Abstract
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.
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References
Aubert, W. G., “Fate and Transport of Low pH Hazardous Materials After Deep Well Disposal”, Ph.D. Dissertation, Louisiana State University, Baton Rouge, Louisiana (1986).
Broadbent, S. R. and Hammersley, J. M., “Percolation Processes”,Cambridge Philosophical Society Proceedings,53, 629 (1957).
Chandler, R., Koplik, J., Lerman, K. and Willemsen, J. F., “Capillary Displacement and Percolation in Porous Media”,J. Fluid Mech.,119, 249 (1982).
Chen, J.-D., “Some Mechanisms of Immiscible Fluid Displacement in Small Networks”,J. Colloid Interface Sci.,110, 488 (1986).
Chen, J.-D. and Koplik, “Immiscible Fluid Displacement in Small Networks”,J. Colloid Interface Sci.,108, 304 (1985).
Chun, M. S., Park, O. O. and Kim, J. K., “Flow and Dynamic Behavior of Dilute Polymer Solutions in Hydrodynamic Chromatography”,Korean J. Chem. Eng,7, 126 (1990).
Chuoke, R. L., VanMeurs, P. and Van derPoel, C, “The Instability of Slow, Immiscible, Viscous Liquid-Liquid Displacements in Permeable Media”,Trans. AIME,216, 188 (1959).
DeGermes, P. and Guyon, E, “Lois generales pour I’ injection d’un fluide dans un milien poreux aléatoire”,J. Mec,17, 403 (1978).
Dussan, V. E. B., “The Moving Contact Line: the Slip Boundary Condition”,J. Fluid Mech.,77, 665 (1976).
Gerstl, Z., Chen., Y., Migelgrin, U. and Yaron, B., “Toxic Organic Chemicals in Porous Media”, Springer-Verlag, New York (1989).
Kim, Y. H. and Lee, E. K., “Comparison of Axial and Radial Flow Chromatography on Protein Separation Speed and Resolution”,Korean J. Chem. Eng,13, 466 (1996).
Greenkorn, R. A., “Flow Phenomena in Porous Media”, Marcel Dekker, Inc., New York (1983).
Larson, R. G., Scriven, L. E. and Davis, H. T., “Percolation Theory of Two Phase Flow in Porous Media”,Chem. Eng. Sci.,36, 57 (1981).
Lenormand, R. and Zarcone, C, “Invasion Percolation in an Etched Network: Measurement of a Fractal Dimension”,Phys. Revu. Let,54, 2226 (1985).
Lesage, S. and Jackson, R. E., eds., “Groundwater Contamination and Analysis at Hazardous Waste Site”, Marcel Dekker, Inc., New York (1992).
Ngan, C. and Dussan, V. E. B., “On the Nature of the Dynamic Contact Angle: an Experimental Study”,J. Fluid Mech.,118, 27 (1982).
Nittman, J., Daccord, G. and Stanley, H. E., “Fractal Growth of Viscous Fingers: Quantitative Characterization of a Fluid Instability”,Nature,314(14), 141 (1985).
Noonan, D. C, Glynn, W. K. and Miller, M. E., “Enhance Performance of Soil Vapor Extraction”,Chem. Eng. Prog.,June, 55 (1993).
Paterson, L., “Diffusion-Limited Aggregation and Two-Fluid Displacements in Porous Media”,Phys. Rev. Lett.,52(18), 1621 (1984).
Saffman, P. and Taylor, G. I., “The Penetration of a Fluid into a Porous Medium or Hele-Shaw Cell Containing a More Viscous Liquid”,Qart Proc. Roy. Soc.,A245, 312 (1958).
Saffman, P. and Taylor, G. I., “A Note on the Motion of Bubbles in a Hele-Shaw Cell and Porous Medium”,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. and Willemsen, J. F., “Invasion Percolation: a New from of Percolation Theory”,J. Phys. A,16, 3365 (1983).
Willemsen, J. F., “Flow Through Porous Materials”, Schlemberger-Doll Research Preprint (1983).
Witten, T. A. and Sander, L. M., “Diffusion-Limited Aggregation, a Kinetic Critical Phenomenon”,Phys. Rev. Lett.,47, 1400 (1981).
Witten, T. A. and Sander, L. M., “Diffusion-Limited Aggregation”,Phys, Rev.5, 27, 5686 (1983).
Yoon, D. Y, Kim, D. S. and Choi, C. K., “Convective Instability in Packed Beds with Internal Heat Sources and Throughflow”,Korean J. Chem. Eng,15, 341 (1998).
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Kim, KH., Chung, CH. A stochastic analysis of the flow of two immiscible fluids in porous media: The case when the viscosities of the fluids are equal. Korean J. Chem. Eng. 18, 796–801 (2001). https://doi.org/10.1007/BF02705599
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DOI: https://doi.org/10.1007/BF02705599