The flux-driven displacement of a wetting fluid by a non-wetting fluid in two-dimensional channels with flat parallel plates and sinusoidal surfaces is investigated via a multi-relaxation time multi-component lattice Boltzmann method. The co-current flow and the capillary filling problems are used for the validation of the numerical method. The results have shown that the present method is effective in simulating the viscous and capillary forces during the displacement process. For flat parallel plates, the effect of wettability on the development of fingering flow and the relationship between the developed finger width ratio and the capillary number are investigated. The results are compared to the previous research. The fingering flow is enhanced with a strongly wetting condition and the finger width ratio decreases as the capillary number is raised. For periodic sinusoidal channels, discontinuous interfaces appear due to the roughness of channel walls. The residual behavior of the wetting displaced fluid is enhanced with the wetting condition, as a strongly wetting condition would lead to a slower contact line motion. Finally, a quantitative study of the combined effect of roughness and wettability on the residual saturation is made.