# Directional Dependency of Relative Permeability in Vugular Porous Medium: Experiment and Numerical Simulation

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

^{−5}–10

^{−4}m). The vuggy pore space consists of a huge amount of vugs, which have length scales ranging from millimeters to several meters (10

^{−3}–10

^{0}m), as shown in Figure 1. The main difficulty for numerical simulation is the coexistence of Darcy flow in the porous region and free flow in the vug region [9]. As shown by previous researchers, the presence of a free-flow region in the surrounding porous region significantly alters the effective permeability of the media, potentially by orders of magnitude. Another difficulty is the large degree of uncertainty related to the shape and location of the interface between the porous region and the vug region [10]. Due to these difficulties, numerical simulation of fluid flow in carbonate reservoirs has always been a challenging problem.

## 2. Displacement Experiment for Vugular Porous Media

#### 2.1. Physical Model Setup and Fluid Characteristics

^{3}. The porous medium is made with 95% glass beads of diameter between 0.2–0.3 mm mixed with 5% epoxy as the cementing material. After the glass beads and the epoxy mixture was filled into an organic glass coating, the model was placed into an oven at 70 °C for 30 min until the epoxy was completely dried. The wettability of the porous medium is controlled by the epoxy and is slightly oil-wet. The ratio between vug volume and total pore volume (which does not include the volume of the two wellbores) is about 0.43. A manufactured sample of the physical model is shown in Figure 2c.

^{3}and a viscosity of 1 cp. A mixture of industrial white oil and kerosene is chosen as the oil phase, which has a density of 0.84 g/cm

^{3}and a viscosity of 18 cp, respectively. To make a visual difference between the two phases, the oil phase was colored red with Sudan Ⅲ.

#### 2.2. Experiment Setup and Procedure

- (1)
- Fully saturate the model with water, then inject oil from the top of the model until no water is produced.
- (2)
- Fully saturate the injection wellbore with water by opening valves V1 and V2 only.
- (3)
- Inject water into the model at a constant rate of 4.5 mL/min by opening valve V1 and one of the valves V3 or V4. This gives an average flow velocity (Darcy velocity) of 6.48 m/day. As shown in Figure 4, the macroscopic flow direction is across the two sides of the wellbores. In this study, we denote the angle between this flow direction and the horizontal plane as θ. For instance, θ equals to −60° in Figure 4. The injection is continued until the water cut reaches 99%, which is usually achieved after 3–5 PV (pore volume) of water is injected.
- (4)
- During step (3), constantly record phase distribution within the model with a camera, as well as the pressure drop across the model and liquid production.
- (5)
- Change a new sample. Set θ equal to −90° (vertically downward), 0° (horizontal), and 90° (vertically upward), respectively, and repeat step (1)–(4).

#### 2.3. Experiment Results

#### 2.3.1. Case 1: Vertically Upwards (θ = 90°)

#### 2.3.2. Case 2: Horizontal (θ = 0°)

#### 2.3.3. Case 3: Vertically Downwards (θ = −90°)

#### 2.3.4. Discussion

## 3. Numerical Scheme

#### 3.1. Mathematical Model

^{4}times the permeability of the matrix. The above large permeability method is adopted and examined in this study, as the permeability of the vug can be estimated by ${K}_{v}={L}_{y}^{2}/12=8.3\times {10}^{-6}{\mathrm{m}}^{2}$, which is larger than 10

^{4}times the matrix permeability (about 2 × 10

^{−11}m

^{2}).

#### 3.2. Model Discretization

^{4}times the permeability of the porous medium and have a porosity of 1.0. These vug grids also have zero capillary pressure, straight line relative permeability, as shown in Figure 11, and zero residual oil, as well as irreducible water saturation. In addition, the two wellbores are also discretized as n × 1 grids, which have parameters identical to the vug grids. Fluid is injected from the bottom of the injection wellbore and is produced from the top of the production wellbore. The other boundaries within the model are all set as no-flow boundaries. Figure 10 shows the case where the flow direction in the model is horizontal. The model can be rotated in the x-z plane by any degree of θ, where −90° ≤ θ ≤ 90°.

#### 3.3. Model Validation

## 4. Directional Dependent Relative Permeability

#### 4.1. Numerical Simulation

^{5}s; the total injected volume of water is 230 times the pore volume.

#### 4.2. Transmissibility Weighted Upscaling

#### 4.3. Directional Relative Permeability Model

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Outcrop pictures from Tahe carbonate oilfield, China, showing a large amount of isolated vugs of millimeters to decimeters in size embedded in the porous medium.

**Figure 2.**Schematic of the single-vug model. (

**a**) Frontal and (

**b**) side cross-sectional view, and (

**c**) a manufactured sample of the physical model.

**Figure 4.**Definition of the angle θ in this study. The angle is taken between the flow direction in the single-vug model and the horizontal plane and has a range of [−90°, 90°].

**Figure 5.**Experiment results in downward (θ = −90°), horizontal (θ = 0°) and upward (θ = 90°) direction. (

**a**) Volume of cumulative produced oil, (

**b**) water cut of the production well, and (

**c**) pressure drop across the model.

**Figure 9.**The four distinct phases of the displacement process during the displacement experiment in the single-vug model in downward (θ = −90°), horizontal (θ = 0°), and upward (θ = 90°) direction.

**Figure 12.**Upscaled absolute permeability of the single-vug model (the vug diameter is taken as 0.6 times the model length in accordance with the reference) with different values of n compared to the result obtained by Huang et al. [30] based on homogenization theory.

**Figure 13.**Comparison of the result of experiment and numerical simulation for Case 2 (flow direction is horizontal): (

**a**) Pressure drop across the model and (

**b**) volume of cumulative produced oil.

**Figure 14.**Comparison of the oil saturation distributions in the experiment and numerical simulation at four different PVIs for Case 2 in Section 2. Water (transparent) displaces oil (red) from left to right.

**Figure 15.**Numerical simulation results of the single-vug model for five different flow directions: (

**a**) Oil recovery ratio and (

**b**) water-cut of the production well.

**Figure 17.**Upscaled relative permeability of the single-vug model for five different flow directions.

**Figure 18.**Upscaled relative permeability of the single-vug model at four different dimensionless water saturations in a polar coordinated system. Note that if the relative permeability is isotropic, then the points should be located on the same circle.

**Figure 19.**The effect of shape factor A in the directional relative permeability model Equation (18).

**Figure 20.**Comparison of water phase relative permeability of the single-vug model obtained via the upscaling method and predicted by the proposed directional relative permeability model.

**Figure 21.**Comparison of oil phase relative permeability of the single-vug model obtained via the upscaling method and predicted by the proposed directional relative permeability model.

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

K | Absolute permeability | 2.4 × 10^{−11} | m^{2} |

ϕ | Porosity | 0.315 | - |

S_{wi} | Irreducible water saturation | 0.358 | - |

k_{ro}(S_{wi}) | Oil phase relative permeability at irreducible water saturation | 1.0 | - |

S_{or} | Residual oil saturation | 0.15 | - |

k_{rw}(S_{or}) | Water phase relative permeability at residual oil saturation | 0.4 | - |

λ | Shape factor for Brooks-Corey relative permeability model Equations (7)–(9) | 3.0 | - |

P_{D} | Pore entry pressure | 0.6 | KPa |

ρ_{o} | Oil density | 840 | kg/m^{3} |

μ_{o} | Oil viscosity | 18.0 | mpa·s |

ρ_{w} | Water density | 1000 | kg/m^{3} |

μ_{w} | Water viscosity | 1.0 | mpa·s |

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

K | Absolute permeability | 1.0 × 10^{−11} | m^{2} |

ϕ | Porosity | 0.3 | - |

S_{wi} | Irreducible water saturation | 0.2 | - |

k_{ro}(S_{wi}) | Oil phase relative permeability at irreducible water saturation | 1.0 | - |

S_{or} | Residual oil saturation | 0.2 | - |

k_{rw}(S_{or}) | Water phase relative permeability at residual oil saturation | 0.6 | - |

λ | Shape factor for Brooks-Corey relative permeability model Equations (7)–(9) | 3.0 | - |

P_{D} | Pore entry pressure | 0 | KPa |

ρ_{o} | Oil density | 800 | kg/m^{3} |

μ_{o} | Oil viscosity | 5.884 | mpa·s |

ρ_{w} | Water density | 1000 | kg/m^{3} |

μ_{w} | Water viscosity | 1.0 | mpa·s |

Q | Injection rate | 2.0 | ml/min |

Gr | Gravity number | 0.1 | - |

Ca | Capillary number | 0.0 | - |

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**MDPI and ACS Style**

Song, S.; Di, Y.; Guo, W. Directional Dependency of Relative Permeability in Vugular Porous Medium: Experiment and Numerical Simulation. *Energies* **2023**, *16*, 3041.
https://doi.org/10.3390/en16073041

**AMA Style**

Song S, Di Y, Guo W. Directional Dependency of Relative Permeability in Vugular Porous Medium: Experiment and Numerical Simulation. *Energies*. 2023; 16(7):3041.
https://doi.org/10.3390/en16073041

**Chicago/Turabian Style**

Song, Shihan, Yuan Di, and Wanjiang Guo. 2023. "Directional Dependency of Relative Permeability in Vugular Porous Medium: Experiment and Numerical Simulation" *Energies* 16, no. 7: 3041.
https://doi.org/10.3390/en16073041