In this study, we present ternary modeling and simulation of interactions between two immiscible liquid drops suspended in a third immiscible liquid. Initially two drops are brought into intimate contact, and their dynamic behaviors are investigated. In particular, we present three different cases of drop-drop interaction, namely engulfing, partial engulfing and non-engulfing, by applying different sets of free energy parameters of the three-phase system. Each phase is assumed to be incompressible Newtonian fluid and inertia is neglected. The Cahn-Hilliard model for a general ternary mixture and Stokes equations are coupled to describe the thermodynamics of the phase-field variables and hydrodynamics. A finite element method is used for the spatial discretization of the governing equations. In addition, we employ a grid deformation method for the local mesh refinement near the diffuse interface. Numerical simulation is carried out in a two-dimensional framework due to axisymmetric nature of the problem. Effects of the initial drop size, viscosity and surface tension on the dynamical behaviors of the drop-drop interactions are investigated. Particularly in engulfing case, penetration behavior of one drop into the other drop is investigated in detail. Also the equilibrium configurations depending on the surface tension are compared with the theoretical and experimental ones. Numerical results, especially the equilibrium configurations, agree qualitatively with the experimental results reported previously in the literature. Results of the penetration behavior in engulfing case also shows similar tendency as reported in the literature.