Linear and non-linear analyses on the onset of miscible viscous fingering in a porous medium

Won Sun Ryoo; Min Chan Kim

Korean Journal of Chemical Engineering, Vol.35, No.7, 1423-1432, July, 2018

DOI10.1007/s11814-018-0046-4 Export Citation

Linear and non-linear analyses on the onset of miscible viscous fingering in a porous medium

Won Sun Ryoo , and Min Chan Kim^{1, †}

Department of Chemical Engineering, Hongik University, Seoul 04066, Korea
¹Department of Chemical Engineering, Jeju National University, Jeju 63243, Korea

E-mail:

The onset of miscible viscous fingering in porous media was analyzed theoretically. The linear stability equations were derived in the self-similar domain, and solved through the modal and non-modal analyses. In the nonmodal analysis, adjoint equations were derived using the Lagrangian multiplier technique. Through the non-modal analysis, we show that initially the system is unconditionally stable even in the unfavorable viscosity distribution, and there exists the most unstable initial disturbance. To relate the theoretical predictions with the experimental work, nonlinear direct numerical simulations were also conducted. The present stability condition explains the system more reasonably than the previous results based on the conventional quasi-steady state approximation.

Keywords:Viscous Fingering;Non-modal Analysis;Modal Analysis;Linear Stability Analysis;Direct Numerical Simulation

[References]

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Slobod RL, Thomas RA, Soc. Pet. Eng. J., 3, 9 (1963)
Perkins TE, Johnston OE, Hoffman RN, Soc. Pet. Eng. J., 5, 301 (1965)
Tan CT, Homsy GM, Phys. Fluids, 29, 3549 (1986)
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Wit AD, Bertho Y, Martin M, Phys. Fluids, 17, 054114 (2005)
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Fujita T, Ohara S, Sugaya T, Saigenji K, Hotta K, Comp. Biochem. Physiol. B, 126, 353 (2000)
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Broyles BS, Shalliker RA, Cherrak DE, Guiochon G, J. Chromatogr. A, 822, 173 (1998)
Dickson ML, Norton TT, Fernandez EJ, AIChE J., 43(2), 409 (1997)
Manickam O, Homsy GM, J. Fluid Mech., 288, 75 (1995)
Azaiez J, Singh B, Phys. Fluids, 14, 1557 (2002)
Hejazi SH, Trevelyan PMJ, Azaiez J, Wit AD, J. Fluid Mech., 652, 501 (2010)
Mishra M, Trevelyan PMJ, Almarcha C, Phys. Rev. Lett., 105, 204501 (2010)
Yortsos YC, Zeybek M, Phys. Fluids, 31, 3511 (1988)
Tan CT, Homsy GM, Phys. Fluids, 31, 1330 (1988)
Doumenc F, Boeck T, Guerrier B, Rossi M, J. Fluid Mech., 648, 521 (2010)
Wooding RA, ZAMP, 13, 255 (1962)
Ben Y, Demekhin EA, Chang HC, Phys. Fluids, 14, 999 (2002)
Pritchard D, Eur. J. Mech. B/Fluids, 28, 564 (2009)
Pramanik S, Mishra M, Phys. Fluids, 25, 074104 (2013)
Pramanik S, Mishra M, Chem. Eng. Sci., 110, 144 (2014)
Kim MC, Choi CK, Phys. Fluids, 23, 084105 (2011)
Kim MC, Choi CK, Phys. Fluids, 24, 044102 (2012)
Kim MC, Adv. Water Resour., 35, 1 (2012)
Kim MC, Transp. Porous Media, 97(3), 395 (2013)
Kim MC, Korean J. Chem. Eng., 35(2), 364 (2018)
Farrell BF, Ioannou PJ, J. Atmos. Sci., 53, 2041 (1996)
Barenblatt GI, Scaling, Self-Similarity and Intermediate Asymptotics, Cambridge University Press (1996).
Zimmerman WB, Homsy GM, Phys. Fluids A, 4, 2348 (1992)
Schmid PJ, Annu. Rev. Fluid Mech., 39, 129 (2007)
Tilton N, Daniel D, Riaz A, Phys. Fluids, 25, 092107 (2013)
Andres JTH, Cardoso SSS, Chaos, 22, 037113 (2012)
Daniel D, Tilton N, Riaz A, J. Fluid Mech., 727, 456 (2013)

[Referenced By]

[Related Articles]

[1] Hill S, Chem. Eng. Sci., 1, 247 (1952)

[2] Slobod RL, Thomas RA, Soc. Pet. Eng. J., 3, 9 (1963)

[3] Perkins TE, Johnston OE, Hoffman RN, Soc. Pet. Eng. J., 5, 301 (1965)

[4] Tan CT, Homsy GM, Phys. Fluids, 29, 3549 (1986)

[5] Homsy GM, Ann. Rev. Fluid Mech., 19, 271 (1987)

[6] Wit AD, Bertho Y, Martin M, Phys. Fluids, 17, 054114 (2005)

[7] Rousseaux G, Wit AD, Martin M, J. Chromatogr. A, 1149, 254 (2007)

[8] Bhaskar KR, Garik P, Turner BS, Bradley JD, Bansil R, Stanley HE, LaMont JT, Nature, 360, 458 (1982)

[9] Fujita T, Ohara S, Sugaya T, Saigenji K, Hotta K, Comp. Biochem. Physiol. B, 126, 353 (2000)

[10] Plante LD, Romano PM, Fernandez EJ, Chem. Eng. Sci., 49(14), 2229 (1994)

[11] Broyles BS, Shalliker RA, Cherrak DE, Guiochon G, J. Chromatogr. A, 822, 173 (1998)

[12] Dickson ML, Norton TT, Fernandez EJ, AIChE J., 43(2), 409 (1997)

[13] Manickam O, Homsy GM, J. Fluid Mech., 288, 75 (1995)

[14] Azaiez J, Singh B, Phys. Fluids, 14, 1557 (2002)

[15] Hejazi SH, Trevelyan PMJ, Azaiez J, Wit AD, J. Fluid Mech., 652, 501 (2010)

[16] Mishra M, Trevelyan PMJ, Almarcha C, Phys. Rev. Lett., 105, 204501 (2010)

[17] Yortsos YC, Zeybek M, Phys. Fluids, 31, 3511 (1988)

[18] Tan CT, Homsy GM, Phys. Fluids, 31, 1330 (1988)

[19] Doumenc F, Boeck T, Guerrier B, Rossi M, J. Fluid Mech., 648, 521 (2010)

[20] Wooding RA, ZAMP, 13, 255 (1962)

[21] Ben Y, Demekhin EA, Chang HC, Phys. Fluids, 14, 999 (2002)

[22] Pritchard D, Eur. J. Mech. B/Fluids, 28, 564 (2009)

[23] Pramanik S, Mishra M, Phys. Fluids, 25, 074104 (2013)

[24] Pramanik S, Mishra M, Chem. Eng. Sci., 110, 144 (2014)

[25] Kim MC, Choi CK, Phys. Fluids, 23, 084105 (2011)

[26] Kim MC, Choi CK, Phys. Fluids, 24, 044102 (2012)

[27] Kim MC, Adv. Water Resour., 35, 1 (2012)

[28] Kim MC, Transp. Porous Media, 97(3), 395 (2013)

[29] Kim MC, Korean J. Chem. Eng., 35(2), 364 (2018)

[30] Farrell BF, Ioannou PJ, J. Atmos. Sci., 53, 2041 (1996)

[31] Barenblatt GI, Scaling, Self-Similarity and Intermediate Asymptotics, Cambridge University Press (1996).

[32] Zimmerman WB, Homsy GM, Phys. Fluids A, 4, 2348 (1992)

[33] Schmid PJ, Annu. Rev. Fluid Mech., 39, 129 (2007)

[34] Tilton N, Daniel D, Riaz A, Phys. Fluids, 25, 092107 (2013)

[35] Andres JTH, Cardoso SSS, Chaos, 22, 037113 (2012)

[36] Daniel D, Tilton N, Riaz A, J. Fluid Mech., 727, 456 (2013)