To address the limitations of the traditional Forchheimer equation in describing the nonlinear seepage behavior of single fracture rock mass under high confining pressure and water pressure coupling

the two-part Hooke’s model (TPHM) was used to describe the relationship between the hydraulic aperture and the effective normal stress of single fracture. Linear coefficient A and nonlinear coefficient B were both characterized as functions of the effective normal stress. A modified Forchheimer model

which accounts for the coupled effect of nonlinear seepage and normal stress in single fractures

was presented

and the validity of the model was verified through experimental data and numerical simulations. Furthermore

the variation of linear coefficient A and nonlinear coefficient B under normal stress and fracture water pressure was analyzed. The hydro-mechanical coupling effect is significant for a single fracture under high confining pressure and high water pressure. The pressure gradient-flow test data increasingly deviate from the traditional Forchheimer equation

while the modified Forchheimer model remains consistent with the experimental data. When the fracture water pressure p is zero

the modified model reduces to the traditional Forchheimer equation without considering water pressure effects. The coefficients A and B are sensitive to the effective normal stress

increasing nonlinearly with the normal stress

and decreasing nonlinearly with increasing fracture water pressure. Compared to simulations that do not account for hydraulic coupling

the numerical model that considers hydraulic coupling requires a smaller pressure gradient for the same flow rate

and the simulation results are accurately described by the modified model.