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\DOI{10.5802/crgeos.315}
\datereceived{2025-06-13}
\daterevised{2025-10-07}
\dateaccepted{2025-10-14}
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\dateposted{2025-11-05}
\begin{document}

\begin{noXML}

\CDRsetmeta{articletype}{research-article}

\TopicFR{Hydrologie, hydrog\'eologie}
\TopicEN{Hydrology, hydrogeology}

\title{An innovative experimental device to quantify the water relative
permeability and in situ water retention curves of unconsolidated
porous media}

\alttitle{Dispositif exp\'{e}rimental innovant permettant de quantifier
la perm\'{e}abilit\'{e} relative \`{a} l'eau et les courbes de
r\'{e}tention d'eau \textit{in situ} des milieux poreux non consolid\'{e}s}

\author{\firstname{Kevin} \lastname{Hernandez-Perez}\CDRorcid{0009-0007-4992-5046}\IsCorresp}
\address{Institut Terre et Environnement de Strasbourg (ITES) UMR 7063,
Universit\'{e} de Strasbourg, CNRS, Strasbourg, France}
\email[K. Hernandez-Perez]{hernandezperez@unistra.fr}

\author{\firstname{Gerhard} \lastname{Sch\"{a}fer}\CDRorcid{0000-0002-7800-4429}}
\addressSameAs{1}{Institut Terre et Environnement de Strasbourg (ITES)
UMR 7063, Universit\'{e} de Strasbourg, CNRS, Strasbourg, France}

\author{\firstname{Fran\c{c}ois} \lastname{Lehmann}\CDRorcid{0000-0002-1028-9907}}
\addressSameAs{1}{Institut Terre et Environnement de Strasbourg (ITES)
UMR 7063, Universit\'{e} de Strasbourg, CNRS, Strasbourg, France}

\author{\firstname{Mohammad}\nobreakauthor\lastname{Piri}}
\address{Center of Innovation for Flow through Porous Media (COIFPM),
Department of Energy and Petroleum Engineering, University of Wyoming,
Laramie, Wyoming, USA}

\author{\firstname{Renaud} \lastname{Toussaint}\CDRorcid{0000-0002-3698-1934}}
\addressSameAs{1}{Institut Terre et Environnement de Strasbourg (ITES)
UMR 7063, Universit\'{e} de Strasbourg, CNRS, Strasbourg, France}
\address{PoreLab, The Njord Centre, Department of Physics, University
of Oslo, Oslo, Norway}

\keywords{\kwd{Water relative permeability}\kwd{In situ water
content}\kwd{Column experiment}\kwd{Unsaturated porous medium}}

\altkeywords{\kwd{Perm\'{e}abilit\'{e} relative \`{a} l'eau}\kwd{Teneur
en eau \textit{in situ}}\kwd{Exp\'{e}rience en colonne}\kwd{Milieu poreux non
satur\'{e}}}

\shortrunauthors

\begin{abstract} 
Accurate determination of water relative permeability
(\tralicstex{\textit{k}\tsub{rw}}{$k_{\mathrm{rw}}$}) is essential for modelling flow in porous media.
Traditional methods often rely on bulk averaging and assumed unit
hydraulic gradients, which limit spatial resolution and introduce
uncertainties. This study presents a novel experimental column setup
that integrates a PICO TDR probe for in situ water content measurement
and dual pressure transducers for direct pressure gradient monitoring
under steady-state flow conditions. Experiments were conducted on two
water-wet quartz sands (P100 and P2040), which differed in grain sizes
and porosities. A stepwise drainage and imbibition protocol enabled the
derivation of both water retention curves and water relative
permeability (\tralicstex{\textit{k}\tsub{rw}--\textit{S}\tsub{w}}{$k_{\mathrm{rw}}$--$S_{\mathrm{w}}$}) curves, using
Darcy's law and matching-point normalization. Initial saturation was
achieved through upward imbibition using distilled water, preceded by a
nitrogen injection to minimize air entrapment and accurately determine
porosity. Each experiment was repeated under identical conditions to
verify the reproducibility of the results. Results show minimal
hysteresis between drainage and imbibition curves and excellent
agreement with the Mualem--van Genuchten predicted \tralicstex{\textit{k}\tsub{rw}}{$k_{\mathrm{rw}}$},
using parameters independently derived from the water retention curves.
The approach demonstrates high reproducibility and predictive accuracy,
providing a strong platform for future studies involving pore-scale
simulations, wettability alteration, or hysteresis modelling.
\end{abstract}

\begin{altabstract} 
La d\'{e}termination pr\'{e}cise de la perm\'{e}abilit\'{e} relative
\`{a} l'eau (\tralicstex{\textit{k}\tsub{rw}}{$k_{\mathrm{rw}}$}) est essentielle pour mod\'{e}liser
l'\'{e}coulement dans les milieux poreux. Les m\'{e}thodes
traditionnelles reposent souvent sur des moyennes globales des
propri\'{e}t\'{e}s des milieux poreux et des hypoth\`{e}ses de gradient
hydraulique unitaire, ce qui limite la r\'{e}solution spatiale et
introduit des incertitudes. Cette \'{e}tude pr\'{e}sente un dispositif
exp\'{e}rimental innovant bas\'{e} sur une colonne verticale
int\'{e}grant une sonde TDR PICO pour la mesure locale de la teneur en
eau, ainsi que deux capteurs de pression pour le suivi direct du
gradient de pression en r\'{e}gime permanent. Les exp\'{e}riences ont
\'{e}t\'{e} men\'{e}es sur deux sables de quartz (P100 et P2040),
pr\'{e}sentant des tailles de grains contrast\'{e}es. Un protocole par
\'{e}tapes de drainage et d'imbibition a permis de d\'{e}river \`{a} la
fois les courbes de r\'{e}tention d'eau et les courbes de
perm\'{e}abilit\'{e} relative \`{a} l'eau
(\tralicstex{\textit{k}\tsub{rw}--\textit{S}\tsub{w}}{$k_{\mathrm{rw}}$--$S_{\mathrm{w}}$}), en appliquant la loi de Darcy et
une normalisation par rapport \`{a} un \og point de r\'{e}f\'{e}rence
\fg. La saturation initiale a \'{e}t\'{e} obtenue par imbibition
primaire avec de l'eau distill\'{e}e, pr\'{e}c\'{e}d\'{e}e d'une
injection d'azote afin de limiter les pi\'{e}geages d'air et de
d\'{e}terminer avec pr\'{e}cision la porosit\'{e}. Chaque
exp\'{e}rience a \'{e}t\'{e} reproduite dans des conditions identiques
afin de v\'{e}rifier la reproductibilit\'{e} des r\'{e}sultats. Les
r\'{e}sultats montrent une hyst\'{e}r\'{e}sis minimale entre les
courbes de drainage et d'imbibition, et un excellent accord avec la
pr\'{e}diction de \tralicstex{\textit{k}\tsub{rw}}{$k_{\mathrm{rw}}$} obtenue par le mod\`{e}le de
Mualem--van Genuchten, \`{a} partir de param\`{e}tres
ind\'{e}pendamment extraits des courbes de r\'{e}tention. L'approche
propos\'{e}e d\'{e}montre une reproductibilit\'{e} \'{e}lev\'{e}e et
une pr\'{e}cision pr\'{e}dictive remarquable, constituant une base
solide pour de futures \'{e}tudes sur la simulation num\'{e}rique \`{a}
l'\'{e}chelle des pores, l'alt\'{e}ration de mouillabilit\'{e} ou la
mod\'{e}lisation de l'hyst\'{e}r\'{e}sis.
\end{altabstract} 

\thanks{French National Centre for Scientific Research (CNRS) through
the International Research Project \og{} CONTINUUM \fg{}, University of
Strasbourg, Doctoral School (ED 413), Research Council of Norway
through its Centers of Excellence funding scheme (project number 262
644), CNRS, IRP D-FFRACT}

\maketitle

{\pagebreak}

\twocolumngrid

\end{noXML}

\section{Introduction}\label{sec1}

Accurate numerical modelling of multiphase flow processes in
unconsolidated porous media plays a fundamental role in addressing
environmental challenges such as groundwater contamination, soil
remediation, and sustainable water resource management. Reliable
predictions of subsurface flow are crucial for practical applications
including pollutant transport, remediation of contaminated aquifers,
and CO\tsub{2} sequestration in geological formations 
\citep{Helmigetal2013,Celiaetal2015,Maoetal2020,Benischetal2020}. 
Effective numerical simulation of these processes requires well-defined
constitutive relationships, notably the water relative permeability
($k_{\mathrm{rw}}$) and capillary pressure--saturation
($P_{\mathrm{c}}$--$S_{\mathrm{w}}$) curves \citep{Schaferetal2020}.
These relationships directly link fluid pressures and saturations,
governing fluid distribution and mobility in porous media, thus playing
a critical role in the accurate modelling of fluid displacement in
soils and aquifers.

Capillary pressure ($P_{\mathrm{c}}$) represents the pressure
difference across the interface between two immiscible fluids in a
porous medium, and is defined by the relationship  $P_{\mathrm{c}} =
P_{\mathrm{nw}} - P_{\mathrm{w}}$ \citep{Morrow1976}, where
$P_{\mathrm{nw}}$ is the pressure of the non-wetting fluid and
$P_{\mathrm{w}}$ is the pressure of the wetting fluid. In hydrological
applications, this relationship between water and air is commonly
described as the water retention curve. Drainage corresponds to a
decrease in the saturation of the wetting fluid (in our case, water),
whereas imbibition refers to the increase in the saturation of this
wetting phase. As found in the early 1960s, water retention curves may
also depend on the direction and on the ``displacement history'' of the
water--air front, further referred to as hysteresis effects
\citep{Poulovassilis1962,Mualem1984,Parkeretal1987,StaufferKinzelbach2001}. 
Later studies have demonstrated that hysteresis effects are not the
only additional influence on this relationship and they show that there
are considerable dynamic effects on the measured capillary pressure curves
\citep{HassanizadehGray1987,BourgeatPanfilov1998,Hassanizadehetal2002,
Dahleetal2005,Lovolletal2011}. Laboratory measurement of
$P_{\mathrm{c}}$--$S_{\mathrm{w}}$ relationships traditionally involves
macroscopic methods, including the widely used hanging water column
technique \citep{Haines1930,Faybishenko1995,Klute2018}. Although
effective, these methods inherently assume global equilibrium
conditions, averaging saturation across samples, thus neglecting local
saturation heterogeneities and potentially introducing substantial
uncertainty into measured data \citep{Mouraetal2015,Ayazetal2020}.

The relative two-phase permeability of a porous medium can be
determined using various techniques, including methods based, for
example, on mathematical modelling of the two-phase flow or
experimentally in the laboratory by measuring the permeability of the
porous medium in controlled one-dimensional two-phase flows.
Constitutive relationships for water relative permeability
($k_{\mathrm{rw}}$) are often derived indirectly from capillary
pressure--saturation data using empirical models such as those of
\citet{VanGenuchten1980} and \citet{BrooksCorey1964}. Alternatively,
$k_{\mathrm{rw}}$ curves can be obtained experimentally. Laboratory
methods for determining relative permeability include flow tests in
one-dimensional columns of porous media, centrifugal methods and
analogue methods with capillary pressure measurements
\citep{Esmaeilietal2019}. Direct hydraulic conductivity  measurements
are based on steady-state as well as transient flow tests. The focus of
this work is on steady-state flow tests, where water is the only moving
fluid phase and the air present is considered immobile. During vertical
steady-state water flow through an unsaturated porous medium, the
measurement of relative water permeability, as shown by
\citet{Duryetal1998} and \citet{Fischeretal1996}, is based on
bulk-averaged measurements of water content and a fixed hydraulic
gradient of one (dimensionless, expressed as $\Delta h/\Delta L$ in
m/m), which is equivalent to the assumption of a uniform vertical water
pressure distribution in the porous medium. These assumptions, however,
limit spatial resolution and introduce significant uncertainties.

Historically, multiphase flow characterization has relied on separate
experiments for obtaining retention and permeability data. Such an
approach is experimentally intensive and may introduce inconsistencies
due to differences in boundary conditions and sample preparations
between tests \citep{Duryetal1999,Fischeretal1996}. Some studies,
highlight the importance of capturing local heterogeneities and dynamic
effects influencing capillary pressures and relative permeabilities, as
these parameters significantly affect flow simulations 
\citep{Hassanizadehetal2002,Dahleetal2005,Lovolletal2011,Toussaintetal2012}. 

In response to these methodological challenges, this study introduces a
novel experimental setup designed explicitly to measure simultaneously
and precisely both the relative permeability and water retention
curves. We employ a column system integrating in situ water content
measurements through a Time Domain Reflectometry (PICO TDR) probe,
combined with direct, continuous monitoring of local pressure gradients
via dual pressure transducers interfaced through semi-permeable porous
ceramic cups. This integrated approach allows simultaneous, spatially
resolved characterization of both retention curves 
($P_{\mathrm{c}}$--$S_{\mathrm{w}}$) and relative permeability
($k_{\mathrm{rw}}$--$S_{\mathrm{w}}$), thereby addressing limitations
identified in previous studies \citep{Duryetal1998,Fischeretal1996}.

The innovation of this experimental setup lies in its ability to
concurrently measure the hydraulic relationships under realistic
subsurface conditions. By combining in situ measurement technology
(PICO TDR and pressure transducers) with rigorous experimental
protocols (primary and secondary drainage and imbibition cycles), we
provide high-quality, reproducible datasets suitable for reliable model
calibration. Moreover, each experiment was independently replicated
under identical conditions to rigorously validate reproducibility and
minimize uncertainty.

\section{Experimental setup}\label{sec2}
\subsection{Column design}\label{ssec21}

The experimental setup consists of a vertical transparent acrylic glass
column, 20~cm in height and 3~cm in internal diameter, which allowed
visual inspection of the wetting and drainage fronts. A semi-permeable
fritted glass porous membrane (DURAN$^{\text{\textregistered}}$,
sintered borosilicate glass 3.3, \O{} 40~mm, thickness 3~mm, edge not
fused) is placed at the base to ensure uniform outflow and support the
granular material. Two porosity grades were used depending on the sand:
Por.~3 (pore size 16--40~$\upmu$m, ISO 4793) for the finer P100 sand,
and Por.~2 (pore size 40--100~$\upmu$m, ISO 4793) for the coarser P2040
sand. The column is filled with an unconsolidated porous medium, which
can be either a fine sand (P100) or a coarse sand (P2040). Although the
sands are described in detail later \mbox{(Section~\ref{ssec23}),} their
mention here is necessary to provide context for the packing and
instrumentation.

The granular material is compacted in uniform layers to minimize
preferential flow and ensure consistent packing. To ensure
compatibility with the hydraulic properties of each sand, porous
membranes with different air-entry pressures were selected. For the
finer P100 sand, a less permeable membrane with an air-entry pressure
of approximately 100~mbar was used to sustain capillary forces and
prevent premature air intrusion. Alternatively, for the coarser P2040
sand, a more permeable membrane with an air-entry pressure of
approximately 50~mbar was chosen, ensuring adequate drainage that is
compatible with its larger pore structure.

To replicate realistic hydraulic boundary conditions, a constant
mechanical overload was applied at the top of the column using small
stainless-steel spheres. These spheres, open to atmospheric pressure,
effectively mimicked natural stress conditions and maintained
consistent boundary conditions at both ends of the column. A perforated
PVC plate placed directly beneath the spheres allowed water to enter
uniformly while ensuring even distribution of the mechanical load
across the surface. This configuration also limited grain displacement
during upward imbibition, preventing free movement of particles visible
to the eye.

Water flow is imposed at the top of the column using two types of
pumps, selected based on the desired flow rate range. This choice was
made to ensure a stable and uniform inflow boundary condition under
steady-state flow. The columns were first saturated by upward
imbibition, which minimized air entrapment and guaranteed full
saturation. Once the porous medium was fully saturated, subsequent
injection from the top did not generate fingering, as the pore space
was already water-filled and flow simply displaced water downward under
gravity, capillary forces and imposed boundary conditions toward the
outlet. For standard flow conditions, a precision-controlled
peristaltic pump is employed. For experiments requiring lower flow
rates (less than 1~mL/min), a two-piston Pharmacia P-500 high-precision
pump is utilized. This pump offers a flow rate range of 1--499~mL/h
with the accuracy of ${\pm}$1.5\% of the set value.

At the base of the column, water exits through the porous membrane into
a small outflow reservoir, which allows precise control of the outlet
pressure. By raising or lowering the reservoir, the pressure at the
base of the column can be actively adjusted, allowing the imposition of
drainage or imbibition conditions as needed. From the reservoir, water
is channeled through tubing to a second collection reservoir, which is
placed in an analytical balance connected to a computer. The balance
transmits its signal via the RS232 protocol, which is converted to USB
using a RS232-USB interface for computer reading. The balance records
the cumulative mass of discharged water at 10-second intervals,
allowing real-time calculation of the flow rate. This approach improves
accuracy by accounting for potential discrepancies between the
programmed and actual flow rates, which may result from tubing
elasticity, head loss, or pump calibration drift.

To measure the pressure gradient across the porous medium, two
high-precision Keller Series PR-41X piezoresistive pressure transducers
are embedded in the side wall of the column at fixed heights (5~cm and
10~cm above the base). These sensors, with a total error band of 0.2\%
of the full scale: ${\pm}$0.4~mbar or ${\pm}$40 Pa and a measurement
range of ${-}$100 to ${+}$100 mbar, are well suited to capturing the
low-pressure differentials typical of unsaturated flow in granular
media. The sensors measure relative pressure, meaning all readings are
referenced to atmospheric pressure, allowing for accurate tracking of
both suction and positive pressure conditions within the column. Each
sensor is connected to a fully saturated, semi-permeable, ceramic
porous cup (Soil Moisture: 0652X03-B01M3, outside diameter ${=}$\ 6~mm,
total length ${=}$\ 27~mm, length in the porous medium ${=}$\ 20~mm,
ceramic Type: 1 bar High flow), which establishes hydraulic continuity
with the pore water and prevents direct contact between air and the
sensor diaphragm. This configuration ensures that only the pore water
pressure is recorded. The air phase was open to the atmosphere at the
column inlet, so the air pressure remained constant at atmospheric
conditions throughout the experiments. This setup is therefore ideal
for capturing capillary-driven processes in unsaturated conditions. The
numerical signal from the piezoresistive pressure transmitter is
transmitted via the RS485 communication protocol, allowing it to be
read by a computer.

The system is allowed to equilibrate at each experimental step until
the pressure gradient displays steady-state behaviour, ensuring signal
stability. This high temporal resolution and sensitivity enable the
calculation of the hydraulic gradient across the soil column and, in
combination with the measured flow rate, allow for the precise
estimation of hydraulic conductivity ($K_{\mathrm{w}}$) and water
relative permeabilities ($k_{\mathrm{rw}}$).

To monitor the volumetric water content within the soil column, a
TRIME$^{\text{\textregistered}}$-PICO TDR soil moisture probe is
installed horizontally at the mid-height of the column. According to
the manufacturer, the TRIME$^{\text{\textregistered}}$-PICO probe has
a small measurement volume (on the order of a few cubic centimeters
surrounding the probe rods), which provides high spatial resolution and
makes it particularly suitable for laboratory column experiments (IMKO,
2022). Moreover, the probe was positioned at the mid-height of the
column, where spatial variations in water content were expected to be
minimal under steady-state flow. This probe uses the IMP-Bus, a
specific communication protocol developed by IMKO, to transmit data.
The USB-Interface IMP-Bus converts the IMP-Bus signal to USB, making it
compatible with a PC. This sensor operates on Time Domain Reflectometry
(TDR) technology, emitting high-frequency electromagnetic pulses
(${\sim}$1~GHz) to measure the dielectric permittivity of the surrounding
medium and infer the volumetric water content. Its measurements are
minimally affected by temperature or salinity, making it reliable
across various soil types. The total sensor rod length is 50~mm;
however, to minimize disturbance within the packed column, only 20~mm
of the probe is inserted into the soil. We highlight that the
reproducibility observed across replicate experiments suggests that the
disturbance was limited. To accommodate this configuration, two hollow
PVC extensions were attached to the exterior of the column to securely
hold the sensor while preserving the integrity of the soil structure.
Data from the TDR sensor are logged concurrently with pressure and flow
measurements, providing spatially and temporally resolved insights into
the column's hydraulic behaviour. A Python script on the computer reads
all the USB inputs and converts them into numerical values representing
the data.

A schematic representation of the experimental column setup is provided
in Figure~\ref{fig1}, illustrating the positioning of the sensors, flow
path, and water recovery system. During the initial imbibition stage,
we observed that the water front progressed through the column in a
uniform manner. Minor irregularities were visible, particularly in the
coarser P2040 sand, but no fingers were developed, which is consistent
with the stabilizing effect of capillary forces in well-sorted sands,
and of gravity with the denser fluid injected from the bottom. Once the
column was fully saturated, direct observation of flow behavior was no
longer possible, although desaturation fronts could still be seen
during subsequent drainage steps. These observations are consistent
with the measured pressure and water content dynamics, supporting that
fingering did not influence the experiments.

\begin{figure}
\includegraphics{fig01}
\caption{\label{fig1}Schematic diagram of the experimental column setup
used for the measurement of water retention and relative permeability
curves. The system includes a vertical PVC column filled with sand,
equipped with two lateral pressure sensors, a PICO TDR probe at
mid-height, a fritted glass porous membrane at the base, and a
mechanical surcharge at the top applied via stainless-steel spheres.
Prior to the flow experiments, the column was saturated from the bottom
by upward imbibition to ensure homogeneous initial conditions and
minimize air entrapment. Water then enters from the top through a
peristaltic pump during the measurement stage of hydraulic conductivity
($K_{\mathrm{w}}$) at successive saturation states, and exits into a
water recovery reservoir connected to an analytical balance for flow
rate monitoring.}
\end{figure}

\subsection{PICO TDR probe calibration}\label{ssec22}

Accurate interpretation of water content data from the
TRIME$^{\text{\textregistered}}$-PICO TDR probe requires calibration
tailored to the specific granular medium in which it is embedded.
Although the sensor is factory-calibrated with a universal soil curve,
we performed a material-specific two-point linear calibration for each
sand type. This is necessary because the dielectric response varies
with grain size, packing structure, and mineral composition.

The calibration assumes a linear relationship between transient time
($t_{\mathrm{p}}$) and volumetric water content
(${\theta}_{\mathrm{w}}$), where $t_{\mathrm{p}}$ corresponds to the
radar travel time, i.e.\ the time taken by the high-frequency
electromagnetic pulse to propagate along the probe rods through the
surrounding soil measured in nanoseconds (ns). This assumption was
confirmed experimentally in specific calibration studies. To evaluate
the sensor response and determine the appropriate calibration model, we
performed a detailed calibration on a reference fine sand (N34) under
controlled ex situ conditions. NE34 is a high-purity quartz sand
(SiO\tsub{2} {${>}$} 99.7\%) with a median grain diameter ($D_{50}$) of
206~$\upmu$m. Its grain size distribution is relatively narrow, with
over 75\% of particles retained between 200 and 315~$\upmu$m. The sand
has a soil density of 2.65 g/cm\tsup{3}, and a bulk density of the
porous medium of approximately 1.7 g/cm\tsup{3}. The calibration
procedure involved preparing a series of samples at known volumetric
water contents, ranging from dry (0\%) to full saturation
(approximately 35\% volumetric water content), in increments of 5\%.
The sand was first air-dried, and known masses of distilled water were
then added to the dry sand mass to reach the target water contents. The
mixtures were thoroughly homogenized before each measurement. The
volumetric water content (${\theta}_{\mathrm{w}}$) was determined
gravimetrically from the ratio of the added water mass to the bulk
volume of sand, calculated from the known dry mass and the packing
density used in the experiments. At each water content step, the probe
was inserted and the corresponding radar travel time ($t_{\mathrm{p}}$)
was recorded. In this study, water saturation ($S_{\mathrm{w}}$) is
defined as the ratio of volumetric water content
(${\theta}_{\mathrm{w}}$) to porosity ($n$), such that $S_{\mathrm{w}}
= {\theta}_{\mathrm{w}}/n$. Therefore, full saturation corresponds to
${\theta}_{\mathrm{w}}=n$, which is approximately 35\% of the sand
used.\looseness=1

Each mixture was packed uniformly into a container, and the
corresponding radar time ($t_{\mathrm{p}}$) was measured using the PICO
TDR probe. The results revealed a strong linear relationship between
$t_{\mathrm{p}}$ and ${\theta}_{\mathrm{w}}$, with a coefficient of
determination $R^{2}>0.99$, confirming the suitability of a linear
calibration model within the relevant saturation range
(Figure~\ref{fig2}). This finding justified the use of a two-point
linear calibration for other sands, eliminating the need for full
multi-point characterization. For P100 and P2040 sands, the two
calibration points were determined in situ during the primary
imbibition phase of the column experiments. The column was initially
dry, and water was slowly introduced from the bottom until full
saturation was reached. The PICO sensor readings at the dry state
(${\theta}_{\mathrm{w}} \approx 0\%$) and at full saturation
(${\theta}_{\mathrm{w}}=$ porosity) were used to define a linear
calibration equation:    
{\begin{equation}\label{eq1}
\theta _{\mathrm{w}} = m * t_{\mathrm{P}} + b,
\end{equation}}\unskip
where $m$ and $b$ are calibration constants specific to each sand. The
in situ calibration ensured that dielectric measurements directly
reflected the actual packing and boundary conditions of each column. 

\begin{figure}
\includegraphics{fig02}
\caption{\label{fig2}Exsitu calibration curve of the PICO-TDR probe
using fine sand NE34. The plot shows the linear relationship between
the transit time ($t_{\mathrm{p}}$) and volumetric water content
(${\theta}_{\mathrm{w}}$). Calibration was performed by preparing sand
samples at known moisture levels and measuring the corresponding
$t_{\mathrm{p}}$ values. The resulting fit exhibits a high coefficient
of determination ($R^{2}>0.99$), confirming the suitability of a linear
model within the tested saturation range.}
\end{figure}

The choice of a linear calibration model is further supported by the
manufacturer's calibration curves, which show that for a wide range of
granular materials (from fine to coarse sands, silts, and lightweight
aggregates) the relationship between moisture content and radar time
remains approximately linear between 0\% to 50\% volumetric water
content. Since the porosity of our test sands (P100 and P2040) limits
their maximum volumetric water content to approximately 33\%--37\%,
all measurements fall within this linear regime. Additionally, the NE34
reference sand used in the ex situ calibration shares similar grain
size and mineral composition with P100, reinforcing the applicability
of the observed linear trend. As both test sands exhibit similar
physical characteristics and operate within the same moisture range,
the use of a two-point linear calibration is both technically sound and
practically efficient.

\subsection{Porous media}\label{ssec23}

The porous media selected for this study were two quartz sands, P100
and P2040, which were also used in previous research on the influence
of wettability on water retention curves in unconsolidated porous media
\citep{Schaferetal2025}. In our work, only the water-wet versions of
these materials were studied. Both sands consist primarily of
high-purity silica with minimal fines, and were chosen to represent
contrasting grain size distributions and pore structures while
maintaining well-sorted textural properties.

The grain size distributions of P100 and P2040 sands, characterized
using laser diffraction and reporting diameters as volume-equivalent
spheres, are shown in Figure~\ref{fig3}. The P100 sand has a median
grain diameter ($d_{50}$) of 0.21~mm and is classified as a fine sand
according to the United States Department of Agriculture (USDA) soil
classification system. The P2040 sand, with a larger $d_{50}$ of
0.65~mm, is considered coarse sand. The uniformity coefficients
($d_{60}/d_{10}$) are approximately 1.59 for P100 and 1.45 for P2040,
indicating that both materials are well sorted, with P2040 displaying a
slightly narrower size distribution. Both sands are naturally water-wet
and were used without any surface treatment or aging. The grain
morphology is angular to sub-angular, promoting a stable packing
structure. The main physical properties of both sands, including
porosity, particle density, and hydraulic conductivity, are summarized
in Table~\ref{tab1}.

\begin{figure}
\includegraphics{fig03}
\caption{\label{fig3}Grain size distribution curves for P2040 and P100
sands. The finer P100 exhibits a left-shifted distribution compared to
the coarser P2040, reflecting differences in median grain diameter and
sorting relevant for permeability analysis. According to the particle
size analysis, P100 is composed entirely of grains between 50 and
500~$\upmu$m, with 0.86\% between 50--100~$\upmu$m, 41.07\% between 100
and 200~$\upmu$m, and 58.07\% between 200 and 500~$\upmu$m. In
contrast, P2040 is dominated by coarser fractions, with 82.26\% between
200 and 500~$\upmu$m, 13.63\% between 500 and 1000~$\upmu$m, and 4.11\%
between 1000 and 2000~$\upmu$m.}
\vspace*{3pt}
\end{figure}

%tab1
\begin{table}
\caption{\label{tab1}Physical properties of the sands used in the study
\citep{Schaferetal2025}} 
\tabcolsep=3pt
\begin{tabular}{ccc}
\thead
Porous media & P100 & P2040 \\
\endthead
\parbox[t]{8pc}{\centering Hydraulic conductivity (K)
(m${\cdot}$s$^{-1}$)}\vspace*{2pt} & $1.52 \times 10^{-4}$ &  $2.26\times 10^{-3}$ \\
$K_{\mathrm{w}}$ ($S_{\mathrm{w},\mathrm{mp}}$)
(m${\cdot}$s$^{-1}$)$^{*}$ & $1.44\times 10^{-4}$ &  $1.5\times 10^{-3}$ \\
Permeability (m\tsup{2}) & $1.48\times 10^{-11}$ & $2.20\times
10^{-10}$ \\
\parbox[t]{7pc}{\centering Soil particle density (g${\cdot}$cm\tsup{3})}\vspace*{2pt}
& 2.55 & 2.59 \\
Porosity (-) & 0.368 & 0.366
\botline
\end{tabular}
\tabnote{$^{*}$Value obtained during this study.}
\end{table}

\section{Experimental methodology}\label{sec3}

This section outlines the experimental protocol and methodology adopted
to accurately determine relative permeability curves in the
unconsolidated porous media under study. The setup and sensors
described previously (Section~\ref{sec2}) were used under controlled
steady-state conditions, complemented by additional elements described
below.

\subsection{Preparation of experimental series}\label{ssec31}

Each experimental series began with the preparation of sand columns
(P100 and P2040) using a dry-packing procedure in uniform layers. The
sand was compacted consistently to avoid preferential flow paths and
ensure reproducible packing density across experiments. Prior to
saturation with distilled water, nitrogen gas was gently injected from
the column base to effectively replace the air. The subsequent upward
injection of distilled water enabled the efficient dissolution of
nitrogen, resulting in thorough and uniform water saturation throughout
the column. The total distilled water volume required to reach this
fully saturated condition was used to calculate the in situ porosity,
which served as a reference for determining water saturation throughout
the experiments.

\subsection{Drainage and imbibition protocol}\label{ssec32}

Following the initial saturation of the column with water (primary
imbibition), the experimental protocol proceeded with a primary
drainage step. The purpose of this primary drainage was to determine
the irreducible water saturation ($S_{\mathrm{wi}}$), representing the
lowest achievable water saturation under experimental conditions. To
accomplish this, the outflow reservoir was abruptly lowered, applying
immediate suction at the column base. The final height to which the
reservoir was lowered depended on the porous medium used, as each sand
type was paired with a porous membrane of a specific air-entry value
(approximately 100~cm for P100 and 50~cm for P2040, see
Section~\ref{ssec21}). These values constrained the maximum suction
that could be imposed without causing a gas breakthrough. This drainage
condition was maintained for an extended period of time to ensure
complete drainage, thereby accurately defining $S_{\mathrm{wi}}$.

Subsequently, a secondary imbibition was initiated. During this step,
water was slowly reintroduced into the column by gradually raising the
outflow reservoir, allowing saturation to progressively increase until
reaching a new maximum saturation value
($S_{\mathrm{w},\max}=1-S_{\mathrm{gr}}$), limited by trapped residual
gas ($S_{\mathrm{gr}}$). Pressure and saturation were continuously
recorded, permitting the construction of the secondary imbibition
branch of the water retention curve.

At the condition of maximum saturation, a steady-state flow regime was
established by injecting distilled water from the top of the column at
a constant rate, while simultaneously lowering the outflow reservoir to
the level of the column base. This imposed a controlled pressure
gradient across the medium. Steady-state conditions were verified by
two criteria: (i) the inflow rate matched the outflow rate, as
continuously recorded by the analytical balance; and (ii) the pressure
readings from the sensors stabilized over time. Also, a visible water
film was maintained at the top of the column. The hydraulic
conductivity ($K_{\mathrm{w}}$) was then calculated based on these
conditions. This value of $K_{\mathrm{w}}$, obtained at
$S_{\mathrm{w},\max}$ water saturation, is referred to as the matching
point conductivity $K_{\mathrm{w}}$ ($S_{\mathrm{w},\mathrm{mp}}$). It
serves as the reference used to normalize all subsequent hydraulic
conductivity measurements at lower saturation levels.
$S_{\mathrm{w},\mathrm{mp}}$ is defined as the matching point
saturation, and corresponds to the maximum water saturation
$S_{\mathrm{w},\max}$ obtained after this secondary imbibition.

A subsequent main drainage was then executed by incrementally
decreasing the saturation through controlled stepwise lowering of the
outflow reservoir, allowing water to exit while also reducing the water
injection rate at the top. At each drainage step, the column was
allowed to reach full equilibrium, defined operationally as a variation
of less than 1\% in both pressure difference (${\Delta}p$) and water
rate over a 10-min interval, facilitating a precise determination of
hydraulic conductivity.

Following the main drainage, a corresponding main imbibition cycle was
performed by gradually raising the outflow reservoir and water
injection rate, repeating the steady-state hydraulic conductivity
measurements at increasing saturation\break levels.

Finally, after completing these main cycles, a last secondary drainage
step was conducted to establish the drainage branch of the water
retention curve. For this drainage curve, the column was first fully
saturated and then subjected to a single-step depression: the outlet
reservoir was rapidly lowered, and suction was applied at the base of
the column. The system was left undisturbed for an extended period,
allowing the internal pressure to equilibrate. Capillary pressure
($P_{\mathrm{c}}$) was determined from the mean of the two pressure
transducer readings ($P_{1}$ and $P_{2}$) providing a reliable
representation of the average pressure within the soil profile. This
was then paired with water saturation data from the PICO TDR probe to
construct the drainage branch of the retention curve. Although true
equilibrium conditions are challenging to attain \citep{DiCarlo2003},
the membrane demonstrated efficient pressure transmission, enabling
reliable capillary pressure--saturation measurements at low water
content.

This systematic protocol ensured the generation of comprehensive
datasets for both water retention curves and water relative
permeability characterization under controlled and reproducible
experimental conditions.

\subsection{Data repeatability and error analysis}\label{ssec33}

To verify reproducibility and assess the reliability of the
experimental method, each test was repeated at least twice under
identical conditions. Variability in measured pressure, saturation, and
flow rates between repeated tests was quantified to estimate
measurement uncertainty.

Pressure measurements exhibited standard deviations typically below
5\%, while the volumetric water content measured by the PICO TDR probe
consistently showed variations of less than 1\% between replicate
experiments. Flow rates, continuously validated by analytical balance
measurements at the column outlet, displayed minor variations due to
regular pump calibration.

The most significant sources of uncertainty identified included slight
differences in column packing and sensor alignment. To mitigate these,
strict procedural consistency was maintained for all experiments,
including careful pre-saturation of porous ceramic cups connected to
pressure sensors, standardized packing procedures, and consistent TDR
insertion positions. 

\subsection{Quantification of hydraulic conductivity and relative
permeability}\label{ssec34}

Hydraulic conductivity ($K_{\mathrm{w}}$) was determined from
steady-state flow conditions at each drainage or imbibition step,
applying Darcy's law as follows:
{\begin{equation}\label{eq2}
K_{\mathrm{w}} = (Q*L) / [A*(\Delta z + \Delta p / (\rho*g))].
\end{equation}}\unskip
In this expression, $Q$ is the actual volumetric flow rate
(m\tsup{3}/s), calculated in real time from the mass of 
\mbox{water} collected
at the outlet and recorded continuously using an analytical balance.
The term $L$ represents the vertical distance between the two pressure
sensors (in meters), and $A$ is the internal cross-sectional area of
the column (in m$^{2}$). The hydraulic gradient driving the flow is
composed of two components: the elevation difference ${\Delta}z$,
which in this setup is equal to $L$, and the pressure difference
${\Delta}p$, converted to head units by dividing by ${\rho}^{*}g$,
where ${\rho}$ is the water density (kg/m\tsup{3}) and $g$ is the
gravitational acceleration (9.81~m/s$^{2}$).

The relative permeability ($k_{\mathrm{rw}}$) was then calculated by
normalizing each measured hydraulic conductivity to the one at the
matching point saturation ($S_{\mathrm{w},\mathrm{mp}}$), which
corresponds to the highest water saturation achieved after secondary
imbibition, accounting for the residual gas saturation
($S_{\mathrm{gr}}$). The relation is expressed as:
{\begin{equation}\label{eq3}
k_{\mathrm{rw}}(S_{\mathrm{w}}) =
K_{\mathrm{w}}(S_{\mathrm{w}})/
K_{\mathrm{w}}(S_{\mathrm{w},\mathrm{mp}}).
\end{equation}}\unskip
This normalization strategy ensured a consistent comparison of relative
permeability values across different saturation states and sand types,
providing a reliable framework for analyzing flow behavior in
unsaturated porous media.

\vspace*{-2pt}

\section{Results and discussion}\label{sec4}

\vspace*{-2pt}

\subsection{Real-time data acquisition and grain size influence}
\label{ssec41}

\vspace*{-2pt}

During each drainage step, both pressure and saturation values reached
clear steady states within the defined 1\% fluctuation threshold. The
pressure sensors exhibited stable, low-noise signals, and the PICO TDR
probe delivered continuous water content readings. This stability
ensured confidence in the calculated values of hydraulic conductivity
at each saturation level. No sensor drift or transient flow behaviour
was observed during the steady-state plateaus, confirming the
reliability of the experimental protocol.

The coarser P2040 sand exhibited higher permeability and a smoother
transition in saturation compared to the finer P100 sand, which
displayed steeper gradients and lower water mobility. This behaviour is
consistent with the influence of grain size on pore structure: coarser
media tend to have wider pores, reduced capillary forces, and enhanced
fluid flow. The finer P100 sand, by contrast, has a more constricted
pore network, resulting in greater resistance to drainage and lower
$K_{\mathrm{w}}$ values. These observations underscore the strong
influence of grain size distribution on relative permeability behavior.

\subsection{Experimental determination of relative permeability curves}
\label{ssec42}

The experimentally determined water relative permeability curves 
$k_{\mathrm{rw}}(S_{\mathrm{w}})$ for P100 and P2040 sands are shown in
Figures~\ref{fig4}a and~\ref{fig4}b, respectively. Both drainage and
imbibition branches were captured. The curves were normalized using the
matching point hydraulic conductivity ($K_{\mathrm{w},\mathrm{mp}}$)
(Table~\ref{tab1}), and plotted as a function of effective water
saturation $S_{\mathrm{we}}$. For both sands, the permeability
decreased monotonically with decreasing saturation.

\begin{figure*}
\includegraphics{fig04}
\caption{\label{fig4}Experimental data on water relative permeability,
$k_{\mathrm{rw}}$, as a function of effective water saturation,
$S_{\mathrm{we}}$. (a) P100 sand: water relative permeability curves
for drainage (red circles) and imbibition (blue triangles), normalized
by the matching-point hydraulic conductivity
$K_{\mathrm{w},\mathrm{mp}}$. (b) P2040 sand: corresponding water
relative permeability curves for the coarser porous medium. For both
sands, the curves show a monotonic decrease of $k_{\mathrm{rw}}$ with
decreasing $S_{\mathrm{we}}$, and limited hysteresis between drainage
and imbibition.}
\end{figure*}

Interestingly, P100 exhibited higher $k_{\mathrm{rw}}$ values at a
given effective saturation compared to the coarser P2040 sand. While
hydraulic conductivity ($K_{\mathrm{w}}$) is indeed greater in P2040
due to its larger pore sizes, the relative permeability reflects the
ease with which the wetting phase flows relative to the total
permeability. The P100 sand likely supports better wetting-phase
continuity, with more uniform pore spaces that promote film flow and
reduce flow resistance. In contrast, the broader pore size distribution
in P2040 may hinder the connectivity of the water phase, especially at
intermediate saturations.

For both sands, the difference between drainage and imbibition branches
was minimal, indicating limited hysteresis under water-wet conditions.
This minimal hysteresis is attributed to the water-wet nature of the
sands, the absence of wettability alteration, and the careful control
of boundary conditions during flow reversal. Under these conditions,
pore-scale displacement processes appear largely reversible, and
capillary entrapment effects are minimized. These results support the
assumption that, in homogeneous water-wet media, $k_{\mathrm{rw}}$ can
be described as a near single-valued function of effective saturation.
This observation is consistent with the results reported by
\citet{Moghadasietal2015}, who also observed minor hysteresis in water
relative permeability. Notably, in both studies, the imbibition curve
lies slightly below the drainage curve, reinforcing the general trend
that hysteresis effects are modest and saturation-driven under
water-wet conditions.

\subsection{Predictive modelling using van Genuchten parameters from
water retention curves}\label{ssec43}

To assess the predictive capability of the Mualem--van Genuchten model
for relative permeability, we used ${\alpha}$ and $n$ parameters
derived independently from water retention measurements
(Figure~\ref{fig5}) \citep{VanGenuchten1980,Luckneretal1989}. These
measurements were performed on the same sands under steady-state
conditions during secondary drainage and secondary imbibition cycles.

\begin{figure*}
\includegraphics{fig05}
\caption{\label{fig5}Main drainage capillary pressure curves and van
Genuchten model fits for (a) P100 sand and (b)~P2040 sand. Experimental
data are shown as red dots, fitted van Genuchten curves as solid blue
lines, and shaded areas represent the 95\% confidence bands. Fitted
parameter values (${\alpha}$, $n$) and their uncertainties are
indicated in the figure legends.}
\end{figure*}

The fitted retention curve provided the ${\alpha}$ and n parameters,
which were then used directly in the Mualem--van Genuchten model for
predicting relative permeability \citep{VanGenuchten1980}. The water
retention behaviour was described using the following van Genuchten
equation:
{\begin{equation}\label{eq4}
P_{\mathrm{c}} = (1/{\alpha})*
[S_{\mathrm{we}}^{(-n/(n-1))}-1]^{(1/n)},
\end{equation}}\unskip
where $P_{\mathrm{c}}$ is the capillary pressure (Pa), ${\alpha}$ (${-}$)
is related to the inverse of the air-entry pressure, $n$ is a pore-size
distribution parameter, and $S_{\mathrm{we}}$ is the effective water
saturation defined as:
{\begin{equation}\label{eq5}
S_{\mathrm{we}} = (S_{\mathrm{w}} - S_{\mathrm{wi}}) / 
(S_{\max} - S_{\mathrm{wi}}).
\end{equation}}\unskip
In this expression, $S_{\mathrm{w}}$ is the volumetric water
saturation, $S_{\mathrm{wi}}$ is the irreducible water saturation, and
$S_{\max}$ is the maximum observed water saturation during the
experiment.

\begin{figure*}
\includegraphics{fig06}
\caption{\label{fig6}Water relative permeability to water
($k_{\mathrm{rw}}$) as a function of effective water saturation
($S_{\mathrm{we}}$) for (a)~P100 sand and (b) P2040 sand. Experimental
data are shown for drainage (red circles) process. Solid lines
represent the predicted $k_{\mathrm{rw}}$ curves obtained from the
Mualem--van Genuchten model using the fitted capillary pressure
parameters presented in Figure~\ref{fig5}. Shaded areas indicate the
95\% confidence intervals of the model predictions; for P100, the
uncertainty band is extremely narrow and thus barely visible.}
\vspace*{-2pt}
\end{figure*}

The same ${\alpha}$ and $n$ values obtained from fitting the retention
curves were then inserted into the Mualem--van Genuchten model for
relative permeability:
{\begin{equation}\label{eq6}
k_{\mathrm{rw}} (S_{\mathrm{we}}) = 
S_{\mathrm{we}}^{(1/2)} * [1 - (1 - 
S_{\mathrm{we}}^{(1/m)})^{m}]^{2},
\end{equation}}\unskip
where $m=1-(1/n)$. These predicted $k_{\mathrm{rw}}$ curves matched the
experimentally measured permeability data with high fidelity for both
P100 and P2040 sands (Figure~\ref{fig6}). The obtained ${\alpha}$ and n
parameters for P100 closely align with those reported by
\citet{Schaferetal2025} under similar water-wet conditions, reinforcing
the consistency and robustness of the retention--permeability
framework. For P2040, a larger discrepancy in the fitted parameters was
observed relative to Sch\"{a}fer et~al.'s values. This difference is
primarily attributed to limited retention data at high saturation
($S_{\mathrm{w}}>0.5$) in the present study, which reduced the accuracy
of the curve fitting near saturation and impacted the stability of the
parameter estimation. Nonetheless, the model's ability to reproduce the
relative permeability behaviour of both sands confirms that reliable
predictions can still be achieved from independently derived retention
data, thereby validating the theoretical assumptions of the Mualem--van
Genuchten model in granular porous media. It is worth noting that the
95\% confidence band in the P100 plot (Figure~\ref{fig5}a) is nearly
invisible due to the high density of experimental data (over 1400
points), which greatly constrained the model and reduced uncertainty in
the fitted parameters (${\alpha}$, $n$). This low parameter uncertainty
is also reflected in the predictive relative permeability curve
(Figure~\ref{fig6}a), where the confidence band is similarly narrow. In
contrast, the P2040 dataset included fewer retention data points
(particularly at high saturations) which led to slightly larger fitting
uncertainties. As a result, the confidence bands are more visible in
both the \mbox{retention} curve (Figure~\ref{fig5}b) and the corresponding
permeability prediction (Figure~\ref{fig6}b).

\section{Conclusions}\label{sec5}

Experimental determination of capillary pressure and water relative
permeability in unconsolidated porous media remains a critical
challenge for accurately modelling multiphase flow. While previous
studies often relied on separate or indirect measurements of these
constitutive relationships, this work introduced an integrated
experimental setup capable of simultaneously quantifying both
properties under steady-state flow conditions.

Our innovative experimental setup successfully combined in situ water
content measurements via a PICO TDR probe with direct hydraulic
gradient monitoring using dual pressure transducers. This approach
yielded precise in situ capillary pressure--saturation curve 
($P_{\mathrm{c}}$--$S_{\mathrm{w}}$) and relative permeability
($k_{\mathrm{rw}}$--$S_{\mathrm{w}}$) data for two unconsolidated
quartz sands (P100 and P2040), differing significantly in grain size
and permeability. The experimental protocol, including primary drainage
cycle, secondary imbibition cycle, main drainage-imbibition cycles, and
a last drainage step allowed trustworthy quantification of the key
water relative permeability without significant hysteresis effects.
These cycles were essential for defining irreducible water saturation
($S_{\mathrm{wi}}$), maximum saturation ($S_{\mathrm{w},\max}$), and
the matching-point conductivity, thereby enabling a consistent
normalization of $k_{\mathrm{rw}}$. In addition, the secondary
imbibition cycle and the last drainage step sequence were used to
establish the $P_{\mathrm{c}}$--$S_{\mathrm{w}}$ curve.

A key feature of this methodology was the use of in situ volumetric
water content measurements to obtain saturation values, rather than
relying on full-column gravimetric weighing. While this represents a
local measurement, two main factors ensure its reliability in our
setup: (i) the sands studied (P100 and P2040) are well-sorted and
homogeneous; (ii)~the local TDR readings were consistent with porosity
values independently determined from total water volume at saturation.
Therefore, the local measurements can be considered representative of
the column under the conditions studied. In contrast, the global
balance method, which calculates water \mbox{content} from cumulative
outflow, may slightly overestimate the average water content for a
given pressure drop. This is explained by gravitational redistribution,
as water tends to accumulate near the base of the column, especially
above the porous membrane, leading to higher apparent values than
inferred from local pressure readings. This approach permitted
monitoring throughout the experiment without disrupting the packing or
flow.

In contrast to studies constrained by a unit hydraulic gradient (i.e.,
$i=1$), our experiments allowed the gradient to evolve naturally.
Hydraulic gradients in our setup ranged from approximately 0.5--2,
depending on boundary conditions and saturation state. As a result, it
was not necessary to enforce pressure equality between the two
transducers (i.e., $P_{1}\neq P_{2}$), and the full pressure
differential was used in conductivity calculations.

Analysis of the resulting retention curves confirmed the suitability of
the van Genuchten model, with independently derived parameters
(${\alpha}$, $n$) closely matching literature values reported by
\citet{Schaferetal2025}. This agreement confirms the consistency of our
protocol, emphasizing the accuracy and predictive value of in situ
measurements in porous media studies.

The reproducibility of our measurements was systematically validated
through replicated experiments conducted under identical conditions,
demonstrating minimal variability and thus ensuring confidence in the
reliability and accuracy of the dataset. Such reproducibility is
essential, particularly when applying experimental data to refine
numerical simulations of fluid transport and contaminant migration in
unconsolidated porous systems.

From a methodological perspective, the integrated setup offers
significant potential for future studies. Extending the current
methodology to explore scenarios of different wettability, such as
oil-wet or mixed-wet conditions, would significantly advance
understanding of how wettability affects water relative permeability
and retention properties. Such studies would have direct implications
for environmental engineering, particularly in soil remediation
applications involving contaminants like hydrocarbons, where
wettability alterations are commonly observed and can greatly influence
remediation efficiency.

Ultimately, the experimental data obtained in this study provides a
foundation for the development and validation of advanced numerical
models. Accurate in situ characterization of both retention and
relative permeability properties will greatly enhance the predictive
capability of numerical codes used for environmental and subsurface
flow management. Future numerical implementations could specifically
incorporate dynamic and wettability-dependent constitutive
relationships derived from this experimental approach, significantly
improving model realism and reliability for practical applications in
groundwater management, contaminant transport modelling, and multiphase
flow simulations.

\section*{Acknowledgements}

This research was funded by the French National Centre for Scientific
Research (CNRS) through the International Research Project
\og~{}CONTINUUM~\fg{}, established between ITES and COIFPM. We are also
grateful for the financial support provided by the University of
Strasbourg and for the administrative and technical support from the
University of Wyoming. We would like to express our sincere thanks to
the Doctoral School (ED 413) for the funding of the PhD thesis of KH. 
RT also acknowledges the support of the Research Council of Norway
through its Centers of Excellence funding scheme, project number 262
644, and the support of the CNRS and IRP D-FFRACT. Special thanks are
also extended to Pascal Friedmann, for the development of experimental
cells, and to Martine Trautmann of the EOST soil analysis laboratory,
University of Strasbourg, for measuring the particle size distribution
of the sands using the laser diffraction method, and to Michelle Church
for the proofreading of the manuscript.

\CDRGrant[RCN]{262 644}

\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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