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Article

Effect of Dynamic Injection Velocity and Mixed Wettability on Two-Phase Flow Behavior in Porous Media: A Numerical Study

by
Wei Hui
1,
Le Wang
2,*,
Xurui Liu
3 and
Yueshe Wang
1
1
State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China
2
College of Civil Engineering, Xi’an Shiyou University, Xi’an 710065, China
3
School of Engineering and Technology, China University of Geosciences (Beijing), Beijing 100083, China
*
Author to whom correspondence should be addressed.
Energies 2025, 18(4), 879; https://doi.org/10.3390/en18040879
Submission received: 17 December 2024 / Revised: 23 January 2025 / Accepted: 6 February 2025 / Published: 12 February 2025
(This article belongs to the Special Issue Recent Advances in Reservoir Simulation and Carbon Utilization and Storage—2nd Edition)
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Abstract

:
Immiscible displacement in porous media is a crucial microscale flow phenomenon in many fields, necessitating an understanding of the flow mechanisms under dynamic injection velocity and mixing wettability to predict and affect this flow accurately. Initially, a dynamic injection velocity method and a computational domain model considering non-dominant/dominant wetting angles were proposed. Then, microscale flow phenomena were modeled in a pore throat structure and doublet geometry under mixed wetting conditions. Finally, the influence of dynamic injection velocity and mixed wettability on microscale flow were investigated using numerical simulations. The results indicate that when stepwise and piecewise linear changes in injection velocity are observed, unlike continuous injection, two preferential displacement pathways are predominantly formed in the porous media. As the difference between the maximum and minimum injection velocity increases, the recovery efficiency initially decreases and then increases. Recovery efficiency is higher under piecewise linear injection velocity changes. The non-dominant wetting angle determines the distribution and flow of oil-water two-phase systems in porous media. With a dominant controlling wetting angle of 45°, as the non-dominant wetting angle increases, the flow phenomenon changes from one preferential pathway in the back region (30°, 45°) to two preferential pathways (60°, 90°, 120°) and then to one preferential pathway in the middle porous media (150°). As the degree of the non-dominant wetting angle increases, the recovery efficiency first increases and then decreases, with a maximum and minimum difference of 13.6%.

1. Introduction

Immiscible two-phase fluid flow in porous media is a widely studied phenomenon with applications in diverse fields, such as the petroleum industry, groundwater remediation, chemical engineering, underground hydrogen storage, and others [1,2,3,4,5,6,7,8,9,10]. In the context of oil extraction, the microscale dynamic behavior of oil–water two-phase flow within the pore throats of porous media is governed by a complex interplay of capillary forces, viscous forces, and gravity. Particularly, the flow behavior in rock porous media can be highly intricate, depending on the dominant force regime. The oil–water two-phase flow can exhibit various patterns, such as viscous fingering, capillary fingering, and stable displacement, and it involves several underlying mechanisms, including snap-off, corner flow, piston flow, and pinch-off [11,12,13,14,15,16].
The volume of fluid (VOF) model has been widely employed to study the microscale dynamics of immiscible two-phase fluids in porous media [5,17,18,19,20]. In the realm of numerical models and algorithms, research efforts in this area have focused on improving numerical modeling capabilities, such as the accurate calculation of interfacial tension, the development of advanced two-dimensional VOF models, and the coupling of VOF with chemical reactions. For instance, Pavuluri et al. (2018) [17] assessed the performance of different VOF formulations, including continuous surface force (CSF), sharp surface force (SSF), and piecewise linear interface calculation (PLIC), for the numerical simulation of oil–water two-phase flow in porous media. Yin et al. (2019) [21] proposed a corrected two-dimensional VOF model to investigate the two-phase flow patterns and dynamics in porous media with varying geometric complexities, and they validated the model against experimental data. Maes et al. (2018) [22] developed a new numerical model that combines two-phase flow, solute reactive transport, and wettability alteration, based on the direct numerical simulation of the Navier–Stokes equations and surface complexation modeling, to simulate salinity water flooding processes in porous media.
Some studies have also focused on the heterogeneity of the geometric structure of porous media, exploring the effects of factors such as the capillary number, capillary force, and oil–water viscosity ratio on the displacement process of immiscible fluids. Rabbani et al. (2018) [23] reported that a gradual and monotonic variation in pore sizes along the flow path can suppress viscous fingering during immiscible displacement, as the gradual reduction in pore sizes acts to restrain this phenomenon. Ambekar et al. (2020) [19] used the VOF model to analyze the impact of water-flooding velocity and interfacial tension on pore-scale oil recovery mechanisms, finding that the drainage process is dominated by Haines jump events at low capillary numbers. Patel et al. (2019) [24] employed the VOF-IBM (immersed boundary method) coupled approach to investigate the effects of capillary number, wetting angle, and water/oil viscosity ratio on oil–water flows in porous media.
Furthermore, the flow mechanisms in microscale single and dual channel systems can provide valuable insights into the behavior of two-phase immiscible flow in porous media [2,25,26]. Pavuluri et al. (2020) [27] analyzed the spontaneous imbibition in a two-dimensional synthetic pore geometry under different parameters, such as the shape of the transition zone, contact angle, and fluid properties, and they provided analytical and semi-analytical expressions to determine the critical contact angle and the position of capillary barrier zones. Lei et al. (2023) [18] studied the snap-off phenomenon in a three-dimensional single pore channel, and they found that the instability of the immiscible fluid interface can be controlled by adjusting the depth of the pore channel. Zhang et al. (2023) [28] believe that, as the viscosity ratio of the two-phase fluids increases, the mathematical model considering the dynamic contact angle better matches the simulation results when modeling the microscale flow of two-phase fluids in a single capillary tube. Mansouri et al. (2023) [29] used a VOF model to capture interface dynamics and discovered a new crossover flow regime in the pore doublet geometry.
The two-phase flow in rock porous media exhibits highly complex microscale dynamic behavior, influenced by factors such as the topological structure of the porous media, the spatial distribution of rock wettability, and the dynamic changes in injection velocity. The impact of these heterogeneities on the flow mechanisms in rock porous media is not yet fully understood. To address this, it is essential to establish a physical model of real rock porous media and employ numerical simulation methods to study the influence of dynamic injection velocity and mixed wettability on the microscale flow behavior. Unlike traditional numerical studies, we take into account homogeneous conditions in our studies, including unsteady injection velocity and uniform wetting angle.
This paper is structured as follows. Section 2 provides a detailed description of the VOF model and its verification. In Section 3, the physical porous model is established, and the different parameter case settings (including dynamic injection velocity and mixed wettability) are presented. Then, in Section 4, the results are analyzed clearly. Conclusions are provided in Section 5.

2. Mathematical Model and Verification

2.1. Mathematical Model

The volume of fluid (VOF) mathematical model was employed to numerically simulate the oil–water two-phase flow behavior in real porous media [30]. Within this mathematical framework, the interfacial tension between the oil and water phases was modeled using the continuum surface force (CSF) approach [31,32], and the flow regime was characterized as laminar. This mathematical model can effectively capture the interface between the oil and water phases with high computational efficiency and has been extensively utilized in numerical simulation studies of two-phase immiscible displacement processes in porous media [5,16,33,34,35,36,37,38].

2.2. Capillary Tube Verification

To validate the accuracy of the mathematical model, the spontaneous imbibition of gas–liquid immiscible fluids in a capillary tube was simulated. Analytically, when a capillary tube is submerged in water, the capillary force acts to elevate the liquid level within the tube until it reaches a maximum height where it achieves equilibrium with the gravitational force. This maximum height of the liquid column is influenced by factors such as the wetting angle of the tube wall, the coefficient of the gas–liquid two-phase capillary force, and the capillary radius. The analytical solution for determining this maximum height is expressed as follows:
H = 4 σ cos θ ρ l g R
In the formula, H represents the maximum liquid column height, θ denotes the wetting angle, σ is the interfacial tension coefficient between gas and liquid, ρ l stands for the density of the liquid phase (water), g is the value of gravitational acceleration, and R represents the radius of the capillary. A rectangular capillary physical model with a height of 0.02 m and a width of 0.001 m was utilized for the simulation. As presented in Table 1, the numerical simulation results and the analytical results are provided. The results reveal that the error between the simulated height value and the analytical solution is minimal, thereby demonstrating the suitability of the VOF model for the requirements of the numerical simulation.

2.3. Gas Bubble Formation Verification

We further validated the numerical model presented in this paper by analyzing the dynamic behavior of rising bubbles and numerically simulating the experiments reported by Walters and Davidson (1963) [39]. The fluid domain was initialized as a two-dimensional rectangle, measuring 0.254 m in width and 0.3048 m in height. The bubble had a diameter of 0.0508 m, with its center positioned 0.1016 m above the bottom of the domain. A structured grid was employed, resulting in a total of 768,000 grid points. The physical properties of the fluid were consistent with those documented in the literature. Figure 1 illustrates the shapes of the bubbles at various time intervals. Our findings indicate that the bubbles rise continuously due to buoyancy and undergo shape changes, with the simulated bubble shapes at different times closely aligning with the experimental results.

3. Physical Model and Parameter Setting

3.1. Physical Model and Mesh

The process of establishing the physical model of the porous media includes the following steps: First, the real core grayscale image is obtained by scanning the rock core using an industrial-grade CT machine. Second, threshold segmentation is performed on the one of the grayscale images to construct the 2D physical model. Figure 2 shows the geometric model shape, mesh, and physical boundary conditions of the real rock porous media, with dimensions of 3 mm in length and 2 mm in width. The physical model contains multiple micron-level complex, tortuous, and interconnected pore channels. The different particles are distinguished by gray and white, and it can be observed that the white particles occupy most of the area, which can be considered as the dominant rock particles. Their wetting angle is defined as the dominant wetting angle. Subsequently, the computational domain is obtained by dividing the geometric model into unstructured grids. The total number of grids is 235,032, with a minimum scale of 1.069 × 10−7 mm3 and a maximum scale of 2.018 × 10−8 mm3. This ensures that there are sufficient grids at any interface within the finest pore channel for the reconstruction of two-phase interfaces.

3.2. Case Settings

The primary purpose of the case settings is to analyze and study the influence mechanisms of dynamic injection velocity and mixed wettability on the two-phase dynamic behavior in real porous media. In the water flooding process, the injection velocity is a dynamic variable that changes over time. This change can occur in two ways: gradually or abruptly. Consequently, we utilize a piecewise linear boundary condition to describe the gradual variation of the injection velocity and a step-wise boundary condition to characterize the abrupt changes. The parameters for Case a are a continuous constant injection velocity of 0.003 m/s. Cases b, c, and d investigate the stepwise injection velocity changes, where the injection velocity is forced to switch to a high level after a period of low-velocity injection, and this process is continuously cycled. The injection velocity is divided into three levels: low, medium, and high. Cases e, f, and g examine the piecewise linear variation of injection velocity, with the initial injection velocity also divided into three intervals: low, medium, and high. Additionally, Cases h–l consider the mixed wettability caused by differences in rock types in various regions. The specific parameters used in the numerical simulation are presented in Table 2.

3.3. Boundary and Simulation Parameter Settings

The open-source CFD toolbox OpenFOAM v2106 is utilized as the foundational platform for conducting numerical tests. During the modeling process, the time step is dynamically adjusted to ensure that the Courant number remains below 0.1. In the numerical simulation, the coupling of pressure and velocity is achieved using the PISO algorithm (Pressure Simplicity with Splitting of Operators). The residuals of physical quantities such as velocity, pressure, and volume fraction are set to 10−6. Dynamic time steps are used in the model to ensure that the maximum Courant number calculated at each step is less than 0.1. The density of water is 1000 kg/m3, the density of oil is 800 kg/m3, the viscosity ratio of oil–water is 10, the viscosity of water is 0.001 kg·m−1·s−1, and the tension coefficient of the oil–water interface is 0.07 kg·m−2. All cases in our study do not consider the effect of gravity. The above parameters do not change during the numerical simulation. During the numerical simulation, the time step size changes dynamically to ensure that the Courant number for each calculation step remains below 0.1. At the initial state, the computational domain is filled with oil, and water is injected into the inlet. When the injection volume reaches 0.5 PV, the simulation results of displacement are statistically analyzed. The Gamma Scheme presented in work [40] is used to discretize the convection term, and the Crank–Nicolson scheme is used for the time term [41]. It is crucial to adjust the inlet velocity after each time step simulation in order to accurately capture the step-wise or piecewise linear change in velocity. The main physical quantities on the physical boundary of the computational domain are subject to the numerical boundary conditions in Table 3.

3.4. Grid Independence Test

The grid utilized in numerical simulation calculations is critical to the accuracy of the simulation results. In this section, we perform grid independence testing to determine the most suitable grid partitioning for the examined pore structure. As illustrated in Figure 3, this study considers five different grid configurations. The number of grids is 99,928, 177,510, 235,032, 412,116, and 567,102, respectively. Based on the numerical simulation results, we analyze the velocity value at a distinct point (0.00195 m, 0.00123 m, 0 m). Ultimately, we select 235,032 grids for calculation, as this configuration balances accuracy and computational efficiency.

4. Results and Discussion

4.1. Dynamic Injection Velocity and Microscale Flow Mechanism

The two-phase flow in porous media is primarily controlled by both capillary and viscous forces [42,43,44]. The dynamic variation of injection velocity can manifest as a change in the equilibrium between viscous force and capillary force. Figure 4a and Figure 4b–d respectively show the oil–water two-phase distribution under continuous injection (Case a, log10Ca = −4.37) and step-wise injection (Cases b–d, −5.15 < log10Ca < −4.11) with a total injection volume of 0.5 PV. In the Figures 4, 5, 9–11, the red represents the water and the blue represents oil. Under the step-wise injection condition, the imbibition and drainage displacement are relatively uniform at the inlet position in Cases b and d, and there are two distinct preferential paths in the upper and lower regions of the porous media. In Case c, the oil–water two-phase distribution is similar to Case a, with uniform displacement in the front and middle area of the porous media, while there is a clear fingering above the rear section of the pore medium. Based on the research of Zakirov and Khramchenkov (2019) [13], the flow patterns of Case a and d are generally capillary fingering (abbreviated as CF), while the flow patterns of Case b and c can be classified as crossover zone (CZ).
Figure 4e–g show the oil–water two-phase distribution when the injection velocity varies in a piecewise linear manner (Cases e–g) and the final total injection volume is 0.5 PV. The displacement areas in Figure 4e–g are similar to Figure 4b,d, with significant lateral invasion in the upper and lower regions. Notably, the large pore throat structure in the upper right region of the porous media is where the remaining oil is displaced. The flow pattern can be classified as the crossover zone (CZ) type. As shown in Figure 4g, with the highest initial injection velocity, there is still no water at the outlet. Therefore, the displacement area is the largest among Cases a–g. When the initial injection velocity is 0.0005 m/s (Case e) and 0.001 m/s (Case g), there is a significant change in the spread area compared to the continuous injection of Case a. In summary, the dynamic variation of injection velocity is also an important factor affecting the distribution of oil and water in porous media.
As depicted in Figure 5, in order to elucidate the oil–water flow mechanism inside the real non-homogeneous porous media, the distribution of oil–water two-phase flow in the porous media at different time points (a–e) under a continuous injection velocity of 0.003 m/s is analyzed. Due to the influence of capillary force, the oil–water interface is located at the junction of pore channels and throats at different time points. From 0.488 s to 0.492 s, the front of the oil–water interface at point a breaks through, and the water invaded the a-b channel. By 0.5 s, the water enters the a-c channel. This is primarily due to the significant influence of the opening angle from the throat to the pore on the capillary force under wetting conditions. When the driving force exceeds the capillary force, the remaining oil in the throat is quickly displaced. From 0.513 s to 0.531 s, the remaining oil in the middle and lower throats of the porous media is displaced.
In Figure 6, the changes in pressure at different points (a–e) in Figure 5 over time are analyzed, from 0.488 s to 0.531 s. The pressure exhibits pulsating changes, presenting multiple peaks and valleys. This is primarily due to the Haines jump that occurs when the fluid migrates from a narrow pore throat to its wide body [45,46]. In the adjacent channels a-c and d-e, the pressure changes are relatively consistent, i.e., the changes at points a, b, and c are roughly the same, and the changes at points d and e are roughly the same. It is worth noting that the events f, g, and h marked in the figure all occur within the peak pressure range and are accompanied by the displacement of the remaining oil in the pore.
As shown in Figure 7, the relationship between the oil–water miscible front and the injection volume is presented to help understand the dynamic movement of the two-phase interface during the invasion process. In Figure 7a, the curves for Case a and d, as well as the curves for Case b and c, exhibit relatively consistent changes. This may be due to the smaller fluctuations in capillary numbers in Case d compared to Case a, while the changes in capillary numbers in Case b and c are larger.
At the beginning of the water injection stage (0–0.1 PV), the value of the miscible front changes linearly with the injection volume. After the injection volume reaches 0.1 PV, the length of the oil–water interface differentiates due to differences in injection speed and uneven pore channel size. After the injection volume reaches 0.1 PV, the length of the oil–water front varies in different cases due to the dynamic changes in injection velocity and the heterogeneity of the pore media topology. At 0.3 PV, the maximum length of the oil–water front in Case a is greater than or equal to Cases b, c, and d. Notably, between 0.25 and 0.35 PV, there is no significant change in the oil–water front position of Case d for an extended period. This is primarily because the variation amplitude of the velocity in Case d is the largest, and the oil–water front quickly reaches a relatively long length. The reason is that a directional fingering phenomenon has already formed, making it difficult for water injections to provide sufficient displacement power along the x-direction, and water drive mainly spreads laterally. During the injection volume between 0.4 and 0.5 PV, in Case c, water flows out first from the outlet.
The curve in Figure 7b shows the relationship between the maximum length of the oil–water front and the injection amount when the injection velocity varies linearly. We found that a larger PV value (>0.45) is required to break through the porous media. In Case g, when the PV number reaches 0.5, there is still no water at the outlet. Both Case e and f exhibit the maximum oil–water front length at approximately 0.47 PV. It is worth mentioning that, around 0.28–0.32 PV, when the injection velocity changes from high to low and from low to high, there is no significant change in the maximum length of the oil-–water front. Currently, the driving force contributes to the spreading direction, which is beneficial for improving the recovery efficiency (the proportion of oil effectively recovered from the porous medium during the waterflood process). This may be because a high injection velocity leads to a rapid fingering flow pattern, and when the velocity is below a certain threshold, lateral spread is more likely to occur.
As illustrated in Figure 8, a comparison of recovery efficiency is presented for different cases with an injection volume of 0.5 PV. The horizontal axis represents the difference in injection velocity (maximum injection velocity minus minimum injection velocity), and the vertical axis represents the recovery efficiency. It is important to note that the injection speed is influenced not only by time but also by the degree of change in the injection speed. This variation serves as the basis for using the difference in injection speed to represent the x-axis in Figure 8. Regardless of whether the change in velocity is piecewise linear or step-wise, as the difference in injection velocity increases, the recovery efficiency exhibits a characteristic of first decreasing and then increasing. Moreover, the recovery efficiencies of Cases e, f, and g are higher than that of Case a. Additionally, the recovery efficiency of the injection velocity piecewise linear variation is higher than that of the injection velocity step-wise change under the same velocity difference. In Case c and Case f, the recovery efficiency is relatively the lowest. Furthermore, Cases g and d show that, by adjusting the injection velocity slightly and slowly, the final recovery efficiency can be increased compared to the continuous injection in Case a. This indicates that, in actual oil extraction, gradually increasing the injection rate can significantly enhance the oil recovery rate.

4.2. Mixed Wettability

Due to the heterogeneous rock types in actual geological formations, from a microscale perspective, this can cause heterogeneity in the wettability of pore media [47,48]. Therefore, we perform threshold segmentation on the rock pore media images obtained from CT scans to distinguish between two distinct rock types. In the numerical simulation, we assign different wetting angles for the two rock types. The particles in the gray area occupy less space within the pore medium, exhibiting a non-dominant wetting angle. In contrast, the rock particles in the white area occupy more space in the pore medium, with their wetting angle defined as the dominant wetting angles.
As shown in Figure 9, the distribution of oil–water two-phase flow in the pore media at 0.5 PV injection under different non-dominant wetting angles is presented. When the non-dominant wetting angle is 30 degrees and 45 degrees (Case h and a), the displacement area is primarily concentrated in the front and middle regions of the porous media, and a preferential water flow pathway is observed in the back region. In contrast, when the non-dominant wetting angles are 60, 90, and 120 degrees (Case i, j, k), the displacement area mainly advances in the upper and lower regions of the porous media, exhibiting two preferential pathways. In the case of a non-dominant wetting angle of 150 degrees, only one preferential pathway is formed in the middle of the porous media. It should be noted that, at the non-dominant wetting angles of 120 and 150 degrees, the displacement area bypasses the gray oil/wet rock particles. This indicates that the non-dominant wetting angle significantly influences the two-phase flow distribution under oil–wet conditions.
To characterize the microscale flow mechanism of immiscible fluids inside porous media, as demonstrated in Figure 10, we present the simulation results of the water flooding process in the pore throats of a simple pore throat structure (ignore the influence of irregular topological structures) under different wetting angles and opening angles from the throat to the pore. The different simulation parameters are shown in Table 4.
In Figure 10a–e, the oil–water two-phase distribution at different time points in Case 1 is presented. When the sum of the opening angle from the throat to the pore and the wetting angle is less than 90 degrees, the oil in the pore throat channel can be invaded by water, with capillarity being the sole driving force. This finding is consistent with the numerical simulation research results of Pavuluri et al. [27]. In Figure 10f–j, unlike Case 1, although the opening angle and the wetting angle of the upper and lower boundaries are not consistent, they jointly determine the curvature of the oil-water interface. At this time, the oil–water interface can still smoothly pass through the pores and throats.
In Figure 10k–o, the opening angle is further increased to 55 degrees, and it is found that the surface shape of the oil–water interface undergoes a transformation during the passage of the throat, from concave menisci in Figure 10k to convex menisci in Figure 10n. More specifically, the oil–water interface no longer changes over time, and its shape remains as shown in Figure 10o.
Figure 10p shows the schematic diagram of the variation of capillary force through the throat during spontaneous imbibition in Cases 1, 2, and 3; the small image in Figure 10p provides a schematic illustration of the morphology of the oil–water interface passing through the throat. Through the throat to the pore, the capillary force first decreases, then increases, and then remains constant. The embedded image in Figure 10p highlights the approximate shapes of the two-phase interface through the throat to the pore for Cases 1 and 2 (curves a, b, c, and d). The menisci have been consistently maintained concave. Although the capillary force varies according to the shape of the throat and pore, the value is still greater than 0, indicating that the capillary force is the driving force under the imbibition process. In Case 3, due to the lower boundary opening angle and wetting angle being greater than 90 degrees, the bending direction of the meniscus of the two-phase interface d* formed is different from that of the initial interface a. The capillary force changes from a driving force to a resistance force, hindering the movement of the two phases.
As shown in Figure 11a, we design a pore doublet geometry to explain the preferential pathway selection in porous media under the imbibition process, where the dimensions of the two pores are consistent, but the lengths are different. Therefore, the capillary force is the same at the beginning, and then the viscous force controls the two-phase fluid flow. In theory, the two-phase spontaneous imbibition dominated by viscous forces can be described by the following formula [28]:
F v = − 8 π μ 1 x + μ 2 L − x d x d t ,
where μ 1 is wetting phase viscosity, μ 2 is non-wetting phase viscosity, L is capillary length, and x is imbibition distance of the wetting phase in the capillary. At the initial time, due to the length of pore 1 being smaller than pore 2 (R1 < R2), the viscous force in pore 1 is smaller than that in pore 2 (F1 < F2). Simultaneously, the capillary force acts as the driving force. In Figure 11b, at 0.00215 s, according to the above Equation (2), the viscous resistance in pore 2 is greater than that in pore 1, resulting in a significantly lower total flow rate of water from the main pore to pore 2 compared to pore 1. With the influence of the spontaneous imbibition effect, the amount of water in pore 1 gradually increases. This suggests that the dynamic viscosity heterogeneity effect increases, causing a decrease in viscous resistance in pore 1, leading to further water flow into pore 1 and causing viscosity fingering phenomena. This is also an important reason for the differences in the invading sequence of different pores within the porous media. In Figure 11c, at 0.00365 s, due to the combined effect of viscosity and capillary force (decreasing capillary force upon entering pore 1), the invading fluid (water) fills pore 1.
As depicted in Figure 12, the change in recovery efficiency when injecting 0.5 PV under different non-dominant controlled wetting angles is presented. It is found that when the dominant control wetting angle is 45 degrees, as the non-dominant control wetting angle increases, the recovery efficiency first increases and then decreases. When the non-dominant control wetting angle is 150 degrees, the recovery efficiency value is the smallest, at 36.41%. Conversely, at a non-dominant wetting contact angle of 60 degrees, the maximum oil recovery efficiency is 50.01%. Notably, in our previous studies [35], when there was only a single uniform wetting angle in the porous media, the curve of the change in recovery efficiency with respect to wetting angle was similar to the trend observed in this paper. This implies that the non-dominant controlling wetting angle is an important factor affecting the magnitude of the recovery efficiency in the porous media.
As shown in Figure 13, we present the change in the oil–water front position with the injection volume at 0.5 PV under mixed wettability conditions. In the case where the difference in wetting angle is the largest in the porous media (Case l), the length of the oil–water front always changes the most rapidly with the injection volume. This phenomenon is also observed in Case k, where only the slope of the rising curve is slightly smaller than in Case l. In Cases h, i, and j, the slope of the rising curve is roughly consistent between 0 and 0.3 PV. Additionally, when the PV number reaches 0.5 in Case i, the length of the miscible front still does not reach the maximum value. The inset picture in Figure 13 shows the required PV number for the first appearance of water at the outlet (with a length of 0.003 m at the oil–water front) in different cases. As the difference in wetting angle increases, the required PV number to reach the maximum length of the oil–water front shows a trend of first increasing and then decreasing, which is consistent with the observed change in recovery efficiency.

5. Conclusions

This paper reveals the imbibition and drainage mechanisms of capillary force and viscosity effects in single pore throats and dual pore throats from a microscale perspective. We then focus on the influence of dynamic changes in injection velocity and rock mixed wettability on the microscale flow phenomena of oil–water two-phase flow in porous media. This article is different from traditional numerical studies that consider homogeneous conditions (constant injection velocity and uniform wetting angle). By establishing a dynamic boundary condition algorithm for injection velocity and defining a non-uniform wetting angle, we explore the impact of these factors on the micro-dynamic behavior of the waterflooding process in porous media.
When stepwise and piecewise linear changes in injection velocity are observed, unlike continuous injection, two preferential displacement pathways are predominantly formed in the porous media. Additionally, regardless of whether the change in velocity is linear or stepwise, as the difference in injection velocity increases, the recovery efficiency first decreases and then increases. When the injection velocity changes linearly, the recovery efficiency is greater than that of continuous water injection.
The non-dominant wetting angle affects the distribution of the oil–water phases. When the non-dominant wetting angle is less than 60 degrees, the oil and water are uniformly spread in the middle and front regions, and a preferential path is formed in the middle and back regions. When the non-dominant wetting angle is between 60 and 120 degrees, two preferential paths are formed in the back region of the porous medium. The non-dominant wetting angle also influences the recovery efficiency, and as the non-dominant wetting angle increases, the recovery efficiency first increases and then decreases. When the non-dominant wetting angle is 150 degrees, the oil–water interface changes most rapidly with injection volume. As the degree of non-dominant wetting angle increases, the recovery efficiency first increases and then decreases, with a maximum and minimum difference of 13.6%.
Although this study reveals the influence of dynamic changes in injection velocity and rock mixed wettability, there are still many scientific factors to be investigated in future research, including 3D porous media, different fluid properties, chemical reactions, underground gas storage, etc.

Author Contributions

Conceptualization, Y.W., L.W. and W.H.; methodology, L.W. and Y.W.; software, L.W. and X.L.; investigation, X.L. and L.W.; writing—original draft preparation, L.W. and W.H.; writing—review and editing, L.W. and W.H.; visualization, L.W. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the Scientific Research Program Funded by Education Department of Shaanxi Provincial Government (Program No. 22JT036), Key Research and Development Program of Shaanxi (Program No. 2023-YBSF-234), Science and Technology Plan Project of Yulin (2024-CXY-170), and the State Key Laboratory of Loess and Quaternary Geology, Institute of Earth Environment, CAS (SKLLQG2234).

Data Availability Statement

All data are given in this paper.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. Bubble shapes at different times: (a) t = 0.0574 s; (b) t = 0.0898 s; (c) t = 0.1724 s; (d) t = 0.2125 s.
Figure 1. Bubble shapes at different times: (a) t = 0.0574 s; (b) t = 0.0898 s; (c) t = 0.1724 s; (d) t = 0.2125 s.
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Figure 2. Physical model, boundary conditions, and mesh.
Figure 2. Physical model, boundary conditions, and mesh.
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Figure 3. Grid independence test.
Figure 3. Grid independence test.
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Figure 4. Oil–water two-phase distribution under different injection velocity modes: (a) case a; (b) case b; (c) case c; (d) case d; (e) case e; (f) case f; (g) case g.
Figure 4. Oil–water two-phase distribution under different injection velocity modes: (a) case a; (b) case b; (c) case c; (d) case d; (e) case e; (f) case f; (g) case g.
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Figure 5. The oil–water two-phase distribution of case 1 at different times: (a) 0.488 s; (b) 0.492 s; (c) 0.5 s; (d) 0.513 s; (e) 0.519 s; (f) 0.523 s; (g) 0.531 s.
Figure 5. The oil–water two-phase distribution of case 1 at different times: (a) 0.488 s; (b) 0.492 s; (c) 0.5 s; (d) 0.513 s; (e) 0.519 s; (f) 0.523 s; (g) 0.531 s.
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Figure 6. The pressure at different points in Figure 5a varies over time.
Figure 6. The pressure at different points in Figure 5a varies over time.
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Figure 7. The oil–water front changes over time: (a) step-wise changes of injection velocity; (b) piecewise linear variation of injection velocity.
Figure 7. The oil–water front changes over time: (a) step-wise changes of injection velocity; (b) piecewise linear variation of injection velocity.
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Figure 8. Recovery efficiency under 0.5 PV in different cases.
Figure 8. Recovery efficiency under 0.5 PV in different cases.
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Figure 9. The oil–water two-phase distribution in mixed wettability: (a) 30° (case h); (b) 45° (case a); (c) 60° (case i); (d) 90° (case j); (e) 120° (case k); (f) 150° (case l).
Figure 9. The oil–water two-phase distribution in mixed wettability: (a) 30° (case h); (b) 45° (case a); (c) 60° (case i); (d) 90° (case j); (e) 120° (case k); (f) 150° (case l).
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Figure 10. The oil–water two-phase distribution (flowing from left to right) and the effect of capillary forces. (a) 0.00037 s, (b) 0.000395 s, (c) 0.00059 s, (d) 0.001235 s, (e) 0.00132 s, (f) 0.000354 s, (g) 0.00036 s, (h) 0.0007 s, (i) 0.00102 s, (j) 0.00119 s, (k) 0.00027 s, (l) 0.00029 s, (m) 0.000295 s, (n) 0.0003 s, (o) 0.0015 s, (p) schematic diagram of capillary force passing through the throat to pore under different cases.
Figure 10. The oil–water two-phase distribution (flowing from left to right) and the effect of capillary forces. (a) 0.00037 s, (b) 0.000395 s, (c) 0.00059 s, (d) 0.001235 s, (e) 0.00132 s, (f) 0.000354 s, (g) 0.00036 s, (h) 0.0007 s, (i) 0.00102 s, (j) 0.00119 s, (k) 0.00027 s, (l) 0.00029 s, (m) 0.000295 s, (n) 0.0003 s, (o) 0.0015 s, (p) schematic diagram of capillary force passing through the throat to pore under different cases.
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Figure 11. Oil–water distribution in a pore doublet at different times affected by viscous effects. (a) 0 s, (b) 0.00215 s, (c) 0.0036 s.
Figure 11. Oil–water distribution in a pore doublet at different times affected by viscous effects. (a) 0 s, (b) 0.00215 s, (c) 0.0036 s.
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Figure 12. Recovery efficiency under mixed wettability conditions.
Figure 12. Recovery efficiency under mixed wettability conditions.
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Figure 13. The length of oil–water front varies with PV.
Figure 13. The length of oil–water front varies with PV.
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Table 1. Comparison of numerical simulation results and analytical results of capillary height.
Table 1. Comparison of numerical simulation results and analytical results of capillary height.
Wetting Angle26°36°46°56°66°
Maximum height value0.01177 m0.01061 m0.00911 m0.00733 m0.00533 m
Analytical solution0.01261 m0.01115 m0.00956 m0.00771 m0.00558 m
error7.12%5.09%4.89%5.17%4.62%
Table 2. Case settings.
Table 2. Case settings.
No.Initial Velocity (m/s)Initial Injection VolumeFinal Velocity (m/s)Final Injection VolumeVelocity Change ModeWetting Angle (White)Wetting Angle (Gray)
Case a0.003 0.5 PV--constant45
Case b0.0005 0.05 PV0.0055 0.05 PVstep-wise45
Case c0.001 0.05 PV0.005 0.05 PVstep-wise45
Case d0.002 0.05 PV0.004 0.05 PVstep-wise45
Case e0.0005 -0.0055 0.05 PVpiecewise linear45
Case f0.001 -0.005 0.05 PVpiecewise linear45
Case g0.002 -0.004 0.05 PVpiecewise linear45
Case h0.003 0.5 PV--constant4530
Case i0.003 0.5 PV--constant45 60
Case j0.003 0.5 PV--constant4590
Case k0.003 0.5 PV--constant45 120
Case l0.003 0.5 PV--constant45150
PV: Pore volume number, the ratio of the injected fluid volume to the total pore volume in the porous medium.
Table 3. Numerical boundary settings.
Table 3. Numerical boundary settings.
Physical BoundaryInletOutletOthers
Physical Quantity
PressureFixedFluxPressurefixedValueFixedFluxPressure
Volume fractionFixedValuezeroGradientconstantAlphaContactAngle
VelocityFixedValueZeroGradientFixedValue
Table 4. Mixed wettability cases.
Table 4. Mixed wettability cases.
No.Opening Angle (°)Upper Boundary Wetting Angle (°)Lower Boundary Wetting Angle (°)PropertiesAnalysis
1354545Density: 1000 kg/m3; Viscosity: 1 × 10−6 m2/s; Interface tension coefficient: 0.01 kg/s2Figure 10a–e
2353045Figure 10f–j
3553045Figure 10k–o
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Hui, W.; Wang, L.; Liu, X.; Wang, Y. Effect of Dynamic Injection Velocity and Mixed Wettability on Two-Phase Flow Behavior in Porous Media: A Numerical Study. Energies 2025, 18, 879. https://doi.org/10.3390/en18040879

AMA Style

Hui W, Wang L, Liu X, Wang Y. Effect of Dynamic Injection Velocity and Mixed Wettability on Two-Phase Flow Behavior in Porous Media: A Numerical Study. Energies. 2025; 18(4):879. https://doi.org/10.3390/en18040879

Chicago/Turabian Style

Hui, Wei, Le Wang, Xurui Liu, and Yueshe Wang. 2025. "Effect of Dynamic Injection Velocity and Mixed Wettability on Two-Phase Flow Behavior in Porous Media: A Numerical Study" Energies 18, no. 4: 879. https://doi.org/10.3390/en18040879

APA Style

Hui, W., Wang, L., Liu, X., & Wang, Y. (2025). Effect of Dynamic Injection Velocity and Mixed Wettability on Two-Phase Flow Behavior in Porous Media: A Numerical Study. Energies, 18(4), 879. https://doi.org/10.3390/en18040879

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