We consider drainage processes in model geometries that represent a matrix block-fracture system. Flow visualization in etched glass micromodels was carried out for various pairs of fluids, injection rates (capillary numbers), and viscosity ratio values. It was found that for a flow rate below a critical value (capillary number threshold) displacement occurs only in the fracture, with matrix invasion occurring after the critical value is exceeded. The experiments were modeled with the use of a pore network simulator based on meniscus displacement. Numerical and experimental results were compared and found to be in good agreement. A theory for the critical capillary number and the invasion process was subsequently developed. The process shares many of the features of invasion in homogeneous systems, with the notable exceptions that the role of capillary pressure is played here by the capillary number, and that the process is dynamic rather than quasi-static. Effective relative permeabilities for the matrix-fracture system were also calculated. Contrary to homogeneous systems, these curves depend on the mobility ratio. The results find application to fractured systems or to systems containing high permeability streaks.