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• A natural multimodal carbonate rock is imaged and processed by a high- resolution micro-CT to extract a 3-D geometry model. • A multiple-relaxation-time color-gradient LB model is developed to simulate two-phase flow in carbonate porous media. • The effects of rock wettability, capillary number and viscosity ratio on displacement efficiency are explored. • The underlying mechanisms of pore-scale oil droplets' mobilization are understood by integral geometry. Marine carbonate reservoirs widely distributed in the Middle East are characterized by multimodal pore structures and complex wettability, usually causing a great difference in pore-scale oil displacement efficiency. However, the underlying mechanisms are still ambiguous. In this study, a natural rock sample selected from a typical carbonate reservoir in the Middle East is imaged with a high-resolution X-ray micro-CT, and the three-dimensional geometry model of microstructures are extracted. A multiple-relaxation-time color-gradient lattice Boltzmann model is developed and validated to simulate oil–water two-phase flow in the carbonate porous media. The influences of rock wettability, oil-wet heterogeneity, capillary number, and oil–water viscosity ratio on oil displacement efficiency and fluid distribution in multimodal carbonate pore space are ultimately explored, following by analysis of pore-scale oil droplets mobilization with integral geometry. Results show that, the rock wettability, capillary number and oil–water viscosity ratio have significant impacts on the pore-scale oil displacement efficiency and fluid distribution in multimodal carbonate pore space while oil-wet heterogeneity has little effect. The oil displacement efficiency usually becomes larger with the increase of capillary number and the water-wetting degree as well as the decrease of oil–water viscosity ratio. Due to dynamic competition between capillary pressure and viscous force, the continuous oil droplets are fragmented into a large quantity of isolated oil droplets in the early stage of water flooding, showing a sharp decrease in the volume of continuous oil droplets, a rapid increase in the volume of isolated oil droplets, and a poor topological connectivity; In the middle and late stage of water flooding, the isolated oil droplets are gradually stripped and mobilized, leading to a decrease in the volume of oil droplets and an obvious improvement in topological connectivity. [ABSTRACT FROM AUTHOR]