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

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.Graphical Abstract