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Field evidence indicates that cavities often occur in fractured rocks, especially in a Karst region. Once the immiscible liquid flows into the cavity, the cavity has the immiscible liquid entrapped and results in a low recovery ratio. In this paper, the immiscible liquid transport in cavity-fractures was simulated by Lattice Boltzmann Method (LBM). The interfacial and surface tensions were incorporated by Multicomponent Shan-Chen (MCSC) model. Three various fracture positions were generated to investigate the influence on the irreducible nonwetting phase saturation and displacement time. The influences of fracture aperture and wettability on the immiscible liquid transport were discussed and analyzed. It was found that the cavity resulted in a long displacement time. Increasing the fracture aperture with the corresponding decrease in displacement pressure led to the long displacement time. This consequently decreased the irreducible nonwetting phase saturation. The fracture positions had a significant effect on the displacement time and irreducible saturation. The distribution of the irreducible nonwetting phase was strongly dependent on wettability and fracture position. Furthermore, this study demonstrated that the LBM was very effective in simulating the immiscible two-phase flow in the cavity-fracture.

The problems of immiscible two-phase flows in fractures are frequently encountered in environmental protection and petroleum industries, such as enhanced oil recovery, geological sequestration of carbon dioxide, and polluted groundwater remediation. Research on the influence of fluid properties on immiscible two-phase flow has been ongoing for decades and has shown that fluid properties (e.g., the capillary number, bond number, viscosity ratio, saturation, and wettability) are very important for immiscible two-phase flows [

Past research dealing with immiscible two-phase flow has focused primarily on fluid properties. Joekar-Niasar and Hassanizadeh [

The well-known fractured karst reservoir is composed of many cavity-fractures. Those cavities are much bigger than the fractures that connect with them. For the typical cavity-fracture, the cavity is settled between inlet and outlet fractures (see Figure

Illustration of cavity-fracture.

Modeling immiscible two-phase flow in fractures is a frontier challenge for petroleum resource science. Work by various researchers has presented different numerical methods to couple fluid flow with other phenomenon. Kordilla et al. [

In this paper, the immiscible two-phase flow in cavity-fracture was simulated using lattice Boltzmann method. The interfacial and surface tensions were incorporated by Multicomponent Shan-Chen model. Three various positions of the inlet and outlet fractures (so-called “lower case,” “middle case,” and “upper case”) were generated to investigate the influence on irreducible non-wetting phase saturation and displacement time. The effect of the aperture of the inlet and outlet fractures on the immiscible liquid transport was discussed and analyzed in detail. In addition, by adjusting the interfacial and surface tensions, the wettability of the cavity-fracture wall varied from strongly water-wet to weakly water-wet and then the effect of the wettability was investigated.

The MCSC model is based on the basic lattice Boltzmann equations. In this study, two distribution functions were used to represent wetting phase and nonwetting phase in cavity-fractures, respectively. The lattice Bhatnagar-Gross-Krook (LBGK) model was implemented for the collision term approximation in each evolution equation. The LBGK model is most widely used model due to its simplicity and accuracy in solving the collision term approximation [

In the MCSC model, the effect of total force is incorporated through adding acceleration into the velocity field. In the typical MCSC model, the total force includes the fluid-fluid cohesive force

The fluid-fluid cohesive force is calculated in (

The fluid-solid adhesive force,

In this paper, the contact angle for a water droplet was changed from 0 to 90 degrees, implying the solid surface ranged from strongly water-wet to weakly water-wet. The solid surface was assumed to be hydrophilic. However, if the contact angle is greater than 90 degree, the solid surface becomes hydrophobic.

Many boundary conditions are available in the literatures [

In this work, the algorithms for MCSC model were developed as a C++ code. All the variables are in lattice units such that one lattice unit (

For the immiscible two-phase flows in fractures, the wetting-phase covers and moves along the fracture wall, while the nonwetting phase flows in the center of fracture (see Figure

Schematic of immiscible two-phase flow in a 2D smooth fracture.

To validate the MCSC model, the velocity distributions of the immiscible two-phase flow in a 2D smooth fracture was investigated. A computational domain of ^{2} representing the 2D smooth fracture was used for model validation of the MSCS model implementation. The periodical boundary condition was applied to the outlet and inlet of the fracture. A constant body force ^{2}/ts and 0.1667 lu^{2}/ts, respectively. The contact angle between the nonwetting phase and wetting phase was set to

Assuming the wetting phase flows along the fracture in the region

Comparison of MCSC model results and analytical solution for the velocity distribution at

^{2} grid resolution

^{2} grid resolution

^{2} grid resolution

Moreover, to test the grid independence for MCSC model, three different lattice grid resolutions were generated. For the three different lattice grid resolutions, the body force, kinematic viscosity, contact angle, and boundary conditions were constant. The initial region of the wetting phase was set as ^{2}, ^{2}, and ^{2} lattice grid resolutions. The MCSC model simulation results are very consistent with the analytical solution under the different lattice grid resolutions. It means that the MCSC model solution is not gird sensitive. In this study, the sets of grid were chosen depending on the capacity of computer. In Figure

To investigate the effect of inlet and outlet fractures on the irreducible nonwetting phase saturation and displacement time, the immiscible liquid transport in cavity-fracture with various fracture positions and apertures was simulated by the MCSC model. In this paper, the cavity was assumed to be a simple ^{2} square that previously was occupied by the nonwetting phase. The inlet and outlet fractures had the smooth and paralleled fracture walls and symmetrically distributed on the right and left sides of the cavity, respectively. The cavity aperture was set to 98 lu. For all the simulations, the maximum wetting phase velocity was ^{2}/ts. Thus, the dimensionless Re number was less than 23, indicating that there was a laminar flow in the cavity-fracture [

Firstly, the processes of the wetting phase displacing the nonwetting phase (imbibition process) are simulated with three various positions of the symmetrical inlet and outlet fractures. The so-called “lower case,” “middle case,” and “upper case” for the three various fracture positions can be found in Figures ^{2}.

The displacement process in the cavity-fracture for the lower case (blue domain represents solid phase, red domain represents nonwetting phase, and green domain represents wetting phase).

The irreducible nonwetting phase distributions in cavity-fractures.

Middle case,

Upper case,

As shown in Figure

In Figure

The effect of aperture on the irreducible nonwetting phase saturation is investigated. Other two apertures (7 lu and 9 lu) for the fractures are employed, respectively. Figure

The distributions of the irreducible nonwetting phase for the various fracture positions and apertures with weakly water-wet fracture wall.

Figure

The relationship between the irreducible nonwetting phase saturation and aperture for three various cases.

The wettability plays an important role on the immiscible two-phase flow and is investigated in the following section. By adjusting the surface tension and keeping the interfacial tension constant, the solid surface becomes strongly water-wet (the contact angle is equal to 85°). The domain of the cavity is the same as that in Section

Figure

The distributions of the irreducible nonwetting phase for the various fracture positions and apertures with strongly water-wet fracture wall.

In this paper, a numerical model based on the lattice Boltzmann method has been developed to study the immiscible two-phase flow in the cavity-fracture. The effects of fracture apertures and positions on the irreducible nonwetting phase saturation and displacement time were discussed in detail. By adjusting the interfacial and surface tensions, the immiscible two-phase flow in the cavity-fracture with the strongly water-wet and the weakly water-wet fracture walls was simulated. This study demonstrated that the LBM was very effective in simulating the immiscible two-phase flow in the cavity-fracture.

Although the computational domain of cavity was constant, the cavity resulted in a long displacement time. For the pressure boundary condition on the inlet and outlet fracture, the capillary pressure in cavity-fractures increased with time. The displacement pressure decreased as the fracture aperture increased. This consequently decreased the irreducible nonwetting phase saturation. The fracture positions had a significant effect on the displacement time and irreducible saturation. The distribution of the irreducible nonwetting phase was strongly dependent on wettability and fracture position.

Lattice sound speed

Local aperture in fractures

Fluid particle distribution function

Local equilibrium distribution function

Fluid-fluid cohesive force

Fluid-solid adhesive force

Parameter that controls the strength of fluid-fluid interaction

Parameter that controls the strength between each fluid and a wall

Body force

Contact angle

Pressure difference between the inside and outside of bubbles or droplets

Displacement pressure

Whole fluid pressure

Macroscopic fluid density

Wetting phase density

Nonwetting phase density

Interfacial tension

Indicator function

Wetting phase saturation

Nonwetting phase saturation

Time step

Dimensionless relaxation time

Common averaged velocity of all the fluid components

Change in velocity field

Macroscopic velocity

Actual whole fluid velocity

Nonwetting phase kinematic viscosity

Wetting phase kinematic viscosity

Kinematic viscosity

Weights

Lattice spacing.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The study is financially supported by the National Natural Science Foundation of China (Grant nos. 41402217, 41202177, and 51109139) and the Natural Science Foundation of Jiangsu (Grant nos. BK2011110 and BK2012814). The authors thank the two anonymous reviewers for their suggestions in improving the paper.