In this paper, the element-free Galerkin (EFG) method is applied to study the vertically averaged multiphase flow of CO2 and brine in the deep saline aquifer. The methodology involves solving two governing partial differential equations (PDEs), namely pressure and saturation equations sequentially using implicit pressure and explicit saturation (IMPES) solution strategy. The governing PDEs are obtained by combining mass conservation (continuity equation) and momentum conservation (Darcy’s law of multiphase extension)equations. In order to capture the asymptotic pressure behaviour near injection well, a local nodal refinement zone was introduced around the well. The computed pressure and average saturation were compared with an existing approximate grid-based FEM and extended FEM (XFEM) numerical solutions to demonstrate the robustness of the developed model.