Open Phys. 2016; 14:610–616
Research Article
Open Access
Yongfei Yang*, Pengfei Liu, Wenjie Zhang, Zhihui Liu, Hai Sun, Lei Zhang, Jianlin Zhao,
Wenhui Song, Lei Liu, Senyou An, and Jun Yao*
Effect of the Pore Size Distribution on the
Displacement Eflciency of Multiphase Flow in
Porous Media
DOI 10.1515/phys-2016-0069
Received Jun 27, 2016; accepted Nov 21, 2016
Abstract: Due to the complexity of porous media, it is difficult to use traditional experimental methods to study the
quantitative impact of the pore size distribution on multiphase flow. In this paper, the impact of two pore distribution function types for three-phase flow was quantitatively
investigated based on a three-dimensional pore-scale network model. The results show that in the process of wetting phase displacing the non-wetting phase without wetting films or spreading layers, the displacement efficiency
was enhanced with the increase of the two function distribution’s parameters, which are the power law exponent in
the power law distribution and the average pore radius or
standard deviation in the truncated normal distribution,
and vice versa. Additionally, the formation of wetting film
is better for the process of displacement.
Keywords: power law distribution, truncated normal distribution, three-dimensional pore network model, displacement efficiency, wettability
PACS: 47.56.+r
Introduction
Porous media exists in most areas of science and engineering, and the multiphase flow phenomenon has impor-
*Corresponding Author: Yongfei Yang: School of Petroleum Engineering, China University of Petroleum (East China), Shandong
Qingdao 266580, China; Email: [email protected]
*Corresponding Author: Jun Yao: Research Centre of Multiphase
Flow in Porous Media, China University of Petroleum (East China),
Shandong Qingdao 266580, China; Email: [email protected]
Pengfei Liu, Wenjie Zhang, Zhihui Liu, Hai Sun, Lei Zhang, Jianlin Zhao, Wenhui Song, Lei Liu, Senyou An: School of Petroleum
Engineering, China University of Petroleum (East China), Shandong
Qingdao 266580, China
tant application significance in scientific research and engineering technology development. Example applications
include reservoir exploration and development, protection and pollution control of soil and groundwater, drying
approaches in various industrial processes, and so on [1].
The porous media with a changeable microcosmic pore
structure as well as interfacial properties results in complicated multiphase flow in porous media and a microscopic
distribution of multiphase fluid in pores. Therefore, it is
extremely difficult to quantitatively describe and calculate the multiphase flow caused by the mutual action of a
multi-interface at the microscopic level. Based on the pore
network model, pore-scale modeling has become an effective method to study multiphase flow in porous media [2–
4]. The pore network model has simplified pore geometry
and real physical topology characteristics. As a result, calculation and simulation of microscopic multiphase flow in
porous media using a pore network model has gradually
become a hot topic in various fields, such as energy, chemical industry, environmental protection, civil engineering,
material fields and more.
In the three-dimensional pore-scale network model,
the pore size and its distribution rule have an important
influence on multiphase flow [5–7]. The pore distribution
function is the relation function between the size of the
pore radius and the occupation proportion of the radius
pore. In the three-dimensional pore-scale network model,
the power law distribution function and truncated normal
distribution function are commonly used. The power law
distribution function is the proportion of the pore radius
size produced by the power law and the truncated normal
distribution function is the randomly produced pore radius size according to the normal distribution [8].
Ioannidis and Chatzis (1993) used regular cubic lattices, with consideration for different pore size distributions, to discuss the effect of the shape of the network elements on the model property predictions [9]. Vogel and
Roth (2001) emphasized that the pore size distribution obtained using serial sections and topology defined by the
3D-Euler number are sufficient to predict the hydraulic
© 2016 Y. Yang et al., published by De Gruyter Open.
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.
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Effect of the Pore Size Distribution on the Displacement Eflciency of Multiphase Flow |
properties in soil [10, 11]. Lehmann et al. (2008) constructed packs of overlapping ellipsoids using Minkowski
functions and concluded that the porosity and surface
area dominate permeability [12]. Garcia et al. (2009) studied the effects of the particle shape and polydispersity
on permeability [13]. Only the type of pores with an angular cross section can form intermediate wetting phase
layers, which is crucial for multiphase flow in the network. Man and Jing (1999) studied the pore geometry effect on the electrical resistivity and capillary pressure, and
they posited that networks of non-circular cross-sectional
pores would be more physically accurate [14]. Van Dijke
et al. (2007) deduced both the thermodynamic and geometrical criteria for the layer in a star-shaped pore [15].
Raoof and Hassanizadeh (2010) developed a semi-regular
multi-directional network, with a maximum coordination
number of 26, to study the effect of the network structure
on transport in porous media [16]. Many researchers have
studied the relationship between the pore structure and
the flow, electric or magnetic properties [17–20]. Due to the
heterogeneity of porous media, a general regulation is difficult to obtain when studying a specific real core sample.
Therefore, it is necessary and meaningful to study sample
models with given pore size distributions.
This paper first established a pore-scale network
model simulating the random wettability in porous media
and then quantitatively studied the effect of the pore size
distribution on multiphase flow.
1 Model establishment
According to the invasion flow principle, the flow is controlled by the capillary force independent of the gravity
and viscous forces. The established pore-scale network
model has a node size of 15 × 15 × 15, and the detailed
description can be found in literature [2]. We assume that
the fluid cannot be compressed, the outlet pressure is constant, and the inlet pressure changes. The other main parameters of the three dimensional network models are as
follows:
1. The pore size is randomly distributed. This paper
mainly study two types of commonly used pore size
distributions, which are the power law distribution
function and truncated normal distribution function. According to the published experimental and
simulation data, the coordination number (z) in the
pore-scale network model of this paper is set to
3 [21]. The pore radius r size distribution is 1 × 10−7 ≤
r ≤ 1 × 10−5 m.
θ ij
611
i
j
γ
Figure 1: Two-phase occupancy at the corner of a pore.
2. In the established network model, the fluid interfacial tension can be set randomly, which means it can
simulate various displacement types. In this article,
the interfacial tensions during the simulation process were set according to literature as follows [2]:
σgo = 24 mN/m, σow = 32 mN/m, σgw = 48 mN/m.
3. Two types of displacement events occurred in the
pores, including piston-like displacement and snapoff. The corresponding capillary entry pressure can
be obtained by the Laplace equation as:
Pc, ij = η
σij cos θij
r
(1)
Where η = 1 indicates the snap-off event and η = 2
indicates the piston-like displacement.
4. The wetting films and spreading layers were evaluated by the wetting degree of the pore and throat in
the network model.
When the section of the pore and throat in the network has a corner, the general geometry condition
of film existence is
θ ij ≤
π
−𝛾
2
(2)
The threshold value (cos θ* ) was introduced to determine the existence of wetting films and spreading layers and four criteria were determined as follows [22]:
– cos θow < cos θ*ow oil film surrounding the water phase;
– cos θgo > cos θ*go oil film or oil phase spreading
level surrounding the gas phase;
– cos θow > cos θ*ow water film surrounding the
oil phase;
– cos θgw > cos θ*gw water film surrounding the
gas phase.
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cos θgo
1
=
{C
cos θow + CS, o +2σgo }
2σgo S, o
cos θgw =
Power law,n=0.5
Power law,n=0.2
Uniform,n=0
Power law,n=-0.2
Power law,n=-0.5
0.05
Power law,n=-0.8
0.00
20
40
60
80
(3)
(4)
6. Multiphase displacement process: The pore-scale
network model is first saturated 100% in the oil
phase; then, it carries on primary water drive, which
is followed by gas drive. Also, the target saturation
of the water drive and gas drive were set as 1, which
results in the greatest displacement degree.
7. Saturation computational method:
Definite the saturation for phase j
∑︁
Sj =
V(r)f (r)
(5)
In the formula, rmax and rmin are the largest and smallest pore radii, respectively; n is the power law exponent;
for n > 0, there are large pores; for n < 0, there are small
pores; and for n = 0, the power law function changes into a
uniform distribution function wherein pores with different
radius sizes have the same proportion. Setting the power
law exponent as 1, 0.5, −0.5, 0.2, −0.2, and −0.8 and substituting these exponents into equation (7) results in the map
of the pore size distribution, as shown in Fig. 2.
2.2 Truncated normal distribution function
Definition of the truncated normal distribution function:
1 r−r̄ 2
)
σ
f (r) = N(rmax − r)(r − rmin )e− 2 (
r=r j
Where r j is the pore radius that phase j occupies;
V(r) is the pore volume function; and f (r) is the pore
size distribution function.
V(r) = ar v
Where a is the normalized constant,
(6)
∑︀
ar v f (r) = 1;
r
r is the sum in all pores, and the degree of saturation
normalizes into 1.
2 Pore size distribution function
2.1 Power law distribution function
n+1
rmax
rn
n+1
− rmin
(8)
where r̄ and σ are the average pore radius and standard
deviation, respectively. Setting the r̄ and σ values as 3, 2;
3, 5; 3, 10; 5, 2; 5, 5; 5, 10; 8, 2; 8, 5; and 8, 10, respectively,
and substituting these values into equation (8) generates
the pore size distribution curve (Fig. 3).
Comparing the first three curves in Figure 3 (red, green
and blue lines with square symbols), the average pore radius r̄ is the same and equal to 3, the curves become flatter with the increase of the standard deviation σ, and the
pore size distribution becomes more non-uniform. When
the standard deviation σ is constant (with σ equal to 2 as
an example, the red group lines up with the square, triangle and star symbols), the curves move to the right with the
increase of the average pore radius r̄ wherein the number
of large pores increases.
Power law distribution function definition:
f (r) = (n + 1)
100
Pore radius r (*10^-7m)
Figure 2: Power law pore size distribution.
1
{(CS, o + 2σow ) cos θow + CS, o
2σgw
+ 2σgo }
Power law,n=1
0.10
f(r)
5. The wettability of each pore and throat can be arbitrarily set in the network model. Each pore or
throat is possibly completely water or oil wet and
may be situated between two situations, which is determined by the cosine value of the oil/water contact angle (cos θow ). In this article, when the pore
size distribution follows the power law distribution,
water wet pores: cos θow = 0.9 and oil wet pores:
cos θow = −0.9;at the same time,the parameter’C’ is
used to represent the ability of liquid spreading in
the solid interface. When the pore size distribution
follows a truncated normal distribution, cos θow =
1. The contact angles of gas/oil and gas/water were
calculated by the equations below [23]:
(7)
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613
0.9
average pore radius standard deviation
3,2
3,5
3,10
5,2
5,5
5,10
8,2
8,5
8,10
f(r)
0.2
Remaining oil saturaon, So
0.3
0.8
0.7
0.6
0.5
0.4
0.3
-0.8
0.1
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Power law exponent
A!er water drive
Figure 4: Comparison of the remaining oil saturation in water wet
system with different power law exponents.
0.0
20
40
Pore radius r (*10^-7m)
Figure 3: Truncated normal pore size distribution.
Table 1: Phase saturations in the water wet system with different
power law exponents
Power law
exponent
−0.8
−0.5
−0.2
0
0.2
0.5
1
A!er gas drive
After water drive
Sw
So
0.218 0.782
0.272 0.728
0.321 0.679
0.348 0.652
0.370 0.630
0.395 0.605
0.424 0.576
After gas drive
Sw
So
Sg
0.077 0.458 0.465
0.111 0.477 0.412
0.147 0.470 0.383
0.169 0.461 0.369
0.188 0.458 0.354
0.211 0.435 0.354
0.237 0.418 0.345
3 Analysis and discussion of the
simulation results
3.1 Pore size distribution follows power law
distribution function
3.1.1 Water wet system
The cosine value of the oil/water contact angle is 0.9 in the
system; 100% oil saturation followed by one water drive
and one gas drive in the systems with different power law
exponents as mentioned above. The saturation is recorded
in Table 1.
It can be observed from Figure 3 and Table 1 that the remaining oil saturation after the water drive process gradually decreases with the increase of power law exponent. In
the water wet system, the capillary force is the driving force
during the water displacing oil process, which is an imbibition process. The threshold cosine value of the oil/water
contact angle for water film existence is set at 0.95, which
is larger than the cosine value of the oil/water contact angle of 0.9; as a result, there is no continual water film in
the system. Therefore, the overall displacement efficiency
is not very good, and certain discontinuous oil phase clusters are difficult to be displaced in the system.
Comparing two extreme cases, the one is that large
pores accounting for a larger proportion (power law exponent n equals 1) and the other is that small pores occupying a larger proportion (power law exponent n equals
−0.8), when the smallest and largest pore radii are set as
the same in the two situations, the large pores example
will have a larger porosity. Because it is difficult to drive oil
in corners without water film, the total pore volume occupied by the remaining oil is almost the same for the above
two cases. However, because the two cases have different
porosities, the remaining oil saturation levels are different. After water drive the remaining oil saturation declines
with increase in the power law exponent.
During the gas drive process, the cosine values of the
gas/oil contact angle and gas/water contact angle can be
obtained by substituting interfacial tensions into eq. (3)
and eq. (4); they are cos θgo = 0.692 and cos θgw = 0.942.
Because the two values are smaller than the wetting film’s
existence condition cos θ* = 0.95, the oil and water films
that can surround gas phase would not appear in the system during the gas drive process. Table 1 shows that the
gas saturation reduces gradually along with the increase
in the power law exponent after the gas drive process. The
reason is that in the water-wet system, water is the wetting
phase, oil is the intermediate wetting phase, and gas is the
non-wetting phase; as a result, the capillary force is the resistance in the gas drive process. In the case with a larger
pore volume (i.e., power law exponent is large), under the
effect of the capillary force, massive fluid in the pore could
not be displaced by gas and the gas saturation is smaller.
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Table 2: Phase saturations in oil wet systems with different power
law exponents
Power law
exponent
−0.8
−0.5
−0.2
0
0.2
0.5
1
Water drive process
Sw
So
0.811 0.189
0.784 0.216
0.751 0.249
0.730 0.270
0.712 0.288
0.689 0.311
0.661 0.339
Gas drive process
Sw
So
Sg
0.666 0.040 0.293
0.617 0.045 0.338
0.569 0.063 0.368
0.543 0.069 0.387
0.521 0.070 0.409
0.495 0.073 0.432
0.454 0.072 0.474
Remaining oil satura!on, So
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Power law exponent
A"er water drive
A"er gas drive
Figure 5: Comparison of the remaining oil saturation in oil wet systems with different power law exponents.
The volume of water entering into the pores increases with
the power law exponent increasing during the water drive
process, and the volume of gas entering into the pores decreases along with the power law exponent increasing during the gas drive process, so the remaining oil saturation
is almost the same after gas drive. (Fig. 4).
It can be concluded that, in the displacement process, when the wetting phase displaces the non-wetting
phase, the displacement efficiency increases along with
the power law exponent increases. When the non-wetting
phase displaces the wetting phase, the displacement efficiency increases along with the power law exponent increases.
3.1.2 Oil wet system
For the oil-wet cases, the fluid parameters and running parameters are the same as the above water-wet cases and
only the contact angle is different (cos θow = −0.9). The saturations after the water and gas drive processes are listed
in Table 2.
With respect to the oil wet system, the water displacing oil is a non-wetting phase displacing the wetting phase
process. The remaining oil saturation after the water drive
process gradually increases with the power law exponent
increases (Figure 5).
The gas drive can be calculated using eq. (3) and
eq. (4): cos θgo = 0.983, cos θgw = −0.108, cos θgo > cos θ* .
The oil film surrounding the gas phase would be formed
in the gas drive process. During the gas drive process, because of the existence of oil film, the gas phase can displace the remaining oil left in the water drive process. With
the increase in the power law exponent, the large pores increase and the pore volume increases; as a result, the remaining oil will be mostly displaced by gas for the existence of the continuous oil film around gas, and the absolute level of displaced oil increases along with the power
law exponent increases (Table 2 and Fig. 5).
Moreover, because a water film surrounding the gas
phase does not exist during the gas drive process, gas displacing water regulation still follows the above rule. In
the oil wet system, oil is the wetting phase, gas is the intermediate wetting phase, and water is the non-wetting
phase; as a result, the gas driving water process is the wetting phase displacing non-wetting phase process. The displacement efficiency increases along with the increasing
power law exponent; namely, the water saturation alteration between terminal points of two displacement processes increases gradually (Table 2), which is the same as
that obtained in the water wet system.
3.2 Pore size distribution follows the
truncated normal distribution function
Because wettability does not affect the law in which
the pore size distribution influences flow in porous media(section 3.1.2), only the completely water wet situation
was studied, namely, cos θow = 1, for which the pore
size distribution follows the truncated normal distribution
function case.
The remaining oil saturation So1 and water saturation
Sw1 were recorded after the water drive process, as shown
in Table 3.
Based on Table 3, when the average pore radius is a
fixed value (red, green and blue lines with square symbols
in Fig. 3), the remaining oil saturation increases gradually with increasing standard deviation. Namely, the larger
standard deviation is not beneficial for water driving oil.
One explanation is that the greater the standard deviation,
the smoother the curve of the pore distribution function
and higher the pore size distribution scattering, which is
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Table 3: Comparison of the remaining oil saturation and terminal
water saturation after water injection
Average
pore
radius/um
3
3
3
5
5
5
8
8
8
Standard
deviation
2
5
10
2
5
10
2
5
10
Sw1
0.372
0.356
0.353
0.408
0.366
0.356
0.474
0.381
0.361
So1
0.628
0.644
0.647
0.592
0.634
0.644
0.526
0.619
0.639
not beneficial for displacement. Conversely, the smaller
the standard deviation, the steeper the pore distribution
function curve such that the pore size distribution mostly
surrounds the average pore radius, which is better for displacement.
For a fixed standard deviation (same color group lines
with square, triangle and star symbols in Fig. 3), with an
increase in the average pore radius, the remaining oil saturation becomes smaller. This is because that water drive is
a wetting phase that displaces the non-wetting phase process in the water wet system; with increases in the average pore radius, the pore distribution function curve shifts
to the right and the total porosity increases, which is better for the displacement process. This conclusion is completely in agreement with the conclusions obtained in the
power law distribution function cases.
After the gas driving process, the saturation values
and changes in the saturation values are as shown in Table 4. Subscript number 2 indicates the state after the gas
drive, i.e., Sw2 is the water saturation after gas driving; So2
is oil saturation after gas driving; Sw1 − Sw2 is the water saturation alteration displaced by gas; and So1 − So2 is the oil
saturation alteration displaced by gas.
Fig. 6 shows that when the average pore radius is
fixed, with increasing standard deviation, the displacement efficiency becomes increasingly low in the process of
gas displacing water; however, the process of gas displacing oil is not significantly related to the standard deviation.
At a constant standard deviation, it can be calculated
that cos θgo = 0.667, cos θgw = 1; as a result, the water
film surrounding the gas phase exists during the gas displacing water process and the displacement efficiency was
greatly improved compared with no spreading layer case.
This process is similar to the before mentioned gas flood-
615
Table 4: Comparison of the phase saturation and change in the
saturation values after gas injection
Num
r̄, σ
Sw2
So2
Sg
1
2
3
4
5
6
7
8
9
3, 2
3, 5
3, 10
5, 2
5, 5
5, 10
8, 2
8, 5
8, 10
0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.373
0.392
0.388
0.372
0.378
0.392
0.390
0.401
0.390
0.624
0.605
0.609
0.625
0.619
0.605
0.607
0.596
0.607
Sw1 −
Sw2
0.369
0.353
0.350
0.405
0.363
0.353
0.471
0.378
0.358
So1 −
So2
0.255
0.252
0.259
0.220
0.257
0.251
0.136
0.217
0.249
Figure 6: The displaced oil saturation and water saturation after gas
injection.
ing oil case in the oil wet system with changing power law
exponents, and it shows that Sw1 − Sw2 increases gradually
(Fig. 6). In addition, gas displacing oil in the water wet system is a process of the non-wetting phase displacing the
wetting phase. The displacement efficiency decreases with
increasing average pore radius, which is in agreement with
the conclusion obtained for the water wet system process
with changing power law exponents (Figure 6).
4 Conclusions
1. During the wetting phase displacing the nonwetting phase, when the systems do not form a wetting film or spreading layers, the displacement efficiency will be improved with an increase of the
power law exponent in the power law distribution
function, or increase of the average pore radius and
standard deviation in the truncated normal distribution function. Additionally, the process of the non-
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616 | Y. Yang et al.
wetting phase displacing the wetting phase has the
opposite effect.
2. When the pore size distribution follows a truncated
normal distribution function, the smaller the standard deviation (more even pore size distribution),
the better it is for displacement. Conversely, the
greater the standard deviation (less uniform pore
size), the worse it is for displacement.
3. Pores with wetting films can increase the continuity of the various phases in the network and decrease the number of disconnected phase clusters
from which low oil saturations can be obtained after
long drainage times with a sufficient high capillary
pressure. As a result, the formation of a wetting film
or spreading layer is better for displacement.
Acknowledgement: We would like to express appreciation to the following financial support: the National Natural Science Foundation of China (No. 51304232, 51674280,
51490654, 51234007, 51274226), Applied basic research
projects of Qingdao innovation plan (16-5-1-38-jch), the
Fundamental Research Funds for the Central Universities
(No. 14CX05026A), Introducing Talents of Discipline to
Universities (B08028), and Program for Changjiang Scholars and Innovative Research Team in University (IRT1294).
And Yongfei Yang thanks M. I. J. van Dijke in Heriot-Watt
University providing the 3D pore network model.
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
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