In naturally fractured reservoirs, fractures are the main flowing channels, while matrix is the dominant storage space. The oil/water relative permeability curve for the fracture in this kind of reservoir is very important for water-injection field development. In this study, we conducted experiments on the oil/water relative permeability of carbonate cores from Kenkiyak oil field and compared the differences in relative permeability curves between natural matrix cores and artificial-fractured cores. After the fracturing process, the two-phase flow area of tested cores becomes narrower, the permeability of the equal-permeability point gets higher, the relative permeability curve rises or drops more rapidly, and the displacement recovery efficiency decreases. The stress-sensitivity characteristics of the relative permeability curves were also studied on the basis of experiments on naturally fractured cores. With increasing effective confining pressure, the irreducible water saturation increases, the residual-oil saturation changes slightly, the equal-permeability point moves downward, and the displacement recovery efficiency declines. Numerical-simulation results indicate that for a given recovery factor, the water cut would increase more slowly but ultimate recovery factor would decrease using the relative permeability curve under higher confining pressure. Therefore, the water injection should be operated when the reservoir pressure is relatively higher to maintain formation pressure during waterflooding and lower the impact of stress sensitivity accordingly.