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\DOI{10.5802/crphys.225}
\datereceived{2024-05-26}
\daterevised{2024-11-27}
\dateaccepted{2024-12-02}
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\dateposted{2024-12-16}
\begin{document}

\begin{noXML}

\makeatletter
\def\TITREspecial{\relax}
\def\cdr@specialtitle@english{Geophysical and astrophysical fluid dynamics in the laboratory}
\def\cdr@specialtitle@french{Dynamique des fluides g\'eophysiques et astrophysiques aulaboratoire}
\makeatother

\CDRsetmeta{articletype}{review}

\title{Quantifying soil surface erosion}

\alttitle{Quantifier l'\'{e}rosion de surface des sols}

\author{\firstname{Pierre} \lastname{Philippe}\CDRorcid{0000-0003-1889-0841}\IsCorresp}
\address{RECOVER, INRAE, Aix Marseille University, Aix-en-Provence, France}
\email[P. Philippe]{pierre.philippe@inrae.fr}

\author{\firstname{Nadia} \lastname{Benahmed}\CDRorcid{0000-0002-3004-2485}}
\addressSameAs{1}{RECOVER, INRAE, Aix Marseille University, Aix-en-Provence, France}
\email[N. Benahmed]{nadia.benahmed@inrae.fr}

\begin{abstract}
This review first introduces the general context of soil erosion, an
omnipresent natural process involving a wide variety of spatial and
temporal scales. While this loss of solid matter induced by all kinds
of fluid flows on the surface of soils is often a source of beauty in
the landscapes that surround us, it also constitutes a major risk and
poses growing threats in the context of global change as loss of
agricultural soil fertility, decline in coastal ecosystems or increase
in safety standards for flood protection structures. It is therefore
essential to be able to predict future soil removal rates at multiple
scales, in order to anticipate or mitigate their impacts. This requires
experimental quantification of soil's vulnerability to erosion. In
order to describe this type of measurement in deeper detail, we
restricted the scope of our review to situations of concentrated
surface erosion of homogeneous soils, on moderate spatial and temporal
scales, as typically studied in the fields of civil or environmental
engineering. Measuring erosion itself is challenging, both in terms of
determining the rate of mass removal from a surface, and in terms of
selecting and quantifying a relevant hydrodynamic quantity to reflect
the strength exerted by the flow. A conceptual framework is then
required to correctly define soil's erodibility, which is defined as an
intrinsic property of the material through an empirical erosion law. In
the most commonly accepted approach, this erodibility combines two
distinct parameters: an erosion initiation threshold, generally chosen
as a critical shear stress, and an erosion kinetic coefficient. The
various types of erodimeters found in literature are then presented and
compared, with a specific and more complete description of the three
main devices used in geomechanics (EFA, HET and JET). Finally, before
concluding and suggesting some perspectives on the topic, we outline
the various advantages and applications of the erodibility values
derived from experimental tests, while showing the limitations of the
approach and the questions raised by them.
\end{abstract}

\begin{altabstract}
Cet article de synth\`{e}se introduit tout d'abord le contexte
g\'{e}n\'{e}ral de l'\'{e}rosion des sols, un processus naturel
omnipr\'{e}sent impliquant des \'{e}chelles spatio-temporelles
extr\`{e}mement vari\'{e}es. Si cette perte de mati\`{e}re solide
induite par toutes sortes d'\'{e}coulements fluides \`{a} la surface
des sols est souvent source de beaut\'{e} des paysages qui nous
entourent, elle n'en constitue pas moins un risque majeur et fait peser
des menaces croissantes dans le contexte du changement global comme la
perte de fertilit\'{e} des sols agricoles, le d\'{e}clin des
\'{e}cosyst\`{e}mes littoraux ou le besoin accru de suret\'{e} des
ouvrages de protection contre les inondations. Par cons\'{e}quent, il
est essentiel de pouvoir pr\'{e}dire aux diff\'{e}rentes \'{e}chelles
les niveaux futurs de retrait des sols afin de mettre en oeuvre de
possibles mesures d'anticipation ou d'att\'{e}nuation. Il convient pour
cela de conna\^{\i}tre la sensibilit\'{e} \`{a} l'\'{e}rosion d'un sol via
une quantification exp\'{e}rimentale. Pour d\'{e}crire plus en
d\'{e}tail ce type de mesures, le choix est fait ici de restreindre le
cadre consid\'{e}r\'{e} aux situations d'\'{e}rosion de surface
concentr\'{e}e de sols homog\`{e}nes, sur des \'{e}chelles spatiales et
temporelles restant mod\'{e}r\'{e}es, et telle que typiquement
\'{e}tudi\'{e}e dans les domaines de l'ing\'{e}nierie civile ou
environnementale. La mesure de l'\'{e}rosion en elle-m\^{e}me est
d\'{e}licate et complexe, aussi bien pour d\'{e}terminer le taux de
mati\`{e}re \'{e}rod\'{e}e \`{a} une surface que pour s\'{e}lectionner
et quantifier une grandeur hydrodynamique pertinente, capable de rendre
compte de la force exerc\'{e}e par l'\'{e}coulement. Un cadre
conceptuel est ensuite n\'{e}cessaire pour d\'{e}finir correctement une
\'{e}rodabilit\'{e} du sol, condid\'{e}r\'{e}e comme une
propri\'{e}t\'{e} intrins\`{e}que du mat\'{e}riau \`{a} travers une loi
empirique d'\'{e}rosion. Dans l'approche la plus commun\'{e}ment
accept\'{e}e, cette \'{e}rodibilit\'{e} regroupe deux param\`{e}tres
distincts : un seuil d'initiation de l'\'{e}rosion,
g\'{e}n\'{e}ralement choisi comme une contrainte de cisaillement
critique, et un coefficient cin\'{e}tique. Les diff\'{e}rents types
d'\'{e}rodim\`{e}tres issus de la lit\'{e}rature sont ensuite
pr\'{e}sent\'{e}s et compar\'{e}s, avec un descriptif sp\'{e}cifique et
plus complet sur les trois principaux dispositifs utilis\'{e}s en
g\'{e}om\'{e}canique (EFA, HET et JET). Enfin, avant de conclure et de
sugg\'{e}rer quelques perspectives futures sur la th\'{e}matique, nous
pr\'{e}sentons les nombreux avantages et utilisations possibles des
valeurs d'\'{e}rodabilit\'{e} issues de ces essais, tout en montrant
les limites de l'approche et les questions que celles-ci soul\`{e}vent.
\end{altabstract}

\keywords{\kwd{Erosion}
\kwd{Measurement}
\kwd{Soil's erodibility}}

\altkeywords{\kwd{Erosion}
\kwd{Mesure}
\kwd{Erodabilit\'{e} des sols}}

\maketitle

\end{noXML}

\section{Context of soil erosion and related issues}\label{sec:intro}

\subsection{A ubiquitous process on a variety of scales}

Quoting
Wikipedia\footnote{\url{https://en.wikipedia.org/wiki/Erosion}},
\textit{erosion is the action of surface processes (such as water flow
or wind) that removes soil, rock, or dissolved material from one
location on the Earth's crust and then transports it to another
location where it is deposited.} Adopt this definition, which is
already rather broad, nevertheless excludes several adjacent mechanisms
likely to lead to the dislocation and crumbling of solid matter but
which involve no motion, such as
weathering~\cite{Gaillardet1999,Dinis2020} or others physico-biological
actions~\cite{Murray2008}. We therefore strictly define our object of
study as being the erosion generated by the detachment and entrainment
of solid matter from the surface of a soil under the action of a fluid
flow. Soil is understood here as a geomaterial in the broadest sense,
encompassing all types of soil, including clay, sand or silt, as well
as coarser materials, from pebbles to boulders, or even snow. The
variety is rather vast, with highly contrasting levels of internal
cohesion between constituent particles that also vary greatly in size:
from cohesion-less granular soils to coherent materials or soft rocks. 


As defined above, the erosion of solid matter is difficult to
dissociate from, and in fact includes, the subsequent transport induced
by the eroding flow. This complementary removal process enabling the
loosened soil to be evacuated, more or less rapidly, thereby allowing
erosion to possibly continue. Thus, the major problems associated with
soil erosion stem directly from the combination of conditions that
initiate soil removal and enable its transport, wether over a few
millimeters or thousands of kilometers. There is a wide variety of soil
erosion types~\cite{Merritt2003,Aksoy2005,Poesen2018} and the main
natural processes involved include: Sediment transport (in mountain
streams, rivers, oceans, or
deserts)~\cite{VanRijn1993,Soulsby1997,Merritt2003,Pahtz2020,Shao2009}; 
Coastal erosion~\cite{Mentaschi2018,Williams2018}; Rain splash and
runoff erosion (including rill erosion, gullies, etc) on farmland or
hill-slope~\cite{Bryan2000,Aksoy2005,Castillo2016}, with essential role
of plant covers~\cite{Gyssels2005,Zuazo2009}; Piping in natural
soils~\cite{Bernatek2018,Wilson2018}; Scouring of
riverbanks~\cite{Lawler1993,Julian2006} or bridge pier
foundations~\cite{Wang2017,Gao2023}; Internal and external erosion of
earthen hydraulic
structures~\cite{Foster2000,Richards2007,Wan2008,Bonelli2013}. From a
geomorphological point of view, the different spatial scales involved
vary tremendously, from a simple sand ripple to mountains, from a drop
of water to the confluence of the Amazon. Similarly, the timescales of
erosion range from a few seconds to several million
years~\cite{Ahnert1998}! Mountains are thus erased or valleys carved
out on geological periods, while violent events such as a massive
landslide may last a very short time. 


As a result, erosion leaves its omnipresent imprint on relief and
landscape, through riverbeds and deltas, cliffs and coastlines, or
inspiring country fields (see for instance
Figure~\ref{fig:ex-farmland})\,\ldots  We enjoy this imprint daily as we
contemplate the exceptional beauty of nature. But we also endure it by
living under the threat of natural events more or less directly
responsible for various erosional processes, such as floods,
landslides, coastal storms or torrential flows. Soil erosion thus has a
feedback effect on our environment through the increasing construction
of protective systems, as it poses a major hazard to many sensitive
natural and man-made elements: beaches, dunes, farmland, building
foundations, bridge piers, earthen engineering structures\,\ldots


\begin{figure}
\includegraphics{fig01}
\caption{\label{fig:ex-farmland}Example of gullies sculpted by erosion in farmland as
strikingly painted by Alexandre Hogue in ``The crucified Land'' (1939).
Licence: CC BY-NC-SA 4.0.}
\end{figure}


\subsection{A primary vector of risks}

Sediment transport is part of a natural geological cycle, based on land
subduction balanced by gradual relief degradation and subsurface
erosion. However this equilibrium is increasingly disrupted by human
activity~\cite{Poesen2018}. The global average rate of soil erosion by
water is estimated to be somewhere between 0.1 and 0.15~mm/year,
exhibiting significant local
disparities~\cite{Wuepper2020,Montgomery2007}, with a strong upward
trend in erosion rate due to climate change, land use and other
socio-economic
factors~\cite{Boardman2003,Li2016,Borrelli2017,Borrelli2020}. In
itself, this acceleration threatens the geological balance between
formation and removal of natural soils, endangering both human food
security and global ecosystem sustainability~\cite{Wuepper2020}. Soil
erosion of agricultural fields, mainly due to rainfall and surface
runoff, is another threat suspected of reducing agricultural potential,
particularly fertility, or inducing soil thinning~\cite{Batista2023}.
For instance, in the US, the Department of Agriculture (USDA)
recommends a soil loss rate below 1~mm/year for soil
conservation~\cite{Montgomery2007}, but research work continues on this
soil loss tolerance~\cite{Li2009}. Agricultural erosion also has an
impact on the global carbon cycle, but whose effect is still uncertain
and debated~\cite{VanOost2008,Kuhn2009,Doetterl2016}.

As mentioned earlier, the erosion of sediments can lead to the
transport of solid or dissolved matter over long distances, posing the
risk of pollution through the dispersion of contaminants. These
dangerous elements include radioactive particles~\cite{Evrard2015},
heavy metals~\cite{Brady2015,Kowalska2018}, or
microplastics~\cite{Lebreton2017,Zhang2017}, whose proliferation is of
particular concern due to their notoriously harmful effects on
virtually all ecosystems.

The desertification of entire regions, such as the Sahel and the
Sahara, has been greatly intensified by anthropogenic climate
change~\cite{Burrell20205}, and is partly induced by wind erosion and
subsequent transport~\cite{Shao2009}, which contribute to both the
degradation of agricultural soils and the invasion of land through the
displacement of large volumes of sand.

The sharp decline in coastline due to increased shoreline and beach
erosion is also a cause for great
concern~\cite{Williams2018,Mentaschi2018}, particularly in the current
and, above all, future context of rising sea levels and more frequent
ocean storms~\cite{Zhang2004,Pang2023}. The main threats concern the
coastal ecosystem and human activities, the sea, which is gradually
moving closer to natural or anthropic habitats, as illustrated in
Figure~\ref{fig:ex-coastal}.

\begin{figure}
\includegraphics{fig02}
\caption{\label{fig:ex-coastal}Photographs from an artistic perspective of Le Signal building
(Soulac-sur-Mer, France). This construction, built between 1967 and
1970, was definitely abandoned in 2014 under the threat of coastal
erosion and finally demolished in 2023. Credit: Le Signal@Olivier
Crouzel (\url{https://www.oliviercrouzel.fr}).}
\end{figure}

In addition, the safety of man-made structures in permanent or
intermittent contact with water is constantly threatened by the erosion
that can be generated by subsequent flows. This hazard seems to be
growing, particularly in the alarming context of global change, which
is very likely to lead to an intensification of extreme climatic events
(storms, precipitation, flooding). Major concerns include scouring
around rigid structures such as bridge piers~\cite{Brandimarte2012} or
concrete dikes, or direct water flow erosion of earthen hydraulic
structures, such as flood defence works, earthen dams and levees, river
and canal
embankments~\cite{Foster2000,Richards2007,Wan2008,Bonelli2013}. 

\subsection{A need for predictive modeling}

Given these severe risks posed, directly and indirectly, by soil
erosion, the human response cannot be solely focused on protection,
although this is an essential element. For effective management, we
should also be able to predict the effects of erosion over time as
reliably as possible, which requires the development of models capable
of incorporating to varying degrees the complexity of the processes
involved. Before attempting any modeling, the scales to be considered
must first be defined, both in terms of time and space. These scales
are constrainted by the global dimensions of the system in question and
the total duration targeted, and require the introduction of elementary
time and space steps at the scale of the basic processes under
consideration.

For example, erosion of the Earth's crust is a central feature of
long-term landform modelling, combining tectonics and surface
processes~\cite{Ahnert1998,Bishop2007}. However, many soil erosion
models are built on a catchment scale, focusing on shorter durations,
typically decades, to assess over time both sediment production and
transport capacity. In an extremely active field of research, a number
of operational tools have been proposed~\cite{Pandey2016,Borrelli2021}.
Such distributed models are specifically designed for a particular
area, on a regional, national or continental scale, and include, among
others: the European Soil Erosion Model EUROSEM~\cite{Morgan1998}, the
Water Erosion Prediction Project WEPP~\cite{Flanagan2001}, the Pan
European Soil Erosion Risk Assessment PESERA coarse scale
model~\cite{Kirkby2008}, the (Revised) Universal Soil Loss Equation
(R)USLE~\cite{Panagos2015,Borrelli2017} (see Figure~\ref{fig:RUSLE} as
an illustration), or else the Limburg Soil Erosion Model
LISEM~\cite{DeRoo2016a}. Empirical, conceptual and physical
process-based approaches can be distinguished, all requiring a large
number of input data explicitly accounting as much as possible for
climate variation (rainfall, temperature, etc.) and vegetation
characteristics. A highly critical parameter is land erodibility,
mostly derived from soil
characteristics~\cite{LeBissonnais1996,Alewell2019} and used as an
integrated coefficient at the model mesh scale. However, despite
significant progress in model spatial resolution, the elementary mesh
incorporates a large number of heterogeneities and spatial
variabilities in subsoil characteristics, remaining well above the
micro-scale where a homogeneous material behavior can be found, known
as Representative Elementary Volume (REV) in the field of soil
mechanics~\cite{AlRaoush2010}. Despite the obvious interest and need
for these large-scale models, they still have significant
limitations~\cite{Alewell2019}. In particular, they suffer from the
wide dispersion of available data and their usually limited calibration
domain, with no effective validation outside of
it~\cite{Favis-Mortlock2001}. 


\begin{figure}
\includegraphics{fig03}
\caption{\label{fig:RUSLE}An example of results achievable with large-scale models is
shown here with the map of soil loss rates in the European Union (for
reference year 2010) based on
RUSLE2015~\cite{Panagos2015}.}
\end{figure}

Another approach consists in studying and modeling soil surface erosion
at the representative material scale that is considerably smaller,
typically on soil sections of a few hundred square centimetres. The
number of input parameters for these models is significantly reduced,
and calibration is generally simpler, often based on well-controlled
laboratory experiments. This line of research was particularly
initiated to analyze river erosion and sediment transport, especially
in the case of granular (i.e.\ non-cohesive) sediments, for which an
empirical modeling framework has long been established. This framework
is essentially based on the dimensionless Shields number, which
compares fluid stress and buoyancy weight for grains or pebbles at the
riverbed interface. The Shields number governs the sediment transport
and a specific curve empirically determining its critical value at the
threshold of erosion
initiation~\cite{Buffington1999,Pahtz2020,Dey2019}. The case of
cohesionless sediments corresponds to a straightforward situation where
the physical origin of grain movement is clear. The understanding is
much more complex for cohesive soils, where soil composition, grain
size distribution and internal interactions within the material play
major role, rendering the previous Shields'approach inadequate in most
cases~\cite{Righetti2007,Jacobs2011}. Moreover an erosion threshold
alone is not sufficient to describe the phenomenon, which has its own
kinetics controlling the rate of soil removal once erosion has
initiated. The mechanisms responsible for material entrainment appear
obviously more complex and varied for cohesive soils, necessitating a
better physical and physico-chemical understanding of the
processes~\cite{Winterwerp2004}. This is all the more true as the
cohesion of a soil can have several possible origins: macroscopic
expression of microscopic inter-molecular interactions within very fine
soils, but also emerging property from the presence of binding agents
between larger grains (partially saturated granular soils, cemented or
bio-cemented soils, etc.).  To the best of the authors' knowledge, it
is currently impossible to present an exhaustive and coherent physical
picture of the mechanisms of cohesive soil erosion, as understanding of
the phenomenon is still very limited in the community concerned. A few
fragments of theory do exist, often quite disconnected from each other,
and will be mentioned in this article. Advances in numerical modeling
have been much more significant, and a recent review article provides
an up-to-date synthesis of modeling of soil surface
erosion~\cite{Feng2024}, but again with very few elements specific to
cohesive soils. While awaiting a clearer understanding, developments
mainly relied on empirical approaches, particularly through the
formulation of erosion laws at material scale~\cite{Knapen2007}. 

Whatever the scale considered, modeling requires establishing a
relationship between the quantity of soil eroded, typically described
by an erosion rate or a sediment transport capacity, and the force
exerted by the hydraulic flow. Such relationships effectively introduce
parameters that account for the soil's resistance to concentrated water
flow~\cite{Knapen2007}. These parameters can be seen as integrated
erodibility coefficients or indexes in large-scale models, deduced from
many input parameters, by contrast with intrinsic soil properties at
the material level. A critical aspect concerns the selection of a proxy
indicator for the hydraulic strength~\cite{Knapen2007}. For instance,
runoff erosion intensity on a ground plot is generally expressed in
terms of the transport capacity, as a function of shear
stress~\cite{Flanagan2001,Zhang2009} or unit stream
power~\cite{DeRoo2016a,Govers1986}. However, alternative choices exist,
such as flow rate, water head or Froude number~\cite{Knapen2007}. These
integrated or more local erosion laws are primarily empirical, with
some being based on a conceptual framework~\cite{Winterwerp2012},
proposing a unified formulation~\cite{Sanford2001,Walder2016}, or
including a stochastic component~\cite{Sidorchuk2005,VanProoijen2010},
as will be discussed further in Section~\ref{sec:erosion-laws}.

However, as a prerequisite to any modeling, obtaining reliable and
varied experimental data, both in field and from the lab, is essential
for supplying, calibrating and validating the various types of erosion
and transport models. With this objective in mind, and in order to
build up the most reliable predictions possible, methodologies have
been developed to measure soil and sediment erosion or, in other words,
to determine the erodibility parameters of the material or elementary
ground unit under consideration with maximum precision. Given the vast
range of different mechanisms at the root of soil loss by water
flow~\cite{Bryan2000}, adopting a global approach appears impractical
and, as Knapen and co-authors have pointed out~\cite{Knapen2007},
\textit{it is obvious that soil erosion should be considered as a
process-specific concept with different erodibility indices}. The
forward corollary is that specific measurement approaches are obviously
needed for each of these erodibility quantifications.


\subsection{Scope of the present review}\label{sec:intro-scope}

In this review, we have chosen to focus on experimental methods
developed to determine the resistance of consolidated cohesive soils
against surface erosion, subject to submerged and concentrated flow
conditions, over reduced time and space scales, excluding any
additional impact of slope with respect to the horizontal in our
discussion. The main practical applications concern scouring and
interfacial erosion on coherent geomaterials such as seabed sediment,
mostly constituted of sand-mud mixtures~\cite{Soulsby1997,Jacobs2011}
and subject in particular to intense and increasing coastal
erosion~\cite{Pang2023}, or compacted clayey soils in earthen hydraulic
structures, typically levees or embankment dams~\cite{Bonelli2013}. In
this particularly sensitive application, two types of erosion can
occur: external erosion, where there is overflow on the face of the
structure, and internal erosion, associated with mechanisms induced by
infiltration flows within the structure and its foundations. These two
causes are responsible, in nearly the same proportion, for almost 100\%
of reported earth dam failures~\cite{Foster2000,Richards2007}. The
mechanisms of internal erosion, which have been extensively studied
over the last twenty years and are now well identified, involve, for
some, zones within or below the structure where cohesive constituent
soils can be superficially eroded by a head of water. This is notably
the case for the so-called concentrated leak erosion, contact erosion
or backward piping erosion in its development phase~\cite{Bonelli2013}.
Within soils with a granular structure, there also exist internal
erosion processes that occur not at the surface but within the volume
of the material~\cite{Bonelli2012}, although these will not be
discussed here.

In what follows, we will present the different methods to assess, for a
soil subjected to a concentrated submerged hydraulic flow along an
initially flat and horizontal surface, the same parameters, hereinafter
referred to as \textit{erodibility}. These parameters will be defined
conceptually in greater detail below.


\section{Experimental determination of soil's erodibility} \label{sec:measuring}
\subsection{Difficulties and challenges}\label{sec:measuring-challenge}

Developing a device and protocol adapted to the study of soil erosion
is a major experimental challenge. It requires simultaneous and
independent quantification of both the rate of solid matter loss and
the hydrodynamic load exerted by the eroding flow. However, each of
these measurements proved difficult to implement for several reasons.

The rate of erosion at the surface of a soil sample is
straightforwardly defined as the mass of solid matter lost per unit of
time and per unit of surface area. The most direct method of
determining it would be to measure the gradual loss of mass over time
for the sample under test but this, however, poses major practical
difficulties. While weighing a given initial mass, it is necessary to
detect very small variations in mass, particularly when identifying an
erosion threshold. As a result, weighing accuracy is severely limited,
especially considering significant noise expected due to the
supplementary force generated by the water flow, usually in a turbulent
regime with substantial hydrodynamic fluctuations. Consequently, the
use of direct mass measurements for erosion monitoring remains
marginal.


Assuming soil homogeneity and measuring the corresponding density, the
change in mass of the eroded sample can be deduced from the change in
volume and therefore, more directly, from the progressive deepening of
the surface exposed to the flow. Since the long-existing point gauge
measurements~\cite{Hanson2004,Nguyen2017}, the techniques used have
become increasingly sophisticated, particularly with the use of 2D or
3D laser systems~\cite{Witt2003,Lee2004,Foti2011} and imaging
processing methods such as digital photogrammetry~\cite{Rieke-Zapp2005}
or light attenuation~\cite{Sutherland2014}. However, there are several
major obstacles to accurately tracking the eroded area: the upper
surface can rapidly develop a complex and evolving morphology; the
rapid build-up of suspended matter in the flow makes the water turbid,
preventing the use of certain probes, particularly optical ones, or
even direct visualization; the presence of heterogeneity or defects at
the scale being investigated compromises the validity of the approach.

In the end, the quantity of soil removal is often deduced from other
complementary experimental data rather than being directly measured at
the eroded area. The resulting sediment loss can be straightforwardly
determined from the concentration in eroded material inside the flow
emerging from the active zone. Erosion rate is then obtained through
the conservation of solid matter exchanges. However, this procedure may
suffer from a time lag induced by the distance imposed between the
measurement zone and the erosion zone. The concentration of eroded
material can be determined by regular sampling and weighing of the
downward flow, but solely on a high periodicity basis, or, for fine
sediment transported in suspension, by continuous time monitoring using
a turbidimeter, which in practice requires prior calibration for each
test. In an even more indirect manner, and subject to strong
simplifying assumptions, the measurement of the flow-pressure
relationship in the erosion zone can be used to estimate the soil
removal rate, based on the increase in flow cross-section and therefore
in the associated hydraulic radius~\cite{Bonelli2013HET}. However, this
method is rather approximate and is also directly linked to
hydrodynamic measurements of the eroding flow, unlike previous
approaches.

Measuring soil erosion rates involves considering factors, including
the history and aging of the soil being studied. There are typically
two options for real soil assessment: conducting tests on-site using
portable devices or collecting soil samples for laboratory testing.
This second procedure generally enables more precise measurements under
better-controlled conditions, with sophisticated equipment. However, in
some cases, the original sampled soil cannot be tested intact, due to
damage or the presence of interfering elements (large particles, roots,
etc.). The soil is then reconstituted under the same conditions as
those found on the site. However, this reconstituted soil may lose the
memory it has gained from in situ consolidation and, as will be
discussed later, its resistance to erosion tends to be consistently
reduced. It should be noted that for specific comprehensive studies,
model soil samples can be prepared on demand using the same methodology
for systematic testing.



Linking erosion rate with flow intensity requires accurately measuring
the flow and deriving appropriate and relevant characteristic
quantities. This also poses a number of difficulties, firstly in
selecting the flow features that closely reflect the physics of
erosion, and, secondly, in implementing the accompanying measurements
in practice, particularly considering that the flow regime is most
often turbulent. In such turbulent flow, the movement of fluid is
chaotic, characterized by irregular fluctuations in velocity and
pressure. This turbulence complicates the measurement process as
traditional techniques may not capture all relevant flow features
accurately.

Many competing quantities can be used to characterize a flow, generally
presenting direct or semi-empirical relationships between them, with a
strong dependence on the specificity of the hydrodynamic configuration
considered, which can vary widely (flow in a channel, flow in a pipe
with cross-sectional variation, submerged impact jet\,\ldots). Classically
accepted quantities for characterising flow include discharge, stream
power, flow shear stress, or shear velocity~\cite{Knapen2007}. As
already mentioned, a key challenge lies in ensuring that the chosen
hydrodynamic indicator adequately represents the erosion mechanisms
which, by definition, take place at the interface between soil and
water flow. A volume-averaged magnitude is therefore unlikely to be
straightforwardly relevant for describing this type of superficial
phenomenon and requires prior calibration to correlate with a more
local characterization. In this respect, the bottom shear stress (or
equivalently shear velocity) at the soil surface appears to be a flow
predictor with significant potential. Consequently, it has often been
selected, although it's not the most straightforward to determine, as
will be discussed below.

Once a hydrodynamic reference variable has been selected, it is
essential to consider whether its mean value is representative of the
effective flow loads generating erosion. While this question does not
arise in the case of a perfectly stationary ideal flow, the same is no
longer true in more realistic situations where the hydrodynamic regime
is generally turbulent, inducing more or less pronounced fluctuations
within the flow~\cite{Cheng2003,Keirsbulck2012}. A moderate average
value can therefore mask much higher extreme values, likely to induce
localized erosion puffs. To take these fluctuations into account, it is
then necessary to characterize the complete statistical distribution,
particularly in the upper tail~\cite{VanProoijen2010}. Note that a
perfectly similar approach can be adopted in the presence of spatial
variations in flow along the erosion surface~\cite{Beguin2013}.
Furthermore, variability in erosion does not solely arise from flow
dynamics but also from the eroded material itself, which is rarely
homogeneous, particularly in field conditions~\cite{Tolhurst2009}.

Monitoring basic hydrodynamic quantities involves measuring the overall
discharge rate, using a suitable flowmeter, and the pressure difference
between upstream and downstream of the erosion zone, using either two
absolute pressure transducers or directly a differential pressure
gauge. This standard instrumentation can be exceptionally accompanied
by more sophisticated measurements, such as for calibration purposes,
in particular to reconstruct the velocity field in space, whether using
normal profiles at the interface of interest, 2D maps in the vertical
plane, or even full 3D velocity fields. To achieve this, precision
instruments are required, such as laser or acoustic Doppler
velocimeter, particle image velocimetry systems, or hot-wire
measurements~\cite{Khoo2001,Nimmo2002,Musumeci2018}. Most
interestingly, it is next possible to derive the time-averaged shear
stress from this mean velocity profile using various correlation
strategies with established laws~\cite{schlichting2000,Klewicki2007}.
Direct measures of bottom shear stress are also
available~\cite{Klewicki2007}, either at a single point with a Preston
tube~\cite{Patel1965,Ackerman2001} or over a surface with MEMS-based,
oil-film interferometry, floating element or hot wire
techniques~\cite{Sturzebecher2001,Naughton2002,Baars2016}. In all
cases, the measurement of bottom shear stress relies on inferring it
from another quantity, necessitating the use of these inventive and
sometimes unexpected methods, such as for example shear-induced
bioluminescence~\cite{Foti2010}, electrochemical
technique~\cite{Keirsbulck2012} or ferrofluid spike
deformation~\cite{Musumeci2018}. It is important to note that high time
resolution is crucial when fluctuations, rather than mean value, have
to be obtained, especially in turbulent flow conditions where
fluctuations play a major role in erosion process.

In practice, from the perspective of routinely carrying out tests to
quantify soil erosion resistance, implementing high-tech measurement
systems to tackle all the key issues raised above can be impractical.
To circumvent this limitation, interpretation models, specific to each
test and its own hydraulic configuration, have been developed. Based on
a number of simplified (or even oversimplified) and/or known empirical
correlations, such models make it possible to propose reduced
relationships between key hydrodynamic and geometric quantities,
related to flow and erosion. This approach helps limit the number of
input quantities that need to be measured, rendering testing more
feasable in practical settings. Pre-calibration is also usually
required. For example, in a test involving flow through a pipe, the
lateral shear stress can be deduced from either overall head loss or
discharge rate~\cite{Bonelli2013HET}. Similarly, in a test involving an
impinging submerged jet, the tangential shear stress can be expressed
as a function of both height and flow rate of the
jet~\cite{Hanson2004}.


\subsection{Conceptual framework for intrinsic soil's erodibility}
\label{sec:erosion-laws}

Since knowledge of the elementary physics involved in the removal and
transport of cohesive soils by flow is still limited to date, erosion
laws, i.e.\ laws relating erosion rate to relevant hydrodynamic
quantities, have primarily been formulated empirically. Several
competing formulations have been proposed, mostly but not exclusively
based on fluid shear
stress~\cite{Partheniades1965,Ariathurai1978,Parchure1985,Krone1999,Sanford2001,Knapen2007,Tolhurst2009,Marot2011,Sharif2012,Winterwerp2012,Walder2016,Wilson2018,Bora2023}. 
As the existence of a threshold for erosion is fairly widely accepted
in the scientific community, although this is still open to
debate~\cite{Lavelle1987}, excess shear stress models are most
frequently proposed, using as control variable the amount of shear
stress exerted by the flow in excess with respect to a threshold
indicative of the material's resistance. For simplicity's sake, these
expressions take typically the form of power laws. This approach is
reminiscent of long-established empirical laws describing solid
transport, as initially suggested by Duboys~\cite{Duboys1879} and later
refined by Meyer-Peter M\"{u}ller equation for the bedload of sand and
gravel~\cite{VanRijn1993,Soulsby1997}. This type of law is also often
proposed for the stream capacity in large-scale integrated approaches
of landscape evolution models, as illustrated in~\cite{Davy2009}.

Although there are a few other alternative
formulations~\cite{Parchure1985,Marot2011,Wilson2018}, the most common
surface erosion law is generically expressed as: 
{\begin{eqnarray}\label{eq:erosionlaw-neq1}
E_{\mathrm{m},\mathrm{v}}=\begin{cases}
 K_{\mathrm{m},\mathrm{v}}(\tau-\tau_{\mathrm{c}})^n &amp; \mbox{if } \tau\geq\tau_{\mathrm{c}}\\
 0 &amp; \mbox{if } \tau&lt;\tau_{\mathrm{c}}
\end{cases}
\end{eqnarray}}\unskip
where the erosion rate $E_{\mathrm{m}}$ (respectively $E_{\mathrm{v}}$) is defined as the
soil mass (respectively volume) loss by surface and time units (see
previous Figure~\ref{fig:sketch-E}) while $\tau$ is the bottom shear
stress exerted by the flow. The new parameters introduced in this
formulation are: (i) the proportionality coefficient $K$ (being
equivalently $K_{\mathrm{m}}$ or $K_{\mathrm{v}}$), called the coefficient of soil erosion,
the detachment rate coefficient, the concentrated flow soil
erodibility, or simply erodibility according to various authors; (ii)
the stress threshold $\tau_{\mathrm{c}}$, which quantifies the soil's resistance
to erosion and is commonly referred to as critical shear stress; (iii)
the exponent $n$ of the power law, where $n&gt;0$.


\begin{figure}
\includegraphics{fig04}
\caption{\label{fig:sketch-E}Sketch of a soil surface $\Sigma$ subject to erosion by a flow
characterised by a given mean velocity $\overline{u}$ profile and
related bottom shear stress $\tau_b$. The mass (respectively volume)
loss is denoted $\delta m$ (respectively $\delta V$) and the erosion
rate $E_{\mathrm{m}}$ in mass (respectively $E_{\mathrm{v}}$ in volume) is directly
derived.}
\end{figure}


In practical terms, especially concerning the units of the
proportionality coefficients, this erosion law can also be conveniently
written in dimensionless form as follows:
{\begin{eqnarray}\label{eq:erosionlaw-n}
E_{\mathrm{m},\mathrm{v}}=\begin{cases}
E^*_{\mathrm{m},\mathrm{v}}\left(\frac{\tau}{\tau_{\mathrm{c}}}-1\right)^n &amp; \mbox{if }
\tau\geq\tau_{\mathrm{c}}\\
 0 &amp; \mbox{if } \tau&lt;\tau_{\mathrm{c}}
\end{cases}
\end{eqnarray}}\unskip
where $E^*_{\mathrm{m},\mathrm{v}}=K_{\mathrm{m},\mathrm{v}}\tau_{\mathrm{c}}^n$ appears as a standard (mass or
volume) erosion rate, which coincides with the actual erosion rate
obtained for a flow shear stress equal to twice the soil's critical
threshold $\tau_{\mathrm{c}}$. This quantity is also occasionally referred to as
the rate coefficient~\cite{Black2002}.

The exponent $n$ of the power law is certainly of practical interest as
it accounts for non-linear relationship between erosion rate and excess
stress, both for concave curvature ($0&lt;n&lt;1$) and convex curvature
($n&gt;1$). However, unlike the other two parameters, $K$ and $\tau_{\mathrm{c}}$,
which will be discussed just beyond, it is much more difficult to
provide for $n$ a straightforward physical interpretation, or even a
more indirect one. While Walder proposed a generalized erosion law
where $n=7/4$~\cite{Walder2016}, other authors have shown that a value
strictly greater than 1 leads to a divergence\footnote{More precisely,
in a pipe flow configuration assuming uniform radial extension by
erosion, the authors of this study demonstrated that the pipe radius
diverges in finite time if ever $n&gt;1$.} in erosion kinetics and
therefore appears non-physical~\cite{Bonelli2008}. Consequently, the
linear excess-shear stress erosion law, where $n=1$, is the one
commonly used in practice. The preference for $n=1$ is most likely due
to several factors. Firstly, it offers conceptual simplicity.
Additionnally, determining an exponent $n\neq 1$ that would be used as
a free fitting parameter is challenging, given the wide dispersion
frequently observed in most erosion rate measurements, as illustrated
in Figure~\ref{fig:ex-comparison} by several examples involving natural
or reference soils~\cite{Laflen1991,Briaud2001,Zhang2002,Haghighi2013}.
Note that selecting $n=1$ also provides convenient units for $K_{\mathrm{m}}$ and
$K_{\mathrm{v}}$, which are $\mathrm{s}{\cdot}\mathrm{m}^{-1}$ and 
$\mathrm{m}^2{\cdot}\mathrm{s}{\cdot}\mathrm{kg}^{-1}$, respectively.


\begin{figure}
\includegraphics{fig05}
\caption{\label{fig:ex-comparison}Some illustrative comparisons of the linear excess shear
stress model with experimental data of either the mass $E_{\mathrm{m}}$ or volume
$E_{\mathrm{v}}$ erosion rate obtained for: (a) cropland soils~\cite{Laflen1991};
(b) low plasticity clay~\cite{Briaud2001}; (c) sandy-clay silty
soil~\cite{Zhang2002}; (d) kaolinite-sand
mixture~\cite{Haghighi2013}.}
\end{figure}

In the simplified modelling of surface erosion, the physical
interpretation of the $K$ and $\tau_{\mathrm{c}}$ parameters is relatively
straightforward. Assuming there exists a threshold for initiating the
erosion process, this threshold is directly reflected by $\tau_{\mathrm{c}}$ in a
bottom shear stress-based representation of hydrodynamic loading. From
a phenomenological perspective, it is natural to introduce the excess
shear stress $\tau-\tau_{\mathrm{c}}$ more precisely. Therefore, under constant
imposed shear stress $\tau&gt;\tau_{\mathrm{c}}$, the proportionality coefficient $K$
will control the intensity of the erosion rate and thus directly the
kinetics of the process. Each of these two parameters contributes to
quantifying a soil's sensitivity to erosion, both in terms of
resistance, with an initiation threshold that must not be exceeded if
the risk of erosion is to be inhibited, and in terms of kinetics once
the process has been initiated, in relation to the survival time of a
structure in a crisis situation. In summary, $K$ and $\tau_{\mathrm{c}}$
collectively capture a soil's sensitivity to erosion, that is referred
to as \textit{soil's erodibility}. This conceptual framework, as
defined by this local linear law, implicitly assumes that the action
exerted by the flow on the soil surface is fully described by the
bottom shear stress alone, regardless of the hydrodynamic configuration
generating the erosion. The corollary leads directly to considering the
erodibility parameters $K$ and $\tau_{\mathrm{c}}$ as intrinsic material
properties, independent of all other external influences. Undoubtedly,
this question is highly debatable. While it is obvious that erodibility
depends on soil properties~\cite{Mitchener1996,Grabowski2011}, it is by
no means evident that it depends exclusively on them. In reality, there
are likely other external factors that can influence erosion
sensitivity, such as environmental conditions, vegetation cover, land
use, variations in the hydraulic regime, etc. These external influences
can interact with the properties of the soil to affect the erosion
rates. Therefore, it may be more accurate to consider erodibility as a
dynamic property that can vary depending on the specific conditions and
context in which erosion occurs.

Despite the simplistic assumption of applying a minimalist model with a
linear relationship between erosion rate and flow stress, the
comparison with experimental measurements is generally acceptable and
sometimes even very satisfactory, as demonstrated in previous
Figure~\ref{fig:ex-comparison} with results from studies using a wide
variety of devices and
soils~\cite{Laflen1991,Briaud2001,Zhang2002,Haghighi2013}. Considering
in addition the broad to very broad dispersion almost systematically
observed in these erosion data, the excess shear stress model appears
to be reasonably acceptable. This explains why little effort has been
made to complicate further the conceptual framework starting from this
consistent although minimalist basis. However, there is a more
sophisticated modeling approach in which this excess shear stress law
is used as part of a probabilistic description of soil
erosion~\cite{VanProoijen2010,Beguin2013,Bonelli2013HET}. Such
probabilistic approaches allow to consider substantial deviations from
the average phenomenon represented by a unique mean field erosion law.
This variability can be temporal and is particularly important with
respect to flow stress in turbulent regimes. It can also occur in
space, either due to intrinsic variability in soil properties, or if
the eroding flow at the surface of the material is not uniform, as in
the case of porous flow for instance~\cite{Beguin2013}. In this case,
this also means that there may be a dependence on the size of the
eroded sample, as long as the latter is too small to fully reflect the
whole range of variation of the key parameters. These probabilistic
models enable for instance to address the structure
scale~\cite{Andreini2019} or to question the existence of a threshold
on a macroscopic scale~\cite{Lavelle1987,Beguin2013,Bonelli2013HET}.


\subsection{Developments of a wide range of dedicated erodimeters}

The systematic evaluation of the two \textit{erodibility} parameters,
allowing quantification of the absolute value of erosion resistance and
relative classification of soils, has driven the development of
numerous specific erosion devices. These devices involve various types
of flow, with the tested soil being submerged or not, and where the
flow features can vary widely: free surface or confined flow,
tangential or impingement flow, stationary or time evolving regime. The
main hydrodynamic configurations of these erosion tests include
longitudinal channels, annular flume, pipe flow, submerged impinging
jet and, more seldom, rotating flow. For example, as suggested
by~\cite{Karim2018} and sketched in Figure~\ref{fig:configs}, these
devices can be broadly classified into four different categories: flume
erosion
tests~\cite{Partheniades1965,Parchure1985,Shaikh1988,Amos1992,Maa1993,McNeil1996,Houwing1998,Ravens1999,Tolhurst2000,Briaud2001,Witt2003,Aberle2004,LeHir2008,Jacobs2011,Crowley2012,Crowley2012b,Sutarto2014,Shan2021,Krishnappan2022,Yao2022},
through-pipe erosion
tests~\cite{Sherard1976,Hjeldnes1980,Lefebvre1985,Reddi2000,Wan2004,Wan2004b,Benahmed2012,Haghighi2013,Fattahi2017,Xie2018,Sharma2022},
jet erosion
tests~\cite{Moore1962,Aderibigbe1996,Hanson2001,Hanson2004,Sutherland2014,Nguyen2017,Briaud2017,Dong2020} 
and rotating apparatus
tests~\cite{Moore1962,Arulanandan1975,Chapuis1986,Lim2009,Bloomquist2012}.
These different types of erosion tests offer a variety of experimental
setups to study erosion processes under different hydrodynamic
conditions. Each type of test has its advantages and limitations, and
the choice of test method depends on the specific research objectives
and conditions being\unskip\break studied.

\begin{figure}
\includegraphics{fig06}
\caption{\label{fig:configs}Sketches of the most typical flow configurations in erosion
testing devices: (a)~tangential flume flow; (b) through-pipe flow; (c)
jet flow; (d) rotation flow.}
\end{figure}

Measurements of erosion and sediment transport in channels represent
some of the first devices used to study these processes. Flows are
generally free surface, although confined configurations also exist,
especially for in-situ purpose as will be discussed later. Erosion
occurs at the bottom of the channel, either over a large area of
sediment, or limited to the upper surface of one or more soil samples,
which are generally cylindrical in shape and vertically oriented. In
the latter case, the sample must be raised either manually or by a
piston, as regularly as possible to ensure that its eroded top remains
aligned with the rest of the floor. Depending on the device, this is
achieved either by visual criteria (operator-dependent) or by more
sophisticated feedback systems~\cite{Witt2003,Shan2021}. These channels
are usually straight and rectangular in cross section, with a flow rate
regulated by a pump, within a recirculating closed-loop
system~\cite{Partheniades1965,Houwing1998,Shaikh1988,Tolhurst2000,Briaud2001,Ternat2008,Crowley2012,Crowley2012b,Shan2021} 
or
not~\cite{Maa1993,McNeil1996,Ravens1999,Witt2003,Aberle2004,LeHir2008,Jacobs2011}.
However, flow establishment length in such flumes can be relatively
large, a condition which is not always met in practice or may require
large dimensions for the device to achieve. From this perspective,
another common geometry for studying erosion and sediment transport is
the annular flume, where flow, confined or not, is generated from above
using a speed-controlled rotation drive system, sometimes called
carousel~\cite{Amos1992}. Symmetry (i.e., invariance by rotation) and
the absence of a pump result in a periodic condition in the flow
direction, as a replica of infinite river, enabling the investigation
of long-time
processes~\cite{Parchure1985,Amos1992,Maa1993,Krishnappan1993,Krishnappan2022,Yao2022}.
However, this is of interest mainly for sediment transport issues, but
less suitable for studying erosion alone.


As mentioned in previous Section~\ref{sec:intro-scope}, one of the main
mechanisms of internal erosion in an earthen hydraulic structure is
driven by concentrated flow along a through pipe creating a
preferential path for erosion. To reproduce this phenomenon in the
laboratory, the Pinhole test was the pioneer test for characterising
qualitatively the soil dispersivity~\cite{Sherard1976}. Afterwards, for
quantitative assessment  of erosion, the Drill Hole test was
proposed~\cite{Lefebvre1985}. Later, Wan and Fell developed the
original version of the device referred to as Hole Erosion Test
(HET)~\cite{Wan2004,Wan2004b}. In this type of test, a hole is first
drilled longitudinally through a soil sample before immersion in water.
Then, to generate erosion, a controlled flow is imposed along this
pipe. The soil must possess specific cohesion properties to withstand
the drilling of the hole and not subsequently collapse. To avoid this
problem and, above all, to enable the erosion phenomenon to be
visualized, few variants of the device have been proposed, with,
instead of a pipe, a slot carved in the soil at the interface with a
transparent wall, and called either Slot Erosion Test
(SET)~\cite{Wan2004}, viewable HET~\cite{Xie2018} or Groove
test~\cite{Sharma2022}. Intrinsically, all these test configurations
based on a pipe across the sample induces a strong and immediate
feedback between soil surface erosion and the eroding flow through the
progressive geometric evolution of the pipe, or slot.

The impinging jet tests are inspired by the hydraulic configuration of
scour as observed in field. These devices were mainly developed in
North America~\cite{Moore1962,Aderibigbe1996,Sutherland2014}, where
significant advancements have been made, notably by Hanson's group with
their Jet Erosion Test
(JET)~\cite{Hanson2001,Hanson2004,Hanson2007,AlMadhhachi2013,Fox2022}.
Generally speaking, these tests consist in completely submerging the
soil zone or sample under study, then applying in the upper water
column a downward jet flow of adjustable velocity impinging, usually
perpendicular, on the surface to be eroded. A scour crater is thus
gradually excavated and its dimensions regularly measured, requiring
the jet to be temporarily stopped, or deflected, for measurement. The
latter is difficult to achieve, and is often limited to the depth
beneath the jet. Nevertheless, more sophisticated techniques have been
developed to determine the overall shape of the scour
crater~\cite{Houwing1998,Sutherland2014,Nguyen2017}. However, the
volume or mass of soil eroded can only be an estimate. In the same
category of tests relying on jet flow perpendicular to the soil, is the
Borehole Test~\cite{Briaud2017,Shidlovskaya2023}, which differs
significantly from the other impinging tests in that the flow takes
place through a vertical tube previously introduced into the soil. Last
but not least, this cratering configuration is also highly relevant to
the case of high-speed gas jets, particularly in the context of
extraterrestrial landings (Moon, Mars)~\cite{Metzger2009,LaMarche2017,Guleria2020}.


Devices based on rotating flows are relatively limited in number,
probably due to their rather complex application. Two examples of such
devices are the Rotating Cylinder Test
(RCT)~\cite{Moore1962,Chapuis1986,Lim2009} and the Rotating Erosion
Testing Apparatus (RETA)~\cite{Bloomquist2012}. In principle, a flow is
generated by an external rotating cylinder while a soil sample is
placed at the center, around the axis. Shear stress at the soil surface
is deduced from a direct or indirect measurement of the torque exerted
on the sample, while erosion is assessed by weighing the remaining soil
and/or the eroded soil, requiring regular pauses in the spinning system
to make these measurements. While these devices offer a unique approach
to studying erosion under rotating flow conditions, their complexity
and the need for frequent interruptions in testing may limit their
widespread use compared to other erosion testing methods. 

There are several fundamental differences between these various devices
and configurations. Apart from the variable degree of turbulence in the
flows, they are distinguished by their permanent or evolving features.
Flume configurations typically maintain a relatively constant flow
throughout the test, with the eroded soil sample being continuously
re-positioned at the bottom of the channel. In contrast, the other
configurations involve a regular receding of the water--soil interface,
leading to a gradual evolution of the flow geometry that may be more or
less pronounced depending on the testing method, such as through-pipe
or jet erosion compared to flume erosion. Flow control methods also
differ among the configurations, with some using regulated discharge
(by pump or rotary drive for carousel configuration) and others
imposing constant pressure difference (in the case of pipe flow). The
hydrodynamic configurations encountered present a variety of
complexities, requiring different degrees of reliance on empirical laws
to determine relevant flow quantities serving as input variables for
the interpretation models (see Section~\ref{sec:measuring}). This is
particularly the case with the impinging jet whose flow is slowed down
on its axis and sharply deflected radially until it tangents the ground
surface~\cite{Beltaos1974}. Combined with the additional dependence on
several jet parameters (nozzle size and shape, distance from the soil
surface, etc.), the direct determination of flow features is a real
challenge, especially in the presence of a developing
crater~\cite{Benseghier2020}. Furthermore, differing situations can be
highlighted with regard to flow direction in relation to soil intrinsic
anisotropy. This anisotropy, directly linked to a gradient of soil
properties, results either from natural consolidation under gravity, or
from static pressure or dynamic layer compaction during the soil sample
reconstitution in the laboratory. Erosion resistance will therefore
differ depending on whether the flow stress is exerted perpendicularly
(i.e.\ flume tests) or tangentially (i.e.\ through-pipe and rotating
tests) to the consolidation or compaction orientation of the soil.
Additionally, there exist various methods in soil mechanics practice to
prepare a specimen under predetermined conditions~\cite{Head2011}, and
the choice made for an erosion test can have an impact on the
erodibility values\unskip\break measured.


Conducting measurements on site is also a critical point, particularly
in the context of submarine hydraulics, necessitating the development
of in-situ devices that prioritize portability and
robustness~\cite{Tolhurst2009}. This is especially important when
dealing with mechanically fragile or environmentally sensitive soils,
such as soft/loose muds or soils subject to biological activity which
may have a strengthening (bio-stabilization) or weakening
(bio-turbation) effect on erosion~\cite{Black2002,Tolhurst2009}. These
on-site devices include the Jet Erosion Test~\cite{Hanson2001}, the In
Situ Erosion Flume (ISEF)~\cite{Houwing1998}, the IFREMER
erodimeter~\cite{LeHir2008,Jacobs2011}, the Mobile Recirculating Flume
(MORF)~\cite{Black1995}, the VIMS Sea Carousel~\cite{Maa1993}, the
Adjustable Shear Stress Erosion and Transport (ASSET)
flume~\cite{Roberts2003} and several versions of the Cohesive Strength
Meter (CSM)~\cite{Tolhurst1999,Tolhurst2000}. However, in the cases
where soil cores can be extracted under good conditions and without
risking material disturbance, there is an obvious advantage in carrying
out erosion measurements in the laboratory, on a set-up that generally
provides far better possibilities in terms of instrumentation,
metrology and control of ambient conditions allowing for more precise
and controlled experimentation. Additionally, aside from in-situ tests,
it is worth noting that erosion tests can also be conducted on real
structures or specific physical models of comparable size, particularly
in overflowing or wave overtopping
configurations~\cite{Hanson2005,Thornton2012,Kakinuma2014,Olschewski2014,Cantre2017,Bonelli2018,Picault2023}.


To further illustrate some of these different types of erodimeters, the
following section describes in more detail the three main erodimeter
systems encountered in the field of geomechanics: Erosion Function
Apparatus (EFA), Hole Erosion Test (HET), and Jet Erosion Test (JET). 


\section{Focus on principal erosion devices in geomechanics}\label{sec:devices}

\subsection{Erosion function apparatus (EFA)}

A well-known illustration of linear erosion channel (i.e.\ configuration
of Figure~\ref{fig:configs}a) is the Erosion Function Apparatus (EFA)
developed in the early 1990's by Briaud and collaborators from Texas
A\&amp;M University (College Station, Colorado, US)~\cite{Briaud2001}. This
apparus has been standardized and commercialized by a company
specialized in construction materials testing
equipment\footnote{\url{https://www.humboldtmfg.com/erosion-function-apparatus.html}}.
It has become widely used in geotechnical engineering research and soil
erosion studies, providing valuable insights into erosion mechanisms
and helping to inform engineering practices related to erosion control.


Typically, the soil sample to be tested is collected in the field using
a Shelby tube and then placed in a circular opening at the bottom of
the hydraulic flume. The latter, depicted in Figure~\ref{fig:EFA}a, is
a rectangular channel with pump-driven re-circulation. In the setup
illustrated in Figure~\ref{fig:EFA}b, it is crucial to maintain
alignment between the top of the sample and the bottom of the channel.
This alignment is achieved by regularly pushing up the Shelby tube with
a piston, following as closely as possible the rate at which the sample
erodes. A side window allows visual monitoring of the sample to
determine when the piston should be raised incrementally, typically in
1~mm steps under manual control. The progression of the piston provides
the volumetric erosion rate $E_{\mathrm{v}}$. A global test protocol was developed
and adapted for the critical visual detection step to ensure accurate
monitoring of erosion progression.   
A flow-meter installed in the flume measures the average flow velocity,
from which the bottom shear stress $\tau$ can be determined using the
Moody's chart. The average velocity can be varied across a wide range,
from 0.2 to $6~\mathrm{m}{\cdot}\mathrm{s}^{-1}$, enabling for multiple
data points to be collected for a single sample in order to plot the
relationship between erosion rate and either flow velocity or bottom
shear stress. The test provides a classification of the soil's
resistance to erosion, as detailed in forthcoming
Section~\ref{sec:charts}. Additionally, through regression analysis,
quantitative values for the critical shear stress $\tau_{\mathrm{c}}$ and erosion
coefficient $K_{\mathrm{v}}$ can be determined.


\begin{figure}
\includegraphics{fig07}
\caption{\label{fig:EFA}(a) Picture of the Erosion Function Apparatus in operation at
ESTP (Paris)~\cite{Bennabi2012,Youssef2023}. (b) Sketch of the soil
sample in the Shelby tube and of typical erosion curves
from~\cite{Briaud2017chart}.}
\end{figure}


\subsection{Hole erosion test (HET)}
There is no real standard for the Hole Erosion Test (HET), and several
research teams have developed their own devices. All of them can be
considered direct descendants of the original HET device developed by
Wan and Fell at University of New South Wales (Sydney, Australia) in
the early 2000's~\cite{Wan2004,Wan2004b}. Figure~\ref{fig:HET} shows
pictures of the first HET device alongside several examples of later
versions that incorporate various improvements. These improvements may
include enhancements to the testing methodology, modifications to the
setup for increased accuracy or efficiency, or adaptations to
accommodate specific research requirements. Overall, these developments
reflect ongoing efforts within the research community to refine and
optimize erosion testing techniques for better understanding of soil
erosion.


\begin{figure}
\includegraphics{fig08}
\caption{\label{fig:HET}(a) Original HET by Wan and Fell at University of New South
Wales (Sydney, Australia)~\cite{Wan2004}. (b) HET developed at US
Bureau of Reclamation (Denver, Colorado, US)~\cite{Erdogan2008Report}.
(c) HET developed at INRAE (Aix-en-Provence,
France)~\cite{Bonelli2013HET,Youssef2023}. (d) HET developed at
University Gustave Eiffel (Champs-sur-Marne,
France)~\cite{Haghighi2013}. (e) HET developed at University of
Lorraine (Nancy, France)~\cite{Mehenni2016}. (f) HET-P developed at
University of British Columbia (Vancouver, Canada)~\cite{Luthi2011}.
(g) HET developed at Shiraz University of Technology (Shiraz,
Iran)~\cite{Fattahi2017}.}
\end{figure}

The distinctive feature of the HET test is the presence of a
through-hole in the soil sample as sketched in
Figure~\ref{fig:configs}b.  By imposing flow along this pipe, high
stresses can be generated in a straightforward manner. However, the use
of undisturbed soils is challenging since the drilling phase, even in a
reconstituted soil specimen, is delicate, as some materials may be too
fragile to withstand the drilled pipe, and may experience slaking
either before or during the test~\cite{Lim2006}. Depending on the
protocol, the initial diameter of the hole varies from 3 to 6~mm. Once
the tube containing the soil sample is assembled with the rest of the
device, the whole system is saturated and a flow is applied, either at
a imposed flow rate or imposed pressure. As erosion occurs and the
diameter of the hole increases with mass loss, interpolation laws are
used to estimate its size at any time, assuming a cylindrical geometry
and, the final diameter is measured or estimated post-test to refine
the interpolation. The measured quantities typically include flow rate,
downstream turbidity and pressure at various positions. The pressure
measurements are particularly crucial for accurately determining the
pressure drop induced by the flow through the pipe. This involves
considering singular pressure losses at the pipe inlet and outlet. For
instance, the HET-P device uses a Pitot static tube for local pressure
measurement, positioned right at the exit of the
hole~\cite{Luthi2012,Fattahi2017}. Interpretation models may vary among
research teams, but all manage to derive the erosion rate and wall
shear stress (in a hole geometry assumed to be cylindrical) from
measurements (pressure, flow rate and turbidity if available), in order
to deduce the erodibility parameters $\tau_{\mathrm{c}}$ and $K$ by linear
regression. 


\subsection{Jet erosion test (JET)}
The Jet Erosion Test (JET) device is probably the most widely used
erosion testing device due to its versatility for both laboratory and
field experiments. Inspired by the practical case of soil scour, the
JET device is the eponymous and certainly the best-known among the jet
erosion tests. It was originally developed in the late 1980's by Hanson
and his colleagues at the Agricultural Research Service Hydraulic
Engineering Research Unit in Stillwater, Oklahoma, US. Since then, it
has undergone a succession of upgrades and
improvements~\cite{Hanson2001,Hanson2004,Hanson2007}. The testing
apparatus, as shown in  Figure~\ref{fig:JET}b, and its accompanying
interpretation method are described in ASTM Standard D5852 (2003).
Additionally, there are two field versions: the original one and a more
compact mini-JET~\cite{AlMadhhachi2013}, as depicted in
Figure~\ref{fig:JET}b. These field versions allow for erosion testing
in real-world environments, providing valuable insights into soil
erosion processes under natural conditions.

\begin{figure}
\includegraphics{fig09}
\caption{\label{fig:JET}(a) Picture of the Jet Erosion Test at ESTP
(Paris)~\cite{Erdogan2008Report}. (b) Picture of the
mini-JET~\cite{McNichol2017}.}
\end{figure}

In practice, after full immersion of a soil sample or the ground area
to be tested on site, a submerged jet is generated at the outlet of a
delivery tube, positioned perpendicular to the soil surface, in the
configuration shown in Figure~\ref{fig:configs}c. The jet is supplied
by a pump, either via an adjustable head tank or by direct connection
when high stresses are required. The circular nozzle at the bottom of
the tube is fixed at 6.4~mm, but the distance to the soil surface is
adjustable. The estimated maximum erosion stress can thus be varied
over a very wide range, from 4 to 1500~Pa. Variations in maximum bed
scour depth are measured using a point gauge. The latter is tightly
fitted to pass exactly through the nozzle, temporarily interrupting the
flow while the measurement is being taken when the gauge comes into
contact with the bottom of the erosion crater. 

There have been several developments and improvements in the
interpretation method for the JET, which ultimately provides the two
expected parameters $\tau_{\mathrm{c}}$ and $K_{\mathrm{v}}$. However, despite using
consistent input variables such as jet velocity, presumably constant
during the test, and measured depth, there are significant disparities
in the erodibility parameters obtained, depending on the method
employed~\cite{Daly2015,Karamigolbaghi2017}. This underscores the
importance of refining interpretation methods and ensuring consistency
in experimental procedures to obtain reliable and accurate results in
soil erosion testing.

\subsection{Comparative analysis of the three erosion devices: EFA, HET, JET}
The comparison of the three erosion testing devices, JET, EFA, and HET,
reveals various advantages and drawbacks in terms of their
applicability, erosion rate measurement methods, and hydrodynamic
complexities. This provides the following comparative
overview~\cite{Erdogan2008Report,NAP2019}.

JET and EFA offer the possibility of testing all types of soil, even
those that are weakly or even non-cohesive, although in the case of
JET, the grains tend to fall back into the crater and may distort the
scour depth measurement. On the other hand, HET, which requires hole
drilling, cannot be implemented on soils that are not cohesive enough
or are too soft, as they risk collapsing~\cite{Bonelli2013}. Similarly,
swelling soils should be avoided. Even on resistant sample, drilling
can cause some damage to the soil.  Regarding erosion rate, it is
measured directly with the EFA, but using a visual procedure that is
thus operator-dependant and rather critical. In the other two devices,
the erosion rate is simply deduced from other measurements, such as
turbidity or scour depth, with varying degrees of accuracy. Indeed,
assuming that the cylindrical geometry is maintained during the test,
HET modelling enables volume loss to be quantified, whereas JET
interpretation is limited to the cavity depth under the jet in a
simplistic 1D approach. With regard to the possibility of testing
natural soils, while the HET and EFA are only laboratory tests, the JET
can also be used in-situ, on flat or inclined (and even vertical)
surfaces, especially its compact version mini-JET that is easy-to-use
and inexpensive. It should be noted, however, that unlike HET where
soil samples generally have to be reconstituted, those tested at EFA
can be preserved in conditions close to those on site, thanks to the
use of a Shelby tube.

In terms of hydrodynamics, the three devices display very different
flow configurations of increasing complexity. The simplest situation is
clearly that of the EFA, which is based on a steady channel flow whose
characteristics have long been known, even if they require the use of
empirical laws for the friction factor. It should be noted, however,
that the transition between the bottom of the channel and the flush
surface of the soil sample can create local flow disturbances, which
may compromise the assumption of flatness preservation on the eroded
interface, as well as the critical visual detection controlling the
1~mm piston increment. The hydrodynamics of the HET are more complex,
firstly because of the flow reduction and expansion at the inlet and
outlet of the hole, which generate singular head losses and intensified
erosion. Secondly, while flows in cylindrical pipes are also well
known, this is an atypical case where the length is small and the
radius grows during the test. As already mentioned, the modelling of
the HET flow has been developed and perfected in great depth to take
account of these different aspects, although it requires the
assumption, which is therefore questionable, that the hole remains
cylindrical. However, the experimental validation of analytical scaling
laws definitely supports the interpretation model~\cite{Bonelli2008}.
In contrast to the other two systems, it's also worth noting the
ability to impose either flow or pressure hydraulic conditions.
Finally, the JET configuration seems to be the most complex, from the
initial situation of a jet impacting a flat surface, but even more so
as the scouring crater develops. Approximate flow modelling therefore
requires drastic simplifications, as is done for JET test
interpretation.


\section{Benefits and limitations of quantifying soil erodibility} 

\subsection{Classification of soil's erodibility}\label{sec:charts}

Several classification rules have been proposed, based on the extensive
databases of experimental values of erodibility parameters that have
been progressively built up with the results of erosion tests on a very
wide range of soils with highly variable properties and compositions.
Each classification rule likely offers insights into soil erosion
behavior under specific testing conditions and may be tailored to
different applications or soil types. Basically, there are three
competing proposals, originating from EFA, JET and HET data,
respectively.

\begin{figure}
\includegraphics{fig10}
\caption{\label{fig:charts}The different charts for soil's erodibility classification:
(a) Chart proposed by Briaud based on EFA test
results~\cite{Briaud2017chart}; (b) Chart proposed by Hanson based
on JET tests~\cite{Hanson2001}; (c) Erosion rate index based
classification as suggested by Wan and
Fell~\cite{Wan2004b}.}
\end{figure}

Briaud's classification system, based on EFA tests carried out on a
wide variety of soils, categorizes soils into six distinct categories
of materials, ranging from non-erosive to very high
erodibility~\cite{Briaud2017chart}. As can be seen in
Figure~\ref{fig:charts}, this classification, with various soil
categories, is developed using a log-log diagram displaying the rate of
erosion as a function of stress, in some words the same type of
representation as that of the erosion law. In this diagram, each test
corresponds to multiple data points, with the eventually of straddling
two categories, which can complicate classification. The boundaries
between the categories are represented as linear, implying power law
relationships between erosion rate $E_{\mathrm{v}}$ and exerted shear stress
$\tau$, with increasing exponent. This type of relationship is
therefore substantially different from the linear excess shear stress
erosion law of Equation~(\ref{eq:erosionlaw-n}) (with $n=1$). It is also
important to note that this classification may have limitations,
particularly in cases where soils exhibit behavior that falls between
two categories as already mentioned.

A second classification has been proposed by Hanson on the basis of the
very large number of tests carried out with JET device, primarily
through in-situ testing across various locations in the United
States~\cite{Hanson2001,Hanson2004,Fox2022}. This classification is
based on a log--log representation of the erosion coefficient $K_{\mathrm{v}}$
(here denoted $k_{\mathrm{d}}$) as a function of critical shear stress $\tau_{\mathrm{c}}$,
with each soil tested corresponding to a single point. As shown in
Figure~\ref{fig:charts}, the chart is divided into five zones by linear
boundaries, representing power law relationships, which categorize
soils from resistant to very erodible. It is noteworthy that $K_{\mathrm{v}}$ is
expressed in unconventional units, in 
$\mathrm{cm}^3{\cdot}\mathrm{N}^{-1}{\cdot}\mathrm{s}^{-1}$.

Alternatively, Wan and Fell introduced the erosion rate index $I_{\mathrm{m}}$,
derived from HET test results, which is defined as $I_{\mathrm{m}}=-\log (K_{\mathrm{m}})$
with $K_{\mathrm{m}}$ expressed in
$\mathrm{s}{\cdot}\mathrm{m}^{-1}$~\cite{Wan2004b}. They proposed a
classification into six categories for the expected soil erosion
ranging from extremely rapid, for $I_{\mathrm{m}}&lt;2$, to extremely slow, for
$I_{\mathrm{m}}&gt;6$ (See Figure~\ref{fig:charts}). 

Although these classifications are all derived from test-specific
databases and established by rather arbitrary boundaries between soil
categories, they appear reasonably applicable across various types of
test. Overall, these classification systems provide a framework for
categorizing soils based on their erosion characteristics, offering
insights into their behavior under different stress conditions. They
serve as practical tools for understanding and assessing soil erosion
potential, aiding in the planning and management of erosion control
measures.


\subsection{Erodibility parameters in practice}
The various erosion tests presented and discussed in previous
Section~\ref{sec:devices} enable positioning a given soil in the
various classifications proposed above, and hence, offer valuable tools
for qualitatively assessing soil's resistance against erosion. By
knowing the erosion characteristics of a soil, particularly one present
on a construction site, engineers and researchers can position it
within the classifications proposed earlier, which enables them to
identify the soil's category in terms of erosion sensitivity.
Therefore, they can make informed decisions regarding erosion risk
management. This knowledge helps ensure that appropriate measures are
taken to prevent erosion-related issues during and after the
construction or repair of hydraulic structures.

From a more quantitative point of view, critical shear stress is the
key parameter of interest as regards soil's resistance against erosion
in civil engineering conception, including design and safety
requirement. It serves as a crucial threshold, defining the maximum
bottom shear stress that a soil can withstand before experiencing
erosion. This parameter is essential for ensuring the stability and
longevity of earthen structures subjected to hydraulic forces such as
infiltration or overflow. The main difficulty lies in accurately
estimating the actual stresses potentially generated by water flows on
the surface and within the hydraulic structure under consideration. 

On the other hand, erosion coefficient $K$ becomes crucial in crisis
situations or emergency scenarios, when erosion is occurring within an
hydraulic structure. In fact, as a variable representative of the
kinetics of soil removal, $K$ controls the rate of pipe expansion by
erosion. For example, in the case of a through pipe with constant
upstream water level and continuous erosion, and considering that the
soil's critical shear stress is virtually negligible compared to that
exerted by the flow, the expansion rate of a through pipe can be shown
to be highly dependent on the erosion index $I_{\mathrm{m}}$. For instance, the
time required for a pipe to expand from 25~mm to 1~m in diameter can vary
over four orders of magnitude, from a few tens of minutes to several
months depending on the value of $I_{\mathrm{m}}$ which ranges from 2 to
6~\cite{Fell2013b}. Going a step further, Bonelli and Benahmed proposed
a simple model, with several mechanical quantities inferred from
fourteen case studies, to estimate the remaining time until breaching
for an earthen dam once a downstream leak due to piping flow erosion
has been detected~\cite{Bonelli2011,Bonelli2013HET}. From this model,
the following relation was obtained:
{\begin{eqnarray}\label{eq:time-to-breach}
\Delta t_{\mathrm{c}}=t_{\mathrm{c}}-t_{\mathrm{d}}&lt;\frac{2\rho_{\mathrm{dry}}}{K_{\mathrm{m}} \nabla P} \ln 
\left(\frac{H_{\mathrm{dam}}}{2R_{\mathrm{d}}}\right)
\end{eqnarray}}\unskip
where $\Delta t_{\mathrm{c}}$ is the remaining time to dam roof collapse with
$t_{\mathrm{d}}$ and $t_{\mathrm{c}}$ the time of initial leak detection and the time of
ultimate collapse, $\rho_{\mathrm{dry}}$ is the soil's dry density, $\nabla P$
denotes the pressure gradient (assumed constant) through the pipe,
$H_{\mathrm{dam}}$ is the dam height and $R_{\mathrm{d}}$ is the estimated radius of the
pipe at detection. As the upper limit is inversely proportional to
$K_{\mathrm{m}}$, it potentially varies over 4 to 5 orders of magnitude. However,
this variation may be less pronounced for soils that are sufficiently
clay-rich and compacted during construction, such as those found in
dams (through perhaps less so in a levee). 

Erodibility parameters can also be useful indicators for validating and
quantifying soil reinforcement achieved through various treatments,
such as biocementing~\cite{Jiang2017}, chemical
stabilization~\cite{Indraratna2009} and cement or lime
addition~\cite{Indraratna2008,Herrier2014,Mehenni2016,Konstadinou2023}.
Figure~\ref{fig:reinforcement1} shows two illustrations of erodibility
strengthening obtained by lignosulfonate~\cite{Indraratna2009} and
lime~\cite{Mehenni2016} treatment with obvious critical shear stress
increase in both cases while erosion coefficient substantially
decreases as shown on the left graph. Another example depicted in
Figure~\ref{fig:reinforcement2} shows the progressive increase in both
critical shear stress $\tau_{\mathrm{c}}$ and erosion index $I_{\mathrm{m}}$ during curing
period after a lime treatment~\cite{Fell2013,Herrier2014}. These
illustrations highlight the effectiveness of various soil treatments in
enhancing resistance to erosion, as evidenced by changes in erodibility
parameters.

\begin{figure}
\includegraphics{fig11}
\caption{\label{fig:reinforcement1}Examples of quantitative evidence of enhanced soil erosion
resistance through (a) lignosulfonate
treatment~\cite{Indraratna2009} and (b) lime
treatment~\cite{Mehenni2016}.}
\end{figure}

\begin{figure}
\includegraphics{fig12}
\caption{\label{fig:reinforcement2}Example of gains in erodibility, 
i.e.\ critical shear stress
$\tau_{\mathrm{c}}$ and erosion index $I_{\mathrm{m}}$, as a function of curing time during
lime treatment of a Rh\^{o}ne river soil specimen (with $t=0$
corresponding to untreated soil). Figure adapted
from~\cite{Fell2013,Herrier2014}.}
\end{figure}


\subsection{Lack of predictability from soil properties}
It is still a widely debated question whether there is a sufficiently
conclusive connection between soil's susceptibility to erosion and
usual properties, as commonly characterized in the laboratory  (see
e.g.\ in~\cite{Head2011} for details on definitions and measurement
methods for the usual characteristics of soils). All the more so since
behind this issue is that of the intrinsic character of erodibility
parameters, as postulated in the conceptual framework commonly proposed
and which would be directly confirmed by such an immediate correlation
with other inherent soil features. 

Much effort has been devoted to finding quantifiable relationships,
mostly on an empirical basis, between erodibility and the main
measurable properties in natural soils, including particle size
distribution, density, water content, plasticity index, consolidation
time and many other
quantities~\cite{Parchure1985,Black2002,Mostafa2008,Grabowski2011,Jacobs2011,Regazzoni2011,Kimiaghalam2016,Fell2013,NAP2019,Shafii2023}.
A wide variety of empirical relationships have emerged in the
literature, partially listed for instance by
Winterwerp~\cite{Winterwerp2004,Winterwerp2012},
Grabowski~\cite{Grabowski2011} and Kimiaghalam~\cite{Kimiaghalam2016},
but further enriched since then. These relationships mainly concern the
critical shear stress $\tau_{\mathrm{c}}$, with numerous correlations, more or
less convincing, the essential ones of which have been synthesized by
various
authors~\cite{Grabowski2011,Winterwerp2004,Fell2013,NCHRP2019,Chen2021,Shafii2023}.
It is also a parameter for which theoretical approaches are frequently
proposed, most often by extension of known results on the threshold for
incipient motion of a purely frictional grain without cohesion or
adhesion~\cite{Righetti2007,Jacobs2011,Chen2018,Chen2021,Brunier-Coulin2020}. 
A power law with the plasticity index is also
suggested~\cite{Winterwerp2004,Winterwerp2012,Kimiaghalam2016}. By
comparison, the kinetic erodibility parameter $K$ has been less
studied, which is most likely due to the greater experimental
difficulty in determining it with confidence, particularly in-situ.
Nonetheless, a theoretical model linking $K$ to some consolidation
parameters for fine soils can be reported~\cite{Winterwerp2004} while
some empirical or semi-empirical laws can be also
mentioned~\cite{AlMadhhachi2013,Daly2015}. What all these studies have
in common is the wide dispersion in erodibility data, which severely
restricts the extent of the empirical relationships they attempt to
support. The main reasons for this are undoubtedly the difficulties in
measuring soil erosion already mentioned in
Section~\ref{sec:measuring-challenge}, and the high degree of spatial
variability in soil properties due to lack of uniformity, which is
particularly acute for field measurements.  A further consideration is
the additional impact of ambient conditions such as
temperature~\cite{Gularte1980,Hoomehr2018,Akinola2019}, for which an
Arrhenius-type activation factor on erosion rate has been proposed and
experimentally validated~\cite{Gularte1980}, or, more specifically for
fine soils with clay fractions, salinity or
pH~\cite{Gularte1980,Reddi2000,Hoomehr2018,Ravisangar2001}. 


Independently of the investigation of direct relationships with other
soil properties, some authors have also argued the existence of an
underlying correlation between $\tau_{\mathrm{c}}$ and $K$, thereby reducing the
characterization of soil erodibility to the knowledge of a unique
parameter, in line~with the previous classification by Wan and
Fell~\cite{Wan2004b}, which is based solely on $K_{\mathrm{m}}$ (see
Section~\ref{sec:charts}). Thus, supported by many JET test results,
Hanson was the first to suggest the existence of an inverse power
relationship of the form $K\propto\tau_{\mathrm{c}}^{-\gamma}$ with
$\gamma\approx0.5$~\cite{Hanson2001}. Interestingly, it turns out that
such an exponent equal to 1/2 can be recovered by dimensional
analysis~\cite{Andreotti2013}. However, a broader review of the
literature reveals a much wider range of exponent values, from
$\gamma\approx0.4$ to $\gamma\approx2.4$, as reported by several
authors~\cite{Daly2015,Julian2006,Indraratna2008,Karamigolbaghi2017,Layzell2014,Nguyen2017,Thoman2008}.
Together with the huge dispersion of the data almost systematically
observed and the dependence of the erodibility parameters on the choice
of protocol or interpretation model (at least for JET
measurements)~\cite{Daly2015,Khanal2016,Karamigolbaghi2017}, it appears
more than hazardous to make any attempt of generalization. Especially
since some contradictory analyses seem to conclude that there is no
universal correlation between $\tau_{\mathrm{c}}$ and $K$~\cite{Knapen2007}. All
in all, it seems difficult to go beyond the simple qualitative
observation that a soil with less (more) resistance in terms of
critical shear stress $\tau_{\mathrm{c}}$ generally undergoes  higher (lower)
erosion rates under equivalent hydrodynamic conditions, i.e.\ when
exposed to the same excess shear stress, thus corresponding to a
greater (lower) erosion coefficient $K$.


In short, the compilation of all these correlation analyses leads to
the conclusion that there are no standardized relationships from a
quantitative point of view, and only very few general trends from a
more qualitative point of view.

\subsection{Ongoing questions, difficulties and limits}
Additionally to the lack of strong correlations, several results
highlight problems and raise questions about some limitations of the
proposed conceptual framework.


One of the first issues encountered is beyond the simple scope of
laboratory erosion tests, and relates more broadly to the ability of a
reconstituted soil sample to reflect the effective hydro-mechanical
behavior of the same soil under its natural conditions on site, leading
to significant discrepancies in erosion tests results. In particular,
it is known that one key issue is the loss of strength exhibited by
reconstituted soil sample compared with that measured on undisturbed
state~\cite{Hoeg2000}. In the specific case of erosion, detachment
rates of disturbed samples were measured to be more than an order of
magnitude higher than those observed for natural soil~\cite{Zhang2003},
although approximately proportional, as shown in
Figure~\ref{fig:intact-recons}. This discrepancy is illustrated in the
same Figure where data by an extract from HET database of INRAE reveals
in the previous Wan and Fell's classification that intact soils
consistently placed in the most erosion-resistant soil categories,
contrasting with reconstituted samples. Consequently, the observed
disparity underscores the challenge of accurately assessing soil
erodibility in laboratory settings, where reconstituted samples may not
faithfully represent the behavior of soils in their natural state. As a
result, the extrapolation of laboratory-derived erodibility parameters
to real-world conditions may be subject to considerable uncertainty and
may not capture the true erosion susceptibility of soils in the field. 

\begin{figure}
\includegraphics{fig13}
\caption{\label{fig:intact-recons}(a) Comparison of the erosion rates measured for disturbed
and natural soil samples~\cite{Zhang2003}. (b) Distinct ranking in
Wan and Fell's chart according to the intact or reconstituted nature of
several soil samples from INRAE's HET
database.}
\end{figure}

The variability in erodibility parameters across different experimental
devices presents another significant challenge in erosion testing.
Studies have highlighted wide disparities in erodibility parameters
such as critical shear stress and erosion coefficient when the same
soil sample is tested using different
devices~\cite{Lim2006,Erdogan2008Report,Bennabi2012,Fell2013,Regazzoni2013,McNichol2017,Mahalder2022,Youssef2023}.
For instance, from a comparison of results obtained with the Rotating
Cylinder and the Hole Erosion Test (HET), Lim (2006) has shown a clear
correlation between the two sets of data, but without achieving
equality between the respective values of critical shear stress and
erosion coefficient~\cite{Lim2006}. This discrepancy was particularly
pronounced when distinguishing between dispersive\footnote{A dispersive
soil is structurally unstable when exposed to water, and particularly
susceptible to erosion, due to a high concentration of exchangeable
sodium ions and a large specific surface area~\cite{Vakili2024}.} and
non-dispersive soils. In cases involving non-dispersive soils, this
work has shown that a much greater difference in erosion indexes $I_{\mathrm{m}}$
is observed for the second category, the erosion index $I_{\mathrm{m}}$ derived
from the HET systematically yielded higher values as can be seen in
Figure~\ref{fig:comp}. Similarly, extensive analyses comparing JET and
HET have consistently shown that HET tends to yield slower erosion
rates $K$, or equivalently higher erosion indexes $I_{\mathrm{m}}$, and larger
critical shear stresses $\tau_{\mathrm{c}}$, up to more than an order of magnitude
as observed in Figure~\ref{fig:comp}, thus overestimating soil
resistance to erosion compared to
JET~\cite{Erdogan2008Report,Fell2013,Regazzoni2013,Luthi2011}. Erdogan
and collaborators (2008) found particularly large discrepancies for
soils with heterogeneous structures, composed for example of clay
aggregates while the difference is still significant, but much smaller,
for more homogeneous materials~\cite{Erdogan2008Report}. It was
suggested that the complex hydraulic configuration of the JET test,
along with the simplifying assumptions required for interpretation, may
contribute to the discrepancies observed. However, comparisons between
mini-JET and flume erosion test suggest that variations are more
moderate on average despite substantial dispersion, and perhaps a
weaker correlation between erodibility
values~\cite{McNichol2017,Mahalder2022}. In addition, a 2D
axisymmetrical numerical modelling of surface erosion by a submerged
turbulent impinging jet showed that there was no major inconsistency
between measured erosion rates and those calculated on the basis of the
erodibility parameters provided by the corresponding JET
interpretation~\cite{Mercier2014}. Furthermore, comparisons between HET
and EFA have similarly indicated an overestimation of soil resistance,
and therefore an underestimation of its erodibility, with the HET
results~\cite{Bennabi2012,Youssef2023}. Note however that the same type
of overall agreement between simulation and experiment is obtained for
the HET configuration~\cite{Mercier2015}. 

\begin{figure}
\includegraphics{fig14}
\caption{\label{fig:comp}(a) Confrontation between the erosion indexes $I_{\mathrm{m}}$
obtained from HET and RCT tests, with distinction between
non-dispersive and dispersive erosion~\cite{Lim2006}. (b)
Confrontation between the critical shear stresses $\tau_{\mathrm{c}}$ obtained
from HET and JET tests~\cite{Regazzoni2013}.}
\end{figure}

In conclusion, using the HET rather than another device therefore can
lead to category changes in Fell's classification based on erodibility
parameters, even though the relative position between two given soils
may remain the same. However, note that a more global energy-based
method enables identical categories to be found between HET and JET
results~\cite{Marot2011}. Overall, as suggested by Mahalder and
co-authors~\cite{Mahalder2022}, $\tau_{\mathrm{c}}$ and $K$ estimates appear to be
significantly influenced by the specific hydrodynamics induced by the
testing device configuration. This raises a number of fundamental
concerns. To be, or not to be intrinsic, that is the question for
soil's erodibility if we are to paraphrase Shakespeare in light of
these inconsistent comparative results! One may indeed wonder whether a
unique and generic framework can effectively apply for all types of
soils and whatever the nature of the eroding flow at their exposed
surface. The underlying question behind this line of thought is what
erosion really is, particularly in terms of small-scale processes.
There remains considerable exploration work to be continued, given the
great variety of situations in terms of soil composition,
physico-bio-chemical aspects and soil/runoff interactions, with widely
varying lengths and characteristic times, including ageing.

\section{Summary and perspectives} 
At a very broad scope, we can conclude that each disciplinary field has
its own perception and definition of soil erosion, with quite specific
scales of time and space. This vision derives organically from the
object of study under consideration, the elementary mechanism of one
discipline corresponding sometimes to the ultimate scale of another. In
view of this vast spectrum of phenomena with many associated challenges
and threats, we have chosen to emphasize the field of soil mechanics,
i.e.\ for typical eroded soil surfaces of a few tens of centimetres with
durations of minutes to hours. 

Throughout this review, the reader has been able to appreciate the vast
amount of work carried out on the subject, but also the intrinsic
difficulties of soil erosion. The importance of quantifying the erosion
resistance of soils has been demonstrated, in order to characterize
their performance or predict topological changes in heterogeneous
ground. We have shown how challenging it is to measure, both in terms
of accurately determining the mass loss at an exposed soil interface,
and in terms of correctly evaluating the most relevant quantities
accounting for the load exerted by rather complex flows. Despite some
limitations, the development of a conceptual framework for surface
erosion has proved its worth and validity, enabling erodibility
parameters to be defined explicitly for any coherent soil, for the
purposes of own characterization, relative classification or even
modeling. Yet questions remain as to the assumed intrinsic nature of
this new material property. Finally, the description and comparison of
the three leading erodimeters used in the field of soil mechanics,
namely EFA, HET and JET, provided practical insights into these
erodibility assessment tests.

From a purely fundamental point of view, a research perspective
consists in improving our knowledge of the elementary processes
involved in soil erosion, since this generic term actually covers
different local mechanisms, some of which have already been identified
but are still poorly understood~\cite{Philippe2017}. Accordingly,
Winterwerp and Kesteren propose the existence of 4 erosion modes for
cohesive marine sediments: entrainment, floc erosion, surface erosion
and mass erosion~\cite{Winterwerp2004}. This approach represents a
major scientific bottleneck that precludes the development of
non-empirical models or realistic scenarios to explain how, and through
which successive steps, a fragment of soil is ultimately removed from
the surface of a compact material under hydraulic stress of sufficient
intensity and/or duration. These mechanisms should be able to provide
us with valuable information on the size (or more likely the size
distribution) of the elementary soil aggregates thereby eroded and on
the characteristic time, or even on the hydraulic conditions, required
for such a process. This effort needs to be based on theoretical
approaches, but must also involve specific experimental investigations,
which require a high degree of customization in order to explore small
scales, particularly on natural geomaterials. An interesting
alternative is to substitute real soils by purely artificial materials,
specifically developed, whose key characteristics are well under
control and capable of facilitating this local
exploration~\cite{Brunier-Coulin2020,Philippe2017}. The contribution of
numerical simulation could also be essential, particularly with the
Discrete Element Method (DEM) integrating both microscopic laws
modeling internal cohesion and coupling between fluid and grains, which
requires significant methodological
developments~\cite{Tran2017,Benseghier2020b,Benseghier2023}. However,
this raises the question of how representative such over simplified
systems can be, given that erosion appears to be highly
material-dependent, as revealed by the poor correlation between
erodibility and soil properties.

From a more practical perspective, there is still room for further
improvement in soil erodibility tests, both from a technical
perspective and in terms of interpretation models, possibly by
extending the conceptual framework used. Progress in this respect could
hopefully address some of the questions raised in the preceding
section. At the same time, it is essential to promote the
implementation of routine tests of this kind in geotechnical and
environmental engineering, particularly when safety considerations
prevail. Beyond the usual properties associated with composition and
structure of a soil, erodibility could then simply become an additional
standard parameter for material identification, on an equal basis with
other parameters such as shear strength or plasticity index. Many of
the researchers involved in some of the works summarized in this review
have been working towards this goal in their respective networks for
several years, with notable advances and real prospects. Undoubtedly,
erosion testing devices have proved their merits and are increasingly
finding their place in the hydraulic engineering community. EFA is
widely used, mainly in the United States, and notably played an
important role in the post-crisis expertise carried out on New Orleans
levees after Hurricane Katrina in 2005~\cite{Briaud2008}. JET also
remains a very common test, particularly because of its ability to be
conducted on site. A recent workshop brought together a large number of
current operators to draw lessons and propose future prospects for this
instrument~\cite{Fox2022}. As for HET, this device has undergone
several improvements since its original version and is increasingly
being used in civil engineering. In France, for example, several
companies have licensed the version developed by INRAE, and a new
standard on HET testing (PR XP
P94-065\footnote{\url{https://www.boutique.afnor.org/fr-fr/norme/xp-p94065}})
has just been certified at the end of 2023, including soil sample
preparation and testing protocol. Progress is definitely on the right
course!

\section*{Declaration of interests}
The authors do not work for, advise, own shares in, or receive funds
from any organization that could benefit from this article, and have
declared no affiliations other than their research organizations.

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