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Gravitational and Finite‐Size Effects On Pressure Saturation Curves During Drainage

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We experimentally and numerically study the influence of gravity and finite-size effects on the pressure-saturation relationship in a given porous medium during slow drainage. The effect of gravity is systematically varied by tilting the system relative to the horizontal configuration. The use of a quasi two-dimensional porous media allows for direct spatial monitoring of the saturation. Exploiting the fractal nature of the invasion structure we obtain a relationship between the final saturation and the Bond number SF =Bo^0.097 using percolation theory. Moreover the saturation, pressure and Bond number are functionally related, allowing for pressure-saturation curves to collapse onto a single master curve, parameterized by the Representative Elementary Volume size and by the Bond and Capillary Numbers. This allows to upscale the pressure-saturation curves measured in a laboratory to large representative elementary volumes used in reservoir simulations. The large-scale behavior of these curves follow a simple relationship, depending on Bond and Capillary number, and on the flow direction. The size distribution of trapped defending fluid clusters is also shown to contain information on past fluid flow, and can be used as a marker of past flow speed and direction.
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Gravitational and Finite-Size Effects On Pressure
Saturation Curves During Drainage
Monem Ayaz1,2,3 , Renaud Toussaint1,3,4 , Gerhard Schäfer2, and Knut Jørgen Måløy3
1University of Strasbourg, CNRS, Institut de Physique du Globe de Strasbourg, UMR 7516, Strasbourg, France,
2Université de Strasbourg, CNRS, LHyGeS UMR 7517, Strasbourg, France, 3PoreLab, The Njord Centre, Department of
Physics, University of Oslo, Oslo, Norway, 4Lavrentev Institute of Hydrodynamics, Siberian Branch, Russian Academy
of Sciences, Novosibirsk, Russia
Abstract We experimentally and numerically study the influence of gravity and finite-size effects on
the pressure-saturation relationship in a given porous medium during slow drainage. The effect of gravity
is systematically varied by tilting the system relative to the horizontal configuration. The use of a quasi
two-dimensional porous media allows for direct spatial monitoring of the saturation. Exploiting the fractal
nature of the invasion structure, we obtain a relationship between the final saturation and the Bond
number SF
nw =Bo0.097 using percolation theory. Moreover, the saturation, pressure, and Bond number are
functionally related, allowing for pressure-saturation curves to collapse onto a single master curve,
parameterized by the representative elementary volume size and by the Bond and capillary numbers. This
allows to upscale the pressure-saturation curves measured in a laboratory to large representative
elementary volumes used in reservoir simulations. The large-scale behavior of these curves follows a
simple relationship, depending on Bond and capillary numbers, and on the flow direction. The size
distribution of trapped defending fluid clusters is also shown to contain information on past fluid flow and
can be used as a marker of past flow speed and direction.
1. Introduction
Displacement of one immiscible fluid by another in a porous medium is an important research topic both
from a fundamental and an applied perspective. An increased understanding of the basic mechanisms that
govern the pore-scale description is of interest for multiple disciplines of science such as soil-science, hydrol-
ogy, physics, and biology and has shown industrial importance through applications such as enhanced
oil recovery, CO2sequestration and by mapping and controlling of migrating ground water contaminants
(Nsir et al., 2012). Furthermore, it has revealed a variety of pattern-forming processes emerging from the
pore scale up to the system scale (Lenormand et al., 1983; Måløy et al., 1985; Méheust et al., 2002; Løvoll
et al., 2004; Toussaint et al., 2005), typically governed by the interplay between viscous, capillary, and gravita-
tional forces. The structures have shown to exhibit a complex behavior, characterized by its rich intermittent
dynamics (Clotet et al., 2016; Furuberg et al., 1988; Måløy et al., 1992; Moura et al., 2020; Moura, Måløy,
Flekkøy, et al., 2017; Moura, Måløy, & Toussaint, 2017; Planet et al., 2009).
We study two phase flow in a quasi two-dimensional (2-D) porous confinement and look at the simple case
of drainage at pore scale, where a nonwetting phase displaces a wetting one. Such experiments have shown
to generate displacement structures that depend on the density and viscosity contrast between the fluids,
surface tension, and the flow rates at which the system is driven (Lenormand et al., 1983; Løvoll et al., 2004;
Toussaint et al., 2005). Furthermore, such structures are assembled by trapped regions of wetting phase,
completely surrounded by the invading phase. The trapped islands are characterized by their power law
distribution in size with an exponential cutoff directly related to the gravitational forces (Blunt & King, 1990)
and the system's finite size.
The forces at play dominate at different length scales and their interplay gives rise to separate scaling regimes
allowing for an upscaling of the system which consists of relating the pore-scale description to properties
defined at the Darcy scale or even at the macroscopic scale (Toussaint et al., 2011; Vasseur et al., 2013;
Wilkinson, 1986). In the limit of very slow flow rates, the emerging structure is seen to depend entirely
on capillary and gravitational fores (Birovljev et al., 1991; Lenormand et al., 1983; Méheust et al., 2002),
RESEARCH ARTICLE
10.1029/2019WR026279
Key Points:
During slow drainage, the residual
saturation can be obtained as a
scaling law of the Bond number or
the system size
A general pressure-saturation law
is obtained for primary drainage
arising from hydrostatic effects,
fractal properties, and the
system size
The final saturation after drainage
allows to obtain information on the
flux while it took place
Correspondence to:
R. Toussaint,
renaud.toussaint@unistra.fr
Citation:
Ayaz, M., Toussaint, R., Schäfer, G.,
& Måløy, K. J. (2020). Gravitational
and finite-size effects on pressure
saturation curves during drainage.
Water Resources Research,56,
e2019WR026279. https://doi.org/10.
1029/2019WR026279
Received 16 SEP 2019
Accepted 10 SEP 2020
Accepted article online 16 SEP 2020
©2020. The Authors.
This is an open access article under the
terms of the Creative Commons
Attribution License, which permits
use, distribution and reproduction in
any medium, provided the original
work is properly cited.
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and the obtained structure is seen to be well modeled by invasion percolation (Chandler et al., 1982;
Wilkinson, 1984). When a low viscous fluid displaces a high viscous one at large enough rates, the unsta-
ble displacement pattern that emerges is called viscous fingering. The analogy between diffusion-limited
aggregation (DLA) and viscous fingering was first proposed by Paterson (1984) and is based on the equiva-
lence between the probability field of diffusing particles and the pressure field in the viscous fluid. On the
other hand, when a highly viscous fluid is injected into a lowly viscous fluid, a stable displacement will be
observed. This situation is analog to the anti-DLA as proposed by Paterson (1984).
Numerical modeling of two-phase displacement in water-saturated porous media using continuum mod-
els has been performed by many researchers (Aggelopoulos & Tsakiroglou, 2009; Kokkinaki et al., 2013;
Nayagum et al., 2004; Schneider et al., 2015; Sleep et al., 2015; Zhang & Smith, 2001). However, the detailed
local description of the invasion front in a porous medium cannot be simulated with these models because
they do not explicitly consider the capillary and viscous pressure fluctuations at the pore scale. To over-
come this shortcoming, microscopic approaches such as Stokes' solution for flow in single pores with a
specified geometry or lattice Boltzmann methods (LBM) have been investigated to reproduce experiments
(Aursjø et al., 2010; Fiorentino et al., 2017; Misztal et al., 2015). Similarly, pore-scale network models have
been developed to study a wide range of displacement processes, including drainage and imbibition (Aker
et al., 1998; Joekar Niasar et al., 2009; Kallel et al., 2017; Nordhaug et al., 2003; Nsir et al., 2012, 2018; Sheng
& Thompson, 2013; Singh & Mohanty, 2003; Tørå et al., 2012; Yang et al., 2019). Ewing and Berkowitz (1998)
developed a generalized growth model based on invasion percolation to simulate immiscible displacement in
saturated porous media. Capillary, viscous, and gravity forces were incorporated in their three-dimensional
(3-D) algorithm, where viscous forces were expressed stochastically rather than explicitly. Glass et al. (2001)
developed a macromodified invasion percolation (MMIP) model, including also the effects of all three forces
within the invading phase in a macroheterogeneous porous medium.
In macroscale modeling of two-phase flow, the porous medium and the flow are described by continuous
mathematical fields using an empirical functional relationship, coupling the saturation of the wetting fluid
Swto the capillary pressure pnw pw=p(Sw), which is the main assumption at hand, where pnw and pwis
the pressure in the nonwetting fluid and pwthe pressure in the wetting one. In practice, this relation can
be found by conducting experiments on a given porous medium and fluid pair on representative volumes.
It is usually parameterized using mathematical models such as Van-Genuchten or Brooks-Corey model
(Brooks & Corey, 1964; van Genuchten, 1980). In the case of primary drainage, the Brooks-Corey model
introduces several model parameters such as pore size index, entry pressure needed by the invading fluid
to displace water, and the residual saturation of the wetting phase. The latter one is an important model
parameter because the description of two-phase flow at the macroscopic scale requires also the use of a
second constitutive relationship: relative permeabilities as function of saturation of the wetting phase.
In this paper we study the capillary pressure-saturation (PS) relationship during drainage in a system of
finite size, taking into account gravitational effects. We examine how this finite size and gravitational forces
influence this PS relationship and how it can be upscaled. First, we propose a simple numerical scheme
based on invasion percolation (IP) in a gradient which produces PS curves. In this numerical scheme we take
into consideration that we have a gravitational field acting on the system, a distribution of capillary pressure
thresholds linked to the pore-scale geometry and the fact that the system has a finite size. Next, we study the
relative pressure of the wetting fluid measured at the outlet of a transparent quasi two-dimensional exper-
imental setup with respect to the imposed constant atmospheric pressure of the nonwetting air phase as a
function of the saturation, ΔP=Papw=𝑓(Sw). Comparing experimental and numerical results allows to
analyze what part of the experimental pressure signal is mostly controlled by the approximate description
underlying the IP in gradient algorithm. It also allows to express analytically the pressure field in the numer-
ical model system, to perform geometrical analysis on the invasion structure and hence to perform further
approximations that cast light on the origin of the shape of the macroscopically measured capillary-pressure
relationship. From the PS relation obtained during the detailed gravity-stabilized experimental and numer-
ical primary drainage studies, we can then directly derive one of the significant characteristics of water
retention curves which is the residual saturation of the wetting phase Sw=1Snw. We further deduce a
functional relationship, which couples the final saturation with the pore-scale description under the influ-
ence of gravity using percolation theory. This allows to upscale the pressure saturation relationship for a
given porous medium, with the help of laboratory drainage tests. We also show how the residual saturation
depends on the Bond number and the conditions of drainage flow, in terms of flow angle and flux.
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2. Theory
When the viscous forces are negligible for the invasion process (capillary regime), the obtained displacement
pattern can be understood by the relative importance of gravity to capillary forces, which is convention-
ally defined by the Bond number Bo, corresponding to the ratio of buoyancy forces to capillary forces at a
pore scale,
Bo =Δpgrav
Δpcap
=Δ𝜌gasin 𝛼
𝛾a,(1)
where Δ𝜌is the density contrast between the two fluids, gis the gravitational acceleration, 𝛼is the angle
between the average flow direction and the horizontal plane, ais the typical pore size, and 𝛾is the surface
tension between the two fluids. Δ𝜌ag sin 𝛼is the hydrostatic pressure drop on a length scale a, parallel to the
flow direction. The typical ratio between viscous and capillary forces over a pore is defined by the capillary
number
Ca =Δpvisc
Δpcap
=𝜇va2
𝛾𝜅 ,(2)
where 𝜇is the viscosity of the most viscous fluid, vis the Darcy velocity, and 𝜅is the permeability of the
porous medium.
When the least viscous fluid displaces the most viscous one, as considered here, viscous forces destabilize
the system and promote fingering. When the least dense fluid invades the system from the top, as also con-
sidered here, gravity forces stabilize the system and promote a horizontal front, on average. Variations of
the capillary pressure thresholds promote a small-scale roughness of the front. The separation between sta-
bility and instability is set by comparing the Bond and capillary numbers (Toussaint et al., 2005): A stable
situation corresponds to Bo >Ca (Méheust et al., 2002), an unstable one to Bo <Ca (Løvoll et al., 2011).
We consider situations where the nonwetting invading fluid has negligible viscosity in front of the defend-
ing one. The presence of disorder in the porous medium gives a distribution of capillary thresholds with a
well-defined spread. In order to capture the relative relevance of these fluctuations, we utilize the fluctuation
number Fl as suggested by Méheust et al. (2002) and Auradou et al. (1999).
Fl =C(Bo Ca),(3)
where the dimensionless prefactor is C=𝛾∕(aWt), with 𝛾/athe characteristic average capillary pressure
threshold and Wtthe standard deviation of the random capillary thresholds in the porous medium. For most
media, the average value and the standard deviation of the capillary pressure thresholds are of the same
order of magnitude (Toussaint et al., 2005), and Cis of order 1. The invasion front is defined as the set of
pores bordering the nonwetting phase to the bulk of the defending fluid. From percolation theory the front
width 𝜎for a gravity-stabilized drainage front is found to scale with the fluctuation number Fl as (Birovljev
et al., 1991; Wilkinson, 1984, 1986)
𝜎aFl𝜈∕(1+𝜈).(4)
Here 𝜈is a critical exponent of the correlation length from percolation theory, 𝜈=43in 2-D (Stauffer &
Aharony, 2014) and 𝜈=0.88 in 3-D (Adler et al., 1990). Since the fluctuation number depends on the width
of the capillary threshold distribution, an increase in its width results in an increase of the front width.
Throughout our study in sections 3, 4, and 5, we will consider Ca Bo; hence, we will utilize the reduced
expression Fl =[𝛾∕(aWt)]Bo Bo, so that
𝜎aBo𝜈∕(1+𝜈).(5)
3. Experimental Setup
The experiments are performed in a quasi two-dimensional porous medium (cf. Figure 1), which is made
up by randomly distributing a monolayer of glass beads (of diameters ranging from 0.9 to 1 mm) between
the sticky sides of two sheets of adhesive contact paper in a transparent Hele-Shaw cell with thick walls.
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Figure 1. Schematics of the experimental setup. The porous medium is sandwiched between two sheets of adhesive
contact paper and kept rigid by clamping it between a water-filled pressure cushion pressed against a rigid transparent
plate and a plexiglass plate. A pulley connected to a light box is used to provide the desired inclination 𝛼.
The boundaries are sealed off with a rectangular layer of silicon glue. Hence, the porous medium can be
considered as a rectangular box of length L=14.5cm, width W=10 cm, and thickness b=1mm.
Appended to the system is a filter, positioned between the fixed glass beads and the outlet. The filter consists
of a sponge with a typical pore size much smaller than the pores in the main porous matrix. The resulting
difference in entrance pressure makes it necessary to invade all clusters connected to the outlet filter, before
invading the outlet filter.
To prevent the potential displacement of beads, the porous matrix is kept rigid by a thick plexiglas plate from
above with milled inlet and outlet channels allowing for the injection and extraction of fluid phases. From
below a mylar film is kept under pressure by a 3.5 m high water column, acting as a pressurized cushion
forcing the glass beads above to stay in place. The rigid model is mounted onto an illuminating light box that
can be tilted by an angle 𝛼over the horizontal, allowing for tuning of the gravitational component gsin 𝛼.
The invading nonwetting phase used in all the experiments is air. The wetting phase used is glycerol-water
solution with a 90% glycerol by weight, dyed with a 0.1% of negrosine for visual contrast. The wetting fluid
has a density of 𝜌w=1,205 kg ·m3and kinematic viscosity of 𝜈=4.25 105m2·s1. The surface tension
between the two fluids is 𝛾=6.4·102N·m1.
During the experiments the relative pressure difference between the nonwetting and wetting phase is mea-
sured by two Honeywell 26PCAFG6G flow-through pressure sensors, connected to the outlet of the model.
The saturation of the two phases is extracted by thresholding the grayscale map from images taken at fixed
time lapse with a NIKON D7100 single reflex camera (SLR) camera mounted perpendicular to model, sim-
ilarly to the procedure used by Moura et al. (2017). A syringe pump is used to provide constant flow rates,
with a discharge rate of Q=6.67 ·1011 m3·s1, corresponding to a capillary number of Ca =1.2·104. The
experiments are terminated once the pores boarding the filter have all been invaded.
4. The Network Model
IP algorithms (Chandler et al., 1982; Moura et al., 2015; Wilkinson & Willemsen, 1983) have been exten-
sively used to model drainage in the capillary regime between immiscible fluids when the viscous pressure
drop and compressibility of the wetting fluid are negligible for the displacement of the fluid-fluid interface,
contrarily to cases where viscous forces (Løvoll et al., 2011) or compressibility (Jankov et al., 2010) play an
important role.
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In such models the invasion is represented by a cluster growth process on a random lattice, dependent
solely on the global homogeneous pressure in each phase and on the local capillary pressure thresholds in
the pore throats along the interface. Such arrangements have been shown to generate displacement struc-
tures that are quantitatively similar to what is observed during drainage (Blunt et al., 1992; Lenormand
& Zarcone, 1985). In this study we will utilize a bond invasion percolation model. The porous medium is
modeled as a completely connected network, where all the pore sites are interconnected to their neigh-
boring pores through pore throats. The pore throats are oriented at 45from the main flow direction on a
rectangular lattice with periodic boundary conditions perpendicular to the flow direction.
Disorder is introduced in the model by letting pore throats sizes be drawn from a distribution corresponding
to the experimental one. First, a random variable is picked from an uniform distribution in [0,1]. Next, a
mapping is made transforming the flat distribution to the experimental one, obtained utilizing an imaging
based algorithm. The size distribution of the pore throats channel radii is characterized on the experimental
porous matrix, utilizing a Delaunay triangulation over all the constricting beads, as suggested by Bryant
and Blunt (1992) and detailed by Moura et al. (2017). For each pore throat, the channel radius Rfixes the
corresponding capillary threshold pressure needed to be overcome in order to invade the adjacent pore site,
from Young Laplace law, ΔPt
c=𝛾2R.
The algorithm of IP in a Gradient to describe drainage in a gravitational field follows the principles intro-
duced by Sapoval et al. (1985) and Birovljev et al. (1991). Gravity is accounted for by linearly weighting
the capillary throat thresholds with the hydrostatic pressure difference between the two fluids, taking into
account that the pressure in the wetting phase at elevation zabove the outlet is Pw=Poutlet 𝜌wgz, where
Poutlet is the pressure in the wetting fluid at the outlet position.
Initially, the network is fully saturated with a wetting fluid. For each iteration the interface of the nontrapped
part of the defending structure moves where the capillary pressure threshold adjusted by the hydrostatic
pressure drop is lowest (see Equation 9). This movement involves both the invasion of the pore throat and
of the connecting pore. The parts of the interface that enclose (trap) entirely a connected part of the defend-
ing fluid will be locked due to the incompressibility assumption of the defending fluid: No further invasion
is allowed into these trapped clusters. Once breakthrough is reached, that is, once a pore along the outlet
is invaded, the simulation is continued, but the pore throats between the model and the outlet cannot be
invaded, reflecting the presence of the semipermeable filter at the outlet. The other pore throats, internal
to the model and along the boundary of the invasion structure, can still be invaded if they are connected
to the outlet via the wetting phase, similarly to the rule applied before breakthrough. Hence, after break-
through, the invasion of pores inside the model and connected to the outlet continues. Finally, the invasion
is stopped once all the sites bordering the outlet of the model have been fully invaded: These sites prevent
the connection of any remaining wetting fluid filled pores and the outlet, effectively blocking any further
displacement of the wetting phase. The aspect ratio of the IP lattice was equal to the experimental system
for all the simulations performed. The size of the lattice used was 232 ×330 sites.
The previous paragraph describes the numerical algorithm. It is worth noting that the numerical inva-
sion events are ordered, but the ordering indexes are not directly proportional to physical time. Physically,
when a pore throat and the adjacent pore are invaded, such movement may eventually give access to one
or more neighboring pores of lower capillary pressure threshold, which the interface will invade at once
(in simulations where the outlet pressure is not reduced), and repetition of this process can lead to the col-
lective displacement known as a Haines jump (Haines, 1930). If, by contrast, the outlet pressure is allowed
to decrease to have a hydrostatic pressure in the defending fluid during this series of numerical invasion
events, a Haines jump corresponds to a set of successive invasion events during an avalanche, defined by the
outlet pressure series: An avalanche starts when the outlet pressure starts to decrease and finishes when it
rises again above the value it had prior to this decrease. Such avalanches in experimental systems have been
studied, for example, by Måløy et al. (1992) and Moura et al. (2020).
5. Results
Figure 2 shows a selected set of snapshots captured for different Bond numbers, displaying the growth of the
displacement structure from the start of the invasion up to the end of the experiments, when all the pores
bordering the outlet channel have been invaded and the final saturation SF
nw is reached. The gray map images
have been thresholded. In the resulting binarized images, air and beads are displayed in white; the wetting
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Figure 2. Snapshots from four different experiments. The air invades the saturated porous media from top to bottom;
the time evolution for each Bond number goes from left to right. In these binarized snapshots, air and beads are white,
and the wetting fluid is black. The first two columns show the displacement structure after, respectively, 8% and 16% of
air has saturated the medium taking 2.7 and 5.4 hr; the two last ones show the Breakthrough and final configurations.
The duration of the experiments with increasing inclination is 18.9, 25.7, 26.1, and 26.6 hr.
fluid is displayed in black. For comparison, snapshots from the simulations are presented in Figure 3. To first
order, simulations and experiments shown in Figures 2 and 3 are qualitatively similar. The main difference
noted is the detailed shape of the patterns at small scale, which could appear slightly more ramified in the
experimental case than in the numerical one. This could be due to the fact that several pores can be invaded
simultaneously in the experiment, which is not the case in the simulations. During such an invasion, the
capillary number becomes locally larger in the experiments, whereas it stays strictly zero in the simulations.
Also, the shape of the pore networks is different between the two cases (rectangular in one case and between
a monolayer of spheres placed at random in the other).
We observe that an increase of the gravitational component leads to a denser displacement structure, that
is, a higher SF
nw. The invasion front, which refers to the set of pores bordering the bulk of the wetting phase,
shows a reduced spread 𝜎along the flow direction; that is, when the Bond number increases, the system
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Figure 3. Snapshots from network simulations with Bond numbers comparable to those set by the inclinations in the
experiments. Invading air is in white; dense wetting defending fluid is in black. The first two columns show the
displacement structure after, respectively, 8% and 16% of air has saturated the medium; the two last ones show the
breakthrough and final configurations.
tends to become more stabilized and the fronts are hence closer to a horizontal flat structure. The hydrostatic
pressure gradient suppresses the maximum height difference between points along the front.
The changes observed in the structure in Figures 2 and 3 between the experiments (or simulations) in hor-
izontal cells, at Bo =0, and the other cases at nonzero angle are more important than the changes between
the other cases. This is coherent with the fact that the interface width predicted by Equation 5 diverges for
Bo 0. As will be detailed later, and as was shown by Moura et al. (2015), the interface width is set by finite
size effects for the horizontal case at Bo =0. The large areas of trapped wetting fluid observed for Bo =0
are also related to this large interface size.
In these experiments, the cells were cleaned and resaturated in wetting fluid prior to each experiment, but
the same bead configuration was used for every experiment. Tests done after rebuilding the cell with other
bead configurations changed the exact position of the fluctuations in pressure saturation curves and location
of the trapped wetting fluid islands but did not change the overall behavior reported in Figures 2 and 6.
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Figure 4. Pressure-saturation curves obtained for five different inclinations
of the experimental model. Here Snw is the saturation of nonwetting phase,
and ΔPcorresponds to the pressure difference between the outlet and the
atmospheric pressure.
5.1. Measuring PS Curves
The saturation of the nonwetting phase Snw is extracted by image analysis
from the sequence of captured images. The saturation increases linearly
with time, in accordance with the constant flow rate imposed on the
system. The pressure drop across the model is at any time such that it
maintains the constraint by the flow rate. The pressure ΔPis the differ-
ence between the atmospheric pressure Paand the pressure in the wetting
phase at the outlet
ΔP=PaPoutlet.(6)
Combining the extracted saturation with the pressure measurements, a
series of PS curves are obtained, as shown in Figure 4. The equivalent
measurement is done on the IP simulations and shown in Figure 5. The
first data point on these curves of pressure difference between the two
phases, in this system is nonzero, which is related to the varying hydro-
static pressure ΔPhacross the system, which scales as the gravitational
term gsin 𝛼for different inclinations. The respective initial values of the
pressure offsets are followed by a pressure buildup without any pore inva-
sion. As the initially flat interface begins to fill the local geometry of the
pore throats, the capillary pressure ΔPcstarts to build up. Pore throats
have widths that are spatially uncorrelated, and thereby, each pore throat exhibits an associated capillary
threshold pressure Pt
cwhich is the minimum pressure needed for the interface to penetrate its respective
pore throat. For a given porous structure, fluid pair, and wettabilities there will be a distribution of capillary
pressure thresholds associated to the system. The condition for invasion is that the local pressure difference
between two phases along the front satisfies ΔPc=Pt
c. In the presence of gravity, height differences along the
invasion front give rise to a hydrostatic term. When accounted for, this gives the following invasion criterion:
ΔPc=Pa−(Poutlet −Δ𝜌gz)=ΔP𝜌gz =Pt
c,(7)
where z=zsin 𝛼is the elevation of the pore throat considered above the cell bottom, with zthe linear
upward coordinate along the cell. zvaries between 0 at the outlet, at the bottom of the cell, and Lat the
inlet at the top of the cell. This can also be expressed as
ΔP=Pt
c−Δ𝜌(gsin 𝛼)z.(8)
Figure 5. The pressure saturation curves obtained numerically, for a
system size with the same aspect ratio L×Was the experimental system.
Therefore, the front will move at the place with the lowest modified
capillary pressure threshold
Pt
c−Δ𝜌(gsin 𝛼)z(9)
When this criterion is met, the smallest capillary threshold is invaded fol-
lowed by a rapid decrease in pressure during the unstable Haines jump
where pores are invaded until the interface stabilizes on smaller pore
throats. As the porous boundary is horizontal, the initial pressure buildup
associated with the invasion of the first pores is independent of gravi-
tational effects, that is, it is invariant with respect to the Bond number.
Once the lowest capillary threshold is reached by ΔPcin Equation 7,
it fluctuates as the pressure drops during these Haines jumps and rises
again. It fluctuates during the whole process, since the subsequent cap-
illary thresholds of invaded pores Pt
care distributed. As the invasion
front progresses downward, the values of zwhere a pore invasion takes
place slowly decrease. Consequently, the pore pressure difference ΔP,
from Equation 8, slowly increases on average with the invading fluid
saturation, in addition to the fluctuations associated to those of Pt
cand
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the Haines jumps. These fluctuations consist in many buildups followed by sudden relaxations (occurring
at many time scales), allowing for the invasion of multiple pores during one burst (Furuberg et al., 1996;
Haines, 1930; Måløy et al., 1992). Some differences can be noted about the shape of the pressure fluctuations
between the experimental (Figure 4) and numerical (Figure 5) results: The pressure signal seems more reg-
ular in the experiments, with less high frequency content. Although pores are invaded one by one in terms
of numerical steps in the simulations whereas several pores are invaded in a Haines jump in the experi-
ments, this is unlikely to be the source of the discrepancy: Multiple pore invasion (in real physical time) is
easy to define in an IP model: It is enough to consider the number of pores necessary to reach again the
same pressure level and define this as an avalanche. This is the usual definition, as in fiber bundle model
dynamics, reviewed, for example, by Pradhan et al. (2010). These avalanches are present in our simulations.
Three other factors can make these fluctuations different between the simulations and experiments: (1) The
sampling rate of the experimental sensor is not so fast as to capture the fastest fluctuations. (2) The capillary
pressure thresholds distribution was not perfectly matched in the simulations due to limited knowledge on
it in the experiments: The right width Wtwas used, but the shape might differ, which leads to different fea-
tures for these fluctuations. (3) Viscous effects are not negligible during the unstable phases; they exist in
the experiments, for which results depend on where the pressure is measured but not in the simulations.
These differences are subtle, and the latter is certainly the main source of discrepancy in the shape of the
jumps (rounded aspect at small scale in Figure 4, and not in Figure 5).
The hydrostatic pressure gradient tends to suppress any height differences between two points along the
front. Two distinct features of the PS curve are observed. The pressure difference is seen to fluctuate, cor-
responding to Haines jumps, arising from the fact that the aforementioned capillary threshold pressure
displays disorder at the invasion front. We further see that the pressure difference ΔPin Equation 8 increases
because the hydrostatic pressure difference Δ𝜌gsin(𝛼)z' decreases linearly as the invading phase approaches
the outlet. The slopes observed in the PS curves during experiments, in Figure 4, and simulations, in Figure 5,
are consistent with each other and reflect this increase of ΔPas the invasion phase approaches the outlet
and the nonwetting phase saturation Snw increases. The increase of the slope with Bo is also consistent with
the sin(𝛼)term in Equation 8.
Once the invading phase reaches breakthrough, the invasion structure is in contact with the filter prolongat-
ing the cell, presenting much smaller pore throats, so that this filter is not invaded, while pores with higher
capillary thresholds are invaded in the cell. This leads to an increase of the capillary pressure as the invasion
goes on. Once the pores bordering the filter have all been invaded, the system is said to have reached its final
saturation SF
nw. We observe that there is a consistency between the numerical (Figure 5) and the experimen-
tal (Figure 4) final saturation values as the inclination increases. A larger difference between experimental
and numerical final saturations SF
nw can be noticed in the case of Bo =0. This case is on the verge of marginal
instability, which happens at Bo =0and Ca =0. This case is very slightly unstable, corresponding to a very
low capillary number Ca. This can be interpreted from the work of Moura et al. (2015), done in such a con-
figuration. In this case, a characteristic correlation length of saturation diverges, and finite-size effects come
into play to determine the final saturation. Large fluctuations are present in the final saturation between
different realizations of experiments, or of simulations, or between different subsets of the system. Figures 4
and 5 correspond to one realization each, and the fact that they differ significantly from each other at Bo =0
presumably reflects this dispersion around the average value for this configuration. As will be detailed later
on, the correlation length is smaller at larger Bo, so that the macroscopic final saturation reflects the average
of a larger number of independent subsystems of size determined by the correlation length. Consequently,
the system-size finite saturation is less dispersed between realizations when Bo increases. This is consis-
tent with the better match observed for the final saturations SF
nw between experiments and simulations in
Figures 4 and 5 at larger Bond numbers.
5.2. Master Curve: Relating the Pressure Drop and Saturation for Different Bond Numbers
For the case Bo >0, that is, when the lightest fluid penerates from the top and gravity stabilizes the sys-
tem, and the system is in mechanical equilibrium, the total pressure difference across the porous model
is expressed as ΔPPc−ΔPh. Here ΔPcis the capillary pressure across the interface, given by the
Laplace pressure, Equation 10, and ΔPhis the hydrostatic pressure difference across the wetting phase.
For slow drainage where Ca 1, this pressure difference is essentially hydrostatic and can be expressed as
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Figure 6. Schematics of a fully saturated porous medium, which has been
drained until the average front position is in a Distance l from the outlet.
ΔPh𝜌gl sin 𝛼. Here, Δ𝜌is the density contrast between the flu-
ids, the term gsin 𝛼is the component of gravity parallel to the cell, and
lzis the distance between the point considered along the two fluids
boundary and the outlet, projected along the steepest descent direction.
Since gravity stabilizes the front, one can consider a characteristic dis-
tance las the mean in-plane distance between the outlet and the invasion
front. Thereby, the measured pressure across the cell can be expressed as
follows:
ΔP=2𝛾R−Δ𝜌gl sin 𝛼. (10)
Furthermore, a functional dependency of the saturation with respect to l
can be expressed, by considering the volume conservation of the nonwet-
ting phase. Saturation is size and region dependent. To distinguish here
between local values and global one, we note the global saturation, which
is the one measured in most experiments, as the system size or total sat-
uration Stot
nw. It corresponds to the average saturation in the entire cell of
size Atot =L×W, that is, to the saturation measured in Figure 4. We
approximate the system as locally transiting from an initial saturation S0
nw
to a characteristic residual final saturation SF
nw after the passage of the
drainage front, that is, above it, compare Figure 6, which is a simplified
representation of the saturation maps of Figures 2 and 3. Fluctuations
along the invasion front are assumed to average out, and the thickness of
this front is neglected in front of the system size L, that is, the local satu-
ration is supposed to jump from the initial value S0
nw, under the front, to
the final value SF
nw, above the front. We denote AF=(LlWthe area of
the region where the invasion front has passed. By volume conservation,
we get an expression for the total saturation
Stot
nwAtot =SF
nwAF+So
nw(Atot AF).(11)
where Atot =LW is the total area, i.e.
Stot
nwLW =SF
nw(Ll)W+So
nwlW .(12)
Dividing this equation by SF
nwLW leads to
l
L=
Stot
nw
SF
nw
1
S0
nw
SF
nw
1
.(13)
Figure 7. Data collapse of pressure as function of the relative saturation,
for the experimentally obtained data sets.
Since the system is initially fully saturated, S0
nw =0. This gives an expres-
sion for the distance to the outlet as l=L(1Stot
nwSF
nw). Inserting this in
Equation 10 leads to the following saturation pressure relation:
ΔP=2𝛾
R𝛾L
a2Bo 1Stot
nw
SF
nw .(14)
This equation has been expressed with the Bond number from
Equation 1. In Figure 7, the rescaled pressure is plotted as a function of
the relative saturation obtained from the experiments shown in Figure 2,
whose raw saturation pressure measurements are shown in Figure 4. Sim-
ilar results obtained from the numerical modeling whose snapshots are
shown in Figure 3 are displayed in Figure 8: we here show the enve-
lope of the PS curves, that is, the maximum of the pressure difference
reached up to a certain saturation, without displaying the fluctuations
related to the Haines jumps. The term 2𝛾/Rcorresponds approximately
to the entrance pressure; that is, it is approximated as the rise in ΔPasso-
ciated to the invasion of the first pore. More precisely, the term 2𝛾/R
to consider corresponds to the capillary pressure at the characteristic
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Figure 8. Data collapse of pressure as function of the relative saturation,
for the numerically obtained data sets.
position of the invasion front, separating the initial saturation zone and
the final saturation one. The position of this front can be evaluated exper-
imentally from its average position in the snapshots in Figure 3. At a given
time, the boundary of the fully saturated cluster is extracted from image
analysis. The average of the zpositions of the pixels belonging to this
boundary defines the average front position l. At the same time, ΔPis
measured and 2𝛾/Ris evaluated using Equation 10, by linear regression
of the central (straight) part of the Plcurve at non zero saturation.
Theoretically, the characteristic position of the front can alternatively be
defined as the position where the occupancy probability corresponding
to the capillary pressure drop is equal to the percolation threshold for the
type of network defined by the porous matrix. As demonstrated by Moura
et al. (2015), this corresponds to
2𝛾R=𝑓1(pc),(15)
with pcthe percolation threshold in terms of occupancy probability,
𝑓(P)=P
0g(P)dPis the cumulative distribution of the normalized
capillary threshold distribution (probability density function) g(P), and
f1is the reciprocal function of f. In other words, the constant term in
Equation 14 corresponds to the capillary pressure threshold corresponding to the percolation threshold in
the pore network. This is the procedure used to determine 2𝛾/Rin the simulations, where the distribution of
capillary pressure thresholds f(p) and square network connectivity allowing to determine pc=12(Moura
et al., 2015; Stauffer & Aharony, 2014) are known.
We see that by inserting pore-scale description of the system, and taking into account the gravity effect across
the cell, we are able to collapse the PS curves onto a single master curve: This equation, Equation 14, cap-
tures properly the entrance pressure, the rise of the pressure drop with the drainage, and the final pressure
buildup when the invader is limited by the semipermeable membrane. This is the case both for the exper-
iments (Figure 7) and for the simulations (Figure 8), in all probed effective gravities (i.e., cell angles over
the horizontal). This reduced pressure has two terms: One is related to the disorder in the system, and the
second term is related to the hydrostatic height difference, which decreases as the drainage takes place and
the average interface moves down, closer to the outlet.
5.3. Final Saturation as Function of the Bond Number
Previous research has allowed to explicit a functional dependency between final air saturation and the spatial
extension of the system size in the absence of gravity, that is, for the flat case, in the capillary regime (Moura
et al., 2015). Løvoll et al. (2011) extended the analysis to take dynamical effects into account, allowing a
broad range of capillary numbers to be permitted, in a viscosity-destabilizing situation, that is, when the
least viscous fluid invades the system, without stabilizing effects of gravity. In this section we propose a
scaling relation between the residual saturation (at the end of the drainage) and the Bond number for a
gravity-stabilized situation, that is, when the lighter fluid is initially on top of the denser one.
The invaded region in the capillary regime is shown to display a fractal structure (Lenormand et al., 1983;
Wilkinson, 1984), with a well-established fractal dimension (Mandelbrot, 1982) of Dc=1.83 for a two-
dimensional (2-D) system (Lenormand & Zarcone, 1985). Exploiting together this fractal property of the
invaded area and the scaling properties of the invasion front, a functional relationship between the final
saturation SF
nw and the Bond number or system size can be obtained. The final saturation of the system SF
nw
is attained once all the pores closest to the outlet have been invaded. We first note that the largest connected
clusters of defending fluid, that stay trapped after the invasion interface moved through the system, are built
by the motion of this interface. Before they are disconnected of the large cluster connected to the outlet,
the boundary of these clusters is part of the mobile interface. Their maximum size is reached when they
become disconnected, after what their size does not evolve significantly. Hence, the maximum linear size of
the trapped clusters corresponds to the typical size of the interface in the average flow direction. This cutoff
length scale lccorresponds thus to the root-mean-square width of the mobile front, that is, lc𝜎, which
follows the scaling law from Equation 5, set by IP in a gradient, as established by (Birovljev et al., 1991):
lc𝜎aBo𝜈∕(1+𝜈).(16)
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Figure 9. The final saturation as a function of the Bond number is plotted
both for the experimental and numerical results together with the predicted
2-D relation SF
nw =Bo0.097.
Up to this size, the system is fractal with a fractal dimension correspond-
ing to capillary fingering and IP, Dc. Above this size, the mobile front
looks like a straight horizontal line. The fractal dimension of the set of
trapped clusters at scales above this size is thus not set any more by the
selection of smallest thresholds but by the geometry of the straight line
leaving trapped clusters everywhere behind it: It presents the trivial frac-
tal dimension of the space left behind the front, that is, D=2in two
dimensions (2-D). If the system size is smaller than this characteristic size
aBo𝜈/(1 +𝜈), the system is fractal up to the system's shortest size, Wor L.
To summarize, the system is fractal with dimension Dcup to a cutoff size
lc=Min(aBo𝜈∕(1+𝜈),W,L).(17)
The functional relationship between the final saturation and lcis found by
considering a partitioning of the model of dimensions L×Winto square
boxes of length lc. For every such box the displacement structure occupies
on average (lca)Dcpores. Above this size, the fractal dimension is the
trivial fractal dimension D=2, and the number of boxes of size l2
cwith
trapped clusters required to cover the system is simply LW l2
c. Moreover,
a pore area is a2, the system's area is LW , and the final saturation is the
fraction of the total area occupied by the nonwetting fluid. Hence,
SF
nw =LW
l2
clc
aDca2
LW =lc
aDc2
.(18)
Using Equation 17 hence leads to
SF
nw Bo[−𝜈∕(𝜈+1)](Dc2),(19)
where 𝜈is a critical percolation exponent, having the value 4/3 in 2-D (Stauffer & Aharony, 2014). This is
the case when gravity sets up the cutoff scale, that is, when aBo𝜈/(1+𝜈)<Min(W,L). By contrast, when the
system size or gravity is negligible, we predict
SF
nw Min(Wa,La)(Dc2).(20)
This last relation is satisfied by experiments and simulations in horizontal cells where Bo 1 for slow
drainage where Ca 1, as demonstrated by Moura et al. (2015).
This relation, Equation 19, holds for Bo >Ca, where the invading fluid from the top has the lower density. In
this case, corresponding to the experiments and simulations done here, we obtain a functional dependency
between the final saturation and the Bond number. In order to check this predicted dependency, the final
saturation SF
nw was measured experimentally (circles) and numerically (stars) for a range of Bond numbers
and shown in Figure 9. From our theoretical prediction we expect
SF
nw Bo0.097,(21)
This corresponds to the dashed line in this figure. The agreement between both simulations and experiments
and this theoretically predicted scaling seems satisfactory. The spread of the final saturation is seen to narrow
in as the Bond number increases, which is associated with the finite size effects being suppressed and the
increasing dominance of the width 𝜎as the limiting factor.
6. Discussion
6.1. Extensions in Three and One Dimensions
This type of argumentation can be extended to three dimensions (3-D). Instead of considering partitioning
of square boxes, we now consider cubes which on average occupy (lca)Dcpores. This leaves us with the
general relation
SF
nw Bo[−𝜈∕(𝜈+1)](DcD),(22)
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Figure 10. Final saturation and corresponding Bond number for four
different soil samples, for different fluid pairs in a given medium. The Bond
number was here varied by changing the wetting phase. The soil types in
these experiments by Nouri et al. (2014) are Persian Gulf Sand (PGS), sandy
loam (SL), silty clay (SiC), and silty loam (SiL). The striped lines indicate
the power law slope obtained from the theoretical prediction.
where Dis 2 or 3 depending on the case considered, 2-D or 3-D. Here
the critical exponent 𝜈=0.88 in 3-D from percolation theory (Adler
et al., 1990), and the fractal dimension obtained from IP is found to be
Dc=2.5(Wilkinson & Willemsen, 1983). Comparative experimental
measurements have also been made (Chen et al., 1992; Nsir et al., 2012;
Yan et al., 2012) ((Toussaint et al., 2012) for a review), finding compatible
Dcvalues. Hence, we expect in 3-D,
SF
nw Bo0.23.(23)
Data in 3-D with detailed capillary-pressure relationships and systemat-
ically changing Bond number are more scarce than in 2-D, as it is easier
to change the orientation of the 2-D cell than it is to perform 3-D experi-
ments where the gravitational field is changed and the wetting properties
are kept fixed. Doing so would require to perform the experiments in a
given medium and with a given fluid pair in a centrifuge, which is an
interesting perspective for future work. To check nonetheless the level
of consistency or inconsistency between the predicted 3-D relation from
Equation 23 and the experiments, the proposed prediction was compared
with measurements conducted by Nouri et al. (2014). Here saturation and
pressure were measured using a hanging column apparatus for four dif-
ferent soil types: namely, persian gulf sand (PGS), sandy loam (SL), silty
clay (SiC), and silty loam (SiL). For every soil type drainage experiments
were performed for four different wetting phases. We determined the final volumetric content of each wet-
ting fluid as function of the soil texture by digitizing the presented data and extracting the resulting final
air saturation. We note that the displacement of petroleum by air was extremely low and resulted in very
small saturation variations even by applying very high capillary pressure values. The reason for this abnor-
mal behavior is not yet known; we therefore used only the well-documented experimental data obtained
for kerosene, diesel, water, and gasoline. The Bond number was calculated for each fluid pair and soil type,
using the given surface tension for each of the four wetting fluids and the density contrast between the wet-
ting fluid and air, the mean grain size diameter (d50, the median value of the diameters) as the typical pore
size, and setting 𝛼=𝜋2. The experimental data from Nouri et al. (2014) were then analyzed for each soil
type separately as shown in Figure 10. This is justified by the large degree of nonuniformity of the differ-
ent grain size distributions. The only uniform porous medium is that of the Persian Gulf Sand whose linear
trend gives a slope which is the closest to the theoretical one (0.23), although the number of points is lim-
ited to allow a precise agreement—only a trend can be evaluated. This limitation is due to the difficulty of
changing the fluid pair for every point related to an experiment performed.
As the process relies on disorder in thresholds, a large dispersion from experiment to experiment is expected.
Further, the capillary number is assumed to be negligible in the derived relation. In these experiments, this
approximation is not necessarily perfectly valid. In addition, the wetting angle is probably different between
fluid pairs. Furthermore, the finite size of the sample will influence the final saturation (Moura et al., 2015).
Some limitations of classical soil physics experiments to determine large-scale multiphase flow have been
discussed in Hunt et al. (2013). Hence, this comparison between this experimental data and the theoreti-
cal prediction shows some moderate consistency but stays limited. This does not validate nor invalidate the
proposed model in 3-D but rather shows the limitations to apply it in small-scale laboratory experiments.
This also shows that large enough systems have to be used experimentally to determine a capillary-pressure
relation that can be used in large-scale simulations, or alternatively, that the experimental tests performed
should allow to extract the parameters in the model proposed here, such as the average capillary pressure
threshold term 2𝛾/Rand use the proposed model for the large-scale simulations. Checking better the suit-
ability of the proposed model in 3-D in its range of validity could be done using experiments in a centrifuge
where gravity is systematically changed, which is a perspective of future work.
Eventually, the general Equation 22 also holds in one dimension (1-D), which is an academically trivial case
but interesting to look at for the sake of checking the generality of the arguments. Indeed, in this case, all
pores are along a single line and are invaded one after the other from the top. Hence, the final saturation is
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1. In such case, D=1, and since all cells are occupied behind the front (composed of one pore), the fractal
dimension of the invaded cluster is also Dc=1. In this case, 𝜈=1(Stauffer & Jayaprakash, 1978) and
Equation 22 predicts SF
nw =1independently of Bo, which is in agreement with the fact that all cells along
the linear network are eventually occupied with the invader, that is, that indeed SF
nw =1.
6.2. Average Capillary Pressure
Here, we have considered the capillary pressure as determined experimentally, between the outlet at the
wetting fluid and the inlet pressure in the nonwetting one. By contrast, in theoretical or numerical studies,
the pressure saturation relationship is often expressed for a quantity defined as the difference between the
average pressure in one fluid at the representative elementary volume (REV) scale and the average pressure
in the other fluid at the same scale (Hassanizadeh & Gray, 1990, 1993a, 1993b; Hassanizadeh et al., 2002;
Reeves & Celia, 1996). This average pressure difference can also be approximated for the current simula-
tions and experiments, in order to use in simulations the quantities determined experimentally. This would
allow to use experiments to determine the characteristic capillary threshold (2𝛾/R) present in Equation 14.
That could be obtained from experiments, or from the characterization of the capillary threshold distribu-
tion f(P) and determinations of the percolation threshold pc: either approximate ones or precise ones by
characterization of the pore connectivity and IP simulations on such a network. One could then use the
model proposed below in Equation 30 or Equation 31 as a large-scale average capillary-pressure relation-
ship in reservoir models, rather than using directly the measured relation ΔP(Snw) in hanging columns as a
direct large-scale model of average capillary-saturation relationship.
Since the variation of the pressure under the approximately horizontal interface is hydrostatic, from
Equations 9 and 15, the pressure at distance zfrom the bottom is thus
P(z)=Poutlet −Δ𝜌g(sin 𝛼)z=Pa−ΔP−Δ𝜌g(sin 𝛼)z=Pa𝑓1(pc)+Δ𝜌g(sin 𝛼)(lz),(24)
with ΔP=PaPoutlet and lthe front average position.
Hence, averaging over the region of the defending fluid connected to the outlet, one can define the average
wetting fluid pressure in this zone as ̄
Pand the difference between the average pressures in the nonwetting
fluid and the wetting one, that is, in these experiments between the atmospheric pressure and the average
wetting fluid pressure Δ̄
P=Pā
P, obtaining:
̄
P=1
ll
0
P(z)dz=Pa𝑓1(pc)+Δ𝜌g(sin 𝛼)l
2=Pa𝑓1(pc)+ 𝛾LBo
2a21Snw
SF
nw (25)
and
Δ̄
P=𝑓1(pc)+ 𝛾LBo
2a2Snw
SF
nw
1.(26)
Eventually, another way to perform the space average of the pressure in both phases is to consider the
whole space, including the trapped fluid region behind the front, and not only the connected fluid con-
sidered above. This alternative definition of average wetting fluid pressure can then be defined as ̄
P=
1
LL
0ΔP(z)dz. In order to evaluate this average, one can approximate the wetting fluid pressure in the
trapped clusters above the front, in the zone where l<z<L, as the percolating pressure threshold, P(z)=
Pa𝑓1(pc). Thus,
̄
P=l
L̄
P+1l
L(Pa𝑓1(pc)) = Pa𝑓1(pc)+Δ𝜌g(sin 𝛼)l2
2L(27)
and
Δ̄
P=Pā
P=𝑓1(pc)−Δ𝜌g(sin 𝛼)l2
2L=𝑓1(pc)+ 𝛾LBo
2a2Snw
SF
nw
12
.(28)
6.3. Dynamic Effects
Dynamic effects also affect the average pressure gradient and the shape of the interface during drainage. In
a situation similar to the one studied so far, that is, with a low viscous fluid invading the cell, so that viscosity
destabilizes the front, and the lightest fluid entering on top of the densest fluid, so that gravity stabilizes the
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Figure 11. Figure shows our modeled general pressure-saturation law
represented by three linear branches.
front, Méheust et al. (2002) showed that the front root-mean-square width
still follows the same scaling law, with a generalized Bond number Bo=
Bo Ca. All the other arguments developed above are expected to hold.
Since the results only depend on the average pressure gradient and the
presence of the disorder, one thus expects that
SF
nw ∝(Bo Ca)[−𝜈∕(𝜈+1)](DcD),(29)
when Bo >Ca, that is, when the situation is intrinsically stable. This is
expected to hold both in 2-D and 3-D, replacing Dby2or3andDcand 𝜈
by their proper values for each dimension.
In three dimensions, the Bond number is defined as Bo pgrav∕Δpcap ,
where Δpgrav is evaluated in the average flow direction, that is, with vthe
average flow velocity, Δpgrav =𝜌(g·v)∕v, so that Bo =𝜌(g·v)a∕(v𝛾).
Conversely, when Ca >Bo, the situation becomes unstable, and the front
splits in fingers that grow in a ballistic way. In such case, a scaling law
can also be obtained between the residual saturation and the system size,
as shown in Løvoll et al. (2011). This study also proposed a relationship
between pressure and saturation related to this unstable situation.
Eventually, the scaling law in Equation 29 allows to propose a track for future applications: Since the residual
saturation depends on the capillary number at which a drainage takes place, it can be used as a marker
of the flux that took place during this drainage: Measuring the residual saturation and knowing the Bond
number (characterizing the fluid densities, surface tension, and pore sizes), this equation allows to estimate
Ca during the drainage. In situations where a drainage took place, this can give a precious estimator to
evaluate the conditions at which the flow took place. This could lead to interesting applications in geology,
where fluids mix in porous media, as, for example, around a magma reservoirs, in hydrology, or to determine
the speed of a past flow during an underground contamination. We propose this as a perspective of future
research.
Some limits of this estimation will naturally affect this estimator. Notably, in a few hours, film flow can lower
further the residual saturation (as, e.g., shown by Moura et al. 2019), and for longer times, evaporation and
transport as vapor phase (Or et al., 2013) will also affect it. This estimator could thus be suitable for fluid
pairs where the evaporation and reconfigurations after the first drainage stay limited, such as cases where
one of the fluids hardens due to cooling after an initial multiphase flow or in laboratory experiments where
the measurements are done shortly after the primary drainage.
6.4. General PS Law Expected for Drainage
Combining Equations 14, 29, and 20, one obtains a general approximate law for primary drainage, arising
from the hydrostatic effects in the system, the fractal properties of the trapped clusters and of the invasion
front, that can be represented by three linear branches, as displayed in Figure 11.
When ΔP<Pt
c𝛾L
a2(Bo Ca), there is no pore invaded and
Stot
nw 0,(30)
where Pt
c=𝑓1(pc)is the capillary pressure threshold whose occupancy probability 𝑓(Pt
c)is equal to the
percolation threshold pcfor the porous media network considered, depending on its connectivity.
When Pt
c𝛾L
a2(Bo Ca)<ΔP<Pt
c,
ΔP=Pt
c+𝛾L
a2(Bo Ca)Stot
nw
[Min(Wa,La,(Bo Ca)𝜈∕(𝜈+1))](DcD)1,(31)
When ΔP>Pt
c, eventually,
Snw =SF
nw ≃[Min(Wa,La,Bo𝜈∕(𝜈+1))](DcD),(32)
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where 𝜈and Dcare the critical percolation exponent and the capillary fingering dimension for the spatial
dimension D(3-D or 2-D).
This behavior corresponds to the observed one in the experiments and simulations, in Figures 4, 5, and to
the behavior studied in Moura et al. (2015) in the case of Bo =0,Ca =0.
In addition, for a system with straight boundaries and exit filters, there is also a curvature at the beginning of
the Pressure-Saturation curve and at the end, related to the deformation of the invasion front from an initial
straight shape to a shape characteristic of percolation invasion in a gradient close to the inlet and conversely,
from such shape to a straight shape against the semipermeable exit filters. Consequently, the entrance pres-
sure, 2𝛾
Rmin with Rthe minimum curvature of the interface in the largest pore throat size, is always smaller
than Pt
c. Such effects of curvature are nonetheless related to the initial and final imposed shape of the inter-
face in tests and should be absent in a REV in an open system, with no imposed shape of the front at the
entrance and exit of that REV. This would be the natural situation for an REV in a reservoir simulation,
where neither semipermeable filters are present between REVs nor initial saturation jumps can be found
along artificial planar boundaries between REVs. The asymptotic behavior described above in Equations 30,
31, and 32 should then be better adapted for such natural situations. Experimentally determined pres-
sure saturation relationships, usually obtained using hanging columns or setups with semipermeable filters
at the exit and initially totally saturated, as in the experiments shown here, are subject to such artifacts.
They should thus be corrected for these curvatures related to the boundary effects in order to be used in
reservoir models.
7. Conclusions
PS relationships are essential in soil science, hydrology, and for other communities who model multiphase
flow in reservoirs and porous media using Darcy equations. To quantify PS curves in the laboratory with
the porous plate (or diaphragm) method, researchers generally use soil samples with a height of only sev-
eral tens of centimeters. The obtained relationships are thus potentially strongly affected by the boundary
effects introduced by the semipermeable membrane, depending on the pore sizes and on the Bond number
associated to it. In our work, we highlight the influence of gravity on the form of PS curves during primary
drainage. The overall pressure difference between the two fluids is therefore influenced by the hydrostatic
pressure difference across the wetting fluid phase. For a system unaffected by boundary effects, the PS rela-
tion during such drainage is then entirely specified by a characteristic capillary pressure that can be obtained
from percolation threshold pcfor the occupancy probability and the cumulative distribution of capillary
pressure thresholds fas f1(pc), a straight line with a slope given by (𝛾L/a2)BoSF
nw, and a final saturation,
SF
nw =Bo[(nu∕(1+nu))(DDc)]. We further show in this study that an IP scheme generates PS curves consistent
with experimental measurements on a REV, for varying Bond numbers. By using a pore-scale description of
the system under the influence of gravity, we are able to collapse the PS curves onto a single master curve.
Furthermore, a functional dependency between the final saturation, the strength of the gravitational com-
ponent (Bond number), the correlation length 𝜈from percolation, and the fractal dimension of capillary
fingering is obtained for 2-D and 3-D. For the 2-D case we find a good fit with our experimental and numer-
ical results. For the 3-D relation we compared with experiments conducted by Nouri et al., which showed a
limited compatibility with the predicted relationship and hinted to possible limitations of small-scale exper-
iments to obtain large-scale capillary-pressure relationships. This leads to tracks to possible future works
in centrifuge to probe further the model in 3-D in its range of validity. Furthermore, knowing the final air
saturation obtained from water retention curves, using the theoretical relation of Equation 22, it is pos-
sible to predict the final nonwetting saturation when other wetting fluids than water are displaced by air
during primary gravity-stabilized drainage. With these elements together, one predicts an approximate cap-
illary pressure-saturation relation as function of dimension, capillary and Bond numbers, and of the system
dimension. Such prediction is done between the air pressure and wetting fluid pressure measured at the
bottom. The theoretically corresponding difference between the average pressures in both phases is also
derived. Knowing the pore size distribution and the characteristics of the fluid pair and the wetting proper-
ties of the system, our results give a guideline for simulations and upscaling of the saturation from small to
large scales.
AYAZ ET A L . 16 of 19
Water Resources Research 10.1029/2019WR026279
Data Availability Statement
The simulation code can be found online (https://zenodo.org/badge/latestdoi/273205082 [https://doi.org/
10.5281/zenodo.3899977]). The experimental raw data are found online (https://doi.org/10.5281/zenodo.
3900005 [https://doi.org/10.5281/zenodo.3900004]).
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Acknowledgments
This work was partly supported by the
Research Council of Norway through
its Centres of Excellence funding
scheme, Project 262644 (PoreLab), the
LIA France-Norway D-FFRACT, the
University of Strasbourg PDI program,
and the CNRS INSU ALEAS program.
We thank Marcel Moura and
Eve-Agnes Fiorentino for fruitful
discussions.
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AYAZ ET A L . 19 of 19
... 2 of 16 notable role (Ayaz et al., 2020;Kuang et al., 2011;Luo et al., 2020). Moreover, there are now published datasets consisting of stacks of many tomograms taken at different elevations of the sample (Jackson, 2019) resulting in very tall images. ...
... As long as the vertical extent H of the domain of immiscible displacement satisfies , fluid invasion patterns are not affected by gravitational forces, with the result that capillary pressure and saturation distribution are spatially uniform. In practice this condition is often not met, such that fluid saturation is not uniform but varies with vertical position in a way determined by the equilibrium between capillary and gravity forces (Ayaz et al., 2020;Clement et al., 1987;Ioannidis et al., 1996;Shahidzadeh-Bonn et al., 2004;Zhao et al., 2013). Until now there did not exist a means of including gravity effects in porous media drainage simulations via an image-based modeling technique. ...
... The Python package, PoreSpy (J. Gostick et al., 2019) was used to generate 2D and 3D porous material images with specified properties by stochastically generating overlapping disks (2D) or spheres (3D), as well as non-overlapping spheres to compare to the micromodel experiments of Ayaz et al. (2020). For 2D, a 3 cm wide by 8 cm tall (1,500 px by 4,000 px) domain was generated with disks of diameter 1 mm to create a porous medium with a target porosity of 65% and a resolution of = 0.02 (i.e., one pixel = 0.02 mm). ...
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... Let L be the length of the porous model and assume L > w. The final saturation behind the front of the invading fluid and its dependence on the pressure across the model has been studied in Ref. [81]. In this study, we considered the volume of the invaded fluid in boxes with size corresponding to the width of the invading front. ...
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We present a theoretical and experimental investigation of drainage in porous media. The study is limited to stabilized fluid fronts at moderate injection rates, but it takes into account capillary, viscous, and gravitational forces. In this theory the work applied on the system, the energy dissipation, the final saturation and the width of the stabilized fluid front can all be calculated if we know the dimensionless fluctuation number, the wetting properties, the surface tension between the fluids, the fractal dimensions of the invasion front and the invading structure, and the exponent describing the divergence of the correlation length in percolation. This theoretical description explains how the Haines jumps' local activity and dissipation relate to dissipation on larger scales.
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We give experimental grounding for the remarkable observation made by Furuberg et al. in Ref. \cite{furuberg1988} of an unusual dynamic scaling for the pair correlation function $N(r,t)$ during the slow drainage of a porous medium. The authors of that paper have used an invasion percolation algorithm to show numerically that the probability of invasion of a pore at a distance $r$ away and after a time $t$ from the invasion of another pore, scales as $N(r,t)\propto r^{-1}f\left(r^{D}/t\right)$, where $D$ is the fractal dimension of the invading cluster and the function $f(u)\propto u^{1.4}$, for $u \ll 1$ and $f(u)\propto u^{-0.6}$, for $u \gg 1$. Our experimental setup allows us to have full access to the spatiotemporal evolution of the invasion, which was used to directly verify this scaling. Additionally, we have connected two theoretical results from the literature to explain the functional dependency of $N(r,t)$ and the scaling exponent for the short-time regime ($t \ll r^{D}$). A new theoretical argument was developed to explain the long-time regime exponent ($t \gg r^{D}$).
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