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An integral equation for immiscible fluid isplacement in a two-dimensional porous medium or Hele-Shaw cell

Published online by Cambridge University Press:  17 February 2009

M. R. Davidson
M. R. Davidson
Affiliation:
CSIRO Division of Mineral Physics, Lucas Heights Research Laboratories, Private Mail Bag 7, Sutherland, N.S.W. 2232.
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Abstract

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An integral equation for the normal velocity of the interface between two immiscible fluids flowing in a two-dimensional porous medium or Hele-Shaw cell (one fluid displaces the other) is derived in terms of the physical parameters (including interfacial tension), a Green's function and the given interface. When the displacement is unstable, ‘fingering’ of the interface occurs. The Saffman-Taylor interface solutions for the steady advance of a single parallel-sided finger in the absence of interfacial tension are seen to satisfy the integral equation, and the error incurred in that equation by the corresponding Pitts approximating profile, when interfacial tension is included, is shown. In addition, the numerical solution of the integral equation is illustrated for a sinusoidal and a semicircular interface and, in each case, the amplitude behaviour inferred from the velocity distribution is consistent with conclusions based on the stability of an initially flat interface.

Type
Research Article
Information
The ANZIAM Journal , Volume 26 , Issue 1 , July 1984 , pp. 14 - 30
Copyright
Copyright © Australian Mathematical Society 1984

References

[1]Bretherton, F. P., “The motion of long bubbles in tubes”, J. Fluid Mech. 10 (1961), 166–188.CrossRefGoogle Scholar
[2]Chuoke, R. L., van Meurs, P. and van der Poel, C., “The instability of slow, immiscible, viscous liquid-liquid displacements in permeable media”, Trans. AIME 216 (1959), 188–194.CrossRefGoogle Scholar
[3]de Josselin de Jong, G., “Singularity distributions for the analysis of multiple-fluid flow through porous media”, J. Geophys. Res. 65 (1960), 3739–3758.CrossRefGoogle Scholar
[4]Fairbrother, F. and Stubbs, A. E., “The ‘bubble-tube’ method of measurement”, J. Chem. Soc. 1 (1935), 527–529.CrossRefGoogle Scholar
[5]Gupta, S. P., Varnon, J. E. and Greenkorn, R. A., “Viscous finger wavelength degeneration in Hele-Shaw models”, Water Resources Res. 9 (1973), 1039–1046.CrossRefGoogle Scholar
[6]Gupta, S. P. and Greenkorn, R. A., “An experimental study of immiscible displacement with an unfavourable mobility ratio in porous media”, Water Resources Res. 10 (1974), 371–374.CrossRefGoogle Scholar
[7]McLean, J. W. and Saffman, P. G., “The effect of surface tension on the shape of fingers in a Hele-Shaw cell”, J. Fluid Mech. 102 (1981), 455–469.CrossRefGoogle Scholar
[8]Pitts, E., “Penetration of fluid into a Hele-Shaw cell: the Saffman-Taylor experiment”, J. Fluid Mech. 97 (1980), 53–64.CrossRefGoogle Scholar
[9]Saffman, P. G. and Sir Taylor, Geoffrey, “The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid”, Proc. Roy. Soc. London Ser. A 245 (1958), 312–329.Google Scholar
[10]Saffman, P. G., “Exact solutions for the growth of fingers from a flat interface between two fluids in a porous medium or Hele-Shaw cell”, Quart. J. Mech. Appl. Math. 12 (1959), 146–150.CrossRefGoogle Scholar
[11]Stakgold, I., Boundary value problems of mathematical physics, Vol. 2 (Macmillan, New York, 1968).Google Scholar
[12]Taylor, G. I., “Deposition of a viscous fluid on the wall of a tube”, J. Fluid. Mech. 10 (1961), 161–165.CrossRefGoogle Scholar
[13]Wooding, R. A. and Morel-Seytoux, H. J., “Multiphase fluid flow through porous media”, Ann. Rev. Fluid Mech. 8 (1976), 233–274.CrossRefGoogle Scholar