As an insulator, dielectric composites should be guaranteed the specific level of insulation performance to maintain the electric device. It is essential to understand the physical phenomena at the interface of the dielectric composites, where the electric field and charge distribution are concentrated. We, therefore, focused on the surface discharge propagated on the interface forming with dielectric composites, composed of nanocomposites as a solid region and oil as a liquid region. Nanocomposites have a broad range of conductivities depending on the content of nanoparticles. Thus, the conductivities of nanocomposites were analyzed from 0 to 10 ^-3 , 10 ^-6 , 10 ^-7 , 10 ^-8 , and 10 ^-9 S/m with the experimentally verified numerical model. Our numerical model was expanded incorporating with the migration-ohmic model to link the physical phenomena between liquid and solid. With extending the simple model that handles only one charge carrier, three charge carriers were employed to analyze the temporal discharge dynamics in the liquid. Using the fully coupled finite element method, the patterns of the streamer were investigated with different conductivities. The higher the conductivity, the slower the propagation speed of the streamer. Conductivity changes the amount of the surface charge accumulated on the surface. This surface charge affects the speed of the streamer. Furthermore, the equivalent circuit model was implemented to explain the relations between the propagation speed and conductivity of solid.