The thermal conductivity of commonly used phase change materials (PCM) for thermal energy storage (TES), such as, fatty acids, paraffin etc., is relatively poor, which is one of the main drawbacks for limiting their utility. In the recent past, few attempts have been made to enhance the thermal conductivity of PCM by mixing different additives in the appropriate amount. Graphene nanoparticles, having higher thermal conductivity may be a potential candidate for the same, when mixed appropriately with different PCM. In present study authors have carried out the numerical investigation for the melting of graphene nano-particles dispersed PCM filled in an aluminum square cavity heated from one side. In this work, the graphene nanoparticles are mixed in three different volumetric ratios (1%, 3%, and 5%), with three different commonly used categories of organic, inorganic and paraffin PCM (namely, Capric Acid, CaCl2·6H2O, and n-octadecane) to see the effect on melting of composite PCM developed. The resulting transient isotherms, velocity fields, and melting front and melt fractions thus have been deliberated in detail. These results clearly indicate that the addition of graphene nanoparticles increases melting rate but can also hamper the convection heat transfer within large cavities. The study also shows that such enhanced PCM can be effectively used for different TES applications in different fields. The prediction of temperature variation and rate of melting or solidification may be found useful especially for designing such TES devices.