Study on fracture propagation law of fractured deep coal‑rock containing gangue based on 3D discrete lattice method

Effective hydraulic-fracture propagation strongly affects gas production from deep coalbed methane (CBM) reservoirs. This study establishes a three-dimensional discrete lattice model for a coal-gangue-roof/floor system and validates it against the analytical solution for a penny-shaped hydraulic fracture. This study proposes two complementary indices: the interlayer penetrability index (IPI), used to quantitatively describe the ability of fractures to penetrate through the gangue layer into adjacent strata; and the fracture propagation uniformity index (FPUI), used to evaluate the balance of propagation among perforation clusters. Unlike conventional metrics based on fracture height, length, or total area, IPI and FPUI characterise cross-layer penetration efficiency and multi-cluster uniformity, respectively. Using these indices, the effects of cluster spacing, cluster number, injection rate, fluid viscosity, perforation number, and injection-rate-viscosity coupling were investigated. Results show that moderate cluster spacing reduces stress interference and promotes penetration, whereas excessive clusters or perforations intensify fracture competition. Increasing injection rate enlarges fracture area and improves IPI, but the benefit diminishes at high rates. Appropriate fluid viscosity improves pressure retention and fluid partitioning among fracture clusters, thereby enhancing both IPI and FPUI. Overall, a suitable combination of injection rate and viscosity promotes interlayer penetration while maintaining uniform multi-cluster fracture propagation.