Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-18T18:53:23.400Z Has data issue: false hasContentIssue false
  • English
  • Français

A boundary-integral method for two-phase displacement in Hele-Shaw cells

Published online by Cambridge University Press:  21 April 2006

A. J. Degregoria  and
L. W. Schwartz
A. J. Degregoria
Affiliation:
Corporate Research-Science Laboratories, Exxon Research and Engineering Co., Clinton Township, Route 22 E., Annandale, New Jersey 08801, USA
L. W. Schwartz
Affiliation:
Corporate Research-Science Laboratories, Exxon Research and Engineering Co., Clinton Township, Route 22 E., Annandale, New Jersey 08801, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We develop a time-dependent numerical algorithm, using a boundary-integral approach, to investigate fingering in Hele-Shaw cells. Starting from a sinusoidal variation in the initial interface, stable fingers quickly form for a wide range of the dimensionless surface-tension parameter. For very low values of the parameter, the incipient finger bifurcates. The stable fingers are clearly the same as those obtained by McLean & Saffman (1981) using a steady-state algorithm. These steady-state solutions were found to be linearly unstable. We resolve this apparent discrepancy regarding stability by tracing the fate of small disturbances placed on and about the finger tip. We show that some small disturbances do, indeed, grow initially; however, they reach a maximum amplitude and decay as they convect backward from the tip of the finger to regions where stabilizing surface tension is the major physical force. Relatively large imposed disturbances, on the other hand, cause a finger to bifurcate; the critical disturbance amplitude decreases as the surface tension is reduced

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 164 , March 1986 , pp. 383 - 400
Copyright
© 1986 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Botha, J. F. & Pinder, G. F. 1983 Fundamental Concepts in the Numerical Solution of Differential Equations. Wiley.
Chouke, R. L., van Meurs, P. & van Der Poel, C. 1959 The instability of slow, immiscible, viscous, liquid—liquid displacements in permeable media. Trans. AIME 216, 188–194.Google Scholar
DeGregoria, A. J. & Schwartz, L. W. 1985 Finger breakup in Hele-Shaw cells. Phys. Fluids 28, 2313–2314.Google Scholar
Hindmarsh, A. C., Livermore Solver for Ordinary Differential Equations (L8ODE) Sept. 23, 1980 version, Lawrence Livermore Laboratory, Livermore CA.
Jaswon, M. A. & Symm, G. T. 1977 Integral Equation Methods in Potential Theory and Elastostatics. Academic.
Jebatjd, J. R., Davis, H. T. & Scriven, L. E. 1984 SPE/DOE Preprint No. 12691.
McLean, J. W. & Saffman, P. G. 1981 The effect of surface tension on the shape of fingers in a Hele-Shaw cell. J. Fluid Mech. 102, 455–469.Google Scholar
Meng, J. C. S. & Thomson, J. A. L. 1978 Numerical studies of some nonlinear hydrodynamic problems by discrete vortex element methods. J. Fluid Mech. 84, 433–453.Google Scholar
Nittmann, J., Daccord, G. & Stanley, H. E. 1985 Fractal growth of viscous fingers: quantitative characterization of a fluid instability phenomenon. Nature 314, 141–144.Google Scholar
Park, C.-W. & Homsy, G. M. 1984 Two-phase displacement in Hele-Shaw cells: theory. J. Fluid Mech. 139, 291–308.Google Scholar
Park, C.-W. & Homsy, G. M. 1985 The instability of long fingers in Hele-Shaw flows. Phys. Fluids 28, 1583–1585.Google Scholar
Paterson, L. 1981 Radial fingering in a Hele-Shaw cell. J. Fluid Mech. 113, 513–529.Google Scholar
Peters, E. J. & Flock, D. L. 1981 The onset of instability during two-phase immiscible displacement in porous media. SPE J. 21, 249–258.Google Scholar
Pitts, E. 1980 Penetration of fluid into a Hele-Shaw cell: the Saffman—Taylor experiment. J. Fluid Mech. 97, 53–64.Google Scholar
Reshotko, E. 1976 Boundary-layer stability and transition. Ann. Rev. Fluid Mech. 8, 311–348.Google Scholar
Romero, L. 1982 Ph.D. thesis, California Institute of Technology.
Saffman, P. G. & Taylor, G. I. 1958 The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. Lond. A 245, 312–329.Google Scholar
Taylor, G. I. & Saffman, P. G. 1959 A note on the motion of bubbles in a Hele-Shaw cell and porous medium. Q. J. Mech. Appl. Maths 12, 265–279.Google Scholar
Tryggvason, G. & Aref, H. 1983 Numerical experiments on Hele-Shaw flow with a sharp interface. J. Fluid Mech. 136, 1–30.Google Scholar
Tryggvason, G. & Ahef, H. 1985 Finger interaction mechanisms in stratified Hele-Shaw flow. J. Fluid Mech. 154, 287–301.Google Scholar
Vanden-Broeck, J. M. 1983 Fingers in a Hele-Shaw cell with surface tension. Phys. Fluids 26, 2033–34.Google Scholar
Wehausen, J. & Laitone, E. 1960 Surface waves. Handbuch der Physik, vol. 9 (ed. S. Flügge), pp. 446–778.