A Conceptual Model Of People S Vulnerability To Floods

RESEARCH ARTICLE
10.1002/2014WR016172
A conceptual model of people’s vulnerability to floods
Luca Milanesi1, Marco Pilotti1, and Roberto Ranzi1
1
DICATAM, Universit
a degli Studi di Brescia, Brescia, Italy
Abstract Hydraulic risk maps provide the baseline for land use and emergency planning. Accordingly,
they should convey clear information on the potential physical implications of the different hazards to the
stakeholders. This paper presents a vulnerability criterion focused on human stability in a flow specifically
devised for rapidly evolving floods where life, before than economic values, might be threatened. The
human body is conceptualized as a set of cylinders and its stability to slipping and toppling is assessed by
forces and moments equilibrium. Moreover, a depth threshold to consider drowning is assumed. In order to
widen its scope of application, the model takes the destabilizing effect of local slope (so far disregarded in
the literature) and fluid density into account. The resulting vulnerability classification could be naturally sub-
divided in three levels (low, medium, and high) that are limited by two stability curves for children and
adults, respectively. In comparison with the most advanced literature conceptual approaches, the proposed
model is weakly parameterized and the computed thresholds fit better the available experimental data sets.
A code that implements the proposed algorithm is provided.
1. Introduction
The increasing density of population in flood prone areas and the intensification of the hydrological cycle
triggered by climate change are contributing to make floods the most devastating and costly hazard world-
wide. According to the European Environment Agency [2010], the overall losses for floods recorded in Europe
in the period 1998–2009 added up to about EUR 52 billion. Examples of the destructive power of the fast
propagation of a wave are provided by well-documented cases of dam break [e.g., Pilotti et al., 2011], as
well as by the recent occurrence of two dramatic tsunamis (26 December 2004, Indonesia; 11 March 2011,
Japan). Therefore, there is a growing need of taking effective flood protection measures.
As usual in the literature [e.g., Varnes, 1984], risk is a representation of the expected damage, that can be
defined as a combination of hazard, vulnerability, and exposure. Hazard represents the probability of a haz-
ardous event to verify in a place; vulnerability is a function of the potential damage to a target due to the
occurrence of the hazardous event; finally, exposure quantifies the presence of the target.
Flood hazard and risk maps are fundamental tools dictated, for the European countries, by the European
Flood directive 2007/60/CE. In the U.S., the National Flood Insurance Program (NFIP) was introduced in
1968, to financially protect property owners from the destructive violence of floods. NFIP requires flood
insurance on properties that are located in areas at high risk of flooding and suggests it as optional in
moderate-to-low risk areas. At the same time, communities must agree to enforce ordinances that meet
FEMA (Federal Emergency Management Agency) requirements to mitigate the risk of flooding. Accordingly,
it is clear that risk zoning has strong implications on the economic value of properties, it sets severe con-
straints for land use planning and must be based on the most updated and rational models and criteria.
This is true, a fortiori, when the primary aim is the reduction of the risk for human life. To this purpose, the
choice of proper vulnerability criteria, whose physical meaning is clearly understandable by stakeholders
and policy makers, is fundamental. Indeed, shared conceptual models enhance the participation of stake-
holders to decision-making processes [Buchecker et al., 2013; Giupponi et al., 2013], making protection meas-
ures more effective.
In a review of flood hazard mapping criteria across European countries, de Moel et al. [2009] documented
strong heterogeneities: there is a significant fragmentation on regulations and a fully satisfying rational
methodology is apparently still missing. In general, hazard maps can be associated to one or more return
periods and be computed on the basis of one or more flow variables, such as the extension of the flood or
Key Points:
An improved physically based model
of human stability in a flow is
proposed
Reduction of stability due to sloping
terrain and fluid density is accounted
The model best matches the
available literature experimental data
sets
Correspondence to:
L. Milanesi,
luca.milanesi@unibs.it
Citation:
Milanesi, L., M. Pilotti, and R. Ranzi
(2015), A conceptual model of people’s
vulnerability to floods, Water Resour.
Res., 51, doi:10.1002/2014WR016172.
Received 24 JUL 2014
Accepted 14 NOV 2014
Accepted article online 19 NOV 2014
MILANESI ET AL. V
C 2014. American Geophysical Union. All Rights Reserved. 1
Water Resources Research
PUBLICATIONS

the separate representation of maximum flow depth or velocity. Only a suitable combination of flow depth
and velocity through properly devised vulnerability curves represents the potential effects of the flow on a
target. As a result, flood risk maps account for flow effects on a target considering its exposure and the
frequency of the events. Limiting the attention to direct losses, the targets can be either economic values
or human lives [e.g., Thieken et al., 2005]. Although the economic values are dominant in the case of slowly
evolving floods, typical for larger watersheds, in the case of rapid processes (such as dam breaks, torrential
processes and levee breaches) [e.g., Pilotti et al., 2010, 2011; Mazzoleni et al., 2014], human safety is the pri-
mary objective to be safeguarded.
Human safety in a flow was experimentally studied at first by Foster and Cox [1973]. They tested the stability
of children with different height and mass combinations in a laboratory flume and found that human stabil-
ity is affected by a wide set of physical, emotional, and dynamic factors. They observed that failure was
mainly caused by slipping. Further tests by Abt et al. [1989] showed that the mechanism of toppling should
also be accounted. More recently, several experimental analyses were performed on real adults and children
[Takahashi et al., 1992; Keller and Mitsch, 1993; Karvonen et al., 2000; Yee, 2003; Russo et al., 2013] considering
different training, wearing, environmental conditions, and definitions of instability. These studies provide an
experimental basis for an inversely proportional linear relationship between mean flow velocity and depth
[e.g., Cox et al., 2010], that are often introduced as a reference in several national regulations on flood haz-
ard zoning. These empirical approximating functions [e.g., HR Wallingford et al., 2006] are however purely
regressive and do not allow to establish an effective link between hazard level and physical effect, such as
instability for children or instability for adults.
In order to overcome some of the limitations of experimental activities and to provide an interpretative
framework, in the last 2 decades some conceptual models were introduced to describe the human stability
as a function of flow velocity U and water depth h. These models are based on different assumptions
regarding the shape of the body, the involved forces, and the failure mechanisms. Love [1987] modeled a
rectangular monolith and recognized the role of the buoyancy force and of toppling instability. Lind et al.
[2004] tested both conceptual and empirical formulas finally calibrating a relation based on the concept of
the product number (Uh) on literature data. Walder et al. [2006], studying a tsunami induced by a debris
flow, developed a simplified approach to represent slipping, disregarding toppling instability and the role
of the buoyancy force. He identified depth thresholds for adult men, women, and children. A different
approach was suggested by Ishigaki et al. [2008] in order to identify the conditions for a safe evacuation
from underground spaces. Their research used a maximum admissible water depth to study the prob-
lem of doors opening and the specific force per unit width to model the movement on stairs and in
corridors. More advanced approaches were pursued by Jonkman and Penning-Rowsell [2008] and,
recently, by Xia et al. [2014], who coupled experimental and theoretical analyses. Jonkman and Penning-
Rowsell [2008] tested an adult stuntman in real channel in conditions of low depth and significant
velocity. Moreover, they calibrated a simplified model for adults accounting for both slipping and top-
pling but neglecting the buoyancy force. The resulting equations were calibrated on the basis of Abt
et al. [1989] and Karvonen et al. [2000] data sets and fit well the experimental points by Jonkman and
Penning-Rowsell [2008], that actually are related to low depths where buoyancy force plays a minor role
on slipping stability. Recently, Xia et al. [2014] conducted experiments on a model of the human body
and developed a strongly parametric scheme, introducing buoyancy force and considering both top-
pling and slipping failure mechanisms. They extensively calibrated the model with literature experimen-
tal data, disregarding the slipping failure mechanism on the ground that it is limited to the ‘‘rare
occurrence of low depth and high velocity.’’
In addition to the mentioned simplifications, it is important to observe that these researches disregarded
the role of slope on human stability, so limiting the validity of their results to floodplains. Only Abt et al.
[1989] and Jonkman and Penning-Rowsell [2008] considered values of slope in the very limited range below
1.5%. However, slope should not be neglected because it affects instability by modifying the relative direc-
tion between the weight and the impacting dynamic forces. This conclusion is supported by the results of
Russo et al. [2013], who conducted a systematic experimental analysis of the effect of terrain inclination on
adults stability. Unfortunately, no quantitative data on slope were reported in their paper.
Another aspect that has seldom been discussed in the literature is the role of fluid density. For instance,
floods in some rivers (e.g., the middle and lower reaches of the Yellow River) are characterized by
Water Resources Research 10.1002/2014WR016172
MILANESI ET AL. V

hyperconcentrated flow with
sediment concentrations even
higher than 400 kg m23
[Pier-
son, 2005; Wan and Wang,
1994; van Maren et al., 2009].
Similar conditions are present
in areas characterized by
unconsolidated volcanic ash
deposits. Similarly, in mountain
areas, mudflows and debris
flows are characterized by high
equivalent density. Risk map-
ping in these areas should
address these peculiarities.
The present paper aims to con-
tribute to this debate, provid-
ing a simple but
comprehensive conceptual
model of human stability
through the description of the
involved forces, considering
slipping, toppling, and drown-
ing related to high water levels.
The model is based on a syn-
thesis of the geometrical con-
ceptualization of the human
body and of the involved forces, so far presented in the literature only for horizontal surfaces. It introduces
some novelties, by explicitly considering local slope and fluid density, widening the scope of application of
the model to both floodplain and mountain environments, in situations where floods could threaten human
life. Differently from most of the literature studies, only one parameter of the model was calibrated within a
narrow experimental range and the others were identified on the basis of literature data. The physical basis
of the model allows to identify two stability thresholds, derived respectively for children and adults. These
two thresholds could be used to naturally separate low, medium, and high vulnerability levels. In spite of its
simplicity, generality, and parsimonious parameterization, the model matches the literature data better
than the other conceptual schemes available in the literature. In particular, the performance of this model is
compared with the results from Xia et al. [2014] that represent the state of the art in this field.
2. The Conceptual Model
The assessment of the actions exerted by the flow on a human body is a complex problem that should
account for the relative motion and the position of the body, its exact shape modified by clothes, the pres-
ence of supplemental loads (e.g., backpacks), and the actual sole-soil interaction. Moreover, psychological
factors and environmental conditions (e.g., lighting, presence of debris, water temperature, obstacles)
should be considered. Clearly, these details cannot be modeled and only a conceptualized representation
of the process, that encompasses the most relevant aspects, is proposed.
2.1. Simplified Representation of the Human Body
In order to analytically describe the interaction between the flow field and the human body, a conceptual
scheme representing its submerged fraction as a function of the water depth h (m), measured orthogonally
to the bed, is needed. The frontal area, the volume, and the center of mass of the submerged fraction of
the body are required to compute forces and to define equilibrium conditions.
As originally suggested by Lind et al. [2004], the human body, of mass m (kg) and height Y (m), is conceptual-
ized by a set of cylinders standing in vertical position on a slope that is inclined at an angle # with respect to
the horizontal direction (see Figure 1a). A person might assume different postures (e.g., aligned with the flow
Figure 1. Simplified representation of the human body in (a) lateral and (b) frontal views.
MILANESI ET AL. V

and with splayed legs) depending on its experience and capability to cope with the current. In the following,
the body will be considered in vertical position, inclined at an angle a 5 p/2 - # with respect to the terrain
and impacted frontally by the flow. This position provides the most cautionary stability thresholds because it
minimizes the postural adaptation to the flow and maximizes the destabilizing actions.
The legs are described by two paired cylinders of diameter d (m), spaced d/2, while the torso is sche-
matized by a single cylinder of diameter D 5 2d (Figure 1b). Both the legs and the torso have height
Y/2. The body’s center of gravity is identified by its coordinates (xG, yG). On the basis of the idealized
geometry and assuming a uniform weight distribution, xG is located at 5/6 d from the heel and yG is
located at 7/12 Y on the vertical from the ground. These results are supported by medical literature on
adults and children [e.g., Winter, 1995; Santschi et al., 1963; Hasan et al., 1996; Palmer, 1944]. Accord-
ingly, the projection of the center of mass of the body falls within the footprint for every value of the
local slope. The submerged volume of the body Vs (m3
) and its wetted frontal area As (m2
) can be
described as:
Vs5
2
h
cos #
p
d2
4
if
h
cos #

Y
2
2
Y
2
p
d2
4
1
h
cos #
2
Y
2

p
D2
4
if
h
cos #

Y
2
8

:
(1)
As5
2
h
cos #
d if
h
cos #

Y
2
2
Y
2
d1
h
cos #
2
Y
2

D if
h
cos #

Y
2
8

:
(2)
The center of gravity of the submerged volume has coordinates (xGs, yGs) and is computed on the basis of
the geometry of the body as:
xGs5
d
2
if
h
cos #

Y
2
2a
Y
2
d
2
1A
h
cos #
2
Y
2

D
2
2a
Y
2
1A
h
cos #
2
Y
2
if
h
cos #

Y
2
8

:
(3)
yGs5
h
2cos #
if
h
cos #

Y
2
2a
Y2
8
1A
h
cos #
2
Y
2

1
2
h
cos #
2
Y
2

1
Y
2

2a
Y
2
1A
h
cos #
2
Y
2
if
h
cos #

Y
2
8

:
(4)
where A 5 pD2
/4 and a 5 pd2
/4 are, respectively, the cross-sectional areas of the torso and of each leg.
2.2. The Flow Field and the Acting Forces
The following considerations are based on the simplifying assumption that the body is standing in a
steady uniform flow where the pressure distribution along the orthogonal direction to the sloping bed
is linear under the action of the gravity component gcos#, being g (m s22
) the acceleration due to
gravity [Chow, 1959]. The flow is characterized by density q (kg m23
) and average velocity U (m s21
).
The forces acting on the body (see Figure 2) are its weight W (N), the fluid dynamic force R (N), the
buoyancy force BN (N), and the friction T (N) between the soles and the ground. The static head differ-
ential due to the partial conversion of kinetic head into potential energy that occurs in front of the
body is ignored, as suggested by Love [1987].
The weight of the body is defined as the product of its mass m and the gravity acceleration g. Considering
the slope of the bed, the weight is geometrically decomposed into two components normal (WN) and paral-
lel (WP) to the slope:
MILANESI ET AL. V

WN5W cos# (5)
WP5W sin# (6)
The weight is applied in the center of mass of the body, whose coordinates (gG, nG), in a frame of reference
normal to the slope (Figures 1a and 2a), are defined as:
gG5
xG
cos #
1yG sin # (7)
nG5yG cos # (8)
According to the hypothesis that the pressure distribution is that of a parallel flow, the buoyancy force BN
on the submerged body acts in the direction normal to the bed [e.g., Christensen, 1995; Armanini and Gre-
goretti, 2005], and it can be expressed as a function of the submerged volume from equation (1) as:
BN5qgVs cos # (9)
As customary in other engineering applications (e.g., the settling velocity of sediments in a fluid), here the
buoyancy force takes into account only the ambient pressure field (i.e., in the absence of the body), because
the shear stress field and the deviation of the pressure field due to the presence of the body is specifically
accounted in the fluid dynamic forces. BN is applied in the center of gravity of the submerged volume (Fig-
ure 2c) that can be defined as:
gGs5
xGs
cos #
1yGssin # (10)
nGs5yGscos # (11)
The fluid dynamic force R is the integral of the dynamic stresses exerted on the surface of the body when the
fluid moves past it. In this configuration, the human body stands in vertical position on the slope so that the flow
field is not perpendicular to the axis of the body. Accordingly, due to the asymmetry of the flow field around the
body, one can expect that the direction of the dynamic force is not parallel to the approaching flow velocity. As
shown by Hoerner [1965, 1985], in case of long cylindrical elements inclined against the flow with acute angle of
attack a, the skin friction is almost negligible, so that the resulting force R (N) is normal to the body frontal area
(Figure 2b) and it can be written as a function of the horizontal component u5 U sina of the mean velocity:
R5
1
2
qCcu2
As5
1
2
qCcU2
sin2
a As (12)
In turn, the dynamic force can be decomposed in directions parallel and orthogonal to the slope (Figure
2b), giving respectively the drag D (N) and the lift L (N):
D5R sin a5
1
2
qCcsin 3
aU2
As (13)
Figure 2. Acting forces and their application points.
MILANESI ET AL. V

L5R cos a5
1
2
qCcsin2
acosaU2
As (14)
where CC (-) is the drag coefficient for a circular cylinder measured for uniform flow profile orthogonal to
the body frontal area. Under the simplifying hypothesis that the dynamic pressures are uniformly distrib-
uted on the body frontal area, the drag and the lift forces are applied at distance nL,D from the ground and
gL,D from the heel:
gL;D5
xGs
cos #
1
h
2
tan # (15)
nL;D5
h
2
(16)
The friction force T between the soles of the human body and the ground (Figure 2c) can be described fol-
lowing a Mohr-Coulomb approach as the product of the friction coefficient l (-) and the effective weight w
(N), that is the algebraic sum of the forces normal to the slope:
T5lw (17)
w5WN2BN2L (18)
2.3. Failure Mechanisms
The following considerations are based on the reasonable assumption that human safety in a flow is
assured when there is static stability without risk of drowning. In turn, stability is controlled by two failure
mechanisms: slipping and toppling. By using a simplified conceptual scheme, Love [1987] demonstrated
that slipping limits stability in the range of high velocities and low depths, whilst toppling dominates in the
range of high water depths and low velocities. Slipping and toppling can be theoretically modeled by equi-
librium of forces and of moments, respectively. The resulting stability thresholds cannot account for the
adaptive capacity of a person in a flow: however, whilst the stability against toppling can be increased
through postural changes (especially by trained people), slipping cannot be significantly counteracted,
because the spatial distribution of the body mass does not significantly affect frictional resistance.
Failure due to slipping depends on the equilibrium of the forces in the direction parallel to the slope. Stabil-
ity is satisfied if friction T overcomes the destabilizing role of drag force D and of the projection of weight
parallel to the slope WP:
D1WP T (19)
Toppling instability occurs when the moment calculated with respect to the pivot point P (see Figure 2)
of the normal component of the weight is exceeded by the destabilizing moments due to the hydrody-
namic forces (drag and lift), the buoyancy force, and the component of the weight WP. Since the friction
force is applied at the interface between the soles and the ground, it does not contribute to toppling.
The pivot point P is always located at the downstream edge of the sole. Accordingly, the orientation of
the person with respect to the flow (i.e., facing upstream or downstream) affects the arm of the
involved forces with respect to P. Although in the tests by Xia et al. [2014], the stability of the model
seems independent from its orientation, the static scheme proposed in the present study suggests
changes in toppling stability since the arm of the forces normal to the terrain would be different in the
two configurations. In particular, a scheme considering the body facing upstream is more cautionary
since the pivot point P in this case is represented by the heel and the arm gG of the stabilizing weight
component is minimized (Figure 2a). The equilibrium condition with respect to toppling can be
expressed as:
DnL;D1WPnG1BNgGs1LgL;D WNgG (20)
Finally, a third condition, independent from the flow velocity, should be introduced in order to account
for the risk of drowning. A similar constraint was suggested by Cox et al. [2010] who proposed a maxi-
mum admissible water depth of 1.2 for adults and 0.5 m for children. The present model introduces a
maximum admissible water depth hd (m) that might be assumed as a function of the height of the neck.
MILANESI ET AL. V

In conclusion, the limiting safety depth, as a function of the flow velocity U, is provided by the minimum
among the slipping (hs), toppling (ht), and drowning depths (hd) :
h5min hs U
ð Þ; ht U
ð Þ; hd
½ (21)
3. Identification of the Parameters of the Model
The proposed model needs several parameters to identify the geometry of the body and the dynamic actions.
However, the quantities regarding human body dimensions and the drag coefficient can be obtained from
the literature. Only the friction coefficient was calibrated within a narrow experimental range.
An advantage of the parameterization with respect to the human body is that it naturally allows to adapt
the method to different geographic areas, where people may have different average mass and height.
Accordingly, one can obtain different thresholds for persons with different features and obtain a subdivi-
sion of vulnerability in two or more classes. In the following, considering a three-levels vulnerability classifi-
cation, the lowest stability threshold might be calibrated on a 7 years old child (since it might be assumed
that, at this age, children can be occasionally involved in a flood without the support of adults) and the
upper stability threshold will be referred to a mean sized adult. The parameters summarized in Table 1 rep-
resents the size of children, from worldwide statistics [Fryar et al., 2012; de Onis et al., 2007], and of Euro-
pean adults, whose average mass (71 kg) represents a median value across the World distribution ranging
from 58 kg (Asia) to 81 kg (Northern America) [Garcia and Quintana-Domeque, 2007; Walpole et al., 2012].
The drowning depth limits hd can be set as a function of the size of the body, so that the head is completely
above the water surface. In particular, the head can be considered as 1/8 of the entire body height [e.g.,
Drillis et al., 1964]; in order to keep a freeboard, the head height was assumed 3/16 Y. Hence, on the basis of
the data of Table 1, the maximum admissible water depth can be set at approximately 1.4 and 1 m, respec-
tively, for adults and children.
According to the geometrical schematization of the human body assumed in section 2.1, its drag coefficient
CC is that of a circular cylinder. In the expected range of Reynolds number (Re), Blevins [1984] suggested val-
ues of CC ranging from 1 to 1.2 that can be further reduced due to the effect of surface roughness. Blevins
[1984] also documents the effect of two paired vertical cylinders, that might represent the legs of a person,
as a function of their relative distance. Considering a distance between the legs ranging from d/2 to d, Cc
might range from 0.9 to 1.1. Within their conceptual researches on human stability in a flow, Lind et al.
[2004] and Jonkman and Penning-Rowsell [2008] introduced drag coefficients respectively equal to 1.2 and
1.1. Accordingly, in order to take into account the roughness that would characterize a dressed human
body, a value Cc 5 1 was adopted, independently from Re.
The friction coefficient l has a wider range of variation in the literature. Love [1987] modeled the friction
force for a prismatic monolith assuming 0.3 l 0.7. Takahashi et al. [1992] estimated friction coefficients
in correspondence of different combinations of soles and ground surfaces by measuring the force exerted
by the flow on three persons equipped with load cells. The results showed that in case of a wet floor, the
coefficient may range from 0.38 to 1.49. Accordingly, Takahashi et al. [1992] assumed the value of 0.4 to
model concrete surfaces covered by algae or seaweed and 0.6 to represent all the other surfaces. Keller and
Mitsch [1993] based their analysis on the cautionary value l5 0.3 and Jonkman and Penning-Rowsell [2008]
assumed l50.5. Accordingly, the friction coefficient l was calibrated within this range by using the least
squares method with respect to the data sets of Foster and Cox [1973], Abt et al. [1989], Takahashi et al.
[1992], Karvonen et al. [2000], Yee [2003], and Jonkman and Penning-Rowsell [2008]. The significant variability
of the velocity values for each assigned water depth in the data set
by Abt et al. [1989] may be partly justified by postural adjustments of
the tested people. Accordingly, the original data set was reclassified
by computing the mean velocity and depth values within depth
ranges of width 0.1 m. Depending on the data set, the model was
applied by using the parameters for adults or children and account-
ing for the slope associated to each experimental test. The friction
coefficient l 5 0.46 was estimated.
Table 1. Physical Parameters Assumed
for the Model of Children and Adults
Parameter Children Adults
m (kg) 22.4 71
Y (m) 1.21 1.71
D (m) 0.17 0.26
d (m) 0.085 0.13
MILANESI ET AL. V

Figure 3 shows the thresholds for slipping (a), toppling (b), and drowning (c) on the U-h plane, in the
case of clear water and horizontal terrain, for the prototypal child (thick line) and adult (thin line) of
Table 1. These thresholds are monotonically decreasing, so that the stability condition expressed by
equations (19) and (20) confirms the general trend presented in the literature. The maximum admissible
velocity tends to 0 when h 5 Y both for the slipping and the toppling thresholds, since the weight is
almost completely counterbalanced by the buoyancy force. The three stability thresholds curves are
combined by equation (21) to yield the curves in Figure 3d, representing the overall adults’ and child-
ren’s stability in a flow.
The discontinuities of the first derivative of the curves in Figure 3d allow to identify the velocity
ranges associated to each instability mechanism. For instance, regarding the children curve, the A
range is controlled by drowning, the B range by toppling instability and the C range by slipping
instability. There is a decreasing influence of depth and an increasing role of velocity passing from
A to C. The stability thresholds for children and for adults could be used to separate low, medium
and high vulnerability regions, providing a three class map. Moreover, the transition from the child-
ren’s to the adults’ curve might be regarded as a sensitivity analysis of the model with respect to
the adopted body’s size parameters (m, d, D, and Y), whose values are strictly connected together.
Figure 3e shows the resulting curves and vulnerability levels overlapped to the experimental points
from literature carried out on adults and children, including the original data set by Abt et al.
[1989]. The curves in Figure 3 are computed for a horizontal terrain and considering a fluid density
of 1000 kg m23
. Figure 4 shows a sensitivity analysis accomplished for an adult with respect to the
friction coefficient l ranging from 0.3 to 0.6. This parameter controls the slipping threshold, whose
intersection with the toppling threshold controls the extension of the B and C ranges.
Figure 3. Stability thresholds for adults (thin line) and children (thick line): (a) slipping, (b) toppling, and (c) drowning. Ya and Yc
represent, respectively, the height of the adults and of the children from Table 1. Figure 3d is their combination according to equa-
tion (21). (e) The experimental data from the literature and the thresholds by Xia et al. [2014] for both adults and children (dashed
lines). High, medium, and low vulnerability areas are, respectively, identified by red, orange, and yellow colors. Here q 5 1000 kg m23
and # 5 0
.
MILANESI ET AL. V

4. Comparison With
Experimental Data
In order to assess the reliability
of the proposed model, the
resulting stability thresholds
were compared to the avail-
able experimental data sets,
obtained from tests on the sta-
bility of adults and children in
clear water flows mostly in hor-
izontal condition. The data sets
provided by Foster and Cox
[1973], Abt et al. [1989], Taka-
hashi et al. [1992], Karvonen
et al. [2000], Yee [2003], and
Jonkman and Penning-Rowsell
[2008] were used. Unfortu-
nately, it was not possible to
accomplish a traditional validation using different data sets for calibration and validation because of the
scarcity of experimental data (33 points for children and 118 for adults). This is true especially for the chil-
dren stability region and in the field of low depth and high velocity. Figure 5 shows, in correspondence to
the depth associated to each experimental point, the correlation between the threshold experimental
velocity UE and the calculated velocity UT provided by the conceptual model. In order to present a
Figure 4. Sensitivity analysis of the stability curve to the variation of the friction coefficient
l in case of adult person, clear water (q 5 1000 kg m23
), and horizontal terrain (# 5 0
).
Figure 5. Correlation between the model results and the experimental literature data.
MILANESI ET AL. V

quantitative assessment of the fit between the model and the n experimental data, the relative root mean
square error (rRMSE) was calculated:
rRMSE5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
n
X
n
i51
UTi
2UEi
UEi
2
v
u
u
t (22)
The model partly underestimates the experimental values by Foster and Cox [1973] who, however, tested
heavier and taller children with respect to the prototype child adopted in the current study. In their experi-
ments only a single type of sole was tested and the children wore swimsuits, which minimized the interac-
tion between clothes and flow. For this comparison, only the instability points were selected.
In spite of the rearrangement of the data set by Abt et al. [1989], the rRMSE value (0.323) highlights a diffi-
culty in reproducing these data, as observed also by Jonkman and Penning-Rowsell [2008] and Xia et al.
[2014]. A possible explanation of this underestimation is that the experimental data set accounts for pos-
tural adaptation of the trained adults involved in the experiments that is not reproduced in conceptual
schemes. According to Abt et al. [1989], the data set might be also affected by several bias such as the pres-
ence of safety equipment and the training acquired by each subject during each tests session. This data set
is mostly representative of toppling, as shown by the typical depth values, ranging between 0.6 and 1.2 m
(range B in Figure 3d) and by the irrelevance of ground materials on the results.
The study by Takahashi et al. [1992] tested adult people on a horizontal terrain in different configurations of
flow, ground, and shoes. The comparison with the conceptual model was accomplished using the data
regarding people facing the approaching flow and with a combination of ground and soles with friction
coefficient l close to 0.4. The model presented in the current study tends to overestimate these experimen-
tal points, probably because of the heavier prototype adult with respect to the tested ones.
In the study by Karvonen et al. [2000], adults were made familiar with the facilities and were asked to move
on a horizontal platform; instability was registered as soon as manoeuvrability was lost. Their data set con-
siders a wide range of depths and velocities so that it is representative of both the slipping and the toppling
domains.
The experimental points by Yee [2003], regarding children impacted by a current on a horizontal terrain,
agree well with the presented conceptual model (rRMSE 5 0.243) with a maximum error lower than 5% for
almost 50% of the experimental values. A partial overestimation of the data set could be explained consid-
ering that the experimental instability criterion was based on both perceived and effective limits of stability.
The tests performed by Jonkman and Penning-Rowsell [2008] in a real-scale channel with constant slope of
1% involved a trained adult stuntman that was asked to walk in the flow without safety harnesses. Since
the water depth was in the order of 0.3 m and the velocities greater than 2.4 m s21
, the data set might be
considered a representative benchmark for slipping instability conditions. Since the size of the person is
similar to the one assumed in the current model, the excellent match with this experimental data set con-
firms the reliability of the model.
A final consideration regards the comparison with two recent data sets. The data set by Xia et al.
[2014], obtained with a physical model of a man in a horizontal flume, provides a lower bound between
all the available experimental points. The measurements accomplished in their physical model cannot
account for the capability of weight redistribution that would characterize the feet of a real person, con-
tributing to increase its stability. Observing this discrepancy, Xia et al. [2014] calibrated two separate
sets of thresholds (both for adults and children), using literature data and their own experimental data
respectively. The fitting between their experimental data and their model calibrated on literature data
provides rRMSE 5 0.777 while the present calibrated model performs slightly better with rRMSE 5 0.637.
Finally, the study by Russo et al. [2013] tested adults stability in a urban environment, where floods may
be characterized by shallow flows and high velocities, mainly representative of the slipping-controlled
area. Russo et al. [2013] investigated slopes up to 10% for which they did not provide quantitative data.
Accordingly, their data set cannot be used for a quantitative comparison, although one may observe that
their points systematically lie under the stability threshold for children at 0% slope (Figure 3e) so that the
relevance of slope in reducing stability is confirmed.
MILANESI ET AL. V

The same assessments can be accomplished separately on the experimental data sets of adults and of chil-
dren. The resulting clusters represent heterogeneous samples and can be considered representative of a
wide population. Accordingly, these data are an effective benchmark to test the reliability of conceptual
models. The rRMSE values obtained by applying both the model presented in this paper and the scheme by
Xia et al. [2014] to the literature data can be compared. As shown in Figures 6a and 6b, rRMSE 5 0.219
(0.317) for adults and rRMSE 5 0.208 (0.378) for children, where the values between brackets make reference
to Xia et al. [2014] model. The rRMSE of the data set comprising all the experimental data of the cited stud-
ies is 0.216 (0.338) (Figure 6c). A similar comparison accomplished with the model by Jonkman and Penning-
Rowsell [2008] and limited to the clustered data set of adults, provides rRMSE 5 0.357. The difficulty in repro-
ducing literature data confirms the importance of an accurate assessment of the forces governing stability,
including the buoyancy force. Accordingly, the proposed model systematically provides a better perform-
ance in the identification of stability limits of children and adults.
5. Discussion
A flood vulnerability criterion should satisfy some fundamental requirements. First, it should be physically
based, representing simple processes which can be understood by stakeholders. Second, it should be easy
to use and provide graded vulnerability thresholds, with the possibility of adaptation to different environ-
ments (from floodplain to mountain areas), where floods have different behavior. Finally, it should reason-
ably fit experimental data with a limited calibration. Unfortunately, most criteria do not match these
requirements [e.g., Ranzi et al., 2011]. Several national regulations (e.g., Austrian, Japanese, and Swiss) intro-
duce intensity thresholds that should better be regarded as vulnerability curves of one or more targets.
Accordingly, these thresholds can be compared to the model presented in this paper. The Austrian regula-
tion [Faber, 2006] classifies hazard levels associated to torrential floods through the flow-specific energy.
High hazard areas are characterized by a total head greater than 1.5 m. As one can observe, the total head
might represent the static head of the flow at a stagnation point. Although according to Kreibich et al.
[2009] this parameter is suitable to predict structural damages on a residential building, its physical mean-
ing in terms of vulnerability is not clearly understandable by stakeholders. Moreover, although applied
mostly in mountain regions, this criterion does not account for local slope and fluid density. According to
the same regulation, in case of fluvial floods, a linear relation between water depth and velocity is estab-
lished, with a maximum admissible velocity of 2 m s21
. The Swiss regulation [BWW, BRP, BUWAL, 2008] for
flood hazard mapping suggests intensity thresholds based on both the maximum water depth and on the
product number (Uh), presumably derived from literature studies on human or vehicles stability. In case of
debris flows, intensity levels are calculated using an empirical combination of the thickness of deposits and
the maximum flow velocity. On the contrary, physically based hazard maps are provided by the Japanese
criterion for debris flows that moves from the comparison of the flow dynamic pressure with the resistance
of typical wooden buildings [Mizuno and Terada, 2004].
Figure 6. Comparison between the model results and the clustered experimental literature data for (a) adults, (b) children, and (c) including all the data.
MILANESI ET AL. V

The proposed model provides a physically based and quantitative description of the vulnerability related to
slipping, toppling, and drowning of a human body in the flow field, fulfilling all the mentioned require-
ments. Its parameterization is essential, flexible, and strictly related to the adopted body schematization: by
varying few geometrical parameters, the stability of an adult rather than of a child can be described. To the
authors’ knowledge, this is the only model where local slope is explicitly taken into account within the equi-
librium conditions. Its completeness widens its applicability to different morphological situations and it
allows a graded classification of hydraulic risk. Other models presented in the literature are not equally com-
plete and simple. For instance, the recent model by Xia et al. [2014] meets some of the mentioned require-
ments but it does not take into account the effect of local slope and needs a substantial calibration.
Moreover, their suggested thresholds disregard sliding, a severe limitation in the range of low depths and
high velocities, particularly when the local slope is not negligible.
Usually, the reduction of stability due to the slope of the terrain is indirectly considered through velocity,
which is a function of local slope. However, another important and independent effect of slope is the
decomposition of the body weight in two contributions, with an overall stability reduction. The effect of
local slope is evident in Figure 7a that shows the stability thresholds for an adult impacted by a water flow
(q 5 1000 kg m23
) on a slope with inclination s ranging from 0% to 30%. Instability due to toppling and slip-
ping becomes increasingly more restrictive and the drowning depth limit disappears for s 6%. Eventually,
for slope higher than about 20%, slipping (C range of Figure 3d) dominates the entire domain. This result is
in contrast with the conclusions by Xia et al. [2014] who postulated the ‘‘rare occurrence’’ of the dynamic
conditions leading to slipping instability. Several experimental studies clearly show the role of slipping in
triggering instability in shallow flows [e.g., Jonkman and Penning-Rowsell, 2008]. This is also clearly docu-
mented in real events: for instance, Figure 8 shows a sequence of screenshots that documents a case of
instability at McTavish street in Montreal (CA). The progression of images clearly shows the loss of balance
due to slipping and the following dragging exerted by the flow. The structure in the background allows to
estimate a local slope of about 15%, whilst the kinetic head of the jet against the woman’s body allows to
evaluate a local water velocity between 3 and 4 m s21
with a water depth of about 0.1 m. This set of hydro-
dynamic parameters identifies a limiting condition since the woman can resist for a short period in standing
position. The corresponding experimental point is shown on Figure 7a, demonstrating that the proposed
model fits the documented instability. In particular one can observe that the point is well below the 0%
curve, so that neglecting slope would imply an unsafe risk zoning [e.g., Milanesi et al., 2014].
Another advantage of the presented model is that the stability curves are directly influenced by the density of
the fluid, affecting both the buoyancy force and the flow dynamic action. The additional drag force caused by
the increased sediment concentration and experienced by a person standing in a hyperconcentrated flow is
quite remarkable (T. Pierson, personal communication, 2014). Figure 7b shows the stability curves for an adult
Figure 7. The effect of (a) slope and (b) density on the stability threshold derived for an adult person. The effect of slope (Figure 7a) is studied for clear water (q 5 1000 kg m23
) while the
role of density (Figure 7b) is tested for the case of horizontal terrain. The labels indicates the slope value s (%) and q (kg m23
) associated to the curves. The intermediate curves are defined
for values of s spaced of 6% and q spaced of 200 kg m23
. The dot in the lower right corner in Figure 7a is representative for the situation of Figure 8. The synergic effect of density and slope
with characteristic values of hyperconcentrated flows in alpine areas (in this case, q 5 1400 kg m23
, that corresponds to a mass concentration of 24%) can be observed in (c).
MILANESI ET AL. V

impacted by currents of increasing density (from 1000 to 1600 kg m23
) on a horizontal terrain. The drowning
dominated area disappears for q 5 1400 kg m23
and the role played by slipping becomes increasingly more
relevant with respect to toppling. In this case, the mixture is conceptualized as an equivalent continuum fluid
whose density corresponds to the average depth-integrated density of the prototypal flow. This hypothesis is
close to reality in the case of hyperconcentrated flows (which may contain concentrations up to 60% by vol-
ume of sediments), like those that characterize lahars in volcanic terrain or in loess. Clear evidence of this
claim is provided by pictures of deposits of these flows [e.g., Rodolfo et al., 1996]. The continuum approxima-
tion is crude in the case of granular debris flows. In this case the impulsive and localized nature of impacts
apparently defies any possibility of description. Accordingly, in this case the use of the proposed approach
has only the advantage of providing a rational scheme for risk zoning in areas affected by torrential processes
where both the density and the slope have a growing effect on vulnerability (Figure 7c). Clearly, the presence
of localized stones and large debris could not be accounted so that the computed vulnerability curves should
be regarded as an upper bound [Chanson and Brown, 2013]. However, Milanesi et al. [2014] showed that this
approach provides more cautionary flood risk maps with respect to the ones provided by the national regula-
tions of Austria, Japan, and Switzerland in case of debris flows mapping.
The comparative analysis accomplished with the literature conceptual models of Xia et al. [2014] can be
extended to some of the cited national regulations. In particular, the thresholds Uh 5 0.5 m2
s21
of the
Swiss method [BWW, BRP, BUWAL, 2008], that separates low and medium intensity, can be tested on the
children data sets providing rRMSE 5 0.223, while the curve Uh 5 2 m2
s21
, that is the lower limit to high
intensity, can be tested on the adults data sets providing rRMSE 5 1.138. The Austrian curves for torrential
processes and fluvial floods may be regarded as adult stability thresholds and provide rRMSE 5 1.879 and
rRMSE 5 0.251, respectively. In both cases, there is an overestimation of the stability of adults.
The Swiss criterion contains an interesting idea that is the dependence of the hazard levels on the return
period of the event. This idea is suggested by the consideration that, for given values of velocity and depth,
events with a low return period are more dangerous than events characterized by a higher return period.
Eventually, in order to identify the final hazard level for each cell of the flooded area, the maximum hazard
identified by three flood simulations with growing return period must be considered.
This approach is not common in national flood regulations, which often prescribe a single event with fixed
return period. In such a case, the stencil in Figure 3d can be used in a straightforward way to produce a
map with three grades of risk, on the basis of the computed local slope, velocity, depth and density. How-
ever, in case of analyses based on events with different return period, the vulnerability levels may be varied
dynamically as shown in the following.
In case of high probability events (with return period Tmin) and of low probability events (with return period
Tmax), only high or low vulnerability thresholds are identified: the separation is represented by the stability
curve for children and for adults, respectively. For a return period s, such that Tmin s Tmax, an intermedi-
ate vulnerability level is added on the basis of a linear interpolation between the previous thresholds:
hhigh U; s
ð Þ5
h U; Tmax
ð Þ2h U; Tmin
ð Þ
Tmax2Tmin
2
s2Tmin
ð Þ1h U; Tmin
ð Þ if s
T
h U; Tmax
ð Þ if s
T
8

:
(23)
Figure 8. Slipping instability and dragging of a woman washed away by shallow flood waters on a sloping terrain. (Source http://www.youtube.com/watch?v5YAv_yUsAvgc. Last access
22 September 2014.)
MILANESI ET AL. V

hlow U; s
ð Þ5
h U; Tmin
ð Þ if s
T
h U; Tmax
ð Þ2h U; Tmin
ð Þ
Tmax2Tmin
2
s2
T
ð Þ1h U; Tmin
ð Þ if s
T
8

:
(24)
where
T5 Tmin1Tmax
ð Þ=2 and hhigh and hlow represent the critical stability depth for each assigned value of
velocity related respectively to the transition from medium to high vulnerability and from low to medium
vulnerability. As one may observe, in this case the vulnerability curves associated to s5
T are the ones
shown in Figure 3d.
6. Conclusions
Hydraulic risk maps are fundamental tools for risk mitigation and their main attributes should be clarity and
robustness. Therefore, risk maps should be based on the representation of physical phenomena with quan-
titative criteria that must be clearly understandable by stakeholders. In turn, this would increase awareness
and preparedness that are fundamental elements to make non structural protection measures effective. In
rapidly evolving hydraulic processes, the most relevant target is to preserve human life, thereby risk maps
should represent the combination of the flow features that threaten it. Human stability in a flow is a funda-
mental requirement for safety and, accordingly, this topic has been repeatedly explored in the past. How-
ever, as observed by Jonkman and Penning-Rowsell [2008], most of the existing criteria for instability are
based on a purely empirical analysis of available experimental data, thus there is a need for a physical inter-
pretation of the process and of an unifying approach, as shown in the recent paper by Xia et al. [2014].
The present paper introduced a comprehensive conceptual vulnerability model based on human stability in a
flow. The main failure mechanisms are represented by slipping and toppling, to be modeled respectively
through the equilibrium of the forces in the direction parallel to the slope and by a moment equilibrium
around a pivot point. A third failure mode was introduced to account for the risk of drowning due to high
water depth, that was modeled by imposing a maximum admissible water level as a function of person’s
height. The proposed approach accounts for the local slope of the terrain as well as the density of the fluid.
Accordingly, the scope of the proposed approach ranges from floodplains to mountain areas so that it might
be applied, with increasing degrees of approximation, in fluvial and torrential floods, hyperconcentrated flows,
tsunamis and debris flows. The results provided by the model were compared to the available data sets from
experimental studies on human stability in a flow. Although the model is simple and required the calibration
of a single parameter, its performance, in terms of rRMSE, is significantly better than the one provided by the
state of the art models and by criteria applied by national regulations in the European alpine area.
Unfortunately, there is a scarcity of experimental data regarding human stability in case of significant slopes.
Accordingly, there is a need of further experimental analyses to fill this gap as well as the development of
comprehensive approaches that account for the vulnerability of structures and the mobilization of vehicles.
In order to allow a straightforward application of the proposed algorithm, a simple program, along with an
instruction file, can be freely downloaded at http://www.ing.unibs.it/hydraulics/?page_id52904.
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We wish to acknowledge Junqiang Xia
for providing the experimental data
measured with the physical model of
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