2.1

Lattice Boltzmann model

By means of the discrete space-time continuity equation, the evolution equation of the particle distribution function is as follows¹¹:

where f_i(x,t) denotes the distribution function of particles, denotes the discrete particles located at each node of the network that migrates to position x + e_iδt on collision at position x. D2Q9 discrete velocity model is shown in Figure 1.

Figure 1.

D2Q9 discrete velocity model.

The equilibrium distribution function can be expressed as:

2.2

SC multi-component multiphase model

The model mentioned in the previous section is single-component model. However, in nature and in engineering, there are far more multi-component fluids than single-component fluids. The transport processes of multi-component fluids are more complex. Nowadays, SC model is widely used in multicomponent fluid systems. The SC model is described below.

Each type of component in the multi-phase model has its own particle distribution function and evolution equation. The evolution equation of the class σ component in the multi-component multiphase model is shown below¹²:

where σ denotes the different components, σ=1,2. S,S denotes the fraction of groups in a fluid system, and α denotes the velocity of a particle, α =1,2. b. The particle equilibrium state particle distribution function for the type σ component fluid is:

The interaction force between particles of class σ can be written as:

where represents the strength of the interaction between component σ and component , and . The fluid-solid surface force is:

where represents the strength of the action between the fluid and the solid wall. s is used to identify the solid lattice points in porous media.

When gravity is considered in a fluid system, gravity is expressed as:

The total force F_σ(x) on the particle of the component σ is :

3.1

Calculation model and parameter settings

To facilitate the analysis of the influencing factors, a dimensionless quantity is used to describe the effect of different forces on the two-phase transport, that is the capillary number (Ca), which is defined as follows: Ca = vpu / λ.

This article studies the two-phase transport of CO₂/brine in porous media. Considering the randomness and complexity of porous media. A two-dimensional porous medium is constructed by QSGS, and the model of porous medium is shown in Figure 2.

Figure 2.

Schematic diagram of porous media model.

In the schematic diagram of the porous media model, blue is the solid skeleton, and green and red are the pore channels. The left side of the porous medium is the inlet and the right side is the outlet. When t=0, the pore space is filled with CO₂ with velocity u in the region 0<x<20, while in the region x≥20, it is filled with brine with velocity 0.

3.2

Influence of capillary number

Two-phase transport of CO₂/brine in porous media is studied with different capillary number Ca. According to Ca=vpu/γ, it is known that changing the injection rate u of CO₂ can change the magnitude of Ca. The injection rate u of CO₂ increases and changes by 0.001 from 0.001 to 0.004, and the capillary number Ca increases from 0.000513 to 0.00205.

Figure 3 shows the evolution of the two-phase transport of CO₂/brine with time at different Ca. In Figure 3, (a) Ca=0.000513, (b) Ca=0.00102, (c) Ca=0.00154, and (d) Ca=0.00205. When Ca is small, there is a clear phase interface between CO₂/brine, and CO₂ gradually displaces the brine in the porous medium; With the increase of Ca, the two-phase interfaces of CO₂/brine become unstable and the finger-in phenomenon is strengthened.

Figure 3.

Two-phase transport process with different Ca.

In order to examine the efficiency of CO₂/brine replacement, two physical quantities are defined in this article. One is the CO₂ breakthrough time (t), and the other is CO₂ saturation (Sco₂). Figure 4 gives the changes of breakthrough time t and CO₂ saturation Sco₂ with different Ca. It can be seen that as Ca increases, the time for CO₂ to reach the outlet gradually decreases. The capillary number Ca increases from 0.003125 to 0.0156, the breakthrough time t decreases by about 18%-21% in left of Figure 4; As Ca increases, the volume of fluid occupied by CO₂ in the porous medium at the completion of repulsion gradually increases. The capillary number Ca increases from 0.003125 to 0.0156, the CO₂ saturation decreases by about 7%-9% in the right of Figure 4.

Figure 4.

Breakthrough time and CO₂ saturation with different Ca.

3.3

Influence of wettability

Two-phase transport of CO₂/brine in porous media is also studied with different rock wettability. The injection velocity of CO₂ is constant with a size of u=0.001. The simulations are given equal size and opposite signs of the wettability characteristics of CO₂ and brine, that is G_w1=-G_w2. Four sets of values of G_w1 are given as -0.05, -0.03, 0.03, 0.05, the corresponding contact angles θ are about 47.76°, 64.47°, 114.62°, 131.31°.

Figure 5 shows the evolution of the two-phase transport of CO₂/brine with time at different θ. In the Figure 5, (a) θ=47.76°, (b) θ=64.47°, (c) θ =114.62°, and (d) θ =131.31°. From Figure 5, it can be seen that when θ<90°, CO₂ exhibits wettability. CO₂ tends to flow along the solid wall surface. The formation of the finger leading edge of the phase interface is inhibited; As θ increases, CO₂ wettability gradually decreases. When θ >90°, the formation of finger-like leading edge at the phase interface is promoted and the finger-in phenomenon is enhanced.

Figure 5.

Two-phase transport process with different θ.

Figure 6 gives the changes of breakthrough time t and CO₂ saturation Sco₂ with different contact angles θ. As θ increases, the finger-in phenomenon strengthens, and breakthrough time t decreases. The contact angle θ increases from 47.76° to 131.31°, and the breakthrough time t decreases by about 8%-12% in the left of Figure 6; At the end of the repulsion, with the increase of θ, the CO₂ saturation decreases. The contact angle θ increases from 47.76° to 131.31°, the CO₂ saturation decreases by about 10%-13% in the right of Figure 6.

Figure 6.

Repetition efficiency with different θ.