Journal of Contaminant Hydrology 74 (2004) 61 – 81
www.elsevier.com/locate/jconhyd
Secondary imbibition in NAPL-invaded
mixed-wet sediments
Ahmed Al-Futaisi a,b, Tad W. Patzek b,*
a
The Civil Engineering Department, Sultan Qaboos University, Oman, Saudi Arabia
b
Department of Civil and Environmental Engineering, Berkeley, CA 94720, USA
Received 12 February 2003; received in revised form 16 January 2004; accepted 9 February 2004
Abstract
A previously developed pore network model is used here to study the spontaneous and forced
secondary imbibition of a NAPL-invaded sediment, as in the displacement of NAPL by
waterflooding a mixed-wet soil. We use a 3D disordered pore network with a realistic
representation of pore geometry and connectivity, and a quasi-static displacement model that
fully describes the pore-scale physics. After primary drainage (NAPL displacing water) up to a
maximum capillary pressure, we simulate secondary imbibition (water displacing NAPL). We
conduct a parametric study of imbibition by varying systematically the controlling parameters:
the advancing contact angles, the fraction of NAPL-wet pores, the interfacial tension, and the
initial water saturation. Once the secondary imbibition is completed, the controlling displacement
mechanisms, capillary pressures, relative permeabilities, and trapped NAPL saturations are
reported. It is assumed that NAPL migrates into an initially strongly water-wet sediment, i.e., the
receding contact angles are very small. However, depending on the surface mineralogy and
chemical compositions of the immiscible fluid phases, the wettability of pore interiors is altered
while the neighborhoods of pore corners remain strongly water-wet-resulting in a mixed-wet
sediment. Here, we compare three different levels of wettability alteration: water-wet (advancing
contact angles (20j to 55j), intermediate-wet (55j to 120j), and NAPL-wet (120j to 155j). The
range of advancing contact angles and the fraction of NAPL-wet pores have dramatic effects on
the NAPL-water capillary pressures and relative permeabilities. The spatially inhomogeneous
interfacial tension has a minor impact on the trapped NAPL saturation and relative permeability
to NAPL, and a slight effect on the relative permeability to water. The initial water saturation
has a slight effect on the two-phase flow characteristics of water-wet sediments; however, with
* Corresponding author. University of California, Department of Civil and Environmental Engineering, 437
Davis Hall, Berkely, CA 94720, USA.
E-mail addresses: [email protected] (A. Al-Futaisi), [email protected] (T.W. Patzek).
0169-7722/$ - see front matter D 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.jconhyd.2004.02.005
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A. Al-Futaisi, T.W. Patzek / Journal of Contaminant Hydrology 74 (2004) 61–81
more NAPL-wet pores in the sediment, it starts to have a profound effect on the water and
NAPL relative permeabilities.
D 2004 Elsevier B.V. All rights reserved.
Keywords: Pore network; Mixed-wet; Relative permeability; Capillary pressure; Secondary imbibition; Initial
water saturation; Trapping
1. Introduction
Understanding NAPL migration in the air- and water-saturated zones is important in
contamination management projects. The spatial distribution of sediment wettability
and the extent of wettability alteration of individual pores influence the two-phase flow
characteristics (Al-Futaisi and Patzek, 2003b; Blunt and Scher, 1995; Blunt, 1998,
2001; Bradford and Leij, 1995; Morrow et al., 1986; Piri and Blunt, 2002). Although
most soils and rocks are naturally strongly water-wet, NAPL invasion may alter their
wettability to intermediate-wet or NAPL-wet (Dekker and Ritsema, 1994; Dwarakanath
et al., 2002; Powers et al., 1996; Villaume, 1985). In fact, changing the soil wettability
has been proposed in remediation and contaminant migration control schemes through
the emplacement of organic-wet zones, or with the in situ chemical reactions (Burris
and Antworth, 1992; Hayworth and Burris, 1997; Xu and Boyd, 1995). The degree to
which the sediment wettability is altered depends on its mineralogy (Anderson, 1986),
surface roughness and charge (Hirasaki, 1991; Morrow, 1975), contaminant aging
(Buckley et al., 1995; Freer et al., 2002; Powers and Tamblin, 1995), presence of
organic acids and bases (Dubey and Doe, 1993; Powers and Tamblin, 1995; Thomas et
al., 1993), chemical composition (Brown and Neustadter, 1980; Buckley and Liu,
1998; Murphy et al., 1992), and aqueous chemistry (Demond et al., 1994; Wang and
Guidry, 1994; Zheng and Powers, 1999). With all of these variables that are difficult to
quantify, the sediment wettability should be modeled as a spatially heterogeneous
random variable.
Solid wettability is heterogeneous macroscopically in the entire soil volume, and
microscopically inside each pore. Those parts of the pore-wall surface that are exposed to
by NAPL (usually the pore center) may have different wettability from those in contact
with water (usually the pore corners). We then say that the soil is mixed-wet, as suggested
by Salathiel (1973). In the petroleum literature, evidence points toward mixed-wettability
as the most likely state of the majority of oil reservoirs. Salathiel reported experiments in
which the initially water-saturated permeable rock was invaded with a crude oil, and the
solid’s wettability was altered by adsorption of the crude oil components. He concluded
that the parts of the rock surface in contact with connate water were protected from
adsorption. Since then, efforts have been directed towards explaining physical and
chemical aspects of mixed-wettability (Anderson, 1986; Buckley et al., 1995; Melrose,
1982; Morrow, 1990). Mixed-wettability can be broadly defined to include any degree of
wetting alteration at the parts of the rock surface exposed to crude oil (Buckley et al.,
1998). The rock wettability is altered following adsorption of asphaltenes or other surfaceactive compounds that essentially coat the mineral surface with an organic layer (Demond
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63
et al., 1994; Kovscek et al., 1993; Powers and Tamblin, 1995; Zheng et al., 2001). In soils,
one should expect a wider range of wettability alteration caused by the presence of
complex organic and inorganic compounds, and high level of chemical and biological
activity.
Researchers differ as to how to induce or model a mixed-wet porous medium.
Kovscek et al. (1993) were the first to propose a pore network model that produces
mixed wettability. They suggested that the stability and thickness of water films between
the oil and the solid surface determine wetting in oil/water/rock systems. When the
water films rupture, the middle part of the pore wall becomes oil-wet, h = 180j, whereas
the near-corner parts of the pore wall remain water-wet, h = 0j. The wettability alteration
model by Kovscek et al. has been adopted in several pore-network models (Blunt, 1997,
1998, 2001; Dixit et al., 1999; Hui and Blunt, 2000a,b; Man and Jing, 1999; Piri and
Blunt, 2002). The main difficulty of this approach is quantifying the disjoining pressure
at which the water films rupture. How exactly a water film ruptures is less important
than the altered value of advancing contact angle after the rupture. Dixit et al. (1999)
represented their mixed-wet system using a pore network with different fractions of oilwet pores (big pores) and water-wet pores (small pores). Water displaces oil first from
the water-wet pores and then from the oil-wet pores. Dixit et al. performed several
simulations with different fractions of oil-wet pores. In their representation, each pore
has a single wettability state but the rock wettability varies in space. Masalmeh (2002),
in his laboratory experiments, let primary drainage establish the mixed-wet rock. He
conducted primary drainage up to different values of maximum capillary pressure, Pmax
c ,
i.e., each value of Pmax
resulted in a rock with a different fraction of oil-wet pores. Then
c
the rock samples were aged for 4 weeks, and secondary imbibition was performed. This
approach produces different mixed-wet rock samples. Finally, Bradford and Leij (1996)
obtained their fractionally wet media by combining untreated and OTS-treated1 sand
using 0%, 25%, 50%, 75% and 100% of OTS-treated sand. The untreated sands were
strongly water-wet, while the OTS-treated sand was oil-wet. The two types of sand were
then mixed in different proportions in a shaker for 5 h. Notice that this approach
produces pores with different combinations of completely oil-wet walls and completely
water-wet walls in a single pore.2
Despite all the progress to date, explaining and reproducing mixed-wettability
remains a significant challenge. The objective of this paper is, first, to develop several
realistic microscopic distributions of altered wettability of mixed-wet porous media.
Then, we study secondary imbibition in these mixed-wet porous media. In our analysis,
we adopt the Kovscek et al. (1993) framework with two major differences. First, we
assume that during primary drainage some of the water films break at random for
whatever reason.3 Second, the random advancing contact angles are not restricted to 0j
1
OTS is octadecyltrichlorosilane.
Therefore, the porous media obtained by Bradford and Leij (1996) are not mixed-wet and do not reproduce
the naturally altered rocks.
3
We argue that in drainage to a moderately high capillary pressure, the rupture disjoining pressure is
commonly exceeded. We do not know what is the new advancing contact angle.
2
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and 180j, but are allowed to vary with any value between 0j and 180j. The flow
characteristics of the various secondary imbibition processes are calculated using our
two-phase pore network model (Al-Futaisi and Patzek, 2003b; Patzek, 2001) which has
been developed to account for any degree of wettability alteration. In a recent paper (AlFutaisi and Patzek, 2003b), this model has been used to study the effect of wettability
alteration on the two-phase flow characteristics using very narrow ranges of advancing
contact angles. In this paper we consider a wider range of advancing contact angles by
focusing on three different types of wetting: water-wet (contact angles 20j to 55j),
intermediate-wet (55j to 120j), and NAPL-wet (120j to 155j). We provide a detailed
comparison among these three wettability ranges, and study the impact of the fraction of
NAPL-wet pores, the inhomogeneous NAPL/water interfacial tension, and the initial
water saturation.
2. Pore network model
The network used in this analysis is a realistic representation of a sample of Bentheimer
sandstone reconstructed from the 3D micro-focused X-ray CT image. The complex
topology and geometry of the network is detailed in Øren et al. (1998), Patzek (2001).
The two-phase flow model is quasi-static, i.e., capillary forces dominate, the effects of
gravity are incorporated, and the effects of viscous forces are ignored. The flow
simulations proceed in the following sequence: (1) fully saturate the network with water,
(2) perform primary drainage (NAPL displaces water) up to a predefined maximum
capillary pressure, (3) alter wettability, and (4) perform secondary imbibition (water
displaces NAPL). The resulting capillary pressure curves are calculated using percolation
algorithms (Al-Futaisi and Patzek, 2003b) with piston-type, snap-off, and cooperative
pore-body filling mechanisms (Øren et al., 1998; Patzek, 2001). The relative permeabilities to water and NAPL are calculated using Darcy’s law and accurate expressions for the
hydraulic conductances in bulk flow (Patzek and Silin, 2001), corner flow (Patzek and
Kristensen, 2001), and intermediate-layer flow (Al-Futaisi and Patzek, 2003c). The spatial
distribution of trapped NAPL clusters is determined using the Hoshen – Kopelman
algorithm (Hoshen and Kopelman, 1976) extended by us to disordered networks (AlFutaisi and Patzek, 2003a).
3. Development of mixed-wettability
A wettability alteration scenario that produces both water- and oil-wet regions in a
single pore has been proposed by Kovscek et al. (1993), and has been used to compute
the macroscopic capillary pressure curve for a bundle of star-shaped pores. Kovscek et
al. argued that upon initial invasion of oil into a water-filled pore, thin water films
separate the oil from the pore walls. The water films remain stable as long as the
capillary pressure does not exceed the system-dependent rupture disjoining-pressure. As
more oil is introduced into the reservoir, the oil –water capillary pressure increases, the
rupture disjoining-pressure may be exceeded, and the water films may break. According
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65
to Kovscek et al., where the thin water films break, the walls of the pores become
strongly oil-wet (180j water advancing contact angle), whereas the water-filled corners
of these pores remain strongly water-wet (the advancing contact angle and receding
contact angle are both equal to 0j). Recently, Freer et al. (2002) provided a detailed
discussion of deficiencies in the conceptual model of Kovscek et al. (1993). In
particular, the assumption of zero receding contact angle and 180j advancing contact
angle seems to be unrealistic. Based on the experiments with a crude oil droplet on
mica, Freer et al. (2002) have postulated that following breakage of the thin water films,
a range of receding and advancing contact angles is possible. This possible variability of
receding and advancing contact angles results in a much richer behavior of secondary
imbibition than that outlined by Kovscek et al. (1993). In this paper, we extend the
Kovscek et al. scenario to any degree of wettability alteration (arbitrary values of contact
angles), and study the ensuing NAPL/water flow characteristics during secondary
imbibition.
The important role of the water-receding and advancing contact angles, hr and ha, in
drainage and imbibition of a single pore can be illustrated with the idealized cylindrical
duct shown in Fig. 1. Initially, this duct is completely filled with water, and the solid
walls are naturally water-wet, Fig. 1(a). Primary drainage starts by increasing capillary
pressure, which allows the NAPL to migrate into the duct when the capillary entry
pressure is exceeded. To avoid confusion in defining the wetting and nonwetting fluid
phase, we define capillary pressure as the difference between the NAPL and water
pressures:
Pc :¼ PNAPL Pw
ð1Þ
The contact angles are always measured through water. When NAPL invades, it
occupies the center of the duct, and water resides in the corners of the duct. After the
NAPL recedes some distance toward the duct corners, the water films rupture and parts
of the duct wall become NAPL-wet as indicated in Fig. 1(b) by the thick lines. It is
usually assumed that the primary drainage receding contact angle is very small, around
zero (Ma et al., 1996). However, Freer et al. (2002) suggest that, when appropriately
measured with aged interfaces, water receding contact angles can reach much larger
values (up to 50j). These large values of receding contact angle may have important
implications on the NAPL/water configurations in the pores during primary drainage.
For example, the corner water can now be completely displaced from the pore corners,
Fig. 1. Possible two fluid configurations in a single pore: (a) an initially water-filled pore, (b) spontaneous
imbibition, (c) force imbibition, (d) forced imbibition with NAPL layers (gray is water and white is NAPL).
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and water trapping is possible. Until further experiments verify the large values of
receding contact angles reported by Freer et al. (2002), we will continue to assume that
hr ranges from 0j to 10j.
Now we describe secondary imbibition in which water displaces NAPL. Due to
aging, the mineralogy of the solid surfaces, and the chemical composition of the fluids,
these surfaces can change their wettability, i.e., the receding contact angle in drainage
can change to a larger advancing contact angle in imbibition (Buckley et al., 1995;
Buckley and Liu, 1998; Freer et al., 2002; Hirasaki, 1991; Morrow et al., 1986).
Because the advancing contact angles are greater than (or equal to) the receding contact
angles, the NAPL/water interface remains pinned until the advancing contact angle is
reached. Once this happens, the interface starts sliding. If the sum of the advancing
contact angle, ha, and the corner half-angle, b, is less than 90j, the imbibition is
spontaneous ( Pc>0); Fig. 1(b). If, on the other hand, this sum is greater than 90j, the
imbibition is forced ( P V 0); Fig. 1(c). For very large advancing contact angles, we may
create the intermediate NAPL layers near the pore corners as shown in Fig. 1(d). Further
description of the two-fluid configurations during primary drainage and secondary
imbibition processes is provided in the works of Al-Futaisi and Patzek (2003b), Øren
et al. (1998) and Patzek (2001).
In this paper, we use the mixed-wet displacement scenario described above for a single
pore to study the effects of changing the advancing contact angles, the fraction of the
NAPL-wet pores, the spatially inhomogeneous NAPL/water interfacial tension, and the
initial water saturation. We report the macroscopic transport properties of two-phase flow,
the displacement regime, the capillary pressure curve, the relative permeabilities, and the
trapped NAPL saturation.4 Since an accurate procedure for quantifying the contact angles
is yet to be provided, we believe that the best we can do at this moment is to assume ranges
of uniformly random contact angles. We consider three different types of wetting: waterwet (advancing contact angles 20j to 55j), intermediate-wet (55j to 120j), and NAPLwet (120j to 155j). The advancing contact angles are drawn from a uniform probability
distribution and correlated with the duct sizes (i.e., larger ducts are assigned larger contact
angles).
4. Effects of advancing contact angles
Al-Futaisi and Patzek (2003b) analyzed the effects of narrowly varying advancing
contact angles on the two-phase flow characteristics. Here we compare the flow behavior
in the three types of network wettability: water-wet (advancing contact angles 20j to 55j),
intermediate-wet (55j to 120j), and NAPL-wet (120j to 155j). We assume that the
receding contact angles during primary drainage are between 0j and 10j. We also assume
that the NAPL/water interfacial tension, rN/w, is 35 mN/m; the initial water saturation in
imbibition, Swi, is 6% (the maximum capillary pressure during primary drainage, Pmax
c , is
4
In the subsequent discussion, we will be using the word ‘‘network’’ to represent a soil, rock, or any
sediment. The parts of each pore, pore throats and pore body, will be replaced with the angular cylindrical ducts of
small and large cross-sectional areas, respectively.
A. Al-Futaisi, T.W. Patzek / Journal of Contaminant Hydrology 74 (2004) 61–81
67
188 kPa); and the no-slip boundary condition at the NAPL/water interfaces. Fig. 2 shows
that the three different wetting conditions result in different capillary pressures and relative
permeabilities and, hence, in different two-phase flow behavior.
4.1. Water-wet
In this wetting condition, secondary imbibition is spontaneous ( Pc>0). As described
by Al-Futaisi and Patzek (2003b), this type of invasion begins inside the network by
snap-off in the narrowest ducts (pore throats). When advancing contact angles are close
to zero, snap-off results in high NAPL trapping. Here the advancing contact angles are
large (20j to 55j), and the initial snap-offs are followed by the piston-like cooperative
pore-body fillings that form growing compact clusters of water-filled ducts. The growth
of compact water clusters reduces the NAPL trapping caused by snap-off. The trapped
NAPL saturation for this wetting condition is 25%. This percent increases if the
contact angles are less than 20j, and decreases if contact angles are between 55j and
90j.
4.2. Intermediate-wet
Here, secondary imbibition is first spontaneous ( Pc>0), and then forced ( Pc < 0) if
capillary pressure can be lowered sufficiently. The spontaneous imbibition occurs in
ducts with advancing contact angles less than 90j (water-wet pores), whereas the
forced imbibition occurs in ducts with advancing contact angles greater than 90j
(NAPL-wet pores). As demonstrated by Al-Futaisi and Patzek (2003b), the range of
contact angles considered in this wetting condition, 55j to 120j, results in a complex
mixture of different displacement regimes. The contact angles between 55j and 90j are
characterized by some snap-off and growth of compact water clusters that reduce the
NAPL trapping. In contrast, the advancing contact angles between 90j and 100j favor
piston-type invasion from the network inlet into the largest ducts, without creation of
the intermediate NAPL layers near the duct corners, Fig. 1(c). The latter range of
advancing contact angles results in strong NAPL trapping due to bypassing. Lastly, the
contact angle range from 100j to 120j results in piston-type invasion from the
network inlet and creation of the intermediate NAPL layers, Fig. 1(d). These layers
enhance NAPL connectivity in the network and reduce the NAPL trapping. With these
different flow regimes in mixed-wet networks, the ratio of NAPL-wet ducts and waterwet ducts has a dramatic effect on two-phase flow. The trapped NAPL saturation in the
mixed-wet case is about 36%.
4.3. Oil-wet
For this wetting condition, secondary imbibition is entirely forced piston-type displacement ( Pc < 0). Since advancing contact angles are now very large, it is easy to create
the stable intermediate NAPL layers near most of the duct corners. These numerous layers
are well connected, and trapping the NAPL decreases. The trapped NAPL saturation is
about 7%. Smaller trapped NAPL saturation could be reached if larger contact angles were
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69
used (e.g., 160j to 180j). However, we should emphasize that the connectivity of the
intermediate NAPL layers is complex and not yet fully modeled. We probably overestimate the layer connectivity from one duct to another by assuming that the layers are
connected as long as at least one layer exists in each pair of joined ducts. Notice that the
shape of the capillary pressure curve in this case is similar to that obtained in primary
drainage when NAPL migrates into the water-filled network.
With all these displacement regimes, it becomes clear why the respective relative
permeabilities are so different for the three wetting conditions; Fig. 2(b) and (c). In
the NAPL-wet condition, one observes that the water relative permeability is the
highest and the NAPL relative permeability is the lowest. In this wetting condition,
NAPL moves in the thin intermediate layers, whereas water moves along the corners
and the centers of the ducts (cf. Fig. 1(d)). The lowest water relative permeability is
observed for the intermediate-wet condition, because water remains pinned near the
duct corners without invading the duct centers. The NAPL relative permeability in the
latter case is at first similar to that in the water-wet condition, but then it becomes
lower because of significant NAPL trapping. For the water-wet condition, the water
relative permeability is between those in the intermediate-wet and NAPL-wet conditions, while the NAPL relative permeability is the highest because the NAPL flows
predominantly in the duct centers. One also observes that close to the trapped
saturation, the NAPL relative permeability curve has a positive slope for the waterwet condition and close to zero slope for the other two conditions.
5. Effects of the fraction of NAPL-wet pores
In this section, we analyze secondary imbibition in the intermediate-wet network.
Because this type of network contains both water-wet and NAPL-wet ducts, the ratio
of the NAPL-wet ducts to the water-wet ducts is important. We therefore vary this
fraction to study its effect on the flow of water and NAPL. We assign to the water-wet
and the NAPL-wet ducts advancing contact angles from 55j to 90j and 90j to 120j,
respectively. An array of one random advancing contact angle per duct in the network
is generated from a uniform probability distribution. Then, for each duct, a contact
angle is drawn from this array with the magnitude proportional to the radius of a circle
inscribed in the duct cross-section, i.e., the largest ducts are assigned the largest
advancing contact angles and the smallest ducts are assigned the smallest contact
angles. The fraction of the NAPL-wet ducts, f, is defined as the ratio of the number of
NAPL-wet ducts to the total number of ducts in the network (Heiba et al., 1983) ( f = 0
means that the entire network is water-wet, ha = 55– 90j, whereas f = 1 means that the
network is entirely NAPL-wet, ha = 90 – 120j). Again, we assume hr = 0j to 10j, rN/w =
35 mN/m, Swi = 6%, Pmax
= 188 kPa, and the no-slip NAPL/water interface boundary
c
condition.
Fig. 2. Secondary imbibition capillary pressures and relative permeabilities for the three wetting conditions:
water-wet, intermediately wet, and NAPL-wet.
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Fig. 4. Effects of the interfacial tension on the secondary imbibition capillary pressures and relative permeabilities
in water-wet network.
Fig. 3(a) –(c) shows the capillary pressures and relative permeabilities as functions of
the fraction of NAPL-wet ducts. Small changes in the fraction of NAPL-wet ducts result in
dramatic changes of flow characteristics of the network. Except for the cases f = 0 and f = 1,
the capillary pressure curves in Fig. 3(a) are composed of the spontaneous imbibition part
and the forced imbibition part. For f = 0, the displacement is entirely spontaneous, whereas
for f = 1 it is entirely forced. As f increases, the forced imbibition part of the capillary
pressure curve grows and the spontaneous imbibition part shrinks. The trapped NAPL
saturation, Fig. 3(a), increases as we increase the fraction of NAPL-wet ducts up to some
critical value between 0.4 and 0.5, after which the trapped NAPL saturation starts to
Fig. 3. Effects of the fraction of NAPL-wet ducts on the secondary imbibition capillary pressures and relative
permeabilities.
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decrease. The increase of the trapped NAPL saturation is due to the introduction of some
NAPL-wet ducts in which the intermediate NAPL layers cannot exist and, thus, NAPL
bypassing increases. On the other hand, the decrease in the trapped NAPL saturation is due
to the existence of the NAPL-wet ducts that allow the creation of continuous, stable
intermediate NAPL-layers near the duct corners.
The fraction of NAPL-wet ducts has great impact on the NAPL and water
relative permeabilities; see Fig. 3(c). In general, the water relative permeability
decreases as we increase f up to 0.4. Above f = 0.4, the water relative permeability
increases for water saturations less than 0.5 and decreases for the saturations greater
than 0.5. The NAPL relative permeability, in general, decreases as the fraction of
NAPL-wet ducts in the network increases. However, the NAPL relative permeability
Fig. 5. Effects of the interfacial tension on the secondary imbibition capillary pressures and relative permeabilities
in intermediate-wet network.
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73
increases for f between 0.8 and 1.0. The calculated significant dependance of the
relative permeability functions on the fraction of NAPL-wet ducts raises questions
about the validity of the relative permeability curves used in the field-scale
simulators.
6. Effects of NAPL/water interfacial tension
In the previous sections, row was spatially uniform and equal to 35 mN/m. Here, we fix
the network wettability property and vary the NAPL/water interfacial tension, rN/w. We
perform secondary imbibition displacements on the water-wet, intermediately wet, and
Fig. 6. Effects of the interfacial tension on the secondary imbibition capillary pressures and relative permeabilities
in NAPL-wet network.
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Fig. 7. The secondary imbibition capillary pressure and relative permeability curves for the variable interfacial
tension between 15 and 75 mN/m (dotted lines), and the constant value of 45 mN/m (solid lines).
NAPL-wet networks using seven different values of rN/w: 15, 25, 35,45, 55, 65, 75 mN/m.
These values are assumed to be constant in the network during each simulation. The
capillary pressure and relative permeability curves for each wetting condition are plotted in
Fig. 4 for the water-wet, Fig. 5 for intermediate-wet, and Fig. 6 for NAPL-wet case.
Because in the calculation of the capillary entry pressures, rN/w is a common factor (see
Al-Futaisi and Patzek, 2003b), a new value of rN/w will simply rescale the single
dimensionless capillary pressure curve for primary drainage and secondary imbibition,
e.g., the capillary pressure curve for rN/w = 55 mN/m can be obtained by multiplying the
Fig. 8. Influence of the initial water saturation on the secondary imbibition capillary pressure and water and
NAPL relative permeabilities in water-wet network.
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35 mN/m capillary pressure curve by 55 = 35. The tiny differences in the trapped NAPL
saturation and the relative permeabilities to water NAPL are caused by the roundoff error
in the simulator.
Now, let us use a range of rN/w values spatially distributed in the network. Fig. 7(a)
compares the secondary imbibition capillary pressure curves for the three types of wet
ting using constant rN/w = 45 mN/m (solid lines) and the spatially random rN/w between
15 and 75 mN/m (dotted lines). The primary drainage and the NAPL-wet secondary
imbibition capillary pressure curves show more variability than the corresponding ones
for the water-wet and intermediate-wet conditions. However, practically, these differences
are negligible. Notice that in the water-wet and intermediate-wet conditions there is a
slight difference in the trapped NAPL saturation. Fig. 7(b) plots the secondary imbibition
relative permeabilities for water and NAPL using constant rN/w = 45 mN/m (solid lines)
and variable rN/w between 15 and 75 mN/m (dotted lines). There is a slight difference in
the two curves. In general, except for the NAPL relative permeability in the intermediatewet condition, the constant rN/w resulted in a slightly higher water and NAPL
permeabilities than those obtained using the constant rN/w. Again, this difference is
small.
7. Effects of initial water saturation
The maximum capillary pressure during primary drainage, Pmax
c , establishes the initial
condition (the initial water saturation and interface configurations) for the secondary
imbibition. In this section, we study the effect of Pmax
on the secondary imbibition flow
c
characteristics. We vary Swi for the three wetting conditions considered in present analysis:
water-wet (contact angles 20j to 55j), intermediate-wet (55j to 120j), and NAPL-wet
(120j to 155j). We assume hr = 0j to 10j, rN/w = 35 mN/m, and the no-slip NAPL/water
interface boundary condition. The secondary imbibition displacements are performed with
the initial water saturations, Swi = 6%, 7%, 10%, 13%, 18%, 24% and 38%. The maximum
primary drainage capillary pressures that correspond to these saturations are, Pmax
= 188, 20,
c
10, 8, 6, 5 and 4 kPa, respectively. Figs. 8– 10 plot the secondary imbibition capillary
pressures and relative permeabilities in the water-wet, NAPL-wet, and NAPL-wet networks,
respectively.
The results show strong effects of initial water saturation on the capillary pressure and the
relative permeability curves. The respective capillary pressure curves are simply rescaled
and shrunk. For water-wet and intermediate-wet networks, the trapped NAPL saturation
does not experience large changes as Swi increases ( F 5%). However, in the NAPL-wet
network, the trapped NAPL saturation increases noticeably with the increasing Swi. In waterwet networks, both water and NAPL relative permeabilities are slightly affected by the
different initial water saturations. However, introducing some NAPL-wetness into the
network results in a tremendous variation in the NAPL and water relative permeabilities.
Fig. 9. Effects of initial water saturation on secondary imbibition capillary pressures and water and NAPL relative
permeabilities in intermediate-wet network.
A. Al-Futaisi, T.W. Patzek / Journal of Contaminant Hydrology 74 (2004) 61–81
77
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A. Al-Futaisi, T.W. Patzek / Journal of Contaminant Hydrology 74 (2004) 61–81
79
In the intermediate-wet network, as Swi increases, the NAPL and water relative permeabilities increase. The increase in the water relative permeability is more dramatic than in the
NAPL one. The behavior of the NAPL-wet network is somewhat different. The effect of Swi
is dramatic for both NAPL and water relative permeabilities. As the Swi increases, the water
and NAPL relative permeabilities increase for the average water saturation below 0.5, and
decrease for the average water saturation above 0.5.
8. Conclusions
Our full quasi-static physics pore network model was used to study the effect of
advancing contact angle, fraction of the NAPL-wet pores, interfacial tension, and initial
water saturation on the secondary imbibition capillary pressures and relative permeabilities. We have used a single network geometry and connectivity, but with three different
levels of mixed-wetting: water-wet (contact angles 20j to 55j), intermediate-wet (55j to
120j), and NAPL-wet (120j to 155j).
For water-wet condition and with a reasonably high capillary pressure after the initial
drainage, the spontaneous secondary imbibition begins in the volume of the sediment
by snap-off. The initial snap-off events are followed by the piston-like cooperative
pore-body fillings that form growing compact water clusters.
For the intermediate-wet condition, the flow is a mixture of spontaneous and forced
secondary imbibition processes. Snap-off and growth of compact water clusters reduce
NAPL trapping. Invasion from the sediment’s boundary into the largest pores without
creation of the intermediate NAPL layers near the pore corners increases NAPL
trapping. Finally, piston-type invasion from the sediment’s boundary that creates
intermediate NAPL layers reduces trapping.
For the NAPL-wet condition, secondary imbibition is forced piston-type displacement
that creates stable intermediate NAPL layers near most of the pore corners in the
sediment. This wetting condition results in least NAPL trapping.
The fraction of the NAPL-wet pores strongly influences two-phase flow in the
sediment. The trapped NAPL saturation increases as the fraction of the NAPL-wet
pores increases up to some critical value around 0:5, after which the trapped NAPL
saturation starts to decrease. A wide range of possible shapes of the water and NAPL
relative permeabilities is observed.
Spatially variable interfacial tension between 15 and 75 mN/m has negligible effect on
the flow characteristics.
The initial water saturation, Swi, has some effect on the two-phase flow
characteristics of water-wet sediments. With more NAPL-wet pores in the sediment,
Swi starts to have a strong effect on the water and NAPL relative permeabilities. In
intermediate-wet sediments, increasing Swi increases the NAPL and water relative
permeabilities. The increase of the water relative permeabilities is more dramatic. In
Fig. 10. Effects of initial water saturation on secondary imbibition capillary pressures and water and NAPL
relative permeabilities in NAPL-wet network.
80
A. Al-Futaisi, T.W. Patzek / Journal of Contaminant Hydrology 74 (2004) 61–81
NAPL-wet sediments, increasing Swi increases the water and NAPL relative
permeabilities for the average water saturation below 0.5 and decreases them for the
average water saturation above 0.5.
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