Transport in Porous Media

, Volume 104, Issue 2, pp 407–433 | Cite as

Linear and Nonlinear Analyses of the Onset of Buoyancy-Induced Instability in an Unbounded Porous Medium Saturated by Miscible Fluids

  • Min Chan KimEmail author
  • Dhananjay Yadav


This study analyzes the stability of an initially sharp interface between two miscible fluids in a porous medium. Linear stability equations are first derived using the similarity variable of the basic state, and then transformed into a system of ordinary differential equations using a spectral expansion with and without quasi-steady-state approximation (QSSA). These transformed equations are solved using the eigenanalysis and initial value problem approach. The initial growth rate analysis shows that initially the system is unconditionally stable. The stability characteristics obtained under the present QSSA are quantitatively same as those obtained without the QSSA. To support these theoretical results, numerical simulations are conducted using the Fourier-spectral method. The results of theoretical linear stability analyses and the numerical simulations validate to each other.


Gravitational fingering Porous media Linear stability analysis  Nonlinear numerical simulation 

List of Symbols



Dimensionless wavenumber, \(\sqrt{a_x^2 +a_y^2 }\)


Modified dimensionless wave number, \(a\sqrt{\tau }\)


Concentration (M)


Dimensionless concentration, \({\left( {C-C_+ } \right) }/{\left( {C_{-} -C_+ } \right) }\)


Effective diffusion coefficient \(\left( {{\hbox {m}^{2}}/\hbox {s}} \right) \)


Gravitational acceleration vector \(\left( {\hbox {m}/{\hbox {s}^{2}}} \right) \)


Permeability \(\left( {\hbox {m}^{2}} \right) \)


Diffusional operator in \(\left( {\tau ,\zeta } \right) \)-domain, \({\partial ^{2}}/{\partial \zeta ^{2}}+\left( {\zeta /2} \right) \left( {\partial /{\partial \zeta }} \right) \)


Pressure \(\left( {\hbox {Pa}} \right) \)


Time \(\left( \hbox {s} \right) \)


Velocity vector \(\left( {\hbox {m}/\hbox {s}} \right) \)


Dimensionless vertical velocity component

\(\left( {x,y,z} \right) \)

Dimensionless Cartesian coordinates

Greek Symbols

\(\mu \)

Viscosity \(\left( {\hbox {Pa}\, \hbox {s}} \right) \)

\(\rho \)

Density \(\left( {{\hbox {kg}}/{\hbox {m}^{3}}} \right) \)

\(\sigma \)

Dimensionless growth rate

\(\tau \)

Dimensionless time \(\left( {{\mathcal{D}t}/K} \right) \)

\(\zeta \)

similarity Variable \(\left( {z/{\sqrt{\tau }}} \right) \)



Critical state


Basic quantity


Perturbed quantity



This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A2038983). The authors are grateful to all the reviewers for their lucid comments which have served to greatly improve the status of the present article.


  1. Ben, Y., Demekhin, E.A., Chang, H.-C.: A spectral theory for small-amplitude miscible fingering. Phys. Fluids 14, 999–1010 (2002)CrossRefGoogle Scholar
  2. Boyd, J.P.: Chebyshev and Fourier Spectral Methods. Dover Publications Inc., Mineola (2000)Google Scholar
  3. Caltagirone, J.-P.: Stability of a saturated porous layer subject to a sudden rise in surface temperature: comparison between the linear and energy methods. Q. J. Mech. Appl. Math. 33, 47–58 (1980)CrossRefGoogle Scholar
  4. Daniel, D., Tilton, N., Riaz, A.: Optimal perturbations of gravitationally unstable, transient boundary layers in porous media. J. Fluid Mech. 727, 456–487 (2013)CrossRefGoogle Scholar
  5. Ennis-King, J., Preston, I., Paterson, L.: Onset of convection in anisotropic porous media subject to a rapid change in boundary conditions. Phys. Fluids 17, 084107 (2005)CrossRefGoogle Scholar
  6. Farrel, B.F., Ioannou, P.J.: Generalized stability theory. Part II: Nonautonomous operators. J. Atmos. Sci. 53, 2041–2053 (1996)CrossRefGoogle Scholar
  7. Fernandez, J., Kurowski, P., Petitjeans, P., Meiburg, E.: Density-driven unstable flows of miscible fluids in a Hele–Shaw cell. J. Fluid Mech. 451, 239–260 (2002)CrossRefGoogle Scholar
  8. Hassanzadeh, H., Pooladi-Darvish, M., Keith, D.W.: Stability of a fluid in a horizontal saturated porous layer: effect of non-linear concentration profile, initial, and boundary conditions. Transp. Porous Med. 65, 193–211 (2006)Google Scholar
  9. Hidalgo, J., Carrera, J.: Effect of dispersion on the onset of convection during \(\text{ CO }_{2}\) sequestration. J. Fluid Mech. 640, 441–452 (2009)CrossRefGoogle Scholar
  10. Horton, C.W., Rogers, F.T.: Convection currents in porous medium. J. Appl. Phys. 16, 367–370 (1945)CrossRefGoogle Scholar
  11. Kim, M.C.: Onset of radial viscous fingering in a Hele–Shaw cell. Korean J. Chem. Eng. 29, 1688–1694 (2012)CrossRefGoogle Scholar
  12. Kim, M.C.: Onset of buoyancy-driven convection in a fluid-saturated porous medium bounded by a long cylinder. Transp Porous Med. 97, 395–408 (2013)CrossRefGoogle Scholar
  13. Kim, M.C., Choi, C.K.: The stability of miscible displacement in porous media: nonmonotonic viscosity profiles. Phys. Fluids 23, 084105 (2011)CrossRefGoogle Scholar
  14. Kim, M.C., Choi, C.K.: Linear stability analysis on the onset of the buoyancy-driven convection in liquid saturated porous medium. Phys. Fluids 24, 044102 (2012)CrossRefGoogle Scholar
  15. Kneafsey, T.K., Pruess, K.: Laboratory experiments and numerical simulation studies of convectively enhanced carbon dioxide dissolution. Energy Procedia 4, 5114–5121 (2011)CrossRefGoogle Scholar
  16. Lapwood, E.R.: Convection of a fluid in a porous medium. Proc. Camb. Philos. Soc. 44, 508–521 (1948)CrossRefGoogle Scholar
  17. Manickam, O., Homsy, G.M.: Fingering instabilities in vertical miscible displacement flows in porous media. J. Fluid Mech. 288, 75–102 (1995)CrossRefGoogle Scholar
  18. Morton, B.R.: On the equilibrium of a stratified layer of fluid. J. Mech. Appl. Math. 10, 433–447 (1957)CrossRefGoogle Scholar
  19. Neufeld, J.A., Hesse, M.A., Riaz, A., Hallworth, M.A., Tchelepi, H.A., Huppert, H.E.: Convective dissolution of carbon dioxide in saline aquifers. Geophys. Res. Lett. 37, L22404 (2010)CrossRefGoogle Scholar
  20. Pramanik, S., Mishra, M.: Linear stability analysis of Korteweg stresses effect on miscible viscous fingering in porous media. Phys. Fluid 25, 074104 (2013)CrossRefGoogle Scholar
  21. Pritchard, D.: The linear stability of double-diffusive miscible rectilinear displacements in Hele–Shaw cell. Eur. J. Mech. B 28, 564–577 (2009)CrossRefGoogle Scholar
  22. Rapaka, S., Chen, S., Pawar, R., Stauffer, P.H., Zhang, D.: Non-modal growth of perturbations in density-driven convection in porous media. J. Fluid Mech. 609, 285–303 (2008)CrossRefGoogle Scholar
  23. Riaz, A., Hesse, M., Tchelepi, H.A., Orr Jr, F.M.: Onset of convection in a gravitationally unstable, diffusive boundary layer in porous media. J. Fluid Mech. 548, 87–111 (2006)CrossRefGoogle Scholar
  24. Rees, D.A.S., Selim, A., Ennis-King, J.P.: The instability of unsteady boundary layers in porous media. In: Vadász, P. (ed.) Emerging Topics in Heat and Mass Transfer in porous media, pp. 85–110. Springer, Berlin (2008a)CrossRefGoogle Scholar
  25. Rees, D.A.S., Postelnicu, A., Bassom, A.P.: The linear vortex instability of the near vertical line source plume in porous media. Transp. Porous Med. 74, 211–238 (2008b)Google Scholar
  26. Selim, A., Rees, D.A.S.: The stability of a developing thermal front in a porous medium. I Linear theory. J. Porous Media 10, 1–15 (2007)CrossRefGoogle Scholar
  27. Selim, A., Rees, D.A.S.: Linear and nonlinear evolution of isolated disturbances in a growing thermal boundary layer in porous media. AIP Conf. Proc. 1254, 47–52 (2010)CrossRefGoogle Scholar
  28. Tan, C.T., Homsy, G.M.: Stability of miscible displacements in porous media: rectilinear flow. Phys. Fluids 29, 3549–3556 (1986)CrossRefGoogle Scholar
  29. Taylor, G.I.: The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. Proc. R. Soc. London. A 201, 192–196 (1950)CrossRefGoogle Scholar
  30. Trevelyan, P.M.J., Alamarcha, C., de Wit, A.: Buoyancy-driven instabilities of miscible two-layer stratifications in porous media and Hele–Shaw cells. J. Fluid Mech. 670, 38–65 (2011)CrossRefGoogle Scholar
  31. Wessel-Berg, D.: On a linear stability problem related to underground \({CO}_{2}\) storage. SIAM J. Appl. Math. 70, 1219–1238 (2009)CrossRefGoogle Scholar
  32. Wooding, R.A.: Rayleigh instability of a thermal boundary layer in flow through a porous medium. J. Fluid Mech. 9, 183–192 (1960)CrossRefGoogle Scholar
  33. Wooding, R.A.: The stability of an interface between miscible fluids in a porous medium. ZAMP 13, 255–266 (1962)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of Chemical EngineeringJeju National UniversityJeju Republic of Korea

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