Urban Flood Drifters (UFDs): Onset of Movement

. Despite their catastrophic implications in flood events, the mobilization and transport of large, loose objects—termed Urban Flood Drifters (UFDs)—are often overlooked in flood management. This oversight stems from our limited understanding of how flowing water interacts with these heterogeneous objects. To bridge this knowledge gap, we introduce a mechanistic stability model that predicts the onset of UFD mobilization across a diverse array of loose objects, from plastics to heavy vehicles. Built on an inventory of key physical properties of UFDs, this model is also validated against existing mobilization studies. Our model generates stability curves that delineate flow conditions leading to their mobilization. We further enhance the reliability of our model by incorporating a Monte Carlo-based probabilistic framework that accounts for uncertainties and interdependencies among the input parameters. These probabilistic stability curves enable us to estimate the likelihood of mobilization for specific categories of UFDs under certain flow conditions. When integrated with flood maps or two-dimensional (2D) hydrodynamic models, our stability curves can guide urban planning efforts to predict and mitigate the impacts of UFDs during extreme flood events.


Introduction
Floods exert significant impacts on human life, economic stability, environmental health, and cultural heritage (Hickey and Salas, 1995;Jonkman and Vrijling, 2008;Hammond et al., 2015;Arrighi, 2021).Between 2000 and 2019, floods affected 1.65 billion people globally, resulting in over 100,000 fatalities (Browder et al., 2021;United Nations, 2023).Floods account for more than half of the total damage attributed to natural disasters since the 1980s (European Environment Agency, 2022).
According to the World Economic Forum (2022), floods are one of the top three most frequent and severe weather events, predicted to displace over 200 million people by 2050 (Clement et al., 2021).Particularly in the European Union, approximately two-thirds of the aggregate damage caused by natural disasters since the 1980s can be attributed to hydrometeorological events (European Environment Agency, 2022).Moreover, a single flood event can incur direct costs running into thousands of millions of euros (European Environment Agency, 2022).
Low-and middle-income countries are acutely susceptible (Kahn, 2005), and even within countries with long tradition in flood defenses, a disparity exists in flood protection preparedness across urban areas of varying resources (Lindersson et al.,Figure 1. Examples of Urban Flood Drifters (UFDs) categorized according to Table 1, based on Bayón et al. (2023): A Sanliurfa, Turkey (credit: AFP, 2023), B Cedar Rapids, USA (credit: Roberson, 2008), andC Laishui, China (credit: Getty Images, 2012) A critical yet underexamined aspect of floods is their ability to mobilize Urban Flood Drifters (UFDs, Figure 1), comprising loose objects commonly found in urban settings (Bayón et al., 2023).The resultant hazards range from structural damage to buildings (Jalayer et al., 2018;Zhang et al., 2018) to the obstruction of critical flood relieving infrastructure (Kramer et al., 2015;Davidson et al., 2015).Despite growing awareness of the role of UFDs in floods (Dewals et al., 2021;Mohr et al., 2023;Ludwig et al., 2022), this phenomenon remains largely underrepresented in urban flood research and impact evaluations.A comprehensive hydro-mechanical framework to gauge the mobilization and transport of UFDs is still lacking, thus limiting the precise forecasts of their impact in flood scenarios.Existing research has largely concentrated on the stability of vehicles (Keller and Mitsch, 1992;Martínez-Gomariz et al., 2018;Bocanegra et al., 2020) and trash containers (Martínez-Gomariz et al., 2020), often under controlled, experimental conditions.Such mechanistic, laboratory scale studies, have only recently been extended to incipient motion of plastic debris (Waldschläger and Schüttrumpf, 2019;Goral et al., 2023b;Mellink et al., 2023), and their transport (Valero et al., 2022;Lofty et al., 2023).
This preprint is currently under revision other investigations on urban flooding (Mignot et al., 2019;Smith et al., 2019a;O'Donnell and Thorne, 2020;Zevenbergen et al., 2020;Wing et al., 2022;Bates et al., 2023;Sanders et al., 2023) indicates that the stability characteristics of the majority of UFDs outlined in Table 1 remain largely unassessed.
Earlier research frequently employed experimental setups, often involving scale models of cars, to analyze stability curves.
These curves are derived from varying water depths (h) and depth-averaged flow velocities (v) under controlled hydraulic laboratory conditions (see, for example, Kramer et al., 2016;Smith et al., 2019b, for 1:1 scale models).Furthermore, theoretical models grounded in first principles have been developed and calibrated against experimental data to predict unstable conditions for specific car models (Xia et al., 2014;Shah et al., 2021) and trash bins (Martínez-Gomariz et al., 2020).Following Martínez-Gomariz et al. (2020) and Smith et al. (2019b), among others, we identify several key characteristics of UFDs that influence their dynamics during flood events: 1.The bounding box dimensions (L x , L y , L z , Fig. 2A-D), which tightly contain a UFD, with L x denoting the longest horizontal length, L y the shortest horizontal length, and L z the vertical height (floor-normal).
2. The total mass (M ) of the UFD, which can be reciprocally defined using the UFD's bulk density (ρ b ): where V = L x L y L z is the enclosed volume of the UFD bounding box.
3. The submerged volume (V ′ ), which is defined as the volume contributing to buoyancy when submerged up to a water depth h.This is inherently smaller than V and is formulated as a fraction thereof (see Fig. 2E): This ratio closely resembles porosity.A value of f V ≈ 1 suggests a fully water-tight cubic volume occupying the entirety of the volume defined by L x L y h.
4. The flow-exposed area of the bounding box, given by A = L x h.For two-wheelers like scooters and bikes, which are assumed to lie on their sides (Fig. 2B), this assumption is based on the expectation that during the initial stages of flooding, two-wheelers may often topple, settling into a more stable, horizontal position.

5.
The UFD effective drag area, which is the actual side area contributing to drag and is only a fraction (f A ) of the flowexposed bounding box area (Fig. 2E): Here, an f A value close to 1 implies that the UFD effectively obstructs most of the water flowing through A, while an f A close to 0 suggests high porosity, permitting free flow of water.
6.The drag coefficient C D , which is a dimensionless factor that quantifies how hydrodynamic the flow-exposed shape of the UFD is, and affects the fluid's drag force exerted on the effective area A ′ .
This preprint is currently under revision 7. The ground clearance z c , which is vertical distance from the floor to the lower chassis in vehicles.Both f A and f V may dramatically change when water depth exceeds this clearance height, such as the chassis height in vehicles.
8. The friction coefficient µ, which quantifies the ratio of frictional to normal forces between two contacting surfaces.To address the complexity of these parameters for UFDs, we compile a comprehensive inventory, presented and detailed in Appendix A, for over 100 typified UFDs.This inventory utilizes data from online catalogues for dimensions and mass.For clearance height, an image-based manual identification method is employed.We also adopt an image-based routine to estimate the effective drag area (A ′ ).This routine, applied to publicly available side images of UFDs (sources detailed in Supplementary Material), employs a binarization technique to extract the silhouette of the UFD.The fraction f A is then calculated as the ratio between this silhouette and the flow-exposed area, under the assumption that the UFD is submerged to a specified water depth (Appendix A).For parameters like drag coefficient and submerged volume fraction, we either rely on empirically-derived values from existing literature or make simplifying geometric assumptions.
For UFDs of heterogeneous composition, our estimates of physical parameters are based on reasonable assumptions such as a range of densities and expected sizes, the latter observed in images from Bayón et al. (2023) (additional details and justification is included in Appendix A).
This preprint is currently under revision 3 Mechanistic model

Limit states of stability
Aligned with previous studies (Smith et al., 2019b;Martínez-Gomariz et al., 2020), we establish a framework that considers the balance among four forces, drag (S y ) versus friction (R y ), buoyancy (S z ) versus weight (R z ), and two moments -toppling load (S yz ) versus toppling resistance (R yz ), as illustrated in Figure 3 for a garbage bin, on the previously commonly identified UFDs.These correspond to three different limit states of stability.These states are associated to the types of load (S) and resistance (R), as specified in Fig. 3B: i) flotation (S z ≥ R z , buoyant force exceeding weight), ii) sliding (S y ≥ R y , drag force exceeding the UFD-floor friction) and iii) toppling (S yz ≥ R yz , drag/buoyancy moment exceeding friction/weight moment).

Flotation
This preprint is currently under revision

Flotation
When buoyancy forces surpass the weight of an UFD, the object starts floating.For this failure mode, the destabilising load (S z ) can be estimated as the buoyancy given by the Archimedes' principle, with ρ w being the water density and g the gravity acceleration: The counteracting force or resisting component in the flotation limit state is the weight (mass

Sliding
Sliding occurs when the drag force, acting in the y-direction, destabilizes the UFD.This load can be estimated considering the UFD effective drag area (A ′ = f A L x h) and the drag coefficient (C D ): We neglect the horizontal hydrostatic force as a destabilizing component.At low flow velocities, net hydrostatic forces are likely to cancel out due to compensation between the upstream and downstream sides.For high flow velocities and in supercritical conditions, the drag force is the predominant factor influencing the y-direction forces.
The resisting component in this limit state is given by the contact friction between the UFD and the floor, and can be expressed through the friction coefficient (µ) as: which implies that, when the normal component cancels (R z → S z ),the horizontal as well (R y → 0).In other words,

Toppling
Toppling may occur when the sum of destabilizing moments around an axis is larger than the sum of stabilizing moments opposing them.Taking the downstream rim of an UFD in the direction x (Fig. 3B) allows the estimation of moments: with z p and y p as the lever arms at which forces S y and S z are applied, respectively (Fig. 3A).These lever arms depend on the distribution of pressures over the wet surface.We make a justifiable approximation by considering: This preprint is currently under revision and: with the z p definition acknowledging that most of the flow impacts the UFD over z c , once it is submerged over the clearance height (h > z c ).
The toppling resistance will be given by: 4 Probabilistic framework

General remarks
Utilizing the classification scheme introduced by Bayón et al. (2023), as outlined in Table 1, we categorize the properties of UFDs into three main groups: i) Vehicles (UFD-V), ii) Furniture (UFD-F), and iii) Heterogeneous UFD (UFD-H).We also incorporate all the subgroups presented in Table 1.This categorization is useful because it allows us to define ranges of realistic values for all parameters used in the mechanistic model discussed in Section 3.
To identify realistic parameters for the UFDs, serving as inputs for the mechanistic model detailed in Section 3, we conduct a comprehensive examination.This examination draws upon various sources, contingent on the specific model of the typified UFDs under investigation.The survey of key properties -covering those identified in Section 2-is further elaborated in Appendix A and Supplementary Material, where all sources are included.For heterogeneous UFDs, we base our parametrization on a comprehensive literature review, complemented by justified boundaries.
In the following, we explain how we leverage this key properties inventory for UFDs and define functional relationships and probabilistic density functions (Section 4.2 and Appendix A).These serve to capture the diversity inherent to each category of UFDs.The ultimate goal is to generate random, yet realistic, samples from a UFD subcategory for the purpose of performing Monte Carlo analyses of their stability, as demonstrated through the mechanistic model in Section 3.This approach is illustrated in Figure 4.

Structural relationships for UFDs' key variables and their variability
This preprint is currently under revision

Structural relationships for UFDs' key variables and their variability
Each UFD, whether it be a car, a trash bin, or another object, exhibits distinct characteristics, a variability that our model aims to capture.Certain characteristics, such as dimensions L x and L y in vehicles, often scale together --these are identified as structural relationships.These co-dependencies across multiple variables need to be considered, to ensure that, when we generate a synthetic UFD to study its stability, we are generating a realistic UFD.
In contrast, other UFD properties might exhibit randomness within defined limits and are categorized as randomistic.For such properties, we utilize Probability Density Functions (PDFs).The choice of PDF for each variable is based on an empirical evaluation of the Cumulative Density Function, typically supplemented by physical reasoning.For example, if a characteristic must be strictly positive, a U (uniform distribution) or T (triangular distribution) is favored over a N (Normal distribution), which carries a non-zero probability of producing negative values.
Either structural or randomistic, the characteristics of each UFD subcategory can be explained through the following equation:

Monte Carlo analysis
This preprint is currently under revision Here, x is the dependent variable, while y is the independent variable.Υ(x) is a function that portrays the trend -or structural relationship-of y relative to x, while y ′ denotes a stochastic component that introduces additional variability into y.The functional form of Υ(x) is flexible, but we primarily investigate two functional relationships-linear and saturation curve-besides a null function: The parameters a and b are optimized through error minimization algorithms to best fit the observed UFD data, as described in Appendix A. For certain co-dependencies, Eq. 14 offers a superior fit.For example, within all vehicles subcategories (UFD-V), an increase in L x is accompanied by an increase in L y , but only asymptotically up to 2.50 m, as constrained by regulations such as the European Union Council Directive 96/53/EC.
When no correlation is observed among pairs of variables, Υ(x) ≡ 0 and only the randomistic component (y ′ ) in Eq. 12 defines an UFD characteristic (y).This is the dominant situation in UFD-H, and incorporates further variability (uncertainty) into the stability curves.For y ′ , we consider three potential PDFs: U, T , or N .
A comprehensive review of the functional relationships for Υ(x) and the PDFs for y ′ across all UFD subcategories is provided in Appendix A. All relevant data and analysis codes are available in the Electronic Supplementary Material.

Monte Carlo analysis
A Monte Carlo analysis is performed to probabilistically estimate the stability curves for different UFD categories, as outlined in Table 1.This analysis is building upon the mechanistic model of Section 3 (Fig. 3) and the input parametrization defined across Section 4 and Appendix A. The procedure for the analysis is conducted for each UFD subcategory and follows the protocol here described (Fig. 4): 1. We generate 1,000 different samples from a given UFD subcategory (e.g., UFD-F2), based on structural and stochastic characteristics of each group derived in Section 4.2, thereby simulating a realistic range of physical properties.
2. Stability limit state equations -for flotation, sliding, and toppling-are solved for each synthetic UFD generated.
3. For each synthetic UFD, the minimum depth h required to initiate motion at a given velocity v is identified by considering all three instability modes and is designated as the value for the stability curve.
4. Results are visualized and data is stored by subcategory in the Electronic Supplementary Material.
This preprint is currently under revision 5 Results

Application and verification of the mechanistic framework
In this section, we utilize various case studies from literature to evaluate the accuracy and applicability of the proposed mechanistic model proposed in Section 3. We focus the verification study on cars (UFD-V2, Table 1), owing to the wealth of available research data.Figure 5 presents both the experimental data from literature and the predicted limit states of our mechanistic model (Section 3).(Xia et al., 2014;Kramer et al., 2016;Smith et al., 2019b).

Probabilistic assessment of stability curves
This preprint is currently under revision Table 2. Key characteristics of the UFD-V2 models used for verification.The fA used is that parameterized in Appendix A for UFD-V2, Eq.A3, with stochastic term f ′ A = 0. fV 1 corresponds to the submerged volume contribution of the wheels, assumed constant, and fV 2 corresponds to the above-clearance contribution.Values with (*) correspond to best fit (calibrated), which is only conducted for missing data.
Some parameters (M , L x , L y , L z , z c ), necessary for the application of the model (Eqs.4 to 11), are directly incorporated from the original case studies.In all instances, we consider the friction coefficient (µ) as 0.3, in line with reporting from laboratory experiments (see study of Smith et al., 2019b).
For the evaluation of f A at each depth, we use the data-driven polynomial approximation specified for UFD-V2 in Appendix A (Eq. A3, excluding the stochastic term f ′ A ). Unspecified parameters such as f V,2 and, occasionally, C D are calibrated iteratively.Given that C D impacts stability at high flow velocities, while f V,2 is influential at low velocities, these parameters can be calibrated independently.We recognize that the drag coefficient (C D ) varies among the considered case studies.This variation is reasonable and can be justified on geometrical differences in shapes of car models, given that Xia et al. (2014).
Inspection of Figure 5 reveals that the stability limit state curves generated by our model, in conjunction with physical properties, are physically consistent and describe the laboratory data in good detail.At near-zero velocities, flotation becomes the governing instability mode.For non-zero velocities, buoyancy lowers the threshold for sliding and toppling by reducing the effective normal force (R y − S y , see Equations 7 and 8).The stability curves exhibit a notable flattening as water levels drop below the clearance height (z c ), which corresponds to a diminishing effect of f A and f V .In all our verification cases, toppling proves to be a less likely failure mode than either sliding or flotation, occurring at relatively lower water depths.

Probabilistic assessment of stability curves
In this subsection, we extend the application of the mechanistic model to encompass all UFD subcategories, deploying the previously outlined Monte Carlo simulation approach (see Fig. 4) that follows the inherent variability of UFDs.For clarity,

Probabilistic assessment of stability curves
This preprint is currently under revision formula (Chow, 1959).This enables the estimation of pairs of h-v for a given slope and roughness coefficient.These highlighted regions within the plots intersect the stability curves, thereby identifying the conditions under which UFD mobility is likely to be initiated.For that, we consider a wide array of slopes and surface roughness, ranging from smooth cement to gravel bed, with corresponding Manning numbers n = 0.011 to 0.025.These are representative of reasonable surfaces and slopes found in urban setups.The highest value of the Manning coefficient (n = 0.025) is also representative of short grass.
Our probabilistic analysis of vehicles' stability reveals several critical insights (Figure 6): -Two-wheelers (UFD-V1) are generally less stable than other vehicle types.However, some V1 vehicles possess higher bulk density than water (due to small internal volumes, metal construction and no cabin).This results in a threshold velocity below which they remain unconditionally stable; i.e., sinking regardless of the flow depth.
-Cars (UFD-V2) exhibit stability conditions very similar to those seen in the verification cases, with the dispersion of the data aligning with the variability observed in the verification cases.This reaffirms that this type of UFD is well represented in the literature.
-Vans (UFD-V3), caravans and RVs (UFD-V4) demonstrate a behavior comparable to UFD-V2, although caravans exhibit a considerably narrower distribution of stability curves.This is because, with increasing size, bulk density tends to reduce for habitable vehicles.Therefore, both vans and caravans tend to be less stable than cars on flatter slopes when flotation becomes the dominant mode of failure.This early onset of mobility, coupled with their larger volumes, increases flood hazard due to a higher potential for infrastructure clogging and damage.Given that these larger vehicles are commonly observed in flood events (Bayón et al., 2023), mitigating flood hazards would require additional focus on UFD-V3 and UFD-V4.
-Heavy vehicles (UFD-V5), owing to their massive scale, exhibit substantially higher stability.However, this category exhibits a wider dispersion due to the considerable range of properties; for instance, L x ranges from 6 to 18 m.Our findings suggest that these vehicles could contribute to flood hazards under flood situations with depths exceeding 1 m, particularly when also exposed to high velocities (v > 2 m/s).UFD-V5, with masses one to two orders of magnitude greater than other UFD-Vs, may substantially amplify potential infrastructure damage due to impacts (Zhang et al., 2018;Jalayer et al., 2018), in addition to the also increased flood-infrastructure clogging hazard.

Probabilistic assessment of stability curves
This preprint is currently under revision When considering furniture (UFD-F), it is discernible that both UFD-F1 and UFD-F2 are highly movable.Urban fixtures (UFD-F1) become unstable at approximate depths of 25 cm, while household equipment (UFD-F2) exhibits instability at depths significantly below 10 cm.This signals an earlier onset of transport as compared to vehicles.Nonetheless, UFD-F2 are commonly installed within private properties, potentially attenuating the direct impact of flooding.
UFD-F, frequently visible during and post flood episodes (see Figure 1), constitutes approximately 7.4% of UFDs relative frequency (Bayón et al., 2023).This statistic, together with the disclosed early mobility, suggests that the role of furniture in urban floods might have been consistently underestimated.For heterogeneous debris and drifters (UFD-H), our analysis (Figure 8) indicates that stability characteristics are closely related to the material density.Plastic debris is the least stable, with the vast majority of items becoming mobilized at water depths of below a few centimeters.This is followed closely by other litter, which shows a very similar behaviour.Wood remains considerably more stable up to flood depths of 10 cm for low velocities (< 1 m/s).Velocities and depths observed are consistent with the wood stability study of Braudrick and Grant (2000).The most stable types within the UFD-H category are construction materials and metal, which require exceptionally large velocities for the inception of mobility --of at least 3 to 5 m/s--typically becoming dislodged only after impacts from other UFDs (such as trucks or cars) with infrastructure or when located near construction sites adjacent to streets.6 Discussion: sequential mobilization of UFDs Our study sheds light on the mobility of UFD, enabling inferences regarding their sequential mobilization during a flood event.
Figures 6, 7 and 8 presented the h − v pairs for which a given UFD is mobilised.Assuming a low slope configuration, we can extract and order which UFDs are mobilised.Assuming that a limited supply of UFDs is available, this allows sketching the expected UFD-graphs presented in Fig. 9. First to mobilize is typically plastic debris, which is highly mobile and remains so until supply exhaustion (Figure 9).This aligns with previous hydrological observations on plastic pollution (van Emmerik This preprint is currently under revision et al., 2019;Van Emmerik et al., 2019;van Emmerik et al., 2022;Bayón et al., 2023) and mechanistic studies (Goral et al., 2023a;Mellink et al., 2023), also including those focusing on early transport (Kuizenga et al., 2022;Valero et al., 2022;Lofty et al., 2023) that show mild settling velocities.

Supply may be exhausted soon
Supply may be linked to: i) natural availability ii) construction's flood-induced damage Next to mobilize are loose wooden pieces, more specifically, those that are buoyant.During the falling limb of the hydrograph, denser wood may settle sooner, while buoyant pieces may continue to travel farther, provided they are not obstructed, as long as the water depth remains large enough.This is also consistent with observations of Ruiz Villanueva et al. (2014) and Ruiz-Villanueva et al. (2016b).
The behavior of UFD-F during a flood is more nuanced.The stability curves obtained (Figure 7) suggest early mobilization, similar to smaller wood debris and other highly-mobile UFD-H.However, some UFD-F may remain anchored or weighted down (for example, by a water tank or a trash bin), as suggested by Martínez-Gomariz et al. (2020), thus introducing variability in their mobilization patterns, which might deviate from our predictions.Large, loose containers may float at shallow water depths, exhibiting behaviours similar to barges.Household equipment (UFD-F2), despite being highly mobile, may remain sheltered as they are frequently situated in backyards or indoor areas.

This preprint is currently under revision
As flood depths and velocities increase, vehicles begin to mobilize, initially with two-wheelers (UFD-V1) on steep slopes.In areas with gentle slopes, the velocity may not reach the required threshold for mobilization (Figure 6A).As the flood progresses towards the peak, two scenarios may unfold: in low-slope areas (low v, high h), vans (UFD-V3), caravans and RVs (UFD-V4) may destabilize more easily than cars, whereas the opposite scenario is expected for high-slope urban areas (Figure 6B-D,F).
As flow intensifies, larger vehicles like trucks (UFD-V5) may also eventually be mobilized, assuming water depths clearly exceed one meter.
Lastly, construction and metal debris are typically mobilized at high velocities and often occur after other UFDs have caused infrastructural damage (Jalayer et al., 2018;Zhang et al., 2018).Even when fully submerged, very high velocities are needed to transport these debris (Fig. 8A,B) and are likely to settle as soon as floodwaters begin to recede.

Conclusions
This research provides a comprehensive analysis of the mobility of a wide range of Urban Flood Drifters (UFDs), a significant but under-studied contributor to flood risk (Dewals et al., 2021;Mohr et al., 2023;Bayón et al., 2023).In contrast to prior work focused mainly on vehicle stability during floods, we offer a broader assessment of UFD stability.Our contributions are as follows: This preprint is currently under revision Appendix A: Characterization and uncertainties in typified UFDs' key properties This appendix explains in detail the parametrization of the variables that determine the stability of Urban Flood Drifters (UFDs), as outlined in Section 3.These variables are addressed one-by-one and by UFD subcategory (c.f.Table 1), thus precisely tailoring the parametrization (see Eq. 12) to the distinct variability and uncertainties inherent to each category.The parametrization of Eq. 12 admits two terms for each dependent variable (e.g., y(x)): the structural function (Υ(x)) and the stochastic component (y ′ ).For the structural dependency, two functions are considered (besides the null function): linear (Eq.13) and saturation (Eq.14).For the stochastic component, the probability density functions considered are: C (Constant, single value), U (Uniform), T (Triangular) and N (Normal); whose parameters are defined in this Appendix.
The parametrization process is systematically organized for each variable as follows: -Dimensions of the bounding box (L x , L y and L z ), in Section A1.
-Fraction of submerged volume (f V ), in Section A5.
A1 Dimensions of the bounding box (L x , L y and L z ) and their co-dependency The bounding box of an UFD is defined by three length scales: L x (largest horizontal length), L y (shortest horizontal length), and L z (height), as depicted in Figure 2A-C.
In this analysis, L x is selected as the independent random variable for UFD-Vs, with its variability serving as the foundation for deriving other physical characteristics.In other words, when running the Monte Carlo procedure (Fig. 4), the first step done to generate a synthetic UFD sample is to generate a L x value, which is governed by the primary descriptors presented in Table A1.
A1 Dimensions of the bounding box (L x , L y and L z ) and their co-dependency This preprint is currently under revision Table A1.Components of Eq. 12 for the description of the length Lx for UFD-V and UFD-F.Probabilistic distributions based on observations presented in Figure A1A,B.Min: minimum value of the variable (y) represented by the probabilistic distribution, Mode: mode (only for triangular), Max: maximum value.(*) This upper bound corresponds to the length of an articulated bus (see Table 5.3 of Vuchic, 2007).
Figure A1 illustrates the codependency of the geometrical properties of UFD-V (L x , L y , L z ).A clear correlation is noted between L y and L x (Fig. A1A), and between L z and L x (Fig. A1B).With an increase in vehicle length L x , all other dimensions tend to grow until a limit is reached at approximately 2.5 m for L y and around 4 m for L z .These limits may be attributed to road regulation dimensions and potential clearance constraints for bridges.Based on these observations, separate functional relationships (Eq.12) are produced for L y and L z for V1, V2 and V3-V5 (Table A2, and also presented in Figure A1A,B).
A1 Dimensions of the bounding box (L x , L y and L z ) and their co-dependency This preprint is currently under revision Table A2 provides the error-minimizing functional relationships (Eq.12) for L y and L z for each UFD-F, which are also illustrated in Fig. A2A,B.
Lastly, UFD-H (encompassing categories DC, DM, DP, DW, and DO in Table 1) are considered.As commercial catalogues with dimensions and characteristics are not available for UFD-H (mainly: DC, DM), an order of magnitude analysis (Table A3) is performed to establish their dimensions (L x , L y , and L z ), albeit with a larger degree of uncertainty than values considered for UFD-V and UFD-F.To be representative of the debris carried by floods, we have reanalyzed the over 16,000 litter elements inventoried by de Lange et al. ( 2023) (Fig. A3A,B).Based on that data, we suggest triangular probability density functions for L x and L y covering of 90% confidence intervals for min-max and median assumed as the mode of the T distribution, see Fig. A3A.L z of DP and DO is chosen as a fraction of L y , therefore assuming that the UFD rests in the most stable configuration.
For wood dimensions, values of Table A3 are inspired in the data review of Gurnell (2013),   2020)) strategy to depict potential structural relationships between key properties.

A2 Clearance height (z c )
This preprint is currently under revision For all UFD samples, once independent random values of L x , L y , and L z -within the predefined ranges-are generated, these are arranged such that L x > L y > L z to ensure the most stable configuration is considered in our stability analysis.

A2 Clearance height (z c )
The clearance height z c represents a critical geometric parameter as it sets the inundation height threshold where the submerged volume (Section A5) -and consequently flotation due to internal voids-of an UFD, significantly escalates.The clearance height of UFD-V (z c ) is directly determined from images (obtained from media and social media data available online, featuring side view shots for specific UFD-V models) for a selected set of vehicles (listed in the ESM), and show in Fig. A4.The z c value extends from the ground level up to the undercarriage of the UFD-V, with the following two exceptional cases considered: 1. UFD-V1 are presumed to be lying down (considering this as their most stable position).Thus, z c is assumed to be zero.
2. For UFD-V5, which include buses and trucks, we note that large voids within the vehicle start at heights greater than the lower chassis levels observed in images (see Fig. 5.3 top of Vuchic, 2007).In light of the review by Vuchic (2007), we propose a z c range for UFD-V5 from 0.32 m (low-floor buses) to 0.74 m (conventional buses).From images, we observe that the distribution between the minimum and maximum tends to appear uniform (linear in cumulative terms, see Fig. A4D), and we therefore extend this PDF to the min-max values extracted from Vuchic (2007).
In the case of UFD-V, the definition of the clearance height is straightforward.However, for urban and private furniture (UFD-F), such objects seldom have wheels, and even when they do, the diameters are just a few centimeters.Thus, for simplicity, we assign a clearance height of z c = 0 to both UFD-F1 and UFD-F2.For UFD-H (Table 1), no wheels are expected, and we embrace the same hypothesis.All the chosen values are presented in Table A4, with their justifications detailed below.
A3 Bulk density (ρ b ) or mass (M ) This preprint is currently under revision ).Note that for UFD-V5, the ground clearance might be visually hidden and we therefore adopt ranges from Vuchic (2007), as clarified in Table A4.
A3 Bulk density (ρ b ) or mass (M ) The bulk density of UFDs is a key parameter involved in the calculation of the balance of forces and moments to evaluate the stability of UFDs for the three modes defined in Fig. 3. Density across UFDs, nonetheless, varies largely.In order to estimate this bulk density, ρ b , we consider Eq. 1, for which the total UFD mass M is necessary, together with the bounding box volume defined by its dimensions of Section A1 (L x L y L z ).These variables, readily available in commercial repositories, enable the calculation of ρ b .
When inspecting the inventory of UFD-V (Figure A5), a consistent clustering of the bulk density can be observed for different types of vehicles.Larger variability is observed in UFD-V1, given the heterogeneity within this category (e.g., this includes bikes and e-scooters as well as grand-tour motorbikes).Fig A5 reveals that density is relatively homogeneous across groups UFD-V2 to UFD-V5, except for the UFD-V4, which includes caravans and recreational vehicles (RVs), where the trend is reversed and bulk density decreases.This fact makes these vehicles relatively more susceptible to flotation.In the case of UFD-V4, an increase of volume is associated to an increase on habitability by increasing the internal space; thus leading to a larger increase of volume (V ) without a significant increase of mass (M ).A3 Bulk density (ρ b ) or mass (M ) This preprint is currently under revision Functional relationships are proposed to predict the bulk density of each UFD-V type (UFD-V2 to UFD-V5, Table A5), based on the dimension L y .For UFD-V1, a min-max uniform distribution is assumed covering the range of densities observed.This is based on a decorrelation of the data for this specific UFD.The root mean square error (RMSE) between functional relationships' predictions (Table A5) and the data is minimized, and this RMSE is subsequently used to enforce a stochastic component (y ′ = ρ ′ b , c.f. Eq. 12) with equivalent standard deviation.The mass of UFD-F, however, is found to be better explained by the bounding box horizontal area, defined by the product L x • L y (Fig. A6).This correlation may be attributed to the stability considerations, as a larger footprint (bounding box base area) often corresponds with greater height (Fig. A6B).Within the furniture category, it is also clear that urban furniture (UFD-F1) is considerably denser than private fixtures (UFD-F2), as shown in Fig. A6.A3 Bulk density (ρ b ) or mass (M ) This preprint is currently under revision For heterogeneous UFDs, we combine the raw material density (ρ s ) with a plausible volume fraction to define the bulk density (ρ b = ρ s f V ), which then allows obtaining the mass (M ), as defined by Eq. 1. Plastics (DP) and others (DO) are an exception, since they can hold a very wide range of densities (van Emmerik and Schwarz, 2020), as well as a wide range of internal voids.To accurately represent the debris found in waterways, we have reanalyzed the over 16,000 litter elements inventoried by de Lange et al. ( 2023) (Fig. A3B).Based on statistical properties of that data, we define the parameters of a T distribution as follows: min-max based on 5 to 95% percentiles and mode assumed as the median of the data (Fig. A3B).This is also extended to DO, under the consideration that both groups represent litter.For wood, table A6 presents the parametrization of the PDFs describing the raw density (ρ s ) of each material and the f V (Table A10), later discussed in Section A5, which allow the estimation of the bulk density via The density referred to here is the oven-dry density (ρ d ), which represents the mass after water has been evaporated divided by the original (green) volume.The actual (green) density (ρ s ) can be calculated as follows: where W C is the water content, defined as a function of green mass (M ) and oven-dry mass (M d ) as follows: We take the dry density of Chave et al. (2009) and assume that the WC varies between 0.04 and 0.30, with a most common value (mode) at around 0.15 (WC ∼ T [0.04, 0.15, 0.30]).The result is presented in Fig. A7 together with a Gaussian fit for ρ d that is later used to estimate wood density in our probabilistic analysis.This wood density estimation (mean: 713.39 kg, std: 207.27 kg) lays in between densities estimated by Ruiz- Villanueva et al. (2016a) for green wood (mean: 800 kg, std: 170 kg) and instream wood (mean: 660 kg, std: 200 kg).

A4 Fraction of effective drag area (f A )
The effective area of an UFD that is exposed to the flow depends on the water depth h.With increasing water level, larger areas will become wet on the impinging face (upstream UFD face, A ′ in Fig. 2D).For convenience, we describe the frontal area of A4 Fraction of effective drag area (f A ) This preprint is currently under revision the UFD through the parameter f A (Eq. 3), which relates the effective drag area A ′ to the flow-exposed area of the bounding box; i.e., L x h.
The estimation of f A for different UFDs follows different strategies depending on their category.For typified UFDs, UFD-V and UFD-F, we implement the following image-based analysis procedure over selected items of the UFD-inventory (see ESM for completeness): 1. We collect images of vehicles and furniture from online commercial repositories.These images are selected only for side looking angles, corresponding to the x − z plane (flow-exposed area of the bounding box).
2. We binarize the images, removing any potential background (commonly white) and enhancing the UFD (bringing it to black).This is checked visually for adequacy.
3. We assume different elevation levels h at which the water level may reach (c.f.Fig. 2C,D).For a given h, we count the number of black pixels underneath, and divide over the total number of pixels (black and white together).This ratio provides a direct estimation of f A .This procedure facilitates the empirical determination of f A for various UFD-V and UFD-F types across an extensive range of potential water levels to which the UFD might be exposed.From the obtained data, a polynomial expression is fitted to facilitate the evaluation of f A at any depth during the application of the model of Section 3.This polynomial fit is obtained for each group within UFD-V and UFD-F, as defined in Table 1, and follows the form: where a i are error-minimizing coefficients (presented in Table A7).The resulting f A curves are presented in Fig. A8, together with the data retrieved from the image-based procedure proposed.

ID a
A5 Fraction of submerged volume (f V ) This preprint is currently under revision As shown in Figure A8, excluding UFD-V1 (bikes, motorbikes, and e-scooters), the f A curves for various UFD-V types remain remarkably consistent.
For UFD-H, f A values are assigned in a broad range to reflect considerable uncertainties.These values often start around 0.4 to 0.6, can take values considering specific shapes (circular areas, 0.78), and may go up to nearly filling the entire flow-exposed area of the bounding box (f A of approximately 0.95).This higher f A value is generally expected when the UFD-H shapes are more stylized, such as bars or bricks.The proposed f A values for each UFD-H class are presented in Table A8  A5 Fraction of submerged volume (f V ) The buoyancy force is proportional to the submerged volume (refer to Eq. 4).In our model, we represent internal volumes using the factor f V (as defined in Eq. 2).This factor represents the fraction of the bounding box volume submerged (L x L y h) contributing to buoyancy.
For f V , we introduce a two-step function to reflect changes in UFD-V's buoyancy with increasing depth: In this function, f V,1 accounts for the buoyancy contribution below the clearance height z c , which we assume to be small and constant for h < z c .The term f V,2 represents the sudden buoyancy increase when water level surpasses the under-chassis.
We select f V,1 to represent the volume occupied by wheels under the clearance (f V,1 ∼5%, as shown in Table A9).For f V,2 of cars (UFD-V2), we decide its value based on calibration of the mechanistic model on real-scale data (described in section 5.1).For other vehicles, we rely on simple geometrical assessments based on images, which may be regarded as approximate.
This attribute carries higher uncertainty compared to other physical quantities, warranting a uniform PDF adoption, which endorses large uncertainty across a give min-max range.The adopted values for f V,2 are introduced and briefly justified in Table A9.

A6 Friction coefficient (µ)
This preprint is currently under revision 1.2ψ,3ψ] *Assuming that all volume is due to aluminium (2,700 kg/m 3 ), the mass M is converted to a volume (V solid ), then: ψ = V solid /V .We then assume an extra 20% volume due to light components (plastic) and larger voids, which we assume that can go up to 3ψ.Table A9.Values of the void fraction coefficient fV 1 and fV 2 (Eq.A5).
In the case of UFD-F, we derive the value of f V,2 (as there's no clearance, hence no f V,1 ) by linking it to f A .We make the approximation f A (prismatic shaped).For UFD-H, we assume sensible values that span a wide range of plausible possibilities.This incorporates larger uncertainty but ensures that we cover realistic combinations.The following assumptions were incorporated when defining UFD-H distributions of f V : for wood, we typically assume cylindrical shapes (f V ≈ 0.78).
However, wood can sometimes take larger volume fractions without fully becoming prismatic (upper limit of f V < 0.95).Since wood can also break into anisotropic pieces, we consider a lower limit of f V > 0.50.These considerations justify a triangular distribution for f V for wood (T [0.5,0.78,0.95]),as presented in Table A10.

A6 Friction coefficient (µ)
The friction coefficient describes the relationship between normal and frictional forces in the UFD-floor contact (see Eq. 7).
This coefficient is varies according to the material and roughness properties of the two contact elements.Based on literature review, we define a minimum, mode, and maximum value for each material pair (characteristic of different UFD-floor types), always factoring in wet contact conditions.We list these values in Table A11.
This preprint is currently under revision  To model the friction coefficient (µ), we make the assumption that all UFD-V types are placed over asphalt/concrete (see Table A11).We represent this via a triangular distribution T [0.30, 0.50, 0.75].
For UFD-F, we consider a reasonably wide range within the material pairs, using a uniform distribution: U[0.30, 0.90].For UFD-H, we adopt a triangular distribution, grounded on defined ranges dependent on the pair of materials in contact (refer to Table A11).Consequently, the distributions adopted for UFD-H are described in Table A12.

A7 Drag coefficient (C D )
The drag coefficient directly scales the drag force exerted by the water upon the vehicle (Eq.6).Some drag coefficients are known for vehicles under flooding conditions (see for instance Smith et al., 2019b).In this investigation, the drag coefficients This preprint is currently under revision of UFD-V, UFD-F and UFD-H are assimilated from simpler geometrical shapes (see Table 7. 2-7.3 of White, 2016).A broad review of C D for arbitrary shapes is presented in Table A13, together with the justification on their assimilation to specific types of UFD.

A7 Drag coefficient (C D )
This preprint is currently under revision In our analysis, a certain additional variability is allowed for C D to account for a wide range of uncertainties beyond the shape assimilation.A triangular probability distribution is assumed for C D in all UFD-V and UFD-F, considering a minimum, a mode and a maximum value (Table A14).The same rationale is used to inform the selection of the probabilistic distributions for UFD-H.However, for plastic (UFD-DP) and other debris (UFD-DO), a uniform PDF is chosen thereby acknowledging a larger uncertainty in the shapes defining the drag.When a UFD could be assimilated to several geometric forms, we take the widest range comprehended by these several shapes' values of C D , hence transferring larger uncertainty to the resulting stability curves' predictions.Drag coefficients for UFD-H are presented in Table A15 Figure 2. Definition of the bounding box, with dimensions Lx, Ly and Lz, over an UFD of types (A) UFD-F1 (urban fixture), (B) UFD-V1 (two-wheeler), (C) UFD-F2 (household equipment) and (D) UFD-V5 (large heavy vehicle).Definition of the bounding box's total volume V (= LxLyLz) and submerged volume V ′ (= fV LxLyh), its bounding box flood-prone area A (= Lxh) and submerged area exposed to the flow A ′ (= fALxh), and its area fraction (fA) and volume fraction (fV ).

Figure 3 .
Figure 3. Definition of (A) stabilizing and destabilizing forces and moments, and (B) limit states for the three destabilization modes.

Figure 5 .
Figure 5. Stability curves obtained with the mechanistic model (Section 3) for specific car models tested in laboratory in literature(Xia et al.,

Figs
Figs. 6F, 7C and Fig 8F incorporate flow conditions based on uniform flow estimations, which are based on the Manning

Figure 7 .
Figure 7. Stability curves for furniture: A urban furniture (UFD-F1), B private furniture (UFD-F2).C Furniture stability curves intersection with uniform flow over different urban surfaces (from smooth cement to excavated gravel, Chow, 1959) at different slopes.

Figure A3 .
Figure A3.Main characteristics derived from the reanalysis of the litter data of de Lange et al. (2023).(A) Main length (Lx) and width (Ly) and (B) mass (M ) distribution.Linear fit both using a error-minimizing (as implemented in Harris et al., 2020) and robust (Theil-Sen estimator, as implemented by Virtanen et al. (2020)) strategy to depict potential structural relationships between key properties.

Figure A4 .
Figure A4.Cumulative density function of the clearance height (zc) in vehicles (V1-V5).Note that for UFD-V5, the ground clearance might
Based on the reanalysis of the data of Smith et al. (2019b), Kramer et al. (2016) and Xia et al.Large closed volumes, assumed water-tight. Exp.

Table A7 .
Coefficients for the fA fit (Eq.A3) obtained from image processing of pictures of side UFDs-V and UFDs-F with plain background. .

Table A11 .
Friction coefficients for different pairs of materials under wet contact conditions.Selection of UFDs potentially lying in each material-pairs configuration.

Table A12 .
Parametrization of friction factors for UFD-H.
(Johnson Jr et al., 1982)C D ) Comments Min and max based on pine and spruce(Johnson Jr et al., 1982)and mode based on cylinder with length to diameter ratio of ten.

Table A13 .
White (2016)ients proposed for different UFDs (vehicles, containers and litter).Coefficients based on simplified geometrical shapes are extracted fromWhite (2016).Flow impinging traverse to the vehicle.Bikes, motorbikes and scooters are considered laying on the floor.

Table A14 .
. Drag coefficients (CD) based on real data and simplified geometrical shapes assimilation, see TableA13.

Table A15 .
Parametrization of heterogeneous UFD characteristics (II).Plastics (DP) and others (DO) mass (M ) based on reanalysis of the data of de Lange et al. (2023).