Discontinuous and continuous Galerkin methods for compressible single-phase and two-phase flow in fractured porous media

•Discontinuous and continuous Galerkin approximations for compressible single-phase and two-phase flow in fractured porous media were developed.•Conducting and blocking fractures were modelled as reduced-dimensional interfaces for flow in porous media.•The continuity and jumps in pressure and saturation between the two sides of the matrix-fracture interface were captured.