Study on two-phase displacement law considering mass transfer and diffusion

The influence of channeling on viscous fingering instability of CO2 miscible displacement is studied in the paper. Due to the viscous fingering not easily observed in porous media, channeling is used to simplify the viscous fingering instability. We adopted nonlinear simulation to investigate the development of viscous fingering instability during the displacement of Newtonian fluids in a channel by miscible fluids, and the influence of different Pe and different viscosity ratio R was studied. Under homogeneous conditions, when R is the same, the larger the Pe is, the more obvious the convection in the process of CO2 displacement is, the earlier the viscous fingering occurs, and the shorter the time for the finger to break through to the right boundary is. When Pe is the same, the larger the R is, the more unstable the contact area of miscible displacement becomes. More finger structures appear, and more complex fingering phenomena occur. The larger the R is, the earlier the fingering phenomenon appears, and the earlier it breaks through to the right boundary. In addition, we studied the change in Relative Mixing Length (RML) during the diffusion process quantitatively. Finally, our investigation delved into the impact of heterogeneity on viscous fingering. We observed that under significant heterogeneity, viscous fingering tends to manifest preferentially in the direction of increasing permeability.


Introduction
Within porous media, the displacement of fluid flow poses a significant challenge in the realms of fluid mechanics and chemical engineering.This challenge has spurred researchers to explore and analyze the matter through both theoretical frameworks and experimental investigations.In recent years, many researchers have carried out relevant research and practice on CO2 displacement and achieved certain economic and environmental benefits.CO2 displacement can not only improve the efficiency of oil and gas development but also reduce the pollution caused by large amounts of CO2 emissions.Therefore, the study of CO2 displacement law has great economic value and environmental benefits.When employing CO2 as a solvent for oil displacement, the miscible nature of CO2 with crude oil becomes pivotal.Molecular diffusion plays a key role as CO2 dissolves into the crude oil, leading to the extraction of some or all the oil.Therefore, it is very important to study the diffusion mechanism.The main diffusion modes include molecular diffusion, convection-diffusion, and viscosity differential diffusion.Molecular diffusion is a diffusion mode based on the motion of molecules of CO2 and crude oil mixture.Convective diffusion is based on the real velocity difference of liquid particles in the pores of the reservoir and is related to the average velocity of the interfacial movement of the mixture.The viscosity differential diffusion is related to the viscosity difference between CO2 and oil and is also influenced by molecular diffusion and convective diffusion.
Chen and Meiburg [1] illustrated the resonance between heterogeneity and mobility-driven instability at inherent length by comparing the front thickness and the relative length of porous media at a constant Peclet number.DeWit and Homsy [2][3] undertook nonlinear simulations and linear stability analyses of miscible displacement within a medium featuring spatially periodic heterogeneity.The aim was to comprehend the interaction mechanism between viscous fingering and heterogeneity.The outcomes of their linear stability analysis validated the resonance phenomena observed in nonlinear simulations.Meng and Guo [4] employed the Lattice Boltzmann Method (LBM) to simulate the viscous fingering phenomenon at a large Pe.The simulation results confirmed that the viscosity ratio exerted a more significant influence on viscous fingering, with the Peclet number determining the number of fingers.Wang et al. [5] proposed a new Euleran-Lagrange local adjoint method and mixed finite element method to solve complex fluid flow problems in porous media.The nonlinear convection-diffusion equation of concentration is solved by the finite element method, and the pressure equation for calculating pressure and Darcy velocity is given.Saghir et al. [6] studied dual convection diffusion based on the finite element method, which is the first numerical simulation study of non-isothermal miscible displacement in the world, but they failed to reveal any difference in flow structure under isothermal and non-isothermal conditions.Islam and Azaiez [7] successfully carried out a nonlinear simulation of the thermal viscous fingering phenomenon based on the spectral method, and the simulation results showed that the viscous fingering instability on either of the two fronts would cause disturbance on the other and affect the fingering instability.Mishra et al. [8] studied the double convective diffusion in the miscible displacement process, analyzed its stability, and studied the change of viscosity under the logarithmic change of the two parameters of solute diffusion coefficient and thermal diffusion coefficient.Mejia et al. [9] explored two-phase displacements, where the aqueous phase displaces the oleic phase, under favorable mobility ratios that are typically anticipated to be stable.Their investigation revealed that the introduction of lowviscosity irreducible water facilitates the development of viscous instabilities.Yang et al. [10] developed a pore network model to simulate non-Newtonian two-phase flow.Their results indicated that a stronger shear-thinning behavior in the Ellis fluid makes it easier for the injected fluid to invade smaller-radius pipes.This effect suppresses viscous fingering and enhances displacement efficiency.Zhao and Mohanty [11] explored the influence of wettability on viscous fingering and observed unstable displacements by using 2D micromodels.Norouzi et al. [12] utilized an Arrhenius equation of state to describe the viscosity's temperature dependency, examining the impacts of viscous dissipation on thermal viscous fingering instability.Singh et al. [13] reviewed theoretical and experimental works related to viscous fingering in radial Hele-Shaw cells.Devkare et al. [14] introduced anisotropies on the plate surface, such as pits or holes, to control the formation of viscous fingers, explaining the phenomenon through Saffman-Taylor instability.Jangir et al. [15] focused on the dynamics of viscous fingering in a miscible flow displacement, considering an exothermic reaction between the displacing and displaced fluids that results in the production of nanoparticles.
This paper is based on COMSOL to simulate the effect of mass transfer and diffusion on CO2 flooding.Based on Darcy's law coupled with the convection-diffusion equation, the oil displacement effect of CO2 in porous media was studied.Considering the difference in viscous fingering phenomenon of different Pe and viscosity ratio R, the variation of RML is quantitatively analyzed.In addition, we also discuss the influence of the parameters  in the permeability equation.

Governing equations
The preservation of mass is articulated through the continuity equation, while Darcy's law serves as the momentum equation.Additionally, the convection-dispersion equation is employed to describe the concentration of the displacing fluid.
where u is velocity, p is pressure, D is dispersion coefficient tensor, k is the permeability of the medium, and c is the concentration of the displacing fluid.
Many experimental and numerical studies on the channel effect have shown that the exponential change permeability equation can well simulate the channel effect under the condition of a no-slip wall.Therefore, in addressing the channeling phenomenon, the assumption is made, which exists a confined channel with varying permeability adjacent to the wall.The permeability of the medium is then defined by means of an exponential function.
) where 0 k is the permeability coefficient; and  are constants.The relationship between the maximum and minimum permeability adjusts with changes in  and  .

Dimensionless analysis
The Peclet number provides a physical insight into the balance between convection and diffusion during the diffusion process.A higher Peclet number signifies a more substantial contribution of convection to the overall flow dynamics.Mathematically expressed as Equation (5), where u represents the characteristic velocity of the fluid, l is the characteristic size of the flow field, and D is the diffusion coefficient, the Peclet number serves as a dimensionless parameter characterizing the dominance of convective transport relative to diffusion.In practical terms, when the Peclet number is larger, the influence of fluid advection becomes increasingly prominent compared to molecular diffusion in shaping the behavior of the flow.
ul Pe D

=
(5) where the parameter R is associated with the viscosity ratio between the displacing and displaced fluids;

Physical model
Within the COMSOL simulation platform, a porous medium is defined with dimensions: a length (L) of 0.04 m and a width (W) of 0.02 m, as illustrated in Figure 1 (the left side is the pure CO2 zone, the inflow velocity of CO2 is u, the viscosity is 1  , the right side is the pure oil zone, the viscosity of oil is 2  , and the contact area between the two is the miscible-phase zone).Oil with Viscosity 2  is dislodged by injection of CO2 at the rate u, and fluids are considered incompressible and are perfectly miscible.

Results
In this section, we simulated and presented the miscibility of CO2 and crude oil under different Pe and R conditions when =0, focused on the influence of these two parameters on viscous fingering, and quantitatively analyzed the variation of RML.In addition, we also discuss the influence of the parameters  in the permeability equation.
Figure 2 describes the concentration distribution of two-phase displacement at a constant diffusion coefficient of R=3, Pe=500, and =0, and the occurrence of viscous fingering and the subsequent growth of the finger structure is evident.At t=30 s, the two-phase interface begins to appear instability, and at t=50 s, the viscous fingering phenomenon begins to appear obviously and is accompanied by the formation of finger structure.When t=120 s, the number of fingers decreases, indicating that finger merging occurred.After that, there is a longer finger growth stage, and at 200 s the finger breaks through the right boundary.

The influence of the Pe number under homogeneous conditions
The concentration distribution of different Pe at R=3 and =0 is shown in Figure 3.When Pe=300 and t=60 s, no obvious viscous fingering phenomenon occurs.As time progresses, the width of the stable diffusion region within the contact zone of the two components expands, marking the predominant phase of diffusion.When t=150 s, obvious disturbances occur, the contact area becomes unstable, the finger phenomenon becomes more obvious, and then it turns into the convective dominant period.At Pe=800, it is clear that the finger has undergone a series of changes over the miscible displacement time, resulting in some finger merges, which will break out to the right side at t=110 s.
It can be analyzed from Figure 3 that with the increase of Pe, the dominant period of diffusion is shortened, while the corresponding dominant period of convection is advanced.With the increase of Pe, the time of fingering becomes earlier, and the number of fingering appears significantly increased at the beginning.Meanwhile, the length of the fingering structure also increases, the evolution of the fingering structure becomes progressively more intricate, and the time required for fingering to break through the right boundary is shorter and shorter.

The influence of R under homogeneous conditions
The influence of different viscosity ratios on the displacement rule with other parameters unchanged was studied.When Pe=500, the concentration distribution of different viscosity ratios R is shown in Figure 4.It is distinctly noticeable that as R ascends from 1 to 3 (R=3 shown in Figure 2), the contact region of miscible displacement becomes more and more unstable with the increase of R, and a more complex viscous fingering phenomenon occurs.The larger the R is, the earlier the time of the fingering phenomenon appears, that is, the shorter the dominant period of diffusion is.As can be seen from Figure 4, the time of the fingering occurrence decreases from t=120 s when R=1 to t=20 s when R=3.This is because a higher viscosity ratio leads to a larger perturbation.In addition, it can be observed that the larger the R is, the more the fingers appear when the finger phenomenon occurs.At the same time, it can be observed that under different R, the advancing distance of CO2 in crude oil is the same at the same time point, which can also be observed in Figure 3, indicating that changing Pe and viscosity ratio R will complicate the viscous fingering phenomenon, the overall advancing speed of CO2 in crude oil is mainly controlled by the inflow speed of CO2, and Pe and R have little influence on it.It also can be observed from Figure 4 that if the fingers are of similar length, the smaller R will take longer.Figure 4 reveals that when the lengths of the fingers are comparable, a smaller R requires more time.Consequently, the two components undergo prolonged mutual diffusion, leading to the formation of a broader diffusion area.As R continues to rise, the sharpness of the contact area does not increase substantially.Instead, the complexity of the fingers experiences a notable increase.Furthermore, as R increases, the breakthrough time at the downstream boundary of the primary finger decreases.

The influence of  under heterogeneous conditions
Figure 5 shows the concentration distribution for α = 0.1 and α = 0.7.When =0.1 and kmax/kmin=1.03,the model is close to the homogeneous medium.The described scenario aligns with the tendency before the viscous fingering phenomenon.However, when =0.7 and kmax/kmin=3.23,heterogeneity is prominent.The permeability at the upper and lower ends is notably higher than that in the middle section, resulting in a pronounced outburst of fingers at these extremities.

The relative mixing length analysis
As evident from the earlier discussion, the initiation of fingering leads to an expansion of the contact area between the displacing and displaced phases, subsequently enhancing mass transfer in miscible displacement.While a distinct interface to separate the two miscible components is absent in the displacement process, a concentration profile featuring a specific concentration can be designated as the representative interface.Consequently, with knowledge of the concentration field, it becomes possible to compute the length of the interface corresponding to a specific concentration profile.To gauge the intricacy of the viscous fingering phenomenon across different flow scenarios, the Relative Mixing Length is employed.This metric is defined as the ratio of the interface length to the width of the area at the concentration profile corresponding to c=0.1 mol/m 3 .In this investigation, a lower value of the Relative Mixing Length (RML) is advantageous for capturing the intricacies of fingering, particularly in the case of unstable flows characterized by more complex interfaces.Consequently, the RML will be employed to examine the impacts of varying Pe and R.
3.4.1.The influence of Pe.At R=3, Figure 5 illustrates the temporal evolution of Relative Mixing Length (RML) for different Pe.RML escalates with the augmentation of Pe, which signifies an escalation in instability and the evolution of intricate finger structures as Pe increases during the displacement process.When Pe=300, RML always maintains a straight line, indicating that the miscible front remains very stable at this time.However, when Pe increases to 500, 800, and 1, 000, the miscible front becomes more and more complex, and the relative contact area curve becomes steeper and steeper.7 shows the change curve of the relative contact area with time under different R conditions when Pe=500.First of all, RML increases with the increase of R, indicating that the instability of the front increases with the increase of R at the same time.However, it can be found that the change curve of their relative contact area maintains a similar rising type under different R conditions.Combined with the concentration field diagram, it is found that with the increase of R, the appearance of an unstable front and the growth rate of fingers are accelerated to a greater extent, but the complexity of viscous finger-pointing is not greatly affected.

Conclusions
In this study, we adopted nonlinear simulation to investigate the development of viscous fingering instability during the displacement of Newtonian fluids in a channel by miscible fluids.
In nonlinear simulations, when R is the same and =0, the larger the Pe is, the more obvious the convection in the process of CO2 displacement is, the earlier the viscous fingering occurs, and the shorter the time for the finger to break through to the right boundary is.When Pe is the same, the larger the R is, the more unstable the contact area of miscible displacement becomes.More finger structures appear, and more complex fingering phenomena occur.The larger the R is, the earlier the fingering phenomenon appears, and the earlier it breaks through to the right boundary.
At the same time, the change law of Relative Mixing Length (RML) during the diffusion process is quantitatively studied.Observations reveal that a higher Pe and R corresponds to a swifter growth of the RML.It also indicates that the fingering phenomenon becomes more obvious and the miscible interface becomes more unstable when the Pe and viscosity ratio R are higher.
In addition, we found that parameter  in the permeability equation has a great influence on viscous fingering.When the parameter  is larger, the heterogeneity of the model is more obvious, that is, the kmax/kmin is larger, and the viscous fingering phenomenon at the upper and lower ends is much more obvious and earlier than that in the middle section.

1  and 2 
represent the viscosities of the displacing and displaced fluids.