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Accurate simulation of the coupled fluid flow and solid deformation in porous media is challenging, especially when the media permeability and storativity are heterogeneous. We apply the enriched Galerkin (EG) finite element method for the Biot's system. Block structure used to compose the enriched space and linearization and iterative schemes employed to solve the coupled media permeability alteration are illustrated. The open-source platform used to build the block structure is presented and illustrate that it helps the enriched Galerkin method easily adaptable to any existing discontinuous Galerkin codes. Subsequently, we compare the EG method with the classic continuous Galerkin (CG) and discontinuous Galerkin (DG) finite element methods. While these methods provide similar approximations for the pressure solution of Terzaghi's one-dimensional consolidation, the CG method produces spurious oscillations in fluid pressure and volumetric strain solutions at material interfaces that have permeability contrast and does not conserve mass locally. As a result, the flux approximation of the CG method is significantly different from the one of EG and DG methods, especially for the soft materials. The difference of flux approximation between EG and DG methods is insignificant; still, the EG method demands approximately two and three times fewer degrees of freedom than the DG method for two- and three-dimensional geometries, respectively. Lastly, we illustrate that the EG method produces accurate results even for much coarser meshes.