1. Introduction
Heavy oil, of which the viscosity ranges from
mPa·s to
mPa·s, has become a prominent class of unconventional oil resources [
1]. Recent studies estimate that the total amount of recoverable heavy oil is almost the same as the remaining conventional oil resources [
2,
3,
4]. Chemical flooding is now widely applied for enhanced heavy oil recovery [
5,
6,
7,
8,
9], and presents several advantages in mature oilfields [
10]. With the presence of chemicals such as surfactants or alkalis acting as emulsifiers, oil-in-water (O/W) emulsions could form in heavy oil reservoirs, carrying out trapping oil [
5,
7,
11,
12]. O/W emulsion is a heterogeneous system, with small oil droplets (with a size of 0.5 μm to 50 μm) dispersed in the continuous water phase. As a dispersed system, the motion of oil-in-water emulsion through the porous medium is significantly different from that of a continuous oil phase or water phase. Thus, the understanding of emulsion flow through a porous medium is crucial for the usage of a chemical flood in heavy oil reservoirs.
Many investigations on O/W emulsion porous flow have been performed through laboratory methods. Uzoigwe and Marsden pioneered the experiments on O/W emulsion flow through glass bead packing models to study the flow behavior of emulsions in porous media [
13]. McAuliffe was the first to recognize the obstructive effect of emulsion droplets on fluid flow in porous media and applied this mechanism to enhance oil recovery [
14]. He suggested that when an emulsion passes through porous media, the Jamin effect is particularly prominent if the oil droplet size is larger than the pore throat diameter, subsequently leading to flow limitation due to capillary resistance. However, the explanation does not accommodate cases where the oil droplet size is smaller than the pore throat diameter. Further experiments conducted by Soo et al. proposed that, during the flow of emulsions through porous media, dispersed phase droplets can be captured and adsorbed by the media, resulting in reduced permeability [
15]. In other experiments, Khambharatana et al. focused on emulsion rheology, concluding that the rheological behavior of emulsions in porous media is not significantly different from their behavior in viscometers [
16]. Alvarado et al. suggested that emulsions can be considered non-Newtonian fluids when the volume fraction of a dispersed phase exceeds 50%, and can be otherwise regarded as Newtonian fluids [
17]. Wang et al. observed an emulsion flood after a water flood through a micromodel and identified two main physical mechanisms for emulsion-enhanced oil recovery [
18]. The first mechanism is altering the pressure distribution in the flow field by blocking the flow channels, thereby reducing residual oil. The second one is dispersed phase droplets exerting a pulling force on residual oil during the flow, causing the oil to deform into smaller droplets.
Many researchers have examined emulsion flow in porous media theoretically, developing a variety of theoretical models to describe their flow behavior. There are three widely recognized conventional mathematical models now: the bulk viscosity model, the retardation model, and the deep filtration model. Alvarado et al. introduced the bulk viscosity model [
17], considering emulsions as homogeneous single-phase fluids. They found that the rheological properties of O/W emulsions in porous media were similar to their properties in capillaries. They concluded that when the dispersed phase ratio exceeded 50%, the emulsions could be considered non-Newtonian fluids and provided parameters to characterize their rheology. Abou-Kassem et al. further refined the bulk viscosity model by presenting a modified Darcy’s law applicable to non-Newtonian emulsions [
19]. However, the bulk viscosity model merely treats emulsions as single-phase fluids, similar to polymer solutions, and does not account for the interactions between dispersed phase droplets and pores. As a result, this model is only applicable in a limited range of situations.
The retardation model proposed by Devereux [
20] was to explain the phenomenon of permeability reduction observed by McAuliffe. Based on the Buckley–Leverett model, he introduced a dispersion phase resistance factor to represent the capillary resistance experienced by the dispersed phase during flow. This model aligns well with the experimental results of emulsion flow in porous media but cannot predict the situation of a water injection following emulsion flow. Yu and Ding et al. [
21,
22] made improvements to the retardation model, considering the porous media as channels composed of larger pores and smaller pore throats and taking into account the blocking phenomenon caused by multiple emulsion droplets. The modified model can better predict the experimental phenomena of water flooding following emulsion flooding. However, the retardation model assumes the properties of emulsions are constant and cannot describe the variations of emulsion droplets during flow.
Soo et al. introduced the deep filtration theory into emulsion porous flow to consider the interaction between droplets and pore throats [
23]. They assumed that dispersed phase droplets would be captured by the porous media such as tiny particles. The captured droplets would block smaller flow channels, resulting in a decrease in permeability. However, the deep filtration model assumes that once droplets are captured by pores, they cannot be remobilized, which means the reduction in permeability is irreversible. This is inconsistent with experimental observations. Although some researchers improved the model by considering the remobilization of droplets [
24], the introduction of new parameters made the modified model excessively complex.
Existing models for emulsion porous flow often represent porous media as a capillary bundle with uniform diameters. They usually use the flow behavior in a single capillary as a substitute for the porous medium. However, many theoretical and experimental research studies have shown that the interaction between dispersed droplets and pores plays a significant role in affecting the flow. Wei et al. [
25] studied emulsion flow in constricted capillaries using a lattice Boltzmann simulation method. They argued the emulsion capillary flow may have complex mechanism, including deformation, snap-off, and the “trap effect”. Thus, we regard the relative size of capillaries and droplets being the most critical factor in this study. Additionally, most of the models mainly focused on the single O/W emulsion porous flow process, which only includes dispersed oil and continuous water. These models are unable to describe the three phases of flow (dispersed oil, continuous water, and continuous oil) that occur during heavy oil recovery adequately. In this work, a new three-phase relative permeability model is proposed, considering the physicochemical properties of emulsions and the pore size distribution of the porous media. The interaction between the dispersed oil droplets and pore system is characterized by a resistance factor, which is carefully derived considering the various relative sizes of pores and droplets.
5. Dimensionless Parameters Analysis
To further comprehend the porous flow behavior in heavy oil emulsion systems, it is essential to examine the influence of the three dimensionless numbers
on the ternary diagram of relative permeability. Upon investigating Equations (35) and (38), it becomes apparent that, within the proposed model, the relative permeability of the continuous oil phase is only related to its own saturation (given the PSD function). This property can be also found in
Figure 8b. As the impact of the PSD function on the relative permeability of the continuous oil phase has been previously illustrated in
Figure 6, the subsequent analysis will focus on the effects of the different values of the three dimensionless numbers on the continuous water phase and the dispersed oil phase.
The PSD function we choose in the following analysis is the same as the function we use in the simulation, which is a log-normal PSD function with a standard deviation
and mean radius
.
Figure 10 shows the relative permeability ternary diagram of the continuous water phase and dispersed oil phase, respectively, while
,
, and
. The maximum value of
in
Figure 10a is 0.026.
Figure 11 shows the relative permeability ternary diagram of the continuous water phase and dispersed oil phase, respectively, while
,
, and
. The maximum value of
in
Figure 11a is 0.167. By comparing
Figure 10b and
Figure 11b, it could be noticed from the contour lines that
can be considered solely influenced by the value of
when the maximum value of
is much less than 1. However, when the value of
is more comparable with 1, the effect of the
. value on
becomes more non-negligible, especially when
is closer to 0, the reason of which can be learned by investigating Equations (36) and (39). The value of
is much less than 1, means that the flow of emulsions in the pores is subject to significant resistance by the oil droplets, causing the contribution of the flow from the emulsion to the production of the continuous water phase is relatively small.
Figure 12 shows the relative permeability ternary diagrams of the dispersed oil phase when
,
, and
, 0.9, and 1.2, respectively. The maximum values of the three
diagrams are 0.0374, 0.0255, and 0.0239, respectively. The effect of the value of the diameter ratio between the dispersed phase droplets and pores on the result of
value is apparent. This pattern could be easy to understand intuitively. As the diameter of the dispersed phase droplets approaches the size of the pores, the likelihood of the droplets becoming trapped by the pores increases, leading to substantial flow resistance. When the droplet size is statistically larger than the average pore size, this probability becomes higher, making less droplets have the chance to pass the pores during the flow.
Figure 13 shows another three diagrams of
with different parameters. Here, we set the volume fraction of the dispersed oil droplets
, and the diameter ratio
. The capillary number
of the three diagrams are set to
,
, and
, respectively. The maximum values of three
diagrams are 0.1619, 0.0255, and 0.004, respectively. It is evident that an increase in the capillary number results in a decrease in the resistance encountered by the dispersed phase droplets when passing through the pores, thereby enhancing their ability to pass through the porous medium. A comparison between
Figure 12 and
Figure 13 reveals that the impact of altering the capillary number on
is significantly more pronounced than that of changing the diameter ratio. In
Figure 13, with the fixed parameters
and
, altering the capillary number
leads to an approximate two orders of magnitude variation in the maximum
value, whereas adjusting the diameter ratio
in
Figure 12 does not induce an order of magnitude change in the maximum
value. This observation can be attributed to the fact that the capillary number serves as a parameter governing the capacity of the dispersed phase droplets to pass through the pores when subjected to blockage. Furthermore, under the conditions presented in
Figure 13, both
and
are relatively large, signifying that the majority of the dispersed phase droplets experience blocking. As a result, alterations in the capillary number’s order of magnitude also provoke corresponding changes in the resistance experienced by the droplets, thereby causing the capillary number to exert a substantial influence on the relative permeability of the dispersed phase droplets.
When the value of is relatively small, the area of the ternary diagram of relative permeability decreases. Investigating the effects of various parameters on relative permeability within the ternary diagram becomes less intuitive. Additionally, as seen from the figures mentioned earlier, the growing patterns in the values of are similar. Consequently, we can substitute the study of endpoint values (the maximum value of ) under different conditions for the entire ternary diagram of relative permeability.
Table 1 is a table of
endpoint values under different parameter scenarios. As evident in the table, the capillary number ca exhibits a more substantial impact on the maximum value of
compared to the diameter ratio lambda, as previously highlighted. When assessing the influence of
on the relative permeability of the dispersed oil droplets, it is essential to note that
serves as the upper boundary for
values. In other words,
signifies the carrying capacity of dispersed oil droplets within the continuous water phase. When the ratio of
to
equals
, it implies a uniform distribution of oil droplets within the emulsion, as dispersed oil droplets participate in all flow channels occupied by the continuous water phase. This corresponds to values of the diagonal line in the ternary phase diagram. Conversely, when the ratio of
to
is less than
, it suggests that the water flow region can be partitioned into an emulsion flow domain and a purely continuous water flow domain. Consequently, dispersed oil droplets only participate in the flow of specific channels, thereby diminishing their capacity to pass through the pore spaces. By analyzing the maximum values of
at varying
values, as well as their ratio, it becomes apparent that the actual ability of dispersed oil droplets to pass through pores weakens, despite the increase in the maximum value of
as
increases. We can consider the case when
and
, and take the maximum value of
as an example. It can be observed that when
= 0.1, 0.2, and 0.4, the maximum values of
are 0.605, 0.538, and 0.403, respectively, showing a decreasing trend when
increases. Based on Darcy’s law for multiphase flow, an increase in
implies that a larger pressure drop is required for the dispersed oil droplets to pass through at the same mass flow rate. Examined from a microscopic perspective, the probability of individual dispersed oil droplets passing through pores decreases, even though the aggregate volume of dispersed oil droplets passing through pores escalates. Statistically, this phenomenon is readily understandable, given that the likelihood of droplets becoming trapped in pores becomes larger as the volume fraction of dispersed oil droplets augments.
Concerning the relationship between the relative permeability of dispersed phase droplets and the three saturation parameters, observations can be made from
Figure 8,
Figure 9 and
Figure 10. Generally, when considering a fixed value of
, as
approaches its upper limit
, the growth rate of
accelerates. When considering a fixed value of
, an increase in the
ratio results in a decline in
values.
To briefly summarize, among the three dimensionless numbers and , the variation of has the most substantial impact on the relative permeability of dispersed phase droplets. This implies that reducing interfacial tension and increasing the injection pressure are vital ways to improve the permeability of dispersed phase droplets. Increases in both and contribute to a reduction in the relative permeability of dispersed phase droplets. It is worth mentioning that in experiments and chemical oil reservoir development, and are usually not independent parameters but are closely related to the physicochemical properties of the reservoir and surfactants. Nevertheless, in this theoretical investigation, these two parameters are provisionally regarded as independent, and their impacts on the flow performance of emulsions are analyzed. In practical reservoir development, it is crucial to distinguish the sources of dispersed phase droplets (whether pre-prepared or in situ generated) and adjust these dimensionless numbers accordingly, to attain the desired relative permeability.
6. Conclusions
A novel relative permeability model applicable to heavy oil chemical flooding emulsion systems is presented in this study. The emulsion system is considered as an O/W emulsion and comprises three phases: continuous oil phase, continuous water phase, and dispersed oil phase. The new relative permeability model takes the physicochemical properties of emulsions and the properties of porous media into account, as well as the interactions between dispersed phase droplets and pores. The validity of the relative permeability model is verified through a sandpack emulsion displacement experiment. The dimensionless numbers determining the relative permeability of dispersed phase droplets are analyzed. The following conclusions are drawn from this study:
1. The emulsion flow behavior through a single capillary was carefully investigated. The flow of emulsion in the capillary, based on the relative size of dispersed phase droplets and the capillary, was divided into three types: (a) the droplet size and capillary diameter are similar; (b) the droplet size is much smaller than capillary diameter; (c) the droplet size is larger than the capillary diameter.
2. The real reservoir is considered as a series of capillary bundles with diameters following the PSD (pore size distribution) function. Assuming the reservoir is hydrophilic, a calculation model for the three-phase relative permeability of the O/W emulsion system was established, by classifying different-sized capillaries as occupied by different fluids. A more realistic log-normal PSD function was used, instead of using the simplified uniform diameter, when calculating relative permeability.
3. A sandpack emulsion displacement experiment was used to verify the effectiveness of the proposed relative permeability model. The experiment process was injecting a pre-prepared emulsion into a sandpack saturated with oil. The injection was at a constant flow rate of 0.5 mL/min. The volume fraction of the dispersed oil phase in the pre-prepared emulsion was 0.2. Based on a one-dimensional three-phase numerical simulator, the simulated cumulative emulsion production using the proposed relative permeability model showed good consistency with the experimental data. The maximum error in the cumulative production did not exceed 10%. Therefore, the new relative permeability model is suitable for the numerical simulation of heavy oil emulsion three-phase systems.
4. Three dimensionless numbers related to the emulsion porous flow process, , , and , were proposed. The effects of these three dimensionless numbers on the relative permeability of dispersed oil droplets were analyzed. The capillary number exerts the most significant influence on the relative permeability among the dimensionless parameters. Alterations in the order of magnitude of correspond to concomitant changes in the order of magnitude for the relative permeability of the dispersed phase. The increase in diameter ratio and dispersed phase volume fraction will reduce the ability of the dispersed phase to pass through porous media. The newly developed three-phase relative permeability model offers valuable insights for enhancing the understanding and prediction of emulsion flow behavior.