Energy 30 (2005) 861–872
www.elsevier.com/locate/energy
Competition of gravity, capillary and viscous forces
during drainage in a two-dimensional porous
medium, a pore scale study
Grunde Løvoll a,b,, Yves Méheust a,b,c, Knut Jørgen Måløy a, Eyvind Aker a,c,d,
Jean Schmittbuhl b
a
b
Department of Physics, University of Oslo, Pb. 1048 Blindern, N-0316 Oslo, Norway
Laboratoire de Géologie, École Normale Supérieure, 24, rue Lhomond, 75231 Paris cedex 5, France
c
Department of Physics, Norwegian University of Science and Technology, Høgskoleringen 5,
N-7491 Trondheim, Norway
d
WesternGeco, Schlumberger House, Solbråv.23, N-1383 Asker, Norway
Abstract
We have studied experimentally and numerically the displacement of a highly viscous wetting fluid by a
non-wetting fluid with low viscosity in a random two-dimensional porous medium under stabilizing gravity. In situations where the magnitudes of the viscous-, capillary- and gravity forces are comparable, we
observe a transition from a capillary fingering behavior to a viscous fingering behavior, when decreasing
apparent gravity. In the former configuration, the vertical extension of the displacement front saturates;
in the latter, thin branched fingers develop and rapidly reach breakthrough. From pressure measurements
and picture analyzes, we experimentally determine the threshold for the instability, a value that we also
predict using percolation theory. Percolation theory further allows us to predict that the vertical extension of the invasion fronts undergoing stable displacement scales as a power law of the generalized Bond
number Bo ¼ Bo Ca, where Bo and Ca are the Bond and capillary numbers, respectively. Our experimental findings are compared to the results of a numerical modeling that takes local viscous forces into
account. Theoretical, experimental and numerical approaches appear to be consistent.
# 2004 Elsevier Ltd. All rights reserved.
Corresponding author. Tel.: +47-228-564-44; fax: +47-228-564-22.
E-mail address: [email protected] (G. Løvoll).
0360-5442/$ - see front matter # 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.energy.2004.03.100
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G. Løvoll et al. / Energy 30 (2005) 861–872
1. Introduction
Two-phase flows in porous media are related to many important industrial and geological
applications, such as oil recovery or ground water flow modeling [1–4]. For immiscible flows, a
wide range of behaviors are observed depending on the wetting properties of the two fluids,
their viscosity ratio, their respective density, and the displacement rate [5].
In this study, we address drainage in a random two-dimensional porous medium. Drainage
consists in the immiscible displacement of a wetting fluid by a non-wetting fluid, with capillary
forces acting against the flow. The role of capillary forces is of special importance in a porous
medium, where the interface between the two fluids consists of many menisci; capillary forces
act at the scale of these menisci, that is, at the pore scale (see Fig. 1) and therefore are significant with respect to other forces that govern the displacement. Indeed, the capillary pressure,
which is the difference between the pressures in the non-wetting and in the wetting phase at a
point of the interface, is defined by the well-known Young–Laplace law:
1
1
;
(1)
þ
pc ¼ pnw pw ¼ c
R1 R2
Where c is the surface tension between the liquids and R1 and R2 are the two principal radii of
curvatures for the interface. For a straight tube with perfect wetting, this reduces to pc ¼ 2c=r,
where r is the tube radius, while, for a ‘‘real porous medium’’ the typical capillary pressure
results from the typical pore size a. Furthermore, in a random porous medium, the random heterogeneity of the capillary pressure distribution greatly affects the local dynamics of the interface, with larger pores being more easily invaded by the displacing fluid, along the interface.
The study by Saffman and Taylor [6] of the displacement of one fluid by another one in a
Hele-Shaw cell has shown how viscous forces can either stabilize or destabilize the interface,
depending on whether the displacing fluid is the more viscous or not, respectively, and how the
presence of the denser fluid below the other one has a stabilizing effect. Destabilization of the
interface results in a fingering of the displacing fluid into the displaced fluid. The Saffman–
Taylor theory is not sufficient to describe the displacement in a porous medium, since it does
not take into account the capillary fluctuations at the pore scale, nor does it account for the
Fig. 1. The difference between the pressure in the non-wetting fluid and that in the wetting fluid is given by the
Laplace law; during drainage, capillary forces act against the displacement, and larger pores are more easily invaded.
G. Løvoll et al. / Energy 30 (2005) 861–872
863
fluctuations in the viscous forces [4,7]. But it still explains the stabilizing or destabilizing tendency of viscous forces and gravity. In what follows, we consider a configuration where a wetting
fluid is being displaced by a fluid placed above it, the viscosity and density of which are much
smaller than those of the wetting fluid. Hence, viscous forces tend to destabilize the interface
against the stabilizing effect of gravity.
The type of displacement observed during drainage in two-dimensional porous medium therefore depends on the relative magnitude of viscous forces and gravity, but also on their relative
magnitude with respect to the heterogeneous capillary forces. A set of dimensionless numbers is
usually defined to quantify these relative magnitudes. The capillary number Ca is the typical
ratio of the viscous pressure drop at pore scale to the capillary pressure, while the ‘‘Bond
number’’ quantifies that of the typical hydrostatic pressure drop over a pore to the capillary
pressure:
Ca ¼
Dpvisc lw va2
¼
Dpcap
cj
(2)
Bo ¼
Dpgrav Dqga2
;
¼
Dpcap
c
(3)
and
Where lw is the viscosity of the wetting fluid, v is the filtration or Darcy velocity, a is the typical
pore size, c is the surface tension, j is the permeability of the porous medium, Dq is the density
difference in the two fluids, and g is the acceleration due to gravity in the direction of flow.
For systems without gravity we expect different flow regimes depending on the capillary number [8]: for very slow (quasi-static) displacements, the displacement is controlled by the heterogeneity of the capillary pressures along the interface [5,9]; this capillary fingering regime is well
modeled by invasion percolation algorithms [10–12]; for fast displacements, where viscous forces
overcome capillary effects, a viscous fingering regime is observed, with a rapid breakthrough of
the non-wetting fluid into the wetting fluid [13]. These two flow regimes have been extensively
studied, in particular, the fractal properties of the corresponding displacement structures.
Experimental studies of slow displacement under gravity have demonstrated how the interface
keeps a finite extension w along the direction of apparent gravity (see Fig. 2a), and how this
width scales as a function of the Bond number [10,12]. Such slow displacement configurations
are well modeled by invasion percolation with an invasion probability gradient. Under conditions of fast displacements where the effect of viscous forces dominate, in the presence of
gravity, an unstable displacement of the interface similar to viscous fingering is observed
(see Fig. 2b).
In this paper, we present experimental and numerical results obtained in configurations where
viscous forces and gravity compete. We study the transition from flow configurations where the
interface is stable with respect to viscous instabilities (Fig. 2a) to flow configurations where viscous fingering occurs (Fig. 2b). As mentioned above, though analogous to the Saffman–Taylor
instability, this transition is not properly described by the Saffman–Taylor theory. Three
approaches are being used to tackle the problem: experiments, numerical simulations, and percolation theory.
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G. Løvoll et al. / Energy 30 (2005) 861–872
Fig. 2. Displacement structure of the invading wetting fluid observed in our experimental setup for configurations of
(a) slow displacement, governed by the competition between capillary forces and gravity, and (b) viscous fingering
under gravity.
2. Experiment
Our experimental synthetic porous medium consists of a single layer of glass beads (diameter
1 mm), which are randomly positioned between two glass plates. It has a porosity / ¼ 0:63 and
a permeability j ¼ 0:0189 103 cm2 ¼ 1915 darcy. See Refs. [13–15] for an extended description of the synthetic porous medium. The boundaries of the porous medium are square
(35 35 cm); two opposite sides of the square are the inlet and outlet for the flow experiment.
The medium is initially filled with a 90%/10% mixture in mass of glycerol and water (viscosity
0.2 Pa s, density 1235 kg m3), dyed with 0.1% Negrosine (black). The temperature of the setup
is measured to correct for any possible change in the fluid viscosity due to temperature changes.
The drainage experiment consists in extracting the mixture from the outlet, letting air invade the
porous medium through the inlet. The surface tension c between the two phases is
6:4 102 N m1 . The plane of the porous medium can be tilted by an angle h with respect to
its horizontal position so as to vary the apparent gravity along the flow direction (see Fig. 3),
while varying the extraction speed allows to tune the intensity of viscous forces.
G. Løvoll et al. / Energy 30 (2005) 861–872
865
Fig. 3. Sketch of the experimental setup. The porous medium is a single random layer of glass beads. The wetting
fluid is extracted from the bottom, allowing air to invade the medium from the top. Tilt angle of the model and
extraction speed can be varied. Pressure is recorded at the outlet channel of the porous medium. Pictures of the displacement structure are taken at regular intervals during the experiment.
Various recordings are made during the experiments. Pressure is recorded at the outlet. The
displacement structures are analyzed from pictures recorded at regular intervals during the
experiment. These pictures contain 1536 1024 pixels, which corresponds to a spatial resolution
of 2.56 pixels per pore; the color scale in made of 256 gray levels. These raw images are
filtered in order to clearly separate the wetting from the non-wetting phase and to extract the
invasion front. Successive steps of the filtering process are presented in Fig. 4 (see [15] for a
Fig. 4. Image filtering process: the raw image (a) is thresholded so as to obtain a black and white image (b) where the
two phases are clearly separated. This displacement image is best viewed in reverse video (c) (see also Fig. 2). Removing the trapped wetting fluid clusters, one obtains two domains, the border line of which is the invasion front. The
front itself is plotted in black on top of the raw image in (a).
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G. Løvoll et al. / Energy 30 (2005) 861–872
complete description of the filtering process). We refer to displacement structures as the structure
shown in black in Fig. 4c, and to invasion front as the line painted in black on top of the raw
image in Fig. 4a. The invasion front is thus the continuous line separating the air cluster from
the glycerol/water mixture. We define the front width w as the root mean square value of the
front extension in the direction of flow.
3. Numerical model
In the simulation, the porous medium is represented by a two-dimensional square network of
v
tubes inclined at a 45 angle. Each tube has a fixed length l and radius ri that is drawn from a
defined distribution. A fragment of the porous network is presented in Fig. 5, with the non-wetting fluid entering the model at constant rate from the top.
The network model takes both capillary fluctuations and local viscous pressure field into
account when solving the flow field [16–18]. It has previously been used to successfully simulate
various aspects of the drainage processes in configurations more general than that studied here.
Viscous stabilization, viscous fingering and capillary fingering have been addressed [16,17,19].
The model is also able to account for the change in capillary pressure inside the tube, and thus
to describe the displacement of the meniscus inside the tube. It can therefore be applied to studies of local burst dynamics [20]. A detailed description of the model can be found in [16,17].
For the present study, the model has been extended to include a tunable gravity field. To
allow for simulations on larger system, the following simplification of the model has also been
introduced. With respect to the capillary pressure pc of a meniscus inside a tube, we consider the
tube as straight and cylindrical, so that
2c
;
(4)
r
which is independent of the position of the meniscus in the tube. To ensure numerical stability,
the capillary pressure forced linearly to zero over a small region e at each end of the tube. By
doing so the menisci in the network will either rest at the entrance of the tube or move with
constant velocity through the tube during a time step. This coarsening of the capillary pressure
pc ¼
Fig. 5. The pore network considered in the simulations is a regular network of tubes with random radii. 6 9 nodes
are drawn here, with the invading non-wetting fluid painted black.
G. Løvoll et al. / Energy 30 (2005) 861–872
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Fig. 6. Geometry of a tube in the simulation. Viscous and capillary pressures are computed using this straight
geometry. In order to ensure numerical stability, the capillary pressure goes linearly to zero in a small region e close
to the ends of the tube.
allows for longer time steps. In the simulation, each time step corresponds to the time needed
for one more tube in the network to be completely filled. The tube geometry is presented in
Fig. 6. The coarsening of the model typically reduces the number of time steps used in one simulation by a factor 102 in comparison to the complete model described in [16]. The accuracy
of the simplified model has been verified by comparing its results to those provided by the complete model. For sufficient high displacement rates, the consistency between the two models is
satisfactory.
In the simplified model, the flow q in each tube is written as [21]
q¼
pr2 k
ðDp pc Þ;
leff
(5)
where k ¼ r2 =8 is the permeability of the tube, leff ¼ lnw x þ lw ðl xÞ (see Fig. 6 for the signification of l and x) is the effective viscosity, and Dp ¼ pj pi is the pressure difference between the
two nodes at the opposite ends of the tube.
Since the geometry of the porous network in the simulation is different from that of the
experimental porous medium, the Bond and capillary numbers have to be defined carefully in
order for the comparison to the experimental values to be valid. The Bond number is defined as
Bo ¼
Dpgrav Dqglr
;
¼
2c
Dpcap
(6)
where l is the tube length and r is the average tube radius over the system (excluding e and l e
regions). The capillary number is estimated as
Ca ¼
Dpvisc DPvisc lr
;
¼
2cL
Dpcap
(7)
where hPvisc is the total viscous pressure drop over the whole model when it is completely filled
with wetting fluid (initial configuration), and L is the length of the system.
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G. Løvoll et al. / Energy 30 (2005) 861–872
In what follows we present results from a series of numerical experiments on a system of 80 160 nodes where we keep the capillary number constant and vary the apparent gravity. The
length of the tubes in the network is l ¼ 1:0 mm; the radii ri are drawn from a flat distribution
in the interval [0.1, 0.4 mm]. The liquids are incompressible and immiscible with a viscosity contrast (lnw ¼ 0:0014 Pa s, lw ¼ 0:140 Pa s), a surface tension (c ¼ 6:4 N m1 ) and a density difference (qnw ¼ 1:3 kg=m3 , qw ¼ 1230 kg=m3 ) chosen to match the liquid pair used in the
experiments.
In the simulations, the invasion front is defined as the position of the interface in the tubes
separating the continuous wetting phase from the non-wetting phase. We define the front width
in the same way as in the experiment.
4. Results
For drainage with high viscosity contrast (lnw 5 lw ), where gravity stabilizes the invasion
front, theoretical arguments based on percolation theory [10,22] in a stabilizing gradient could
be developed [12,15,23]. The theory predicts the following scaling law for the displacement from
which w under stable displacement [15]
w ðBo CaÞa ¼ ðBo CaÞðv=1þvÞ
(8)
where m is the correlation length exponent in percolation theory, for two-dimensional systems
m ¼ 4=3 [22], and thus a ’ 0:57. For convenience, we introduce the generalized Bond number
Bo Bo Ca. Since the front width w in Eq. (8) diverges when Bo ! 0, we expect a criterion
for interface stabilization in the form
Bo Ca ¼ Bo > 0
(9)
This criterion is analogous to that found by Saffman and Taylor [6] for the displacement in an
Hele-Shaw cell (qg ðvlÞ=j > 0). But it should be noted that the dynamics of the instability is
radically different from what is observed for Hele-Shaw cell experiments; this is obvious when
considering the finite width of the invasion front or the trapping of wetting fluid behind the
front.
4.1. Stabilization of the invasion front
In order to check if stabilization occurs and under which circumstances, we investigate the
evolution of the width of the displacement front in the flow direction as a function of time. In
Fig. 7, the front width is plotted as a function of time normalized by the time at breakthrough,
tb, for both the experiments and the numerical simulations. From these curves, it is apparent
that the invasion fronts reach saturation (stabilize) if they correspond to a generalized Bond
number Bo > 0.
The value Bo ¼ 0 is that for which the average hydrostatic pressure drop and viscous pressure drop over a pore are equal. As a consequence, for this value of the Bond number, the overall pressure difference over the model is expected to be constant. We have checked this by
plotting the evolution of the total pressure difference over the model as a function of normalized
G. Løvoll et al. / Energy 30 (2005) 861–872
869
Fig. 7. Evolution of the front vertical extension as a function of time in (a) the experiments and (b) the simulations.
time. The pressure difference over the model, P, consists of a hydrostatic contribution DPhyd
and a viscous contribution DPvisc:
P ¼ Poutlet Pfront ¼ DPhyd þ DPvisc :
(10)
Experimental pressure measurements are plotted in Fig. 8a, and numerical results are presented
in Fig. 8b. Both curves demonstrate that pressure reaches a constant level for Bo ’ 0.
4.2. Scaling of the front width
Fig. 9 presents the measured front width w as a function of the generalized Bond number
Bo, using a log–log scale, for configurations of stable displacement. The results from both the
experiments (Fig. 9a) and numerical simulations (Fig. 9b) are consistent with the predicted
scaling:
w Bo0:57
(11)
The vertical shift between the data in Fig. 9 is due to a different width of the capillary threshold
distribution in the experiments and the simulations [15,23].
5. Discussion and conclusion
When gravity is stabilizing and larger than the viscous forces, the invasion of a non-wetting
liquid into a wetting liquid is stable. This problem could be understood by using a combination
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G. Løvoll et al. / Energy 30 (2005) 861–872
Fig. 8. Evolution of the pressure difference between the outlet and the invasion front, P ¼ Poutlet Pfront , as a function of time, (a) in the experiments and (b) in the simulations. P consists of two contributions, a hydrostatic pressure
drop (positive) over the model and a viscous pressure drop (negative) over the model: P ¼ DPhyd þ DPvisc . If
DPhyd > DPvisc , pressure decreases with time. The pressure difference over the model at breakthrough is used as the
reference pressure: Pðtb Þ 0.
of Darcy’s law and a mapping to percolation theory, which allowed us to predict the scaling of
the front width w (Eq. (8)) and the stabilization criterion (Eq. (9)) [15].
It is perhaps not surprising that our criterion for stabilization is indeed the same as the Saffman–Taylor criterion [6]. Despite this, the geometry in our system is radically different from the
geometry in the Hele-Shaw cell. Our system is a ‘‘real porous medium’’ where the relevant
length scale is the pore scale; in contrast, the relevant size for Hele-Shaw experiments is the
overall width of the channel. The differences between the two systems are clearly seen if we look
at two effects. First, in a real porous medium we have a finite width w of the invasion front even
for stable displacement, while in the Hele-Shaw geometry the front is flat as long as the displacement is stable. Second, the finite front gives rise to another characteristic feature of drainage in a porous media, which is the trapping of clusters of wetting liquid behind the front. This
G. Løvoll et al. / Energy 30 (2005) 861–872
871
Fig. 9. Scaling of the front width as a function of the generalized Bond number Bo, for the experiments and numerical simulations. The two scaling are consistent, and in good agreement with the law predicted by the percolation
theory, Eq. (11).
phenomenon is absent in the Hele-Shaw geometry but prominent in our experiments (see
Fig. 2a). The process of trapping might even lead to a violation of the Saffman–Taylor stabilization criterion for systems with a lower viscous contrast since trapping decreases the permeability
for the non-wetting fluid [14].
We currently working on extending our results to situations where the displacements
is unstable. Preliminary results seem to indicate that the mapping to percolation theory in a
gradient is no longer valid, as characteristic length scales exhibit scaling laws that differ from
what we observed in configurations of stable displacement.
Acknowledgments
The work was supported by NFR, the Norwegian Research Council, VISTA, the Norwegian
academy of science and letters’ research program with Statoil and the French/Norwegian collaboration PICS contract number 1373.
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