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

^{1}

^{2}

^{*}

## 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

## References

- Wei, C.; Song, H.; Li, Y.; Zhang, Q.; Song, B.; Wang, J. Production Characteristics with Different Superimposed Modes Using Variogram: A Case Study of a Super-Giant Carbonate Reservoir in the Middle East. Energies
**2017**, 10, 250. [Google Scholar] [CrossRef] [Green Version] - Alzayer, H.; Mehran, S. A New Approach to Simulate Near-Miscible Water-Alternating-Gas Injection for Mixed-Wet Reservoirs. In Proceedings of the SPE Kingdom of Saudi Arabia Annual Technical Symposium and Exhibition, Dammam, Saudi Arabia, 23–26 April 2018. [Google Scholar] [CrossRef]
- Tian, F.; Di, Q.; Jin, Q.; Cheng, F.; Zhang, W.; Lin, L.; Wang, Y.; Yang, D.; Niu, C.; Li, Y. Multiscale geological-geophysical characterization of the epigenic origin and deeply buried paleokarst system in tahe oilfield, tarim basin. Mar. Pet. Geol.
**2019**, 102, 16–32. [Google Scholar] [CrossRef] - Xie, X.; Weiss, W.W.; Tong, Z.; Morrow, N.R. Improved oil recovery from carbonate reservoirs by chemical stimulation. SPE J.
**2005**, 10, 276–285. [Google Scholar] [CrossRef] - Burchette, T.P. Carbonate rocks and petroleum reservoirs; a geological perspective from the industry. Geol. Soc. Lond. Spec. Publ.
**2012**, 370, 17–37. [Google Scholar] [CrossRef] - Qing, S.S.; Sloan, R. Quantification of uncertainty in recovery efficiency predictions: Lessons learned from 250 mature carbonate fields. In Proceedings of the SPE Annual Technical Conference and Exhibition, Denver, CO, USA, 5–8 October 2003. [Google Scholar] [CrossRef]
- Agada, S.; Chen, F.; Geiger, S.; Toigulova, G.; Agar, S.; Shekhar, R.; Benson, G.S.; Hehmeyer, O.; Amour, F.; Mutti, M.; et al. Numerical simulation of fluid-flow processes in a 3D high-resolution carbonate reservoir analogue. Pet. Geosci.
**2014**, 20, 125–142. [Google Scholar] [CrossRef] - Lucia, F.J. Carbonate Reservoir Characterization: An Integrated Approach, 2nd ed.; Springer: Berlin/Heidelberg, Germany, 2007; pp. 29–38. ISBN 978-3-540-72740-8. [Google Scholar] [CrossRef]
- He, J.; Killough, J.E.; Fadlelmula, M.M.; Fraim, M. A unified finite difference model for the simulation of transient flow in naturally fractured carbonate karst reservoirs. In Proceedings of the SPE Reservoir Simulation Symposium, Houston, TX, USA, 23–25 February 2015. [Google Scholar] [CrossRef] [Green Version]
- Popov, P.; Efendiev, Y.; Qin, G. Multiscale modeling and simulations of flows in naturally fractured karst reservoirs. Commun. Comput. Phys.
**2009**, 6, 162–184. [Google Scholar] [CrossRef] - Liu, J.; Bodvarsson, G.S.; Wu, Y.S. Analysis of flow behavior in fractured lithophysal reservoirs. J. Contam. Hydrol.
**2003**, 62–63, 189–211. [Google Scholar] [CrossRef] [Green Version] - Camacho-Velázquez, R.; Vásquez-Cruz, M.; Castrejón-Aivar, R.; Arana-Ortiz, V. Pressure-transient and decline-curve behavior in naturally fractured vuggy carbonate reservoirs. SPE Res. Eval. Eng.
**2005**, 8, 95–112. [Google Scholar] [CrossRef] - Wu, Y.S.; Ehlig-Economides, C.; Qin, G.; Kang, Z.; Zhang, W.; Ajayi, B.; Tao, Q. A Triple-Continuum Pressure-Transient Model for a Naturally Fractured Vuggy Reservoir. In Proceedings of the SPE Annual Technical Conference and Exhibition, Anaheim, CA, USA, 11–14 November 2007. [Google Scholar] [CrossRef]
- Kang, Z.; Wu, Y.S.; Li, J.; Wu, Y.; Zhang, J.; Wang, G. Modeling Multiphase Flow in Naturally Fractured Vuggy Petroleum Reservoirs. In Proceedings of the SPE Annual Technical Conference and Exhibition, San Antonio, TX, USA, 24–27 September 2006. [Google Scholar] [CrossRef]
- Wu, Y.S.; Di, Y.; Kang, Z.; Fakcharoenphol, P. A multiple-continuum model for simulating single-phase and multiphase flow in naturally fractured vuggy reservoirs. J. Pet. Sci. Eng.
**2011**, 78, 13–22. [Google Scholar] [CrossRef] - Yao, J.; Huang, Z.; Li, Y.; Wang, C.; Lv, X. Discrete fracture-vug network model for modeling fluid flow in fractured vuggy porous media. In Proceedings of the International Oil and Gas Conference and Exhibition, Beijing, China, 8–10 June 2010. [Google Scholar] [CrossRef]
- Zhang, N.; Yao, J.; Xue, S.; Huang, Z. Multiscale mixed finite element, discrete fracture–vug model for fluid flow in fractured vuggy porous media. Int. J. Heat Mass Transf.
**2016**, 96, 396–405. [Google Scholar] [CrossRef] - Zhang, X.; Huang, Z.; Lei, Q.; Yao, J.; Gong, L.; Sun, S.; Li, Y. Connectivity, permeability and flow channelization in fractured karst reservoirs: A numerical investigation based on a two-dimensional discrete fracture-cave network model. Adv. Water Resour.
**2022**, 161, 104142. [Google Scholar] [CrossRef] - Beavers, G.S.; Joseph, D.D. Boundary conditions at a naturally permeable wall. J. Fluid Mech.
**1967**, 30, 197–207. [Google Scholar] [CrossRef] - Saffman, P.G. On the boundary condition at the surface of a porous medium. Stud. Appl. Math.
**1971**, 50, 93–101. [Google Scholar] [CrossRef] - Chen, J.; Sun, S.; Wang, X. A numerical method for a model of two-phase flow in a coupled free flow and porous media system. J. Comput. Phys.
**2014**, 268, 1–16. [Google Scholar] [CrossRef] - Chen, J.; Sun, S.; Chen, Z. Coupling two-phase fluid flow with two-phase darcy flow in anisotropic porous media. Adv. Mech. Eng.
**2014**, 6, 871021. [Google Scholar] [CrossRef] [Green Version] - Huang, Z.; Gao, B.; Zhang, X.Y.; Yao, J. On the coupling of two-phase free flow and porous flow. In Proceedings of the 15th European Conference on the Mathematics of Oil Recovery, Amsterdam, The Netherlands, 29 August–1 September 2016. [Google Scholar] [CrossRef]
- Xie, H.; Li, A.; Huang, Z.; Gao, B.; Peng, R. Coupling of two-phase flow in fractured-vuggy reservoir with filling medium. Open Phys.
**2017**, 15, 12–17. [Google Scholar] [CrossRef] - Yan, X.; Huang, Z.; Yao, J.; Zhang, Z.; Liu, P.; Li, Y.; Fan, D. Numerical simulation of hydro-mechanical coupling in fractured vuggy porous media using the equivalent continuum model and embedded discrete fracture model. Adv. Water Resour.
**2019**, 126, 137–154. [Google Scholar] [CrossRef] - Liu, L.; Huang, Z.; Yao, J.; Di, Y.; Wu, Y. An efficient hybrid model for 3D complex fractured vuggy reservoir simulation. SPE J.
**2020**, 25, 907–924. [Google Scholar] [CrossRef] - Liu, L.; Huang, Z.; Yao, J.; Lei, Q.; Di, Y.; Wu, Y.; Zhang, K.; Cui, S. Simulating two-phase flow and geomechanical deformation in fractured karst reservoirs based on a coupled hydro-mechanical model. Int. J. Rock Mech. Min. Sci.
**2021**, 137, 104543. [Google Scholar] [CrossRef] - Arbogast, T.; Lehr, H.L. Homogenization of a darcy-stokes system modeling vuggy porous media. Comput. Geosci.
**2006**, 10, 291–302. [Google Scholar] [CrossRef] [Green Version] - Arbogast, T.; Brunson, D.S. A computational method for approximating a darcy-stokes system governing a vuggy porous medium. Comput. Geosci.
**2007**, 11, 207–218. [Google Scholar] [CrossRef] - Huang, Z.; Yao, J.; Li, Y.; Wang, C.; Lv, X. Numerical calculation of equivalent permeability tensor for fractured vuggy porous media based on homogenization theory. Commun. Comput. Phys.
**2011**, 9, 180–204. [Google Scholar] [CrossRef] [Green Version] - Popov, P.; Qin, G.; Bi, L.; Efendiev, Y.; Kang, Z.; Li, J. Multiphysics and Multiscale Methods for Modeling Fluid Flow Through Naturally Fractured Vuggy Carbonate Reservoirs. SPE Res. Eval. Eng.
**2009**, 12, 218–231. [Google Scholar] [CrossRef] - Qin, G.; Bi, L.; Popov, P.; Efendiev, Y.; Espedal, M.S. An efficient upscaling process based on a unified fine-scale multi-physics model for flow simulation in naturally fracture carbonate karst reservoirs. In Proceedings of the International Oil and Gas Conference and Exhibition, Beijing, China, 8–10 June 2010. [Google Scholar] [CrossRef]
- Golfier, F.; Lasseux, D.; Quintard, M. Investigation of the effective permeability of vuggy or fractured porous media from a darcy-brinkman approach. Comput. Geosci.
**2015**, 19, 63–78. [Google Scholar] [CrossRef] - Pal, M. A unified approach to simulation and upscaling of single-phase flow through vuggy carbonates. Int. J. Numer. Methods Fluids
**2012**, 69, 1096–1123. [Google Scholar] [CrossRef] - Li, Y.; Yao, J.; Li, Y.; Yin, C.; Pan, B.; Lee, J.; Dong, M. An equivalent continuum approach for modeling two-phase flow in fractured-vuggy media. Int. J. Multiscale Comput. Eng.
**2017**, 15, 79–98. [Google Scholar] [CrossRef] - Huang, Z.; Yao, J.; Wang, Y. An efficient numerical model for immiscible two-phase flow in fractured karst reservoirs. Commun. Comput. Phys.
**2013**, 13, 540–558. [Google Scholar] [CrossRef] [Green Version] - Wang, L.; Golfier, F.; Tinet, A.; Chen, W.; Vuik, C. An efficient adaptive implicit scheme with equivalent continuum approach for two-phase flow in fractured vuggy porous media. Adv. Water Resour.
**2022**, 163, 104186. [Google Scholar] [CrossRef] - Pairoys, F.; Lasseux, D.; Bertin, H. An experimental and numerical investigation of water-oil flow in vugular porous media. In Proceedings of the International Symposium of the Society of Core Analysis, Pau, France, 21–24 September 2003. [Google Scholar]
- Wang, J.; Liu, H.; Ning, Z.; Zhang, H.; Hong, C. Experiments on water flooding in fractured-vuggy cells in fractured-vuggy reservoirs. Pet. Explor. Dev.
**2014**, 41, 74–81. [Google Scholar] [CrossRef] - Yuan, D.; Hou, J.; Song, Z.; Wang, Y.; Luo, M.; Zheng, Z. Residual oil distribution characteristic of fractured-cavity carbonate reservoir after water flooding and enhanced oil recovery by N2 flooding of fractured-cavity carbonate reservoir. J. Pet. Sci. Eng.
**2015**, 129, 15–22. [Google Scholar] [CrossRef] - Wang, Y.; Hou, J.; Tang, Y.; Song, Z. Effect of vug filling on oil-displacement efficiency in carbonate fractured-vuggy reservoir by natural bottom-water drive: A conceptual model experiment. J. Pet. Sci. Eng.
**2019**, 174, 1113–1126. [Google Scholar] [CrossRef] - Yang, W.; Zhang, D.; Lei, G. Experimental study on multiphase flow in fracture-vug medium using 3D printing technology and visualization techniques. J. Pet. Sci. Eng.
**2020**, 193, 107394. [Google Scholar] [CrossRef] - Lu, G.; Zhang, L.; Liu, Q.; Xu, Q.; Zhao, Y.; Li, X.; Deng, G.; Wang, Y. Experiment analysis of remaining oil distribution and potential tapping for fractured-vuggy reservoir. J. Pet. Sci. Eng.
**2022**, 208, 109544. [Google Scholar] [CrossRef] - Krotkiewski, M.; Ligaarden, I.S.; Lie, K.; Schmid, D.W. On the importance of the stokes-brinkman equations for computing effective permeability in karst reservoirs. Commun. Comput. Phys.
**2011**, 10, 1315–1332. [Google Scholar] [CrossRef] - Zhou, D.; Fayers, F.J.; Orr Jr, F.M. Scaling of multiphase flow in simple heterogeneous porous media. In Proceedings of the SPE/DOE Improved Oil Recovery Symposium, Tulsa, OK, USA, 17–20 April 1994. [Google Scholar] [CrossRef]
- Durlofsky, L.J. Numerical calculation of equivalent grid-block permeability tensors for heterogeneous porous media. Water Resour. Res.
**1991**, 27, 699–708. [Google Scholar] [CrossRef] [Green Version] - Durlofsky, L.J. Coarse scale models of two phase flow in heterogeneous reservoirs: Volume averaged equations and their relationship to existing upscaling techniques. Comput. Geosci.
**1998**, 2, 73–92. [Google Scholar] [CrossRef] - Chen, Y.; Li, Y. Local-global two-phase upscaling of flow and transport in heterogeneous formations. Multiscale Model. Simul.
**2009**, 8, 125–153. [Google Scholar] [CrossRef] - Darman, N.H.; Sorbie, K.S.; Pickup, G.E. Development of pseudo functions for gravity-dominated immiscible gas displacements. In Proceedings of the SPE Reservoir Simulation Symposium, Houston, TX, USA, 14–17 February 1999. [Google Scholar] [CrossRef]
- Darman, N.H.; Pickup, G.E.; Sorbie, K.S. A comparison of two-phase dynamic upscaling methods based on fluid potentials. Comput. Geosci.
**2002**, 6, 5–27. [Google Scholar] [CrossRef] - Azoug, Y.; Tiab, D. The performance of pseudofunctions in the upscaling process. In Proceedings of the SPE Production and Operations Symposium, Oklahoma City, OK, USA, 22–25 March 2003. [Google Scholar] [CrossRef]
- Hashemi, A.; Shadizadeh, S.R.; Zargar, G. Upscaling of relative permeability using pseudo functions. Energy Sources Part A Recovery Util. Environ. Eff.
**2014**, 36, 2227–2237. [Google Scholar] [CrossRef]

**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 | - |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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