

Tobin, G. A. 1995. "The Levee Love Affair: A Stormy Relationship?" *JAWRA Journal of the American Water Resources Association* 31, no. 3: 359–367. https://doi.org/10.1111/j.1752-1688.1995.tb04025.x.

Toronto Region Conservation. 2010. "Etobicoke and Mimico Creeks Watersheds Technical Update Report." https://trcaca.s3.ca-central-1. amazonaws.com/app/uploads/2016/03/10130030/2.0-StudyArea-Physi calSetting.pdf.

Tufano, R., L. Guerriero, M. Annibali Corona, et al. 2023. "Multiscenario Flood Hazard Assessment Using Probabilistic Runoff Hydrograph Estimation and 2D Hydrodynamic Modelling." *Natural Hazards* 116: 1029–1051. https://doi.org/10.1007/s11069-022-05710-3.

Vick, S. G., and R. A. Stewart. 1996. "Risk Analysis in Dam Safety Practice." In *ASCE: Uncertainty in the Geologic Environment: From Theory to Practice*, 586–603. Reston, VA: Geotechnical Special Publications.

Visser, P. J., A. W. Kraak, W. T. Bakker, et al. 1995. "A Large-Scale Experiment on Breaching in Sand-Dikes." In *Coastal Dynamics' 95-Proceedings of the International Conference on Coastal Research in Terms of Large Scale Experiments*, 583–594. Gdansk, Poland: Amer Society of Civil Enginers.

Vorogushyn, S., B. Merz, K. E. Lindenschmidt, and H. Apel. 2010. "A New Methodology for Flood Hazard Assessment Considering Dike Breaches." *Water Resources Research* 46: W08541. https://doi.org/10. 1029/2009WR008475.

Vuijk, V., W. Van Balen, and S. van Vuren. 2017. "Fully Probabilistic Assessment of Safety Against Flooding Along the Dutch Coast." *Journal of Flood Risk Management* 10, no. 3: 349–360. https://doi.org/10.1111/ jfr3.12153.

Water Office. 2023. "Hydrometric Data: Daily Discharge Graph for Etobicoke Creek at Brampton Station 02HC017." https://wateroffice.ec. gc.ca/report/historical_e.html?stn=02HC017.

Winsemius, H. C., L. P. H. Van Beek, B. Jongman, P. J. Ward, and A. Bouwman. 2013. "A Framework for Global River Flood Risk Assessments." *Hydrology and Earth System Sciences* 17, no. 5: 1871– 1892. https://doi.org/10.5194/hess-17-1871-2013.

Wu, W. 2016. *Inputs and Outputs of DL Breach-A Simplified Physically Based Dam/Levee Breach Model*. Technical Report version 2016.4, 1-17. NY: Clarkson University.

Yilmaz, K., Y. Darama, Y. Oruc, and A. B. Melek. 2023. "Assessment of Flood Hazards due to Overtopping and Piping in Dalaman Akköprü Dam, Employing Both Shallow Water Flow and Diffusive Wave Equations." *Natural Hazards* 117, no. 1: 979–1003. https://doi.org/10. 1007/s11069-023-05891-5.