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Growth of radial viscous fingers in a Hele-Shaw cell

Published online by Cambridge University Press:  26 April 2006

Jing-Den Chen
Jing-Den Chen
Affiliation:
Mead Imaging, 3385 Newmark Drive, Miamisburg, OH 45342, USA
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Abstract

A series of experiments are performed in a Hele-Shaw cell, consisting of two parallel closely spaced glass plates. A liquid (oil or water, both of viscosity of 1.0 cP) is injected at a constant volumetric flow rate, q, to radially displace a much more viscous liquid (glycerine. The data presented in this paper are taken mostly at late stages of the fingering process, when the pattern has multiple generations of splitting. Correlations with time are obtained for the finger length and the overall pattern density. The time-and lengthscales have been found for the immiscible case. At the same dimensionless time, immiscible patterns are similar and have the same generation of splitting. The overall density of each pattern decreases with time. The pattern shows fractal behaviour only after a certain number of generations of splitting. The fractal dimension of the immiscible pattern decreases from 1.9 to 1.82 when the pattern goes from the third to the sixth generation of splitting. The fractal dimension of the miscible pattern reaches a constant value after about ten generations of splitting and the fractal dimension ranges from 1.50 to 1.69 for q/Db = 4.8 × 105−7.0 × 106. The miscible patterns are insensitive to dispersion for large q/Db. For immiscible fingers λ/b scales with Ca−0.31 for capillary number Ca ranging from about 8 × 10−4 to 0.05. For miscible fingers, λ/b is insensitive to dispersion and ranges from 5 to 10 for large q/Db. Here D is the molecular diffusion coefficient in glycerine, b the cell gap width and λ the splitting wavelength.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 201 , April 1989 , pp. 223 - 242
Copyright
© 1989 Cambridge University Press

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References

Benham, A. L. & Olson, R. W. 1963 A model study of viscous fingering. Soc. Petrol. Engng J. 138–144.Google Scholar
Blackwell, R. J., Rayne, J. R. & Terry, W. M. 1959 Factors influencing the efficiency of miscible displacement. Petrol. Trans. AIME 216, 1–8.Google Scholar
Burnett, L. J. & Harmon, J. F. 1972 Self-diffusion in viscous liquids: pulse NMR measurements. J. Chem. Phys. 57, 1293–1297.Google Scholar
Chen, J. D. 1987 Radial viscous fingering patterns in Hele-Shaw cells. Exp. Fluids 5, 363–371.Google Scholar
Chouke, R. L., Van Meurs, P. & Van Der Poel, C. 1959 The instability of slow, immiscible, viscous liquid-liquid displacements in permeable media. Petrol. Trans. AIME 216, 188–194.Google Scholar
Daccord, G., Nittmann, J. & Stanley, H. 1986 Radial viscous fingers and diffusion-limited aggregation. Phys. Rev. Lett. 56, 336–339.Google Scholar
Degregoria, A. J. & Schwartz, L. W. 1985 Finger breakup in Hele-Shaw cells. Phys. Fluids 28, 2313–2314.Google Scholar
Hele-Shaw, H. S. 1898 The flow of water. Nature 58, 34–36.Google Scholar
Homsy, G. M. 1987 Viscous fingering in porous media. Ann. Rev. Fluid Mech. 19, 271–311.Google Scholar
Howard, G. C. & Fast, C. R. 1970 Hydraulic Fracturing. Soc. Petrol. Engng. AIME, Dallas, TX.
Howison, S. D. 1986 Fingering in Hele-Shaw cells. J. Fluid Mech. 167, 439–453.Google Scholar
Kopf-Sill, A. R. & Homsy, G. M. 1988 Nonlinear unstable viscous fingers in Hele-Shaw flows. I. Experiments. Phys. Fluids 31, 242–249.Google Scholar
Mandelbrot, M. M. 1983 The Fractal Geometry of Nature. San Francisco: Freeman.
Park, C. W. & Homsy, G. M. 1984 Two-phase displacement in Hele-Shaw cells: theory. J. Fluid Mech. 139, 291–308.Google Scholar
Paterson, L. 1981 Radial fingering in a Hele-Shaw cell. J. Fluid Mech. 113, 513–529.Google Scholar
Paterson, L. 1985 Fingering with miscible fluids in a Hele-Shaw cell. Phys. Fluids 28, 26–30.Google Scholar
Rauseo, S. N., Barnes, P. D. & Maher, J. V. 1987 Development of radial fingering patterns.. Phys. Rev. A 35, 1245–1251.Google Scholar
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
Schwartz, L. 1986 Stability of Hele-Shaw flows: The wetting layer effect. Phys. Fluids 29, 3086–3088.Google Scholar
Stalkup, F. I. 1984 Miscible Displacement. Soc. Petrol. Engng, AIME, Dallas, TX.
Tabeling, P., Zocchi, G. & Libchaber, A. 1987 An experimental study of the Saffman-Taylor instability. J. Fluid Mech. 177, 67–82.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. & Aref, H. 1985 Finger-interaction mechanisms in stratified Hele-Shaw flow. J. Fluid Mech. 154, 287–301.Google Scholar
Van Meurs, P. 1957 The use of transparent three-dimensional models for studying the mechanism of flow process in oil reservoirs. Trans. AIME 210, 295–301.Google Scholar
Witten, T. A. & Sander, L. M. 1983 Diffusion-limited aggregation.. Phys. Rev. B 27, 5686–5697.Google Scholar