Effect of the Pore Size Distribution on the Displacement Efficiency of Multiphase Flow in Porous Media

Yongfei Yang; Pengfei Liu; Wenjie Zhang; Zhihui Liu; Hai Sun; Lei Zhang; Jianlin Zhao; Wenhui Song; Lei Liu; Senyou An; Jun Yao

doi:10.1515/phys-2016-0069

Article Open Access

Effect of the Pore Size Distribution on the Displacement Efficiency of Multiphase Flow in Porous Media

Yongfei Yang , Pengfei Liu , Wenjie Zhang , Zhihui Liu , Hai Sun , Lei Zhang , Jianlin Zhao , Wenhui Song , Lei Liu , Senyou An and Jun Yao

Published/Copyright: December 30, 2016

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From the journal Open Physics Volume 14 Issue 1

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 important 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 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:

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.
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]: σ_go = 24 mN/m, σ_ow = 32 mN/m, σ_gw = 48 mN/m.
Two types of displacement events occurred in the pores, including piston-like displacement and snap-off. The corresponding capillary entry pressure can be obtained by the Laplace equation as:
(1)Pc,ij=ησijcosθijr
Where η = 1 indicates the snap-off event and η = 2 indicates the piston-like displacement.
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
(2)θij≤π2−γ
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]:
1. cos⁡θow<cos⁡θow∗ oil him surrounding the water phase;
2. cos⁡θgo>cos⁡θgo∗ oil film or oil phase spreading level surrounding the gas phase;
3. cos⁡θow>cos⁡θow∗ water film surrounding the oil phase;
4. cos⁡θgw>cos⁡θgw∗ water film surrounding the gas phase.
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 θ_ow). In this article, when the pore size distribution follows the power law distribution, water wet pores: cos θ_ow = 0.9 and oil wet pores: cos θ_ow = -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 θ_ow = 1. The contact angles of gas/oil and gas/water were calculated by the equations below [23]:
(3)cosθgo=12σgoCS,ocosθow+CS,o+2σgo
(4)cosθgw=12σgwCS,o+2σowcosθow+CS,o+2σgo
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.
Saturation computational method:
Definite the saturation for phase j
(5)Sj=∑r=rjV(r)f(r)
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.
(6)V(r)=arv
Where a is the normalized constant, ∑rarvf(r)=1;
r is the sum in all pores, and the degree of saturation normalizes into 1.

Figure 1

Two-phase occupancy at the corner of a pore.

2 Pore size distribution function

2.1 Power law distribution function

Power law distribution function definition:

(7)f(r)=(n+1)rnrmaxn+1−rminn+1

In the formula, r_max and r_min 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.

Figure 2

Power law pore size distribution.

2.2 Truncated normal distribution function

Definition of the truncated normal distribution function:

(8)f(r)=N(rmax−r)(r−rmin)e−12(r−r¯σ)2

where r̄ and σ are the average pore radius and standard deviation, respectively. Setting the r̄and 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).

Figure 3

Truncated normal pore size distribution.

Comparing the first three curves in Figure 3 (red, green and blue lines with square symbols), the average pore radius r 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 r wherein the number of large pores increases.

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.

Table 1

Phase saturations in the water wet system with different power law exponents

Power law	After water drive		After gas drive

exponent	Sw	So	Sw	So	Sg
-0.8	0.218	0.782	0.077	0.458	0.465
-0.5	0.272	0.728	0.111	0.477	0.412
-0.2	0.321	0.679	0.147	0.470	0.383
0	0.348	0.652	0.169	0.461	0.369
0.2	0.370	0.630	0.188	0.458	0.354
0.5	0.395	0.605	0.211	0.435	0.354
1	0.424	0.576	0.237	0.418	0.345

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 η equals 1) and the other is that small pores occupying a larger proportion (power law exponent η 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 θ_go = 0.692 and cos θ_gw = 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.

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).

Figure 4

Comparison of the remaining oil saturation in water wet system with different power law exponents.

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 θ_ow = -0.9). The saturations after the water and gas drive processes are listed in Table 2.

Table 2

Phase saturations in oil wet systems with different power law exponents

Power law	Water drive process		Gas drive process

exponent	Sw	So	Sw	So	Sg
-0.8	0.811	0.189	0.666	0.040	0.293
-0.5	0.784	0.216	0.617	0.045	0.338
-0.2	0.751	0.249	0.569	0.063	0.368
0	0.730	0.270	0.543	0.069	0.387
0.2	0.712	0.288	0.521	0.070	0.409
0.5	0.689	0.311	0.495	0.073	0.432
1	0.661	0.339	0.454	0.072	0.474

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).

Figure 5

Comparison of the remaining oil saturation in oil wet systems with different power law exponents.

The gas drive can be calculated using eq. (3) and eq. (4): cos θ_go = 0.983, cos θ_gw = -0.108, cos θ_go > 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 nonwetting 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 θ_ow = 1, for which the pore size distribution follows the truncated normal distribution function case.

The remaining oil saturation S_o1 and water saturation Swi were recorded after the water drive process, as shown in Table 3.

Table 3

Comparison of the remaining oil saturation and terminal water saturation after water injection

Average	Standard	S_w1	S_ol
pore	deviation
radius/urn
3	2	0.372	0.628
3	5	0.356	0.644
3	10	0.353	0.647
5	2	0.408	0.592
5	5	0.366	0.634
5	10	0.356	0.644
8	2	0.474	0.526
8	5	0.381	0.619
8	10	0.361	0.639

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 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., S_w2 is the water saturation after gas driving; S_o2 is oil saturation after gas driving; S_w1 – S_w2 is the water saturation alteration displaced by gas; and S_o1 – S_o2 is the oil saturation alteration displaced by gas.

Table 4

Comparison of the phase saturation and change in the saturation values after gas injection

Num	r̄, σ	S_w2	S_o2	S_g	S_w1 –	S_ol –
					S_w2	S_o2
1	3, 2	0.003	0.373	0.624	0.369	0.255
2	3, 5	0.003	0.392	0.605	0.353	0.252
3	3,10	0.003	0.388	0.609	0.350	0.259
4	5,2	0.003	0.372	0.625	0.405	0.220
5	5,5	0.003	0.378	0.619	0.363	0.257
6	5,10	0.003	0.392	0.605	0.353	0.251
7	8,2	0.003	0.390	0.607	0.471	0.136
8	8,5	0.003	0.401	0.596	0.378	0.217
9	8,10	0.003	0.390	0.607	0.358	0.249

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.

Figure 6

The displaced oil saturation and water saturation after gas injection.

At a constant standard deviation, it can be calculated that cos θ_go = 0.667, cos θ_gw = 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 flooding oil case in the oil wet system with changing power law exponents, and it shows that S_w1 – S_w2 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

During the wetting phase displacing the non-wetting 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 nonwetting phase displacing the wetting phase has the opposite effect.
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.
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.

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Received: 2016-6-27

Accepted: 2016-11-21

Published Online: 2016-12-30

Published in Print: 2016-1-1

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