# Effect of Wettability Heterogeneity on Water-Gas Two-Phase Displacement Behavior in a Complex Pore Structure by Phase-Field Model

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Models

#### 2.1. Establishment of Heterogeneous Wetting Pore Models

#### 2.2. Case Studies and Boundary Conditions

^{3}·s/kg), $\lambda $ is the mixing energy density (N), and $\epsilon $ is the interface thickness parameter (m). The variable $\Psi $ is referred to as the phase-field help variable. The compressive Navier–Stokes equations [59] were used to describe the transport of mass and momentum for fluids:

^{3}), $\mu $ denotes the dynamic viscosity (Pa $\xb7$ s), $u$ represents the velocity (m/s), $p$ denotes the pressure (Pa), and ${F}_{st}$ is the surface tension force acting at the two-phase interface [52]. Herein, we performed the simulation of static contact angles for immiscible fluids on a solid surface under different wettability conditions. Figure 4 shows various interfacial configurations of immiscible fluids with contact angles from 15° to 165°. The used phase model has been verified, and more detailed theories and governing equations can be found in our previous studies [53,54].

_{v}= 3 mL/min and 10 mL/min were set as a constant at the inlets, and the gas volume fraction was one at the inlets in the simulations. Table 1 summarizes the basic data and fluid properties used in the simulations. The capillary number Ca (= (μ × v)/σ) and mobility ratio M (=μ

_{gas}/μ

_{w}) were used to determine the various flow conditions, where μ and v are the viscosity and Darcy velocity of the fluid, respectively, and σ (=0.0721 N/m) and θ are the interfacial tension and contact angle of the two-phase fluids, respectively. The mobility ratio is 0.0181 and the capillary numbers are 6.28 × 10

^{−5}and 1.26 × 10

^{−4}for cases A1–A6 and B1–B6, respectively. According to the phase diagram (logCa−logM) for the interfacial stability areas indicated by Zhang et al. [60] and Lenormand et al. [61], these scenarios are expected to fall within the crossover range from a capillary to a viscous fingering regime. Figure 5 depicts the computational domain and the initial and boundary conditions used in the simulation. The model used tetrahedral and adaptive meshes to optimize the two-phase interface to ensure the accuracy of the calculations. The detailed results are discussed in Section 3.

## 3. Results and Discussion

_{ss}to characterize the shape of the flooded channels. The fractal dimension D of all cases at the quasi-steady state was computed using the fractal box counter method in ImageJ software. Referring to previous studies [62,63], the specific surface area A

_{ss}is defined as the area per unit length (m

^{−1}) and is calculated using the following formula:

_{ss}increased with the wettability ratio ${r}_{g-w}$ and then decreased, and the types of wettability heterogeneity slightly affected the specific surface area A

_{ss}. This weak impact was also demonstrated in Foroughi et al.’s work [43] (2021), which indicated that spatial correlation in wettability is not significant in the rock samples.

## 4. Conclusions

- (I)
- The surface fraction of gas-wetting and alternating wettability heterogeneity all present an essential effect on the fluid displacement path and invasion patterns, while the injecting flux rate has less influence in the capillary–viscous crossover flow regime.
- (II)
- Among various surface fractions of gas-wetting, a uniformly water-wet medium prefers to produce a higher gas-phase displacement efficiency.
- (III)
- For heterogeneous porous structures with the uniform gas-wetting distribution, the dalmatian wetting structure has a higher displacement efficiency than the mixed wetting structure during the immiscible two-phase displacement process.
- (IV)
- The complexity of the two-phase fluid distribution, characterized by the fractal dimension and specific surface area, is sensitive to the wettability heterogeneity, while the spatial wettability correlation has a weaker impact.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Model pore network consisting of nine identical substructures in a 3 × 3 array, indicated by the red dashed lines. The white areas are pore spaces, and the black areas are the solids; (

**b**) The distribution of pore radius quantified by the local thickness plugin in ImageJ software; (

**c**) The heterogeneity of pore distribution computed by the self-programming codes in Matlab.

**Figure 3.**Wettability distribution used in the simulations; the blue walls are gas-wet (θ = 120°), and the black walls are water-wet (θ = 65.5°).

**Figure 6.**Gas and water distributions (gas-wet solid in gray, water-wet solid in black, and water in blue and gas in red) in porous structures with homogenous wettability at the quasi-steady state: (

**a**) A1—full water-wet, ${r}_{g-w}$ = 0; (

**b**) A4—full gas-wet, ${r}_{g-w}$ = 1; (

**c**) B1—full water-wet, ${r}_{g-w}$ = 0; (

**d**) B4—full gas-wet, ${r}_{g-w}$ = 1.

**Figure 7.**Gas and water distributions (gas-wet solid in gray, water-wet solid in black, and water in blue and gas in red) in porous structures with mixed wettability at the quasi-steady state: (

**a**) A2—continuous gas-wet, ${r}_{g-w}$ = 1/3; (

**b**) A3—continuous gas-wet, ${r}_{g-w}$ = 2/3; (

**c**) B2—continuous gas-wet, ${r}_{g-w}$ = 1/3; (

**d**) B3—continuous gas-wet, ${r}_{g-w}$ = 2/3.

**Figure 8.**Gas and water distributions (gas-wet solid in gray, water-wet solid in black, and water in blue and gas in red) in porous structures with dalmatian wettability at the quasi-steady state: (

**a**) A5—alternating gas-wet, ${r}_{g-w}$ = 1/3 and (

**b**) A6—alternating gas-wet, ${r}_{g-w}$ = 2/3; (

**c**) B5—alternating gas-wet, ${r}_{g-w}$ = 1/3; (

**d**) B6—alternating gas-wet, ${r}_{g-w}$ = 2/3.

**Figure 9.**Variation of gas saturation as a function of time (

**a**) and the ultimate gas saturation (

**b**) in porous structures with various wetting conditions, where A1/B1 represents the full water-wet case with ${r}_{g-w}$ = 0, A2/B2 for continuous gas-wet with ${r}_{g-w}$ = 1/3; A3/B3 for continuous gas-wet with ${r}_{g-w}$ = 2/3, A4/B4 for the full gas-wet with ${r}_{g-w}$ = 1, A5/B5 for alternating gas-wet with ${r}_{g-w}$ = 1/3, A6/B6 for alternating gas-wet with ${r}_{g-w}$ = 2/3.

**Figure 10.**Temporal variation of the gas front position in the horizontal direction: normalized distance of the gas front to the inlet in six cases with various wetting conditions. Note: time and position variables are all dimensionless.

**Figure 11.**Fractal dimension characterization of immiscible fluid distribution in the quasi-steady state at flux rate 3 mL/min (

**a**) and flux rate 10 mL/min (

**b**), where the horizontal axis represents the log-arithm of box size, and the vertical axis represents the logarithm of the number of counted box-es. The slopes of the fitting lines are the fractal dimensions of flooded channels for twelve cases. R2 refers to the adjusted R-square of the fitting lines. (

**c**) depcits the correlation between fractal dimension of fluid distribution and gas-wet area fraction.

**Figure 13.**The effect of wettability heterogeneity on the relative permeability curve. Simulation results are plotted as scatter points representing the water relative permeability (

**a**) and gas relative permeability (

**b**), respectively.

No. | q_{v}(mL/min) | ${\mathit{r}}_{\mathit{g}-\mathit{w}}$ | Wettability | μ_{w}(×10 ^{−3} Pa·s) | μ_{gas}(×10 ^{−5} Pa·s) | ρ_{w}(kg/m ^{3}) | ρ_{a}(kg/m ^{3}) | θ ( ^{ο}) | σ (×10 ^{−2} N/m) |
---|---|---|---|---|---|---|---|---|---|

A1 | 3 | 0 | Mixed | 1 | 1.81 | 1000 | 1.28 | 65.5/120 | 7.21 |

B1 | 10 | ||||||||

A2 | 3 | 1/3 | |||||||

B2 | 10 | ||||||||

A3 | 3 | 2/3 | |||||||

B3 | 10 | ||||||||

A4 | 3 | 1 | |||||||

B4 | 10 | ||||||||

A5 | 3 | 1/3 | Dalmatian | 1 | 1.81 | 1000 | 1.28 | 65.5/120 | 7.21 |

B5 | 10 | ||||||||

A6 | 3 | 2/3 | |||||||

B6 | 10 |

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**MDPI and ACS Style**

Gong, W.; Liu, J.
Effect of Wettability Heterogeneity on Water-Gas Two-Phase Displacement Behavior in a Complex Pore Structure by Phase-Field Model. *Energies* **2022**, *15*, 7658.
https://doi.org/10.3390/en15207658

**AMA Style**

Gong W, Liu J.
Effect of Wettability Heterogeneity on Water-Gas Two-Phase Displacement Behavior in a Complex Pore Structure by Phase-Field Model. *Energies*. 2022; 15(20):7658.
https://doi.org/10.3390/en15207658

**Chicago/Turabian Style**

Gong, Wenbo, and Jinhui Liu.
2022. "Effect of Wettability Heterogeneity on Water-Gas Two-Phase Displacement Behavior in a Complex Pore Structure by Phase-Field Model" *Energies* 15, no. 20: 7658.
https://doi.org/10.3390/en15207658