A fractal model for the starting pressure gradient for Bingham fluids in porous media. (English) Zbl 1388.76019

Summary: We present a fractal model for the starting pressure gradient for Bingham fluids in porous media based on the fractal characteristics of pores in the media and on the capillary pressure effect. Every parameter in the proposed models has clear physical meaning, and the proposed model relates the starting pressure gradient of Bingham fluids to the structural parameters of porous media, the yield stress, the capillary pressure parameters and the fractal dimensions of porous media. The model predictions from the present model for the starting pressure gradient are in good agreement with the available expression Eq. (2). The results also show that at smaller radii \(\bar r < 0.3\)mm) and low porosity (\(\phi < 0.3\)), the capillary pressure has the significant influence on the starting pressure gradient in porous media and thus cannot be neglected. However, at high porosity, the starting pressure gradient is primarily produced by the shear stress and the contribution to the starting pressure gradient from the capillary pressure is negligible.


76A05 Non-Newtonian fluids
76S05 Flows in porous media; filtration; seepage
Full Text: DOI


[1] Bear, J.: Dynamics of fluids in porous media, (1972) · Zbl 1191.76001
[2] Barenblatt, G. E.; Entov, B. M.; Rizhik, B. M.: Flow of liquids and gases in natural formations, (1984)
[3] Vradis, G. C.; Dougher, J.; Kumar, S.: Entrance pipe flow and heat transfer for a Bingham plastic, Int. J. Heat mass transfer 36, 543-552 (1993)
[4] Hammad, K. J.; Vradis, G. C.: Creeping flow of a Bingham plastic through axisymmetric sudden contractions with viscous dissipation, Int. J. Heat mass transfer 39, 1555-1567 (1996) · Zbl 0964.76501
[5] Nascimento, U. C. S.; Macêdo, E. N.; Quaresma, J. N. N.: Thermal entry region analysis through the finite integral transform technique in laminar flow of Bingham fluids within concentric annular ducts, Int. J. Heat mass transfer 45, 923-929 (2002) · Zbl 0991.76515
[6] Khatyr, R.; Ouldhadda, D.; Idrissi, A. Il: Viscous dissipation effects on the asymptotic behaviour of laminar forced convection for Bingham plastics in circular ducts, Int. J. Heat mass transfer 46, 589-598 (2003) · Zbl 1121.76390
[7] Wu, Y. S.; Pruess, K.; Witherspoon, P. A.: Flow and displacement of Bingham non-Newtonian fluids in porous media, Spere 7, 369-376 (1992)
[8] Blackery, J.; Mitsoulis, E.: Creeping motion of a sphere in tubes filled with a Bingham plastic material, J. non-Newton fluid mech. 70, 59-77 (1997)
[9] Roquet, N.; Saramito, P.: An adaptive finite element method for Bingham fluid flows around a cylinder, Comput. meth. Appl. mech. Eng. 192, 3317-3341 (2003) · Zbl 1054.76053
[10] Balhoff, M. T.; Thompson, K. E.: Modeling the steady flow of yield-stress fluids in packed beds, Aiche j. 50, 3034-3048 (2004)
[11] Bird, R. B.; Stewart, W. E.; Lightfoot, E. N.: Transport phenomena, (1960)
[12] Prada, A.; Civan, F.: Modification of Darcy’s law for the threshold pressure gradient, J. petrol. Sci. eng. 22, 237-240 (1999)
[13] Wang, S. J.; Huang, Y. Z.; Civan, F.: Experimental and theoretical investigation of the zaoyuan field heavy oil flow through porous media, J. petrol. Sci. eng. 50, 83-101 (2006)
[14] Mandelbrot, B. B.: The fractal geometry of nature, (1982) · Zbl 0504.28001
[15] Feder, J.: Fractals, (1988) · Zbl 0648.28006
[16] Feder, J.; Aharony, A.: Fractals in physics, (1989) · Zbl 0674.00017
[17] Katz, A. J.; Thompson, A. H.: Fractal sandstone pores: implications for conductivity and pore formation, Phys. rev. Lett. 54, 1325-1328 (1985)
[18] Krohn, C. E.; Thompson, A. H.: Fractal sandstone pores: automated measurements using scanning-electron-microscope images, Phys. rev. B 33, 6366-6374 (1986)
[19] Mandelbrot, B. B.; Passoja, D. E.; Paullay, A. J.: Fractal character of fracture surfaces of metals, Nature 308, 721-722 (1984)
[20] Xie, H.; Bhaskar, R.; Li, J.: Generation of fractal models for characterization of pulverized materials, Miner. metall. Process., 36-42 (1993)
[21] Bayles, G.; Klinzing, G.; Chiang, S.: Fractal mathematics applied to flow in porous systems, Part. part. Syst. char. 6, 168-175 (1989)
[22] Wu, J. S.; Yu, B. M.: A fractal resistance model for flow through porous media, Int. J. Heat mass transfer 50, 3925-3932 (2007) · Zbl 1117.76062
[23] Yu, B. M.; Cheng, P.: A fractal model for permeability of bi-dispersed porous media, Int. J. Heat mass transfer 45, 2983-2993 (2002) · Zbl 1101.76408
[24] Yu, B. M.; Lee, L. J.; Cao, H. Q.: A fractal in-plane permeability model for fabrics, Polym. compos. 22, 201-221 (2002)
[25] Yu, B. M.: Fractal character for tortuous streamtubes in porous media, Chin. phys. Lett. 22, 158-160 (2005)
[26] Comiti, J.; Renaud, M.: A new model for determining mean structure parameters of fixed beds from pressure drop measurements, Chem. eng. Sci. 44, 1539-1545 (1989)
[27] Kong, X. Y.: Advanced mechanics of fluids in porous media, (1999)
[28] Ahn, K. J.; Seferis, J. C.; Berg, J. C.: Simultaneous measurements of permeability and capillary pressure of thermosetting matrices in woven fabric reinforcements, Polym. compos. 12, 146-152 (1991)
[29] Yu, B. M.; Li, J. H.: Some fractal characters of porous media, Fractals 9, 365-372 (2001)
[30] Amico, S.; Lekakou, C.: An experimental study of the permeability and capillary pressure in resin-transfer moulding, Comp. sci. Technol. 61, 1945-1959 (2001)
[31] Zhang, B.; Yu, B. M.; Wang, H. X.; Yun, M. J.: A fractal analysis of permeability for power-law fluids in porous media, Fractals 14, 171-177 (2006) · Zbl 1302.28024
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