Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-15T23:24:20.842Z Has data issue: false hasContentIssue false
  • English
  • Français

Fingering instability in buoyancy-driven fluid-filled cracks

Published online by Cambridge University Press:  24 February 2011

T. TOUVET ,
N. J. BALMFORTH ,
R. V. CRASTER  and
B. R. SUTHERLAND
T. TOUVET
Affiliation:
Département de Physique, École Normale Supérieure de Lyon, Université de Lyon, 46 allée d'Italie, 69364 Lyon CEDEX 07, France
N. J. BALMFORTH
Affiliation:
Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC V6T 1Z2, Canada Department of Earth and Ocean Science, University of British Columbia, 6339 Stores Road, Vancouver, BC V6T 1Z4, Canada
R. V. CRASTER*
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, T6G 2G1, Canada Department of Mathematics, Imperial College London, South Kensington, London SW7 2AZ, UK
B. R. SUTHERLAND
Affiliation:
Department of Physics, University of Alberta, Edmonton, T6G 2G1, Canada
*
‡Email address for correspondence: [email protected] or [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The stability of buoyancy-driven propagation of a fluid-filled crack through an elastic solid is studied using a combination of theory and experiments. For the theory, the lubrication approximation is introduced for fluid flow, and the surrounding solid is described by linear elasticity. Solutions are then constructed for a planar fluid front driven by either constant flux or constant volume propagating down a pre-cut conduit. As the thickness of the pre-cut conduit approaches zero, it is shown how these fronts converge to zero-toughness fracture solutions with a genuine crack tip. The linear stability of the planar solutions towards transverse, finger-like perturbations is then examined. Instabilities are detected that are analogous to those operating in the surface-tension-driven fingering of advancing fluid contact lines. Experiments are conducted using a block of gelatin for the solid and golden syrup for the fluid. Again, planar cracks initiated by emplacing the syrup above a shallow cut on the surface of the gelatin develop transverse, finger-like structures as they descend. Potential geological applications are discussed.

JFM classification

Interfacial Flows (free surface): Fingering instability Geophysical and Geological Flows: Magma and lava flow Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 672 , 10 April 2011 , pp. 60 - 77
Copyright
Copyright © Cambridge University Press 2011

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.)

Footnotes

†

Present address: Département de Mécanique, École Polytechnique, 91128 Palaiseau CEDEX, France.

References

REFERENCES

Abramowitz, M. & Stegun, I. A. 1969 Handbook of Mathematical Tables. Dover.Google Scholar
Adachi, A., Siebrits, E., Peirce, A. & Desroches, J. 2007 Computer simulation of hydraulic fractures. Intl J. Rock Mech. Mining Sci. 44, 739–757.CrossRefGoogle Scholar
Balmforth, N. J., Craster, R. V., Rust, A. C. & Sassi, R. 2006 Viscoplastic flow over an inclined surface. J. Non-Newtonian Fluid Mech. 139, 103–127.CrossRefGoogle Scholar
Bloomer, S. H., Stern, R. J. & Smoot, N. C. 1989 Physical volcanology of the submarine Mariana and Volcano arcs. Bull. Volcanol. 51, 210–224.CrossRefGoogle Scholar
Boschi, C., Dini, A., Dallai, L., Ruggieri, G. & Gianelli, G. 2009 Enhanced CO2-mineral sequestration by cyclic hydraulic fracturing and Si-rich fluid infiltration into serpentinites at Malentrata (Tuscany, Italy). Chem. Geol. 265, 209–226.CrossRefGoogle Scholar
Craster, R. V. & Matar, O. K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81, 1131–1198.CrossRefGoogle Scholar
Desroches, J., Detournay, E., Lenoach, B., Papanastasiou, P., Pearson, J. R. A., Thiercelin, M. & Cheng, A. 1994 The crack tip region in hydraulic fracturing. Proc. R. Soc. Lond. A 447, 39–48.Google Scholar
Economides, M. & Nolte, K. (Eds.) 2000 Reservoir Stimulation, 3rd. edn. John Wiley & Sons.Google Scholar
Freund, L. B. 1990 Dynamic Fracture Mechanics. Cambridge University Press.CrossRefGoogle Scholar
Heimpel, M. & Olson, P. 1994 Buoyancy-driven fracture and magma transport through the lithosphere: models and experiments. In Magmatic Systems (ed. Ryan, M. P.), pp. 223–240. Academic.CrossRefGoogle Scholar
Huppert, H. E. 1982 Flow and instability of a viscous current down a slope. Nature 300, 427–429.CrossRefGoogle Scholar
Ioakimidis, N. I. 1982 Application of finite-part integrals to the singular integral equations of crack problems in plane and three-dimensional elasticity. Acta Mechanica 45, 31–47.CrossRefGoogle Scholar
Jaeger, J. C. 1964 Elasticity, Fracture and Flow, 2nd edn. Methuen.Google Scholar
Krawczynski, M. J., Behn, M. D., Das, S. B. & Joughin, I. 2009 Constraints on the lake volume required for hydro-fracture through ice sheets. Geophys. Res. Lett. 36, L10501.CrossRefGoogle Scholar
Lister, J. R. 1990 a Buoyancy-driven fluid fracture: the effects of material toughness and of low-viscosity precursors. J. Fluid Mech. 210, 263–280.CrossRefGoogle Scholar
Lister, J. R. 1990 b Buoyancy-driven fracture: similarity solutions for the horizontal and vertical propagation of fluid-filled cracks. J. Fluid Mech. 217, 213–239.CrossRefGoogle Scholar
Liu, J., Paul, J. D. & Gollub, J. P. 1993 Measurements of the primary instabilities of film flows. J. Fluid Mech. 250, 69–101.CrossRefGoogle Scholar
Menand, T. & Tait, S. 2001 A phenomenological model for precursor volcanic eruptions. Nature 411, 678–680.CrossRefGoogle ScholarPubMed
Menand, T. & Tait, S. 2002 The propagation of a buoyant liquid-filled fissure from a source under constant pressure. J. Geophys. Res. B 107, 2306–2320.CrossRefGoogle Scholar
Oakenfull, D. G., Parker, N. S. & Tanner, R. I. 1988 Method for determining absolute shear modulus of gels from compression tests. J. Texture Stud. 19 (4), 407–417.CrossRefGoogle Scholar
Oron, A., Davis, S. H. & Bankoff, S. G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 931–980.CrossRefGoogle Scholar
Roper, S. M. & Lister, J. R. 2007 Buoyancy-driven crack propagation: the limit of large fracture toughness. J. Fluid Mech. 580, 359–380.CrossRefGoogle Scholar
Rubin, A. M. 1995 Propagation of magma-filled cracks. Annu. Rev. Earth Planet Sci. 23, 287–336.CrossRefGoogle Scholar
Silvi, N. & Dussan, V., E. B., 1985 On the rewetting of an inclined solid-surface by a liquid. Phys. Fluids 28, 5–7.CrossRefGoogle Scholar
Spaid, M. A. & Homsy, G. M. 1996 Stability of Newtonian and viscoelasticity dynamic contact lines. Phys. Fluids 8, 460–478.CrossRefGoogle Scholar
Spence, D. A., Sharp, P. W. & Turcotte, D. L. 1987 Buoyancy-driven crack propagation: a mechanism for magma migration. J. Fluid Mech. 174, 135–153.CrossRefGoogle Scholar
Spence, D. A. & Turcotte, D. L. 1990 Buoyancy-driven magma fracture: a mechanism for ascent through the lithosphere and the emplacement of diamonds. J. Geophys. Res. 95 (B4), 5133–5139.CrossRefGoogle Scholar
Taisne, B. & Jaupart, C. 2009 Dike propagation through layered rocks. J. Geophys. Res. Solid Earth 114, B09203.CrossRefGoogle Scholar
Taisne, B. & Tait, S. 2009 Eruption versus intrusion? Arrest of propagation of constant volume, buoyant, liquid-filled cracks in an elastic, brittle host. J. Geophys. Res. 114, B06202.CrossRefGoogle Scholar
Takada, A. 1990 Experimental study on propagation of liquid-filled crack in gelatin: shape and velocity in hydrostatic stress condition. J. Geophys. Res. B 95, 8471–8481.CrossRefGoogle Scholar
Troian, S. M., Herbolzheimer, E., Safran, S. A. & Joanny, J. F. 1989 Fingering instabilities of driven spreading films. Europhys. Lett. 10, 25–30.CrossRefGoogle Scholar
Tsai, V. C. & Rice, J. R. 2010 A model for turbulent hydraulic fracture and application to crack propagation at Glacier beds. J. Geophys. Res. 115, F03007.CrossRefGoogle Scholar
Weertman, J. 1971 Theory of water-filled crevasses in glaciers applied to vertical magma transport beneath oceanic ridges. J. Geophys. Res. 76, 1171–1183.CrossRefGoogle Scholar
Weiss, J. 2004 Subcritical crack propagation as a mechanism of crevasse formation and iceberg calving. J. Glaciol. 50, 109–115.CrossRefGoogle Scholar