The applicability of a numerical method is preliminarily examined to determine its suitability for simulations of microscopic immiscible liquid-liquid motion on solid walls with textured surfaces, which is used for evaluating flexible large-area fluidic devices and a novel screen printing process. In the method, a conservation-modified Allen-Cahn equation is adopted for calculations of diffuse-interface advection and formation based on a phase-field model, and a lattice-Boltzmann method is employed as numerical scheme for solving the two-phase fluid-dynamics equations. The simulations demonstrate the ability of the method to successfully capture the shape and motion of a droplet, not only in a liquid flow inside a rectangular micro channel with linearly grooved wall, but also on wettable parallel-wall structured surface under capillary force.