Coupling flow with geomechanical processes at the pore scale in fractured rocks is essential in understanding the macroscopic processes of interest, such as geothermal energy extraction, CO2 sequestration, and hydrocarbon production from naturally and hydraulically fractured reservoirs. To investigate the microscopic (pore-scale) phenomena, we propose an efficient and accurate flow-geomechanics coupling algorithm to advance the fundamental flow mechanism from the micro-continuum perspective. Further, we investigate the stress influence on fluid leakage caused by matrix-fracture interaction. In this work, we employ a hybrid micro-continuum approach to describe the flow in fractured rocks, in which fracture flow is described by Navier-Stokes (NS) equations and flow in the surrounding matrix is modeled by Darcy's law. This hybrid modeling is achieved using the extended Darcy-Brinkman-Stokes (EDBS) equations. This approach applies a unified conservation equation for flow in both media (fracture & matrix). We then couple the EDBS flow model with the Brown-Scholz (BS) geomechanical model, which quantifies the deformation of rock fractures. We demonstrate the accuracy of the coupled flow-geomechanical algorithm, in which the accuracy of the EDBS flow model is validated by a simple case with a known analytical solution. The BS geomechanical model is demonstrated with experimental data collected from the literature. The developed flow-geomechanical coupling algorithm is then used to perform sensitivity analyses to explore the factors impacting the fluid leakage caused by the matrix-fracture interaction. We found that the degree of fluid leakage increases as matrix permeability increases and fractures become rougher. Fluid leakage degree decreases with the increase of inertial forces because of the existence of eddies, which prevents the flux exchange between the matrix and fracture. We also investigate the stress influence on fluid leakage and further on fracture permeability under the impact of matrix-fracture interaction. We conclude the fracture permeability would increase with the consideration of the fluid leakage and exhibits an exponential relation with the effective stress.