Water flooding of fractured reservoirs is risky because water breakthrough can occur early, leading to a prohibitively high water cut. In mixed or oil-wet carbonates, capillary drive is negligible or absent. For this scenario, we investigate fluid-pressure-driven displacement of oil by water in two-phase flow numerical models based on naturally fractured limestone beds mapped along the British Channel coast. These reservoir analogs are represented by unstructured finite-element grids with discrete representations of intersecting fractures. We solve the governing equations for slightly compressible two-phase flow with our original control-volume finite-element method. This permits the direct examination of displacement patterns in fractures and rock matrix. We find that the irreducible saturation in the fractured carbonate is much higher than the value prescribed to the rock matrix. The shape of water invasion fronts is highly sensitive to the viscosity ratio of oil and water. Men the Brooks-Corey relative permeability model is applied to the rock matrix at a viscosity ratio of 1, the total mobility, lambda(t), is low at intermediate saturations. This stabilizes displacement fronts where a girdle of reduced lambda(t) develops, but this effect disappears as the viscosity ratio increases. For an idealized model with a water-wet matrix, we have also evaluated the effect of countercurrent capillary-pressure-driven flow across fracture-matrix interfaces. The rate of this countercurrent imbibition scales with the specific fracture surface area and decays exponentially as intermediate saturation zones develop adjacent to the fractures. The resulting reduced, feeds back into the fluid-pressure-driven displacement process.