Miscible and immiscible injection flows in heterogeneous porous media, for which the permeability is characterized by a log Gaussian distribution, are simulated by a robust diffuse-interface formulation. The robust numerical method enables direct qualitative and quantitative comparisons regarding pattern formations in various fluid miscibility conditions. For miscible injections, the typical size of fingering structures depends strongly on the correlation length and forms tapered fingers with sharper tips. On the other hand, the typical size of immiscible fingers is affected less significantly by the permeability heterogeneity, and wide spreading tips are retained in the fingering patterns. Prominence of fingering instability is quantitatively evaluated by the channeling width and the interfacial length. The channeling width shows strong and monotonic dependences on the heterogeneous variance. On the contrary, maximum channeling width occurs at intermediate correlation length due to local resonant effect between the faster penetrating fingers and permeability heterogeneity. On the other hand, effects of the correlation length and the permeability variance on the interfacial lengths are generally consistent. Longer interfacial length is perturbed by smaller correlation length or higher variance. Interesting invariant evolutions of interfacial lengths are revealed regardless of the permeability variance in sufficiently large correlation length under all miscibility conditions. In addition, the regime of slower growth of interfacial length at later times experimentally observed in homogeneous miscible injection is verified in heterogeneous porous media as well.

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