A fractal model based on a new governing equation of fluid flow in fractures for characterizing hydraulic properties of rock fracture networks

This study presents a fractal length distribution model of fractures in discrete fracture networks (DFNs), adopting a fractal dimension Df that represents the geometric distribution characteristics of fractures and another fractal dimension DT that represents the tortuosity of fluid flow induced by surface roughness of single fractures in DFNs. A new governing equation for fluid flow in single fractures based on the cubic law was incorporated into this fractal model. Fluid flow in 1290 DFNs with different geometric characteristics of fractures and side lengths was simulated and their equivalent permeability was calculated. The results show that the values of a, which is the power law exponent of the fracture size distribution, calculated by the proposed fractal model are consistent with those reported in similar previous studies. The flow rate of a DFN changes proportionally with e6-DT where e is the aperture, which agrees better with the in-situ measurements reported in literature than the prediction of classical cubic law (e3). The equivalent permeability of DFNs is more sensitive to the random number utilized to generate the fracture length than the ones used to generate the orientation and center point of fractures. With the increment of Df, the size of the representative elementary volume (REV) decreases. When the size of a DFN is larger than the REV, the variation of equivalent permeability induced by the random number holds constant. When Df < 1.5, fluid flow in a DFN is dominated by the relatively small fractures with their lengths shorter than the side length of the DFN. With increasing Df, fluid flow becomes more dominated by the longer fractures, especially the ones cutting through the models.