Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

A Simplified Model and Similarity Solutions for Interfacial Evolution of Two-Phase Flow in Porous Media

  • 199 Accesses

  • 5 Citations


For two-phase flows of immiscible displacement processes in porous media, we proposed a simplified model to capture the interfacial fronts, which is given by explicit expressions and satisfies the continuity conditions of pressure and normal velocity across the interface. A new similarity solution for the interfacial evolution in the rectangular coordinate system was derived by postulating a first-order approximation of the velocity distribution in the region that the two-phase fluids co-exist. The interfacial evolution equation can be explicitly expressed as a linear function, where the slope of the interfacial equation is simply related to the mobility ratio of two-phase fluids in porous media. The application of the proposed solutions to predictions of interfacial evolutions in carbon dioxide injected into saline aquifers was illustrated under different mobility ratios and operational parameters. For the purpose of comparison, the numerical solutions obtained by level set method and the similarity solutions based on the Dupuit assumptions were presented. The results show that the proposed solution can give a better approximation of interfacial evolution than the currently available similarity solutions, especially in the situation that the mobility ratio is large. The proposed approximate solutions can provide physical insight into the interfacial phenomenon and be readily used for rapidly screening carbon dioxide storage capacity in subsurface formations and monitoring the migration of carbon dioxide plume.

This is a preview of subscription content, log in to check access.


  1. Bachu S.: Screening and ranking of sedimentary basins for sequestration of CO2 in geological media in response to climate change. Environ. Geol. 44(3), 277–289 (2003). doi:10.1007/s00254-003-0762-9

  2. Bear J.: Dynamics of Fluids in Porous Media. Dover Publications Inc, New York (1972)

  3. Class H., Ebigbo A., Helmig R., Dahle H.K., Nordbotten J.M., Celia M.A., Audigane P., Darcis M., Ennis-King J., Fan Y.Q., Flemisch B., Gasda S.E., Jin M., Krug S., Labregere D., Beni A.N., Pawar R.J., Sbai A., Thomas S.G., Trenty L., Wei L.L.: A benchmark study on problems related to CO2 storage in geologic formations. Comput. Geosci. 13(4), 409–434 (2009). doi:10.1007/s10596-009-9146-x

  4. Dentz M., Tartakovsky D.: Abrupt-interface solution for carbon dioxide injection into porous media. Transp. Porous Med. 51(7), 1–13 (2008)

  5. Dentz, M., Tartakovsky, D.: Response to “Comments on abrupt-interface solution for carbon dioxide injection into porous media by Dentz and Tartakovsky (2008)” by Lu et al. Transport in Porous Media 79, 3 (2009). doi:10.1007/s11242-009-9363-8

  6. Hesse M.A., Tchelepi H.A., Cantwell B.J., Orr F.M.: Gravity currents in horizontal porous layers: transition from early to late self-similarity. J. Fluid Mech. 577, 363–383 (2007). doi:10.1017/s0022112007004685

  7. Hesse M.A., Orr F.M., Tchelepi H.A.: Gravity currents with residual trapping. J. Fluid Mech. 611, 35–60 (2008). doi:10.1017/s002211200800219x

  8. Holloway S., Pearce J.M., Hards V.L., Ohsumi T., Gale J.: Natural emissions of CO2 from the geosphere and their bearing on the geological storage of carbon dioxide. Energy 32(7), 1194–1201 (2007)

  9. Lu C., Lee S.-Y., Han W.S., McPherson B.J., Lichtner P.C.: Comments on “Abrupt-interface solution for carbon dioxide injection into porous media” by M. Dentz and D. Tartakovsky. Transp. Porous Med. 79, 9 (2009). doi:10.1007/s11242-009-9362-9

  10. Liu Y.Z., Wang L., Yu B.: Sharp front capturing method for carbon dioxide plume propagation during injection into a deep confined aquifer. Energy Fuels 24, 1431–1440 (2010). doi:10.1021/ef9010498

  11. MacMinn C.W., Juanes R.: A mathematical model of the footprint of the CO2 plume during and after injection in deep saline aquifer systems. Energy Procedia 1(1), 3429–3436 (2009a)

  12. MacMinn C.W., Juanes R.: Post-injection spreading and trapping of CO2 in saline aquifers: impact of the plume shape at the end of injection. Comput. Geosci. 13(4), 483–491 (2009b). doi:10.1007/s10596-009-9147-9

  13. Nordbotten J.M., Celia M.A.: Similarity solutions for fluid injection into confined aquifers. J. Fluid Mech. 561, 307–327 (2006). doi:10.1017/s0022112006000802

  14. Nordbotten J.M., Kavetski D., Celia M.A., Bachu S.: Model for CO2 leakage including multiple geological layers and multiple leaky wells. Environ. Sci. Technol. 43(3), 743–749 (2009). doi:10.1021/es801135v

  15. Okwen R.T., Stewart M.T., Cunningham J.A.: Analytical solution for estimating storage efficiency of geologic sequestration of CO2. Int. J. Greenhouse Gas Control 4(1), 102–107 (2010)

  16. Pinder G.F., Gray W.G.: Essentials of Multiphase Flow and Transport in Porous Media. Wiley, Hoboken, NJ (2008)

  17. Schnaar G., Digiulio D.C.: Computational modeling of the geologic sequestration of carbon dioxide. Vadose Zone J. 8(2), 389–403 (2009). doi:10.2136/vzj2008.0112

Download references

Author information

Correspondence to Yongzhong Liu.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Wang, L., Liu, Y. & Chu, K. A Simplified Model and Similarity Solutions for Interfacial Evolution of Two-Phase Flow in Porous Media. Transp Porous Med 93, 721–735 (2012). https://doi.org/10.1007/s11242-012-9979-y

Download citation


  • Two-phase flow
  • Porous media
  • Interfacial dynamics
  • Similarity solution
  • Rectangular coordinate system