A Novel Tripod Methodology of Scrutinizing Two-Phase Fluid Snap-Off in Low Permeability Formations from the Microscopic Perspective

Abstract

According to the requirements of carbon-neutral development, this study explores the comparison and new discussion of replacing nitrogen with carbon dioxide in the conventional two-phase microfluid flow. Thus, carbon dioxide application in various fields can be more precise and convenient. This research uses an artificially continuously tapering micro model to mimic the natural rock channel in low permeability formation, where the liquid imbibition process is entirely under surface tension-dominant. The tested capillary number decreased to 8.49 × 10⁻⁶, and the thinnest observed liquid film was reduced to 2 μm. The comparison results in two gas groups (nitrogen and carbon dioxide) show that CO₂ gas fluid in microscopic porous media would have more tendency to snap off and leave fewer residual bubbles blocked between the constrictions. However, the N₂ gas fluid forms smaller isolated gas bubbles after snap-off. By combining the experimental data and numerical output with the theoretical evolution equation by Beresnev and Deng and by Quevedo Tiznado et al., the results of interface radius, temporal capillary pressure, and velocity profiles for axisymmetric and continuously tapering models are presented and validated. Those findings create a paradigm for future studies of the evolution of microscopic multiphase fluid and enhance a deeper understanding of geological underground fluid properties for greenhouse gas storage and utilization in low permeability formations.

1. Introduction

At present, geological, biological, and petroleum engineers are focusing increased research attention on the microscopic domain. When the current macroscopic multiphase flows in the porous medium have acknowledged more understanding, the fluid flow investigation from the microscopic perspective is imminent. As one of the basic multiphase flow patterns, the two-phase fluid flow has been widely studied in the environment and in the petroleum, agricultural, and food industries, such as field irrigation, hydrocarbon recovery, water infiltration in the soil, micro-systems in the food industry, etc. [1,2,3]. The pressure-driven dynamic flow properties for microscopic fluids are different from those of macroscopic fluids, which generally take viscosity and surface tension as the dominant study parameters. In some tiny spaces, the influence of the surface tension during the wetting phase imbibition process can exceed the effect of viscosity [4,5]. Cense and Berg used Stokes’ law to calculate the viscous term and the Young–Laplace equation to calculate a capillary force and to explain the contradiction between macroscope and microscope models [4]. They emphasized that the microscopic equilibrium between two phases is dominated by viscous force and capillary force at the fluid interface meniscus. Their concept can be stated more simply when the fluid velocity is lower than a certain value, the effect of viscosity force will be limited, and the surface tension force will become the dominant force.

In addition, the Canadian government pledged to reduce greenhouse gas (GHG) emissions by 2030 to 30% below the 2005 levels in accordance with the Paris Agreement. Therefore, CO₂ has been considered to be a sequestered subsurface and used in the enhanced hydrocarbon production process. It is vital to explore the liquid-gas two-phase fluid flow characteristic under a microscopic viewpoint when carbon dioxide gas is utilized for energy resource production or a factory supply system. Numerous studies have shown that there are two important processes in the fluid displacement process: (1) pore-filling or piston-like displacement and (2) snap-off. The former acts as the non-wetting or wetting phase fluid and gradually displaces the other fluid with the front curvature pushing forward. The latter does the opposite, which has a higher potential of snap-off occurring as well as non-wetting phase trapping [6,7]. It leads to two possible outcomes in the process of gas-liquid fluid flow: one is that the wetting phase fluid completely displaces and replaces the pore space of the original non-wetting phase fluid. The other possible outcome is that the non-wetting phase fluid is pinched off by the wetting phase fluid and then trapped inside the porous medium. Therefore, understanding the trapping conditions and mechanisms of the non-wetting phase will aid engineers in accurately designing and controlling the subsurface carbon dioxide gas storage technology and its subsequent utilization in improved hydrocarbon extraction.

J. G. Roof developed a classic static snap-off criterion in the 1970s [8]. The model was built in a doughnut-hole pore which could be depicted in a two-dimensional axisymmetric pore throat. It mentioned the capillary pressure imbalance between the front of the oil bulge and the oil-water interface in the pore throat. The result of Roof’s experiment supports the theory that the snap-off occurs when the capillary pressure in the oil bubble is less than that in the pore throat. While the oil bulge is moving far from the pore throat, the decreasing oil phase capillary pressure at the pore throat will cause more water to fill in the pore throat.

In 2010, the research of Beresnev and Deng [9] built a two-phase fluid flow model to describe the fluid-fluid interface’s temporal evolution. They introduced time into the system as one of the dynamic factors. Following the mass conservation principle, they derived an evolution equation for capillary-dominate flow in a two-dimensional coordinate. Later, the research by Deng et al. established mathematical equations by combining a few dynamic factors into a more general dimensionless number [10]. Further research by J. A. Quevedo Tiznado et al. [11] identified snap-off inhibition. They pointed out that the upper limited capillary number analysis is unlike the static criteria, which shows that the snap-off is a dynamic breakup criterion. At the end of the discussion, Quevedo Tiznado et al. denied the feasibility of solving highly nonlinear fourth-order partial differential equations and proposed that simple formulations have the advantage of close connection with various physical parameters and easy simulation.

The improvement made in this research is the combination of real-time visualization results and corresponding simulation results, which will benefit the imbibition prediction in porous medial with a similar scale. Based on the data support from experiments, the numerical simulation can avoid the highly non-linear fourth-order partial differential equations while remaining accurate. Moreover, this research introduces higher accuracy results in real-time visualization by the high-temperature treatment of glass models. Finally, it also proves there is less possibility of snap-off occurring within a continuously tapering pore throat when compared to previous experiments by Tian et al. [7].

This research aims to investigate the mechanisms of non-wetting phase snap-off and bubble formation in the two-phase fluid flow in the continuously tapering constraint pore throat model for carbon dioxide gas application and identify the most significant difference with previous nitrogen gas experiments. The results can be considered as a real-life experiment supporting the application of carbon dioxide to replace the original immiscible gas in various fields at this stage.

2. Materials and Methods

This research contains three parts: real-time experiment, numerical simulation, and mathematical model verification. The experiment results are designed to complement the required input parameters of numerical simulation. The simulation output is used to verify the mathematical criterion of gas snap-off from Beresnev and Deng [9]. The procedure flow chart is shown in Figure 1. The following subsections describe the physical models used in the experiment, the numerical meshing, and the mathematical equations.

In Figure 1, the red lines show the comparison between the three parts, and the blue lines show the source of the input parameters. The input parameters taken from the experiment for the numerical simulation are liquid injected velocity, liquid-gas contact angle, operating pressure and temperature, and liquid-gas surface tension. The input parameters for the mathematical calculation from the simulation results are the pressure gradient for each phase and the corresponding liquid-gas interface radius.

2.1. Micromodel Characterization

The micromodel is made of Schott glass with a uniform pore body and varying constricted areas (pore throats) to imitate the corresponding sand rock porous medium. The glass chip’s maximum pressure limit is 5 MPa, and its upper-temperature range is 500 °C. The glass models have a width of 2 cm, a length of 6 cm, and a thickness of 3 mm. There are 4830 μm straight pipes on the leading and tailing sides, four straight channels (every 9660 μm), and five curved segments, as indicated in Figure 2 (with a radius of 690 μm). These buffers are used to stabilize the flow state of the injected and displaced fluids, as well as the shape of the gas-liquid contact surface. The etched microchannel is 6 μm in depth from top to bottom, with a rectangular cross-section in the middle of the glass model. The radius of the straight channel (R_T) is 75 μm, and the radius of the pore throats (R_g) is 15 μm for the axisymmetric pore throat. For the continuously tapering pore throat, the radius is 25 μm, 15 μm, and 7.5 μm from inlet to outlet. These specific ratios between pore body and pore throats (3, 5, and 10) follow Roof’s snap-off static criteria (R_T ≥ 2R_g). Roof’s theory was proven by the real-time visualization experiment in 2020 by Tian et al. [7], which demonstrates that water film extension and gas snap-off phenomena on the micrometer scale are influenced both by water injection rates and the pore-throat ratio at fixed geometry microchannels. In our experiment, the purpose of using the micromodel is to establish the mathematical relationship between fluid velocity, pore throat radius, and liquid–gas surface tension for a realistic experiment.

In Figure 2, R_T represents the pore body radius, R_T,f represents the converted pore body radius with a circular cross-area, R_g represents the pore throat radius with a rectangular cross-area, and R_g,f represents the converted pore throat radius with a circular cross-area. According to the hypotheses of Beresnev and Deng and Quevedo Tiznado et al. [9,11], a two-dimensional coordinate system is developed based on the pore throat area of microchips, and it stretches out in a 980 μm straight channel horizontally along the fluid flow direction from both intake and outlet. The numerical model is based on the same microchip shape as seen in Figure 2. The correlated factors are calculated from the experiment glass model, and the results are shown in

Table 1. Calculated Factor for the Micromodel.

Factors	Values	Factors	Values
Rgf,1 (μm)	14.12	ᾶ1= (RT,f − Rgf,1)/(l/2)	0.1016
Rgf,2 (μm)	10.02	ᾶ2= (RT,f − Rgf,2)/(l/2)	0.1288
Rgf,3 (μm)	6.33	ᾶ3= (RT,f − Rgf,3)/(l/2)	0.1536
RT,f (μm)	29.36	μ1= μN2 (cp)	0.01767
Linlet/outlet (μm)	980	μ1= μCO2 (cp)	0.01478
l (μm)	300	μ2= μwater (cp)	0.8900
a1= Rgf,1/RT,f	0.4808	Wall λ1(x)	22.535
a2= Rgf,2/RT,f	0.3420	Wall λ2(x)	20.711
a3= Rgf,3/RT,f	0.2154	Wall λ3(x)	19.048

, where λ(x) represents the radial coordinate of the capillary wall. Equation (1) shows the calculation process for λ(x) [9].

$$\begin{gathered} \lambda \left(x\right)=0.5R_{T,f}-\left[\left(1+a\right)-\left(1-a\right)cos\left(\frac{ᾶ\pi x}{R_{T,f}}\right)\right] \\ wherea=\frac{R_g}{R_T};ᾶ=\frac{\left(R_T-R_g\right)}{\left(\raisebox{1ex}{$l$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)} \end{gathered}$$

(1)

In both Equation (1) and Table 1, l represents the length of the defined channel, a is the ratio between the radius of the pore throat and the pore body, ᾶ represents the ratio between the radius of the pore body and half of the defined channel, and μ represents the viscosity of the fluid.

2.2. Equipment Set-Up

The experiment was conducted in the University of Regina laboratory at a stable room temperature and pressure. The liquid-gas displacement mechanism in the micro glass chip models is observed using an Olympus SZX16 wide zoom stereomicroscope. This stereomicroscope can visualize the phenomenon under a micrometer scale and zoom those images out by Olympus Stream Basic software installed on a lab computer. The Olympus SC180 camera video continues to record at 30 frames per second to capture the dynamic changes in liquid-gas displacement processes every 34 milliseconds. Simultaneously, with the Olympus SDF PLAPO 1XPF objective lens, the degree of magnification is from 27× to 430× with a maximum working distance of 60 mm. A Chemyx Fusion 200 micro-syringe pump with a minimum injection rate of 0.0001 mL/min is used for the water injection operation, which has been proved to be the most similar condition for outputting surface tension-dominant fluid condition [7,12]. As a result, the injection velocity range is suitable for simulating the liquid-gas imbibition process in geological faults.

A high-pressure gas tank is used to link a microfluidic valve and a digital pressure gauge for gas injection. The gas injection pressure is held at no more than 5 MPa to avoid fracturing the glass models. The equipment set-up chart is shown in Figure 3.

2.3. Meshing Grid

In the meshing step, the geometric model is divided into fine grids; the Fluent method uses either the finite or infinite method to perform further detail simulation. Because the fluid flow state in the fluid-solid boundary is undetermined, the flow state and the known conditions must be related to each other to obtain other unknown variables at the same time step. The detailed meshing grids are presented in

Table 2. Meshing Grids in Each Model.

	Axisymmetric Pore Throat	Continuously Tapering Pore Throat
Inlet/Outlet Divisions	30	15
Enter/Exit Channel	200	150
Transition Curve	10	8; 7; 5
Pore Throat	30	15; 12; 10
Total elements	18,993	18,103

, where the 830 μm straight channel for inlet and outlet divisions is divided into 200 × 30 and 15 × 150 small rectangular grids in the axisymmetric pore throat and continuously tapering pore throat, respectively. The middle pore throat area is divided into 30 × 30 grids for axisymmetric pore throat and 15 × 15, 12 × 15, and 10 × 15 small grids according to the throat diameters for the continuously tapering pore throats.

The Courant number involves grid size, velocity scale, and time step size. These three values are used to calculate the Courant–Friedrichs–Lewy (CFL) number for a stable solution [13,14,15]. The CFL number can be stated as follows [16]:

$$CFL=\frac{U\Delta t}{\Delta x}\le 1$$

(2)

where U represents the maximum velocity in m/s, and Δx represents the single element’s grid size (in m) in the meshing step. The Global Courant number usually is set at a default value of 2 in the Ansys Fluent system, but it depends on the startup conditions. The Global Courant number is set at 0.25 to control the startup transients [16]. Based on the above CFL equation, the global time-step size Δt can be calculated as [16]:

$$\Delta t=CFL\frac{\Delta x}{U}=\frac{CFL}{max\left({\displaystyle \sum }\frac{outgoing flux}{volume}\right)}$$

(3)

Another point that should be noticed is the transition grid size from the straight pore body to the pore throat areas. It is necessary to ensure that there is a gradual transition between finer mesh to coarser mesh. In the simulation, the meshing size difference is limited to be lower than 0.95 to allow the grid size transformation between the pore body and pore throat to be meticulous and gradual [15]. Along with these aforementioned requirements, the final numerical models for axisymmetric pore throats and continuously taping pore throats after meshing contain 18,993 and 18,103 elements, and the differences between each element are lower than 0.597 and 0.681, respectively [12,17,18].

2.4. Numerical Simulations

The numerical simulations were achieved by the Ansys Fluid VOF method (Volume of Fluid), which has the ability to simulate fluid dynamics with free boundary problems [13,14]. The numerical simulation is based on the Navier-Stokes equations with the Wetting Force Model (WFM) [19,20]. The Wetting Force Model supports that the surface tension and wetting force between liquid and gas work together to affect the movement of the interface in microfluid flow [21]. Malgarinos’ experiments show that the dynamic contact angle model is different from those under static conditions [22,23], which means that the dynamic fluid model needs to contain an additional term that assumes the adhesion force parallel to the wall (Figure 4).

In Figure 4, where θ_rec is the static contact angle, θ_dyn is the dynamic contact angle, and θ_rec represents the static contact angle for the hydrophilic surface. The mathematical equation that describes the adhesion force along the contact line for both a hydrophilic and hydrophobic surface is [22]:

$$d{f}_{WFM}=\sigma \left(cos{\theta }_{eq}-cos{\theta }_{dyn}\right)dl$$

(4)

Following that, experiments by Brackbill et al. with the addition of the WFM term, f_WFM, and surface tension, σ, formed the wall adhesion concept as follows [22]:

$$\widehat{n}={\widehat{n}}_{w}cos{\theta }_{dyn}+{\widehat{n}}_{t}sin{\theta }_{dyn}$$

(5)

where ${\widehat{n}}_w$ and ${\widehat{n}}_t$ are the unit vectors normal and tangential to the wall, respectively (Figure 5). The contact angle θ_dyn is the angle between the wall and the tangent to the interface at the wall. Based on the equation by Brackbill et al. [21], the movement of liquid fluid in the micro-channel was able to be simulated.

The VOF approach took into account the effects of surface tension and wall adherence. A special dynamic contact angle value was inserted as the reference of the wetting force model [15,16], where σ is the surface tension in the momentum equation referenced from Brackbill et al. [21]. The volume fraction, α, is defined as the percentage of liquid volume versus the total volume of the cell [22,24]:

$$\overrightarrow{f_{\sigma }}=\sigma \frac{\rho \kappa \nabla \alpha }{\frac{1}{2}\left({\rho }_{liq}+{\rho }_{gas}\right)}$$

(6)

The combination governing equation for the VOF method became [22,24]:

$$\frac{\partial \rho \overrightarrow{u}}{\partial t}+\nabla (\rho \overrightarrow{u}\otimes \overrightarrow{u}-\overrightarrow{T})=\rho \overrightarrow{T}+\overrightarrow{f_{\sigma }}+\overrightarrow{f_{WFM}}$$

(7)

$$\frac{\partial \alpha }{\partial t}+\nabla (\overrightarrow{u}\alpha )=0$$

(8)

where $\overrightarrow{f_{\sigma }}$ represents the surface tension term (N/m³) in the momentum Equation (4), which is included in the Fluent simulation model. The curvature of the free surface, κ (m⁻¹), is the divergence of the unit surface normal (α perpendicular vector to the surface at a given point), where $\kappa =\nabla \frac{\nabla \alpha }{\left|\nabla \alpha \right|}$, and $\overrightarrow{T}$ represent the stress tensor (kg/ms²). The factors u and υ represent the velocity in the x-axis and y-axis (m/s) in the Cartesian coordinate or r-axis and z-axis in the cylindrical coordinate, respectively. The gravity effect was ignored in this two-dimensional fluid simulation. Simultaneously, the fluid properties will be renewed along with each iteration [16]:

$$\mu =\alpha {\mu }_{liq}+\left(1-\alpha \right){\mu }_{gas}$$

(9)

$$\rho =\alpha {\rho }_{liq}+\left(1+\alpha \right){\rho }_{gas}$$

(10)

where μ represents the fluid viscosity, ρ represents the fluid density, and α is the volume fraction which will renew automatically with each simulated iteration.

3. Results and Discussion

The results from the axisymmetric pore throat and continuously tapering pore throat are shown in this part to demonstrate the properties of gas-liquid fluid flow with a liquid injection rate of 0.0001 mL/min, which gives the closest surface-tension driving fluid for both gas groups. Then, to establish its validity, the radius of the interface (κ_f) measured in an axisymmetric pore throat is compared to the findings of the numerical simulation. The results from the continuously tapering pore throat are used to validate the breakup criteria from the gas-liquid two-phase dynamic fluid condition.

Table 3. Liquid Injection Values.

	With Nitrogen Gas			With Carbon Dioxide Gas
Liquid Injection Rate (mL/min)	Measured Flow Velocity (μm/s)	Corresponding Capillary Number, NCa	θdyn (°C)	Measured Flow Velocity (μm/s)	Corresponding Capillary Number, NCa	θdyn (°C)
0.0001	687.2	8.49 × 10−6	2.33	942.2	1.16 × 10−5	2.89

shows the actual measured velocity and related calculated capillary number for each injection rate in the primary velocity test. The capillary number N_Ca equals ηυ/σ, where η is the dynamic viscosity of the liquid phase, which equals 8.9 × 10⁻⁴ Pa·s (around 25 °C and 1 atm pressure) from the Engineering ToolBox [25,26], υ is the liquid flow velocity in m/s, σ is the interface tension between liquid and gas phases, which is 0.072 N/m for both nitrogen gas and carbon dioxide groups at room temperature and pressure [27,28,29], and θ_dyn represents the contact angle inserted into the wall adhesion options to allow surface tension force to act on the boundaries. The value of θ_dyn is taken from experimental recorded results.

3.1. Axisymmetric Pore Throat

Figure 6 shows the liquid-gas interface radius at the center point of the pore throat (the κ_f value at ΔX = 0 μm) versus the dimensionless time factor τ ($\tau =\frac{t}{{\mu }_1{R}_T/\sigma }$) for both numerical experimental results. The goal of the comparison is to show that the simulation results match the real-time visualization results, allowing the simulation findings to be utilized as a more realistic database in the mathematical discussion. The comparison between the real-time visualization results and the simulation results is shown in Figure 6, where the reference time step is recorded as Δt = 0 ms (τ = 0) when the liquid-gas interface radius starts changing at the center point. The corresponding images for the simulation results are shown in Figure A1 in the Appendix A.

In Figure 6, the liquid-gas interface radius κ_f has a starting point of around 8 μm because the water film thickness lower than 2 μm is hard to be seen from the real-time visualization images. When the experimental and numerical simulation findings are compared, the liquid-gas interface radius has a similar declining tendency, and the simulated data is comparable to the observed interface radius in the same magnitude. Furthermore, the pressure value at the trough of the curve (Figure 7) represents the simulated pressure at the gas-liquid interface, which is the capillary pressure between two phases. The results reveal that when the dimensionless time τ for each group increases, the capillary pressure increases as well. The capillary pressure for nitrogen-water tests is higher than for carbon dioxide-water testing at the same time step. Sun and Santamarina’s research [30] provides an explanation based on a simplified capillary pressure Equation (11):

$$\Delta P_c=P_1-P_2=\frac{4\sigma cos\left[{\theta }_{rec}-{\theta }_{dyn}\right]}{R_g}$$

(11)

In Beresnev and Deng’s simulation analysis [9], they mentioned that the liquid-gas interface showed irregular fluctuations. He later deleted the affected values for aesthetic reasons and created an image that better matched the changes in the interface form. The capillary pressure that is not impacted by the jagged contact is gathered in this investigation to create a representative figure (Figure 7).

The results reveal that the nitrogen group’s capillary pressure is lower than the carbon dioxide group’s (P_c,N2 < P_c,CO2) at first, but the slope of P_c,N2 is steeper than that of P_c,CO2. The dominating factor, according to the capillary pressure Equation (11), is the dynamic contact angles between the pore wall surface and the flow direction when the surface tension difference between the two groups is negligible. Combined with the real-time visualization and interface radius, when the two-phase fluid passes through the pore throat, the liquid fluid thickness accumulates in the pore throat, causing the interface radius κ to decrease until gas snap-off occurs. Simultaneously, the images from real-time visualization show that the fluid flow direction is not perfectly parallel to the pore wall. The interface shape shrinks, which indicates that the dynamic contact angle increases with the decreasing of the interface radius at the center of the pore throat.

The simulated interface radius chart shows that the κ value of the nitrogen group is greater than that of the carbon dioxide group at the beginning. But when the interface radius is lower than 5.0, the two graphs converge to a similar downward trend, and the interface radius in the nitrogen group is smaller. The results give a critical liquid-gas interface radius conjecture that when κ_f > 0.5R_g, the N₂-water group shows a lower capillary pressure between liquid-gas phases than the CO₂-water group. When the gas volume factor is lower than 0.5, the capillary pressure in the CO₂-water group is larger than that in the N₂-water group.

3.2. Continuously Tapering Pore Throat

3.2.1. Experimental Result for Continuously Tapering Pore Throat

Figure 8 depicts the real-time visualization results, with the left side of the pore throat representing the upstream and the right side representing the downstream.

With the 0.0001 mL/min injection rate, the gas snap-off occurs at the narrowest pore throat with a diameter of 15 μm for both gas groups. The thin wetting film continues along the channel boundaries and cuts off the gas phase at the third pore throat, as shown in Figure 8, after which the bulk gas fluid downstream flows together with the newly created wetting film and gas-liquid meniscus. The observation results also show multiple gas snap-off phenomena, as shown in Figure 8a,b. At the same time, a Haines jump occurred downstream, which means that isolated bubbles tend to coalesce to form a large amount of gas fluid [30,31]. The main difference between nitrogen gas and carbon dioxide gas experiments lies in the gas snap-off at the second pore throat and the size of the residual bubble. The results for carbon dioxide show there is another gas snap-off in the middle size pore throat, as shown in Figure 8b(viii,ix), while the nitrogen gas fluid sustains its connectivity when passing through the first (R_g,1 = 25 μm) and second (R_g,2 = 15 μm) pore throat. The results for carbon dioxide gas also show that the isolated bubbles formed by the upstream separated gas fluid maintain a larger volume. However, the newly formed bubble in the nitrogen gas stream has shrunk in size. Moreover, this bubble formation tendency came with an opposite result in the final residual gas bubble: the carbon dioxide gas fluid has a smaller gas residual (Figure 8b(xiv)) compared with the nitrogen gas fluid (Figure 8a(vii)) in the pore throat.

3.2.2. Bubble Formation under the Surface-Dominate Condition

This section emphasizes the properties of gas bubble size, the ratio between bubble weight and length, and optimum gas bubble residual conditions for further snap-off influence discussion. A comparison chart is shown in Figure 9. The curves represent the ratio between the width and length of the gas bubble downstream, and the symbol x represents the gas bubble surface area after the measurement by a program IC Measure. The program IC Measure allows the user to measure length, area, and angle directly from the inserted pictures.

The red curve (CO₂) in Figure 9 has a lower width-length ratio at the beginning, which shows that carbon dioxide gas bubbles have better sphericity than nitrogen gas bubbles, as they have less width and height ratio since the perfect sphericity is achieved when the width-length ratio is equal to 1. This observation magnifies the difference in surface tension between carbon dioxide and nitrogen gas in liquid water. Since Roof’s criterion expresses the pressure imbalance as a derivation of the bubble front curvature and pore throat radius with specific assumptions, the snap-off process is reflected in the downstream bubble size. The size of the bubbles in the capillary depends on the surface tension between the gas phase and the liquid phase, so smaller carbon dioxide bubbles can visually demonstrate the minor surface tension.

Another point of interest is the convergence of two curves. From Figure 9, the two curves (CO₂ and N₂) gradually decrease as the gas phase fluid passes through the pore throat (represented by the value increment of λ_D) and then converges around λ_D = 0.8. The values of width/height for carbon dioxide and nitrogen gas in the subsequent λ_D range are also highly coincident (floating around 1). At the same time, the bubble volumes generated downstream of the two gases are also very similar in this range (λ_D from 0.8 to 1.4). This result inspires a conjecture bound on the volume of the residual upstream bubble from the perfect sphericity bubble formation after snap-off. Adding the bubble areas after the first convergence (values after λ_D,CO2 = 0.78 and λ_D,N2 = 0.76) makes the calculation of the total size of the upstream residual bubbles approximately 59,911.6 μm² for nitrogen and 56,063.5 μm² for carbon dioxide with the absolute difference percentage being around 6.42%. Considering the thickness of the microchip model is 2 mm, the total volume calculated for nitrogen gas is 119.8 mm², and for carbon dioxide gas, 112.1 mm².

Figure 10 demonstrates that the convergence is more obvious as both upstream residual gas volume and downstream gas bubble volume have less difference between nitrogen and carbon dioxide gas. When the value of λ_D climbs to 0.8, the residual gas volume drops below 150 mm³, and the gas bubble volume shrinks to less than 30 mm³. As a result, the gas snap-off condition for carbon dioxide gas fluid tends to approach nitrogen gas fluid. On the other hand, when the detected residual gas volume is more than 150 mm³, the carbon dioxide gas snap-off significantly differs from nitrogen. The newly formed downstream gas bubble will form into a smaller ellipse than the nitrogen gas fluid.

Furthermore, the symbol x, which represents the CO₂ gas bubble volume, shows two significant numerical fluctuations: the first at λ_D = 0.2 and the second at λ_D = 0.6. By reviewing Figure 10, this fluctuation shows up with the secondary snap-off that happens in the middle pore throat (R_g,2 = 15 μm). Whenever carbon dioxide gas loses its continuity in the second pore throat, the gas bubble generated under the third pore throat (R_g,3 = 7.5 μm) will shrink suddenly. The overall results show that replacing nitrogen in the two-phase fluid with carbon dioxide gas produces a higher frequency of snap-offs while simultaneously generating smaller bubbles downstream. Furthermore, based on relatively weaker interfacial tension, the CO₂ gas is discharged more efficiently with the liquid phase fluid, leaving a smaller volume of residual trapped gas. This observation of micro-dispersion supports the advantage of using carbon dioxide gas in two-phase fluid, as it causes a lower possibility of gas-phase trapping.

3.2.3. Numerical Simulation Result for Continuously Tapering Pore Throat

The simulated interface radius for each gas group is presented in Figure 11, where the different color of lines indicates the liquid-gas interface radius (κ_f) in each pore throat. The reference time for dimensionless factor τ is the starting time step when the interface radius κ starts changing in the third pore throat.

The relative contrast curves of the two experimental groups and the liquid-gas interface descending curves in Figure 11 are very similar for each pore throat. When the interface radius value (κ_f) from each pore throat in these two groups is compared, the nitrogen group’s lines are closer together, while the carbon dioxide group’s lines have a steeper decreasing slope. This change is consistent with the results observed by real-time visualization (Figure 10). The gas snap-off of the carbon dioxide group occurred earlier than that of the nitrogen group. At the same time, the liquid imbibition process of the nitrogen group is more stable than that of the carbon dioxide group, which indicates the gas-liquid interface fluctuation is lower. The corresponding images for the simulation results are shown in Figure A2 in the Appendix A.

3.2.4. Two-Phase Dynamic Fluid Flow: Beresnev and Deng Breakup Criterion

The core concept for the models by Beresnev et al., Deng et al., and Quevedo et al. [9,10,11] is to introduce the radius of the liquid-gas interface, κ, as a temporal dynamic factor to connect the pressure gradient in the wetting and non-wetting phases. After checking the reliability and effectiveness of the experimental results and the Ansys simulation results, the volume flux evolution equations from Quevedo Tiznado et al. [11] are introduced:

$$u_1=\frac{Q_1}{A_{cross\text{-}area}}=\frac{2\pi {{\displaystyle \int }}_0^{\kappa }{\mu }_{1}rdr}{\pi R_g^2}=\frac{-\frac{{\kappa }^4}{8{\mu }_1}\frac{\partial P_1}{\partial x}+\frac{1}{4{\mu }_2}\frac{\partial P_2}{\partial x}\left({\kappa }^4-{\lambda }^2{\kappa }^2\right)+\frac{{\kappa }^4}{2{\mu }_2}\left(\frac{\partial P_2}{\partial x}-\frac{\partial P_1}{\partial x}\right)ln\left(\frac{\lambda }{\kappa }\right)}{R_g^2}$$

(12)

$$u_2=\frac{Q_2}{A_{cross\text{-}area}}=\frac{2\pi {{\displaystyle \int }}_0^{\lambda }{\mu }_{1}rdr}{\pi R_g^2}=\frac{-\frac{\pi }{4{\mu }_2}\left(\frac{{\lambda }^4}{2}-{\lambda }^2{\kappa }^2+\frac{{\kappa }^2}{2}\right)\frac{\partial P_2}{\partial x}+\frac{\pi {\kappa }^4}{2{\mu }_2}\left(\frac{\partial P_2}{\partial x}-\frac{\partial P_1}{\partial x}\right)\left[{\lambda }^2+2{\kappa }^4\left(ln\frac{\kappa }{\lambda }-\frac{1}{2}\right)\right]}{R_g^2}$$

(13)

where u₁ and u₂ are velocities for the wetting and non-wetting phases, respectively. The cross-sectional area (A_cross-area) is selected to be the center of the pore throat for gas snap-off detection, where the liquid-gas interface radius declines significantly. Then the overall fluid velocity is calculated as u = u₁ + u₂. The comparison of mathematical results and Ansys simulation results is shown in Figure 12.

In Figure 12, the absolute average relative difference (AARD) percentage is calculated followed by Equation (14):

$$AARD\%=\frac{1}{N}\left(\frac{{{\displaystyle \sum }}_i^N\left|u_{Mathematical}-u_{Simulation}\right|}{u_{Simulation}}\right)\times 100\%$$

(14)

In the axisymmetric and continuously tapering pore throat, the absolute average relative difference (AARD) percentage for the nitrogen group is 4.70% and 8.34%, respectively, followed by 9.84% and 8.82% for the carbon dioxide group. The tiny difference in percentage proves that the simulation results of continuously narrowing pore throats have a certain degree of reliability.

It is worth noting that when the pore throat radius shrank, the AARD values increased. Further discussion is carried out by extracting simulated data for fluid rate, interface radius, and the related capillary pressure to unearth the source of the disparity. The Beresnev and Deng [9] breakup criterion comes from the condition that P_c^throat > P_c^crest when the non-wetting phase pinches off. The improved formula used to judge the standard is obtained by Quevedo Tiznado et al. [11] through a series of algebraic operations:

$$L>2\pi \sqrt{\left(1-{\delta }^{*}\right)\left(\alpha -{\delta }^{*}\right)} where {\delta }^{*}=0.0412lo{g}_{10}\left(C{a}_{local}\right)+0.1475$$

(15)

Beresnev and Deng’s simulation infers that the pinch-off occurs when the fluid condition satisfies the inequality (13). However, the dynamic flow argument by Quevedo Tiznado et al. indicates that the inequality equation is not accurate in the discussion of the capillary-driven flow. This research uses liquid and gas as wetting and non-wetting phases to test the snap-off criterion. The calculated results are in Figure 13. Define the L = l/R_T = 300 μm/75 μm = 4.0 in the inequality Equation (15). The output criteria had the applied capillary number in the range of 1.537 × 10⁻⁵ ≤ Ca ≤ 0.000184, and the ratio between the radius of the pore throat (R_g,1, R_g,2, and R_g,3) and the pore body was ɑ₁ = 0.481, ɑ₂ = 0.342, and ɑ₃ = 0.215.

The comparison result for gas-liquid fluid is consistent with the conclusion by Quevedo Tiznado et al. that the criterion proposed by Beresnev and Deng for judging whether snap-off occurs in the non-wetting phase is not accurate under dynamic flow conditions. However, while inequality (13) cannot pinpoint the precise time step at which the non-wetting phase snaps off, it can indicate that the matching constriction (pore throat) that snaps off is likely to occur. The first two pore throats (R_g,1 (ɑ₁) and R_g,2 (ɑ₂)) that do not meet the inequality (13) conditions in Figure 13 display no liquid-gas interface fluctuations when paired with real-time imaging resulting in constantly lowering pore throats (Figure 12a). The gas snap-off takes place in the third pore throat (R_g,3 (ɑ₃)), satisfying the inequality requirement. In the same way for the carbon dioxide group, the last two pore throats (R_g,2 and R_g,3) that meet the inequality conditions shown in Figure 10 have gas snap-off (Figure 12b), but the first pore throat does not. The comparison findings show that Beresnev and Deng’s evolution equations can forecast whether the gas phase breaks apart in a two-phase gas-liquid fluid under the specified pore throat constraints. In detail, compared with the original capillary number range from 3 × 10⁻⁴ to 1 × 10⁻², the valid capillary number for predicting the gas snap-off occurrence has a lower limit of 1.537 × 10⁻⁵ ≤ Ca. Meanwhile, the microchannel radius also shows the minimum value of 0.342, which is more specific than the original tested range of 0.20 ≤ ɑ.

4. Conclusions

This research is an innovative and comprehensive study that combines real-time visualization experiments, Ansys Fluid numerical simulation, and mathematical calculation for the first time to investigate the two-phase gas-liquid fluid flow from the microscopic perspective. It explores a numerical method that remedies the infeasible fourth-order nonlinear partial differential equation in Beresnev and Deng’s theoretical prediction. The axisymmetric model was designed to reproduce the theoretical fluid configuration from Beresnev and Deng and Quevedo Tiznado et al. with capillary numbers between 3 × 10⁻⁴ ≤ Ca ≤ 1 × 10⁻². The continuously tapering model was developed from rock samples to illustrate the two-phase fluid flow characteristics with a more comprehensive constriction. The gas snap-off criterion verification shows that it is possible to predict the gas snap-off in the gas-liquid dynamic flow according to the algebraic inequality developed by Beresnev and Deng and Quevedo Tiznado et al. within the applied capillary range of 5.083 × 10⁻⁵ ≤ Ca ≤ 6.99 × 10⁻⁵, and the ratio of microchannel radius (ɑ) between 0.3 ≤ ɑ ≤ 0.5.

This study gives reference to three conclusions on the application of CO₂: (1) both experimental and numerical results show that carbon dioxide gas fluid is more susceptible to snap-off during the wetting phase imbibition process than nitrogen gas fluid; (2) the CO₂ gas fluid forms a giant gas bubble following gas snap-off compared to nitrogen gas fluid and it tends to have less gas residual in micromodels, which has a positive effect on CO₂ drive recovery but a negative effect on carbon dioxide underground storage, and (3) the Beresnev and Deng’s has less accuracy in narrow pore throats.