A Stochastic Analysis of the Flow of Two Immiscible Fluids in Porous Media: The Case When the Viscosities of the Fluids are Equal

Kyung-Hoe Kim; Chan-Hong Chung

Korean Journal of Chemical Engineering, Vol.18, No.6, 796-801, November, 2001

Kyung-Hoe Kim^†, and Chan-Hong Chung

Department of Chemical Engineering, Taegu University, Kyungbuk 712-714, Korea

E-mail:

A new stochastic theory is developed to explain the flow of two immiscible fluids in porous medium when the viscosity difference between two fluids is zero. In an individual micropore the radius of curvature of the interface separating the fluids is assumed constant and flow is modeled by the random jumping of microscopic interfaces. A one dimensional model composed of an array of parallel capillary tubes of constant radius is analyzed in detail. For the case in which two fluids have equal viscosity an analytical solution is obtained. The fluid displacement process is Fickian and dispersion is described in terms of a diffusion or spreading constant.

Keywords:Stochastic Analysis;Immiscible Fluids;Porous Media;Microscopic Interfaces;Fickian Dispersion

[References]

Aubert WG, "Fate and Transport of Low pH Hazardous Materials After Deep Well Disposal," Ph.D. Dissertation, Louisiana State University, Baton Rouge, Louisiana (1986)
Broadbent SR, Hammersley JM, Cambridge Philosophical Society Proceedings, 53, 629 (1957)
Chandler R, Koplik J, Lerman K, Willemsen JF, J. Fluid Mech., 119, 249 (1982)
Chen JD, J. Colloid Interface Sci., 110, 488 (1986)
Chen JD, Koplik J, J. Colloid Interface Sci., 108, 304 (1985)
Chun MS, Park OO, Kim JK, Korean J. Chem. Eng., 7(2), 126 (1990)
Chuoke RL, Van Meurs P, Van der Poel C, Trans. AIME, 216, 188 (1959)
De Gennes P, Guyon E, J. Mec., 17, 403 (1978)
Dussan VEB, J. Fluid Mech., 77, 665 (1976)
Gerstl Z, Chen Y, Migelgrin U, Yaron B, "Toxic Organic Chemicals in Porous Media," Springer-Verlag, New York (1989)
Kim YH, Lee EK, Korean J. Chem. Eng., 13(5), 466 (1996)
Greenkorn RA, "Flow Phenomena in Porous Media," Marcel Dekker, Inc., New York (1983)
Larson RG, Scriven LE, Davis HT, Chem. Eng. Sci., 36, 57 (1981)
Lenormand R, Zarcone C, Phys. Rev. Lett., 54, 2226 (1985)
Lesage S, Jackson RE, "Groundwater Contamination and Analysis at Hazardous Waste Site," Marcel Dekker, Inc., New York (1992)
Ngan C, Dussan VEB, J. Fluid Mech., 118, 27 (1982)
Nittman J, Daccord G, Stanley HE, Nature, 314(14), 141 (1985)
Noonan DC, Glynn WK, Miller ME, Chem. Eng. Prog., June, 55 (1993)
Paterson L, Phys. Rev. Lett., 52(18), 1621 (1984)
Saffman P, Taylor GI, Qart. Proc. Roy. Soc., A245, 312 (1958)
Saffman P, Taylor GI, Qart. J. Mech. Appl. Math., 12, 265 (1959)
Wang S, "Computer Assisted Tomography and Its Application in the Study of Multiphase Flow Through Porous Media," Ph.D. Dissertation, Columbia University, New York (1983)
Wilkinson D, Willemsen JF, J. Phys. A, 16, 3365 (1983)
Willemsen JF, "Flow Through Porous Materials," Schlemberger-Doll Research Preprint (1983)
Witten TA, Sander LM, Phys. Rev. Lett., 47, 1400 (1981)
Witten TA, Sander LM, Phys. Rev., B, Condens. Matter, 27, 5686 (1983)
Yoon DY, Kim DS, Choi CK, Korean J. Chem. Eng., 15(3), 341 (1998)

[Referenced By]

[1] Aubert WG, "Fate and Transport of Low pH Hazardous Materials After Deep Well Disposal," Ph.D. Dissertation, Louisiana State University, Baton Rouge, Louisiana (1986)

[2] Broadbent SR, Hammersley JM, Cambridge Philosophical Society Proceedings, 53, 629 (1957)

[3] Chandler R, Koplik J, Lerman K, Willemsen JF, J. Fluid Mech., 119, 249 (1982)

[4] Chen JD, J. Colloid Interface Sci., 110, 488 (1986)

[5] Chen JD, Koplik J, J. Colloid Interface Sci., 108, 304 (1985)

[6] Chun MS, Park OO, Kim JK, Korean J. Chem. Eng., 7(2), 126 (1990)

[7] Chuoke RL, Van Meurs P, Van der Poel C, Trans. AIME, 216, 188 (1959)

[8] De Gennes P, Guyon E, J. Mec., 17, 403 (1978)

[9] Dussan VEB, J. Fluid Mech., 77, 665 (1976)

[10] Gerstl Z, Chen Y, Migelgrin U, Yaron B, "Toxic Organic Chemicals in Porous Media," Springer-Verlag, New York (1989)

[11] Kim YH, Lee EK, Korean J. Chem. Eng., 13(5), 466 (1996)

[12] Greenkorn RA, "Flow Phenomena in Porous Media," Marcel Dekker, Inc., New York (1983)

[13] Larson RG, Scriven LE, Davis HT, Chem. Eng. Sci., 36, 57 (1981)

[14] Lenormand R, Zarcone C, Phys. Rev. Lett., 54, 2226 (1985)

[15] Lesage S, Jackson RE, "Groundwater Contamination and Analysis at Hazardous Waste Site," Marcel Dekker, Inc., New York (1992)

[16] Ngan C, Dussan VEB, J. Fluid Mech., 118, 27 (1982)

[17] Nittman J, Daccord G, Stanley HE, Nature, 314(14), 141 (1985)

[18] Noonan DC, Glynn WK, Miller ME, Chem. Eng. Prog., June, 55 (1993)

[19] Paterson L, Phys. Rev. Lett., 52(18), 1621 (1984)

[20] Saffman P, Taylor GI, Qart. Proc. Roy. Soc., A245, 312 (1958)

[21] Saffman P, Taylor GI, Qart. J. Mech. Appl. Math., 12, 265 (1959)

[22] Wang S, "Computer Assisted Tomography and Its Application in the Study of Multiphase Flow Through Porous Media," Ph.D. Dissertation, Columbia University, New York (1983)

[23] Wilkinson D, Willemsen JF, J. Phys. A, 16, 3365 (1983)

[24] Willemsen JF, "Flow Through Porous Materials," Schlemberger-Doll Research Preprint (1983)

[25] Witten TA, Sander LM, Phys. Rev. Lett., 47, 1400 (1981)

[26] Witten TA, Sander LM, Phys. Rev., B, Condens. Matter, 27, 5686 (1983)

[27] Yoon DY, Kim DS, Choi CK, Korean J. Chem. Eng., 15(3), 341 (1998)