Fractal permeability model of fracture network based on topological graph theory

The complex fracture network provides the main channel for coalbed methane seepage, and its structural characteristics significantly affect the productivity of coalbed methane. In view of the lack of fracture network connectivity in existing coalbed methane seepage models, based on topological graph theory, the connectivity characteristics of fracture network were analyzed, and the expression of power law relation of fracture branch length was derived. Combined with fractal geometry, a permeability model considering fracture network structure characteristics was established on the basis of classical cubic law. The model shows that the permeability of fracture network is a function of fractal dimension, porosity, tortuosity fractal dimension, scale coefficient, connectivity, maximum fracture branch length, azimuth angle, dip angle and feature length. The permeability evolution of fracture network under true triaxial stress was analyzed. The results show that with the increase of the principal stresses σ₁, σ₂ and σ₃, the fractures in coal are gradually compressed and the gas seepage channel becomes narrow. The permeability of fracture network shows a negative exponential decrease with the increase of the principal stresses σ₁, σ₂ and σ₃. Based on the experimental rules and results, the fracture network fractal permeability model and S&D model (Shi-Durucan) were combined to establish the fracture network permeability calculation model under true triaxial stress. The effectiveness of the model was verified by the experimental data. The obtained permeability variation trend is consistent with the process of loading the three principal stresses, which can reflect the influence trend of the change of three-way stress on the permeability. Compared with the S&D model, it can better reflect the anisotropy characteristics of the permeability of the fracture network.