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Estimating permeability of porous media based on modified Hagen–Poiseuille flow in tortuous capillaries with variable lengths

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Abstract

Fractal analysis of permeability of porous media has received much attention over several decades. Many traditional and recently presented fractal models are derived based on Hagen–Poiseuille (H–P) flow in circular-shaped straight/tortuous capillaries that cut through the model. However, the real capillaries in the nature have different lengths and variably-shaped cross sections. In this study, a fractal scaling law for length distribution of capillaries in porous media was derived and its validity was verified by comparisons of fractal dimensions for length distribution between calculated and theoretical values, and then we proposed a modified H–P equation that considers the fractal properties of capillary length and capillary shape. Based on a discrete element method (DEM) code, a series of capillary network models were established and their equivalent permeability was calculated by solving the modified H–P equation. Some empirical expressions were given, which could be utilized to predict the magnitudes of capillary number, equivalent permeability of porous media, the smallest side length of capillary networks at the representative elementary volume size, ratio of capillary network permeability to fracture network permeability, dimensionless permeability, where dimensionless permeability is defined as the ratio of equivalent permeability of models consist of variably-shaped capillaries to that of circular-shaped capillaries. The results show that the proposed fractal length distribution of capillaries, the modified H–P equation, and the developed DEM code can be used to calculate the equivalent permeability of porous media and that the capillaries cannot always be assumed to cut through the model with circular shapes.

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Acknowledgments

This study has been partially funded by National Basic Research 973 Program of China (Grant No. 2013CB036003) and National Natural Science Foundation of China (Grant Nos. 51379117, 51579239, 41427802). These supports are gratefully acknowledged.

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Authors and Affiliations

  1. State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou, 221116, Jiangsu, China

    Richeng Liu & Liyuan Yu

  2. School of Engineering, Nagasaki University, Nagasaki, 852-8521, Japan

    Richeng Liu & Yujing Jiang

  3. Rock Mechanics and Geo-Hazards Center, Shaoxing University, Shaoxing, 312000, China

    Bo Li

  4. State Key Laboratory of Mining Disaster Prevention and Control Co-founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao, 266590, China

    Yujing Jiang

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  1. Richeng Liu
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  2. Yujing Jiang
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  3. Bo Li
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  4. Liyuan Yu
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Correspondence to Yujing Jiang.

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Liu, R., Jiang, Y., Li, B. et al. Estimating permeability of porous media based on modified Hagen–Poiseuille flow in tortuous capillaries with variable lengths. Microfluid Nanofluid 20, 120 (2016). https://doi.org/10.1007/s10404-016-1783-5

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  • DOI: https://doi.org/10.1007/s10404-016-1783-5

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