# Three-Dimensional Interaction of Viscous Fingering and Gravitational Segregation in Porous Media

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}sequestration, saltwater aquifer intrusion, and groundwater pollution [20,21,22].

## 2. Materials and Methods

#### 2.1. Experimental Apparatus

^{−10}m

^{2}. In our previous study [23], a thin layer comprising particles with a smaller diameter was used for the wall-contact, to eliminate a highly permeable layer associated with particle sorting on the inner surface of the cylinder. However, in the present study, although the smaller particle layer is missing, the sorting effect was not observed in the experiments.

#### 2.2. Experimental Procedure

_{1}= 1.54 × 10

^{−4}m/s and the CT scan time is 1.2 cm, where Δρ is the density difference between the MVF and LVF, g is the gravitational acceleration, and μ

_{1}is the viscosity of the injected LVF. This implies the onset of convection requires a long time, and the convection velocity may not reach the Darcy velocity.

#### 2.3. Experimental Conditions

^{2}), to that of in a vertical direction d/(Δρgk/μ

_{1}) due to buoyancy [10,11]. It is calculated as

_{p}is the average particle diameter (338 μm), φ is the porosity, and D

_{m}is the diffusivity of glycerol in water (1.0 × 10

^{−10}m

^{2}/s). Viscous fingering for miscible fluids without buoyancy is characterized by the Péclet number and viscosity ratio M of the displaced MVF to the displacing LVF.

#### 2.4. Image Processing

- Rough-cut: the undesirable black background area around the cylinder was eliminated to allow focus on the porous medium in the next step of image processing.
- Adjustment: to eliminate the drift in brightness of the CT images among the scans; the brightness of the CT images was adjusted to ensure a constant CT value for the cylinder wall.
- Noise Removal: a noise filter was applied to remove local bright noise associated with impurities in the particles.
- Subtract: to enhance the finger structure, the “after” scan image was subtracted from the “before” scan image.

## 3. Results and Discussion

#### 3.1. Three-Dimensional Structure

^{−2}(Figure 4d). The other branch of the finger extends directly below the finger with the gravitational tongue, with negligible influence of gravity. The concentration of LVF in the gravitational tongue was obviously higher compared with that in the finger. At high Gs of 1.18 × 10

^{−1}(Figure 4a) and 5.89 × 10

^{−2}(Figure 4b), clear gravitational tongues extend from the inlet to the outlet, but viscous fingering was only observed around the inlet. Since the injected LVF runs mostly along the gravitational tongue, a smaller amount of LVF contributes in the viscous fingering structure growth. The concentration of LVF increases in the gravitational tongue (Figure 4a,b,d) as the position gets closer to the top wall in the gravity direction.

#### 3.2. Concentration Profiles and the Center of Gravity of LVF

^{−1}and 5.89 × 10

^{−2}is almost constant at 0.2.

#### 3.3. Displacement Efficiency

^{−3}and 1.18 × 10

^{−2}, the displacement efficiency is like that without gravity, showing values of 0.36, 0.37, and 0.34, for Péclet numbers of 3.10 × 10

^{1}, 6.20 × 10

^{1}, and 1.24 × 10

^{2}, respectively, despite the difference in the M of 52.1 without gravity and 119 with gravity. In 2D or 3D numerical simulations [10,11], the displacement efficiency, defined at breakthrough, decreases with increasing gravity number. As the displacement efficiency without gravity depends on many factors, including the viscosity ratio, Péclet number, properties of the porous media, etc., the highest displacement efficiency with gravity changes with conditions. The displacement efficiency obtained for a Hele-Shaw cell with miscible fluids also decreases with increasing gravity number [18]. Berg et al. [19] proposed a novel enhanced oil recovery (EOR) scheme by performing flooding experiments with Berea sandstone cores, using less viscous and dense carbon disulfide to displace n-decane and light crude oil [19]. The displacement efficiency decreases with increasing gravity factor because of the gravity underrun of the carbon disulfide. Han et al. [33] performed CO

_{2}EOR experiments with a Berea sandstone plate at immiscible and near-miscible conditions. They attributed the decrease in displacement efficiency with decreasing gravity number to increasing injection velocity causing reduction in the contact time between the fluids. The displacement efficiency overwhelmingly decreases with increasing gravity number, with the highest value achieved without gravity. However, it depends on many factors, including the viscosity ratio and Péclet number.

## 4. Conclusions

- At a low gravity number, viscous fingering like that without gravity occurs showing nonlinear interactions like tip-splitting, shielding, and coalescence.
- At a moderate gravity number, the gravitational tongue and viscous fingering appeared simultaneously, and the concentration of LVF in the gravitational tongue was higher than that in the finger.
- At a high gravity number, a clear gravitational tongue penetrated from the inlet to the outlet, whereas viscous fingering was only observed around the inlet. In the gravitational tongue, the concentration of LVF increases as the position gets closer to the top wall in the gravity direction.
- The averaged concentration near the inlet reduces with an increase in the gravity number and is characterized by a flat concentration profile. The average concentration at the gravitational tongue was almost constant at 0.2.
- The displacement efficiency decreases with the gravity number from the highest value achieved without gravity but depends on many factors, including the viscosity ratio and Péclet number.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Cross-sectional view of a packed bed of particles. Melamine particles with an average diameter of 338 µm were packed in a tube with an inner diameter of 32 mm and over 90 mm in height. The diameter of the inlet nozzle is 1.59 mm (1/16 in). A sintered glass plate placed at the exit serves as a distributor.

**Figure 2.**Calibration curves between the concentration of LVF and X-ray computer tomography (CT) value.

**Figure 3.**Left: Structure of fingering patterns at the LVF injection of 0.3 PV without density difference. Iso-contour surfaces corresponding to a 20 vol.% concentration of LVF were visible Center: Concentration of LVF in a vertical cross-section through the axis of the cylinder. Right: Concentration of LVF in a vertical cross-section at the center, between the inlet and the outlet.

**Figure 4.**Left: Structure of fingering patterns at the LVF injection of 0.3 pore volume (PV) with density difference. Iso-contour surfaces corresponding to a 20 vol.% concentration of LVF are visible. Center: Concentration of LVF in a vertical cross-section through the axis of the cylinder. Right: Concentration of LVF in a vertical cross-section at the center between the inlet and the outlet.

**Figure 5.**Concentration profiles in a flow direction (

**a**) without gravity for three different Péclet numbers and (

**b**) with gravity at various gravity numbers. The distance from the inlet in a flow direction is represented on the horizontal axis.

**Figure 7.**Displacement efficiency vs. the gravity number. Tchelepi and Orr [10], recovery at breakthrough obtained by two-dimensional and three-dimensional numerical simulations Ruith and Meiburg [11], recovery at the breakthrough obtained by two-dimensional numerical simulations. R is the parameter relates the viscosity and concentration. Jiao and Maxworthy [18], miscible flow in a Hele-Shaw cell. Berg et al. [19], n-decane and light crude oil displaced by carbon disulfide in Berea sandstone core at one PV injection. Han et al. [33], n-decane displaced by CO

_{2}at immiscible and near-miscible condition in a Berea sandstone plate.

Fluids | Composition | Density ρ (kg/m^{3}) | Viscosity ^{1} μ (mPa·s) |
---|---|---|---|

M1 | Glycerol solution 85 wt.% glycerol | 1.21 × 10^{3} | 138 |

L1 | NaI + NaCl solution 7.85 wt.% NaI 21.5 wt.% NaCl | 1.21 × 10^{3} | 2.63 |

L2 | NaI solution 10 wt.% NaI | 1.07 × 10^{3} | 1.16 |

^{1}Estimated by the tuning fork vibration viscometer (A and D Co., Tokyo, Japan, SV-10H) t 23.2 °C.

**Table 2.**Experimental conditions for various sets of more viscous fluids (MVFs) and less viscous fluids (LVFs) tested.

Fluids | Pe | Darcy Velocity v (m/s) | Viscosity Ratio M | Density Difference Δρ (kg/m ^{3}) | G |
---|---|---|---|---|---|

M1-L1 | $3.10\times {10}^{1}$ | $4.68\times {10}^{-6}$ | 52.5 | 0 | 0 |

$6.20\times {10}^{1}$ | $9.35\times {10}^{-6}$ | ||||

$1.24\times {10}^{2}$ | $1.87\times {10}^{-5}$ | ||||

M1-L2 | $3.10\times {10}^{1}$ | $4.68\times {10}^{-6}$ | 119 | 140 | $1.18\times {10}^{-1}$ |

$6.20\times {10}^{1}$ | $9.35\times {10}^{-6}$ | $5.89\times {10}^{-2}$ | |||

$1.24\times {10}^{2}$ | $1.87\times {10}^{-5}$ | $2.95\times {10}^{-2}$ | |||

$1.86\times {10}^{2}$ | $2.81\times {10}^{-5}$ | $1.96\times {10}^{-2}$ | |||

$3.10\times {10}^{2}$ | $4.68\times {10}^{-5}$ | $1.18\times {10}^{-2}$ | |||

$6.20\times {10}^{2}$ | $9.35\times {10}^{-5}$ | $5.89\times {10}^{-3}$ |

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**MDPI and ACS Style**

Suekane, T.; Koe, T.; Barbancho, P.M.
Three-Dimensional Interaction of Viscous Fingering and Gravitational Segregation in Porous Media. *Fluids* **2019**, *4*, 130.
https://doi.org/10.3390/fluids4030130

**AMA Style**

Suekane T, Koe T, Barbancho PM.
Three-Dimensional Interaction of Viscous Fingering and Gravitational Segregation in Porous Media. *Fluids*. 2019; 4(3):130.
https://doi.org/10.3390/fluids4030130

**Chicago/Turabian Style**

Suekane, Tetsuya, Tomotaka Koe, and Pablo Marin Barbancho.
2019. "Three-Dimensional Interaction of Viscous Fingering and Gravitational Segregation in Porous Media" *Fluids* 4, no. 3: 130.
https://doi.org/10.3390/fluids4030130