Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-24T18:42:17.574Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical calculation of unstable immiscible fluid displacement 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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A numerical procedure for calculating the evolution of a periodic interface between two immiscible fluids flowing in a two-dimensional porous medium or Hele-Shaw cell is described. The motion of the interface is determined in a stepwise manner with its new velocity at exach time step being derived as a numerical solution of a boundary integral equation. Attention is focused on the case of unstable displacement charaterised physically by the “fingering” of the interface and computationally by the growth of numerical errors regardless of the numerical method employed. Here the growth of such error is reduced and the usable part of the calculation extended to finite amplitudes. Numerical results are compared with an exact “finger” solution and the calculated behaviour of an initial sinusoidal displacement, as a function of interfacial tension, initial amplitude and wavelength, is discussed.

Type
Research Article
Information
The ANZIAM Journal , Volume 26 , Issue 4 , April 1985 , pp. 452 - 469
Copyright
Copyright © Australian Mathematical Society 1985

References

[1]Ahlberg, J. H., Nilson, E. N. and Walsh, J. L., The theory of splines and their applications (Academic, New York, 1967).Google Scholar
[2]Chuoke, R. L., van Meurs, P. and van der Poel, C., “The instability of slow, immiscible, viscous liquid-liquid displacement in permeable media”, Trans. AIME 216 (1959), 188–194.CrossRefGoogle Scholar
[3]Davidson, M. R. “An integral equation for immiscible fluid displacement in a two-dimensional porous medium or Hele-Shaw cell”, J. Austral. Math. Soc. Ser. B 26 (1984), 14–30.CrossRefGoogle Scholar
[4]Gradshteyn, I. S. and Ryzhik, I. M., Table of integrals, series and products, enlarged edition (Academic, New York, 1980).Google 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]Hamming, R. W., Numerical methods for scientists and engineers (McGraw-Hill, New York, 1962).Google Scholar
[7]Longuet-Higgins, M. S. and Cokelet, E. D., “The deformation of steep surface waves on water. I. A numerical method of computation”, Proc. Roy. Soc. London Ser. A 350 (1976), 1–26.Google Scholar
[8]Outmans, H. D., “Nonlinear theory for frontal stability and viscous fingering in porous media”, Soc. Petrol. Engr. J. 2 (1962), 165–176.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 fluid”, 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