In this dissertation, an effective numerical algorithm is developed for establishing simulation for the two-component two-phase flow with partial miscibility at the pore scale. Many studies in the rock-fluid interaction have been done for immiscible flow, whose components do not mix and separate instantaneously. This paper extends the study to miscible flow, whose components will mix with certain pressure and temperature, and exploits the potential of simulating complex real-life fluid interactions. The mathematical model consists of a set of Cahn-Hilliard equations and a realistic equation of state (i.e. Peng-Robinson equation of state). The numerical challenges lie in the fact that these are highly coupled, fourth-order, nonlinear partial differential equations. For solving the proposed PDEs, a discontinuous Galerkin (DG) method is used for space discretization, and a combination of backward Euler method and convex-concave splitting method is used for time discretizition. The resulting simulation can extract essential characteristics of the digital rock sample, agreeing with conventional lab-based tests but with only a fraction of cost in time and resources. Practically, the proposed algorithm and simulation can help engineers to make more informed decisions, for example in oil industry for enhancing oil recovery.