# The Displacement of the Resident Wetting Fluid by the Invading Wetting Fluid in Porous Media Using Direct Numerical Simulation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}, would increase the surface tension [14], while some miscible pollutants, such as amines, alcohols, and other organic compounds, would reduce the surface tension of water [15].

_{nw}) varied in the stable and unstable states. The results revealed that the stability of trapped air in the porous medium was affected by the consecutive drainage–imbibition cycles. Tavangarrad et al. (2019) [17] evaluated the effect of multiple imbibition–drainage cycles on capillary pressure–saturation curves of thin hydrophilic fibrous layers. The result showed that if a sample was wetted in the first imbibition–drainage cycle, the capillary phenomenon would be more obvious in the next cycles. The discussion mentioned above infers that the drainage–imbibition cycles influence the two immiscible fluid–fluid flows in porous media and the displacement of the resident wetting fluid.

_{1}) by the invading wetting fluid (w

_{2}) in porous media under different conditions were simulated. The effects of the non-wetting fluid (nw), interfacial tension (σ

_{w}

_{1nw}), contact angle (θ), injection rate, and drainage–imbibition cycles on the displacements were evaluated by observing and quantifying the simulation results. Furthermore, the implications and significance of these simulation results for research and applications in the environmental field were discussed in the following sections.

## 2. Methodology

#### 2.1. Navier-Stokes Equations

_{s}denoted velocity, density, viscosity, pressure, and surface force, respectively.

#### 2.2. Volume of Fluid (VOF) Method

_{r}was the relative velocity between two phases.

_{s}) imposed on the interface and changing with volume fraction was expressed as F

_{s}= σk∇α, where σ is the interfacial tension and k is the curvature of the interface. The curvature of the interface k, defined as the divergence of a unit normal vector $\widehat{n}$, was expressed as $\kappa =-\nabla \cdot \widehat{n}=-\nabla \cdot \left(\frac{\nabla \alpha}{\Vert \nabla \alpha \Vert}\right)$. The wettability was expressed by the static contact angle (θ) following the equation $\widehat{\text{}n}={\widehat{n}}_{w}cos\theta +{\widehat{t}}_{w}cos\theta $, where ${\widehat{n}}_{w}$ and ${\widehat{t}}_{w}$ were the unit normal vector to the solid and the unit tangent vector to the solid, respectively [28,32,33].

_{1}) and the invading wetting fluid (w

_{2}), the process was regarded as a solute transport phenomenon and described as the following advection–diffusion equation:

_{1}(or w

_{2}) and D

_{m}is the diffusivity.

#### 2.3. Numerical Domain, Boundary, and Initial Conditions

^{−4}m), 48.39% (N = 64, R = 3.8 × 10

^{−4}m), 49.73% (N = 25, R = 6 × 10

^{−4}m), and 65.09% (N = 25, R = 5 × 10

^{−4}m), respectively.

_{1}by w

_{2}in porous media with or without nw were implemented. All the properties for simulations are listed in Table 1.

_{1}. As for the condition without nw, w

_{2}was directly injected into the system at a constant rate (0.05 m/s). As for the condition with nw, nw was first injected into the system at a constant rate (0.05 m/s). After dynamic balance, w

_{2}was injected into the system at a constant rate (0.05 m/s), and monitoring of the displacement started.

_{w}

_{1nw}) and contact angle (θ) (Section 3.2), three micromodels (Micromodel 1, Micromodel 2, and Micromodel 3) with little difference in the spatial distribution of the grain were adopted to avoid an extremely abnormal result causing misinterpretation. All the simulations were assumed under the condition with nw. At first, nw was injected into the system, which was assumed to be initially filled with w

_{1}, at a constant rate (0.05 m/s). After dynamic balance, w

_{2}was injected into the system at a constant rate (0.05 m/s), and monitoring of the displacement started. During the injection process of nw, all the parameters for simulation were the same as in Table 1 to ensure that the initial conditions for all cases were identical. When w

_{2}started entering the system, the parameters, interfacial tension (σ

_{w}

_{1nw}), or contact angle (θ) were changed for comparison.

_{2}for comparison under the condition with nw. At first, nw was injected into the system, which was assumed to be initially filled with w

_{1}, at a constant rate (0.05 m/s). After dynamic balance, w

_{2}was injected into the system at a constant rate and monitoring of the displacement started. As for the high injection rate condition, the rate was 0.05 m/s for 0.2 s, while for the low injection rate condition, the rate was 0.01 m/s for 1 s.

_{2}into the system. Initially, the system was assumed to be filled with w

_{1}. Then nw, w

_{2}, nw, and w

_{2}were sequentially injected into the system at a constant rate (0.05 m/s) for 0.15 s.

#### 2.4. Quantification of Fluid Saturations

_{w}

_{1}, α

_{w}

_{2}, and α

_{nw}, were recorded. To further investigate the displacement efficiency, the fluid saturations were quantified using the data extracted from the simulation results at each time step. The fluid saturations (S) were calculated as follows:

_{w}

_{1}, S

_{w}

_{2}, and S

_{nw}, mentioned in this study were quantified using the central area (the quadrilateral region was formed by the four centers of circles in four corners, such as the yellow quadrilateral in Figure 1). In addition, the normalized concentration (C/C

_{0}) represented in Figure 2 was defined as the following equation:

## 3. Results and Discussions

#### 3.1. Effect of the Non-Wetting Phase (nw)

_{1}by w

_{2}in porous media under the condition without nw to the condition with nw. From Figure 1a, it can be observed that w

_{1}in the porous system (Micromodel 1) without nw gradually mixed with w

_{2}and was displaced. On the other hand, Figure 1b shows that when nw existed in the porous system, nw would hinder the displacement of w

_{1}. Even though nw was partially flushed out of the system, nw remained in the system and hindered the displacement of w

_{1}either by trapping it, such as the red circle 1 region in Figure 1b (C/C

_{0}remained constant in Figure 2), or by limiting the interaction of w

_{2}with w

_{1}, such as in Figure 1b shown by the red circle 2 region. In the red circle 1 region, w

_{1}was trapped by nw and could not contact w

_{2}. In the red circle 2 region, nw occupied the main flow path of the displacement, thus impeding the mixing of w

_{1}and w

_{2}. This result indicated that the remaining nw in the porous system would affect the displacement of w

_{1}by w

_{2}.

_{1}should be totally displaced by w

_{2}after a long period of flushing. However, based on the column experiments implemented by Gouet-Kaplan et al. (2012) [13], the old water was still released from the unsaturated porous media over a long period of flushing by the new water. The phenomenon can be interpreted from pore scale observation and quantification based on simulation results in this study. Figure 2 shows the normalized concentration (C/C

_{0}) of w

_{1}in Figure 1b red circle regions versus time. In Figure 2, the time span was from t = 0.1 s to t = 0.2 s in Figure 1b, where t = 0.1 s represented the start of monitoring and t = 0.2 s represented the end of monitoring. In the red circle 1 region, C/C

_{0}remained constant, which meant that w

_{1}was totally trapped by nw. In the red circle 2 region, C/C

_{0}decreased to 85.8% (|slope| = 4.73) from t = 0.1 s to t = 0.13 s, decreased to 76.0% (|slope| = 3.25) from t = 0.13 s to t = 0.16 s, decreased to 70.7% (|slope| = 2.68) from t = 0.16 s to t = 0.18 s, and decreased to 68.1% (|slope| = 1.30) from t = 0.18 s to t = 0.2 s. The more and more slow decrease in C/C

_{0}implied that w

_{1}would be continuously released even after a long period of flushing. In the red circle 3 region, C/C

_{0}significantly decreased to 34.2% from t = 0.1 s to t = 0.15 s and continuously decreased to 11.8% from t = 0.15 s to t = 0.2 s. This indicated that w

_{1}would be gradually displaced without the interference of nw. The results of Figure 2 illustrated that when nw occupied the main path for displacement, w

_{1}nearby nw would not be easily displaced completely and only be partially released very slowly.

_{1}, more nw pockets indicated more opportunity to cause the phenomenon of slow release of w

_{1}mentioned above. Table 2 summarizes the displacement results of all six micromodels under the conditions with nw. Micromodel 1, Micromodel 2, and Micromodel 3 (the three micromodels were slightly different in the grain distribution) had different remaining nw pockets and distributions, which resulted in very different displacement and mixing of w

_{1}. The slight differences in geometry could result in significant displacement results when nw is involved.

_{c}is the capillary pressure, r is the mean curvature, σ is the interfacial tension, and θ is the contact angle). Therefore, the snap-off phenomenon occurs more easily in porous media with narrower pore channels due to the larger interfacial instability between two immiscible fluids, leading to more nw pockets trapped in the pores. For the micromodels (Micromodel 1, Micromodel 4, and Micromodel 5) with similar porosity and different average pore throats, the number of remaining nw pockets increased with decreasing the average pore throat. Micromodel 4, with the smallest average pore throat (1.66 × 10

^{−4}m), had the maximum number of the remaining nw pockets, which implied that it was the most difficult to completely displace w

_{1}in Micromodel 4. As for Micromodel 6, whose porosity (65.09%) is much larger than the other five micromodels (49.73% or 48.39%), the number of the remaining nw pockets was basically smaller than others, except Micromodel 5.

#### 3.2. Effects of Interfacial Tension (σ_{w1nw}) and Contact Angle (θ)

_{w}

_{1nw}) between w

_{1}and nw and images of displacement results at t = 0.2 s. All the properties of fluids for simulation were the same as in Table 1 and σ

_{w}

_{1nw}decreased to half of the original value for comparison when w

_{2}started entering the system. When the properties listed in Table 1 were applied to the simulations of Micromodel 1, Micromodel 2, and Micromodel 3, in which w

_{1}and w

_{2}have the same density (ρ), viscosity (ν), and interfacial tension (σ), the S

_{w}

_{1}of Micromodel 1, Micromodel 2, and Micromodel 3 were 4.30%, 6.67%, and 6.85% at t = 0.2 s, respectively. All three cases had a similar phenomenon in that some w

_{1}were trapped by nw and some nw occupied the main flow paths, resulting in the incomplete mixing zones of w

_{1}and w

_{2}(Figure 3b,e,h).

_{w}

_{1nw}decreased to 0.03535 kg/s

^{2}(half of the original value), S

_{w}

_{1}of Micromodel 1 and Micromodel 2 decreased to 1.88% and 2.48% at t = 0.2 s, but that of Micromodel 3 increased to 9.10%, even higher than the original simulation result (6.85%). Based on Figure 3c,f,i, it could be found that, unlike the original simulation results, most of nw were flushed out and could not interfere with the displacements of w

_{1}. However, when nw remained in the system, the displacement of w

_{1}was still significantly hindered by nw.

_{2}started entering the system. Figure 4a,d,g shows that when the contact angle (θ) was adjusted from 45° to 30°, the displacement results were different with no rules. For Micromodel 1, S

_{nw}slightly increased from 7.13% to 9.67%, but S

_{w}

_{1}dramatically increased from 4.30% to 16.24%. For Micromodel 2, S

_{nw}decreased from 7.42% to 5.32%, and S

_{w}

_{1}decreased from 6.67% to 5.34%. For Micromodel 3, S

_{nw}increased from 11.95% to 12.17%, and S

_{w}

_{1}increased from 6.85% to 8.35%. Based on Figure 4b,c,e,f,h,i, it could be found that a change in the contact angle (θ) affected the distribution of the remaining nw in the porous system. This result implied that the contact angle (θ) influenced the immiscible fluids flow pattern, thus causing a different displacement efficiency of w

_{1}.

#### 3.3. Effect of Injection Rate

_{nw}under a low injection rate condition (0.01 m/s) is much larger than under a high injection rate condition (0.05 m/s) because a lower driving force (Capillary number C

_{a}= 1.41 × 10

^{−4}) could not effectively flush out nw. In addition, w

_{1}was not efficiently displaced by w

_{2}under low injection rate conditions. This result implied that low injection rates might cause more of nw to remain in the porous medium, which could hinder the displacement of w

_{1}.

_{2}displaced w

_{1}through contact areas, nw did block some routes (longitudinal direction) and prevent w

_{2}from mixing with w

_{1}. When there was more nw remaining in the porous medium, the displacement of w

_{1}was easier interfered with by the remaining nw. This result demonstrated that a very low injection rate (0.01 m/s) could not effectively displace w

_{1}by w

_{2}in porous media with nw.

#### 3.4. Effect of Drainage–Imbibition Cycles

_{1}, and then two cycles of nw followed by w

_{2}were injected into the system. Based on Figure 6, it could be significantly observed that the S

_{w}

_{1}was lower after the second cycle than after the first cycle in all six micromodels (one cycle represented one drainage–imbibition process). This result inferred that the drainage–imbibition cycles could improve the displacement of w

_{1}. For instance, the S

_{w}

_{1}decreased from 6.84% (first cycle) to 1.46% (second cycle) in Micromodel 1. However, it could be observed that the S

_{nw}, the crucial factor affecting the displacement of w

_{1}, decreased in Micromodel 3 (from 11.80% to 7.45%), Micromodel 4 (from 15.60% to 13.42%), and Micromodel 6 (from 13.26% to 10.57%), but increased in Micromodel 1 (from 7.01% to 12.03%), Micromodel 2 (from 7.51% to 14.28%), and Micromodel 5 (from 5.54% to 9.50%).

_{1}originally trapped or limited by nw after the first cycle re-contacted w

_{2}and were displaced by them during the second cycle. Micromodel 1, Micromodel 2, and Micromodel 3 had the same porosity and similar average pore throats. Although the hindrance patterns of nw on the displacement of Micromodel 1, Micromodel 2, and Micromodel 3 were different, causing significantly different S

_{w}

_{1}after the first cycle, when the regions where nw hindered the displacements of w

_{1}were broken during the second cycle process, the S

_{w}

_{1}of the three micromodels were similar after the second cycle (Figure 6a–c).

_{1}formed during the first cycle would be broken during the second cycle. In Micromodel 4, the one originally with the maximum number of the remaining nw pockets due to its smallest average pore throat, the region where w

_{1}was trapped by nw (the upper middle area of the micromodel) still existed after the second cycle (Figure 6d). Even in a micromodel with a larger average pore throat, it did not mean that nw, which hindered the displacement, would be absolutely removed. In Micromodel 5, the one with similar porosity to Micromodel 1 and Micromodel 4 but with a larger average pore throat, the slightly upper right area in the middle of the micromodel continuously affected the displacement of w

_{1}after the second cycle (Figure 6e) although the S

_{w}

_{1}significantly decreased during the second cycle process.

_{w}

_{1}decreased from 12.96% (first cycle) to 3.19% (second cycle). The decrease in S

_{w}

_{1}was more obvious than those of other micromodels with lower porosity. This was because when one nw pocket, which originally hindered the displacement of w

_{1}after the first cycle, was removed after the second cycle, a large amount of w

_{1}was rapidly displaced by w

_{2}. This result implied that the effect of the drainage–imbibition cycle on the displacement of w

_{1}was more significant in micromodels with larger porosity.

#### 3.5. Environmental Significance

_{1}by w

_{2}in the porous media and successfully investigated the potential mechanism. Furthermore, these simulation findings not only provided new directions for academic research but also contributed to a more comprehensive perspective and a deeper understanding of engineering applications.

_{1}. Since this cycle occurs repetitively naturally in the environment, this finding carries not only theoretical implications but also practical significance in guiding future soil remediation strategies.

## 4. Conclusions

_{1}by w

_{2}in porous media using DNS. A series of displacement simulations were carried out under different conditions. Using observation and quantification based on simulation results, the main conclusions can be summarized as follows:

- (1)
- When nw existed in the porous system, the displacement of w
_{1}by w_{2}would be impeded. By calculating (C/C_{0}) of w_{1}in the regions hindered by nw, it could be observed that w_{1}was displaced very slowly. This result helped explain the “slow-release phenomenon of old water” in previous column experiments. - (2)
- When σ
_{w}_{1nw}decreased to half of the original value, the S_{w}_{1}would decrease because most of nw was flushed out. A change in contact angle (θ) caused a different distribution of nw in the system, which could result in a different displacement efficiency of w_{1}. - (3)
- At a very low injection rate = 0.01 m/s, w
_{2}could not effectively displace w_{1}in porous media because of the remaining nw. - (4)
- The drainage–imbibition cycles could improve the displacement of w
_{1}in porous media because the constrained regions caused by nw were broken during consecutive drainage–imbibition cycles. - (5)
- The simulation results have significantly advanced our understanding of future research and applications. In addition, the DNS method authentically described the immiscible fluid–fluid flow in porous media and could be easily applied to study physical mechanisms in other natural or industrial systems.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

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**Figure 1.**The displacement processes of w

_{1}by w

_{2}at a constant rate (0.05 m/s) in porous media (Micromodel 1) (

**a**) without nw; (

**b**) with nw. Red circle 1 in subfigure (

**b**) is the region where w

_{1}was trapped by nw. Red circle 2 in subfigure (

**b**) is the region where nw hindered the mixing of w

_{1}and w

_{2}. Red circle 3 in subfigure (

**b**) is the region where w

_{1}was displaced without being affected by nw.

**Figure 2.**The normalized concentration (C/C

_{0}) of w

_{1}in Figure 1b red circle regions versus time.

**Figure 3.**Effect of interfacial tension (σ

_{w}

_{1nw}) on the displacement of w

_{1}in porous media with nw. (

**a**–

**c**) Micromodel 1; (

**d**–

**f**) Micromodel 2; (

**g**–

**i**) Micromodel 3.

**Figure 4.**Effect of contact angle (θ) on the displacement of w

_{1}in porous media with nw. (

**a**–

**c**) Micromodel 1; (

**d**–

**f**) Micromodel 2; (

**g**–

**i**) Micromodel 3.

**Figure 5.**Effect of injection rate on the displacement of w

_{1}in the porous medium (Micromodel 2) with nw. w

_{2}entered the system at 0.05 m/s for 0.2 s for the high injection rate condition and at 0.01 m/s for 1 s for the low injection rate condition. (

**a**) Fluid saturations; (

**b**,

**c**) images of displacement results.

**Figure 6.**Effect of drainage–imbibition cycles on the displacement of w

_{1}in porous media. (

**a**) Micromodel 1; (

**b**) Micromodel 2; (

**c**) Micromodel 3; (

**d**) Micromodel 4; (

**e**) Micromodel 5; (

**f**) Micromodel 6.

Parameter | Value | Unit |
---|---|---|

density, ρ_{w}_{1} | 1000 | kg/m^{3} |

density, ρ_{w}_{2} | 1000 | kg/m^{3} |

density, ρ_{nw} | 1 | kg/m^{3} |

kinetic viscosity, ν_{w}_{1} | 10^{−6} | m^{2}/s |

kinetic viscosity, ν_{w}_{2} | 10^{−6} | m^{2}/s |

kinetic viscosity, ν_{nw} | 1.48 × 10^{−5} | m^{2}/s |

interfacial tension, σ_{w}_{1nw} | 0.0707 | kg/s^{2} |

interfacial tension, σ_{w}_{2nw} | 0.0707 | kg/s^{2} |

Diffusivity, D_{w}_{1w2} | 3 × 10^{−9} | - |

contact angle, θ | 45 | ° (degree) |

Micromodel | Porosity (%) | Average Pore Radius (m) | Average Pore Throat (m) | Number of the Remaining nw Pockets | Final Image |
---|---|---|---|---|---|

Micromodel 1 | 49.73 | 3.53 × 10^{−4} | 2.39 × 10^{−4} | 4 | |

Micromodel 2 | 49.73 | 3.53 × 10^{−4} | 2.36 × 10^{−4} | 3 | |

Micromodel 3 | 49.73 | 3.53 × 10^{−4} | 2.38 × 10^{−4} | 6 | |

Micromodel 4 | 48.39 | 2.63 × 10^{−4} | 1.66 × 10^{−4} | 12 | |

Micromodel 5 | 49.73 | 4.41 × 10^{−4} | 2.73 × 10^{−4} | 1 | |

Micromodel 6 | 65.09 | 5.24 × 10^{−4} | 4.37 × 10^{−4} | 3 |

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## Share and Cite

**MDPI and ACS Style**

Wang, Y.-L.; Huang, Q.-Z.; Hsu, S.-Y.
The Displacement of the Resident Wetting Fluid by the Invading Wetting Fluid in Porous Media Using Direct Numerical Simulation. *Water* **2023**, *15*, 2636.
https://doi.org/10.3390/w15142636

**AMA Style**

Wang Y-L, Huang Q-Z, Hsu S-Y.
The Displacement of the Resident Wetting Fluid by the Invading Wetting Fluid in Porous Media Using Direct Numerical Simulation. *Water*. 2023; 15(14):2636.
https://doi.org/10.3390/w15142636

**Chicago/Turabian Style**

Wang, Yung-Li, Qun-Zhan Huang, and Shao-Yiu Hsu.
2023. "The Displacement of the Resident Wetting Fluid by the Invading Wetting Fluid in Porous Media Using Direct Numerical Simulation" *Water* 15, no. 14: 2636.
https://doi.org/10.3390/w15142636