Multiphase flow in porous media is crucial for many energy and environmental processes, including hydrocarbon recovery, CO2 sequestration, groundwater contamination, etc. Prediction of flow properties such as permeability and tortuosity is challenging as internal structures, including pore geometry and connectivity, directly affect fluid flow. The natural porous media may consist of fractures and cracks, etc. Dual-porosity models often explain transport and flow in heterogeneous media, envisioning high- and low-permeability zones with limited pore connections. In our present study, we have studied single-phase and two-phase flows in a dual-porosity model to investigate the effect of heterogeneity on flow properties and immiscible displacement. The pressure drop was identical when the low- and high-permeable regions were parallel to the flow direction. However, when the fluid flows from a high-permeable zone to a low-permeable zone, it gives a higher pressure drop as compared to when fluid flows from the low-permeable zone to the high-permeable zone. The effect of dual porosity on immiscible displacement was also investigated, and a breakthrough change was observed for different dual-porosity models.