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A pore-scale model has been developed to study the gas flow through multiscale porous media based on a two-dimensional self-similar Sierpinski carpet. The permeability tensor with slippage effect is proposed, and the effects of complex configurations on gas permeability have been discussed. The present fractal model has been validated by comparison with theoretical models and available experimental data. The numerical results show that the flow field and permeability of the anisotropic Sierpinski model are different from that of the isotropic model, and the anisotropy of porous media can enhance gas permeability. The gas permeability of porous media increases with the increment of porosity, while it decreases with increased pore fractal dimension under fixed porosity. Furthermore, the gas slippage effect strengthens as the pore fractal dimension decreases. However, the relationship between the gas slippage effect and porosity is a nonmonotonic decreasing function because reduced pore size and enhanced flow resistance may be simultaneously involved with decreasing porosity. The proposed pore-scale fractal model can present insights on characterizing complex and multiscale structures of porous media and understanding gas flow mechanisms. The numerical results may provide useful guidelines for the applications of porous materials in oil and gas engineering, hydraulic engineering, chemical engineering, thermal power engineering, food engineering, etc.

Fluid flow through natural and artificial porous media such as soils, rocks, minerals, sludge, ceramics, textile, food, paper, plants, tissues, organs, and fuel cell plays an important role in daily life and practical applications [

As one of the key macroscopic transport properties of a porous medium, the value of permeability depends on the microscopic structures of the medium. Therefore, pore-scale mathematical models on fluid flow through porous media are significant for predicting the permeability and understanding the physical mechanisms of fluid flow through porous media [

Recently, gas flow through microscale and nanoscale porous media has attracted increasing interests from science and engineering as it is of great significance for fuel cell, open-cell foams, membrane, microelectromechanical system, low-permeability reservoirs, energy storage devices, etc. [^{-3}. However, the influence mechanisms of slippage effect on the permeability of anisotropic porous media are not clear. Therefore, the present work is aimed at developing a pore-scale model for gas flow through multiscale anisotropic porous media with slippage effect based on the Sierpinski carpet model and exploring the relationship between the macroscopic gas permeability and microscopic structures of porous media.

In order to characterize the multiscale structures, an exactly self-similar Sierpinski carpet model is used to generate the geometrical structure of the porous media. The 2D Sierpinski carpet model can be constructed by applying recursive algorithms on a void square with a size of ^{th} generation of the Sierpinski carpet model can be determined by

A schema for the construction process of the 2D Sierpinski carpet with

The relationship between porosity and pore fractal dimension can be gotten by combining Equations (

In order to quantitatively characterize the anisotropic properties of porous media, two anisotropic factors are introduced.

As shown in Figure

Parameters of the isotropic Sierpinski carpet models.

Group no. | ||||||||
---|---|---|---|---|---|---|---|---|

S1 | 3 | 1 | 1.893 | 0.889 | 0.790 | 0.702 | 0.624 | 0.555 |

S2 | 4 | 4 | 1.792 | 0.750 | 0.563 | 0.422 | 0.316 | 0.237 |

S3 | 5 | 1 | 1.975 | 0.960 | 0.922 | 0.885 | 0.849 | 0.815 |

S4 | 5 | 9 | 1.723 | 0.640 | 0.410 | 0.262 | 0.168 | 0.107 |

S5 | 6 | 4 | 1.934 | 0.889 | 0.790 | 0.702 | 0.624 | 0.555 |

S6 | 6 | 16 | 1.672 | 0.556 | 0.309 | 0.171 | 0.095 | 0.053 |

S7 | 7 | 1 | 1.989 | 0.980 | 0.960 | 0.940 | 0.921 | 0.902 |

S8 | 7 | 9 | 1.896 | 0.816 | 0.666 | 0.544 | 0.444 | 0.363 |

S9 | 7 | 25 | 1.633 | 0.490 | 0.240 | 0.118 | 0.058 | 0.028 |

S10 | 8 | 4 | 1.969 | 0.938 | 0.879 | 0.824 | 0.772 | 0.724 |

S11 | 8 | 16 | 1.862 | 0.750 | 0.563 | 0.422 | 0.316 | 0.237 |

S12 | 8 | 36 | 1.602 | 0.438 | 0.191 | 0.084 | 0.037 | 0.016 |

Parameters of the anisotropic Sierpinski carpet models.

Group no. | |||||
---|---|---|---|---|---|

A1 | 3 | 1 | 1.893 | 0 | 1 |

1 | 0 | ||||

1 | 1 | ||||

A2 | 4 | 4 | 1.792 | 0 | 1 |

1 | 0 | ||||

1 | 1 | ||||

A3 | 5 | 1 | 1.975 | 0 | 1 |

0 | 2 | ||||

1 | 0 | ||||

1 | 1 | ||||

1 | 2 | ||||

2 | 0 | ||||

2 | 1 | ||||

2 | 2 | ||||

A4 | 5 | 9 | 1.723 | 0 | 1 |

1 | 0 | ||||

1 | 1 | ||||

A5 | 6 | 4 | 1.934 | 0 | 1 |

0 | 2 | ||||

1 | 0 | ||||

1 | 1 | ||||

1 | 2 | ||||

2 | 0 | ||||

2 | 1 | ||||

2 | 2 | ||||

A6 | 6 | 16 | 1.672 | 0 | 1 |

1 | 0 | ||||

1 | 1 |

The 3rd order of the isotropic and anisotropic Sierpinski carpet models.

For the gas flow through porous media at very low Reynolds numbers, the inertial term in the Navier-Stokes equations can be neglected. Thus, the governing equations for a steady peristaltic flow of the incompressible Newtonian fluid through the Sierpinski carpet models are the continuity equation for the conservation of mass and Stokes equations for the conservation of momentum.

The creeping flow module in COMSOL Multiphysics was used to solve the gas flow through the 2D Sierpinski carpet models. Methane (CH_{4}) with density

In order to validate the present mathematical model, the predicted permeability of the isotropic Sierpinski carpet models was compared with that of the theoretical models and experimental data. As shown in Figure

A comparison of present numerical results with theoretical models and experimental data: (a) permeability without slippage effect and (b) gas slippage factor vs. absolute permeability.

Based on the linear correlation for gas permeability of the Klinkenberg equation, the gas slippage factor can be expressed by

In order to explore the effect of fractal dimension on the permeability, a pore size range (

The effect of pore fractal dimension on the permeability of isotropic porous media.

The velocity contour with streamlines for the 2nd order of the isotropic Sierpinski carpet models: (a) S1 (

Figure

The permeability of anisotropic Sierpinski carpet models: (a) A1 (

The velocity contour with streamlines for the 3rd order of anisotropic Sierpinski carpet models: (a) A1 (isotropic), (b) A3 (isotropic), (c) A1 (

In order to study the slippage effect in microscale porous media, the slippage boundary was performed on the surface of solid particles in the isotropic Sierpinski carpet models. A dimensionless parameter was defined to characterize the gas slippage effect:

Because the pore fractal dimension decreases with the decrease of porosity, it can be found in Figure

The effect of slippage effect on the permeability of the Sierpinski carpet models.

In this work, a two-dimensional Sierpinski carpet model has been adopted to characterize the multiscale microstructures of porous media. And a pore-scale mathematical model has been developed to study the gas flow through both the isotropic and anisotropic porous media. The influence of microstructures and anisotropy as well as slippage effect on the permeability has been discussed. It has been found that the permeability of porous media depends on the porosity and pore fractal dimension as well as pore size range. The value of permeability increases with increased porosity and decreases as pore fractal dimension increases under fixed porosity. The flow field and permeability of anisotropic porous media are different from that of isotropic porous media. The anisotropic factor is beneficial to the vertical fluid flow and can enhance the corresponding permeability. For the microscale porous media, gas slippage phenomena show a significant effect on the effective permeability. The numerical results indicate that the slippage effect strengthens as the pore fractal dimension decreases. However, it may be reduced by increased porosity under certain pore fractal dimensions as two competitive factors (pore size and flow resistance) are involved. The proposed fractal model shows advantages in characterizing the complex and irregular microstructures of porous media and provides a conceptual tool to understand the flow mechanisms of gas flow through the porous media. It should be pointed out that some complications such as dead-end pores, contact and overlap of solid particles, pore/particle configurations, and morphology were neglected in the proposed fractal model. As an extension to this study, it would be helpful to investigate randomly a three-dimensional fractal model.

The numerical data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there is no conflict of interest regarding the publication of this paper.

This work was jointly supported by the Natural Science Foundation of China (grant numbers 51876196, 51736007, and 21873087), the Zhejiang Provincial Natural Science Foundation of China (grant number LR19E060001), and the State Key Laboratory Cultivation Base for Gas Geology and Gas Control of Henan Polytechnic University (grant number WS2018A02).