Transp Porous Med
DOI 10.1007/s11242-016-0815-7
Pore-scale Network Modeling of Three-Phase Flow Based
on Thermodynamically Consistent Threshold Capillary
Pressures. II. Results
Arsalan Zolfaghari1 · Mohammad Piri1
Received: 8 February 2016 / Accepted: 20 December 2016
© Springer Science+Business Media Dordrecht 2017
Abstract We use the model described in Zolfaghari and Piri (Transp Porous Media, 2016)
to predict two- and three-phase relative permeabilities and residual saturations for different
saturation histories. The results are rigorously validated against their experimentally measured counterparts available in the literature. We show the relevance of thermodynamically
consistent threshold capillary pressures and presence of oil cusps for significantly improving
the predictive capabilities of the model at low oil saturations. We study systems with wetting
and spreading oil layers and cusps. Three independent experimental data sets representing
different rock samples and fluid systems are investigated in this work. Different disordered
networks are used to represent the pore spaces in which different sets of experiments were
performed, i.e., Berea, Bentheimer, and reservoir sandstones. All three-phase equilibrium
interfacial tensions used for the simulation of three-phase experimental data are measured
and used in the model’s validation. We use a fixed set of parameters, i.e., the input network
(to represent the pore space) and contact angles (to represent the wettability state), for all
experiments belonging to a data set. Incorporation of the MSP method for capillary pressure
calculations and cusp analysis significantly improves the agreement between the model’s
predictions of relative permeabilities and residual oil saturations with experimental data.
Keywords Three-phase flow in porous media · Pore-scale network modeling ·
Cusp formation · Threshold capillary pressure · Wettability
Abbreviations
EB
G/O
B
Energy balance
Gas/oil
Mohammad Piri
[email protected]
Arsalan Zolfaghari
[email protected]; [email protected]
1
Dept. 3295, 1000 E. University Avenue, Laramie, WY 82071, USA
123
A. Zolfaghari, M. Piri
LC
LF
O/W
SGI
W/
W/o
Layer collapse
Layer formation
Oil/water
Secondary gas injection
With
Without
1 Introduction
Network models are developed to predict two- and three-phase relative permeabilities for
multi-phase fluid flow in porous media. Over the last two decades, authors have presented
several models incorporating various levels of pore-scale physics. A detailed review of the
pore-scale network models and their features is provided elsewhere (Piri and Blunt 2005a;
Zolfaghari and Piri 2016). Here, we briefly list studies in which network models were developed as a tool to predict experimentally measured three-phase relative permeabilities and
residual saturations.
Fenwick and Blunt (1998a, b) used regular pore networks to study three-phase flow in
water-wet porous media. In particular, they investigated the mechanisms relevant to high oil
recovery during various gravity drainage experiments. They used angular pores and allowed
layer drainage in their model to mimic quadratic characteristic form for the oil relative permeability at low saturations observed experimentally. The simulated three-phase relative
permeabilities and saturation paths were compared qualitatively with those from experimental measurements. Authors found that three-phase flow properties are strong functions
of saturation histories, displacement mechanisms, and spreading coefficients. Lerdahl et al.
(2000) stochastically modeled the main processes of sandstone formation and transformed
the results into a pore network. The authors compared the simulated relative permeabilities
successfully against experimental data by Oak (1990). Piri and Blunt (2005b) compared their
predictions of oil, water, and gas three-phase relative permeabilities and the corresponding
saturation histories against the experimental counterparts in Berea sandstone reported by
Oak (1990). Using a Berea sandstone network constructed by Øren et al. (1998), their overall predictions for different phases’ relative permeabilities were favorable. They, however,
systematically overpredicted oil relative permeabilities at low oil saturations where the flow
is dominated by spreading oil layers. As a result, the residual oil saturation at the end of gas
injection was underestimated. The comparison of the model presented by Al-Dhahli et al.
(2013) against the same experimental data continued to overpredict oil relative permeabilities
at low oil saturations. Consequently, the experimental residual oil saturation at the end of
gas injection was not predicted accurately. Three-phase gas relative permeabilities were not
reported for the same experiment. The authors, however, showed improved predictions of oil
relative permeability compared to those presented by Piri and Blunt (2005b). By matching
the measured two-phase relative permeabilities and capillary pressures for a relatively simple pore network, Svirsky et al. (2007) predicted three-phase relative permeabilities for the
reported experimental measurements in water-wet Berea sandstone (Oak 1990). Neglecting
the conductance of spreading oil layers in the model, the authors underpredicted three-phase
oil relative permeabilities during gas injections. Their respective saturation paths did not
agree with the experimentally measured values.
In this study, we use three independent sets of experimental data from literature to rigorously validate the prediction of our model, particularly at low oil saturation where the
123
Pore-scale Network Modeling of Three-Phase Flow Based…
stability of the wetting and spreading oil layers is simply the most important controlling
factor. The impact of thermodynamically consistent threshold capillary pressures and cusp
analysis (Zolfaghari and Piri 2016) is investigated by comparing our results against experimental data for both two- and three-phase systems. For each data set, we use one set of
contact angles to represent the wettability state of the system. No additional parameters are
used for the simulations of different experiments in the same data set. A modified saturation
path tracking technique is used to follow the experimental saturation path during gas injection. For those experiments where water saturation reached lower values than the irreducible
water saturation of the medium, a special evaporation module is developed to follow the
exact saturation path of the experiment. This ensures that relative permeability of each phase
at a particular saturation uniquely corresponds to an actual saturation state reached during
the experiment. We also compare the results against those predicted by other pore network
models. The paper is concluded by a set of final remarks.
2 Validation Against Experimental Data
The network model described in Zolfaghari and Piri (2016) is used to simulate two- and
three-phase relative permeabilities, capillary pressures, and saturation paths for three groups
of experiments performed in different sandstone samples with different wettabilities. Table 1
lists different groups of experimental data used for model validation in this work. We first
examine the effect of wetting layer formation (LF) and collapse (LC) on two-phase relative
permeabilities and residual saturations. This is done by comparing the results against experimental data reported by Valvatne and Blunt (2004) for waterflooding in an oil-wet reservoir
sandstone. We then validate the model against three-phase data, where spreading LF and LC
as well as cusp analysis impact the characteristics of relative permeability curves and residual
oil saturations. We compare the modeling results against two sets of experimental data generated with weakly water-wet Berea and Bentheimer sandstone core samples. Following the
approach proposed by Piri and Blunt (2005b), for each experimental data set, we first match
two-phase steady-state measurements of relative permeabilities while honoring Bartell and
Osterhof (1927) constraint among contact angles and interfacial tensions. After fixing the
system’s wettability state, we simulate various three-phase flow processes and compare the
simulated three-phase relative permeabilities, residual oil saturations, and saturation paths
against their experimental counterparts.
2.1 Two-Phase Flow in an Oil-Wet Reservoir Sandstone
As mentioned earlier, an experimental data set representing an oil-wet reservoir sandstone
(Valvatne and Blunt 2004) is selected to investigate the effect of thermodynamically consistent
threshold capillary pressures and stability of wetting oil layers on two-phase flow properties.
The steady-state relative permeabilities were reported for two core samples with fairly similar
permeabilities (approximately 300 mD). We did not have access to the original rock samples
used in the experiments to generate high-resolution images of the pore space and construct
equivalent pore networks. Therefore, we used the network modification process proposed
by Valvatne and Blunt (2004) to modify a Berea sandstone network used by various authors
(see, for instance, Refs. Valvatne and Blunt 2004; Piri and Blunt 2005a; Valvatne et al. 2005;
Piri and Blunt 2005b; Suicmez et al. 2007, 2008) to reproduce primary drainage capillary
pressure data. We assumed that the main topological features of the Berea pore network and
the sandstone samples used in the experiments are similar.
123
123
Rock type
Reservoir sandstone
Berea sandstone
Bentheimer sandstone
No.
1.
2.
3.
Weakly water wet
Weakly water wet
Strongly oil wet
Wettability
Two- and three-phase
Two- and three-phase
Two-phase
Data set
Table 1 Three different groups of experimental data used for model validation
Spreading oil layers
Spreading oil layers
Wetting oil layers
Type of LF
MSP, geometric, & cusp
MSP, geometric, & cusp
MSP & geometric
Type of LC
Alizadeh and Piri (2014)
Oak (1990)
Valvatne and Blunt (2004)
References
A. Zolfaghari, M. Piri
Pore-scale Network Modeling of Three-Phase Flow Based…
(a)
(b)
Fig. 1 Network modification for the simulation of experimental data. a Pore and throat size distributions for
the original Berea network are modified to simulate, b the experimental capillary pressure data for oil primary
drainage
Experimentally measured primary drainage capillary pressure data were used to modify
inscribed radius of the individual elements. The modified network was then used to simulate
two-phase relative permeability curves from waterflooding. Experimental values of interfacial tension and contact angles were not reported. Following the methodology described by
Valvatne and Blunt (2004), throat size distribution was tuned by a factor representing the
percentage of change. The factor could be fixed or linearly regressed between the maximum
and minimum values determined by the user. This approach ensures that the ranking order
of the throat sizes remains the same after modification. Each pore radius is then modified to
maintain the same aspect ratio as those of the pores in the original network. We performed a
careful analysis of the parameters and selected the best values to match the capillary pressure
data. Figure 1a illustrates the pore and throat size distributions for both initial and modified pore networks. The simulated oil/water capillary pressure generated using the modified
pore network is shown in Fig. 1b. The oil/water interfacial tension and PD contact angles
used in this simulation were 48 mN/m and 0◦ , respectively. After wettability alteration, core
exhibits strongly oil-wet characteristics with low residual oil saturation and Amott water
index of 0. We established an initial water saturation similar to the experimental value of
3%. We use MSP-Geometrical-a as the layer collapse rule (see Zolfaghari and Piri 2016 for
details). The advancing oil/water contact angles were uniformly distributed between 93◦ and
172◦ . According to the two-phase thermodynamically consistent threshold capillary pressure
results presented in Zolfaghari and Piri (2016), stability of wetting oil layers strongly depends
on the wettability. MSP method restricts the span of oil/water contact angles and capillary
pressures over which wetting oil layers can exist. Specifically, the new model predicts that the
wetting layers are stable on more oil-wet surfaces and at higher oil/water capillary pressures
(Zolfaghari and Piri 2016) and, hence, the contact angles used in this work are greater than
those reported in Valvatne and Blunt (2004). This is also consistent with the range of advancing contact angles used by Ryazanov et al. (2010) for the simulation of similar data utilizing
a network of pores and throats with regular cross sections. Figure 2 shows the simulated oil
and water relative permeabilities during waterflooding. As it is shown in Fig. 2b, implementation of thermodynamically consistent threshold capillary pressures for LF and LC not only
captures relative permeability trends but also accurately simulates residual oil saturation at
the end of waterflood. The simulated Amott water and oil indices after wettability alteration
are 0 and 0.145, respectively. These values are in good agreement with their reported exper-
123
A. Zolfaghari, M. Piri
(a)
(b)
Fig. 2 Comparison of the measured (Valvatne and Blunt 2004) and simulated oil and water relative permeabilities during waterflooding in an oil-wet reservoir sandstone: a a linear scale and b a semilog scale to
highlight the prediction of residual non-wetting phase saturation
imental counterparts (i.e., Iw = 0 and Io = 0.14 reported by Valvatne and Blunt 2004).
One should note that in these experiments, the sample used with the centrifuge method for
capillary pressure measurements was different than those used for the relative permeability
tests. That explains the difference between connate water saturations in Figs. 1b and 2.
The prediction of residual oil saturation with our model was possible through MSP analysis of the formation and collapse of wetting oil layers. A network model without MSP
analysis could still simulate relative permeabilities’ trend but would underestimate residual
oil saturation, see, for instance, simulation of the same experiments by Valvatne and Blunt
(2004). The residual oil saturation was underpredicted. In that work, the oil layers were
formed and collapsed based on geometric criteria. In other words, the same set of contact
angles that captures relative permeabilities’ trend do not lead to prediction of residual oil
saturation unless MSP analysis is implemented for the displacements.
2.2 Three-Phase Flow in Berea Sandstone
In order to validate our model for three-phase flow properties, we compare its predictions
against three-phase steady-state relative permeabilities, saturation paths, and residual oil
saturations reported by Oak (1990) and Alizadeh and Piri (2014). Both data sets use steadystate measurement techniques in which simultaneous injections of two or three phases at
different ratios are used to create various experimental saturation paths. In this section, we
focus on the Oak’s experimental data set (Oak 1990). This experimental data set includes twoand three-phase flow data in Berea sandstone core samples and has been previously studied
by Øren et al. (1998), Piri and Blunt (2005b), and Al-Dhahli et al. (2013). Piri and Blunt
(2005b) matched two-phase oil/water and gas/oil imbibition relative permeability curves by
adjusting relevant advancing contact angles. Having advancing oil/water and gas/oil contact
angles, they used Bartell and Osterhof constraint to obtain a distribution for the gas/water
contact angles. They successfully matched two-phase gas/water relative permeability with
the calculated contact angle distribution. In this work, we use oil/water and gas/oil contact
angle distributions similar to those used by Piri and Blunt (2005b). We, however, measured
experimentally three-phase equilibrium interfacial tensions for the fluids used by Oak at the
experimental pressure and temperature (Table 2). We then used the Bartell and Osterhof
123
Pore-scale Network Modeling of Three-Phase Flow Based…
(a)
(b)
Fig. 3 Comparison of the measured (Oak 1990) and simulated two-phase relative permeabilities in Berea
sandstone for the a oil/water and b gas/oil imbibition processes
Table 2 Three-phase equilibrium interfacial tensions and spreading coefficient (mN/m) measured for fluids
used by Oak (1990) in experiments on high-permeability Berea sandstone samples
Fluids
σow
σgo
σgw
Cs
Dodecane with 10 vol% iodooctane–brine–nitrogen
47.3
24.3
69.1
−2.5
Table 3 Contact angles in degree used for the simulation of various two- and three-phase experiments in
Berea sandstone
PD
θow
r
θow
a
θow
r
θgo
a
θgo
r
θgw
a
θgw
0
43–60
63–80
20–50
40–70
33.8–55.4
54.5–76.2
constraint to obtain gas/water contact angle distribution, which is slightly different than that
used by Piri and Blunt (2005b). Table 3 lists the contact angles used in this section. To
represent the pore space in Berea sandstone, we use a similar network as the one in Piri and
Blunt (2005a, b). To construct this network, a process-based technique founded on stochastic
reconstruction of the main sandstone forming processes (Bakke and Øren 1997; Øren et al.
1998; Øren and Bakke 2002) was used.
With these contact angle distributions, we were able to match the experimentally measured
two-phase relative permeabilities (see Figs. 3a, b and 4). For all fluid pairs, initial saturations of
the relevant drainage experiments have been matched. As reported in different experimental
studies (Oak 1990; Alizadeh and Piri 2014), gas/oil imbibition relative permeabilities are
different than those of experiments in which water is present, i.e., oil/water and gas/water
imbibitions. The gas/oil imbibition results were obtained with 80% clay volume of the original
network (see Fig. 3b). We assume a smaller fraction of the clay volume contributes to the
saturations. The clays are reported to have extremely small pore sizes (i.e., in order of several
nanometers). The layer spacing of clay minerals can vary with the clay structure and its water
content. As an example, the layer spacings for Wyoming montmorillonites are reported to
be between 1 and 1.5 nm for different water content values (Boek et al. 1995). Such small
pore sizes enhance water retention in clay minerals even under very small controlled relative
123
A. Zolfaghari, M. Piri
Fig. 4 Comparison of the
measured (Oak 1990) and
simulated two-phase gas/water
imbibition’s relative
permeabilities in Berea sandstone
humidities (Delage et al. 1998; Cui et al. 2002). In other words, the nanopore space within
the clay structure may still be saturated with water even during gas/oil experiments. This is
consistent with the experimental observations of higher wetting phase relative permeabilities
for gas/oil imbibitions in comparison with those of oil/water and gas/water experiments.
We study numerous three-phase experiments reported by Oak (1990). The same set of
contact angles, obtained based on the two-phase results (Table 3), is used. We first present
our results for one of the experiments that has been studied by other authors to compare
the predictions. This would allow us to illustrate the improvement in model’s predictive
capabilities on three-phase oil relative permeability at low oil saturations and residual oil
saturation. We then proceed to validating our results against other experiments in the same
data set. But first, we present equilibrium interfacial tensions measured for the fluid systems
used by Oak (1990) and Alizadeh and Piri (2014).
2.2.1 Measurements of Equilibrium Interfacial Tensions
Accurate measurement of interfacial tensions and consequently spreading coefficient affect
all cusp analysis and MSP calculations for three-phase flow modeling. Therefore, in this
work, we have measured the three-phase equilibrium interfacial tensions for the fluids used
by Oak (1990) at experimental pressure and temperature. The values for the fluid system
used by Alizadeh and Piri (2014) are reported from that reference.
Interfacial tensions were measured by pendant drop/rising bubble technique for dodecane
(with 10 vol% iodooctane), brine (with 5 wt% sodium chloride, 0.5 wt% calcium chloride,
and 10 wt% cesium chloride), and nitrogen. All three phases were pre-equilibrated for 12 h
prior to the measurements. The density values were 63.57, 804.53, and 1151.63 kg/m3 for
the equilibrated gas, oil, and brine phases, respectively. Axisymmetric drop shape analysis
technique was used to analyze the images of the bubbles/drops. More details about the
experimental apparatus can be found elsewhere (see Saraji et al. 2013). The measurement
cell was pressurized with one fluid, while the other fluid was slowly injected into the cell
through a needle. Different needle sizes were used for each pair of fluids to maintain a
bond number of approximately 1. At least 20 independent measurements were performed for
each pair of fluids. The average equilibrium interfacial tension values for dodecane/brine,
nitrogen/dodecane, and nitrogen/brine are listed in Table 2.
123
Pore-scale Network Modeling of Three-Phase Flow Based…
Fig. 5 Comparison of the measured (Oak 1990) and simulated three-phase saturation path of the secondary
gas injection for experiment 10 in sample 14
Fig. 6 Comparison of the measured (Oak 1990) and simulated three-phase oil relative permeabilities of the
secondary gas injection for experiment 10 in sample 14
2.2.2 Three-Phase Relative Permeabilities and Residual Saturations
The new implemented capabilities to handle formation and collapse of layers and cusps make
the model more sensitive to the magnitude of initial oil/water capillary pressures for gas
injection processes. Oil/water capillary pressure can be kept unchanged during gas injection,
if we do not use saturation path tracking steps.
123
A. Zolfaghari, M. Piri
Fig. 7 Impact of different models/parameters on the simulated three-phase oil relative permeabilities for
experiment 10 in sample 14 reported by Oak (1990). In this figure, w/, w/o, EB, and Cs denote with, without,
energy balance, and spreading coefficient, respectively
We start with comparing our results with those of experiment 10 in sample 14 of Oak’s
data set. The saturation path of this experiment indicates that it is a gas injection flow test
after a primary oil drainage. A closer investigation reveals that the oil volume flow rate had
been reduced by a factor of 8 before gas injection was started (i.e., at the end of primary
oil drainage). This explains the higher oil saturation reported for this experiment compared
to those of the other primary drainage tests listed for the same sample with similar oil and
water flow rates. Reducing the oil flow rate at the end of primary drainage reduces oil pressure
drop and consequently reduces the capillary pressure (Pini et al. 2012; Pini and Benson 2013;
Akbarabadi and Piri 2015; Zolfaghari 2014). To compare our simulated relative permeabilities
with those of the experiment, we followed three steps: (1) modeled a primary drainage to an
initial brine saturation similar to that of the experiment, (2) adjusted the capillary pressure
according to oil flow rate changes in the experiment as mentioned earlier, and (3) modeled
gas injection. We then compare the simulated three-phase relative permeabilities with those
of the experiment.
Figure 5 shows the experimental and simulated saturation paths for experiment 10 in
sample 14. The comparison of the simulated oil relative permeability with its experimental
counterpart for this experiment is illustrated in Fig. 6. In this plot, the results by Piri and
Blunt (2005a, b) and Al-Dhahli et al. (2013) are also shown for comparison. As it is seen,
incorporation of thermodynamically consistent spreading oil LF and LC as well as cusp configurations has significantly improved model’s predictive capabilities particularly at low oil
saturations. In addition to an improved relative permeability prediction, the model produces
closer agreement with the experimental residual oil saturation at the end of gas injection process. Similar to two-phase oil relative permeability results (see Sect. 2.1), previous models
tend to underestimate the residual oil saturation by allowing the oil layers collapse at higher
gas/oil capillary pressures.
As mentioned earlier, the agreement shown in Fig. 6 has been achieved through incorporation of LF, LC, and cusp analysis based on energy balance. A set of three-phase equilibrium
interfacial tensions were measured in our laboratory for brine, dodecane, and nitrogen, the
123
Pore-scale Network Modeling of Three-Phase Flow Based…
fluid system that had been used by Oak in the flow experiments. The calculated spreading
coefficient was negative based on the measured interfacial tension (see Sect. 2.2.1). Here,
we investigate the impact of each factor to illustrate the resulting variations in the simulated relative permeabilities for experiment 10 in sample 14. This allows us to demonstrate
that only after all the pertinent features are incorporated, one can properly simulate oil relative permeabilities at low oil saturations. Figure 7 presents simulation results for four cases
designed to demonstrate the sensitivity of the relative permeabilities to presence/absence of
different parameters/capabilities. For comparison, we also include the predictions made by
Piri and Blunt (2005a, b) and Al-Dhahli et al. (2013) for the same experiment. In this figure,
EB stands for energy balance analysis for all displacements including LF and LC. W/ and
w/o denote “with” and “without”, respectively. Cs represents spreading coefficient for the
set of interfacial tensions used. As it is seen, any simulation results without energy balance
or cusp analysis fail to quantitatively produce the three-phase oil relative permeability and
residual oil saturation.
We first turn off both LF and LC based on energy balance and cusp analysis, and at
the same time use the measured interfacial tensions in the simulation (“W/o EB & Cusp,
Cs = measured” or case 1). This means that the layers will form and collapse based on geometric criteria. The goal is to find out whether the measured interfacial tensions and the
resulting negative spreading coefficient would improve the agreement with the data compared to the results presented by Piri and Blunt (2005b). In this case, new LF/LC and cusp
capabilities were not used. Next, we use the interfacial tensions and spreading coefficient
used in Piri and Blunt (2005b) (i.e., σow = 48, σgo = 19, and σgw = 67 mN/m) as well
as new LF/LC capability (based on energy balance) but without cusp formation (“W/ EB &
w/o Cusp, Cs = 0” or case 2). In case 3, we repeat case 2 with this difference that we use
measured interfacial tension and spreading coefficient (“W/ EB, w/o Cusp, Cs=measured”).
And finally, we simulate a gas injection with all new capabilities included but we use a
less negative spreading coefficient (“W/ EB & Cusp, Cs = −0.2” or case 4). In this case,
we run the model with both energy balance and cusp analysis for a non-spreading system
with a slightly negative spreading coefficient, i.e., −0.2 mN/m. A slightly negative spreading
coefficient is used to make the system as close as possible to completely spreading while
using cusp analysis. It shows that cusp analysis has the most prominent impact on oil relative
permeability at low oil saturations and the residual oil saturations at the end of gas injection. According to thermodynamically consistent threshold capillary pressures (Zolfaghari
and Piri 2016), cusp forms and collapses at much lower gas/oil capillary pressures. As it
is concluded from Fig. 7, all new developments including energy balance, cusp analysis,
and representative values of three-phase equilibrium interfacial tensions affect the model’s
prediction of three-phase oil relative permeability as well as residual oil saturation. The best
agreement in Fig. 7 was obtained only after incorporation of all the relevant factors mentioned
earlier. Also, one should note that the simplified representations of pore geometries used in
the model could possibly impact conductance of the layers and, hence, their connectivities.
Layer conductances in real pore bodies could be reduced due to the variable cross sections,
surface roughnesses, and corner geometries.
Figure 8 shows the comparison of simulated and measured three-phase gas relative permeability for the same experiment (experiment 10 in sample 14). Impact of the new methodology
for handling oil layers/cusps is minimal on the relative permeability of the most non-wetting
phase, particularly if gas to water displacements do not compete with gas to oil displacements
during gas injection process, which is the case in this experiment as initial water saturation
was low prior to the injection of gas. As it is seen, all simulated results show good agreement
with those of the experiment.
123
A. Zolfaghari, M. Piri
Fig. 8 Comparison of the measured (Oak 1990) and simulated three-phase gas relative permeabilities of the
secondary gas injection for experiment 10 in sample 14
Next, we study experiments 25 in sample 13, 13 in sample 14, and 7 in sample 14 from
Oak’s data set (Oak 1990). The last two experiments relate to gas injections after primary
oil drainage, waterflood, and secondary oil drainage, but with a rather important difference
with the experiment we have studied earlier. In these experiments, water saturation reaches
below the irreducible values reported for the primary drainage experiments in the same core
samples. This reduction often occurs after the first introduction of gas into the core sample,
i.e., beginning of the gas injection process. After an initial drop, water saturation remains fairly
constant throughout the gas injection experiment. We think this indicates water evaporation
due to the injection of not fully equilibrated gas at the beginning of experiments. Oak had used
fresh gas for the experiments as opposed to recirculation of fully equilibrated gas. In order
to account for this reduction in water saturation prior to gas injection, we have developed a
procedure in which the saturated clay volume is slightly tuned to match experimental water
saturation during gas injection. With this approach, we compare our results against those of
three different gas injection experiments.
First, we simulate a secondary gas injection process with initial oil saturation of 71.9%,
i.e., experiment 25 in sample 13. Since there was no change in oil flow rate before the start of
gas injection, no adjustment is made to the oil/water capillary pressure at the end of simulated
primary drainage. As mentioned earlier, we use a technique to match water saturation during
gas injection process. This allows us to follow the same saturation path as the experiment
(see Fig. 9a). The minimum observed water saturation for experiment 25 in sample 13 is
22.5%, which is slightly below irreducible water saturation at the end of primary drainage
for the same sample, i.e., 24.6%. Figures 9b and 9c illustrate the simulated oil and gas
relative permeabilities for this experiment. As it is seen, we obtain a good agreement with
the experimental data, particularly for oil relative permeability at low oil saturations. Both
MSP and cusp analysis have been used in the simulation of all experiments presented in this
section.
Experiment 13 in sample 14 is selected to test the model for simulation of a tertiary
gas injection process with initial oil saturation of 73.1%. We think this experiment is a gas
injection process after waterflooding followed by a secondary oil drainage, since the closest
123
Pore-scale Network Modeling of Three-Phase Flow Based…
(a)
(b)
(c)
Fig. 9 Comparison of the measured (Oak 1990) and simulated secondary gas injection of experiment 25 in
sample 13 for a the saturation path, and the three-phase b oil and c gas relative permeabilities
previous scan numbers in the experimental data set show water injection into the core. Oil
saturation is then brought to 73.1% by a secondary oil drainage. We follow the same flooding
steps. We then use the procedure mentioned earlier to match the saturation path. Figure 10a
shows the comparison of the simulated and experimental saturation paths. Figures 10b and 10c
show the agreement between simulated oil and gas relative permeabilities and their experimental counterparts for this flow test. The non-monotonic trend of oil relative permeabilities
observed at the end of gas injection is caused by slight variations in the solutions of systems
of linear equations used to compute oil pressure at each pore containing a connected oil phase
location. These equations are written for the connected phase clusters to compute relative permeabilities (Piri and Blunt 2005a). As the gas injection proceeds, formation of oil layers and
cusps creates flow bottlenecks causing significantly different oil pressures in the neighboring
pore bodies. Oil/water capillary pressure was not modified for this experiment since oil flow
rate had not been changed compared to the previous secondary oil injection experiment. One
should note that the capillary pressure reached during a secondary oil drainage is less than
that of the primary drainage for a similar saturation. This might be attributed to the contact
angle hysteresis after a primary drainage. The lower oil/water capillary pressure explains
why oil relative permeabilities fall lower for a gas injection process after a secondary oil
drainage than that of secondary gas injection.
123
A. Zolfaghari, M. Piri
(a)
(b)
(c)
Fig. 10 Comparison of the measured (Oak 1990) and simulated tertiary gas injection of experiment 13 in
sample 14 for a the saturation path, and the three-phase b oil and c gas relative permeabilities
The last experiment we study in this group is experiment 7 in sample 14. The difference between this and the previous experiment is that the oil/water capillary pressure has
been adjusted according to the change in oil flow rate at the beginning of the gas injection
experiment. This adjustment procedure is similar to the one used earlier for experiment 10
in sample 14. As mentioned earlier, experiment 7 is considered as a gas injection after secondary oil drainage. For the simulation purposes, a complete primary drainage establishing
Swi of 24.2% is followed by a complete waterflood (Sor = 25.7%). Subsequently, a secondary
oil injection is performed to match the reported remaining water saturation of 24.7% prior
to the introduction of gas in the experiment. Gas injection is then simulated with the full
MSP and cusp analysis included. Since the first reported experimental data point after gas
injection has lower water saturation (i.e., Sw = 20.3%) than Swirr of the system (24%), the
evaporation module is invoked to adjust saturated clay volume and match the corresponding
water saturation. For the rest of the reported saturation trajectory, water saturation remains
fairly constant. The simulated and experimental saturation paths are compared in Fig. 11a
for this gas injection experiment. Figures 11b and 11c illustrate the comparison of the simulated and measured oil and gas relative permeabilities, respectively. The agreements are
encouraging.
123
Pore-scale Network Modeling of Three-Phase Flow Based…
(a)
(b)
(c)
Fig. 11 Comparison of the measured (Oak 1990) and simulated gas injection of experiment 7 in sample 14
for a the saturation path, and the three-phase b oil and c gas relative permeabilities
2.2.3 Saturation Path Tracking
All the three-phase experiments that we have studied so far, have one thing in common;
they all have relatively low initial water saturations. That makes gas to oil displacements
the dominant displacement group during gas injection. This situation changes if the initial
condition (i.e., before gas injection) has significantly higher water saturation to be displaced
by gas. This means that gas could displace both oil and water leading to rather complicated
saturation paths. As it is explained in Zolfaghari and Piri (2016), we use a saturation path
tracking technique to follow the experimental saturation path in these experiments. Three
gas injection experiments with different initial oil saturations were selected. Cusp analysis
and MSP method were used in conjunction with saturation path tracking technique to gauge
the model’s predictive capabilities for a broader range of gas injection experiments.
The first experiment in this group relates to a gas injection process with initial oil saturation
of 59% established by an oil drainage process (experiment 9 in sample 13). We track the
saturation history of the experiment by simulating a primary oil drainage to create the initial
condition for the gas injection. At this point, a complete gas injection is modeled using
our saturation path tracking algorithm. The algorithm adjusts oil/water capillary pressure to
123
A. Zolfaghari, M. Piri
match the experimentally measured saturation path of the experiment (Zolfaghari and Piri
2016). Adjustments are made at constant gas/oil capillary pressures to prevent any oil to gas
displacements. Figures 12a and 12b illustrate the simulated three-phase oil/water and gas/oil
capillary pressures, respectively, for experiment 9 in sample 13. For comparison purposes,
the corresponding two-phase capillary pressures are also included. Three-phase oil/water
capillary pressure closely follows that of the drainage curve in two-phase flow. This represents
oil to water drainage as proposed in the double-displacement mechanisms, where gas pushes
oil to invade water-filled elements. Three-phase gas/oil capillary pressures (see Fig. 12b), on
the other hand, reflect threshold gas pressures for the most favorable displacements of gas
to oil or water. A gas to water invasion could only happen as a single displacement where
it produces water at the outlet. Two different mechanisms, on the other hand, could trigger
gas to oil displacements. They are single and continuous-continuous double-displacement
mechanisms. The experimental and simulated three-phase saturation paths are compared and
shown in Fig. 13a for experiment 9 in sample 13. Figure 13b shows simulated three-phase oil
relative permeability during the gas injection process. The model captures the sharp reduction
in oil relative permeability as well as the residual oil saturation. Oil/water capillary pressure
at the beginning of gas injection process was not modified since the experimental oil flow rate
had remained unchanged from the preceding oil injection experiment. One, however, should
note that oil/water capillary pressure was modified during gas injection process due to the
saturation path tracking algorithm. Cusp analysis and MSP method always use the latest
value of the oil/water capillary pressure reached in the simulation. In other words, saturation
path tracking impacts energy balance calculation throughout the gas injection process due to
changes in oil/water capillary pressure.
The simulated gas relative permeability for the same experiment is presented in Fig. 13c.
As it is seen, the first experimental gas relative permeability appears at a lower gas saturation
than that of the simulated results. This is attributed to the finite size effect associated with the
use of small pore networks. When it is compared against other sets of similar experimental
data, experiment 9 in sample 13 exhibits much earlier first relative permeability point. The
closest experiment for the same sample would be experiment 8, with the first data point gas
saturation of almost three times as that observed in experiment 9. This might be related to a
lower initial ratio of gas to oil flow rates used in experiment 9 compared to other similar tests
(for instance, see experiments 6, 7, and 8 in sample 13 and experiment 5 in sample 14 reported
by Oak (1990)). Figure 13d shows the simulation results of water relative permeability for this
experiment. All the figures show good agreement with the experimental measurements. The
quality of agreements presented in Fig. 13a–d demonstrates the significance of saturation path
tracking, whereas oil relative permeability agreement in Fig. 13b validates the importance of
cusp analysis and MSP method.
The next flow test we investigate is experiment 8 in sample 13. Similar to the previous
experiment, this is a gas injection process after a primary oil injection. However, initial oil
saturation before gas injection is slightly lower, i.e., 57%, and more importantly the gas-to-oil
flow rate ratio is much higher than that of experiment 9 in sample 13. Figure 14a shows the
experimental and simulated saturation path of the experiment obtained using our saturation
path tracking algorithm. For simulation, the exact saturation history, as explained for the
experiment 9 above, has been followed. The only difference is the initial oil saturation at the
end of primary oil injection (57%). Figure 14b–d illustrates the comparison of simulated oil,
gas, and water relative permeabilities with their experimental counterparts. The agreements
for oil and water relative permeabilities are satisfactory. We, however, slightly overpredict
gas relative permeability. This is attributed to much higher initial gas flow rate (5.52 mL/min)
used in this experiment compared to that in experiment 9 (0.183 mL/min).
123
Pore-scale Network Modeling of Three-Phase Flow Based…
(a)
(b)
Fig. 12 Comparison of the simulated two- and three-phase a oil/water and b gas/oil capillary pressures.
Two-phase results are obtained for the drainage and imbibition cycles of the corresponding pair of fluids.
Three-phase capillary pressures are simulated using the saturation path tracking algorithm for experiment 9
in sample 13
The last flow test studied in this group is experiment 7 in sample 13. This is similar
to the previous experiment, but with lower initial oil saturation prior to gas injection, i.e.,
48%. The lower initial oil saturation translates to lower oil/water capillary pressure at the
beginning of gas injection. As before, we do not modify the oil/water capillary pressure
at the beginning of gas injection process. Figure 15a shows the experimental saturation
path and the simulated counterpart for this experiment. Experimental and simulated relative
permeabilities for oil, gas, and water are presented in Fig. 15b–d. Similar to the previous
experiment, we overpredict gas relative permeability because of the gas flow rate at the
123
A. Zolfaghari, M. Piri
(a)
(c)
(b)
(d)
Fig. 13 Comparison of the measured (Oak 1990) and simulated gas injection of experiment 9 in sample 13
for a the saturation path, and the three-phase b oil, c gas, and d water relative permeabilities
beginning of gas injection (compare Q g = 7.79 mL/min at the beginning of experiment 7 with
Q g = 0.183 mL/min in experiment 9). Figure 16a shows the comparison of the experimental
three-phase gas relative permeabilities for all three experiments presented in this section.
Similar gas flow rate steps in experiments 7 and 8 lead to identical gas relative permeability
trends in these experiments, while experiment 9 shows higher gas relative permeability for a
given gas saturation as it used much lower initial gas flow rates.
By comparing Figs. 13c and 14c, one notices the importance of gas flow rate, particularly at the beginning of gas injection, on the magnitude of gas relative permeability at a
given saturation. This is independently confirmed through experimental measurements in
Bentheimer sandstone in which secondary gas injection (SGI) experiments were performed
with the same initial saturation but different gas flow rates, see Fig. 16b from Alizadeh and
Piri (2014). SGI is used to refer to a process in which gas is injected into a sample that
has previously been subjected to primary oil drainage in order to establish the initial water
saturation. The injection sequences in both experiments presented in Fig. 16b are identical.
The only difference is the capillary numbers, which is about two orders of magnitude higher
for the viscous-dominated experiment, i.e., about 10−4 .
In the experiment with lower capillary number, gas finds the least resistance path from
inlet of the core sample to the outlet. Being the most non-wetting phase, gas occupies the
largest pores in the rock sample spanning from the inlet to outlet with the minimum gas
123
Pore-scale Network Modeling of Three-Phase Flow Based…
(a)
(b)
(c)
(d)
Fig. 14 Comparison of the measured (Oak 1990) and simulated gas injection of experiment 8 in sample 13
for a the saturation path, and the three-phase b oil, c gas and d water relative permeabilities
saturation possible, i.e., 10.8% in Fig. 16b. In the higher capillary number test, however,
viscous pressure drop plays a significant role. Consequently, gas does not necessarily occupy
the largest elements particularly in areas closer to the inlet of the core. This causes the gas
saturation to increase. The increase in gas saturation is not necessarily due to the largest
elements of the rock and does not significantly contribute to the connectivity of the main
flow path. These two factors lead to a lower gas relative permeability as a function of its
saturation for the viscous-dominated flow regime as observed in Fig. 16b.
Comparing experiment 8 with 9 in sample 13 of the Oak data set (Oak 1990), a similar
observation could be made about the gas flow rate at the beginning of gas injection experiments. The flow rates at the beginning of gas injection in experiment 9 are Q w = 0.02,
Q o = 2.56 and Q g = 0.183 mL/min. In experiment 8, water and oil flow rates are the
same as those in experiment 9 but gas flow rate is increased more than 30 times, i.e.,
Q g = 5.52 mL/min. Gas relative permeability is higher when is plotted against its saturation for the experiment with lower initial gas flow rate, i.e., experiment 9, as explained
earlier for the case of experiments presented by Alizadeh and Piri (2014). This explains the
model’s overprediction of gas relative permeability depicted in Figs. 14c and 15c.
The model developed in this work assumes capillary-dominated displacements during any
fluid injection process. As a result, the model’s prediction of gas relative permeability is more
representative of experiments with smaller capillary number, i.e., experiment 9, (see Fig. 13c).
123
A. Zolfaghari, M. Piri
(a)
(b)
(c)
(d)
Fig. 15 Comparison of the measured (Oak 1990) and simulated gas injection of experiment 7 in sample 13
for a the saturation path, and the three-phase b oil, c gas and d water relative permeabilities
(a)
(b)
Fig. 16 Impact of viscous- and capillary-dominated regimes on the measured gas relative permeabilities for
a experiments 7–9 in sample 13 reported by Oak (1990) and b SGI reported by Alizadeh and Piri (2014)
Interestingly, higher gas flow rates show very little impact on oil relative permeabilities in
the experiments studied here. This is supported by the fact that our simulated results agree
with their experimental counterparts in both groups.
123
Pore-scale Network Modeling of Three-Phase Flow Based…
Table 4 Three-phase equilibrium interfacial tensions and spreading coefficient (mN/m) measured for fluids
used by Alizadeh and Piri (2014) in experiments on Bentheimer sandstone samples
Fluids
σow
σgo
σgw
Cs
Soltrol 170-brine-nitrogen
41
20.9
61.7
−0.2
2.3 Three-Phase Flow in Bentheimer Sandstone
Two- and three-phase steady-state measurements of relative permeabilities in Bentheimer
sandstone have been recently reported by Alizadeh and Piri (2014). All experiments were
performed on two core plugs cut from the same block of Bentheimer sandstone. The core
plugs had the same dimensions, 15.24 cm in length and 3.81 cm in diameter. High-resolution
micro-CT images of samples cut from the same block were used to construct the Bentheimer
sandstone pore network used in this study. The extracted Bentheimer network is 2.7 mm in
each direction with 15,664 pores and 28,095 throats. A 67.6% of all pores and throats have
triangular cross sections, while 31.6% and 0.8% are rectangular and circular, respectively.
The network has 0.82% clay volume, and an average coordination number of 3.52. Network
porosity and permeability are 23.8% and 2.66 D, respectively. For both samples, the average
measured X-ray porosity and brine permeability were 24.4% and 2.64 D, respectively. The
fluids used in the experiments were Soltrol 170 with 5 vol% iodooctane, brine (2 wt% calcium
chloride, 12 wt% sodium iodide, and 0.01 wt% sodium nitrate), and nitrogen. Interfacial
tensions were measured by the authors using the pendant drop/rising bubble technique at
experimental pressure and temperature conditions. Table 4 lists the measured equilibrium
interfacial tensions.
2.3.1 Two-Phase Results
In this section, we present the comparison of our simulated two-phase flow properties with
those measured experimentally for primary oil drainage and oil/water, gas/oil, and gas/water
imbibition processes.
For primary oil drainage, we assume receding oil/water contact angle on the original
solid surface is 0◦ . The network is initially fully saturated with water. Oil is then injected to
establish the experimentally measured water saturation. Figure 17a shows the comparison
of the experimental and simulated relative permeabilities for the primary oil/water drainage
process.
To simulate imbibition, we use two imbibition experimental data sets (i.e., oil/water and
gas/oil) to find the relevant advancing contact angles on the altered wettability surfaces. We
adjust the contact angles to match the pertinent relative permeabilities. Similar to the work of
Valvatne and Blunt (2004), two different ranges of contact angles are distributed uniformly
for each imbibition process. Before simulation, we assign a target volume fraction of pore
bodies to mark two distinct groups of elements in the network. The pores are sorted based
on their inscribed radii from the largest to the smallest. From the top of the list, the pores are
added to the first group (hereinafter called “larger” group) until the target volume fraction is
reached. The rest of the pores are automatically categorized under the second group (called
smaller group). Then, the contact angles are distributed uniformly for each group separately
based on the assigned ranges. For each throat, if all neighboring pores belong to a group, we
assign it to the same group. Throats with neighboring pores belonging to different groups
123
A. Zolfaghari, M. Piri
(a)
(b)
Fig. 17 Comparison of the measured (Alizadeh and Piri 2014) and simulated two-phase oil and water relative
permeabilities in Bentheimer sandstone for the a primary drainage and b imbibition processes
would have the average contact angle of the connecting pores. Our investigation shows the
smaller group tends to be less water wet. This is consistent with the theoretically based
pore-level scenarios for wettability alteration proposed by Kovscek et al. (1993). It is also
supported by several experimental observations. Using simple uniform distributions, the
model overestimates both non-wetting phase relative permeabilities and residual saturations
of two-phase imbibition flow tests. This indicates that the non-wetting phase tends to reside in
smaller elements of the intermediate-size pores. These smaller elements keep the non-wetting
phase connected at lower saturations, and as a result, residual non-wetting phase saturations
decrease at the end of imbibition cycles. As mentioned earlier, this is similar to the work
of Valvatne and Blunt (2004), where smallest oil-filled pores were made more oil wet. The
authors correlated advancing oil/water contact angles with the pore sizes to match Amott oil
and water indices for imbibition tests with different initial oil saturations. We, however, use
a uniform distribution for our contact angles in each group. This means that the minimum
value for the smaller group is the same as the maximum value for the larger group. This leads
to a smooth transition in contact angle distribution between the two groups. Contact angle
ranges were adjusted to match the residual non-wetting phase saturation at the end of two
imbibition experiments. The target volume fraction has been mainly changed to match the
magnitude of relative permeabilities.
Figure 17b illustrates the comparison of the simulation of experimental results for oil/water
imbibition process. The experimental irreducible water saturation, i.e., 9.1%, was matched
during PD as the initial condition for the following imbibition. The top 85% of the largest pores
and their connecting throats form the larger group for which contact angles are uniformly
distributed between 45◦ and 55◦ . The rest of the pores and their connecting throats use
uniform distribution of 55◦ –75◦ . The clay volume is assumed to be fully saturated with water
in the model for the simulation of oil/water imbibition.
The simulation and experimental results for gas/oil imbibition are shown in Fig. 18a. To
establish the initial condition for imbibition, a gas/oil primary drainage was performed to
reach the experimental oil saturation of 20.5% with gas/oil receding contact angle of 0◦ .
We used a similar procedure for contact angle distribution as the one used for the oil/water
imbibition test. Target volume fraction for the gas/oil contact angle distribution was 65%.
The larger group was assigned a uniform distribution of advancing contact angle between
30◦ and 40◦ . The smaller group used the 40◦ –65◦ range. For two-phase gas/oil imbibition in
123
Pore-scale Network Modeling of Three-Phase Flow Based…
(a)
(b)
Fig. 18 Comparison of the measured (Alizadeh and Piri 2014) and simulated two-phase relative permeabilities
in Bentheimer sandstone for the a gas/oil and b gas/water imbibition processes
Bentheimer sandstone, the clay is excluded to obtain the results shown in Fig. 18a. A similar
approach was used for simulations in Berea sandstone pore network (see Fig. 3b). This is
consistent with the experimental observations of water retention in clay minerals at different
levels of relative humidity (Delage et al. 1998; Cui et al. 2002).
Since we have established the ranges for oil/water and gas/oil contact angles, gas/water
contact angles can be calculated from Bartell and Osterhof (1927) constraint. There are three
target volume fractions for the gas/water contact angle distribution, as the oil/water and gas/oil
distributions have different target volume fractions. The larger group (pores and connecting
throats with the target volume fraction of 65%) was assigned a uniform distribution of 40.2◦
and 50.1◦ . Minimum and maximum contact angles for the intermediate group (target volume
fraction of 65–85%) are calculated as 43.1◦ and 58.3◦ , respectively. The range for the smallest
group was between 50.1◦ and 71.6◦ . Figure 18b shows the comparison of the simulated
results (obtained with the calculated contact angle ranges) with those of the experiment. Clay
volume is considered for this experiment, since water is present. The simulation results are
encouraging. In particular, residual gas saturation is predicted accurately.
2.3.2 Three-Phase Results
Using the same advancing contact angles obtained from the two-phase validations presented
earlier, we simulate the three-phase relative permeabilities for gas injection processes. The
receding oil/water and gas/oil contact angles for each element are assumed to be 5◦ and 30◦
less than the corresponding advancing values, respectively.
Figure 19a illustrates the comparison of the simulation of steady-state three-phase oil
relative permeabilities with those from different gas injection experiments. The experimental
results are related to various gas injection processes with different initial conditions. The simulated results are for a secondary gas injection (SGI) after a complete primary oil drainage.
Due to the relatively narrow pore size distribution of Bentheimer sandstone, the experimentally measured three-phase oil relative permeabilities show weak sensitivity to changes in
initial oil saturation. Therefore, we compare our secondary gas injection simulation results
with all the three-phase oil relative permeabilities reported by Alizadeh and Piri (2014).
Interestingly, experimental data points of secondary gas injection after a complete primary
drainage follow the simulation results much more closely (see Fig. 19a). Clay mineral micro-
123
A. Zolfaghari, M. Piri
(a)
(b)
Fig. 19 Comparison of the measured (Alizadeh and Piri 2014) and simulated gas injection in Bentheimer
sandstone for the three-phase a oil and b gas relative permeabilities
porosity is assumed to be fully saturated with water for the three-phase simulations. Oil/water
capillary pressure at the end of primary drainage was not changed since oil and water flow
rates were not adjusted in the experiments prior to the start of gas injection. The simulated and
experimental steady-state three-phase gas relative permeabilities are compared and shown in
Fig. 19b. Simulation results at the beginning of gas injection overestimate the corresponding
experimental measurements. We think this may have been caused by finite size effects and
the fact that the pore network constructed from the micro-CT images may have had slightly
different pore size distributions.
Alizadeh and Piri’s (2014) experimental data clearly show two distinct regions for oil
relative permeability: the quartic dependency of oil relative permeability to its saturation at
high oil saturation (region 1) and quadratic dependency at low oil saturations (region 2).
The latter is believed to be due to oil layer drainage (Piri and Blunt 2005a). Experimental
observation of kro ∼So2 confirms a layer drainage-dominated regime at low oil saturations
during gas injection. Similar trend has been reported for oil relative permeability in Clashach
sandstone by Naylor et al. (1995), in bead packs using analog fluids by Grader and O’Meara
(1988), during gravity drainage process in sand packs by Sahni et al. (1998) and DiCarlo et al.
123
Pore-scale Network Modeling of Three-Phase Flow Based…
(2000), and during gas injection into high-permeability Clashach sandstone by Goodyear and
Jones (1993).
There are two independent gas injection experiments in Alizadeh and Piri (2014) indicating the existence of a third region at very low oil saturations in the Bentheimer sandstone
sample. Oil saturations are 5–9% smaller than those in the region 2. Lower gas flow rates,
and at the same time, higher ratio of gas to oil flow rates were used in these experiments to
achieve such low oil saturations. In this region, oil relative permeability drops with a hexic
dependency to its saturation. The sharp drop is caused by oil layer/cusp collapse displacements and, hence, trapping of oil. This is shown by the slope of the line labeled kro ∼So6 in
Fig. 19a for the corresponding experimental data. By incorporating the correct criteria for
layer/cusp formation and collapse (Zolfaghari and Piri 2016), the model’s prediction of residual oil saturation is improved significantly. All three regimes are predicted well by the model
presented here. Regions 1 and 3 show stronger dependencies of the oil relative permeability
to its saturation. These regions correspond to the beginning and end of gas injection where
pore space topology impacts oil relative permeabilities more strongly. Note that Fig. 19a uses
logarithmic scales for both x- and y-axes.
Sensitivity analysis of the model with respect to different parameters such as initial oil
saturation, equilibrium spreading coefficient, and wettability conditions is given elsewhere
(Zolfaghari 2014).
3 Conclusions
A significantly improved three-phase pore-scale network model was successfully used to simulate the trends observed in three independent experimental data sets on two- and three-phase
flow in porous systems with different wettability states and working fluids. The new model
for calculating thermodynamically consistent threshold capillary pressures was considerably
more sensitive to the experimental parameters such as interfacial tensions, contact angles, and
oil/water capillary pressures. As a consequence, all three-phase interfacial tensions used for
the simulation of three-phase data sets were measured under this study. For each set of twoand three-phase experiments, we used one set of fixed contact angles. Different experiments
belonging to each data set were then simulated with no additional adjustments in the parameters. After implementing the appropriate threshold capillary pressures and collapse scenarios
for two-phase flow, the model was validated by accurately predicting residual oil saturation
at the end of waterflooding in an oil-wet reservoir sandstone. The simulation results for both
primary drainage and waterflood relative permeabilities as well as capillary pressure data
were promising. We then focused on validating the model for simulating three-phase flow
systems. Oil/water and gas/oil contact angles were determined based on the simulation results
of the corresponding two-phase imbibition relative permeability and residual saturations in
those systems while honoring their initial saturations at the end of drainage in each experiment. Using Bartell and Osterhof (1927) constraint, we calculated gas/water contact angles.
The simulations of gas/water imbibition relative permeabilities were successfully compared
against the experimental measurements using the calculated values of contact angles. After
determining all the contact angles, the model was used to simulate three-phase flow tests
and gauge its predictive capabilities for relative permeabilities and residual oil saturations at
the end of secondary and tertiary gas injection processes. Initial saturation conditions were
honored for each of the experiments investigated in this work. For experiments in which
oil and water flow rates were reduced prior to the start of gas injection, oil/water capillary
123
A. Zolfaghari, M. Piri
pressures were modified accordingly. The model was extensively validated against various
three-phase experimental data sets in Berea and Bentheimer sandstones. The model uses the
continuous-continuous double-displacement mechanism to follow the exact saturation path
of experiments with relatively high initial oil and water saturations. This enabled the model to
compare relative permeabilities for the exact saturation points reached during experiments.
The model requires the oil/water capillary pressure to increase when the saturation path of an
experiment during gas injection is followed. Predictive capability of the model was significantly improved particularly for oil relative permeability at low oil saturations where the flow
of the intermediate-wet phase is controlled by layer/cusp flow. As a consequence, relative
permeabilities and residual oil saturations at the end of gas injections were estimated accurately. The importance of the initial gas flow rates, and hence, capillary numbers on the pore
fluid occupancies and consequently gas relative permeabilities was illustrated by comparing
various experimental data sets in Berea and Bentheimer sandstones and their corresponding
simulation results.
There was no single new development in the model solely responsible for its improved predictive capabilities. The combined effect of all new developments resulted in the significantly
improved agreements with the experimental data.
Acknowledgements We gratefully acknowledge financial support of Total, Saudi Aramco, EnCana, and the
School of Energy Resources and the Enhanced Oil Recovery Institute at the University of Wyoming. We extend
our gratitude to Hu Dong and Sven Roth of iRock Technologies for generating some of the pore networks for
our samples. We thank Pål-Eric Øren of FEI for sharing his Berea network data. We also thank Henry Plancher
and Soheil Saraji of Piri Research Group for measuring the interfacial tension and density values and for the
details of the IFT experimental procedure used.
References
Akbarabadi, M., Piri, M.: Co-sequestration of SO2 with supercritical CO2 in carbonates: an experimental
study of capillary trapping, relative permeability, and capillary pressure. Adv. Water Resour. 77, 44–56
(2015)
Al-Dhahli, A., van Dijke, M.I., Geiger, S.: Accurate modelling of pore-scale films and layers for three-phase
flow processes in clastic and carbonate rocks with arbitrary wettability. Transp. Porous Media 98(2),
259–286 (2013)
Alizadeh, A.H., Piri, M.: The effect of saturation history on three-phase relative permeability: an experimental
study. Water Resour. Res. 50(2), 1636–1664 (2014)
Bakke, S., Øren, P.E.: 3-D pore-scale modelling of sandstones and flow simulations in the pore networks. SPE
J. 2(02), 136–149 (1997)
Bartell, F.E., Osterhof, H.J.: Determination of the wettability of a solid by a liquid. Ind. Eng. Chem. 19(11),
1277–1280 (1927)
Boek, E.S., Coveney, P.V., Skipper, N.T.: Molecular modeling of clay hydration: a study of hysteresis loops
in the swelling curves of sodium montmorillonites. Langmuir 11(12), 4629–4631 (1995)
Cui, Y.J., Yahia-Aissa, M., Delage, P.: A model for the volume change behavior of heavily compacted swelling
clays. Eng. Geol. 64(2), 233–250 (2002)
Delage, P., Howat, M.D., Cui, Y.J.: The relationship between suction and swelling properties in a heavily
compacted unsaturated clay. Eng. Geol. 50(1), 31–48 (1998)
DiCarlo, D.A., Sahni, A., Blunt, M.J.: Three-phase relative permeability of water-wet, oil-wet, and mixed-wet
sandpacks. SPE J. 5(01), 82–91 (2000)
Fenwick, D.H., Blunt, M.J.: Network modeling of three-phase flow in porous media. SPE J. 3(01), 86–96
(1998a)
Fenwick, D.H., Blunt, M.J.: Three-dimensional modeling of three phase imbibition and drainage. Adv. Water
Resour. 21(2), 121–143 (1998b)
Goodyear, S.G., Jones, P.I.R.: Relative permeabilities for gravity stabilized gas injection. In: 7th European
Symposium on Improved Oil Recovery, Moscow, Russia (1993)
123
Pore-scale Network Modeling of Three-Phase Flow Based…
Grader, A.S., O’Meara, D.J. Jr.: Dynamic displacement measurements of three-phase relative permeabilities using three immiscible liquids. In: Paper SPE 18293, In SPE Annual Technical Conference and
Exhibition. Society of Petroleum Engineers (1988)
Kovscek, A.R., Wong, H., Radke, C.J.: A pore-level scenario for the development of mixed wettability in oil
reservoirs. AIChE J. 39(6), 1072–1085 (1993)
Lerdahl, T.R., Øren, P.E., Bakke, S.: A predictive network model for three-phase flow in porous media. In:
Paper SPE 59311, SPE/DOE Improved Oil Recovery Symposium. Society of Petroleum Engineers (2000)
Naylor, P., Sargent, N.C., Crosbie, A.J., Tilsed, A.P., Goodyear, S.G.: Gravity drainage during gas injection.
In: IOR 8th European Symposium on Improved Oil Recovery (1995)
Oak, M.J.: Three-phase relative permeability of water-wet Berea. In: Paper SPE 20183, SPE/DOE Enhanced
Oil Recovery Symposium. Society of Petroleum Engineers (1990)
Øren, P.E., Bakke, S., Arntzen, O.J.: Extending predictive capabilities to network models. SPE J. 3(04), 324–
336 (1998)
Øren, P.E., Bakke, S.: Process based reconstruction of sandstones and prediction of transport properties. Transp.
Porous Media 46(2–3), 311–343 (2002)
Pini, R., Krevor, S.C., Benson, S.M.: Capillary pressure and heterogeneity for the CO2 /water system in
sandstone rocks at reservoir conditions. Adv. Water Resour. 38, 48–59 (2012)
Pini, R., Benson, S.M.: Simultaneous determination of capillary pressure and relative permeability curves
from core-flooding experiments with various fluid pairs. Water Resour. Res. 49(6), 3516–3530 (2013)
Piri, M., Blunt, M.J.: Three-dimensional mixed-wet random pore-scale network modeling of two-and threephase flow in porous media. I. Model description. Phys. Rev. E 71(2), 026301 (2005a)
Piri, M., Blunt, M.J.: Three-dimensional mixed-wet random pore-scale network modeling of two-and threephase flow in porous media. II. Results. Phys. Rev. E 71(2), 026302 (2005b)
Ryazanov, A.V., van Dijke, M.I., Sorbie, K.S.: Pore-network prediction of residual oil saturation based on oil
layer drainage in mixed-wet systems. In: Paper SPE 129919, In SPE Improved Oil RecoverySymposium.
Society of Petroleum Engineers (2010)
Sahni, A., Burger, J., Blunt, M.J.: Measurement of three phase relative permeability during gravity drainage
using CT scanning. In: Paper SPE 39655, In SPE/DOE Improved Oil Recovery Symposium. Society of
Petroleum Engineers (1998)
Saraji, S., Goual, L., Piri, M., Plancher, H.: Wettability of supercritical carbon dioxide/water/quartz systems:
simultaneous measurement of contact angle and interfacial tension at reservoir conditions. Langmuir
29(23), 6856–6866 (2013)
Suicmez, V.S., Piri, M., Blunt, M.J.: Pore-scale simulation of water alternate gas injection. Transp. Porous
Media 66(3), 259–286 (2007)
Suicmez, V.S., Piri, M., Blunt, M.J.: Effects of wettability and pore-level displacement on hydrocarbon trapping. Adv. Water Resour. 31(3), 503–512 (2008)
Svirsky, D.S., van Dijke, M.I., Sorbie, K.S.: Prediction of three-phase relative permeabilities using a pore-scale
network model anchored to two-phase data. In: Paper SPE 89992, In SPE Annual Technical Conference
and Exhibition. Society of Petroleum Engineers (2007)
Valvatne, P.H., Piri, M., Lopez, X., Blunt, M.J.: Predictive pore-scale modeling of single and multiphase flow.
Transp. Porous Media 58, 23–41 (2005)
Valvatne, P.H., Blunt, M.J.: Predictive pore-scale modeling of two-phase flow in mixed wet media. Water
Resour. Res. 40(7), W07406 (2004)
Zolfaghari, A.: Pore-scale network modeling of two- and three-phase flow based on thermodynamically consistent threshold capillary pressures. Ph.D. dissertation, University of Wyoming (2014)
Zolfaghari, A., Piri, M.: Pore-scale network modeling of three-phase flow based on thermodynamically consistent threshold capillary pressures. I. Cusp formation and collapse. Transp. Porous Media (2016). doi:10.
1007/s11242-016-0815-7
123